180 Days of Problem Solving for Fourth Grade [1 ed.] 9781425895693, 9781425816162

Citation preview

4 Level

Author Chuck Aracich, M.Ed.

For information on how this resource meets national and other state standards, see pages 4–7. You may also review this information by visiting our website at www.teachercreatedmaterials.com/administrators/correlations/ and following the on-screen directions.

Publishing Credits Corinne Burton, M.A.Ed., Publisher; Conni Medina, M.A.Ed., Managing Editor; Emily R. Smith, M.A.Ed., Series Developer; Diana Kenney, M.A.Ed., NBCT, Content Director; Paula Makridis, M.A.Ed., Editor; Stacy Monsman, M.A., Editor; Lee Aucoin, Senior Multimedia Designer; Kyleena Harper, Assistant Editor; Kevin Pham, Graphic Designer

Image Credits All images from iStock and Shutterstock.

Standards © Copyright 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved. (CCSS)

Shell Education

A division of Teacher Created Materials 5301 Oceanus Drive Huntington Beach, CA 92649-1030 www.tcmpub.com/shell-education ISBN 978-1-4258-1616-2 ©2017 Shell Education Publishing, Inc.

The classroom teacher may reproduce copies of materials in this book for classroom use only. The reproduction of any part for an entire school or school system is strictly prohibited. No part of this publication may be transmitted, stored, or recorded in any form without written permission from the publisher.

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Table of Contents Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 How to Use this Book. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Standards Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Daily Practice Pages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Answer Key. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 Practice Page Rubric. . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 Practice Page Item Analysis. . . . . . . . . . . . . . . . . . . . . . 214 Student Item Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . 218 Problem-Solving Framework. . . . . . . . . . . . . . . . . . . . . 219 Problem-Solving Strategies. . . . . . . . . . . . . . . . . . . . . . 220 Digital Resources. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

Introduction The Need for Practice To be successful in today’s mathematics classrooms, students must deeply understand both concepts and procedures so that they can discuss and demonstrate their understanding during the problem-solving process. Demonstrating understanding is a process that must be continually practiced for students to be successful. Practice is especially important to help students apply their concrete, conceptual understanding during each step of the problem‑solving process.

Understanding Assessment In addition to providing opportunities for frequent practice, teachers must be able to assess students’ problem-solving skills. This is important so that teachers can adequately address students’ misconceptions, build on their current understandings, and challenge them appropriately. Assessment is a long-term process that involves careful analysis of student responses from discussions, projects, practice pages, or tests. When analyzing the data, it is important for teachers to reflect on how their teaching practices may have influenced students’ responses and to identify those areas where additional instruction may be required. In short, the data gathered from assessments should be used to inform instruction: slow down, speed up, or reteach. This type of assessment is called formative assessment.

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How to Use This Book 180 Days of Problem Solving offers teachers and parents problem-solving activities for each day of the school year. Students will build their problem-solving skills as they develop a deeper understanding of mathematical concepts and apply these concepts to real-life situations. This series will also help students improve their critical-thinking and reasoning skills, use visual models when solving problems, approach problems in multiple ways, and solve multi-step, non-routine word problems.

Easy-to-Use and Standards-Based These daily activities reinforce grade-level skills across a variety of mathematical concepts. Each day provides a full practice page, making the activities easy to prepare and implement as part of a classroom routine, at the beginning of each mathematics lesson as a warm-up or Problem of the Day, or as homework. Students can work on the practice pages independently or in cooperative groups. The practice pages can also be utilized as diagnostic tools, formative assessments, or summative assessments, which can direct differentiated small-group instruction during Mathematics Workshop.

Domains and Practice Standards The chart below indicates the mathematics domains addressed and practice standards applied throughout this book. The subsequent chart shows the breakdown of which mathematics standard is covered in each week. Note: Students may not have deep understanding of some topics in this book. Remember to assess students based on their problem-solving skills and not exclusively on their content knowledge. Grade-Level Domains

1. Operations and Algebraic Thinking



2. Number and Operations in Base Ten



3. Number and Operations—Fractions



4. Measurement and Data

5. Geometry

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Practice Standards

1. Make sense of problems and persevere in solving them.



2. Reason abstractly and quantitatively.



3. Construct viable arguments and critique the reasoning of others.



4. Model with mathematics.



5. Use appropriate tools strategically.



6. Attend to precision.



7. Look for and make use of structure.



8. Look for and express regularity in repeated reasoning. © Shell Education

How to Use This Book

(cont.)

College-and-Career Readiness Standards Below is a list of mathematical standards that are addressed throughout this book. Each week students solve problems related to the same mathematical topic. Week 1 2 3 4 5 6 7 8 9

10

11

12 13

Standard Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right.  Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons. Use place value understanding to round multi-digit whole numbers to any place. Fluently add multi-digit whole numbers using the standard algorithm. Fluently subtract multi-digit whole numbers using the standard algorithm. Solve multistep word problems posed with whole numbers and having wholenumber answers using addition and subtraction, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding. Find all factor pairs for a whole number in the range 1–100. Determine whether a given whole number in the range 1–100 is prime or composite. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1–100 is a multiple of a given one-digit number. Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 × 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations. Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison. Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Multiply a whole number of up to four digits by a one-digit whole number, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. Multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

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14

15

16

17

18 19 20

21

22

23 24 25 26

6

Find whole-number quotients and remainders with up to two-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding. Explain why a fraction � is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions. Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as M . Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or , =, or m Michael is using X of the memory on his smartphone for pictures. Mitchell is using ��� of the memory on his smartphone for pictures. Who is using more memory for pictures?

Michael

Mitchell

0

1

0

1

_______________________________________________________________________________ 100

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WEEK 18 DAY

Name: _______________ Date: _________ Directio n s:

.

At a dog show, a poodle and a beagle are competing on an obstacle course. The dog that completes the greatest amount of the course within the time limit is the winner. The poodle completes j of the course. The beagle finishes ��� of the course. Which dog completed the greater amount of the obstacle course? Strategy 1

? Solve It Two Ways!

1.

Show two ways to solve the problem.

4

Strategy 2

2.

Which strategy do you think is better? Explain your reasoning. ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

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WEEK 18 DAY

5

Name: _______________ Date: _________

Challenge Yourself!

Directio n s:

Read and solve the problem.

A family orders three equal-sized small pizzas for dinner. The pepperoni pizza is cut into 8 slices, the cheese pizza is cut into 4 slices, and the vegetable pizza is cut into 6 slices. The family eats 5 slices of the pepperoni pizza, 2 slices of the cheese pizza, and 2 slices of the vegetable pizza.

1.

Show the amount of each pizza the family eats. Shade and label each model.

2.

Use your models to write what fraction of each pizza is left over. _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

3.

Choose a strategy to list the fractional amounts of leftover pizza in order from least to greatest. Justify your solution. _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

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WEEK 19 DAY

Name: _______________ Date: _________ Directio n s:

Think about the problem. Answer the questions.

Indoor activity

Board game

Computer

Drawing

Card game

Fraction of students

�

�

�

�

Think About It!

During indoor recess, students choose an activity to play. The table shows the fraction of students that choose each activity. What fraction of students choose to play either a card or board game?

1.

1

What do you think the denominator of 20 represents? _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

2.

What is the problem asking you to find? _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

3.

Which activity is played by O of the class? How do you know? _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

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WEEK 19 DAY

2

Name: _______________ Date: _________ Directio n s:

Solve It!

?

Read and solve each problem.

Problem 1: During indoor recess, students choose an activity to play. The table shows the fraction of students that choose each activity. What fraction of students choose to play either a card or board game? Indoor activity

Board game

Computer

Drawing

Card game

Fraction of students

�

�

�

�

What Do You Know?

What Is Your Plan?

Solve the Problem!

Look Back and Explain!

Problem 2: What fraction of the students from the table in Problem 1 do not play on the computer?

104

What Do You Know?

What Is Your Plan?

Solve the Problem!

Look Back and Explain!

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WEEK 19 DAY

Name: _______________ Date: _________ Directio n s:

3

Look at the example. Then, solve the problems using the information in the table. Fraction

Fruit

Fraction

apples

�

strawberries

�

bananas

�

watermelons

�

Visualize It!

Fruit

Example 1: Write an equation to show the fraction of the fruit that is either apples or bananas.

� + � = M� M� of the fruit is either apples or bananas. Example 2: Write an equation to show the fractional amount of fruit that is not watermelons.

W� – � = WF WF of the fruit is not watermelons. 1.

Write an equation to show the fraction of fruit that is either strawberries or watermelons. _______________________________________________________________________________ _______________________________________________________________________________

2.

Write an equation to show the fractional amount of fruit that is not bananas. _______________________________________________________________________________ _______________________________________________________________________________

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WEEK 19 DAY

4

Solve It Two Ways!

?

Name: _______________ Date: _________ Directio n s : Show two ways to solve the problem.

1.

Myra’s cat Fluffy eats ��� of a bag of cat food for breakfast. Her other cat Scooter eats ��� of the same bag of cat food. What fractional amount of the cat food is left in the bag? Strategy 1

Strategy 2

2.

Which strategy do you like better? Explain your reasoning _______________________________________________________________________________ _______________________________________________________________________________ ______________________________________________________________________________ .

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WEEK 19 DAY

5

Name: _______________ Date: _________ Directio n s:

Read and solve the problem.

?



• There are 15 cards in the deck.



•



• The fraction of blue cards is equivalent to N .

1.

��� of the deck are either red or blue cards.

Challenge Yourself!

Use the clues to find the fraction of red cards, blue cards, and white cards in a board game.

What denominator will you use? Why? _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

2.

Solve the problem. Justify your answers. _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

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WEEK 20 DAY

1

Name: _______________ Date: _________

Think About It!

Directio n s:

Think about the problem. Answer the questions.

?

Annabelle and her two friends share a whole vegetable tray for a snack. Annabelle eats c of the vegetables. Her friends each have the same amount of vegetables. What fraction of the vegetables do each of Annabelle’s friends eat?

1.

What do you need to find in the problem? _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

2.

What fraction represents the whole tray? Explain how you know. _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

3.

Can each of Annabelle’s friends eat w of the tray? Explain how you know. _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

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WEEK 20 DAY

Name: _______________ Date: _________ Directio n s:

2

Read and solve each problem.

What Do You Know?

What Is Your Plan?

Solve the Problem!

Look Back and Explain!

Solve It!

Problem 1: Annabelle and her two friends share a whole vegetable tray for a snack. Annabelle eats c of the vegetables. Her friends each have the same amount of vegetables. What fraction of the vegetables do each of Annabelle’s friends eat?

Problem 2: Four squirrels are collecting nuts for the winter. Three squirrels each collect Q of the nuts needed to get through the winter. What fraction of nuts must the fourth squirrel collect so they have enough food for winter? What Do You Know?

What Is Your Plan?

Solve the Problem!

Look Back and Explain!

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WEEK 20 DAY

3

Name: _______________ Date: _________

Visualize It!

?

Directio n s:

Look at the example. Then, solve the problem.

Example: Write the fractional amount that is shaded in each shape. Write two different expressions that represent the same amount.

Shape

Fraction shaded

Expression 1

Expression 2

w

S+S+S+S

S+m

j

P+P+P

Z+P

�

T+T+T+T+T

n+d

circle

rectangle

triangle

Write the fractional amount that is shaded in each shape. Write two different expressions that represent the same amount.

Shape

Fraction shaded

Expression 1

Expression 2

circle

rectangle

triangle

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WEEK 20 DAY

Name: _______________ Date: _________ Directio n s:

Reggie walks 2� kilometers to school. Show how to represent 2� . Strategy 1

? Solve It Two Ways!

1.

Show two ways to solve the problem.

4

Strategy 2

2.

What strategy do you think is better? Explain your reasoning. _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

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WEEK 20 DAY

5

Name: _______________ Date: _________

Challenge Yourself!

Directio n s:

Read and solve the problem.

Four student artists are painting 3 different rectangular murals. Each mural is partitioned into equal sections. The students must paint an equal fractional amount of each mural.

1.

Complete the mural diagrams by shading the amount each student will paint in a different color.

Mural 1

2.

112

Mural 3

Mural 2

Write an equation to show the fractional amount each student should paint for each mural. Rectangle 1:

+

+

+

=

LW

Rectangle 2:

+

+

+

=

�

Rectangle 3:

+

+

+

=

WA

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WEEK 21 DAY

Name: _______________ Date: _________ Directio n s:

Think about the problem. Answer the questions.

Think About It!

James has two buckets. The first bucket is filled with 2u liters of water. The second bucket contains 1k liters of water. How many liters of water are there in all?

1.

1

What operation will you use to find the solution to the problem? How do you know? _______________________________________________________________________________ _______________________________________________________________________________

2.

How can the whole number 1 in 1k liters be written as a fraction? Explain your reasoning. _______________________________________________________________________________ _______________________________________________________________________________

3.

Will the solution be less than or greater than 3 liters? How do you know? _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

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WEEK 21 DAY

2

Name: _______________ Date: _________ Directio n s:

Solve It!

?

114

Read and solve the problem.

Problem: James has two buckets. The first bucket is filled with 2u liters of water. The second bucket contains 1k liters of water. How many liters of water are there in all? What Do You Know?

What Is Your Plan?

Solve the Problem!

Look Back and Explain!

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WEEK 21 DAY

Name: _______________ Date: _________ Directio n s:

3

Look at the example. Then, solve the problem by using a model.

Visualize It!

Example: The model shows 3c shaded. How much remains after taking away 1� ? Use the model to show your solution.

3c – 1� = 1w The model shows 2� shaded. How much will remain after taking away 1� ? Use the model to show your solution.

2� – 1� = ____________

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WEEK 21 DAY

4

Solve It Two Ways!

?

Name: _______________ Date: _________ Directio n s:

1.

Show two ways to solve the problem.

Daniel’s parents buy 3 whole pizzas for his birthday party. At the end of the evening, 1a pizzas are left over. How much pizza was eaten during the party? Strategy 1

Strategy 2

2.

Which strategy do you think is better? Explain your reasoning. _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

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WEEK 21 DAY

5

Name: _______________ Date: _________ Directio n s:

Read and solve the problem.

?

1.

Challenge Yourself!

Ty sees four boxes of the same type of candy. Each box weighs a different amount. He can choose only two boxes. His parents tell him that he must give away 1M kilograms of the candy he chooses to his friends. Which two boxes should Ty choose so he ends up with the most candy possible for himself? How much candy will Ty have left after he gives the candy to his friends?

Box A

Box B

Box C

Box D

2i kg

1i kg

3O kg

2Y kg

Which two boxes of candy weigh the most? How will knowing this help you solve the problem? _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

2.

Choose a strategy to solve the problem. Explain how you found your answer. _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

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WEEK 22 DAY

1

Name: _______________ Date: _________

Think About It!

Directio n s:

Think about the problem. Answer the questions.

?

Samantha is a sandwich artist at Load It Up Deli. She uses a of a pound of turkey in one sandwich. How many pounds of turkey will she need to make 8 sandwiches?

1.

How can you use addition to solve the problem? _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

2.

How can you use multiplication to solve the problem? _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

3.

Will the answer be greater than or less than one pound? Explain how you know. _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

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WEEK 22 DAY

Name: _______________ Date: _________ Directio n s:

2

Read and solve each problem.

What What Do Do You You Know? Know?

What What Is Is Your Your Plan? Plan?

Solve the Problem!

Look Back and Explain!

Solve It!

Problem 1: Samantha is a sandwich artist at Load It Up Deli. She uses a of a pound of turkey in one sandwich. How many pounds of turkey will she need to make 8 sandwiches?

