Unit 2 Introduction to functions and equations
This unit introduces an important mathematical concept that you will use throughout this course and in future math courses: multiple representations of mathematical relationships. You will represent relationships among quantities using concrete models, tables, graphs, diagrams and pictures, verbal descriptions, and algebraic rules. You will be introduced to the concept of algebraic functions, another important mathematical idea. You will also learn about metacognition, or “thinking about thinking.” Finally, you will continue to use the routines that you started in the previous unit, including the Homework-­‐ and Assessment-­‐processing routines, math journals, and the Mathematical Problem-­‐Solving Routine. Practicing all of these routines will help you continue to grow and develop as a mathematics learner. OUTLINE
Topic 4: Representing mathematical relationships in multiple ways When you encounter a mathematical problem, it almost always helps to look for a pattern. You can usually represent a pattern in different ways. For example, you could use tables, pictures, verbal descriptions, and graphs. Several representations of the same problem are called multiple representations. Each representation can be used to answer questions about mathematical relationships and to solve problems. In this topic, you will: •
Use mathematical models to solve real-­‐world problems •
•
•
Use words, tables, graphs, and algebraic rules to represent mathematical relationships Make connections among different representations of mathematical relationships Identify advantages and limitations of representations in given situations •
•
Find equivalent algebraic expressions and rules Understand basic aspects of linear relationships, including “allowable inputs,” discrete and continuous data, and the relationship between input and output Topic 5: Problem solving and metacognition In this topic, you will continue to solve problems, but you will take more time to reflect on how you think about problems. You will also learn some additional strategies to use when working on challenging problems. These strategies, combined with the Mathematical Problem-­‐Solving Routine from the unit Getting started with Algebra, will help you become a more effective learner. In this topic, you will: •
Understand the role of metacognition in the learning process •
Solve a non-­‐routine problem •
Practice scaling graph axes •
Prepare and present a solution and solution process to a problem •
Explore what makes problems mathematically similar Copyright © 2012 Charles A. Dana Center at the University of Texas at Austin, Learning Sciences Research Institute at the University of Illinois at Chicago, and Agile Mind, Inc. Topic 6: Working with functions and equations Functions and equations play a major role in describing relationships mathematically in order to predict behavior. In this topic, you will use functions and equations to make predictions and answer questions about real-­‐world situations. In this topic, you will: •
Understand the concept of a functional relationship •
Identify independent and dependent variables in functional relationships •
Represent functions using words, tables, graphs, and symbols – including function notation •
Develop equations that arise from functions and solve these equations by inspection or “undoing” •
Distinguish between proportional and non-­‐proportional situations represented by linear functions Copyright © 2012 Charles A. Dana Center at the University of Texas at Austin, Learning Sciences Research Institute at the University of Illinois at Chicago, and Agile Mind, Inc. 131
Topic 4: Representing mathematical relationships in multiple ways
REPRESENTING MATHEMATICAL RELATIONSHIPS
IN MULTIPLE WAYS
Lesson 4.1 Introducing multiple representations
4.1 OPENER
Think back to the Bike and Skateboard Problem you solved in Unit 1. An essential piece of information was the relationship between the number of bikes and the number of wheels on the bikes. Represent this relationship in each of the following ways. Note that the table already contains input values. 1. In an input-­‐output table: 2. In a picture or diagram: Number of bikes Number of wheels 1 2 3 4 5 3. In words: 4. In a graph: 5. In an algebraic rule: (Hint: Use n to represent the number of bikes and w to represent the number of wheels.) 4.1 CORE ACTIVITY
1.
Compare the representations that you created in the Opener with those of your classmates. Then compare them with the representations shown in the animation. How are they similar? How are they different? Copyright © 2012 Charles A. Dana Center at the University of Texas at Austin, Learning Sciences Research Institute at the University of Illinois at Chicago, and Agile Mind, Inc. 132 2.
Unit 2 – Introduction to functions and equations Erlinda and Chris are working with a caterer to rent banquet tables for the ninth grade fall dance. The tables are hexagon shaped, and six chairs will be placed around each table. Erlinda and Chris want to determine how many tables they will need for whatever number of students plan to attend the dance. A picture representation of the situation is shown below. Represent the relationship between the number of tables and the total number of people that can be seated as a table, as a graph, in words, and as an algebraic rule. Picture or diagram Table Algebraic rule 3.
Words Graph Which representation do you think best illustrates the relationship between the number of tables and the number of people that can be seated? Explain your response. 4.
Why might the choice of representation depend upon the question you are trying to answer? Copyright © 2012 Charles A. Dana Center at the University of Texas at Austin, Learning Sciences Research Institute at the University of Illinois at Chicago, and Agile Mind, Inc. 133
Topic 4: Representing mathematical relationships in multiple ways
4.1 CONSOLIDATION ACTIVITY
The dance committee is also responsible for other arrangements for the dance. Create tables, graphs, and algebraic rules for each of the situations described in questions 1-­‐3. Include a process column in your tables to help you find and write the algebraic rules. Then, answer question 4. (Hint: Read through question 4, parts a-­‐f, before making your tables and graphs.) 1.
Situation: There will be a centerpiece on each table. A centerpiece costs $5.50. Represent the relationship between the number of centerpieces and the total cost for the centerpieces. Table: Graph: Process Algebraic rule: 2.
Situation: A DJ will play music. The DJ charges $20 per hour plus a $50 appearance fee. Represent the relationship between the number of hours the DJ works and the total cost of hiring the DJ. Table: Graph: Process Algebraic rule: Copyright © 2012 Charles A. Dana Center at the University of Texas at Austin, Learning Sciences Research Institute at the University of Illinois at Chicago, and Agile Mind, Inc. 134 3.
