Algebra: Connections with Multiple Representations

Algebra: Connections with Multiple Representations A preview of the CPM Algebra Connections Course

For more information about the materials you find in this packet, contact: Chris Mikles (888) 808-4276 [email protected] In this new course, significant emphasis is placed on representing information in four ways: a rule, graph, table and a pictorial or written description of the situation. Students find ways to move from all representations to each of the others, developing deep understanding of multiple ways to see problems and their solutions. This handout contains highlights of the new course, followed by a series of problems from Chapters 3, 4, and 8 that demonstrate the multiple representations focus. Finally, one complete lesson is included that illustrates the new lesson structure.

Highlights of the Algebra Connections Course CPM’s “third generation” algebra course offers exciting, integrated chapters full of connections between topics. What was called a “day” in the Math 1 course is now a complete lesson containing an introduction that “launches” the lesson, problems that develop the topic, and suggestions that give ideas for providing closure. Technology is woven into the course— specifically, the use of graphing calculators. For every central idea, there are investigations and labs, as well as days devoted to consolidating understanding and practice. Course Design •

Each chapter contains two or three sections that may or may not be related. Each section consists of a group of lessons that together study a major concept (such as solving equations).

•

Each chapter contains a closure component with five different options from which teachers may choose.

•

Homework is evenly structured so that 60% is review, 20% is from new material, and 20% previews future learning.

Lesson and Curricular Enhancements •

Significant emphasis is placed on representing information in four ways: a rule, a graph, a table, and a pictorial or written description of the situation. Students find ways to move from all representations to each of the others and use this flexibility to solve problems.

•

Each lesson contains an introduction informing students and parents of the purpose of that specific lesson. Each lesson is labeled with a title that communicates the subject matter and poses an overarching question that the lesson addresses.

•

All topics are eventually highlighted in “Math Notes” boxes (formerly tool kit entries).

•

The students are given questions (called “focus questions”) to ask themselves and their teammates to help advance mathematical discussion. Typical questions include: “How do you see it?”, “Is there another way?”, “Can you see a pattern?”, and “How else can you represent it?”

•

There is a consistent use of algebra tiles, particularly to solve equations in Chapters 2 through 5. Tiles are also used as a tool to introduce systems of equations, factoring, and completing the square in later chapters.

•

The course consistently uses investigations and applications for every central idea.

Teacher Support •

Each lesson is presented with a coherent lesson plan for the teacher that includes detailed suggestions for how to introduce the lesson, conduct the investigation, and bring the lesson to closure.

•

Support for how to set up and manage study teams effectively is embedded in the teacher materials.

•

Large problems (investigations, labs, etc) are written to accommodate teachers who prefer open investigations. However, student guidance is also provided for teachers who prefer a more structured approach.

•

Teachers are provided suggestions for questions to ask while circulating among study teams, such as: “How does changing the equation change the graph?”, “Do you predict that all parabolas behave that way? How do you know?”, and “What happens if you change that value?

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Algebra Connections: Multiple Representations Thread

Chapter 3 starts a focus on the different representations (table, graph, rule, and situation) and helps students start to recognize and understand some connections between them. 3-10.

JOHN’S GIANT REDWOOD, Part One John found the data in the table below about his favorite redwood tree. He wondered if he could use it to predict the height of the tree at other points of time. Consider this as you analyze the data and answer the questions below. Be ready to share (and justify) your answers with the class.

3-11.

3-12.

Page 3 of 17

Number of Years after Planting

3

4

5

Height of Tree (in feet)

17

21

25

a.

How tall was the tree 2 years after it was planted? What about 7 years after it was planted? How do you know?

b.

How tall was the tree the year it was planted?

c.

Estimate the height of the tree 50 years after it was planted. How did you make your prediction?

John decided to find out more about his favorite redwood tree by graphing the data. a.

On the Lesson 3.1.2B Resource Page provided by your teacher, plot the points that represent the height of the tree over time. What does the graph look like?

b.

Does it make sense to connect the points? Explain your thinking.

c.

According to the graph, what was the height of the tree 1.5 years after it was planted?

d.

