2.1 Identifying Patterns

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1 I. Foundations for Functions 2.1 Identifying Patterns: Leaders' Notes 2.1 Identifying Patterns Overview: Objective: s: Materials: Participants represent linear relationships among quantities using concrete models, tables, diagrams, written descriptions, and algebraic forms. Algebra I TEKS (b.1) The student understands that a function represents a dependence of one quantity on another and can be described a variety of ways. (b.2.b) For a variety of situations, the student identifies the mathematical domains and ranges and determines reasonable domain and range values for given situations. (b.2.c) The student interprets situations in terms of given graphs or creates situations that fit given graphs. (b.3) The student understands algebra as the mathematics of generalization and recognizes the power of symbols to represent situations. (c.1.b) The student determines the domain and range values for which linear functions make sense for given situations. function, independent variable, dependent variable building blocks, color tiles, graphing calculators Procedures: Participants should be seated at tables in groups of 3 4. Do the Student Activity depending on the level of participants. Activity 1: Painting Towers Do the Activity together as a whole group, bringing out the following points and asking the indicated questions. 1. Encourage participants to write how they found the number of faces to paint in the process column. This can often be done in several ways, which will lead to different, yet equivalent algebraic expressions. This is a desired outcome. Possible equivalent expressions include: ( ) ( )+ 22 ( ) ( ) ( )+ 32 ( ) ( 4) n( 4)+ 1= 4n+ 1 n 2 n n ( )+ ( )+ 5 ( 1)( 4) Ask participants to use the cubes to physically demonstrate the algebraic rules they found in the table. In this example, they will mainly be pointing to faces on the cubes and relating them to the heights of the towers. Note: encourage participants to obtain the equivalent expressions from the model, not by simplifying the algebraic expressions. TEXTEAMS Algebra I: 2000 and Beyond Spring

2 I. Foundations for Functions 2.1 Identifying Patterns: Leaders' Notes 2. Discuss with participants that choosing appropriate windows and then justifying window choices is a precursor to students learning about domain and range. Graph the function over the scatter plot to verify as shown above x + 1= 25 4x = 24 x = 6 What does the ordered pair (8, 33) mean? [For term number 8 (figure 8 or a tower 8 cubes high), the numerical term value (number of faces to paint) is 33.] Does the ordered pair (12,50) belong to this graph? Why or why not? [A simple answer might be that there will always be that extra top face to paint, and therefore the number of faces to paint will be odd. Thus it is not possible for there to be 50 faces to paint. Also, the symbolic rule suggests odd numbers.] 5. Examples: ( + 1) 26 ( ) ( ()+ 1) 36 ( ) ( ()+ 1) n( 6)+ 2 = 6n+ 2 2( n()+ 3 1)= 23n+ 1 ( ) TEXTEAMS Algebra I: 2000 and Beyond Spring

3 I. Foundations for Functions 2.1 Identifying Patterns: Leaders' Notes 6. Sample answer: For 35 cubes in each column: you need to paint 6 faces times 35 plus the top 2 faces. 7. F = 6n What changed in the rule? The figure? The graph? The domain? The range? Compare. Use the following questions to compare the single tower situation and the double tower situation. What represents the changing quantity and what represents the fixed quantity in the pattern? [The addition of 4 faces with each additional cube represents the changing quantity in the single tower. The addition of 6 faces with each additional level represents the changing quantity in the double tower. The faces on tops of the towers represents the unchanging one and two faces.] What number in the rule affects the slope, steepness, of the line? [The coefficient of x. In this case, it is the number 4 in the single tower and the number 6 in the double tower.] What number in the rule affects the starting point for the scatter plot (y-intercept for the line)? [The constant. In the first case, it is the number 1, and in the second case, the number 2.] Underline the constants in both of the functions, y = 1+ 4 x and y = 2+ 8 x. What do the constants represent in the functions, y = 1+ 4 x and y = 2+ 8 x? [faces to paint] Circle the coefficient of x in both of the functions, y = 1+ 4 x and y = 2+ 8 x. What do the coefficients of x represent? [faces to paint per tower height or faces to paint per figure number.] Repeat the above questions for Activities 2 3. Activity 2: Building Chimneys Have participants do Activity 2 together in groups. Discuss as a whole group, asking several participants to share their different methods of arranging the cubes to find appropriate expressions. 1. Encourage participants to write how they found the number of blocks in the process column. TEXTEAMS Algebra I: 2000 and Beyond Spring