Problem 2: Six students are running one mile in physical education class. Each student has already completed i of the mile. What is the total distance the six students will have run? What Do You Know?

What Is Your Plan?

Solve the Problem!

Look Back and Explain!

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WEEK 22 DAY

3

Name: _______________ Date: _________

Visualize It!

?

Directio n s:

Look at the example. Then, solve the problem by completing the number line.

Example: Show 9 × O on the number line.

+O +O +O +O +O +O +O +O +O 0

O

Y

i

s � � 9×O=�

�

�

�

Show 8 × N on the number line.

0

8 × N = __________

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WEEK 22 DAY

Name: _______________ Date: _________ Directio n s:

Sal walks u kilometers to school. Eva walks 3 times as far to school. How many kilometers does Eva walk to school? Strategy 1

? Solve It Two Ways!

1.

Show two ways to solve the problem.

4

Strategy 2

2.

Which strategy do you think is better? Explain your reasoning. _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

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WEEK 22 DAY

5

Name: _______________ Date: _________

Challenge Yourself!

Directio n s:

Read and solve the problem.

A list of ingredients from a recipe for chili is shown. The recipe serves 6 people. How much of each ingredient is needed to serve chili to 30 people?

1.

Vince says, “All you have to do is multiply each ingredient by 30.” Do you agree with Vince’s reasoning? Why or why not?

Ingredients for Chili 2 pounds of ground beef � can of kidney beans i can of diced tomatoes � tablespoon of chili powder M teaspoon of salt N teaspoon of black pepper

_______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

2.

3.

What is the solution for each ingredient? Show your work. ground beef: __________

chili powder: __________

kidney beans: __________

salt: __________

diced tomatoes: __________

black pepper: __________

Which strategies did you use to find the solution for each ingredient? _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

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WEEK 23 DAY

Name: _______________ Date: _________ Directio n s:

Think about the problem. Answer the questions.

Water Drank by Students

x

x x

O

Y

i

x

x x x x

1

1O

1Y

x x x

x

1i

2

Think About It!

Mr. Walker’s students measure how many cups of water they drank during a school day for a science experiment. They organize the results on a line plot.

0

1

Cups of water

1.

What does each X represent? _______________________________________________________________________________ _______________________________________________________________________________

2.

How many total students are represented in this line plot? Explain how you know. _______________________________________________________________________________ _______________________________________________________________________________

3.

Write a question that can be asked based on this data. _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

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WEEK 23 DAY

2

Name: _______________ Date: _________ Directio n s:

Read and solve the problem.

Solve It!

?

Problem: Mr. Walker’s students measure how many cups of water they drank during a school day for a science experiment. They organize the results on a line plot. How many students drank at least 1O cups of water?

Water Drank by Students

0

x

x x

O

Y

i

x

x x x x

1

1O

1Y

x x x

x

1i

2

Cups of water

124

What Do You Know?

What Is Your Plan?

Solve the Problem!

Look Back and Explain!

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WEEK 23 DAY

3

Name: _______________ Date: _________ Directio n s:

Look at the example. Then, solve the problem.

Vegetable

carrots beets potatoes onions broccoli peppers cauliflower

Cooking time in hours

c � � � c c m

Visualize It!

Example: A cookbook provides the length of time it takes vegetables to bake in an oven. Create a line plot to show the results.

Vegetables

____________________________ (title)

0

S

x x x

x

c

m

x x

w

�

x

�

�

1

____________________________ Cooking time in hours (axis label)

A deli keeps track of the amount of certain cheeses sold during lunch. Create a line plot to show the results. Cheese

Pounds sold

gouda

1O

swiss

O

cheddar

1i

mozzarella

1O

provolone

1i

pepper jack

Y

colby

1i

parmesan

O

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____________________________ (title)

0

1

2

____________________________ (axis label)

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WEEK 23 DAY

4

Solve It Two Ways!

?

Name: _______________ Date: _________ Directio n s:

1.

Write two questions about the line plot.

A lawn service is cutting grass in the neighborhood. The owner keeps track of how many gallons of gasoline are used in each lawn mower. Write two questions that can be answered using the information on the line plot. Provide answers to your questions.

Lawn Mowers

1

x x 1S

x

x x

x x x

x x

1c

1m

1w

1�

1�

x

x

1�

2

Number of gallons of gasoline used Question 1 _______________________________________________________________________________ _______________________________________________________________________________

Question 2 _______________________________________________________________________________ _______________________________________________________________________________

2.

Which question that you wrote do you think is more challenging? Explain your reasoning. _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

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WEEK 23 DAY

5

Name: _______________ Date: _________ Directio n s:

Read and solve the problem.

?

______________________________________________ (title)

Challenge Yourself!

A family of 8 eats sandwiches for lunch. One of the family members tracks the fraction of each sandwich that is eaten and organizes the information on a line plot. However, some important information was left off of the line plot. What is the total amount of sandwiches eaten by the entire family?

x x

x

x

x

x

x

0

x 1

______________________________________________ (axis label)

1.

Complete the line plot by writing a title, a fraction for each tick mark, and a label for the axis.

2.

What is the solution? Use words, numbers, or a picture to show your work.

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WEEK 24 DAY

1

Name: _______________ Date: _________

Think About It!

Directio n s:

Think about the problem. Answer the questions.

?

Mr. McGwire walks from his house to work. He walks � of a mile and stops for coffee. Then, he walks � of a mile and arrives at his office. How far does Mr. McGwire travel?

1.

Colleen says, “I will add 3 and 45 to get 48, so Mr. McGwire must have traveled � of a mile.” Do you agree with Colleen’s reasoning? Why or why not? _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

2.

How can you use the relationship between tenths and hundredths to solve the problem? _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

3.

Did Mr. McGwire walk less than or more than half a mile? How do you know? _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

128

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WEEK 24 DAY

Name: _______________ Date: _________ Directio n s:

2

Read and solve each problem.

What Do You Know?

What Is Your Plan?

Solve the Problem!

Look Back and Explain!

Solve It!

Problem 1: Mr. McGwire walks from his house to work. He walks � of a mile and stops for coffee. Then, he walks � of a mile and arrives at his office. How far does Mr. McGwire travel?

Problem 2: A rabbit enters a garden and eats � of a carrot before it is scared off. The next day, the rabbit returns and eats � of the same carrot. How much of the carrot has the rabbit eaten in all? What Do You Know?

What Is Your Plan?

Solve the Problem!

Look Back and Explain!

© Shell Education

#51616—180 Days of Problem Solving

129

WEEK 24 DAY

3

Name: _______________ Date: _________

Visualize It!

?

Directio n s:

Look at the example. Then, solve the problem using the hundreds grids.

Example: Two teams are participating in a jumping contest. The team with the longest combined jump wins. Which team wins the jumping contest?

Jump length (meters)

Team Participant name student 1

Tigers

student 2

Bears

student 3

Bears

student 4

Tigers

Tigers

Bears

� � � �

89 — 100 Winning team: __________________________________

Tigers with

meter

Four teachers from two grade levels are grading history reports. Which grade level has graded more reports?

Grade level

Fraction of reports graded

4th

�

5th

�

Mr. Garcia

4th

Mr. Stone

5th

� �

Teacher Ms. Lindsey Mrs. Franklin

4th

5th

The grade level that has graded more history reports is ______________ 130

#51616—180 Days of Problem Solving

© Shell Education

WEEK 24 DAY

Name: _______________ Date: _________ Directio n s:

Raquelle ran a total of � mile on Saturday and Sunday. Jaiden ran � mile on Saturday and � mile on Sunday. Jaiden says he ran farther than Raquelle. Is he correct? Use words, numbers, or a picture to show your work. Strategy 1

? Solve It Two Ways!

1.

Show two ways to solve the problem.

4

Strategy 2

2.

Which strategy do you think is better? Explain your reasoning. _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

© Shell Education

#51616—180 Days of Problem Solving

131

WEEK 24 DAY

5

Name: _______________ Date: _________

Challenge Yourself!

Directio n s:

Read and solve the problem.

Mr. Ly asks his students to write addition equations with a sum of � . He tells the class that one addend must have a denominator of 10 and the other addend must have a denominator of 100.

1.

2.

Write three equations. Show your work to prove your equations are correct.

�

+

�

=

�

�

+

�

=

�

�

+

�

=

�

Choose one of your equations. Write a story problem that represents the equation. _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

132

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WEEK 25 DAY

Name: _______________ Date: _________ Directio n s:

Think about the problem. Answer the questions.

Think About It!

Plot 0.78 on the number line. Then, write it as a decimal and as a fraction.

0

1.

1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Maybell reads 0.78 as “zero point seventy-eight.” What advice would you give Maybell as to how to correctly read the decimal? _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

2.

How many tenths are represented in 0.78? How many hundredths? _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

3.

Complete the statement using decimals from the number line. The number 0.78 is greater than ________ , but less than ________.

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133

WEEK 25 DAY

2

Name: _______________ Date: _________ Directio n s:

Solve It!

?

Read and solve each problem.

Problem 1: Plot 0.78 on the number line. Then, write it as a decimal and as a fraction. 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

What Do You Know?

What Is Your Plan?

Solve the Problem!

Look Back and Explain!

Problem 2: Plot 0.41 on the number line. Then, write it as a decimal and as a fraction. 0

134

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

What Do You Know?

What Is Your Plan?

Solve the Problem!

Look Back and Explain!

#51616—180 Days of Problem Solving

© Shell Education

WEEK 25 DAY

Name: _______________ Date: _________ Directio n s:

3

Look at the example. Then, solve the problem.

Fraction

Expanded fraction form using place value

Expanded decimal form using place value

Decimal

0.8 + 0.04

0.84

�

�

�

�+�

0.3 + 0.06

0.36

�

�+�

0.0 + 0.02

0.02

+

�

Visualize It!

Example: Find the decimal equivalent of each fraction by completing the chart.

Find the decimal equivalent of each fraction by completing the chart.

Fraction

Expanded fraction form using place value

�

+

Expanded decimal form using place value

Decimal

�

0.2 + 0.04

�

© Shell Education

#51616—180 Days of Problem Solving

135

WEEK 25 DAY

4

Solve It Two Ways!

?

Name: _______________ Date: _________ Directio n s:

1.

Show two ways to solve the problem.

Write a decimal that is greater than M , but less than � . Use two different strategies to prove that your solution is correct. Strategy 1

Strategy 2

2.

Which strategy do you like better? Explain your reasoning. _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

136

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WEEK 25 DAY

5

Name: _______________ Date: _________ Directio n s:

Read and solve the problem.

?

1.

Challenge Yourself!

Cassidy and Carter are working on their math assignment. So far, Cassidy has completed 0.38 of the assignment. Carter has completed � of the assignment. Which student is closer to being halfway finished?

How can 0.38 be expressed in fraction form? _______________________________________________________________________________

2.

How can � be expressed in decimal form? _______________________________________________________________________________

3.

Plot and label 0.38 and � on the number line.

0

4.

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Explain how you know which student is closer to being halfway finished with the homework assignment. _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

© Shell Education

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137

WEEK 26 DAY

1

Name: _______________ Date: _________

Think About It!

Directio n s:

Think about the problem. Answer the questions.

?

centimeters The lengths, in centimeters, of four ants are 0.07, 0.7, 0.37, and 0.3. Write the lengths in order from least to greatest.

1.

Will comparing only the whole numbers help you write the lengths in order from least to greatest? _______________________________________________________________________________ _______________________________________________________________________________

2.

Are these lengths greater than or less than 1 centimeter? How do you know? _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

3.

Rewrite each length as a fraction with the denominator of 100. _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

138

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WEEK 26 DAY

Name: _______________ Date: _________ Directio n s:

2

Read and solve each problem.

What Do You Know?

What Is Your Plan?

Solve the Problem!

Look Back and Explain!

Solve It!

Problem 1: The lengths, in centimeters, of four ants are 0.07, 0.7, 0.37, and 0.3. Write the lengths in order from least to greatest.

Problem 2: A town in Alaska measures its total number of inches of snow over a four-day period. On Monday, it snows 4.39 inches. On Tuesday, the snowfall is 3.4 inches. The town receives 4.3 inches on Wednesday and 3.58 inches on Thursday. Write the snowfall for each day in order from least to greatest. What Do You Know?

What Is Your Plan?

Solve the Problem!

Look Back and Explain!

© Shell Education

#51616—180 Days of Problem Solving

139

WEEK 26 DAY

3

Name: _______________ Date: _________

Visualize It!

?

Directio n s:

Look at the example. Then, solve the problem.

Example: Plot and label 0.34 and 0.6 on the number line. Then, plot and label three decimals that are greater than 0.34, but less than 0.6.

0.4 0.34 0

0.1

0.2

0.3

0.4

0.52 0.45 0.6 0.5

0.6

0.7

0.8

0.9

1

Plot and label 0.7 and 0.98 on the number line. Then, plot and label three decimals that are greater than 0.7, but less than 0.98.

0

140

0.1

0.2

0.3

#51616—180 Days of Problem Solving

0.4

0.5

0.6

0.7

0.8

0.9

1

© Shell Education

WEEK 26 DAY

Name: _______________ Date: _________ Directio n s:

The student store sells supplies before school. Pencil-top erasers are $0.08 each. Spiral notebooks are $0.80 each. Write a statement using >, < , or = to compare the prices. Then, use two different strategies to prove that your statement is correct.

? Solve It Two Ways!

1.

Show two ways to solve the problem.

4

Strategy 1

Strategy 2

2.

Which strategy do you think is better? Explain your reasoning. _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

© Shell Education

#51616—180 Days of Problem Solving

141

WEEK 26 DAY

5

Name: _______________ Date: _________

Challenge Yourself!

Directio n s:

Read and solve the problem.

Evan’s teacher challenged the class to express 0.57 in as many ways as possible. Evan’s group wrote their answers using a thinking web. If you were Evan’s teacher, would you accept their responses as correct? Explain your reasoning.

1.

0.57 4 tenths and 17 hundredths

50 — 100

+

7 — 100

What is “4 tenths and 17 hundredths” written in fraction form? What result will you get by combining these fractions? Should Evan’s teacher accept this response from the group? _______________________________________________________________________________ _______________________________________________________________________________

2.

What is the sum of � and � in fraction form? What is the sum in decimal form? Should Evan’s teacher accept this response from the group? _______________________________________________________________________________ _______________________________________________________________________________

3.

142

Contribute at least four more unique ways to express 0.57 on the thinking web.

#51616—180 Days of Problem Solving

© Shell Education

WEEK 27 DAY

Name: _______________ Date: _________ Directio n s:

Think about the problem. Answer the questions.

Think About It!

A group of students use wooden sticks to build a tower that is 4M feet tall. How tall is the tower in inches?

1.

1

What information do you need to know that is not stated directly in the problem? _______________________________________________________________________________ _______________________________________________________________________________

2.

Which operations (addition, subtraction, multiplication, and division) will you use to help you solve the problem? _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

3.

Can the answer be 24 inches? Explain how you know. _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

© Shell Education

#51616—180 Days of Problem Solving

143

WEEK 27 DAY

2

Name: _______________ Date: _________ Directio n s:

Solve It!

?

Read and solve each problem.

Problem 1: A group of students use wooden sticks to build a tower that is 4M feet tall. How tall is the tower in inches? What Do You Know?

What Is Your Plan?

Solve the Problem!

Look Back and Explain!

Problem 2: A walkway in front of a school is 9 meters long. How long is the walkway in centimeters?

144

What Do You Know?

What Is Your Plan?

Solve the Problem!

Look Back and Explain!