Can you use your graph to predict the height of the redwood tree 20 years after it was planted? Why or why not?

John is still not satisfied. He wants to be able to predict the height of the tree at any time after it was planted. a.

Find John’s table on your resource page and extend it to include the height of the tree in the 0th year, 1st year, 2nd year, and 6th year.

b.

If you have not already, use the ideas from the Silent Board Game to write an algebraic rule for the data in your table. Be sure to work with your team and check that the rule works for all of the data.

c.

Use your rule to check your prediction in part (c) of problem 3-10 for how tall the tree will be in its 50th year. How close was your prediction? Algebra Connections: Multiple Representations Thread

3-18.

3-19.

3-20.

Page 4 of 17

Find the “Big Cs” pattern shown at right on the Lesson 3.1.3 Resource Page provided by your teacher. a.

Draw Figure 0 and Figure 4 on the grid provided on the resource page.

b.

On the resource page, represent the number of tiles in each figure with: •

An x → y table.

•

An algebraic rule.

•

A graph.

Figure 1

Figure 2

Figure 3

c.

How many tiles will be in Figure 5? Justify your answer in at least two different ways.

d.

What will Figure 100 look like? How many tiles will it have? How can you be sure?

Use the graphing technology provided by your teacher to analyze the pattern further and make predictions. a.

Enter the information from your x → y table for problem 3-18 into your grapher. Then plot the points using a window of your choice. What do you notice?

b.

Find another x → y pair that you think belongs in your table. Use your grapher to plot the point. Does it look correct? How can you tell?

c.

Imagine that you made up 20 new x → y pairs. Where do you think their points would lie if you added them to the graph?

In the same window that contains the data points, graph the algebraic equation for the pattern from problem 3-18. a.

What do you notice? Why did that happen?

b.

Charles wonders about connecting the points of the “Big Cs” data. When the points are connected with an unbroken line or curve, the graph is called continuous. If the graph of the tile pattern is continuous, what does that suggest about the tile pattern? Explain.

c.

Jessica prefers to keep the graph of the tile-pattern data as separate points. This is called a discrete graph. Why might a discrete graph be appropriate for this data?


3-21.

If necessary, re-enter your data from the “Big Cs” pattern into your grapher. Re-enter the rule you found in problem 3-18 and graph the data and rule in the same window. For the following problems, justify your conclusions with the graph, the rule, and the figure (whenever practical). Your teacher may ask your team to present your solution to one of these problems. Be sure to justify your ideas using all three representations.

3-22.

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a.

Frangelica thinks that Figure 5 in the “Big Cs” pattern has 30 tiles. Decide with your team whether she is correct and justify your answer by using the rule, drawing Figure 5, and adding the point to your graph of the data. Be prepared to show these three different ways to justify your conclusion.

b.

Giovanni thinks that the point (16, 99) belongs in the table for the “Big Cs” pattern. Decide with your team whether he is correct and justify your conclusion by examining the graph and the rule.

c.

Jeremiah is excited because he has found another rule for the “Big Cs” pattern! He thinks that y = x + 8 also works. Prove or disprove Jeremiah’s claim. Be prepared to convince the class that your conclusion is correct.

d.

LaTanya has been thinking hard and has found another rule for the same pattern! She is sure that y = 3(2x + 1) is also correct. Prove or disprove LaTanya’s position in as many ways as you can.

Look back at the prediction you made in problem 3-18 for Figure 100 in the “Big Cs” pattern. Decide now if your prediction was correct and be ready to defend your position with all of the math tools you have.


After using the different representations separately (graph, table, rule, and situation), students are challenged with problem 4-1 below at the beginning of Chapter 4. 4-1.

TILE PATTERN TEAM CHALLENGE Your teacher will assign your team a tile pattern (one of the patterns labeled (a) through (e) on the next page). Your team’s task is to create a poster showing every way you can represent your pattern and highlighting all of the connections between the representations that you can find. For this activity, finding and showing the connections are the most important parts. Clearly presenting the connections between representations on your poster will help you convince your classmates that your description of the pattern makes sense. Pattern Analysis: •

Extend the pattern: Draw Figures 0, 4, and 5. Then describe Figure 100. Give as much information as you can. What will it look like? How will the tiles be arranged? How many tiles will it have?