4 I. Foundations for Functions 2.1 Identifying Patterns: Leaders' Notes For example: 6 + 2(1) 2 + 2(2 + 1) 6 + 2(2) 2 + 2(2 + 2) 6 + 2(3) 2 + 2(2 + 3) 6 + 2n 2 + 2(2 + n) 2. Graph the function over the scatter plot to verify as shown. A reasonable domain for the situation is 0 to 10 blocks and a reasonable range is 0 to 26 blocks. 3. You need 6 blocks for the base and then 23 rows of 2 blocks for the chimney. The ordered pair is (23, 52). 4. You need 28 blocks to build a house with a chimney 11 blocks high. The ordered pair is (11, 28). 5. No, the ordered pair (13, 34) does not belong to the graph because if you were building figure 13, you would need ( )+ = blocks, not Examples: (3 + 1) (3 + 2) (3 + 3) 9 + n (3 + n) 7. Sample answer: The total number of blocks equals 9 plus the number of blocks in the chimney. TEXTEAMS Algebra I: 2000 and Beyond Spring

5 I. Foundations for Functions 2.1 Identifying Patterns: Leaders' Notes 9. What changed in the rule? The figure? The graph? Compare. Activity 3: Constructing Trucks 1. Encourage participants to write how they found the number of blocks in the process column. For example: (2) (3) n + 2 2(n+1) + n 2. Graph the function over the scatter plot to verify as shown. 3. You need 152 blocks for the 50 th figure. 4. If you use 242 blocks, you are on the 80 th figure (term number) n = 242 3n = 240 n = 80 TEXTEAMS Algebra I: 2000 and Beyond Spring

6 I. Foundations for Functions 2.1 Identifying Patterns: Leaders' Notes 5. Examples: (2) + 1 2(2) + 3 3(3) + 1 2(3) + 4 3n + 1 2n + (n + 1) 6. Sample answer: The total number of blocks equals 1 plus 3 times the figure number. 8. The second scatter plot, with one block on top of each truck starts (the y- intercept is) lower than the previous plot, with two blocks on the top of each truck. 10. The graph starts higher than the original because now you have a constant 4 blocks on top of each truck. The graph is steeper than the original because you are now adding 6 blocks each time, instead of 3. Compare Activities 1 3 Use the following questions to compare the previous activities. What changed in the rule? The figure? The graph? Compare. What represents the changing quantity and what represents the fixed quantity in each of the patterns? What number in the rule changes the slope of the line? What number in the rule affects the starting point for the line? On graphs of the lines generated in the Activities, draw triangles to show the idea that all of these rules have a constant rate of change, each time the same thing was changing. See below for an example. TEXTEAMS Algebra I: 2000 and Beyond Spring

7 I. Foundations for Functions 2.1 Identifying Patterns: Leaders' Notes Activity 4: Generating Patterns Have half of the groups generate patterns that model surface area, similar to Activity 1: Painting Towers. Have the other half of the groups generate patterns that model volume (number of cubes), similar to Activity 2: Building Chimneys and Activity 3: Constructing Trucks. Sample answers: 1. How many faces to paint: Visual (Figure) Written Description Column Total Faces to Paint 1 Paint 4 lateral faces and the top and bottom. 2 Paint 4 lateral 24 ( ) faces twice and the top and bottom. 3 Paint 4 lateral 34 ( ) faces three times and the top and bottom. n 4n + 2 How many blocks to build: Visual (Figure) Written Description Column of Blocks 1 Base of 4 plus Base grows by 4 plus the 2 on top. 24 ( ) Base grows by 4 more plus the 2 on top. 34 ( ) n 4n + 2 TEXTEAMS Algebra I: 2000 and Beyond Spring