#51616—180 Days of Problem Solving

© Shell Education

WEEK 27 DAY

Name: _______________ Date: _________ Directio n s:

Look at the example. Then, solve the problem by completing the table.

Feet

Inches

Equation

1

12

1 × 12 = 12

6

72 6 36 96

6 × 12 = 72 M × 12 = � = 6 3 × 12 = 36 8 × 12 = 96

3 8

Visualize It!

Example: Convert the measurements from feet to inches. Write an equation to show how you calculated the answer.

M

3

Convert the measurements from yards to inches. Write an equation to show how you calculated the answer.

Yards

Inches

Equation

1

36

1 × 36 = 36

7 9 4

M

© Shell Education

#51616—180 Days of Problem Solving

145

WEEK 27 DAY

4

Solve It Two Ways!

?

Name: _______________ Date: _________ Directio n s:

1.

Show two ways to solve the problem.

A new highway is being built in three sections. The length of the first section is 4,000 meters. The second section is 8 kilometers. The final section is 3,000 meters. What is the total length of the highway? Strategy 1

Strategy 2

2.

Do you think using meters or kilometers to measure the length of a highway is more practical in the real world? Explain your reasoning. _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

146

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WEEK 27 DAY

5

Name: _______________ Date: _________ Directio n s:

Read and solve the problem.

?

Clue: 1 mile = 5,280 feet

1.

Challenge Yourself!

Alexander sets a goal to walk 10,000 feet. In the morning, he walks his dog for one mile. That evening, he walks 150 yards around the park. The next morning, he walks 1,450 feet to school. He walks that same distance back home after school. How many more feet does Alexander need to walk to achieve his goal?

Why is the clue helpful? _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

2.

How many feet are in 150 yards? Show your work.

3.

Choose a strategy to solve the problem.

© Shell Education

#51616—180 Days of Problem Solving

147

WEEK 28 DAY

1

Name: _______________ Date: _________

Think About It!

Directio n s:

Think about the problem. Answer the questions.

?

On Tuesday, Jeff plays basketball for 25 minutes. Then, he works on his homework for 40 minutes. He begins these activities at 3:50 p.m. What time does Jeff finish?

1.

What information do you know? _______________________________________________________________________________ _______________________________________________________________________________

2.

How can the problem be rewritten to combine the time spent playing basketball and doing homework? _______________________________________________________________________________ _______________________________________________________________________________

3.

Why might it make sense to count by tens when beginning to solve the problem? _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

148

#51616—180 Days of Problem Solving

© Shell Education

WEEK 28 DAY

Name: _______________ Date: _________ Directio n s:

2

Read and solve each problem.

What Do You Know?

What Is Your Plan?

Solve the Problem!

Look Back and Explain!

Solve It!

Problem 1: On Tuesday, Jeff plays basketball for 25 minutes. Then, he works on his homework for 40 minutes. He begins these activities at 3:50 p.m. What time does Jeff finish?

Problem 2: Zara ice skates for 1 hour and 35 minutes. She finishes at 6:15 p.m. What time did Zara start skating? What Do You Know?

What Is Your Plan?

Solve the Problem!

Look Back and Explain!

© Shell Education

#51616—180 Days of Problem Solving

149

WEEK 28 DAY

3

Name: _______________ Date: _________

Visualize It!

?

Directio n s:

Look at the example. Then, solve the problem by completing the number line.

Example: A fourth grade field trip begins at 8:25 a.m. It ends at 3:30 p.m. How long is the field trip? 1 hour 1 hour 1 hour 1 hour 1 hour 1 hour 1 hour 5 minutes

8:25

9:25

10:25 11:25 12:25

1:25

2:25

3:25 3:30

1 hour + 1 hour + 1 hour + 1 hour + 1 hour + 1 hour + 1 hour + 5 minutes = 7 hours and 5 minutes A family travels by car to the beach. They leave home at 11:50 a.m. and arrive at the beach at 4:35 p.m. How long is the car trip?

11:50

4:35

__________________________________________________________________________ __________________________________________________________________________ 150

#51616—180 Days of Problem Solving

© Shell Education

WEEK 28 DAY

Name: _______________ Date: _________ Directio n s:

Kassandra wants to find out how many hours are in 9 days. What is another strategy she can use to solve the problem? Strategy 1

Days

Hours

1 2 3 4 5 6 7 8 9

24 48 72 96 120 144 168 192 216

? Solve It Two Ways!

1.

Solve the problem. Then, answer the question.

4

Strategy 2

2.

Which strategy do you think is better? Explain your reasoning. ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

© Shell Education

#51616—180 Days of Problem Solving

151

WEEK 28 DAY

5

Name: _______________ Date: _________

Challenge Yourself!

Directio n s:

Read and solve the problem.

Which of the following time periods are not equivalent to the others? Explain your reasoning. 42 days

58 days 1,008 hours

1.

6 weeks 720 hours

How can you convert the time periods to make them easier to compare? _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

2.

Choose a strategy to solve the problem. Show your work.

3.

Select one time period that is not equivalent to the others. Write a time period that is equivalent to this amount. _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

152

#51616—180 Days of Problem Solving

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WEEK 29 DAY

Name: _______________ Date: _________ Directio n s:

1

Think about the problem. Answer the questions.

1.

Complete the statement: 1 liter = ____________ milliliters

2.

How many times must Braiden fill the measuring cup to get 1 liter of water? Explain your thinking.

Think About It!

A fish bowl holds 4 liters of water. Braiden uses a measuring cup that holds 500 milliliters to fill the bowl. How many times must Braiden fill the measuring cup to fill the bowl?

_______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

3.

How can repeated addition help you solve the problem? _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

© Shell Education

#51616—180 Days of Problem Solving

153

WEEK 29 DAY

2

Name: _______________ Date: _________ Directio n s:

Solve It!

?

Read and solve each problem.

Problem 1: A fish bowl holds 4 liters of water. Braiden uses a measuring cup that holds 500 milliliters to fill the bowl. How many times must Braiden fill the measuring cup to fill the bowl? What Do You Know?

What Is Your Plan?

Solve the Problem!

Look Back and Explain!

Problem 2: A puppy weighs 2 kilograms. The puppy’s weight increases by 2,000 grams after one week. How many grams does the puppy weigh now?

154

What Do You Know?

What Is Your Plan?

Solve the Problem!

Look Back and Explain!

#51616—180 Days of Problem Solving

© Shell Education

WEEK 29 DAY

Name: _______________ Date: _________ Directio n s:

Look at the example. Then, solve the problem by completing the table.

Kilograms

1,000 1

2,000 2

3,000 4,000 5,000 6,000 3 4 5 6

Visualize It!

Example: How many kilograms are in 5,000 grams?

Grams

3

5

5,000

____________ grams is equal to ____________ kilograms.

How many tons are in 6,000 pounds?

Pounds

4,000

Tons

2

____________ pounds is equal to ____________ tons.

© Shell Education

#51616—180 Days of Problem Solving

155

WEEK 29 DAY

4

Solve It Two Ways!

?

Name: _______________ Date: _________ Directio n s:

1.

Show two ways to solve the problem.

A deli sells cheese by the pound. A restaurant owner purchases 6M pounds of cheddar from the deli. He also buys 3M pounds of mozzarella. How many ounces of cheese does the restaurant owner buy in all? Strategy 1

Strategy 2

2.

Which strategy do you prefer? Explain your reasoning. _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

156

#51616—180 Days of Problem Solving

© Shell Education

WEEK 29 DAY

5

Name: _______________ Date: _________ Directio n s:

Read and solve the problem.

?

Clue: 1 quart = 32 fluid ounces

1.

Challenge Yourself!

Elan is celebrating his birthday along with eight of his friends. His mom buys 3 quarts of fruit punch and fills each of nine cups with 6 fluid ounces for Eric and his guests. Will Eric’s mom have enough punch to pour everyone a second full cup of punch? Explain your reasoning.

Why is the clue helpful? _______________________________________________________________________________ _______________________________________________________________________________

2.

How many ounces are in 3 quarts? Write an equation to show your answer. _______________________________________________________________________________ _______________________________________________________________________________

3.

Choose a strategy to solve the problem. Then, justify your answer.

_______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ © Shell Education

#51616—180 Days of Problem Solving

157

WEEK 30 DAY

1

Name: _______________ Date: _________

Think About It!

Directio n s:

Think about the problem. Answer the questions.

?

A farmer is installing a fence around a rectangular chicken coop. The length of the coop is 24 feet and the width is 14 feet. How many feet of fencing does the farmer use?

1.

The farmer’s daughter, Isabel, says, “We can multiply 24 by 14 to find how many feet of fencing we will use.” Should the farmer take Isabel’s advice? Why or why not? _______________________________________________________________________________ _______________________________________________________________________________

2.

How might a drawing help you solve the problem? _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

3.

Will the farmer require more than 100 feet or less than 100 feet of fencing? How do you know? _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

158

#51616—180 Days of Problem Solving

© Shell Education

WEEK 30 DAY

Name: _______________ Date: _________ Directio n s:

2

Read and solve each problem.

What Do You Know?

What Is Your Plan?

Solve the Problem!

Look Back and Explain!

Solve It!

Problem 1: A farmer is installing a fence around a rectangular chicken coop. The length of the coop is 24 feet and the width is 14 feet. How many feet of fencing does the farmer use?

Problem 2: Camilla’s art project is displayed on a rectangular sheet of poster paper measuring 18 inches by 12 inches. What is the perimeter of the poster paper? What Do You Know?

What Is Your Plan?

Solve the Problem!

Look Back and Explain!

© Shell Education

#51616—180 Days of Problem Solving

159

WEEK 30 DAY

3

Name: _______________ Date: _________

Visualize It!

?

Directio n s:

Look at the example. Then, solve the problem.

Example: Elsie is installing a small fence around a flower bed at the park. Complete the table to show three strategies for finding the perimeter of the flower bed.

12 ft. 5 ft.

5 ft. 12 ft.

l+w+l+w

2(l + w)

2l + 2w

12 + 5 + 12 + 5 = 34 ft.

2(12 + 5) = 34 ft.

2(12) + 2(5) = 34 ft.

1.

Elsie is putting border paper around the bulletin board in the park’s 19 in. office. Complete the table to show three strategies for finding the perimeter of the bulletin board.

l+w+l+w

2.

2(l + w)

43 in. 19 in. 43 in.

2l + 2w

What do you notice about the three strategies for finding perimeter? Why do you think this happens? _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

160

#51616—180 Days of Problem Solving

© Shell Education

WEEK 30 DAY

Name: _______________ Date: _________ Directio n s:

?

The manager at Paws and Claws Day Care is installing a fence to create a dog exercise section. The manager has 96 yards of fencing. What are the length and width of two different rectangular sections that can be built using all 96 yards of fencing? Solution 1

Solve It Two Ways!

1.

Show two solutions for the problem.

4

Solution 2

2.

Which solution do you think is better? Explain your reasoning. ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

© Shell Education

#51616—180 Days of Problem Solving

161

WEEK 30 DAY

5

Name: _______________ Date: _________

Challenge Yourself!

Directio n s:

Read and solve the problem. The perimeter of a rectangular television screen is 64 inches. The length of the screen is 19 inches. What is the width of the television screen?

1.

Sketch the television screen. Label your sketch with all four dimensions. Choose a strategy to prove that the screen has a perimeter of 64 inches.

2.

What would be the perimeter of a television screen that is twice as large as the screen in the problem? Give an example of a possible length and width for this larger screen. Prove that these dimensions result in the correct perimeter. _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

162

#51616—180 Days of Problem Solving

© Shell Education

WEEK 31 DAY

Name: _______________ Date: _________ Directio n s:

Think about the problem. Answer the questions.

Think About It!

A construction worker is placing 1-foot by 1-foot vinyl squares on the floor of a rectangular room. The length of the floor is 15 feet. The width is 18 feet. How many square feet of vinyl flooring will the construction worker use?

1.

1

What is the problem asking you to find? _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

2.

Why is the problem asking you to express the answer in square feet? _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

3.

How is area different from perimeter? Is the problem asking you to find the area or perimeter of the floor? _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

© Shell Education

#51616—180 Days of Problem Solving

163

WEEK 31 DAY

2

Name: _______________ Date: _________ Directio n s:

Solve It!

?

Read and solve each problem.

Problem 1: A construction worker is placing 1-foot by 1-foot vinyl squares on the floor of a rectangular room. The length of the floor is 15 feet. The width is 18 feet. How many square feet of vinyl flooring will the construction worker use? What Do You Know?

What Is Your Plan?

Solve the Problem!

Look Back and Explain!

Problem 2: The length of one side of a square table is 48 inches. What is the area of the surface of the table?

164

What Do You Know?

What Is Your Plan?

Solve the Problem!

Look Back and Explain!

#51616—180 Days of Problem Solving

© Shell Education

WEEK 31 DAY

Name: _______________ Date: _________ Directio n s:

Look at the example. Then, solve the problem.

Length × Width = Area Area = 45 square feet

? × 3 = 45 ________________________________________________ 15 × 3 = 45 ________________________________________________ 15 feet ________________________________________________

Visualize It!

Example: Find the length of each missing side.

3 feet

3

Length × Width = Area Area = 100 square feet 20 feet

20 × ? = 100 20 × 5 = 100 ________________________________________________ ________________________________________________ 5 feet ________________________________________________

Find the length of each missing side.

Length × Width = Area 9 feet

Area = 216 square feet

________________________________________________ ________________________________________________ ________________________________________________

Length × Width = Area

© Shell Education

Area = 222 square feet

________________________________________________

37 feet

________________________________________________

________________________________________________

#51616—180 Days of Problem Solving

165

WEEK 31 DAY

4

Solve It Two Ways!

?

Name: _______________ Date: _________ Directio n s:

1.

Show two solutions for the problem.

A rectangle has a length of 1 meter and a width of 4 meters. The area of the rectangle is 4 square meters. Find the dimensions of two rectangles that have double this area. Draw a sketch and prove that your solutions are correct. Solution 1

Solution 2

2.

Will doubling each dimension of the original rectangle, forming a 2 meter by 8 meter rectangle, result in a correct solution for the problem? Why or why not? _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

166

#51616—180 Days of Problem Solving

© Shell Education

WEEK 31 DAY

5

Name: _______________ Date: _________ Directio n s:

Read and solve the problem.

? Challenge Yourself!

Deanne buys a new rectangular rug for her house. The rug has an area of 84 square feet.

1.

What are two possible sets of dimensions for a rug with an area of 84 square feet? _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

2.

Sketch and label the dimensions of each rectangular rug. Use the sketches to calculate the perimeters of each rug.

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WEEK 32 DAY

1

Name: _______________ Date: _________

Think About It!

Directio n s:

Think about the problem. Answer the questions.

?

Label each angle in the polygon as acute, obtuse, or right.

1.

How are the three types of angles different from one another? _______________________________________________________________________________

R

_______________________________________________________________________________ _______________________________________________________________________________

2.

How is an angle formed? How many angles does the figure have?

O

_______________________________________________________________________________

A

_______________________________________________________________________________

R

O

_______________________________________________________________________________

3.

What is the relationship between the number of sides, angles, and vertices in a polygon? _______________________________________________________________________________

B

_______________________________________________________________________________ _______________________________________________________________________________ 168

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A

WEEK 32 DAY

2

Name: _______________ Date: _________ Directio n s:

Read and solve each problem.

What Do You Know?

Solve It!

Problem 1: Label each angle in the polygon as acute, obtuse, or right. What Is Your Plan? R

O

A

R

Solve the Problem!

A

O

B Look Back andR Explain! Z R

A

C O

E

A

D

O

B

Problem 2: Label each angle in the polygon as acute, obtuse, or right. What Do You Know?