•

Generalize the pattern by writing a rule that will give the number of tiles in any figure in the pattern. Show how you got your answer.

•

Find the number of tiles in each figure. Record your data in a table and on a graph.

•

Demonstrate how the pattern grows using color, arrows, labels, and other math tools to help you show and explain. Show growth in each representation.

•

What connections do you see between the different representations (graph, figures, and x → y table)? How can you show these connections?

Presenting the Connections: As a team, organize your work into a large poster that clearly shows each representation of your pattern, as well as a description of Figure 100. When your team presents your poster to the class, you will need to support each statement with a reason from your observations. Each team member must explain something mathematical as part of your presentation. a.

b.

Figure 1

Figure 2

c.

Figure 2

Figure 3

d.

Figure 1 Page 6 of 17

Figure 1

Figure 3

Figure 2

Figure 3

Figure 1

Figure 2

Figure 3


In the next lesson, students begin to study linear tile patterns and begin to use a Representation Web to document the connections between the different representations. 4-9.

For each of the patterns below, i.

What do you notice? After everyone has had a moment on his or her own to examine the figures, discuss what you see with your team.

ii.

Sketch the next figure in the sequence (Figure 4) on your resource page. Sketch the figure that comes before Figure 1 (Figure 0).

iii.

How is the tile pattern growing? Where are the tiles being added with each new figure? Color in the new tiles in each figure with a marker or colored pencil on your resource page.

iv.

What would Figure 100 look like? Describe it in words. How many tiles would be in the 100th figure? Find as many ways as you can to justify your conclusion. Be prepared to report back to the class with your team’s findings and methods.

a.

Tile Pattern #2:

Figure 1 b.

Figure 3

Tile Pattern #3:

Figure 1

4-10.

Figure 2

Figure 2

Figure 3

PUTTING IT TOGETHER Look back at the three different tile patterns in problems 4-8 and 4-9 to answer these questions.

Page 7 of 17

a.

What is the same and what is different between these three patterns? Explain in a few sentences.

b.

Write an equation (rule) for the number of tiles in each pattern.

c.

What connections do you see between your equations and the tile pattern? Show and explain these connections.

d.

Imagine that the team next to you created a new tile pattern that grows in the same way as the ones you have just worked with, but they refused to show it to you. What other information would you need in order to predict the number of tiles in Figure 100? Explain your reasoning. Algebra Connections: Multiple Representations Thread

4-11.

4-12.

Consider Tile Pattern #4, shown below. a.

Draw Figures 0 and 4 on the resource page.

b.

Write an equation (rule) for the number of tiles in this pattern. Use a Figure 1 new color to show where the numbers in your rule appear in the tile pattern.

Figure 2

Figure 3

c.

What is the same about this pattern and Tile Pattern #3? What is different? What do those similarities and differences look like in the tile pattern? In the equation?

d.

How is the growth represented in each equation?

For today’s Learning Log entry, draw a web of the different representations, starting with the diagram below. Draw lines and/or arrows to show which representations you have connected so far. Explain the connections you learned today. Be sure to include anything you figured out about how the numbers in equations (rules) relate to tile patterns. Title this entry “Starting the Web” and label it with today’s date.

Table Graph

Rule Pattern

Students are introduced to the equation y = mx + b and spend several days studying constant rate and y-intercepts. Some problems, like those shown below, are designed to help students make new connections in their web. y

GRAPH → RULE Allysha claims she can find the equation of a line by its graph without a table. How is that possible? Discuss this idea with your team and then try to find the equation of the line at right without first making a table. Be ready to share with the class how you found the rule.

Total Number of Tiles

4-27.

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

x

Figure Number

Page 8 of 17


4-28.

TABLE → RULE Allysha wonders if she can use the idea of m and b to find the equation of a line from its table. a.

For example, if she knows the information about a linear pattern given in the table below, how can she find the equation of the line? Work with your team to complete the table and find the rule. IN (x)

0

OUT (y)

–2

1

+5

b.

ii.

4-29.