Unit 2 – Introduction to functions and equations Situation: Refreshments cost $10 per person, plus a $15 set-­‐up charge. Represent the relationship between the number of people and the total cost for the refreshments. Table: Graph: Process Algebraic rule: 4. Use the tables, graphs, and algebraic rules you created to help you answer the following questions. Tell which representation you used and why, and describe how you found the answer. a. How much will 10 centerpieces cost? What about 20 centerpieces? Representation used and why: b. How many centerpieces can the committee buy for $88? Representation used and why: c. How much will the DJ charge for playing 3 hours? How much will the DJ charge for playing only 2 hours? Representation used and why: The committee has budgeted a maximum of $130 for the DJ. How many hours of DJ services will that amount provide? Representation used and why: The committee is estimating that 60 to 120 people will attend the dance. How much will refreshments cost for that number of people? Representation used and why: If the committee wants to keep refreshment costs under $1000, what is the maximum number of people who can attend the dance (assuming they all want refreshments)? Representation used and why: d. e. f. Copyright © 2012 Charles A. Dana Center at the University of Texas at Austin, Learning Sciences Research Institute at the University of Illinois at Chicago, and Agile Mind, Inc. 135
Topic 4: Representing mathematical relationships in multiple ways
HOMEWORK 4.1
Notes or additional instructions based on whole-­‐class discussion of homework assignment: 1.
Some, but not all, of the representations in each set model the situation described. Answer the questions for each problem. a. Situation Algebraic Rule Table Graph A store is having a sale. Every Number Cost in item in the store costs $2, and of items dollars you get to subtract $1 from your x y total bill. 1 1 2 3 6 11 Does the algebraic rule match the situation? YES / NO Explanation: Does the table match the situation? YES / NO Explanation: Does the graph match the situation? YES / NO Explanation: b. Situation Bananas sell for $0.60 per pound. Algebraic Rule Table Number of pounds x 2 5 6 Graph Cost in dollars y $1.60 $4.00 $4.80 Does the algebraic rule match the situation? YES / NO Explanation: Does the table match the situation? YES / NO Explanation: Does the graph match the situation? YES / NO Explanation: Copyright © 2012 Charles A. Dana Center at the University of Texas at Austin, Learning Sciences Research Institute at the University of Illinois at Chicago, and Agile Mind, Inc. 136 2.
Unit 2 – Introduction to functions and equations Whenever water comes into contact with heated underground rocks, geothermal energy is generated. The underground temperature of the rocks depends on the rocks’ depth below the surface. The temperature, t, in degrees Celsius is estimated by the following algebraic rule: In this rule, d is the depth of the rocks in kilometers. a. Complete the table for this algebraic rule. Note that the table contains a process column. The first row has been completed for you. Depth, d (kilometers) c.
Temperature, t (degrees Celsius) Process 1 55 2 3 5 8 10 Use your table and/or graph to help you answer the following questions. Explain how you found your answers. i. Find the temperature of the rocks at a depth of 3 kilometers. 3.
b. Create a graph for the geothermal energy situation. Use the table you created in part a. ii. Find the depth if the temperature of the rocks is 195 degrees Celsius. Create tabular and algebraic representations of the information shown in the graph. Rule for table/graph: Input (x) Number of tickets sold Process Output (y) Amount of money collected (dollars) Copyright © 2012 Charles A. Dana Center at the University of Texas at Austin, Learning Sciences Research Institute at the University of Illinois at Chicago, and Agile Mind, Inc. 137
Topic 4: Representing mathematical relationships in multiple ways
STAYING SHARP 4.1
Practicing algebra skills & concepts 1.
2.
Write an algebraic expression to represent each phrase: a. 4 more than 7 times a number b. 4 more than a number, then multiply the result by 7 3.
Consider the following magic number puzzle directions: a. Start with a number, n. b. Divide n by 2. c. Then add 11. What is the algebraic rule for this magic number puzzle? 4.
An isosceles triangle has two congruent sides, called legs. The third side is called the base. An isosceles triangle has a base of 5 cm and a perimeter of 27 cm. What is the length of each leg of the triangle? Answer with supporting work: 5.
Bevil and Sona each got a big cookie. Sona gave Alfredo half of her cookie, and Bevil gave Alfredo one-­‐third of his cookie. What fraction of a whole cookie did Alfredo get? Show evidence for your answer. 6.
Match each tile below with the appropriate place on the number line. Preparing for upcoming lessons Reviewing pre-­‐algebra ideas 2
Find the value of b – 4ac when a = −2, b = 5, and c = −1. Show your work. Copyright © 2012 Charles A. Dana Center at the University of Texas at Austin, Learning Sciences Research Institute at the University of Illinois at Chicago, and Agile Mind, Inc. 138 Unit 2 – Introduction to functions and equations Copyright © 2012 Charles A. Dana Center at the University of Texas at Austin, Learning Sciences Research Institute at the University of Illinois at Chicago, and Agile Mind, Inc. 139
Topic 4: Representing mathematical relationships in multiple ways
Lesson 4.2 The Banquet Table Problem
4.2 OPENER
A sequence is shown in picture form. Assume that the sequence continues to grow in the same way. Study the pattern. Then complete the following tasks.                     Figure 1 Figure 2 Figure 3 Figure 4 1. Organize the information for the figures into an input-­‐
2. Write an algebraic rule for the table. Your rule should use output table. the variables n and d. Figure number, n Process Number of dots, d 4.2 CORE ACTIVITY
Erlinda and Chris continue their work on the dance committee. Erlinda just found out the hall where the tables will be located is long and narrow. There is not enough room to spread the tables out. Chris suggests pushing the tables together in a row. He makes diagrams showing arrangements of tables and chairs.
1 hexagonal table 2 hexagonal tables pushed together 3 hexagonal tables pushed together DIRECTIONS: On your own, work on questions 1-­‐4. After you have answered all of the questions, share your answers with your partner. Were your answers the same? Did you see the problem the same way or differently? Copyright © 2012 Charles A. Dana Center at the University of Texas at Austin, Learning Sciences Research Institute at the University of Illinois at Chicago, and Agile Mind, Inc. 140 1.
Unit 2 – Introduction to functions and equations For each number of tables, n, shown below, find the number of people that can be seated, p. Show how you performed each calculation in the process column. Use the diagrams to help you. Then use the information in your process column to find a general algebraic rule for the number of people that can sit around any number of hexagonal tables that are pushed together. Write your rule using the variables n and p. Number of tables, n 1 2 3 4 5 20 100 n Number of people that can be seated, p Process 2.
Write an explanation of how you used the information in the process column to find your algebraic rule. 3.
How many tables will be needed to seat 50 people? Show how you found your answer. 4.