What Is Your Plan?

Solve the Problem!

Look Back and Explain!

© Shell Education

Z

C

E

D

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WEEK 32 DAY

3

Name: _______________ Date: _________

Visualize It!

?

Directio n s:

Look at the example. Then, solve the problem.

Example: Draw line segment BD perpendicular to line segment AB. Connect point D to Point C to create a four-sided figure. C

A

D

B

acute angle

What type of angle is CAB? ___________________________________

right angle

What type of angle is ABD? ___________________________________

1.

Draw line segment MP parallel to line segment NO. Connect point O to point P. M

N

170

O

2.

What type of angle is MNO? _____________________________

3.

What type of angle is OPM? _____________________________

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WEEK 32 DAY

Name: _______________ Date: _________ Directio n s:

Write a set of directions to help someone sketch an equilateral triangle. Write a second set of directions to help someone sketch a scalene triangle. Be sure to describe the sides and angles. Equilateral triangle _______________________________________________________________________________

? Solve It Two Ways!

1.

Show two ways to solve the problem.

4

____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ Scalene triangle _______________________________________________________________________________

____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

2.

What is similar about your two sets of directions? What is different? ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

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WEEK 32 DAY

5

Name: _______________ Date: _________

Challenge Yourself!

Directio n s:

1.

Read and solve the problem.

Draw a three-sided polygon. Describe your polygon using as many of these terms as possible. right angle

acute angle

obtuse angle

line segment

_______________________________________________________________________________ _______________________________________________________________________________

2.

Draw a four-sided polygon. Describe your polygon using as many of these terms as possible. right angle parallel

acute angle perpendicular

obtuse angle

line segment

_______________________________________________________________________________ _______________________________________________________________________________

3.

172

Choose one of the terms from questions 1 or 2 that you did not use. Draw an illustration of the term.

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WEEK 33 DAY

Name: _______________ Date: _________ Directio n s:

Think about the problem. Answer the questions.

Think About It!

Suzanne is making a mosaic with triangular-shaped tiles. Each angle in the triangular tiles measures 60 degrees. As a part of her design, Suzanne is making a full rotation of triangular tiles. How many tiles will Suzanne need to make one 360-degree rotation?

1.

1

What information is given? _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

2.

What are you trying to find out? _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

3.

Can the answer be 10 tiles? Explain how you know. _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

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WEEK 33 DAY

2

Name: _______________ Date: _________ Directio n s:

Solve It!

?

Read and solve each problem.

Problem 1: Suzanne is making a mosaic with triangular-shaped tiles. Each angle in the triangular tiles measures 60 degrees. As a part of her design, Suzanne is making a full rotation of triangular tiles. How many tiles will Suzanne need to make one 360-degree rotation? What Do You Know?

What Is Your Plan?

Solve the Problem!

Look Back and Explain!

Problem 2: If one angle created by turning a page in a book is 94 degrees, what is the measurement of the other angle?

174

?

94º

What Do You Know?

What Is Your Plan?

Solve the Problem!

Look Back and Explain!

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WEEK 33 DAY

Name: _______________ Date: _________ Directio n s:

3

Look at the example. Then, solve the problem.

30o

Visualize It!

Example: Use a protractor to draw an acute angle measuring 30 degrees. Then, draw an angle that is 15 degrees larger than the 30-degree angle. Label each angle.

45o

Use a protractor to draw a right angle. Then, draw an obtuse angle that is 30 degrees larger than a right angle and an acute angle that is 45 degrees less than a right angle. Label each angle.

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WEEK 33 DAY

4

Solve It Two Ways!

?

Name: _______________ Date: _________ Directio n s:

1.

Show two ways to solve the problem.

Sean draws two angles that share a common vertex and a common side. The two angles form a right angle. One of the angles is 70 degrees. What is the measurement of the other angle? Strategy 1

Strategy 2

2.

Which strategy do you prefer? Explain your reasoning. _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

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WEEK 33 DAY

5

Name: _______________ Date: _________ Directio n s:

Read and solve the problem.

castle wall

Challenge Yourself!

A drawbridge at a historical castle forms a right angle when fully open. As the drawbridge opens, it moves and then pauses before moving again. From a closed position, the bridge needs to do this 5 times to fully open. What is the measurement of each angle if the bridge opens the same amount each time it moves?

?

open drawbridge

1.

The drawbridge opens in five equal movements. Sketch the angles in the image above, estimating where the drawbridge will pause.

2.

What are the measurements of the five equal sections? How do you know? _______________________________________________________________________________ _______________________________________________________________________________

3.

Use a protractor to measure the angles in your sketch, checking your estimates. How do your estimates compare with the actual answer? _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

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WEEK 34 DAY

1

Name: _______________ Date: _________

Think About It!

Directio n s:

Think about the problem. Answer the questions.

?

Krista is saving money in a piggy bank. Every day, she doubles the amount from the previous day. She puts 4 cents in the bank on the first day. How much will Krista put in the piggy bank on the eighth day?

1.

How can a table help you solve the problem? How would you organize the table? _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

2.

Is this a repeating pattern or a growing pattern? Explain how you know. _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

3.

Can the answer be 32 cents? Why or why not? _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

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WEEK 34 DAY

Name: _______________ Date: _________ Directio n s:

2

Read and solve each problem.

What Do You Know?

What Is Your Plan?

Solve the Problem!

Look Back and Explain!

Solve It!

Problem 1: Krista is saving money in a piggy bank. Every day, she doubles the amount from the previous day. She puts 4 cents in the bank on the first day. How much will Krista put in the piggy bank on the eighth day?

Problem 2: If the pattern continues, what will be the 16th shape?

What Do You Know?

What Is Your Plan?

Solve the Problem!

Look Back and Explain!

© Shell Education

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WEEK 34 DAY

3

Name: _______________ Date: _________

Visualize It!

?

Directio n s:

Look at the example. Then, solve the problem.

Example: Find the rule for the In and Out Table. Use the rule to write the missing numbers.

In

Out

4

32

9

72

12 15

96 120

20

160

25

200

Multiply by 8

Rule: _____________________________________

Find the rule for this In and Out Table. Use the rule to write the missing numbers.

In

Out

7

28

12 14 17

38

Rule: _____________________________________

45 28

180

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WEEK 34 DAY

Name: _______________ Date: _________ Directio n s:

Start with the number 6. Create a rule that results in all even numbers. Then, create a rule that results in alternating odd and even numbers. Prove each rule works by listing the first five results. Rule 1

? Solve It Two Ways!

1.

Show two ways to solve the problem.

4

6 _________ _________ _________ _________ _________

Rule: ________________________________________________________

Rule 2 6 _________ _________ _________ _________ _________

Rule: ________________________________________________________

2.

Which rule was easier to determine? Explain your reasoning. _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

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WEEK 34 DAY

5

Name: _______________ Date: _________

Challenge Yourself!

Directio n s:

Read and solve the problem.

Gloria is making a necklace with beads in the shapes of stars, suns, and moons. She begins by placing one sun, two moons, and two stars on the necklace. If she continues this pattern, what will be the 100th bead on the necklace?

1.

Sketch the first ten beads in the necklace.

2.

How can your sketch help you determine the 100th bead? Solve the problem and justify your solution.

_______________________________________________________________________________ _______________________________________________________________________________

3.

Using the star, sun, and moon beads, draw a necklace pattern of your own. Determine what the 100th bead on your necklace will be.

_______________________________________________________________________________ 182

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WEEK 35 DAY

Name: _______________ Date: _________ Directio n s:

Think about the problem. Answer the questions.

Think About It!

Chad draws a four-sided polygon. The quadrilateral contains only one set of parallel lines and no right angles. What type of quadrilateral is this? Draw a sketch to prove your answer.

1.

1

What do you know about the shape? _______________________________________________________________________________ _______________________________________________________________________________

2.

What will you do to solve the problem? _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

3.

Is a rectangle a possible solution for the quadrilateral? Why or why not? _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

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WEEK 35 DAY

2

Name: _______________ Date: _________ Directio n s:

Solve It!

?

Read and solve each problem.

Problem 1: Chad draws a four-sided polygon. The quadrilateral contains only one set of parallel lines and no right angles. What type of quadrilateral is this? Draw a sketch to prove your answer. What Do You Know?

What Is Your Plan?

Solve the Problem!

Look Back and Explain!

Problem 2: A rectangle is cut diagonally to form two equal triangles. What type of triangles are formed? Draw a picture to prove your answer.

184

What Do You Know?

What Is Your Plan?

Solve the Problem!

Look Back and Explain!

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WEEK 35 DAY

Name: _______________ Date: _________ Directio n s:

3

Look at the example. Then, solve the problem.

Quadrilateral 1: four equal sides and four right angles

square

Quadrilateral 2: opposite sides parallel and no right angles

parallelogram

Visualize It!

Example: Draw and identify each quadrilateral.

Draw and identify each triangle. Triangle 1: one obtuse angle and two equal sides

Triangle 2: one right angle and no equal sides

Triangle 3: three acute angles and three equal sides

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WEEK 35 DAY

4

Solve It Two Ways!

?

Name: _______________ Date: _________ Directio n s:

1.

Show two ways to solve the problem.

Classify and sort the six polygons into two groups. Then, classify and sort the six polygons a different way. Strategy 1

Group A

Group B

Group A

Group B

Strategy 2

2.

Describe each rule you used to sort the polygons. _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

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WEEK 35 DAY

5

Name: _______________ Date: _________ Directio n s:

Read and solve the problem.

?

1.

How many sides must the shape have? _____________________________________________________________ _______________________________________________________________

2.

Draw the shape. Label the parallel sides and the right angles.

3.

What type of polygon did you draw? Is there another type of polygon you can draw to match this description? Explain your thinking.

Challenge Yourself!

Jermaine is thinking of a shape that has exactly one set of parallel sides and at least two right angles.

_______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ © Shell Education

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WEEK 36 DAY

1

Name: _______________ Date: _________

Think About It!

Directio n s:

Think about the problem. Answer the questions.

?

Paul says a rectangle has three lines of symmetry. His friend Nancy says one of the three lines of symmetry is incorrect. Who do you agree with and why?

1.

What does it mean to have a line of symmetry? _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

2.

How can you prove that a figure has a line of symmetry? _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

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WEEK 36 DAY

Name: _______________ Date: _________ Directio n s:

Read and solve the problem.

What Do You Know?

What Is Your Plan?

Solve the Problem!

Look Back and Explain!

#51616—180 Days of Problem Solving

Solve It!

Problem: Paul says a rectangle has three lines of symmetry. His friend Nancy says one of the three lines of symmetry is incorrect. Who do you agree with and why?

© Shell Education

2

189

WEEK 36 DAY

3

Name: _______________ Date: _________

Visualize It!

?

Directio n s:

Look at the example. Then, solve the problem.

Example: Identify and draw all the lines of symmetry for each triangle.

0 lines of symmetry 3 lines of symmetry

1 lines of symmetry

________________________ ________________________ ________________________

Identify and draw all the lines of symmetry for each quadrilateral.

____________________

____________________

____________________ 190

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WEEK 36 DAY

Name: _______________ Date: _________ Directio n s:

?

Find all the lines of symmetry for the two pentagons. Solution 1

Solve It Two Ways!

1.

Show two solutions for the problem.

4

Solution 2

2.

Do the two pentagons have the same number of lines of symmetry? Why or why not? _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

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WEEK 36 DAY

5

Name: _______________ Date: _________

Challenge Yourself!

Directio n s:

Read and solve the problem.

Half of a figure and a line of symmetry are shown.

1.

How can the line of symmetry help you complete the drawing of the figure? _______________________________________________________________________________ _______________________________________________________________________________

2.

Complete the figure.

3.

Is it possible to draw another line of symmetry on the completed figure? Draw the line and explain your thinking to prove your answer. _______________________________________________________________________________ _______________________________________________________________________________

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Answer Key Week 1: Day 1 (page 13) 1. There are 3,000 pieces of chocolate. The factory wants to put the chocolate into packages of 10. 2. Possible answer: I know that 10 tens is equal to 100. I also know that 100 tens is equal to 1,000. 3. The answer cannot be 30 packages of 10 chocolates. That would equal 300 chocolates, not 3,000. Week 1: Day 2 (page 14) 1. 300 packages; divide to find how many groups of 10 are in 3,000; 3,000 ÷ 10 = 300 2. 180 stacks; divide to find how many groups of 1,000 are in 180,000; 180,000 ÷ 1,000 = 180 Week 1: Day 3 (page 15) 1. Answers will vary but should equal 294. Hundreds Tens Ones

Total

2

9

4

200 + 90 + 4 = 294

2

6

34

200 + 60 + 34 = 294

1

19

4

100 + 190 + 4 = 294

2. In 169, the digit 9 represents 9 ones. In 294, the digit 9 represents 9 tens, or 90. The tens place is one place to the left of the ones place, so the digit represents ten times what it would represent when in the ones place. Week 1: Day 4 (page 16) 1. Possible answer: 1,500 + 120 + 26 = 1,646 2. Possible answer: I liked writing each number according to its place value, and then adding the numbers together. Week 1: Day 5 (page 17) 1. Drawings will vary but should include dollar amounts equal to $2,490. Possible answer: two $1,000 bills, four $100 bills, and nine $10 bills. 2. Player 2 must pay Player 1 $2,490. After multiplying by ten, each digit moves one place value position to the left, and is now worth ten times what it was worth originally. So, 9 ones is now worth 9 tens, 4 tens is now worth 4 hundreds, and 2 hundreds is now worth 2 thousands.

© Shell Education

(cont.)

Week 2: Day 1 (page 18) 1. There are six digits in standard form. The greatest place for this number is hundred thousands, followed by ten thousands, thousands, hundreds, tens, and ones. 2. 300,000; 60 3. Ten thousands and hundreds Week 2: Day 2 (page 19) 1. Standard form: 305,061; Expanded form is 300,000 + 5,000 + 60 + 1. There are six digits in the number. The hundred thousands place is the greatest and the value is 300,000. There are 0 ten thousands. There are 5 thousands (5,000). There are 0 hundreds. There are 6 tens (60) and 1 one (1). 2. Expanded form: 700,000 + 9,000 + 200 + 5; Word form: seven hundred nine thousand, two hundred five. There are six digits in the number. The hundred thousands place is the greatest and the value is 700,000. There are 0 ten thousands. There are 9 thousands (9,000). There are 2 hundreds (200). There are 0 tens and 5 ones (5). Week 2: Day 3 (page 20) Standard form: 84,396; Expanded form: 80,000 + 4,000 + 300 + 90 + 6; Riddle: right Week 2: Day 4 (page 21) 1. Student 1 copied only the first digit in each number, creating a number with the greatest place being thousands. The greatest place should be hundred thousands. This made the remaining place values incorrect, with each digit one place to the right of where it should be. Student 2 wrote 700 thousand instead of 709 thousand. This also made the remaining place values incorrect, with each digit one place to the right of where it should be. 2. 700,000 + 9,000 + 400 + 80 = 709,480 3. Possible answer: Expanded form shows the value of each digit. When these values are added correctly, the standard form is the answer.