+5

3

4

+5

+5

5

+5

6

+5

Use this same idea to find the rule of the linear tile patterns represented by the tables below. i.

c.

2

IN (x)

–1

0

1

2

3

4

5

OUT (y)

3

5

7

9

11

13

15

IN (x)

0

1

2

3

4

5

6

OUT (y)

7

4

2

–2

–5

–8

–11

Write a summary statement explaining how you used your knowledge about m and b to quickly write a rule.

RULE → PATTERN In each problem below, invent your own pattern that meets the stated conditions. Draw Figures 0, 1, 2, and 3 and write the rule (equation) for your pattern.

Page 9 of 17

a.

A tile pattern that has y = 4x + 3 as a rule.

b.

A tile pattern that decreases by 2 tiles and Figure 2 has 8 tiles.


Then, in lesson 4.1.5, students are confronted with a new team challenge. This time, they need to put all their understanding of the representations along with what they know about y = mx + b . 4-37.

CHECKING THE CONNECTIONS: TEAM CHALLENGE Today you are going to apply what you know about the starting point (Figure 0), growth factor, and the connections between representations to answer some challenging questions. The information in each question, parts (a) through (d), describes a different pattern. The graph of each pattern is a line. From this information, generate the rule, x → y table, graph, and tile pattern (Figures 0 through 3) that follow the pattern. You may answer these questions in any order, but make sure you answer each one completely before starting another problem. Work together as a team. The more you listen to how other people see the connections and the more you share your own ideas, the more you will know at the end of the lesson. Stick together and be sure to talk through every idea. Each person will turn in his or her own paper at the end of this activity, showing four complete representations for each pattern. Your work does not need to be identical to your teammates’ work, but you should have talked and agreed that all explanations are correct. y

a.

4 1 Figure 3 –1

b.

1

2

3

x

y 58 31

1

2

3

4

5

6

7

x

Problem continues on next page →

Page 10 of 17


4-37.

Problem continued from previous page.

c.

Figure Number

Number of Tiles

0 1 2 3 d.

y = !3x + 7

Figure 8 12 y 7 6 5 4 3 2 1 –1

1

–3

1

2

3

4

5

x

Multiple representations becomes a tool used in other contexts throughout the course. For example, students are required to use at least two methods to solve the first system of equations problem they have encountered. Since they have not learned an algebraic method yet, students usually choose to solve by graphing and by comparing tables. In this case, multiple representations then helps motivate students to find a method to solve using rules as well, developing the Substitution and Elimination Methods. 4-76.

BUYING BICYCLES Latanya and George are saving up money because they both want to buy new bicycles. Latanya opened a savings account with $50. She just got a job and is determined to save an additional $30 a week. George started a savings account with $75. He is able to save $25 a week. Your Task: Use at least two different ways to find the time (in weeks) when Latanya and George will have the same amount of money in their savings accounts. Be prepared to share your methods with the class.

In addition, a second Representations Web is introduced when students study quadratic equations in Chapter 8. The Water Balloon Problem included in the lesson below requires students to answer a question using a rule, table, graph, and a description of a situation. Lesson 8.2.2 is included here in its entirety, including teacher notes and homework.

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Lesson 8.2.2 What’s the connection? Multiple Representations for Quadratics Lesson Objective:

Students will identify connections between different representations of quadratics: an equation, a table, a context, and a graph. Students will also connect the intercepts and vertex of a parabola to a context: the launch, maximum height, and landing of a water balloon.

Length of Activity: One day (approximately 50 minutes) Core Problems:

Problems 8-51 and 8-52

Ways of Thinking:

Making connections, reversing thinking, applying and extending

Materials:

Lesson 8.2.2 Resource Page, one per student Poster paper and markers for the class quadratic web Colored pencils or pens

Suggested Lesson Activity:

The introduction to today’s lesson reminds students of the multiple representations web they created for linear equations in Chapter 4. For linear equations, the “situations” they studied were tile patterns and various linear contexts. Today students will begin constructing a similar web to keep track of connections they know for quadratic equations. Place the quadratic web (as shown below with no connections made) on a large poster paper in a visible spot in the classroom. During the next four lessons, add arrows on the poster when the class decides it has found the connection between those representations. After reading the lesson introduction, have students work on problem 8-51. There are many possible answers, so encourage students to be creative!