How many tables will be needed to seat 400 people? Show how you found your answer. 4.2 REVIEW END-OF-UNIT ASSESSMENT
Today you will review the end-­‐of-­‐unit assessment. Copyright © 2012 Charles A. Dana Center at the University of Texas at Austin, Learning Sciences Research Institute at the University of Illinois at Chicago, and Agile Mind, Inc. 141
Topic 4: Representing mathematical relationships in multiple ways
HOMEWORK 4.2
Notes or additional instructions based on whole-­‐class discussion of homework assignment: Consider the pattern in the set of figures shown here. Figure 1 1.
5.
Figure 3 Figure 2 Arrange the information from the diagrams into an input-­‐output table. Label the input column Figure number, n. Label the output column Number of squares in figure, s. Label the center column Process. 2.
3.
4.
In the space to the right of the figures above, draw Figure 4. How many squares are needed to build Figure 4? Add this information to your input-­‐output table. In the space to the right of the figures above, draw Figure 5. How many squares are needed to build Figure 5? Add this information to your input-­‐output table. Create a graph of your input-­‐output table. (Remember that the input variable is graphed along the horizontal/bottom axis.) How many squares would be needed to build the tenth figure? Justify your answer. 6.
Write an explanation of how you can calculate the number of squares needed if you know the figure number. 7.
Write an algebraic rule that gives the number of squares needed to build any figure number. Write your rule using the variables n and s. Use the process column in your table to help you if needed. 8.
What figure number has 60 squares in it? Justify your answer. Copyright © 2012 Charles A. Dana Center at the University of Texas at Austin, Learning Sciences Research Institute at the University of Illinois at Chicago, and Agile Mind, Inc. 142 Unit 2 – Introduction to functions and equations Practicing algebra skills & concepts STAYING SHARP 4.2
Let T represent the number of tables in a banquet hall, and let C represent the number of chairs for each table. 1. Is the expression C ·∙ T meaningful in this context? If it is meaningful, what does it mean? If it is not meaningful, explain why. 2. Is the expression C + T meaningful in this context? If it is meaningful, what does it mean? If it is not meaningful, explain why. Reviewing pre-­‐algebra ideas Preparing for upcoming lessons 3.
Teresa babysits for her neighbors. She charges $13 for the first hour and $10 for each additional hour. Write a rule for the total amount of money she charges, T, based on the number of hours she babysits, H. 4.
Angela also babysits for her neighbors. She charges $11 per hour. Which babysitter, Teresa or Angela, will charge less to babysit for 2 hours? 3 hours? 4 hours? Show evidence for your answers. 600 students were surveyed about their favorite sport. Their responses are represented in this circle graph, or pie chart. 5. What percentage of students chose baseball as their favorite sport? 6. How many students chose basketball as their favorite sport? Copyright © 2012 Charles A. Dana Center at the University of Texas at Austin, Learning Sciences Research Institute at the University of Illinois at Chicago, and Agile Mind, Inc. 143
Topic 4: Representing mathematical relationships in multiple ways
Lesson 4.3 Equivalent representations and expressions
4.3 OPENER
1.
In the table, each row contains two numerical or algebraic expressions. Decide whether Expression 1 and Expression 2 show two different ways of representing the same thing. Provide a justification for your answer. Expression 1 1 2 3 Expression 2 Are the two expressions different ways of representing the same thing? Justification YES NO YES NO YES NO 4.3 CORE ACTIVITY
Consider 6 hexagonal tables pushed together in a row. How many people can be seated at the tables? The caterer, Erlinda, Chris, and another member of the dance committee, Pauline, each approached the problem in a different way. The number sentences in the table show how each person approached the problem. 1.
Examine the caterer’s method. Then, fill in the caterer’s row in the table by completing the following steps: •
Explain how the calculation method for 6 tables can be matched to the diagram of the situation. •
Show how the caterer would calculate the number of people that can be seated around 9 tables. •
Generalize from the arithmetic to write an algebraic rule that gives the number of people that can be seated around any number of tables. This rule should link the number of tables (n) to the number of people (p) that can be seated. 2. Complete the rest of the table using the steps in question 1 for Erlinda’s, Chris’s, and Pauline’s calculation methods. Calculation for 6 tables Calculation for 9 tables Explanation Caterer:
Algebraic rule using n and p Erlinda: Chris: Pauline: Copyright © 2012 Charles A. Dana Center at the University of Texas at Austin, Learning Sciences Research Institute at the University of Illinois at Chicago, and Agile Mind, Inc. 144 Unit 2 – Introduction to functions and equations 4.3 CONSOLIDATION ACTIVITY
Part I. In the Banquet Table Problem, the caterer and members of the dance committee each created different algebraic rules to represent the number of people, p, who can be seated around n tables that are pushed together. 1. Rewrite each rule in simpler form to determine whether all the rules are equivalent. a.
Caterer: c.
Chris: 2.
b.
Erlinda: d.
Pauline: Do Kathy’s and Troy’s rules both work for this situation? Find out by simplifying the expressions to see whether they are equivalent to those in question 1. a.
Kathy: b.
Troy: Part II. Now use what you have learned about rewriting expressions to compare algebraic expressions that are not in a problem context. 3.
Decide whether the two expressions are equivalent and explain your reasoning. a.
b.
and
Equivalent: Yes No Equivalent: Yes No Equivalent: Yes No Equivalent: Yes No Equivalent: Yes No Equivalent: Yes No Justification: and
Justification: c.
d.
e.
f.
and Justification: Justification: and Justification: and Justification: and Copyright © 2012 Charles A. Dana Center at the University of Texas at Austin, Learning Sciences Research Institute at the University of Illinois at Chicago, and Agile Mind, Inc. 145
Topic 4: Representing mathematical relationships in multiple ways
HOMEWORK 4.3
Notes or additional instructions based on whole-­‐class discussion of homework assignment: Sarah, the yearbook editor, and Mrs. Moore, the yearbook sponsor, are reviewing costs to produce the yearbook. New yearbook design software costs $800, and it costs $15 to print each yearbook. 1.
Define the variables in this situation: 2.
# of yearbooks produced _______ Let _______ = __________________________________________ Let _______ = __________________________________________ Input variable: ___________________________________ Output variable: ____________________________________ 3.
4.
