Caryn Loves Math 1

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1 Caryn Loves Math 1

2 Piecewise-defined Functions Bundles By Caryn White Table of Contents Copy Right Information:... 3 Worksheets... 4 Evaluating Piecewise Functions (1)... 5 Evaluating Piecewise Functions (2)... 6 Graphing Piecewise Functions (1)... 7 Graphing Piecewise Functions (2)... 8 Graphing Piecewise Functions (3)... 9 Finding the Equations of a Piecewise-defined Function KEY Evaluating Piecewise Functions (1) KEY Evaluating Piecewise Functions (2) KEY Graphing Piecewise Functions (1) KEY Graphing Piecewise Functions (2) KEY Graphing Piecewise Functions (3) KEY Finding a Piecewise-defined Function Foldable: Writing Equations for Piecewise Functions Finding Equations Foldable - Outside Finding Equations Foldable Inside - Completed Finding Equations Foldable Inside - Blank Note Pages Note Page: Finding the Piecewise Function Given the Graph Note Page: How to graph a piece-wise function by hand Note Page: Graphing Piecewise Functions with 3 sections How to graph a piece-wise function using a TI83/84 Graphing Calculator KEY Walk Around Activity: Level Absent Student Sheet Student Response Sheet Version A Student Response Sheet Version A Version B Caryn Loves Math 2

3 Student Response Sheet Version B Student Response Sheet Version B KEYS Walk -Around Activity: Level Version A Absent Student Sheet Student Response Sheet Version A Student Response Sheet Version A Version B Student Response Sheet Version B Student Response Sheet Version B KEYS Copy Right Information: This package is copyrighted 2015 by Caryn White: Caryn Loves Math All rights reserved. Purchase of this unit entitles the purchaser the right to reproduce the pages in limited quantities for classroom use only. You may also duplicate for use in with your students in your special education and English Language Learners support rooms. Duplication for an entire school, an entire school system or commercial purposes is strictly forbidden without written permission from the publisher. Please contact me if you wish to be granted special permission. I can be contacted using the questions section of my TPT store. Copying any part of this product and placing it on the Internet in any form (even a personal/classroom website) is strictly forbidden and is a violation of the Digital Millennium Copyright Act (DMCA). These items can be picked up in a google search and then shared worldwide for free Caryn Loves Math 3

4 Worksheets Caryn Loves Math 4

5 Evaluating Piecewise Functions (1) Name f(x) g(x) h(x) Use the graphs above to evaluate the functions. 1. f(0)= 2. h(-1)= 3. g(0)= 4. f(3)= 5. g(-4)= 6. f(-3)= 7. g(10)= 8. h(0)= 9. h(-10)= 10. g(8)= 11. f(1)= 12. f(5)= 13. h(-2)= 14. g(1)= 15. h(2)= 16. f(-5)= 2x x > 3 f(x) = { x 0 x 3} g(x) = { x x 0 5 x > 1 } h(x) = { x 1 x < 0 x x < 0 2x x 1 } j(x) = { 3 x = 5 2x + 1 x 5 } 1. f(0)= 2. g(0)= 3. h(x)= 4. j(0)= 5. f(3)= 6. f(-3)= 7. g(-4)= 8. h(-3)= 9. f(-5)= 10. j(5)= 11. h(-1)= 12. j(-3)= 13. g(9)= 14. g(1)= 15. h(4)= 16. j(5)= 17. f(4)= 18. h(-3)= 19. j(2)= 20. g(-1)= Caryn Loves Math 5

6 Evaluating Piecewise Functions (2) Name f(x) g(x) h(x) Use the graphs above to evaluate the functions. 1. f(0)= 2. h(-1)= 3. g(0)= 4. f(3)= 5. g(-4)= 6. f(-3)= 7. g(10)= 8. h(0)= 9. h(-10)= 10. g(8)= 11. f(1)= 12. f(5)= 13. h(-2)= 14. g(1)= 15. h(2)= 16. f(-5)= x + 3 x > 2 2 x x 0 x > 1 f(x) = { 2x 0 x 2} g(x) = { } h(x) = {2x 2x 1 x < 0 x x < 0 5 x 1 } j(x) = { 3 x = 5 2x 3 x 5 } 17. f(0)= 18. g(0)= 19. h(x)= 20. j(0)= 21. f(3)= 22. f(-3)= 23. g(-4)= 24. h(-3)= 25. f(-5)= 26. j(5)= 27. h(-1)= 28. j(-3)= 29. g(9)= 30. g(1)= 31. h(4)= 32. j(5)= 33. f(4)= 34. h(-3)= 35. j(2)= 36. g(-1)= Caryn Loves Math 6

7 Graphing Piecewise Functions (1) Name: Graph the following by completing the tables below. 2x + 3 x 2 1. f(x) = { x + 2 x < 2 } If x 2 If x < -2 x y=2x+3 (x, y) x y=-x+2 (x, y) x x 3 2. f(x) = { 3 3 x < 3 } 2 If x 3 If x < 3 x y = 2 3 x (x, y) x y=3 (x, y) x x 0 3. f(x) = { x x < 0 } If x 0 If x < 0 x y = x (x, y) x y = x (x, y) 3x + 1 x 1 4. f(x) = { 2x + 1 x < 1 } If x 1 If x < 1 x y = 3x + 1 (x, y) x y = 2x + 1 (x, y) Caryn Loves Math 7