QUADRATIC WEB Table Rule or

Graph

Equation While debriefing problem 8-51, take the opportunity to point out that not every curve is a parabola or part Situation of a parabola. For instance, to many students, a semi-circle seems like part of a parabola. On the other hand, you can point out the interesting fact that, due to the laws of physics, an object thrown, kicked, or flung into the air on Earth will always rise and fall in the shape of a parabola (assuming no other factors such as wind or spin affect it). This will provide a good lead-in to problem 8-51.

Ask a few student volunteers to read the problem statement for problem 8-51. You may want to have a “ball toss” as a demonstration of a parabolic curve in front of the class. Distribute the resource page and make sure students understand the task. Students will be given information about four water-balloon launches, each in a different Page 12 of 17

Jen

Maggie

30

Imp

25

Al

20 15 10 5

2

4

6

8

10

12

14

16

Solution to part (a)


representation. They will need to analyze the information given and create a table and graph on the resource page for each launch. See solution graph above. This is a full day, and problem 8-53 may be used as an extension if time is available. Closure: (15 minutes)

Pull the class together and revisit the web QUADRATIC WEB (shown again in problem 8-52). Ask the Table class to use arrows to connect any representations they can. Discuss any Rule or discrepancies between different teams’ Graph Equation webs. You can refer to the various contestants in the water-balloon contest as Situation concrete examples for students to test whether they are currently able to move around the web in particular ways. At this point, students are able to make the connections shown above. Students should recognize that there are a few connections still missing. Problem 8-53 reviews the tile pattern problems of Chapter 4 in which students took the situation (the pattern) and wrote a rule to represent the number of tiles in the xth figure. However, if time is not available, you can remind students of their work from problem 4-1 by displaying one of the team posters from that lesson.

Team Roles:

As students work on the resource page, the Recorder/Reporter should make sure that teammates can clearly see each other’s work to enhance communication.

Homework:

Problems 8-54 through 8-58

8.2.2 What’s the connection? •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••

Multiple Representations for Quadratics In Chapter 4 you completed a web for the different representations of linear equations. You discovered special shortcuts to help you move from one representation to another. For example, given a linear equation, you can now draw the corresponding graph as well as determine an equation from a graph.

Table Rule or Equation

Graph Situation

Today you will explore the connections between the different representations for quadratics. As you work, keep in mind the following questions: What representations are you using? What is the connection between the various representations? What do you know about a parabola? Page 13 of 17


8-51.

WATER-BALLOON CONTEST Every year Newtown High School holds a water-balloon competition during halftime of their homecoming game. Each contestant uses a catapult to launch a water balloon from the ground on the football field. This year you are the judge! You must decide which contestants win the prizes for Longest Distance and Highest Launch. Fortunately, you have a computer that will collect data for each throw. The computer uses x to represent horizontal distance in yards from the goal line and y to represent the height in yards.

Jen’s data

The announcer shouts, “Maggie Nanimos, you’re up first!” She runs down and places her catapult at the 3-yard line. After Maggie’s launch, the computer reports that the balloon traveled along the parabola y = ! x 2 + 17x ! 42 . Then you hear, “Jen Erus, you’re next!” Jen runs down to the field, places her catapult at the goal line, and releases the balloon. The tracking computer reports the path of the balloon with the graph at right. The third contestant, Imp Ecable, accidentally launches the balloon before you are ready. The balloon launches, you hear a roar from the crowd, turn around, and…SPLAT! The balloon soaks you and your computer! You only have time to write down the following partial information about the balloon’s path before your computer fizzles: x (yards) y (yards)

2 0

3 9

4 16

5 21

6 24

7 25

8 24

9 21

Finally, the announcer calls for the last contestant, Al Truistic. With your computer broken, you decide to record the balloon’s height and distance by hand. Al releases the balloon from the 10-yard line. The balloon reaches a height of 27 yards and lands at the 16-yard line. a.