Complete the table to explore this relationship. Cost in dollars _________ Process 0 1 2 10 50 100 500 Write an algebraic rule that describes this situation. Create a graph for this problem situation on the grid provided. Be sure to scale and label your axes. Copyright © 2012 Charles A. Dana Center at the University of Texas at Austin, Learning Sciences Research Institute at the University of Illinois at Chicago, and Agile Mind, Inc. 146 5.
Unit 2 – Introduction to functions and equations Use your table, graph, and/or algebraic rule to help you answer the following questions. a. How much does it cost to produce 600 yearbooks? Explain how you found your solution. b. How many yearbooks can be produced for $8,000? Explain how you found your solution. 6.
Translate each of the following verbal descriptions into an algebraic expression. Use n for “any number.” a.
Any number increased by 6 b.
Any number multiplied by −3 c.
Any number multiplied by 2 and then increased by 4 d.
Any number divided by−4, then subtract 7 from the result e.
Any number decreased by 7, then divide the result by −4 7.
Translate each of the following verbal descriptions into an algebraic rule. a.
The total cost of the order, t, is 11 dollars multiplied by the number of pizzas ordered, p. b.
The number of songs that can be downloaded, s, is equal to the number of credits in the account, c, plus an additional 2 bonus credits. 8.
Decide whether the two expressions are equivalent, and explain your reasoning. a. and b. and c. and
d. and Equivalent: Yes No Equivalent: Yes No Equivalent: Yes No Equivalent: Yes No Justification: Justification: Justification: Justification: Copyright © 2012 Charles A. Dana Center at the University of Texas at Austin, Learning Sciences Research Institute at the University of Illinois at Chicago, and Agile Mind, Inc. 147
Topic 4: Representing mathematical relationships in multiple ways
STAYING SHARP 4.3
Practicing algebra skills & concepts 1.
a.
b.
c.
2.
Find the value of 5x – 2y + 3z when x = 7, y = 11, and z = -­‐1. Show your work. 6 • (30 + 7) = (100 – 15) • 5 = 7 (n + 5) = 3.
Speedy Movers charges its customers a flat fee of $300 and an additional $40 per hour. Write an algebraic rule to find the total cost charged in dollars, c, in terms of the number of hours needed for a move, h. 4.
Use the set of shape equations below as clues to find the value of each shape. + + = 30 3 • = 21 — = 3 Answer: = ____ = ____ = ____ 5.
Jocelyn got 19 correct answers out of 25 questions on her last test. What percent of the questions did she answer correctly? Answer with supporting work: 6.
Two fractions can be easily added together when they have a common denominator. a. What is one denominator you could use to add the following two fractions? Preparing for upcoming lessons Reviewing pre-­‐algebra ideas Use the distributive property to write the following expressions in an equivalent form: b. Find the sum of the two fractions. Show your work. Copyright © 2012 Charles A. Dana Center at the University of Texas at Austin, Learning Sciences Research Institute at the University of Illinois at Chicago, and Agile Mind, Inc. 148 Unit 2 – Introduction to functions and equations Copyright © 2012 Charles A. Dana Center at the University of Texas at Austin, Learning Sciences Research Institute at the University of Illinois at Chicago, and Agile Mind, Inc. 149
Topic 4: Representing mathematical relationships in multiple ways
Lesson 4.4 Extending the Banquet Table Problem
4.4 OPENER
Zac is waiting for his friends at Cory’s Coffee Shop. He starts playing with the toothpicks at the table and makes these patterns: Pattern 1 1.
Pattern 2 Pattern 3 How many toothpicks would be needed for Pattern 4? For Pattern 5? 2.
Organize the information for the first five patterns into an input-­‐output table. 3.
4.
Use the process column to help you write an algebraic rule for the table. Use your rule to figure out how many toothpicks would be needed for Pattern 50. Input Pattern number, n Output Number of toothpicks, t Process 4.4 CORE ACTIVITY
Any Occasion Party Rental rents tables that have different shapes. In the rental catalog, the advertising manager wants to include information about the number of people that can sit around the tables, depending on what shape is ordered. Some of the shapes and their arrangements are shown. For questions 1-­‐3, complete the following steps: 1.
•
Fill in the table. For each number of banquet tables, n, show the process you can use to figure out the number of people that can be seated. Then enter the number of people seated, p. •
Write a general rule that gives the number of people, p, that can be seated around n tables. •
Find the number of tables needed to seat 100 people. Triangular tables: Number of tables (n) Process Number of people seated (p) 1 2 3 4 5 10 Rule describing the relationship between the number of tables, n, and the number of people, p: How many triangular tables are needed to seat 100 people? Show your reasoning. Copyright © 2012 Charles A. Dana Center at the University of Texas at Austin, Learning Sciences Research Institute at the University of Illinois at Chicago, and Agile Mind, Inc. 150 2.
Unit 2 – Introduction to functions and equations Trapezoidal tables: Number of tables (n) Process Number of people seated (p) 1 2 3 4 5 10 Rule describing the relationship between the number of tables, n, and the number of people, p: How many trapezoidal tables are needed to seat 100 people? Show your reasoning. 3.
Octagonal tables: Number of tables (n) Process Number of people seated (p) 1 2 3 4 5 10 Rule describing the relationship between the number of tables, n, and the number of people, p: How many octagonal tables are needed to seat 100 people? Show your reasoning. 4.
The “super generalization”: You have created separate rules that describe the relationship between the number of tables, n, and the number of people, p, for tables with different numbers of sides. Now, write a single rule that describes the relationship between the number of tables, n, and the number of people, p, for tables with any number of sides, s. Copyright © 2012 Charles A. Dana Center at the University of Texas at Austin, Learning Sciences Research Institute at the University of Illinois at Chicago, and Agile Mind, Inc. 151
Topic 4: Representing mathematical relationships in multiple ways
4.4 CONSOLIDATION ACTIVITY
1.
Complete the following self evaluation: Effort in class excellent / good / okay / need to improve Effort on homework excellent / good / okay / need to improve Effort on tests excellent / good / okay / need to improve Class participation excellent / good / okay / need to improve Working with my partner excellent / good / okay / need to improve Making presentations excellent / good / okay / need to improve Being a good audience member excellent / good / okay / need to improve Working independently excellent / good / okay / need to improve Writing and showing work excellent / good / okay / need to improve Reflecting on learning excellent / good / okay / need to improve 2.
Think about your performance on the Unit 1 end-­‐of-­‐unit assessment. a.
