problem solving and metacognition · problem(solving(and(metacognition(!! (! problem.!

Problem solving and metacognition 1

Copyright © 2016 Charles A. Dana Center at the University of Texas at Austin, Learning Sciences Research Institute at the University of Illinois at Chicago, Agile Mind, Inc.

PROBLEM SOLVING AND METACOGNITION Lesson 1: Thinking about thinking

LESSON 1: OPENER Consider some of the problems you have tackled in this course.

• The Banquet Table Problem

• The Van Rental Problem

• The Pond Border Problem

1. How did you feel when you first started working on each problem? Were you confident? Confused? Worried?

2. How did you feel after you solved each problem?

LESSON 1: CORE ACTIVITY

The Frozen Yogurt Store Problem

You and your friend go to a frozen yogurt store. You both like to get frozen yogurt cones with different toppings. The store has a sign showing the different kinds of cones, yogurt, and toppings you can buy. You and your friend wonder how many different one-‐topping cones you can make.

Work with your partner to solve the following problems.

1. Find all the different combinations of cones, yogurt, and toppings you can make and explain how you know you have found all of them.

2 Problem solving and metacognition


2. How would your numbers change if the store added a waffle cone? Explain.

Now that you have completed the Frozen Yogurt Store Problem, talk with your partner about what happened while you were solving the problem and answer the following questions.

3. Think about how you began the task.

a. Did you feel confused? Did you know exactly how to get started? Think of a few words to describe your feelings.

b. Did you make a plan or just start making different combinations of cones, yogurt, and toppings? If you came up with a plan, describe it.

c. Did the approach you wrote about in question 1 work? Explain.

d. How did you represent the information in the problem? Did you draw a picture, make a table, or create some other representation? Did you keep the same representation as you worked on the problem or did you change representations?

4. Think about how you and your partner figured out whether you were making progress.

a. Did you stop and think about whether or not what you were doing was working? How?

b. If what you were doing wasn’t working, did you talk about other things you could try instead?

5. Did you use the Mathematical Problem-‐Solving Routine from the unit Getting started with algebra? If so, how did you use it?



LESSON 1: REVIEW ONLINE ASSESSMENT

You will work with your class to review the online assessment questions.

Problems we did well on: Skills and/or concepts that are addressed in the problems we did well on:

Problems we did not do well on: Skills and/or concepts that are addressed in the problems we did not do well on:

Addressing areas of incomplete understanding:

Use this page and notebook paper to take notes and re-‐work particular online assessment problems that your class identifies.

Problem #_____

Work for problem:

Problem #_____

Work for problem:

Problem #_____

Work for problem:



LESSON 1: HOMEWORK Notes or additional instructions based on whole-‐class discussion of homework assignment:

Next class period, you will take a mid-‐unit assessment. One good study skill to prepare for tests is to review the important skills and ideas you have learned. Use this list to help you review these skills and concepts, especially by reviewing related course materials.

Important skills and ideas you have learned so far in the unit Introduction to functions and equations:

• Use words, tables, graphs, and algebraic rules to identify, describe, and analyze patterns and mathematical relationships • Solve problems and model situations using patterns and mathematical relationships • Connect different representations of mathematical relationships • Understand the advantages and limitations of particular representations of mathematical relationships • Analyze and create equivalent algebraic expressions and rules • Understand ideas related to “allowable inputs,” discrete versus continuous data, and proportional versus non-‐proportional

linear relationships

Homework Assignment

Part I: Study for the mid-‐unit assessment by reviewing the key ideas listed above.

Part II: Complete the online More practice in the topic Representing mathematical relationships in multiple ways. Note the skills and ideas for which you need more review, and refer back to related activities and animations from this topic to help you study.

Part III: Complete Lesson 1: Staying Sharp.



LESSON 1: STAYING SHARP Practic

ing skills & con

cepts

1. If Michelle drove 273 miles in 4 hours, what was her speed in miles per hour? Answer with supporting work:

2. Consider the circles in this diagram:

What is the ratio of: Shaded to un-‐shaded? Shaded to total?

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3. Complete the table for this magic number puzzle:

Directions Example For any number,

n

Choose a number

8 n

5 less than the number

Multiply by 3

4. Matchsticks are used to make the pattern shown. To make these 4 squares, you need 13 matchsticks. How many matchsticks do you need to make 20 squares?