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Answer Key Week 2: Day 5 (page 22) 1. 382,905 2. 300,000 + 80,000 + 2,000 + 900 + 5 3. Three hundred eighty-two thousand, nine hundred five Week 3: Day 1 (page 23) 1. Possible answer: All the numbers have six digits. I will need to compare the digits in each place value position. I see that each number has a three in the hundred thousands place, but the Power Drop coaster has a three in the ten thousands place whereas the other numbers have a zero in the ten thousands place. This gives me a clue as to which roller coaster had the greatest number of riders. 2. No, all of the amounts have the digit 3 in the hundred thousands place. 3. Possible answer: The rest of the roller coasters have a 0 in the ten thousands place. I will then compare the thousands, then the hundreds, and then the tens place. Week 3: Day 2 (page 24) 333,033; 303,303; 303,003; 300,330; compare by using place value Week 3: Day 3 (page 25) 1. The numbers should be plotted on the number line in the following order: 136,396; 136,515; 136,714; 137,915. 2. The fifth number should be greater than 136,714, but less than 137,915. Week 3: Day 4 (page 26) 1. Possible answers: 489,378 < 498,782 or 498,782 > 489,378; Possible strategies: place value chart, number line; 489,378 should be placed to the left of 498,782 on a number line 2. Possible answer: I think using a place value chart is better because I can compare the digits in each place value more easily. Each number has six digits and the ten thousands place is the digit that shows which population is greater.

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(cont.)

Week 3: Day 5 (page 27) 1. Answers will vary, but each number should use the digits 1–6 once and only once, and all four numbers must be unique. 2. Possible answer: I can be sure that my numbers are different by comparing the digits in each place value position so that I do not repeat the same number. 3. Answers will vary, but the four numbers should be ordered from least to greatest on the number line. Week 4: Day 1 (page 28) 1. Possible answer: Since the nails come in boxes of 1,000, I can round 26,834 to the nearest thousand to make sure the builder has enough nails. 2. The number must be greater or there will not be enough nails. 3. Possible answer: If the builder uses nails that come in boxes of 100, there would be a lot more boxes, so 1,000 is probably an easier quantity to order. But, boxes of 100 would make it easier to order an amount that is closer to the exact amount of nails needed. Week 4: Day 2 (page 29) 27 boxes of nails; round 26,834 to 27,000; divide to find how many groups of 1,000 are in 27,000 Week 4: Day 3 (page 30) Bus 200,000 miles; truck 100,000 miles; airplane 300,000 miles; train 100,000 miles Week 4: Day 4 (page 31) 1. The number can be rounded to the nearest ten thousand to 330,000, to the nearest thousand to 329,000, or to the nearest hundred thousand to 400,000. Otherwise, not enough tires will be ordered. 2. Rounding to the nearest thousand will save the most money since 1,000 fewer tires will need to be ordered.

© Shell Education

Answer Key Week 4: Day 5 (page 32) 1. Finest: 146; Wonder Foods: 124; Deal Mart: 179; Food Square: 142. Possible explanation: I rounded each amount of apples to the nearest hundred to help find my number of baskets. I can prove that each rounded amount is correct by multiplying the number of baskets by 100: 146 × 100 = 14,600; 124 × 100 = 12,400; 179 × 100 = 17,900; 142 × 100 = 14,200. 2. Food Square; Possible explanation: Food Square will have the most leftover apples because 14,163 is the furthest from its rounded result of 14,200. 14,200 – 14,163 = 37 apples 3. Deal Mart; Possible explanation: Deal Mart will have the fewest leftover apples because 17,889 is the closest to its rounded result of 17,900. 17,900 – 17,889 = 11 apples Week 5: Day 1 (page 33) 1. Monday: 9,000; Tuesday: 13,000; Wednesday: 11,000 2. 33,000 steps 3. The actual steps will be less than the estimate because all three numbers are rounded up to the nearest thousand. Week 5: Day 2 (page 34) 32,410 steps; add the three numbers together to find the total number of steps she walked; 8,563 + 12,954 + 10,893 = 32,410 Week 5: Day 3 (page 35) Places passed

Distance in feet

Home to forest

2,684 feet

© Shell Education

Forest to school

2,309 feet

School to pond

Total distance from home to pond

6,320 feet

2,684 + 2,309 + 6,320 =11,313 feet

(cont.)

Week 5: Day 4 (page 36) 1. 8,788 customers; Possible strategies: Using standard algorithm: 2,397 + 3,178 + 3,213 = 8,788 customers; Using place value: (2000 + 3,000 + 3,000 = 8,000); (300 + 100 + 200 = 600); (90 + 70 +10 = 170); (7 + 8 + 3 = 18); 8,000 + 600 + 170 + 18 = 8,788 customers; Using groupings: 2,397 + 3,178 = 5,575 and 5,575 + 3,213 = 8,788 customers 2. Possible answer: I think using place value is better because I can easily see how the numbers will combine to make a sum. Week 5: Day 5 (page 37) 1. Jacket, sweater, and shoes; $88 + $49 + $30 = $167 2. Possible answer: I can check my solution by adding other combinations of three items to see if there is a combination with a sum closer to $170 without going over. 3. No, $200 is not enough money to buy one of each item: $43 + $18 + $49 + 23 + 30 + $88 = $251. The amount is greater than $200. Week 6: Day 1 (page 38) 1. The population was 115,932 two years ago. The population today is 98,739. 2. Find how much the population decreased in two years. 3. Possible answer: The population has decreased by about 15,000 when I estimate using simpler numbers: 115,000 – 100,000 = 15,000. Week 6: Day 2 (page 39) 17,193 people; 115,932 – 98,739 = 17,193; check by adding 17,193 + 98,739 = 115,932 Week 6: Day 3 (page 40) 606 kilometers; 2,500 – 1,894 = 606 Week 6: Day 4 (page 41) 1. Student 1 added the oranges instead of subtracting them. Student 2 subtracted each of the places incorrectly. Regrouping is required for the ones and hundreds places, and the student did not do this. The student simply subtracted the lesser digit from the greater digit and ignored the correct order. 2. The correct answer is 4,337 oranges. Strategies will vary but may include using a standard algorithm to subtract 8,186 – 3,849 = 4,337 or adding on to 3,849 on an open number line to find the total 8,186.

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Answer Key Week 6: Day 5 (page 42) 1. Possible answer: I can add a number of my choice to 15,723 to get a total or sum. This total minus the number I selected will result in 15,723. 2. The story problem should include the correct information and involve taking away or finding the difference between two numbers. 3. Possible equation: 30,250 – 14,527 = 15,723. Possible story problem: An intersection had 30,250 vehicles go through it in December. In January, only 14,527 vehicles went through the intersection. How many more vehicles went through the intersection in December than in January? I can check my work by adding 14,527 + 15,723 to find 30,250.

(cont.)

Week 7: Day 5 (page 47) 1. There are clues in the problem, the word “receive” means to add packages and the phrase “shipped out” means to subtract packages. 2. 210 packages; 1,240 + 325 – 560 + 325 – 560 – 560 = 210 packages 3. Possible answer: If I change the 1,240 packages to 1,030 packages by subtracting the 210 packages that were left in the original problem, there will be no packages left at the end of the day on Friday. 1,030 + 325 – 560 + 325 – 560 – 560 = 0 packages

Week 7: Day 1 (page 43) 1. Find the total price of the teddy bear and the pogo stick. 2. Find the difference between the total of the items and 50 dollars. 3. Possible answer: Five dollars is a reasonable estimate. I can use simpler numbers to find an estimated answer: 20 + 25 = 45 and 50 – 45 = 5.

Week 8: Day 1 (page 48) 1. Find all the ways to put 20 celery sticks into snack bags so that each snack bag has the same amount of celery sticks. 2. Listing factors of 20 will help find the possibilities for the number of snack bags the cafeteria manager can prepare. 3. Possible answer: The cafeteria manager cannot prepare 7 snack bags because 7 is not a factor of 20. There would not be an equal amount of celery sticks in each of the 7 bags.

Week 7: Day 2 (page 44) $6; add the cost of the pogo stick ($25) and teddy bear ($19); and subtract the sum from $50; $50 – $44 = $6; subtract each amount from $50; $50 – $25 – $19 = $6

Week 8: Day 2 (page 49) 1 bag of 20 celery sticks; 2 bags of 10 celery sticks; 4 bags of 5 celery sticks; 5 bags of 4 celery sticks; 10 bags of 2 celery sticks; 20 bags of 1 celery stick; find all of the factors of 20

Week 7: Day 3 (page 45) 462 pages; 602 + 742 + 323 + 477 + 394 = 2,538 pages; 3,000 – 2,538 = 462

Week 8: Day 3 (page 50) 1.

Week 7: Day 4 (page 46) 1. 6,999 people; Possible strategies: 5,841 – 1,381 + 2,539 = 6,999; 5,841 + 2539 – 1,381 = 6,999 2. Possible answer: I like the first strategy because following the order of the numbers in the story situation helps me understand what is happening in the story problem.

Grade level

4

Students

Number of students in equal rows (factors)

28

1 row of 28 students 2 rows of 14 students 4 rows of 7 students 7 rows of 4 students 14 rows of 2 students 28 rows of 1 student

2. Possible answer: A factor pair provides two answers. For example, 2 rows of 14 and 14 rows of 2 because 2 × 14 and 14 × 2 are 28.

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#51616—180 Days of Problem Solving

© Shell Education

Answer Key Week 8: Day 4 (page 51) 1. Kit is correct; 2 has exactly two factors, 1 and itself, so it is prime; Possible strategies: factor list (1, 2); factor pairs (1 × 2 and 2 × 1); arrays (2 rows of 1 or 1 row of 2) 2. Possible answer: The arrays clearly show that there are exactly two ways to make 2: 1 × 2 and 2 × 1. Prime numbers have exactly two arrays, so 2 must be prime. Week 8: Day 5 (page 52) 1. 25; Factors: 1, 5, and 25 2. Three possible arrays: 1 × 25, 25 × 1, and 5 × 5.

(cont.)

Week 9: Day 3 (page 55) 6 packages; Drawing should include 6 packages of bottled water; Labels: 12, 24, 36, 48, 60, 72 Week 9: Day 4 (page 56) 1. Possible strategies: Using listing or skip counting to prove that 91 is a multiple of 13 (13, 26, 39, 52, 65, 78, 91); using a multiplication problem, such as 7 × 13 = 91; or a labeled sketch of bundles of 13 2. Possible answer: I like skip counting because it makes it easier for me to find the multiples of a number and to check my work. Week 9: Day 5 (page 57) 1. 9, 18, 27, 36, 45, 54 (first six multiples of 9) 2. 99 points; Possible explanation: I would multiply 9 × 11, which is 99, or I would skip count by 9s eleven times until I get to 99. 3. 14 levels; Possible explanation: I would continue skip counting by 9s fourteen times until I get to 126, or multiply 9 × 14, which is 126. Week 10: Day 1 (page 58) 1. Katherine has 8 daffodils and the neighbor has 7 times as many daffodils as Katherine. 2. Find the number of daffodils in the neighbor’s garden. 3. Margo is incorrect. Possible explanation: Margo added 8 + 7 to get 15, but the neighbor has 7 times as many daffodils as Katherine, not 7 more daffodils than Katherine.

Week 9: Day 1 (page 53) 1. Gavin’s mother will take him to the library every 4 days. This month has 30 days. 2. The problem is asking to find multiples. Possible explanation: I need to count by 4s to get the solution, not find out what factors can be multiplied together to make 4. 3. Possible answer: Gavin will not visit the library on day 30 because 30 is not a multiple of 4. If I count by 4s, I will not say 30. Week 9: Day 2 (page 54) 1. 7 times; find the multiples of 4 that are less than or equal to 30; Multiples: 4, 8, 12, 16, 20, 24, 28 2. 5 times; find the multiples of 6 that are less than or equal to 30; Multiples: 6, 12, 18, 24, 30

© Shell Education

Week 10: Day 2 (page 59) 1. 56 daffodils; multiply (8 × 7 = 56); add 8 seven times (8 + 8 + 8 + 8 + 8 + 8 + 8 = 56) 2. 27 meters; multiply (9 × 3 = 27); add 9 three times (9 + 9 + 9 = 27) Week 10: Day 3 (page 60) 4 × 8 = 32; The fruit bowl has 8 pieces of fruit. 32 pieces of fruit Fruit basket:

8

Fruit bowl:

8

8

8

8

#51616—180 Days of Problem Solving

197

Answer Key Week 10: Day 4 (page 61) 1. 12 times as many days; Possible strategies: 7 × 12 = 84; 84 ÷ 7 = 12; a bar model showing 12 groups of 7; skip counting by 7s twelve times to get 84 2. Possible answer: I think, “Seven times what number will equal 84?” I multiply 7 by different factors until I get a product of 84. The factor that makes the multiplication equation work is my solution. Week 10: Day 5 (page 62) 1. 54 books; 6 × 9 = 54 2. 42 books; 6 × 7 = 42 3. 102 books; 6 + (6 × 9) + (6 × 7) = 6 + 54 + 42 = 102 Week 11: Day 1 (page 63) 1. There are 4 elephants and each elephant eats 604 pounds of food a day. 2. Find the estimated number of pounds of food all four of the elephants eat in one day. 3. Multiplication or repeated addition; make 4 groups of 604 or add 604 four times 4. No, 240 pounds is not a reasonable estimate. One elephant eats 604 pounds of food every day, and there are 4 elephants, so 240 pounds is even less than what one elephant eats in one day. Week 11: Day 2 (page 64) 1. 600 × 4 = 2,400 pounds 2. 8,000 ÷ 2 = 4,000 cars Week 11: Day 3 (page 65) Month November December

198

Expression using estimates

Estimated texts per person

800 ÷ 4

200

2,800 ÷ 4

#51616—180 Days of Problem Solving

700

(cont.)

Week 11: Day 4 (page 66) 1. The multiplication problem should include 50 and 70 as factors and the product should equal 3,500. Possible answer: There are 70 passengers on each bus. There are 50 buses leaving the station every day. How many passengers are there altogether? 70 × 50 = 3,500. The division problem should include 3,500 as the dividend with the 50 and 70 as either the divisor or quotient. Possible answer: There are 3,500 apartment units in the city. Each apartment building has 70 units. How many apartment buildings are there in the city? 3,500 ÷ 70 = 50. 2. Possible answer: The product is the number that represents the total. So, the greatest number is the product: factor × factor = product, or 70 × 50 = 3,500. In the division equation, I had to start with the product: product ÷ factor = factor, or 3,500 ÷ 70 = 50. Week 11: Day 5 (page 67) 1. 1 × 12; 2 × 6; 3 × 4 2. Possible answer: If I know the factor pairs of 12, I can use multiples of 10 to help me write multiplication equations with a product of 1,200. 3. Possible answer: I can multiply 1,200 by any factor, find the product, and write the division equation. For example, 1,200 × 2 = 2,400, so 2,400 ÷ 2 must be 1,200. 4. Possible answers: 300 × 4 = 1,200; 30 × 40 = 1,200; 600 × 2 = 1,200; 60 × 20 = 1,200; 1,200 × 1 = 1,200; 9,600 ÷ 8 = 1,200; 6,000 ÷ 5 = 1,200; 4,800 ÷ 4 = 1,200; 3,600 ÷ 3 = 1,200; 2,400 ÷ 2 = 1,200 Week 12: Day 1 (page 68) 1. The hotel charges $189 dollars a night. The guest plans to stay for a week. One week is equal to 7 days. 2. Find the total cost for the whole week. 3. Rounding 189 up to 200 and multiplying by 7 is 1,400. Since the number was rounded up for the estimate, the actual answer must be less than $1,400.