Are you pleased with your performance on the end-­‐of-­‐unit assessment? Circle one: Yes / No b.
Does your performance reflect your understanding of the topics in Unit 1? Circle one: Yes / No If you answered “No”, why do you think this? c.
What might you do differently in order to improve your performance on the next end-­‐of-­‐unit assessment? If you are pleased with your performance on the Unit 1 assessment, what will you do to perform at least as well on the next end-­‐of-­‐
unit assessment? 3.
Set one goal for your work/performance/achievement for Unit 2. Your goal should be specific and realistic. Carefully think through and record some concrete steps that will help you achieve this goal. Goal: Steps to achieve your goal: Copyright © 2012 Charles A. Dana Center at the University of Texas at Austin, Learning Sciences Research Institute at the University of Illinois at Chicago, and Agile Mind, Inc. 152 Unit 2 – Introduction to functions and equations HOMEWORK 4.4
Notes or additional instructions based on whole-­‐class discussion of homework assignment: The Math Club members want to buy shirts to wear during their contests. They get prices from three different shirt companies. They learn that shirt prices may include a printing set-­‐up fee in addition to the cost of each shirt. Each shirt company represents its prices in a different way. 1.
Create the missing representations (table, graph, algebraic rule, and/or verbal description) to describe each company’s price. Use S for the number of shirts and P for the total price. Shirt World Table Graph # of Shirts 1 5 10 15 Price 20 100 200 300 Totally Tees Table Graph Graph Verbal Description: We charge $10 per shirt, plus a one-­‐time print set-­‐up fee of $50. Algebraic Rule P = 15s + 20 # of Shirts Algebraic Rule Price Ts-­‐R-­‐Us Table Verbal Description: # of Shirts Algebraic Rule Price Verbal Description: Copyright © 2012 Charles A. Dana Center at the University of Texas at Austin, Learning Sciences Research Institute at the University of Illinois at Chicago, and Agile Mind, Inc. Topic 4: Representing mathematical relationships in multiple ways
153
Based on the representations in question 1, answer the following questions. Describe how you found your answers, and tell what representation(s) you used. 2. The Math Club currently has 5 members. Which shirt company should they use if they only buy shirts for current members? How did you find your answer? 3.
The Math Club members are also thinking about selling the shirts as a fundraiser. They think that they will be able to sell 100 shirts. Which shirt company should they use? How did you find your answer? 4.
The Math Club treasurer realizes that the club currently has only $120 available to purchase shirts. Which company will allow them to buy the most shirts for that amount of money? How did you find your answer? Copyright © 2012 Charles A. Dana Center at the University of Texas at Austin, Learning Sciences Research Institute at the University of Illinois at Chicago, and Agile Mind, Inc. 154 Unit 2 – Introduction to functions and equations STAYING SHARP 4.4
Practicing algebra skills & concepts 1.
Write an expression for the perimeter of the polygon. Make sure your expression is as simple as possible. 2.
Write a phrase to represent each algebraic expression: a. 2x – 9 b. 2(x – 9) Preparing for upcoming lessons The algebraic rule 6x + 3y + 2z = p can be used to find the total points earned in a carnival darts game, with these variables: • x represents the number of times a player hits the bull's-­‐eye 3.
4.
•
y represents the number of times the player hits the white ring around the bull's-­‐eye •
z represents the number of times the player hits the red ring How many points does Sasha earn if he hits the bull's-­‐eye twice, the white ring once, and the red ring zero times? When Josie plays, she earns a total of 15 points. If she hit the bull's-­‐eye once and the white ring once, how many times did she hit the red ring? The student members of a local community group are listed by age in this bar graph. Reviewing pre-­‐algebra ideas 5.
6.
How many student members are 14 or 15 years old? What fraction of the members are 13 years old? Copyright © 2012 Charles A. Dana Center at the University of Texas at Austin, Learning Sciences Research Institute at the University of Illinois at Chicago, and Agile Mind, Inc. 155
Topic 4: Representing mathematical relationships in multiple ways
Lesson 4.5 The Van Rental Problem
4.5 OPENER
A DVD movie plays at 30 frames per second. Create an algebraic rule, a table, and a graph to represent the relationship between the number of seconds, x, and the number of frames, y. Table Graph Input Number of seconds, x Process Output Number of frames, y Algebraic Rule 4.5 CORE ACTIVITY
Part I. Solving the problem 1.
The Math Club is planning a trip to the state math league competition. The contest is 140 miles from the school. The rental fees charged by two car rental companies are modeled in the graph and table. Which car rental company gives the better deal? Justify your answer. (Don’t forget that the students need to drive to the competition and then return home.) Copyright © 2012 Charles A. Dana Center at the University of Texas at Austin, Learning Sciences Research Institute at the University of Illinois at Chicago, and Agile Mind, Inc. 156 Unit 2 – Introduction to functions and equations After solving the problem, answer the following questions. 2.
3.
Vince’s Vans posted a table of rental fees on its website. a. Was the table useful when you were determining which rental company had the better deal? If so, how? b. Were there any disadvantages to using the table? If so, what were they? c. Would another representation have been more useful? If so, why? Carla's Cars-­‐to-­‐Go posted a graph on its website to represent its rental fees. a.
b.
4.
c.
Was the graph useful when you were determining which rental company had the better deal? If so, how? Were there any disadvantages to using the graph? If so, what were they? Would another representation have been more useful? If so, why? When comparing relationships, it is usually a good idea to represent both relationships in the same way. a. Create a graph relating cost to miles driven for both Vince’s Vans and Carla’s Cars-­‐to-­‐Go. Graph both relationships on the same set of axes to show how they compare. b. Describe in words how the cost of renting from Carla’s Cars-­‐to-­‐Go compares with the cost of renting from Vince’s Vans. Copyright © 2012 Charles A. Dana Center at the University of Texas at Austin, Learning Sciences Research Institute at the University of Illinois at Chicago, and Agile Mind, Inc. 157
Topic 4: Representing mathematical relationships in multiple ways
Part II. Analyzing and extending the problem 5.
The Vince’s Vans table is shown. Complete the process column in the table, then create a verbal description and an algebraic rule to represent the cost of renting from Vince’s Vans. Use these representations, along with the graph for Vince’s Vans that you created in question 4, to answer questions 6 and 7. Tabular Representation Vince's Vans Miles driven Process Rental fee 50 $22.50 100 $45.00 150 $67.50 200 $90.00 250 $112.50 300 $135.00 Verbal Description 6.