Answer with justification:

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5. These statements involve exponents. Complete the blanks to make each statement true.

32 = 2 ·∙ 2 ·∙ 2 ·∙ 2 ·∙ 2 = 2 ____

____ = 2 ·∙ 2 = 2 ____

2 = 2 ____

____ = 20

6. Anna has 2 pairs of pants (brown and gray) and 3 shirts (pink, lavender, and yellow). List the different outfits she can make with one pair of pants and one shirt. (Hint: It may be faster to list if you abbreviate.)





Lesson 2: Checking for understanding

LESSON 2: OPENER Complete the table to show different ways of representing the relationship between the number of pentagonal tables that are pushed together and the number of people that can be seated. Picture or diagram

1 pentagonal table

2 pentagonal tables pushed together

3 pentagonal tables pushed together

Table

Number of tables, n

Number of people seated, p

Words Graph

Algebraic rule

LESSON 2: MID-UNIT ASSESSMENT

Today you will take a mid-‐unit assessment.



LESSON 2: CONSOLIDATION ACTIVITY Recall Matthew, the baseball player from earlier in the course. When he graphed data from a recent baseball drill, he did not get the graph he was expecting.

.

When making a graph, the choice of scales to use for the axes is critical. It determines whether or not the graph will accurately depict a mathematical situation or relationship. In this topic, you will practice scaling graph axes. A learning goal is that you will move closer to automaticity with this important skill. Graph the data for Matthew’s recent baseball drill using the grid provided. Choose an appropriate scale for both the x-‐axis and the y-‐axis. Your choice of scales should allow you to see a good representation of the data. The sketch shown (which does not contain numbers on the axes) is provided as a guide for what your finished graph should look like.

x

y

Time from beginning of throw (seconds)

Heigh

t of b

all from groun

d (fee

t)






Part I. Complete the Consolidation activity if you did not have time to do so in class.

Part II. A critical step in creating a graph is deciding how to scale the axes. In this assignment, you will practice this skill.

1. State an appropriate scale to use to graph the data in the x-‐y table shown:

x −5 −4 −3 −2 −1 0 1 2 3 4 5

y 5 4 3 2 1 0 −1 −2 −3 −4 −5

Minimum x-‐value: Minimum y-‐value: Maximum x-‐value: Maximum y-‐value: Increment for x-‐axis: Increment for y-‐axis:

2. State an appropriate scale to use to graph the algebraic rule 2 3y x= − on the graphing calculator. Consider inputs from 10x = − to 10x = . You might find it helpful to create an x-‐y table for the algebraic rule to assist you in making decisions

about the graph scales.

3. Data for a car traveling on a highway are contained in the table. State an appropriate scale to use to graph the data.

Time (hrs) 0 0.5 1 1.5 2 2.5 3 3.5 4

Distance (miles) 0 30 60 90 120 150 180 210 240




4. An input-‐output table for the algebraic rule 2y x= is provided. State an appropriate scale to use to graph the algebraic rule.

x −5 −4 −3 −2 −1 0 1 2 3 4 5

y 25 16 9 4 1 0 1 4 9 16 25

5. In the study of electricity, the relationship between power, P, (measured in watts), current, I, (measured in amperes), and

voltage, V, (measured in volts) is given by the formula: =P I V⋅ . Graph the relationship between power and current for a 12-‐volt battery. In other words, the goal is to consider what graph scale would make sense for the algebraic rule =P I 12⋅ . Consider inputs for the variable I from 0 to 10 amperes. State an appropriate scale to use for the graph. You might find it helpful to create an input-‐output table for the algebraic rule to assist you in making decisions about the graph scales.





ing skills & con

cepts

1. It took Marty 10 seconds to walk 15 feet. How fast was he walking, in feet per second? Answer with supporting work:

2. A particular shade of purple paint is made of a ratio of 2 pints of red paint to 1 pint of blue paint. List 4 other combinations of red and blue paint that would make the same shade of purple paint.

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3. Complete the table for this magic number puzzle:

Directions Example For any number,

n

Choose a number 7 n

Multiply by 3

Add 1

4. Matchsticks are used to make the pattern shown. To make these 4 squares, you need 13 matchsticks. How many squares in this pattern could you make with 55 matchsticks?