Obtain the Lesson 8.2.2 Resource Page from your teacher. For each contestant, create a table and graph using the information provided for each toss. Determine which of these contestants should win the Longest Distance and Highest Throw contests.

b.

Find the x-intercepts of each parabola (also called roots). What information do the x-intercepts tell you about each balloon toss?

c.

Find the vertex of each parabola. What information does the vertex tell you about each balloon throw?

Page 14 of 17


8-52.

8-53.

Today you have explored the four different representations of quadratics: table, graph, equation, and a description of a physical situation involving motion. Draw the representations of the web as shown below in your Learning Log and label it “Quadratic Web.” a.

Draw in arrows showing the connections that you QUADRATIC WEB currently know how to make between different Table representations. Be prepared to justify a connection for the class. [ We expect students to Rule or Graph connect rule  table, table  graph, graph  Equation situation, and table  situation. In problem 4-1 and in other problems, students found rules Situation for quadratic tile patterns. Also, in Chapter 3, students connected table  rule with the Silent Board Game, but they will continue to strengthen this connection. ]

b.

What connections are still missing? [ table → rule, rule ↔ graph, and rule → situation ]

SITUATION TO RULE Review how to write a rule from a situation by examining the tile pattern below.

QUADRATIC WEB Table Rule or Equation

Graph

Figure 1

8-54.

Figure 2

Situation

Figure 3

a.

Write a rule to represent the number of tiles in Figure x. [ One way to write the rule is y = (x + 1)(x + 2) + 2 . ]

b.

Is the rule from part (a) quadratic? Explain how you know. [ Yes; students can multiply it out to find that y = x2 + 3x + 4 . ]

c.

If you have not done so already, add this pathway to your web from problem 8-52.

Graph y = x 2 ! 8x + 7 and label its vertex, x-intercepts, and y-intercepts. [ vertex: (4, –9), x-intercepts: (1, 0) and (7, 0), y-intercept: (0, 7) ]

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8-55.

What is special about the number zero? Think about this as you answer the questions below. a.

Find each sum: 3+0 =

0 +3=[ 3 ]

What is special about adding zero? Write a sentence that begins, “When you add zero to a number, …” [ …it does not change the value of the number. ]

c.

Julia is thinking of two numbers a and b. When she adds them together, she gets a sum of b. Does that tell you anything about either of Julia’s numbers? [ It tells us that a = 0 . ]

d.

Find each product:

e.

8-58.

0 !6 = [ 0 ]

(!7) "0 = [ 0 ]

0 !("2) = [ 0 ]

What is special about multiplying by zero? Write a sentence that begins, “When you multiply a number by zero, …” [ …the result is always 0. ]

For each rule represented below, state the x- and y-intercepts. [ a: x-intercepts (2, 0), (– 4, 0), and (3, 0), y-intercept: (0, 18); b: x-intercepts (3, 0) and (8, 0), y-intercept: (0, –3); c: x-intercept (1, 0) and y-intercept (0, – 4) ] a.

8-57.

0 + (!2) = [ –2 ]

0+6 =[ 6]

b.

3! 0 = [ 0 ]

8-56.

! 7 + 0 = [ –7 ]

x 2 0 –4 –1 6 3

y 0 18 0 –8 22 0

b.

x 7 3 10 0 8 –7

y –4 0 8 –3 0 –1

c.

x 0 –5 3 1 13 –6

y –4 11 –2 0 27 14

For the line y = 2x + 6 : a.

What is the x-intercept? [ (–3, 0) ]

b.

What is the slope of any line perpendicular to the given line? [ ! 12 ]

Solve the following systems of equations using any method. Check your solution if possible. [ a: no solution, b: (7, 2) ] a.

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6x ! 2y = 10 3x ! y = 2

b.

x ! 3y = 1 y = 16 ! 2x Algebra Connections: Multiple Representations Thread

Lesson 8.2.2 Resource Page Problem 8-51

Water-Balloon Contest

30

Height in Air (in yards)

25

20

15

10

5

2

4

6

8

10

12

14

16

Distance Along Ground (in yards) Maggie’s Toss x y

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Jen’s Toss x y

Imp’s Toss x y

Al’s Toss x y