© Shell Education

Answer Key Week 12: Day 2 (page 69) 1. $1,323; 189 × 7 = 1,323; multiply $189 by 7; estimate to check if answer is reasonable; $200 × 7 = $1,400, answer is reasonable 2. 1,888 tires; 472 × 4 = 1,888; multiply 472 by 4; estimate to check if answer is reasonable; 500 × 4 = 2,000, answer is reasonable Week 12: Day 3 (page 70) 988 sides; 59 × 8 = 472; 34 × 6 = 204; 78 × 4 = 312; 472 + 204 + 312 = 988 Week 12: Day 4 (page 71) 1. $2,701; Possible strategies: multiplication, area model, repeated addition 2. Possible answer: I think it is quicker to multiply than to use repeated addition. It takes longer to add 6 groups of 286 than to multiply 6 and 286. It takes longer to add 5 groups of 197 than to multiply 5 and 197. Week 12: Day 5 (page 72) 1. 54 bananas; 1 box has 9 bunches and there are 6 bananas in each bunch; 9 × 6 = 54 2. 270 bananas; 1 stack has 5 boxes and there are 54 bananas in a box; 54 × 5 = 270 3. 2,160 bananas; 1 crate has 8 stacks and there are 270 bananas in a stack; 270 × 8 = 2,160 4. 6,480 bananas; there are 2,160 bananas in 1 crate; 2,160 × 3 = 6,480 Week 13: Day 1 (page 73) 1. Student 1: add 24 + 24 + 24 + 24 +24 + 24 + 24 + 24 + 24 + 24 + 24 +24 + 24 + 24. Student 2: add 26 + 26 + 26 + 26 + 26 + 26 + 26 + 26 + 26 + 26 + 26 + 26 2. Multiplication can be used instead of repeated addition. 3. Possible answer: It is easier and quicker to multiply two numbers rather than add a long list of numbers. There is a greater chance of making a mistake when adding several numbers together. Week 13: Day 2 (page 74) First student; Student 1: 24 × 14 = 336 stickers; Student 2: 26 × 12 = 312 stickers; 336 > 312 Week 13: Day 3 (page 75) 57 × 39; (50 × 30) + (50 × 9) + (7 × 30) + (7 × 9); 1,500 + 450 + 210 + 63; 39 × 57 = 2,223

(cont.)

Week 13: Day 4 (page 76) 1. 3,264 sheets of paper; Possible strategies: area model, partial products, distributive property: (90 × 30) + (90 × 4) + (30 × 6) + (6 × 4) = 3,264 sheets of paper 2. Answers may include: I like using an area model and then calculating the partial products. This helps me to break down (decompose) the factors, which makes it simpler to calculate the solution. Week 13: Day 5 (page 77) 1. Possible answer: Estimation helps me solve the problem because I know 40 × 40 = 1,600 and 30 × 30 = 900. So, I know that the factors must be greater than 30, but less than 40, to result in a product of 1,444. 2. 38 × 38; Possible strategy and explanation: Since the factors must be greater than 30, but less than 40, I used guess and check until I found the product of 1,444. Week 14: Day 1 (page 78) 1. Division; The total number of cards is being divided among 3 players. 2. Find the number of cards left over, which is the remainder in the division problem. 3. The amount of leftover cards cannot be 3 because each of the three players would be able to receive one more card. Week 14: Day 2 (page 79) 1. 2 cards left over; divide 92 by 3 and find the remainder; 92 ÷ 3 = 30 with a remainder of 2; check solution with addition; 30 + 30 + 30 + 2 = 92 2. 12 strawberries with 3 left over; divide 75 by 6 and find the remainder; 75 ÷ 6 = 12 with a remainder of 3; check solution with multiplication and addition; 12 × 6 = 72; 72 + 3 = 75 Week 14: Day 3 (page 80) 23, remainder 3 4 95

×

4

20

80

95 - 80 = 15

3

12

15 - 12 = 3

20 + 3 = 23

© Shell Education

reminder

#51616—180 Days of Problem Solving

199

Answer Key Week 14: Day 4 (page 81) 1. Each friend will get 8 bracelets, with 2 remaining for Tito. 6 × 7 = 42; 42 ÷ 5 = 8 remainder 2; Possible strategies: multiply, divide, repeated addition, repeated subtraction, equations 2. Possible answer: I like using multiplication and division equations because it is faster for me than using repeated addition and repeated subtraction. Week 14: Day 5 (page 82) 1. Number of lollipops in each group 2. Number of groups 3. Possible answers: 57 ÷ 9 = 6 remainder 3; 57 ÷ 3 = 18 remainder 3; 57 ÷ 6 = 9 remainder 3; 57 ÷ 27 = 2 remainder 3 Week 15: Day 1 (page 83) 1. There are 138 students. Each student needs one slice of pizza. Each pizza has 8 slices. . 2. Possible question: How many pizzas need to be ordered? 3. Possible answer: I will divide 138 by 8 to find how many pizzas to order. Week 15: Day 2 (page 84) 1. Divide 138 by 8 and find the remainder; 138 ÷ 8 = 17 remainder 2; Another pizza must be ordered since there are 2 people who did not receive a slice. The solution is 18 whole pizzas. 2. Divide 210 by 9 and find the remainder; 210 ÷ 9 = 23 remainder 3; Each attraction will have 23 balloons in its display with 3 leftover balloons. Week 15: Day 3 (page 85) 1. For chocolate, there are 8 drawings of whole cones representing 288 gallons (8 × 36 = 288). Since 9 grocery stores receive the ice cream, each store will receive 32 gallons of chocolate ice cream (288 ÷ 9 = 32). 2. For strawberry, there are 3 drawings of whole cones equal to 108 gallons (3 × 36 = 108). There is also a drawing of half of a cone. Half of 36 is 18 (36 ÷ 2 = 18). Therefore, there are 126 gallons of strawberry ice cream (108 + 18 = 126). Since 9 grocery stores receive the ice cream, each store will receive 14 gallons of strawberry ice cream (126 ÷ 9 = 14).

200

#51616—180 Days of Problem Solving

(cont.)

Week 15: Day 4 (page 86) 464; Possible answers: 464 464 6 2,784 6 2,784 – 600 100 – 2,400 400 2,184 384 – 600 100 – 300 50 1584 84 – 600 100 – 60 10 984 24 – 600 100 – 24 4 384 0 – 300 50 84 – 72 12 12 – 12 2 0 2. Possible answer: I like Charlotte’s strategy better because finding greater-sized groups of 6 will allow me to find the solution more quickly. Week 15: Day 5 (page 87) 1. Day

Start

1

2

3

4

5

Number of pieces of candy

16

32

64

128

256

512

2. 1,008 total pieces of candy; 16 + 32 + 64 + 128 + 256 + 512 = 1,008 3. 252 pieces of candy for each person; 1,008 ÷ 4 = 252 Week 16: Day 1 (page 88) 1. Find the cost of one submarine sandwich. 2. Kristen’s niece is incorrect. If the $14 is subtracted from $92, the difference is the cost for all six of the submarine sandwiches, not just one. 3. The cost of 1 sandwich multiplied by 6 is the total for all 6 of the sandwiches. The total cost of the sandwiches plus the cost of the fruit punch must equal $92.

© Shell Education

Answer Key Week 16: Day 2 (page 89) 1. $13; subtract $14 from $92, then divide the difference by 6 to find how much each sandwich costs; $92 – $14 = $78; $78 ÷ 6 = $13 2. 14 pages; subtract 19 from 145 to find how many cards he has left, then divide by 9 to find how many pages are needed; 145 –19 =126; 126 ÷ 9 = 14

(cont.)

Week 17: Day 2 (page 94) 1. Possible answer: a is equivalent to N; partition the pie into 6 equal parts 2. Possible answer: c is equivalent to O; partition the pan of brownies into 8 equal parts Week 17: Day 3 (page 95)

���

O

�

c

Week 16: Day 3 (page 90) Item

Cost

Number of items

Cost to buy

curtains

$162

2

$162 × 2 = $324

table

$245

1

$245 × 1 = $245

chair

$96

4

$96 × 4 = $384

cupboards

$141

6

$141 × 6 = $846

can of paint

$19

3

$19 × 3 = $57

$324 + $245 + $384 + $846 + $57 = $1,856; $2,000 – $1,856 = $144 left Week 16: Day 4 (page 91) 1. Possible strategies: multiplication equations; area model; 22 × 4 = 88 pounds; 88 × 3 = 264 pounds 2. Possible answer: I think using multiplication equations is better since there are fewer calculations to make when multiplying numbers and it is also quicker. It takes longer to use repeated addition, and there are more chances to make mistakes when adding long lists of numbers. Week 16: Day 5 (page 92) 1. 80 bottles; 20 × 4 = 80 2. 192 bottles; 64 × 3 = 192 3. 34 packages of water; 80 + 192 = 272; 272 ÷ 8 = 34; or 80 ÷ 8 = 10; 192 ÷ 8 = 24; 10 + 24 = 34 Week 17: Day 1 (page 93) 1. The denominator represents the total number of parts in the whole. 2. The numerator represents the number of parts that are shaded. 3. Possible answer: If I partition the pie into 6 equal parts, N will be equal to a . © Shell Education

Week 17: Day 4 (page 96) 1. Sabrina and Nina; X = � ; Possible strategies: multiplying the numerator and denominator in X by h to prove the equivalent fractions, or drawing fraction models to show equivalence 2. Possible answers: I think drawing fraction models is better because it helps me visualize how the two fractions are equivalent. Week 17: Day 5 (page 97) 1. Derrick practices soccer M of the time. Derrick practices baseball b of the time. Derrick has an off-day ��� of the time. 2. Sun.

Mon. Tues. Wed. Thurs.

B

Fri.

Sat.

B B

B

3. Accept any equivalent fractions to LE ; Possible equivalent fractions: � = Mz = WI Week 18: Day 1 (page 98) 1. m is less than M . M = w and w is greater than m . 2. u is greater than M; M = k which is less than u . 3. Possible answer: If the monthly goals were different, we could not make a fair comparison. We can only compare the grade levels because the goal, or the whole, is the same.

#51616—180 Days of Problem Solving

201

Answer Key

(cont.)

Week 18: Day 2 (page 99) 1. Fourth grade; u > m ; compare the fractions by using M as a benchmark; u is greater than M and m is less than M 2. Fruity Circles; n < � ; compare the fractions by using M as a benchmark; n is less than M and � is exactly M

Week 19: Day 2 (page 104) 1. � of the students; � + � = � 2. MU or j of the students; WA – � = MU = j

Week 18: Day 3 (page 100) Michael and Mitchell are using the same amount of memory. X = � Michael

Week 19: Day 4 (page 106) 1. � or S ; Possible strategies: add the two fractions and then subtract the sum from the whole; � + � = Lu ; L� – Lu = � = S ; or subtract each fraction from the whole; L� – � – � = � = S 2. Possible answers: I like to solve the problem as it is presented in the story. I added the two fractions together first to find the total amount of cat food the two cats ate. Then, I subtracted this amount from the whole bag.

0

N

1

X

Mitchell 0 � � � �

�

���

���

���

���

LC LM

1

Week 18: Day 4 (page 101) 1. Poodle; j > � . Possible strategies: number line or drawing to compare amounts; verbal explanation describing that j is greater than M and � is less than M . So j must be greater than �. 2. Possible answer: I think using a number line is better because I can compare the amounts on each number line and see which is greater. Week 18: Day 5 (page 102) 1.

pepperoni

cheese

vegetable

2. Pepperoni pizza: m left over; cheese pizza; Y or M left over; vegetable pizza: u or X left over 3. m , Y , u ; Possible strategies: fraction models; number lines; compare each fraction to M as benchmark fraction; m is less than M , Y is exactly M , and u is greater than M .

Week 19: Day 3 (page 105) 1. � + � = � ; � of the fruit is either strawberries or watermelons 2. W� – � = M� ; M� of the fruit is not bananas

Week 19: Day 5 (page 107) 1. 15; there are 15 cards in the deck; 15 represents the whole, or the denominator 2. Blue cards: � ; � = N ; Red cards: � ; � + � = �; White cards: � ; L� – � = � Week 20: Day 1 (page 108) 1. Find what fraction of the vegetable tray each of Annabelle’s friends ate. Each fraction needs to be the same amount. 2. The whole tray is � because the fraction is given in eighths. 3. Possible answer: Each of Annabelle’s friends cannot eat w of the tray. w is equal to M . If each of Annabelle’s friends eats M of the tray, there would be nothing left for Annabelle because the friends ate the whole tray. Annabelle eats c of the tray, so this cannot be true. Week 20: Day 2 (page 109) 1. m of the vegetables; subtract c from the whole � and then find half of the difference; � – c = � ; half of � is m ; check answer by adding all of the fractions to find if they total � ; c + m + m = � . 2. k or M of the nuts; add 3 groups of Q and then subtract the sum from the whole � ; Q + Q + Q = k ; � – k = k or M

Week 19: Day 1 (page 103) 1. The denominator 20 represents the total number of students in the class. 2. Find the fraction of students that played cards or board games. 3. Board game; � = O 202

#51616—180 Days of Problem Solving

© Shell Education

Answer Key Week 20: Day 3 (page 110) Possible answers: Shape

Fraction shaded

Expression 1

Expression 2

circle

�

�+�+�+ �+�+�

�+�

rectangle

�

Q+Q+Q+Q +Q

k+a

triangle

i

O+O+O

Y+O

Week 20: Day 4 (page 111) 1. Possible strategies: fraction model; number line; equations: 1 + 1 + � ; or LA + LA + �

(cont.)

Week 21: Day 2 (page 114) 4Q liters of water; add the whole numbers and then add the fractions; 2 + 1 = 3; u + k = � ; � = � + Q , or 1 + Q = 1Q ; 3 + 1Q = 4Q Week 21: Day 3 (page 115) LC or �

Week 21: Day 4 (page 116) 1. 1u or 1X pizza is left over; Possible strategies: fraction model; number line; equations: 3 = � + � + � = ��� ; 1a = � + a = � ; ��� – � = ��� or 1u or 1X

2. Possible answers: I think drawing a fraction model is better because it helps me to understand what a mixed number represents. Week 20: Day 5 (page 112) 1. Mural 1: 12 equal sections with � shaded for each student; Mural 2: 8 equal sections with c shaded for each student; Mural 3: 20 equal sections with � shaded for each student 2. Mural 1: � + � + � + � = LW ; Mural 2: c + c + c + c = � ; Mural 3: � + � + � + � = WA Week 21: Day 1 (page 113) 1. Addition; The problem is asking to find how many liters of water in all. 2. � ; Since the fraction in 1k is expressed in sixths, it is best to write the whole number using the same denominator. 3. Greater than 3 liters; Possible explanation: 2 + 1 is already 3 liters, and I still need to add the fractions to this sum, so the number of liters of water will definitely be greater than 3 liters. © Shell Education

2. Possible answer: I think drawing a fraction model is better because I can easily sketch the three pizzas, partition each pizza into sixths, and cross off the amount left over. Week 21: Day 5 (page 117) 1. Box A and Box C; The two boxes that weigh the most will give Ty the greatest possible amount of candy after he shares 1M kilograms with his friends. 2. 4Y or 4M kilograms; 2i + 3O = 5s = 6; 6 – 1M = 6 – 1Y = 5s – 1Y = 4Y or 4M . The total for Boxes A and C is 6 kilograms. After Ty gives away 1M kilograms to his friends, he will still have 4Y or 4M kilograms of candy for himself. Week 22: Day 1 (page 118) 1. Use repeated addition by adding a eight times. 2. Multiply a by 8. 3. Greater than one pound; Possible explanation: The answer will be greater than 1 pound because a + a + a is already one pound, and that is only for three sandwiches. #51616—180 Days of Problem Solving

203

Answer Key Week 22: Day 2 (page 119) 1. 2u or 2X pounds of turkey; use repeated addition or multiplication; a + a + a + a + a + a + a + a = ��� or 2u = 2X ; or 8 × a = ��� or 2u = 2X 2. 4Y = 4M miles; use repeated addition or multiplication; i + i + i + i + i + i = ��� or 4Y = 4M ; or 6 × i = ��� or 4Y

Week 23: Day 3 (page 125)

Cheese

x x x

Week 22: Day 3 (page 120) 8×N=� +N 1

+N N

+N X

+N h

0

+N r

(cont.)