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27 = 3 ·∙ 3 ·∙ 3 = 3 ____

____ = 3 ·∙ 3 = 3 ____

3 = 3 ____

____ = 30

6. It's peanut butter and jelly time! Fred has two types of peanut butter (smooth and crunchy), three types of jelly (raspberry, grape, and blueberry), and one type of bread (wheat). How many different peanut butter and jelly sandwiches can he make? Answer with justification:



Lesson 3: Metacognition and the Friendship Club Problem

LESSON 3: OPENER Mike and Ike are working on the following problem: Imagine that a cell splits itself into two cells every 24 hours. How many cells will there be after 5 days?

1. Consider Mike’s plan for solving the problem. Is he on the right track to finding a solution? Explain. Mike’s work

If the cell splits each day, then there will be 2 cells on the second day, 3 cells on the next day, and so on. I can use a table to help me keep track of the situation.

Day 1 2 3 Number of cells 1 2 3

2. Consider Ike’s plan for solving the problem. Is he on the right track to finding a solution? Explain.

3. When solving a problem, why is it a good idea to check to make sure that you are on the right track?

4. If you realize that you are not on the right track, what can you do?

Ike’s work

This diagram shows how the cell splits over the first 3 days.

Day 1 Day 2 Day 3



LESSON 3: CORE ACTIVITY The components of the Mathematical Problem-‐Solving Routine are shown. How are the metacognitive questions related to the problem-‐solving routine?

Mathematical Problem Solving Routine Metacognitive questions

• How is this problem similar to other problems I have solved?

• What questions do I have about the problem? • What are some entry points into this problem? • How can I represent this problem?

• What tools do I have available to help me with this task?

• Which strategies might help me? • What should I do first?

• Am I on the right track? • What should I do next? • Should I try a different approach? • If I’m stuck, what other strategies can I try?

• Does this answer make sense in the context of the problem?

• How close is this answer to my estimate? • Are there things I still don’t understand? • What did I learn from this problem that I could

use in solving other problems?



1. For each step in the Mathematical Problem-‐Solving routine, list additional metacognitive questions that you can ask yourself.

a. Understand

b. Make a plan

c. Do the math

d. Look back

2. How can metacognitive strategies help you be a more effective learner?



3. The Friendship Club Problem: You are a member of a friendship club. There are 8 members in your club. On the 8th day of each month, you have a friendship call. Each member of the club talks to every other member that day by phone. This way everybody expresses his or her friendship with each other.

a. How many conversations occur on the 8th day of each month?

b. How many conversations occur each year? Show how you figured this out.

c. If the club adds four new friends, how many conversations will occur each month? Explain your solution.

d. Another friendship club likes your friendship conversation idea. The members of the other club want to know how they can figure out how many conversations will occur given any number of members. Explain how they can figure out the total number of conversations for each month.



LESSON 3: CONSOLIDATION ACTIVITY

1. Uri and Susan are lab partners in science class. They collect data (shown below) on the heating curve of water.

Time (sec) 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180

Temp (°C) −25 −10 0 0 0 0 0 15 28 43 58 73 88 100 100 100 100 100 100 They want to create a graph of the data on the graphing calculator. State an appropriate scale for the axes for the graph.

2. You want to graph the algebraic rule 3 5y x= + . Suppose you want to consider both positive and negative values for the

input. State an appropriate scale for the axes for the graph. You might find it helpful to create an x-‐y table for the algebraic rule to assist you in making decisions about the graph scales.


3. Noemi is performing a walking experiment in her science class using a motion detector. She walks 20 feet from the motion

detector; this part of the walk takes 10 seconds. She stops for 5 seconds. Finally, she walks back to the motion detector; this part of the walk takes 15 seconds. Noemi needs to create a graph of the walk. Time, in seconds, will be graphed on the x-‐axis. Distance from the motion detector, in feet, will be graphed on the y-‐axis. State an appropriate scale for the axes for the graph.


4. You want to graph the function 2y x= using your graphing calculator. State an appropriate scale for the axes for the graph.

Consider inputs from 10 to −10. You might find it helpful to create an x-‐y table for the algebraic rule to assist you in making decisions about the graph scales.

5. The formula 1.8 32F C= + describes the relationship between the Fahrenheit and Celsius temperature scales. Suppose you

want to create a graph that shows the relationship for Celsius temperatures from 0 to 100 degrees. State an appropriate scale for the axes for the graph. You might find it helpful to create an input-‐output table for the algebraic rule to assist you in making decisions about the graph scales.