Algebraic Rule (Use the variable f to represent the rental fee. Use the variable m to represent the number of miles driven.) The Math Club recently traveled to a different competition. The members rented a van and drove 175 miles round trip. a.
How much did it cost to rent the van from Vince's Vans? b.
Which representation did you use to answer the question in part a? c.
Describe how you could have used one of the other representations to answer the question in part a. 7.
The Math Club wants to visit a science museum that is 180 miles away from the school. If the club has $150 in its field trip account, can the club afford to rent the van from Vince's Vans? Use two representations to answer this question. Describe how you used each representation. Copyright © 2012 Charles A. Dana Center at the University of Texas at Austin, Learning Sciences Research Institute at the University of Illinois at Chicago, and Agile Mind, Inc. 158 Unit 2 – Introduction to functions and equations 4.5 CONSOLIDATION ACTIVITY
1.
Create a math journal entry for the idea of “multiple representations.” The first column of the table lists each type of representation you have used to model problems. In the second column, give an example of that type of representation. In the third column, answer the question: "How is this particular representation useful?” (Keep in mind that some representations allow you to see or do things that other representations do not.) After you’ve completed your entry, compare your journal entry with that of your partner. Representation Verbal description Example How it is useful Picture Table Graph Algebraic rule Copyright © 2012 Charles A. Dana Center at the University of Texas at Austin, Learning Sciences Research Institute at the University of Illinois at Chicago, and Agile Mind, Inc. 159
Topic 4: Representing mathematical relationships in multiple ways
HOMEWORK 4.5
Notes or additional instructions based on whole-­‐class discussion of homework assignment: Part I. Solving and analyzing the MP3 Player Problem 1. Anthony and his sister, Ciara, are saving their money to buy $240 MP3 players. They each plan to buy their own player. Anthony starts off with $50 in his bank account and adds $20 each week. Ciara starts off with $10 in her bank account and adds $30 each week. Complete each of the tables below based on the information given. The first table has been started for you. Anthony 2.
Ciara Total dollars (y) Week (x) Process Total dollars (y) 50 0 1 50 + 20 • 0 50 + 20 • 1 70 1 2 2 3 3 4 4 5 5 8 8 10 10 Week (x) Process 0 State an algebraic rule for Anthony and an algebraic rule for Ciara. Let x stand for the week number, and let y represent the total dollars saved. The process columns in your tables can help you develop your rules. Rule for Anthony: _______________________ Rule for Ciara: _______________________ 3.
Use the information in the tables to create a graph for this situation. Graph the data for Anthony and Ciara on the same set of axes. Scale and label your axes. Find a way to distinguish the data for Anthony and Ciara, and create a legend to show this information. Use your graph and table to answer the follow questions: 4.
Is there a time when Anthony and Ciara have the same amount of money? If so, state when this occurs. 5.
When will Anthony have enough money to purchase the MP3 player? When will Ciara have enough money to purchase the MP3 player? Copyright © 2012 Charles A. Dana Center at the University of Texas at Austin, Learning Sciences Research Institute at the University of Illinois at Chicago, and Agile Mind, Inc. 160 Unit 2 – Introduction to functions and equations Part II. Creating your own multiple representation problem 6. Create a multiple representation problem. Each representation should model the same mathematical relationship. Verbal Description Algebraic Rule Graphical Representation Tabular Representation Copyright © 2012 Charles A. Dana Center at the University of Texas at Austin, Learning Sciences Research Institute at the University of Illinois at Chicago, and Agile Mind, Inc. 161
Topic 4: Representing mathematical relationships in multiple ways
STAYING SHARP 4.5
Preparing for upcoming lessons Practicing algebra skills & concepts 1.
a.
(11 • 6) + (11 • 10) = b.
(8 • 20) – (8 • 1) = c.
(2 • l) + (2 • w) = 2.
Find the value of when p = -­‐8, q = -­‐3, and r = 68. Show your work. Celene needs to buy icing to frost some cupcakes. She writes the rule 0.1c = i to model her situation. In her rule, c represents the number of cupcakes, and i represents the total amount of icing she will need, in ounces. 3. How many ounces of icing will Celene use to frost 1 cupcake? 4. If Celene needs 15 ounces of icing, how many cupcakes is she frosting? 5.
Reviewing pre-­‐algebra ideas Use the distributive property to write each of the following expressions in an equivalent form: On a recent test, Mark got 6 questions out of 40 wrong. Which answer best describes the percent of questions he got correct? a. Less than 50% b. More than 50% c. Less than 25% d. More than 75% Explain or support your choice: 6.
Find a fraction that, when added to , gives a sum of 1. Show your work. Copyright © 2012 Charles A. Dana Center at the University of Texas at Austin, Learning Sciences Research Institute at the University of Illinois at Chicago, and Agile Mind, Inc. 162 Unit 2 – Introduction to functions and equations Copyright © 2012 Charles A. Dana Center at the University of Texas at Austin, Learning Sciences Research Institute at the University of Illinois at Chicago, and Agile Mind, Inc. 163
Topic 4: Representing mathematical relationships in multiple ways
Lesson 4.6 The Pond Border Problem
4.6 OPENER
Anthony Chen’s father is going to put a tile border around this square fish pond. The pond is 1 yard on each side. The tiles are squares that are 1 foot on each side. The tiles fit in the grey border around the pond. How many tiles will Mr. Chen need to make the border? Explain how you solved this problem. 4.6 CORE ACTIVITY
1.
Anthony's model for a 1-­‐yard-­‐square pond (3-­‐foot by 3-­‐foot) and border is shown. What should Anthony's models for the two smaller ponds in this sequence look like (that is, his models for a 1-­‐foot by 1-­‐foot pond and a 2-­‐foot by 2-­‐foot pond)? Draw sketches of the smaller ponds and borders beside Anthony's current model. 2.
3.