8 I. Foundations for Functions 2.1 Identifying Patterns: Leaders' Notes 2. How many faces to paint: Visual (Figure) Written Description Column Faces to Paint 1 Paint the two faces on the end plus the 8 front and back faces. 2 Paint the 2 new 22 ( ) front and back faces plus the two on the end plus the original 8 lateral faces. 3 Add the 2 new 32 ( ) faces to the previous. n 2n + 8 How many blocks to build: Visual (Figure) Written Description Column of Blocks 1 Base of 8 plus Base of 8 plus 2 rows of 2 on top. 8+ 2( 2) 12 3 Base of 8 plus 3 rows of 2 on top. 8+ 3( 2) 14 n 8+ 2n TEXTEAMS Algebra I: 2000 and Beyond Spring

9 I. Foundations for Functions 2.1 Identifying Patterns: Leaders' Notes Note that both exercises in Activity 4 are examples of doing and undoing an important habit of mind for algebraic thinking. Math Note: The manipulative model (using blocks to build figures to represent patterns) has inherent domain and range restrictions. The sequences generated in the table are for whole number input (domain) values. For example, you would not build figures with 0.5 or 0.3 of a block. The algebraic equations developed in this Activity are linear. The domain and range of a line are all real numbers Answers to Reflect and Apply: 1. The first set of figures show adding 3 blocks every time (3x) to a constant 6 blocks, 3x + 6. The second set of figures show 3 groups of adding a block every time, x, to a constant 2 blocks, 3( x + 2). TEXTEAMS Algebra I: 2000 and Beyond Spring

10 I. Foundations for Functions 2.1 Identifying Patterns: Leaders' Notes 2. b 3. d 4. a 5. c As an extension, ask participants to label the axes with units and explain their reasoning. Summary: By using concrete models and the process column, participants model linear patterns and explore constant rates of change. Participants model both input and output questions with equations and solve them using tables and graphs. TEXTEAMS Algebra I: 2000 and Beyond Spring

11 I. Foundations for Functions 2.1 Identifying Patterns: Activity 1 Activity 1: Painting Towers Suppose you are painting a tower built from cubes, based on the pattern below. Use the table to find the relationship between the number of faces to paint and the number of blocks in the tower. (Paint only the sides and the top.) ( of blocks) Visual (Figure) Written Description 1 A 1 cube-high tower has 5 faces to paint. Column Numerical Value of (Faces to Paint) n TEXTEAMS Algebra I: 2000 and Beyond Spring

12 I. Foundations for Functions 2.1 Identifying Patterns: Activity 1 1. Use the process column to write a function that expresses the relationship between the number of faces to be painted and the number of cubes. 2. Graph the data from your table on 1 graph paper and/or create a scatter plot on a graphing calculator. What is a reasonable domain for this situation? A reasonable range? 3. How many faces need to be painted for a 25 cube tower? Explain two ways of getting an answer. 4. If the tower you paint has 25 faces, how many cubes are in the tower? Explain two ways of getting an answer. TEXTEAMS Algebra I: 2000 and Beyond Spring

13 I. Foundations for Functions 2.1 Identifying Patterns: Activity 1 5. Suppose you have two adjacent columns of cubes instead of the one column as before. Use your cubes to build the first four figures and determine the number of faces that need to be painted. 1 Visual (Figure) Written Description Column Numerical Value of (Faces to Paint) n 6. Write a rule in words to describe how to find the total number of faces that need to be painted for two columns of cubes with 35 cubes in each column. 7. Write a rule in symbols that expresses the relationship between the number of cubes in each column and the total number of faces to be painted. 8. Predict how the graph of this data differs from the graph of the original data. Explain. 9. Graph the above data on 1 graph paper and/or create a scatter plot on a graphing calculator and compare to the previous graph. TEXTEAMS Algebra I: 2000 and Beyond Spring

14 I. Foundations for Functions 2.1 Identifying Patterns: Activity 2 Activity 2: Building Chimneys Suppose you are building a house with a chimney, based on the pattern below. Use the table to find the relationship between the number of blocks you need and the term number. Visual (Figure) Written Description 1 A house with a chimney 1 block high takes 8 blocks to build. Column Numerical Value of (number of blocks) n TEXTEAMS Algebra I: 2000 and Beyond Spring