Homework Assignment

Part I: Complete the online More practice in the topic Problem solving and metacognition.

Part II: Complete Lesson 3: Staying Sharp.

Record any questions that you have as you complete the More practice (perhaps as it connects to challenges you encountered with particular online assessment questions) below.




ing skills & con

cepts

1. It took Mary 15 seconds to walk 10 feet. How fast was she walking, in feet per second? Answer with supporting work:

2. Find the ratio of width to length for each rectangle. Simplify each ratio.

Width Length Ratio

2 8

6 24

5 20

3 12

Are the rectangles similar? Explain your answer.

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3. In an algebra class last year, students who earned grades of A spent 4.7 hours more on homework per week than students who earned grades of C. Write an algebraic rule to represent the situation, using A for the hours spent on homework per week for students who earned grades of A, and C for the hours spent on homework per week for students who earned a grade of C. Answer with supporting work:

4. Kory bought 2 boxes of candy and 1 soda at the movies and paid $8.00. Christopher bought 1 box of candy and 1 soda and paid $5.50. How much does 1 box of candy cost? How much does 1 soda cost? Answer with justification:

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4 ·∙ 4 ·∙ 4 ·∙ 4 = 256 = 4 ____

4 ·∙ 4 ·∙ 4 = ___ = 4 ____

4 ·∙ 4 = ___ = 4 ____

4 = 4 ____

6. Three soccer players, Alberto, Ben, and Chike, are forming a "wall" to defend against a penalty kick. How many different ways can they line up side by side?

Make a list and explain how you know you found all of the possible arrangements.





Lesson 4: Friendship Club Problem presentations

LESSON 4: OPENER In this lesson, you and your partner will prepare and present a solution to the Friendship Club Problem. Review the Class Presentation Criteria then answer the following question.

Class Presentation Criteria Speakers Audience Members

1. Include a clear write-‐up of your solution to the problem.

2. Include a clear, concise explanation as to why you believe your answer is correct.

3. Include a clear, concise explanation of solution strategy.

4. Both partners participate in presentation.

5. Both partners use strong, clear voices when making the presentation; both partners employ good posture and make eye contact.

1. Give their full attention and respect to the presenter.

2. Take notes, as needed.

3. Are prepared to summarize the presenters’ mathematical argument or strategy.

4. Ask clarifying questions.

5. Are prepared to make connections between the presenters’ ideas and their own ideas.

1. What is a goal you have for the presentation that you will make to the class?

LESSON 4: CORE ACTIVITY 1. Work with your partner to prepare a write-‐up of the solution to the Friendship Club Problem along with a clear, concise

explanation of your solution strategy.

As different solutions are presented, note how different representations were used to solve the problem. Also note different ways in which the Mathematical Problem-‐Solving Routine and metacognitive strategies were helpful in solving the problem.

My notes:

2. Use the table to find the number of calls each member of the Friendship Club makes. Then use the completed table to find the general rule showing the total number of calls a Friendship Club will make, given any number of members.

3. In the last lesson, you wrote a general rule to find the number of calls a Friendship Club will make given any number of

members. How was the method you used to find this rule similar to or different from the method presented in the animation?

LESSON 4: REVIEW MID-UNIT ASSESSMENT

Person Number of calls

1 n – 1

2 3 4

n – 1 n



Use the Assessment-‐Processing Routine to help review and correct your mid-‐unit assessment.


1. Complete the following math journal.

Term My understanding of what the idea means An example that shows the meaning of the idea

Metacognition

2. You presented the Friendship Club Problem in class today. What are some ways that you used the metacognitive questions to solve this problem?



Ashley and Brittany's Trip Problem

Ashley and Brittany are cousins who are going on a trip with their families to visit their grandparents in Mississippi. The families decide to take two cars on the trip, but both families leave from Ashley’s house and take the same route. Ashley’s family drives in one car at a steady rate of 40 miles per hour. Brittany’s family leaves two hours later and drives at a constant rate of 50 miles per hour. Both families arrive at the grandparents’ house at the same time! What is the distance from Ashley’s house to the grandparents’ house?

3. Use the Mathematical Problem-‐Solving Routine to solve the problem of Ashley and Brittany's Trip. The steps are outlined below. As you apply the Problem-‐Solving Routine to this problem, make note of where you are also using metacognitive strategies. The last question asks you to write the ways you used metacognitive strategies.