+N �

+N �

Y

i

1

1O

1Y

1i

2

Pounds sold

+N �

O

x x x

x x

�

�

Week 22: Day 4 (page 121) 1. ��� or 2 kilometers; Possible strategies: number line; fraction model; repeated addition; multiplication equations; u + u + u = ��� or 2; or 3 × u = ��� or 2 2. Possible answer: I think repeated addition works well here since I only have to add three fractions together. If there were more fractions, I would say multiplication is better. Week 22: Day 5 (page 122) 1. Vince’s reasoning is incorrect. Possible explanation: The recipe feeds six people. To feed 30 people, each ingredient must be multiplied by 5, not by 30. 2. 10 pounds of ground beef; ��� or 4m cans of kidney beans; ��� or 3i cans of diced tomatoes; ��� or 4Q tablespoons of chili powder; � or 2M  teaspoons of salt; � or 1X teaspoons of black pepper 3. Possible strategies: repeated addition; multiplication; fraction models; number lines Week 23: Day 1 (page 123) 1. Each X represents one student. 2. 12 students; There are 12 X’s on the line plot. 3. Possible answer: How many students drank more than 1 cup of water?

���

Week 23: Day 4 (page 126) 1. Possible questions and answers: How many lawn mowers used less than 1w gallons of gas? (5); How many lawn mowers used more than 1w gallons of gas? (4); How many lawn mowers are represented on the line plot? (12) 2. Possible answer: I think that my question asking how many lawn mowers used more than 1w gallons of gas was more challenging. To find the solution, you have to count all the lawn mowers using more than that amount, but not exactly that amount. Week 23: Day 5 (page 127) 1.

Sandwiches

x x x x x x 0

S

c

m

x w

�

�

x �

1

Amount of sandwiches eaten 2. ��� or 3S sandwiches; S + S + c + c + c + m + � + � = ��� or 3S

Week 23: Day 2 (page 124) 8 students; 4 + 3 + 1 = 8; count the X’s that are plotted at 1O , 1Y , 1i , and 2; 4 students drink 1O cups of water, 3 students drink 1i cups of water, and 1 student drinks 2 cups of water.

204

#51616—180 Days of Problem Solving

© Shell Education

Answer Key Week 24: Day 1 (page 128) 1. Colleen is incorrect. Possible explanation: The answer cannot be �. She got this answer by adding the numerators together. Fractions must have the same denominators before adding the numerators. 2. Possible answer: I will have to change � into an equal fraction in hundredths since the other fraction is �. Since 10 × 10 is 100, I will be able to rewrite � as an equivalent fraction with a denominator of 100. 3. Mr. McGwire walked more than half a mile. Possible explanation: Half a mile is �. � is almost a half. When � is added on to this, Mr. McGwire will definitely walk more than half of a mile. Week 24: Day 2 (page 129) 1. � or i mile; � = � ; � + � = � or i 2. � of the carrot; � = �; � + � = � Week 24: Day 3 (page 130) 4th grade: � + � = � ; 5th grade: � + � = � ; The grade level with more history reports graded is 5th grade. Week 24: Day 4 (page 131) 1. Jaiden is incorrect because � is less than �; � = �; � + � = � + � = �; � < �; Possible strategies: hundreds grids; equations, fraction model 2. Possible answer: I think using a hundreds grid is better because it helps me to compare and add fractions with denominators of 10 and 100. Week 24: Day 5 (page 132) 1. Equations will vary, but the sum should equal � ; Possible answers: � + � = � + � = �; � + � = � + � = �; � + � = � + � = � 2. Story problems will vary but should represent an equation written; Possible example: Shelby uses � of a container of iced tea mix to make drinks for her guests. Then, she uses � of that same container of mix to make the iced tea stronger. How much of the iced tea mix has Shelby used altogether?

© Shell Education

(cont.)

Week 25: Day 1 (page 133) 1. Maybell needs to read the number using the correct place value and say, “seventy-eight hundredths.” 2. 7 tenths or � ; 8 hundredths or � 3. The number 0.78 is greater than 0.7, but less than 0.8. Week 25: Day 2 (page 134) 1. The decimal 0.78 or fraction � is between 0.7 and 0.8 on the number line. It is closer to 0.8 because it is more than 0.75, which is halfway between 0.7 and 0.8. 0.78 = � 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

2. The decimal 0.41 or fraction � is between 0.4 and 0.5 on the number line It is closer to 0.4 because it is less than 0.45, which is halfway between 0.4 and 0.5. 0.41 = �

Week 25: Day 3 (page 135) Expanded Expanded fraction form decimal Fraction Decimal using place form using value place value �

�+�

0.4 + 0.07

0.47

�

�+�

0.2 + 0.04

0.24

�

��� + �

0.0 + 0.09

0.09

Week 25: Day 4 (page 136) 1. Answers will vary but should be greater than M or 0.50 and less than � or 0.80; Possible strategies: place value chart; number line; hundreds grids; Possible answers: 0.65 is greater than 0.50, but less than 0.80; � is greater than � , but less than � ; 0.7 is greater than 0.50, but less than 0.80; � is greater than � , but less than � . 2. Possible answer: I like using a place value chart and expressing all of the decimals as fractions with the denominator 10 or 100. When I see all of the numbers written in the same form with the same denominator, I can clearly see that my solution is greater than M and less than � .

#51616—180 Days of Problem Solving

205

Answer Key Week 25: Day 5 (page 137) 1. � 2. 0.4 or 0.40 3. 0.38 � 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

4. Possible explanation: I wrote both amounts in fraction form with a denominator 100 (� and �). � is closer to half (�) than � is; or I wrote both amounts in decimal form (0.38 and 0.4 or 0.40). 0.4 or 0.40 is closer to half (0.5 or 0.50) than 0.38 is; or I can see on my number line that � is closer to half (0.5) than 0.38 is. Week 26: Day 1 (page 138) 1. No, comparing only the whole numbers does not help write the lengths in order from least to greatest because the whole number is 0 for all of the lengths. 2. Less than 1 centimeter. All of them start with the whole number of 0. 3. � ; � ; � ; � Week 26: Day 2 (page 139) 1. 0.07; 0.3; 0.37; 0.7; write the fraction form of each decimal with a denominator of 100 and order the numbers; plot the numbers on a number line 2. 3.4; 3.58; 4.3; 4.39; write the fraction form of each decimal with a denominator of 100 and order the numbers; plot the numbers on a number line Week 26: Day 3 (page 140) Answers will vary, but should include decimals greater than 0.7, but less than 0.98; Possible answers: 0.79, 0.85, and 0.91 0.79 0.7

0

0.91

0.85

0.98

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

Week 26: Day 4 (page 141) 1. $0.08 < $0.80 or $0.80 > $0.08; Possible strategies: hundreds grids, number line, place value chart 2. Possible answer: I think using a place value chart to compare the decimals is the best strategy because it is easy to compare the value of the digits in each place value position. 206

#51616—180 Days of Problem Solving

(cont.)

Week 26: Day 5 (page 142) 1. � and � ; � ; � = � ; � + � = � ; Yes, Evan’s teacher should accept this response because � is equal to 0.57. 2. � ; 0.57; Yes, Evan’s teacher should accept this response because � is equal to 0.57. 3. Answers will vary but should include at least four more unique ways to express 0.57; Possible answers: 5 tenths and 7 hundredths; � ; 57 hundredths; 2 tenths and 37 hundredths; �+� Week 27: Day 1 (page 143) 1. 12 inches = 1 foot; half a foot is equal to 6 inches 2. Use division to find how many inches are in half a foot (12 ÷ 2 = 6) or multiplication 12 × M = 6. Use repeated addition or multiplication to find the number of inches in 4 feet. 3. The answer cannot be 24 inches because 2 feet is equal to 24 inches, and the tower measures 2M feet. Week 27: Day 2 (page 144) 1. 54 inches; convert feet to inches; 12 inches = 1 foot; 4 × 12 = 48 inches; M foot = 6 inches; 48 + 6 = 54 inches 2. 900 centimeters; convert meters to centimeters; 1 meter = 100 centimeters, 9 × 100 = 900 centimeters; 9 meters = 900 centimeters Week 27: Day 3 (page 145) Yards

Inches

Equation

1

36

1 × 36 = 36

7

252

7 × 36 = 252

9

324

9 × 36 = 324

4

144

4 × 36 = 144

M

18

M × 36 = ��� or 18

Week 27: Day 4 (page 146) 1. Strategy 1: 4,000 + 8,000 + 3,000 = 15,000 meters; Strategy 2: 4 + 8 + 3 = 15 kilometers; 4,000 meters = 4 kilometers; 8 kilometers = 8,000 meters; 3,000 meters = 3 kilometers 2. Possible answers: Using kilometers in the real world is more practical because kilometers are used for longer distances, such as highways.

© Shell Education

Answer Key Week 27: Day 5 (page 147) 1. The clue is helpful because knowing how many feet are in one mile will help write all the lengths in feet, making it easier to solve the problem. 2. 450 feet; 3 feet = 1 yard; 150 × 3 = 450 3. 1,370 feet; 5,280 + 450 + 1,450 + 1,450 = 8,630 feet; 10,000 – 8,630 = 1,370 feet Week 28: Day 1 (page 148) 1. Jeff starts his activities at 3:50 p.m. He plays basketball for 25 minutes and works on his homework for 40 minutes. 2. Possible answer: Jeff plays basketball and works on his homework for 65 minutes (or 1 hour, 5 minutes). He begins these activities at 3:50 p.m. What time does he finish? 3. If you count by 10 minutes, the elapsed time will move from 3:50 to 4:00. It will be easier to add the minutes from that point. Week 28: Day 2 (page 149) 1. 4:55 p.m.; Add 25 minutes and 40 minutes (65 minutes), and then use a clock number line to count on from 3:50 to find what time he finished. 2. 4:40 p.m.; Use a clock number line to count back from 6:15 by 1 hour and 35 minutes to find what time she started. Week 28: Day 3 (page 150) The clock number line should include the start time of 11:50 and end time of 4:35. The elapsed time is 4 hours and 45 minutes. Week 28: Day 4 (page 151) 1. Possible strategy: 9 × 24 = 216 hours 2. Possible answer: I like my strategy of multiplying 9 times 24 because the calculation took less time than making a table and writing the number of hours for each number of days.

© Shell Education

(cont.)

Week 28: Day 5 (page 152) 1. Convert all of the times to hours, days, or weeks. 2. 720 hours and 58 days are not equivalent to the other time amounts. The other time amounts are equivalent to each other because 1,008 hours = 42 days = 6 weeks. 42 days × 24 hours is 1,008 hours and 6 weeks × 7 days is 42 days. These amounts are equivalent. 3. 58 days is equal to 1,392 hours (1 day = 24 hours; 58 × 24 = 1,392); or 720 hours is equal to 30 days (720 ÷ 24 = 30) Week 29: Day 1 (page 153) 1. 1,000 milliliters 2. Two measuring cups; 500 + 500 = 1,000 milliliters; 1,000 milliliters = 1 liter 3. Possible answer: I can use repeated addition by adding groups of 500 until I get to 4,000 to solve the problem. Week 29: Day 2 (page 154) 1. 8 times; convert 4 liters to milliliters (4 liters = 4,000 milliliters); multiplication; division; repeated addition; 500 + 500 + 500 + 500 + 500 + 500 + 500 + 500 = 4,000; 8 groups of 500 in 4,000 2. 4,000 grams; convert 2 kilograms to grams (2 kilograms = 2,000 grams); addition; 2,000 + 2,000 = 4,000 Week 29: Day 3 (page 155) Pounds 2,000 4,000 6,000 8,000 10,000 12,000 Tons 1 2 3 4 5 6 6,000 pounds is equal to 3 tons. Week 29: Day 4 (page 156) 1. 160 ounces; Possible strategies: find the number of ounces in each type of cheese and then find the sum; or add the pounds of cheese together first and then convert the sum to ounces; Cheddar: 6 × 16 = 96 ounces and M pound = 8 ounces; 96 + 8 = 104 ounces; Mozzarella: 3 × 16 = 48 ounces and M pound = 8 ounces; 48 + 8 = 56 ounces; 104 + 56 = 160 ounces; or 6M + 3M = 10 pounds of cheese; 10 × 16 = 160 ounces 2. Possible answer: I prefer changing pounds to ounces first. Then, I can add the two amounts together. #51616—180 Days of Problem Solving

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Answer Key Week 29: Day 5 (page 157) 1. The clue is helpful because knowing how many fluid ounces are in one quart will help write all the amounts in fluid ounces, making it easier to solve the problem. 2. 96 fluid ounces; 3 quarts = 96 fluid ounces (3 × 32 = 96) 3. Possible explanation: Elan’s mom will not have enough to pour each person a second full cup. There are 9 people including Elan. For the first cup, each person receives 6 fluid ounces. 9 people × 6 fluid ounces = 54 fluid ounces. There are 3 quarts of punch, which is 96 fluid ounces. 96 fluid ounces – 54 fluid ounces already used = 42 fluid ounces remaining. There are not 54 more ounces remaining. Or, to have two full cups of juice, each person would need 12 ounces. 9 people × 12 ounces = 108 ounces. 3 quarts, or 96 ounces, will not be enough. Week 30: Day 1 (page 158) 1. No, The farmer should not follow Isabel’s advice. Isabel multiplied the dimensions, finding the area. The problem is asking to find the perimeter, not area. 2. Sketching a rectangle and labeling each of the sides with the correct dimensions of the chicken coop may help to visualize the problem. 3. Less than 100 feet; Each of the four dimensions is less than 25 (25 + 25 + 25 + 25 = 100). Week 30: Day 2 (page 159) 1. 76 feet of fencing; add lengths together: 24 + 24 + 14 + 14; multiply each length by two and then add the lengths: 2(24 + 14); or 2(24) + 2(14) 2. 60 inches; add lengths together: 18 + 18 + 12 + 12; multiply each length by two and then add the lengths: 2(18 + 12); or 2(18) + 2 (12)

(cont.)

Week 30: Day 3 (page 160) 1. l+w+l+w

2(l + w)

2l + 2w

43 + 19 + 43 + 19 = 124 in.

2(43 + 19) = 124 in.

2(43) + 2(19) = 124 in.