15 I. Foundations for Functions 2.1 Identifying Patterns: Activity 2 1. Use the process column to write a function that expresses the relationship between the total number of blocks needed to build the house and the term number. 2. Graph the data from your table on 1 graph paper and/or create a scatter plot on a graphing calculator. What is a reasonable domain for this situation? A reasonable range? Explain. 3. Use words to describe how to use blocks to build a house with a total of 52 blocks. What is this ordered pair on the graph? 4. If a house has a chimney that is 11 blocks high, how many blocks will you need to build the house? What is this ordered pair on the graph? 5. Does the ordered pair (13, 34) belong to this graph? How do you know? TEXTEAMS Algebra I: 2000 and Beyond Spring

16 I. Foundations for Functions 2.1 Identifying Patterns: Activity 2 6. Suppose the chimney is made of 1 block instead of two and the house is built of three rows of 3 blocks instead of two rows of 3 blocks. Use your cubes to build the first three figures and record the data below. 1 Visual (Figure) Written Description Column Numerical Value of n 7. Write a rule for this new data that expresses the relationship between the total number of blocks and the number of blocks in the chimney for a house. 8. Predict how the graph of this data differs from the graph of the original data. Explain. 9. Graph the above data on 1 graph paper and/or create a scatter plot on a graphing calculator and compare to the previous graph. TEXTEAMS Algebra I: 2000 and Beyond Spring

17 I. Foundations for Functions 2.1 Identifying Patterns: Activity 3 Activity 3: Constructing Trucks Suppose you are building a truck, based on the pattern below. Use the table to find the relationship between the number of blocks you need and the figure number. Visual (Figure) Written Description 1 The truck has a base of 3 blocks with 2 blocks on top. Column Numerical Value of n TEXTEAMS Algebra I: 2000 and Beyond Spring

18 I. Foundations for Functions 2.1 Identifying Patterns: Activity 3 1. Use the process column to write a function that expresses the relationship between the total number of blocks needed to build the truck and the term number. 2. Graph the data from your table on 1 graph paper and/or create a scatter plot on a graphing calculator. What is a reasonable domain for this situation? A reasonable range? 3. Find the total number of blocks needed for the 50 th figure. 4. If there are a total of 242 blocks, what term number is this? TEXTEAMS Algebra I: 2000 and Beyond Spring

19 I. Foundations for Functions 2.1 Identifying Patterns: Activity 3 5. Suppose there is only one block on the top of each truck. Use your cubes to build the first three figures and record the data below n Visual (Figure) Written Description Column Numerical Value of 6. Write a rule for this new data that expresses the relationship between the total number of blocks and the figure/term number. 7. Predict how the graph of this data differs from the graph of the original data. Explain. 8. Graph the above data on 1 graph paper and/or create a scatter plot on a graphing calculator and compare to the previous graph. What effect did changing the number of blocks on top of the truck have on the graph? 9. Suppose the original trucks (2 blocks on top) were built double-wide. Predict how the graph differs from the original. 10. Build the first three double-wide trucks and graph the data on your graphing calculator. How does this graph compare to the graph of the original? Why? TEXTEAMS Algebra I: 2000 and Beyond Spring

20 I. Foundations for Functions 2.1 Identifying Patterns: Activity 4 Activity 4: Generating Patterns 1. Given the following graph, use blocks to generate a sequence of figures that fits the data. Fill in the table and sketch the figures Visual (Figure) Written Description Column Numerical Value of 2 3 n TEXTEAMS Algebra I: 2000 and Beyond Spring

21 I. Foundations for Functions 2.1 Identifying Patterns: Activity 4 2. Given the function y= 2x+ 8, use blocks to generate a sequence of figures that fits the function. Fill in the table, sketch the figures, and plot the graph. Label the graph. 1 Visual (Figure) Written Description Column Numerical Value of 2 3 n TEXTEAMS Algebra I: 2000 and Beyond Spring

22 I. Foundations for Functions 2.1 Identifying Patterns: Reflect and Apply Reflect and Apply 1. Create a physical model to demonstrate 3x+ 6= 3( x+ 2). Match: 2. y= 3 + x 3. y= x 4. y= 3+ 2x 5. y= 2x a b c d 6. Reflect on the activities. How might you adapt the activities to use with your students? TEXTEAMS Algebra I: 2000 and Beyond Spring