Understand

a. Try to visualize the situation. Consider drawing a diagram to help you picture the situation.

b. State the goal of the problem in your own words.

c. What is the important information in the problem?

Plan

d. What strategy/representation will you try to solve the problem? Why?

Solve

e. Carry out your plan.

Look back

f. Is the answer that you got reasonable?

g. Is there a way to check your answer?

h. What can you learn from this problem? What strategy did you use and why did it work for this problem? For what types of problems would this strategy work in general?

Metacognitive strategies

i. When you worked Ashley and Brittany's Trip Problem, what are some ways that you used the metacognitive questions?




ing skills & con

cepts

1. Johann walked 1.2 feet per second How far, in feet, did he walk in 5 seconds? Answer with supporting work:

2. Two boxes of identical birthday candles are pictured: a 2-‐ounce box and a 6-‐ounce box. There are 96 candles in the large box. How many candles are in the small box?


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3. Two bottles of cough medicine are the same size. The cost of the Store Brand is two-‐thirds the cost of the National Brand. If you use S to represent the cost of the Store Brand and N to represent the cost of the National Brand, which of these equations correctly represents the situation?

3S N= 23S N=

2

3N S= 2N S=

Explain or justify your choice:

4. Using the situation in question 3, if one bottle of the National Brand of cough medicine costs $12, what is the cost of a bottle of the Store Brand? Answer with justification:

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5. These statements involve exponents. Complete the blanks to make each statement true. 54 = ______

53 = ______

52 = ______

51 = ______ 50 = ______

6. There are 6 different ways to arrange the three letters A, B, and C: ABC, ACB, BAC, BCA, CAB, CBA. There are 24 different ways to arrange the four letters A, B, C, and D. The table below contains a list of these arrangements in a systematic order. Complete the table. ABCD BACD CABD DABC ABDC BA___ C____ D____ ACBD BCAD CB___ DB___ AC___ BC___ CB___ D____ ADBC BDAC CDAB D____ AD___ BD___ CD___ _____



Lesson 5: Solving a similar problem

LESSON 5: OPENER Of the three problems below, which two do you think are the most mathematically similar? You do not have to solve the problems, but you should provide an explanation for your selection. Problem 1

Find the length of the hypotenuse of the right triangle shown.

Problem 2

On a baseball diamond, the distance between each of the bases is 90 feet, as shown in the diagram. A player hits a home run and travels around the bases (from home plate to first base to second base to third base to home plate). What is the total distance around the bases?

Problem 3

On a baseball diamond, the distance between each of the bases is 90 feet, as shown in the diagram. A player throws the ball from home plate to second base. What is the distance from home plate to second base? Answer: Problems and are mathematically similar because …

LESSON 5: CORE ACTIVITY 1. Examine the following six problems. Which problems do you think are mathematically alike? Make at least two groups of

problems and record them as Group 1 and Group 2. If you find more groups, you can record them as Group 3 and Group 4. For each group, write the reason you have grouped those problems together.

A. Ella had $40. She spent $15 on music. How much does she have left?

B. North High School is having a charity sale in the gymnasium. If 750 of the gym’s 7,000 sq ft of floor space is used to sell refreshments, how much floor space is available for selling merchandise?

C. Mike’s bedroom is 9 ft by 12 ft. How many square feet of floor space does he have?

D. A teacher bought 5 new storage shelves. Each shelf holds 45 student journals. How many journals can she store on her new shelves?

E. Jan earns $15 per hour tutoring math. How much will she earn if she tutors for 40 hours?

F. 16 of the 25 best-‐selling albums of all time are by groups. How many are not by groups?



Group 1:

Reason:

Group 2:

Reason:

Group 3 (if needed):

Reason:

Group 4 (if needed):

Reason:

2. List ways in which the Frozen Yogurt Store Problem and the Friendship Club Problem are mathematically alike and ways in

which they are different.

Alike:

Different:

3. Apply the Mathematical Problem-‐Solving Routine and metacognitive strategies to solve the Diagonals Problem.

The table shows the relationship between the number of sides of a polygon and the number of diagonals that can be drawn in the polygon. (Remember: A diagonal is a line segment inside a polygon that joins two vertices of the polygon. A diagonal cannot be an edge of a polygon.)

a. Based on this pattern, how many diagonals would a 7-‐sided polygon have? Justify your answer.



b. Draw the diagonals on the 7-‐sided polygon shown.

c. How many diagonals does a polygon with 20 sides have? Justify your answer.

d. How many diagonals does a polygon with any number of sides, n, have? Justify your answer.

e. If a polygon has 65 diagonals, how many sides does that polygon have? Justify your answer.

f. What are some of the ways you used the Mathematical Problem-‐Solving Routine and metacognitive questions when you worked on this problem?

g. Is this problem similar to a problem that you have solved before? If so, name the problem(s) and tell how these problems are similar. Did the similarity help you solve the problem? If so, how?