2. Each strategy results in the same perimeter. Possible explanation: I can add two of each length and width, or add the length and the width and then double the sum, Or, I can double the length, double the width, and then add these results. All of the strategies are related and perform the same calculations in different ways. Week 30: Day 4 (page 161) 1. Answers will vary, but the perimeters should total 96 yards; Possible answers: length = 25 yards, width = 23 yards; 2(25 + 23) = 96 yards; length = 30 yards, width = 18 yards; 2(30 + 18) = 96 yards 2. Possible answer: I think making the section 25 yards by 23 yards is better because it will give the dogs a wider space to exercise. Week 30: Day 5 (page 162) 1. Sketches should have the lengths labeled with 19 inches and the widths labeled with 13 inches; 2(19 + 13) = 64 inches 2. Accept any dimensions with a perimeter of 128 inches; Possible answer: 38 inches by 26 inches; 2(38 + 26) = 128 Week 31: Day 1 (page 163) 1. Find the square feet of vinyl that will cover the floor. 2. The problem is asking for square feet because the worker is covering the floor with 1-foot by 1-foot squares. 3. Area measures the space inside a shape and the perimeter measures the distance around the outside of the shape. The problem requires finding the area. Week 31: Day 2 (page 164) 1. 270 square feet; multiply the dimensions; 18 × 15 = 270 2. 2,304 square inches; multiply the dimensions; 48 × 48 = 2,304 Week 31: Day 3 (page 165) Rectangle 1: 24 feet; Rectangle 2: 6 feet

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© Shell Education

Answer Key Week 31: Day 4 (page 166) 1. The new area must be 8 square meters. Possible dimensions: 1 meter by 8 meters; 2 meters by 4 meters; 4 meters by 2 meters; 8 meters by 1 meter 2. No, a 2 meter by 8 meter rectangle has an area of 16 square meters. Both of the dimensions were doubled and only one dimension needs to be doubled in order to make a rectangle double the size of the original. Week 31: Day 5 (page 167) 1. Possible dimensions: 1 ft. × 84 ft.; 84 ft. × 1 ft.; 2 ft. × 42 ft.; 42 ft. × 2 ft.; 3 ft. × 28 ft.; 28 ft. × 3 ft.; 21 ft. × 4 ft.; 4 ft. × 21 ft.; 14 ft. × 6 ft.; 6 ft. × 14 ft.; 12 ft. × 7 ft.; 7 ft. × 12 ft.; 2. Sketches should have the dimensions labeled and show the area are equal to 84 square feet; perimeters will vary based on rectangle dimensions used; Possible solution: 6 ft. × 14 ft. rectangle has a perimeter of 40 feet; 7 ft. × 12 ft. rectangle has a perimeter of 38 ft. Week 32: Day 1 (page 168) 1. Acute angles are less than 90 degrees, right angles are exactly 90 degrees, and obtuse angles are greater than 90 degrees. 2. An angle is formed when 2 sides meet at a vertex. Since there are six vertices, there are six angles. 3. The number of angles, sides, and vertices are the same. Week 32: Day 2 (page 169) 1. Right (R); Acute (A); Obtuse (O) R

A

R O

R

R

O

OA

O

O

A

A

O

2. Right A (R); Acute (A); Obtuse (O) B

A

O

R

O O

R

R

© Shell Education

Z O E R

B C Z E

D

C

D

(cont.)

Week 32: Day 3 (page 170) 1. Completed figure should be a quadrilateral with vertices labeled as M, N, O, and P. 2. Angle MNO is an obtuse angle. 3. Angle OPM will vary depending on where student placed Point P: less than 90 degrees is acute, exactly 90 degrees is right, and greater than 90 degrees is obtuse. Week 32: Day 4 (page 171) 1. Possible directions for equilateral triangle: 3 sides, all 3 sides equal, 3 angles, all 3 angles are equal; Possible directions for scalene triangle: 3 sides, all 3 sides are different, 3 angles, angles are not all equal 2. Both sets of directions have 3 sides and 3 angles. The equilateral triangle has 3 equal sides and the scalene triangle has 3 sides of different lengths. Week 32: Day 5 (page 172) 1. Triangle: Descriptions will vary based on sketch. Check descriptions based on reference list below. 2. Quadrilateral: Descriptions will vary based on sketch. Check descriptions based on reference list below. 3. Term and illustration will vary based on descriptions in questions 1 and 2. Check illustration based on reference list below. (Reference list: right angle—angle measuring exactly 90 degrees; acute angle—angle measuring less than 90 degrees; obtuse angle—angle measuring greater than 90 degrees; line segment—line with two end points—the side of a polygon; parallel—lines that will never intersect and are always an equal distance apart; perpendicular—lines that intersect at a 90 degree angle) Week 33: Day 1 (page 173) 1. Each angle on the triangular tiles measures 60 degrees. Suzanne is making a 360-degree rotation for her design. 2. Find out how many tiles she will need. 3. The answer cannot be 10 tiles since that would result in 600 degrees, not 360 degrees. Week 33: Day 2 (page 174) 1. 6 tiles; divide 360 degrees by 60 degrees; 360 ÷ 60 = 6 2. 86 degrees; subtract 94 degrees from 180 degrees (straight line); 180 – 94 = 86 degrees #51616—180 Days of Problem Solving

209

Answer Key Week 33: Day 3 (page 175) Verify that students draw a right angle (90 degrees), an obtuse angle (120 degrees), and an acute angle (45 degrees). Week 33: Day 4 (page 176) 1. 20 degrees; 90°– 70° = 20°; Possible strategies: use a protractor to draw a picture; equations

(cont.)

Week 34: Day 1 (page 178) 1. Possible answer: A table could help keep track of the money each day. I would make a column showing the day and a column showing how much money she has on each day. 2. Growing pattern; numbers are increasing and not repeating 3. No, this would be how much money Krista had if she put 4 cents per day into the bank, which is not what the problem states. Week 34: Day 2 (page 179) 1. 512 cents or $5.12; create a table to find how much money she will have on the 8th day; amount doubles each day

20°

70°

2. Possible answer: I prefer to use equations to solve the problem. I know that a right angle is 90 degrees, so I can subtract 70 from 90. The angle measurement of the other angle is 20 degrees. Week 33: Day 5 (page 177) 1. Sketches should show five separate sections or angles comprising the 90-degree angles.

Day

1

2

3

4

5

Money in cents

4

8

16

32

64

6

7

8

128 256 512

2. The sixteenth shape will be a triangle. If the pattern repeats, the shapes are: triangle, star, star, triangle, star, star, triangle, star, star, triangle, star, star, triangle, star, star, triangle. Or, the pattern repeats in groups of three. Five complete groups of three will create a pattern of 15 shapes. The sixteenth shape would start the pattern again, so the shape is a triangle. Week 34: Day 3 (page 180)

18˚ 18˚ castle wall

18˚ 18˚ 18˚

2. There are 18 degrees in each of the five equal sections. 90 degrees divided by five sections is 18 degrees. 3. Possible answer: I was within 5 degrees of 18 degrees on each of the five sections, so my estimates are fairly accurate.

210

#51616—180 Days of Problem Solving

In

Out

12

33

14

35

17

38

24

45

7

28

28 49 Rule: add 21 Week 34: Day 4 (page 181) 1. Possible answers for even numbers: 6, 12, 24, 48, 96, 192; Rule: multiply by 2; Possible answers for alternating odd and even numbers: 6, 9, 12, 15, 18, 21; Rule: add 3 2. Possible answer: The first rule was easier to determine since all numbers had to be even and multiplying by 2 always results in an even number. © Shell Education

Answer Key

(cont.)

Week 34: Day 5 (page 182) 1. Sketch should include: sun, moon, moon, star, star, sun, moon, moon, star, star 2. This can help solve the problem because there are 10 beads in that pattern, and 10 × 10 is 100. The 100th bead must be a star. 3. Answers will vary. Check that the 100th bead is accurate based on the pattern created. Possible answer: In the pattern sun, star, moon, the 100th bead will be a sun because the pattern would start over with the 100th bead.

Week 35: Day 4 (page 186) 1. Possible solutions: Group A: square, rectangle Group B: parallelogram, rhombus, trapezoid, quadrilateral; Group A: rhombus, square, parallelogram, rectangle Group B: trapezoid, quadrilateral 2. Possible answer: For solution 1, I sorted the polygons by deciding which had right angles and which did not. For solution 2, I sorted by whether each polygon had 2 sets of parallel lines or not.

Week 35: Day 1 (page 183) 1. It is a four-sided polygon with only one set of parallel lines and no right angles. 2. Identify the type of quadrilateral and draw a sketch. 3. The polygon cannot be a rectangle because rectangles have four right angles and two sets of parallel sides.

Week 35: Day 5 (page 187) 1. 4 or more sides, depending on the shape 2. Possible shape:

Week 35: Day 2 (page 184) 1. Trapezoid; Sketches may vary, but should show a four-sided polygon with only one set of parallel lines and no right angles. 2. Right scalene triangles; Sketches may vary, but should include a rectangle divided diagonally into two triangles. Week 35: Day 3 (page 185) Obtuse isosceles triangle; Possible sketch:

Right scalene triangle; Possible sketch:

parallel side

parallel side

right angle

right angle

3. Possible polygons: trapezoid, pentagon, or any polygon that has one set of parallel sides and two right angles. Week 36 Day 1 (page 188) 1. A line of symmetry is an imaginary line that divides the figure into two parts that are mirror images of each other. 2. You can prove a figure has a line of symmetry by cutting out the figure and folding it over the line being tested. If it is a line of symmetry, the parts will match.

Acute equilateral triangle; Possible sketch:

© Shell Education

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211

Answer Key Week 36: Day 2 (page 189) Nancy is correct; the diagonal line of symmetry cuts the rectangle in half, but each side is not a mirror image of the other; check by cutting out a shape similar to the one in the drawing and folding to test the lines of symmetry

(cont.)

Week 36: Day 4 (page 191) 1.

Week 36: Day 3 (page 190)

2. No, the sides and angles on the first pentagon are equal so there are five lines of symmetry. There is only one line of symmetry on the second pentagon because the angle measurements are different. Week 36: Day 5 (page 192) 1. The line of symmetry shows me where a mirror image needs to be drawn. 2.

3. Yes, it is possible to draw another line of symmetry. The line can be drawn horizontally to divide the figure into two equal parts.

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© Shell Education

Practice Page Rubric Directions: Evaluate student work in each category by choosing one number in each row. Students have opportunities to score up to four points in each row and up to 16 points total.

Problem-solving strategies

Advanced

Mathematical knowledge

Beginning

Uses appropriate strategies

Demonstrates some form of strategic approach

No strategic approach is evident

Uses a detailed and appropriate visual model

Uses an appropriate visual model

Uses a visual model but is incomplete

No visual model is attempted

4

3

2

1

Provides correct solutions and multiple solutions when relevant

Provides correct solutions

Shows some correct solutions

No solutions are correct

Connects and applies the concept in complex ways

Demonstrates proficiency of concept

Demonstrates some proficiency of concept

Does not demonstrate proficiency of concept

4

Points Explanation

Developing

Uses multiple efficient strategies

Points

Explains and justifies thinking thoroughly and clearly 4

Points Organization

Proficient

Well‑planned, well‑organized, and complete

Points © Shell Education

4

3 Explains and justifies thinking

3 Shows a plan and is complete

3

2 Explains thinking but difficult to follow

2 Shows some planning and is mostly complete

2

1 Offers no explanation of thinking

1 Shows no planning and is mostly incomplete

1

#51616—180 Days of Problem Solving

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214

#51616—180 Days of Problem Solving

Average Class Score

Student Name

Week Week Week Week Week Week Week Week Week 1 2 3 4 5 6 7 8 9

Total Scores

Directions: Record students’ rubric scores (page 213) for the Day 5 practice page in the appropriate columns. Add the totals and record the sums in the Total Scores column. You can view: (1) which students are not understanding the mathematical concepts and problem-solving steps, and (2) how students progress after multiple encounters with the problem‑solving process.

Practice Page Item Analysis

© Shell Education

© Shell Education

Average Class Score

Student Name

Week Week Week Week Week Week Week Week Week 10 11 12 13 14 15 16 17 18 Total Scores

Directions: Record students’ rubric scores (page 213) for the Day 5 practice page in the appropriate columns. Add the totals and record the sums in the Total Scores column. You can view: (1) which students are not understanding the mathematical concepts and problem-solving steps, and (2) how students progress after multiple encounters with the problem‑solving process.

Practice Page Item Analysis (cont.)

#51616—180 Days of Problem Solving

215

216

#51616—180 Days of Problem Solving

Average Class Score

Student Name

Week Week Week Week Week Week Week Week Week 19 20 21 22 23 24 25 26 27

Total Scores

Directions: Record students’ rubric scores (page 213) for the Day 5 practice page in the appropriate columns. Add the totals and record the sums in the Total Scores column. You can view: (1) which students are not understanding the mathematical concepts and problem-solving steps, and (2) how students progress after multiple encounters with the problem‑solving process.

Practice Page Item Analysis (cont.)

© Shell Education

© Shell Education

Average Class Score

Student Name

Week Week Week Week Week Week Week Week Week 28 29 30 31 32 33 34 35 36 Total Scores

Directions: Record students’ rubric scores (page 213) for the Day 5 practice page in the appropriate columns. Add the totals and record the sums in the Total Scores column. You can view: (1) which students are not understanding the mathematical concepts and problem-solving steps, and (2) how students progress after multiple encounters with the problem‑solving process.

Practice Page Item Analysis (cont.)

#51616—180 Days of Problem Solving

217

Student Item Analysis Directions: Record individual student’s rubric scores (page 213) for each practice page in the appropriate columns. Add the totals and record the sums in the Total Scores column. You can view: (1) which concepts and problem-solving steps the student is not understanding and (2) how the student is progressing after multiple encounters with the problem-solving process. Student name: Week 1 Week 2 Week 3 Week 4 Week 5 Week 6 Week 7 Week 8 Week 9 Week 10 Week 11 Week 12 Week 13 Week 14 Week 15 Week 16 Week 17 Week 18 Week 19 Week 20 Week 21 Week 22 Week 23 Week 24 Week 25 Week 26 Week 27 Week 28 Week 29 Week 30 Week 31 Week 32 Week 33 Week 34 Week 35 Week 36 218

Day 1

#51616—180 Days of Problem Solving

Day 2

Day 3

Day 4

Day 5

Total Scores

© Shell Education

Problem-Solving Framework Use the following problem-solving steps to help you:

1. understand the problem 2. make a plan 3. solve the problem 4. check your answer and explain your thinking What Do You Know?

What Is Your Plan?

• read/reread the problem

• draw a picture or model

• restate the problem in your own words

• decide which strategy to use

• visualize the problem

• choose an operation (+, –, ×, ÷)

• find the important information in the problem

• determine if there is one step or multiple steps

• understand what the question is asking

Solve the Problem! • carry out your plan • check your steps as you are solving the problem • decide if your strategy is working or choose a new strategy • find the solution to the problem

© Shell Education

Look Back and Explain! • check that your solution makes sense and is reasonable • determine if there are other possible solutions • use words to explain your solution

#51616—180 Days of Problem Solving

219

Problem-Solving Strategies Draw a picture or diagram.

+

Make a table or list.

10 + 4 = 14 A=1xw

=

Make a model.

Use a number sentence or formula.

Look for a pattern.

Act it out.

5 + 2 4

18

3, 6, 9, 12, 15, _____

Solve a simpler problem.

Work backward.

Use logical reasoning.

7+6

x 3 x 5 = 30

7+3+3 (7+3)+3 10 + 3 = 13

5 4 3 2 1

Guess and check.

2×

0

Monday Tuesday Thursday Create a Wednesday graph.

+ 5 = 13

Use concrete objects.

Family vacation photos

2 × 4 + 5 = 13 13 = 13 Yes! 220

#51616—180 Days of Problem Solving

Number of photos

5 4 3 2 1 0

Monday

Tuesday

Wednesday Thursday

Days of the week

base-ten blocks © Shell Education

Digital ResourceS Teacher Resources Resource

Filename

Practice Page Rubric

rubric.pdf

Practice Page Item Analysis

itemanalysis.pdf itemanalysis.docx itemanalysis.xlsx

Student Item Analysis

studentitem.pdf studentitem.docx studentitem.xlsx

Student Resources Resource

Filename

Problem-Solving Framework

framework.pdf

Problem-Solving Strategies

strategies.pdf

© Shell Education

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221

Notes

222

#51616—180 Days of Problem Solving

© Shell Education

Notes

© Shell Education

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223

Notes

224

#51616—180 Days of Problem Solving

© Shell Education

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