23 I. Foundations for Functions 2.1 Identifying Patterns: Student Activity Student Activity: Perimeter of Rectangles Overview: Objective: s: Materials: Students investigate linear relationships using concrete models, tables, diagrams, written descriptions, and algebraic forms. Algebra I TEKS (b.1) The student understands that a function represents a dependence of one quantity on another and can be described a variety of ways. (b.3) The student understands algebra as the mathematics of generalization and recognizes the power of symbols to represent situations. function, independent variable, dependent variable, pattern color tiles, graphing calculator Procedures: Students should be seated in groups of 3 4. Activity : Perimeter of Rectangles Do the activity together as a whole group, bringing out the following points and asking the indicated questions. 1. Encourage students to write how they found the number of perimeter in the process column. This can often be done in several ways, which will lead to different, yet equivalent algebraic expressions. This is a desired outcome. Possible equivalent expression include: Sample Sample Sample () ( ) () n+ n+ n+ n 4n 2. Justify: The variable x stands for the figure number and xmin=0 to xmax=5 shows the figures 1 4 nicely. The variable y stands for the perimeter and ymin= 2 to ymax=20 shows the perimeters of 4 to 18 nicely. 3. The perimeter of figure 11 is ( )= Figure 12 has a perimeter of 48. 4n = 48 TEXTEAMS Algebra I: 2000 and Beyond Spring

24 I. Foundations for Functions 2.1 Identifying Patterns: Student Activity Ask students to use the tiles to physically demonstrate the algebraic rules they found in the table. In this example, they will mainly be pointing to sides on the tiles and relating them to the numbers in the process column. 5. Sample Sample Sample ( )+ 21 () 41 () ()+ 22 ( ) 42 ( ) ( )+ 23 () 43 ()+ 2 ( n+ 1)+ n+ ( n+ 1)+ n 2( n+ 1)+ 2( n) 4( n) Figure 11 has a perimeter of ( )+ 2= Figure 13 has a perimeter of 54. 4n + 2 = 54. Ask students to use the tiles to physically demonstrate the algebraic rules they found in the table. For example, the rule in the first column above is simply adding each side in order. The rule in the second column above is noting that there are two sides of length n+1 and two sides of length n. The rule in the third column above is based on the idea of adding two additional sides to a square of side n. Ask students to compare the rules, P = 4 n and P = 4n+ 2, and their respective graphs. Note that the lines have the same slope but that the line P = 4n+ 2 is the line P = 4 n shifted up two. The perimeters grow by the same amount each time you change figure numbers by one, but P = 4n+ 2 starts 2 higher than P = 4 n. Summary Using multiple representations, students gain added understanding for the linear relationship of a rectangle s perimeter and the length of a side. TEXTEAMS Algebra I: 2000 and Beyond Spring

25 I. Foundations for Functions 2.1 Identifying Patterns: Student Activity Student Activity: Perimeter of Rectangles Build these squares and the next three squares in the sequence, using color tiles. Figure number 1, 2, 3 Figure 1. Complete the table, using the process column to write a function for figure n, and graph the relation. Figure (length of side) Perimeter n Length of Side 2. On your graphing calculator, make a scatter plot. Graph the function over the scatter plot to confirm. Justify your window choice. Answer the questions and write the equation that represents the question: 3. What is the perimeter of figure number 11? 4. What figure number has a perimeter of 48? TEXTEAMS Algebra I: 2000 and Beyond Spring

26 I. Foundations for Functions 2.1 Identifying Patterns: Student Activity Build these rectangles and the next three rectangles in the sequence, using color tiles. Figure number 1, 2, 3 Figure 5. Complete the table, using the process column to write a function for perimeter of the nth figure, and graph the relation. Figure (length of side) Perimeter n Length of Side 6. On your graphing calculator, make a scatter plot. Graph the function over the scatter plot to confirm. Justify your window choice: Answer the questions and write the equation that represents the question: 7. What is the perimeter of figure number 11? 8. What figure number has a perimeter of 54? TEXTEAMS Algebra I: 2000 and Beyond Spring

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