Anthony knows that the length of the sides of a square pond can vary. The larger a pond is, the more border tiles he will need. Complete the following input-­‐output table and use the process column to help you find a general rule for the situation. Length of side of square pond, s Process Number of border tiles needed, t 1 2 3 10 s State a general rule for the situation that relates the number of border tiles needed for the pond, t, to the length, in feet, of one side of the square pond, s. ____________ Copyright © 2012 Charles A. Dana Center at the University of Texas at Austin, Learning Sciences Research Institute at the University of Illinois at Chicago, and Agile Mind, Inc. 164 4.
Unit 2 – Introduction to functions and equations How is the rule you found connected to the picture representation of the pond? In other words, how is the rule connected to a way of “seeing” the tile pattern for the different size ponds? (Hint: You did something similar to this when you connected rules for the Banquet Table Problem to the pictures of the banquet tables. You might find it helpful to review the Core activity in Lesson 4.3.) 5.
Anthony, his father, and his Uncle Tajil all show different picture representations relating the number of tiles, t, to the side length, s, of a pond. For each of their representations shown in the table, write an algebraic rule that fits the picture. Explain how your rule matches the picture. Picture representation Anthony’s representation: Algebraic rule that fits the representation Rule: Explanation: Anthony’s father’s representation: Rule: Explanation: Uncle Tajil’s representation: Rule: Explanation: 6.
Show that all three of your rules are equivalent. Copyright © 2012 Charles A. Dana Center at the University of Texas at Austin, Learning Sciences Research Institute at the University of Illinois at Chicago, and Agile Mind, Inc. 165
Topic 4: Representing mathematical relationships in multiple ways
4.6 CONSOLIDATION ACTIVITY
Create sets of “matching” table, graph, and algebraic rule cards. “Matching” is defined as different representations of the same relationship. Each set will have a table card, a graph card, and an algebraic rule card. When you and your partner agree on a set of matching cards, tape them together to form a set. Table Card Graph Card Algebraic Rule Card When you have matched all the cards and discussed the activity as a class, answer questions 1-­‐3. Discuss each question with your partner. Then write the answer you agree on in the space provided. 1. What strategy did you use to match the table, graph, and algebraic rule cards? 2. Was there a particular representation that was easiest to use as your starting point? If so, which representation was it, and what made it an easier starting point? 3. Sort your “matched sets” into groups that are similar. You should create at least two groups, but less than five groups. List your groups below; then answer the questions in parts a-­‐c. a. What criteria did you use to sort your sets into groups? b. What is the same about the sets in each group? Consider all three representations: graphs, tables, and algebraic rules. c. How do the groups differ from each other? Again, consider all three representations: graphs, tables, and algebraic rules. Copyright © 2012 Charles A. Dana Center at the University of Texas at Austin, Learning Sciences Research Institute at the University of Illinois at Chicago, and Agile Mind, Inc. 166 Unit 2 – Introduction to functions and equations HOMEWORK 4.6
Notes or additional instructions based on whole-­‐class discussion of homework assignment: For each x-­‐y table in questions 1-­‐4, find the graph and algebraic rule that show the same relationship. 1. 2. x -­‐2 -­‐1 0 1 2 Graph: ______ Rule: ______ A. y -­‐5 -­‐3 -­‐1 1 3 3. x -­‐2 -­‐1 0 1 2 y 3 2 1 0 -­‐1 4. x -­‐2 -­‐1 0 1 2 Graph: ______ Rule: ______ y -­‐2 -­‐1.5 -­‐1 -­‐0.5 0 x -­‐2 -­‐1 0 1 2 Graph: ______ Rule: ______ y -­‐3 -­‐1 1 3 5 Graph: ______ Rule: ______ B. G. y = −x + 1 H. y = x – 1 C. D. I. y = E. F. J. y = 2x – 1 K. y = −x − 1 L. y = 2x + 1 Copyright © 2012 Charles A. Dana Center at the University of Texas at Austin, Learning Sciences Research Institute at the University of Illinois at Chicago, and Agile Mind, Inc. 167
Topic 4: Representing mathematical relationships in multiple ways
Fill in the missing representations.
Table 5.
Graph Algebraic Rule x y -­‐3 -­‐10 -­‐1 -­‐4 0 -­‐1 1 2 2 5 4 11 0 29 _ 6.
7.
x y x y y = −10x + 20 _ Copyright © 2012 Charles A. Dana Center at the University of Texas at Austin, Learning Sciences Research Institute at the University of Illinois at Chicago, and Agile Mind, Inc. 168 Unit 2 – Introduction to functions and equations Reviewing pre-­‐algebra ideas Preparing for upcoming lessons Practicing algebra skills & concepts STAYING SHARP 4.6
Venu goes to a fruit stand and buys some apples and bananas. Let A represent the number of apples she buys, B the number of bananas she buys, CA the cost of each apple, and CB the cost of each banana. 1. a. Is the expression A + B meaningful in this situation? If so, what does it mean? If not, explain why not. b. Is the expression A ·∙ CB meaningful in this situation? If so, what does it mean? If not, explain why not. 2. Write an expression for the total cost of Venu’s purchase. Kevin works at an Internet café. The café charges customers based on the amount of time they use the Internet. There is a flat fee of $3 for all users. In addition, users are charged 20 cents for each 10-­‐minute period they use the Internet. 3. Which of these two statements makes more sense? a. The amount of money charged depends on the amount of time spent on the Internet. b. The amount of time spent on the Internet depends on the amount of money charged. Explain or justify your selection. 4. A customer at the café is charged $4.60 for her time on the Internet. How many hours was she on the Internet? Show your work. This graph illustrates the total number of students accepted to college from Ms. Fredrickson’s class. 5. a. Estimate the value for February. b. Assume the pattern continues, and estimate a value for November. 6. In which value are you more confident? Why? Copyright © 2012 Charles A. Dana Center at the University of Texas at Austin, Learning Sciences Research Institute at the University of Illinois at Chicago, and Agile Mind, Inc. 169
Topic 4: Representing mathematical relationships in multiple ways
Lesson 4.7 Making sense of multiple representations
4.7 OPENER
Rick’s Rentals is a car rental company, and Penny’s Pencils is a company that delivers pencil orders to schools. The algebraic rule used by each company to calculate cost is shown.
1.
For Rick’s Rentals: a.
Algebraic rule for Rick’s Rentals: What does the 0.10 represent in the algebraic rule? b.
where x represents the number of miles driven and y represents the rental cost, in dollars. Give a reasonable explanation for the meaning of the 22 in the algebraic rule. Algebraic rule for Penny’s Pencils: 2.
a.