LESSON 5: CONSOLIDATION ACTIVITY 1. What are some of the ways that you used the Mathematical Problem-‐Solving Routine and metacognitive questions when

you worked on the problems in this topic: the Frozen Yogurt Store Problem, the Friendship Club Problem, Ashley and Brittany’s Trip Problem, and the Diagonals Problem?



2. What benefits did you experience from using the Mathematical Problem-‐Solving Routine and these metacognitive strategies?

3. Think about the problem-‐solving strategies you used to solve the Frozen Yogurt Store Problem, the Friendship Club Problem, Ashley and Brittany’s Trip Problem, and the Diagonals Problem. Record the strategies in your Personal Record Sheet on the appropriate line.

As you continue in this course, you will keep adding problem-‐solving strategies to your math toolbox. The more tools you have, the better equipped you will be to tackle and solve challenging problems.



Personal Record Sheet: Problem-‐Solving Strategies Mark a ✓ for strategies that you used to work on the problem or problem type.

Problem or Problem Type

Strategies Used

Solve a similar

prob

lem

Draw

a diagram

Make a table

Make a grap

h

Make an

organized list

Solve a simpler

prob

lem

Look fo

r a pattern

Set u

p an

d solve

an equ

ation

Gue

ss-‐and

-‐che

ck

Use logical

reason

ing

Work ba

ckwards

Act it o

ut

The Frozen Yogurt Store Problem

The Friendship Club Problem

Ashley and Brittany’s Trip Problem

Diagonals Problem




The Will

The family of Mr. I. M. Rich, an eccentric millionaire, is gathered in his lawyer’s office for the reading of his will. The lawyer reads the following:

I, I.M. Rich, distribute my fortune as follows and in this order:

• The first $5 million goes to my best friend, Sam, my dog;

• Half the remainder goes to my high school algebra teacher; because of her, I learned the math I needed to make my fortune;

• Then, I give $50 million to build a new hospital in town;

• Of the remaining amount, 50% goes to my wife, 40% to my daughter, 9% to my lawyer, and 1% to my son who has never worked a day in his life. It’s time he gets started.

“What!” says the son. “All I get is a measly $10,000!”

1. How much was I.M. Rich’s fortune? Justify your answer.

2. Who receives the most money and how much will that inheritor receive? Justify your answer.

3. What are some of the ways that you used the Mathematical Problem-‐Solving Routine and metacognitive strategies when

you worked on this problem?

4. Is this problem similar to a problem that you have solved before? If so, name the problem(s) and tell how these problems are similar. Did the similarity help you solve the problem? If so, how?




ing skills & con

cepts

1. Luis has two cars. His minivan gets 18 miles per gallon of gasoline and his compact gets 24 miles per gallon. How many gallons of gasoline will he save if he travels 216 miles using the compact instead of the minivan? Answer with supporting work:

2. To get ready for a field trip, students and adults are put into groups. For every 12 students in a group, there are 2 adults. If there are 96 students on the field trip, how many adults are on the field trip? Answer with justification:

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3. Casey, a second-‐grade boy, is raising caterpillars which each have the same number of legs. Let C represent the number of caterpillars Casey has, and L represent the number of legs each caterpillar has. Write an expression for the total number of legs that the caterpillars and Casey have together. Answer with supporting work:

4. Complete the square box problem.

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5. These statements involve exponents. Complete the blanks to make each statement true. 20 = _____

2 ____ = 2

2 ____ = 32

26 = _____

6. This table shows the number of possible arrangements of n letters. For example, there are 3 ·∙ 2 ·∙ 1 = 6 different ways to arrange 3 letters. Complete the table. Number of letters Process

Number of arrangements

1 1 1

2 2 ·∙ 1 2

3 3 ·∙ 2 ·∙ 1 6

4

5

6

-2

-26



problem solving and metacognition · problem(solving(and(metacognition(!! (! problem.!

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