What does the 0.10 represent in the algebraic rule? where x represents the number of pencils ordered and y represents the cost of the order, in dollars.
For Penny’s Pencils: b.
Give a reasonable explanation for the meaning of the 22 in the algebraic rule. 4.7 CORE ACTIVITY
Part I. Finding what input values make sense for a particular algebraic rule 1.
Consider the algebraic rule . Suppose this rule is not related to a real-­‐world situation. a. It makes sense to use 20.5 as an input for the algebraic rule. True / False Explain: b. It makes sense to use −5 as an input for the algebraic rule. True / False Explain: 2.
Again, consider the algebraic rule . Suppose this rule now relates to the real-­‐world situation of Rick’s Rentals, as described in the Opener (where x represents the number of miles driven and y represents the rental cost, in dollars). a. It makes sense to use 20.5 as an input for the algebraic rule. True / False Explain: b. It makes sense to use −5 as an input for the algebraic rule. True / False Explain: 3.
Again, consider the algebraic rule . Suppose this rule now relates to the real-­‐world situation of Penny’s Pencils, as described in the Opener (where x represents the number of pencils ordered and y represents the cost of the order, in dollars). a. It makes sense to use 20.5 as an input for the algebraic rule. True / False Explain: b. It makes sense to use −5 as an input for the algebraic rule. True / False Explain: 4.
What conclusions can you make about “allowable” input values based on questions 1-­‐3? Copyright © 2012 Charles A. Dana Center at the University of Texas at Austin, Learning Sciences Research Institute at the University of Illinois at Chicago, and Agile Mind, Inc. 170 Unit 2 – Introduction to functions and equations Part II. Graphs for different situations 5.
Consider the situations of Rick’s Rentals and Penny’s Pencils from the Opener. One of the graphs below shows a scatterplot (in which only specific ordered pairs are graphed) to represent the relationship between the inputs and outputs. The other graph shows the graph of a line to represent the relationship between the inputs and outputs. (Because this is a puzzle, there are no axis labels on the graphs.) Graph 1 Graph 2 a.
b.
c.
Which graph, Graph 1 or Graph 2, better represents the inputs and outputs for Rick’s Rentals? Explain. Which graph, Graph 1 or Graph 2, better represents the inputs and outputs for Penny’s Pencils? Explain. Explain why both graphs show data only in the first quadrant. Part III. More on the Banquet Table Problem representations You considered two types of Banquet Table Problem, as shown in the table arrangements below. Tables separated Tables pushed together
Copyright © 2012 Charles A. Dana Center at the University of Texas at Austin, Learning Sciences Research Institute at the University of Illinois at Chicago, and Agile Mind, Inc. 171
Topic 4: Representing mathematical relationships in multiple ways
6.
For each question, select either “Tables pushed together” or “Tables separated.” Circle your answer. Which situation does each table represent? a. b. Number of tables Number of people seated 1 Number of tables Number of people seated 6 1 6 2 12 2 10 3 18 3 14 4 24 4 18 Tables separated / Tables pushed together Tables separated / Tables pushed together Which situation does each algebraic rule represent? c. d. Tables separated / Tables pushed together Tables separated / Tables pushed together Which situation does each graph represent? e. f. Tables separated / Tables pushed together Tables separated / Tables pushed together g. Consider the tables, algebraic rules, and graphs for the two situations (“Tables separated” and “Tables pushed together”). How are they the same? How are they different? Copyright © 2012 Charles A. Dana Center at the University of Texas at Austin, Learning Sciences Research Institute at the University of Illinois at Chicago, and Agile Mind, Inc. 172 Unit 2 – Introduction to functions and equations Part IV. Math journal 7. After you have discussed questions 1-­‐6 as a class, complete the following math journal. Idea My understanding of what the idea means An example that shows the meaning of the idea Allowable inputs for a rule that represents an abstract situation Allowable inputs for a rule that represents a real-­‐
world situation Discrete data Continuous data 4.7 ONLINE ASSESSMENT
Today you will take an online assessment. Copyright © 2012 Charles A. Dana Center at the University of Texas at Austin, Learning Sciences Research Institute at the University of Illinois at Chicago, and Agile Mind, Inc. 173
Topic 4: Representing mathematical relationships in multiple ways
HOMEWORK 4.7
Notes or additional instructions based on whole-­‐class discussion of homework assignment: The manager of a school bookstore has found four different companies from which she may order book bags. The algebraic rules shown for each company represent the cost of placing a book bag order, c, based on the number of book bags purchased, b. 1. Write a situation for each algebraic rule, if possible. Company A: 2.
Company B: Company C: Company D: Company A: Company B: Company C: Company D: Do all four algebraic rules fit a book bag situation? Explain your answer. 3.
4.
Make a table for each algebraic rule. Company A Company B b c b c Company C b c Company D b c Graph each algebraic rule. Be sure to scale and label your axes. Company A Company B Company C Company D Copyright © 2012 Charles A. Dana Center at the University of Texas at Austin, Learning Sciences Research Institute at the University of Illinois at Chicago, and Agile Mind, Inc. 174 Unit 2 – Introduction to functions and equations STAYING SHARP 4.7
Preparing for upcoming lessons Practicing algebra skills & concepts 1.
Write an expression for the perimeter of the rectangle. Make sure your expression is as simple as possible. Write an algebraic expression to represent each phrase, using n to represent “a number.” Rewrite the expression more simply if possible. a. A number plus itself b. A number times itself c. A number minus itself d. A number divided by itself Nicole’s mother says to a friend, “If you triple Nicole’s age, then add 16, you get her grandmother Gladys’s age.” 3. Write an algebraic rule to match the sentence. Use n for Nicole’s age and g for her grandmother Gladys’s age. 4. If Gladys is 67 years old, how old is Nicole? Show your work. 5.
Caitlin is 62
inches tall. Her older brother Jaime is 9 inches taller than Caitlin. Jaime is 2
Reviewing pre-­‐algebra ideas 2.
6.
Find two different fractions whose sum is equal to 1. inches shorter than their father. How tall is their father? Show your work. Copyright © 2012 Charles A. Dana Center at the University of Texas at Austin, Learning Sciences Research Institute at the University of Illinois at Chicago, and Agile Mind, Inc.