8 Graphing Piecewise Functions (2) Name: Graph the following by completing the tables below. x + 3 x 3 1. f(x) = { 2x + 2 x < 3 } 3 x < 3 2. f(x) = { x 3 x 3} 3 x > 3 2x x > 1 3. f(x) = { x 1 x 1} 2x x < 1 4. f(x) = { x x 1 x + 1 x < 1} Caryn Loves Math 8

9 Graphing Piecewise Functions (3) Name: Graph the following by completing the tables below. x x 1 1. f(x) = { 2x x < 1 } 4 x < 2 2. f(x) = { x 2 2 x 2} 2x + 5 x > 2 3. f(x) = { (x 2)2 x 0 x 2 3 x < 0 } 1 x x 1 4. f(x) = { 3 x x < 1 } Caryn Loves Math 9

10 Finding the Equations of a Piecewise-defined Function Name: Caryn Loves Math 10

11 KEY Evaluating Piecewise Functions (1) Name f(x) g(x) h(x) Use the graphs above to evaluate the functions. 1. f(0)= h(-1)= 0 3. g(0)= 4 4. f(3)= 3 5. g(-4)= 9 6. f(-3)= 3 7. g(10)= h(0)= 0 9. h(-10)= g(8)= f(1)= f(5)= h(-2)= g(1)= h(2)= f(-5)= -5 2x x > 3 f(x) = { x 0 x 3} g(x) = { x x 0 5 x > 1 } h(x) = { x 1 x < 0 x x < 0 2x x 1 } j(x) = { 3 x = 5 2x + 1 x 5 } 1. f(0)= 0 2. g(0)= 0 3. h(0)= 5 4. j(0)= 1 5. f(3)= 3 6. f(-3)= g(-4)= 2 8. h(-3)= f(-5)= j(5)= h(-1)= j(-3)= g(9)= g(1)= h(4)= j(5)= f(4)= h(-3)= j(2)= g(-1)= Caryn Loves Math 11

12 KEY Evaluating Piecewise Functions (2) Name f(x) g(x) h(x) Use the graphs above to evaluate the functions. 1. f(0)= h(-1)= 5 3. g(0)= 1 4. f(3)= 2 5. g(-4)= 9 6. f(-3)= 2 7. g(10)= h(0)= 4 9. h(-10)= g(8)= f(1)= f(5)= h(-2)= g(1)= h(2)= f(-5)= Not possible x + 3 x > 2 2 x x 0 x > 1 f(x) = { 2x 0 x 2} g(x) = { } h(x) = {2x 2x 1 x < 0 x x < 0 5 x 1 } j(x) = { 3 x = 5 2x 3 x 5 } 17. f(0)= g(0)= h(0)= j(0)= f(3)= f(-3)= g(-4)= h(-3)= f(-5)= j(5)= h(-1)= j(-3)= g(9)= g(1)= h(4)= j(5)= f(4)= h(-3)= j(2)= g(-1)= Caryn Loves Math 12

13 KEY Graphing Piecewise Functions (1) Name: Graph the following by completing the tables below. 2x + 3 x 2 1. f(x) = { x + 2 x < 2 } if x 2 If x < -2 x y=2x+3 (x,y) -2 2(-2)+3=-1 (-2,-1) -1 2(-1)+3=1 (-1,1) 0 2(0)+3=3 (0,3) x y=-x+2 (x,y) -2 -(-2)+2=4 (-2,4) -3 -(-3)+2=5 (-3,5) -4 -(-4)+2=6 (-4,6) x x 3 2. f(x) = { 3 3 x < 3 } 2 If x 3 If x < 3 x y = 2 3 x (x,y) 3 2/3 (3) = 2 (3,2) 6 2/3 (6) = 4 (6,4) 9 2/3 (9) = 6 (9,6) x y=3 (x,y) 3 3 (3,3) 2 3 (2,3) 1 3 (1,3) x x 0 3. f(x) = { x x < 0 } If x 1 If x < 1 x y = x (x, y) 0 0 (0,0) 1 1 (1,1) 2 2 (2,2) x y = x (x, y) 0 0 (0,0) -1 1 (-1,1) -2 2 (-2,2) 3x + 1 x 1 4. f(x) = { 2x + 1 x < 1 } If x 1 If x < 1 x y = 3x + 1 (x, y) 1-3(1)+1=-2 (1,-2) 2-3(2)+1=-5 (2,-5) 3-3(3)+1=-8 (3,-8) x y = 2x + 1 (x, y) 1 2(1)+1=3 (1,3) 0 2(0)+1=1 (0,1) -1 2(-1)+1=-1 (-1,-1) Caryn Loves Math 13

14 KEY Graphing Piecewise Functions (2) Name: Graph the following by completing the tables below. x + 3 x 3 1. f(x) = { 2x + 2 x < 3 } 3 x < 3 2. f(x) = { x 3 x 3} 3 x > 3 2x x > 1 3. f(x) = { x 1 x 1} 2x x < 1 4. f(x) = { x x 1 x + 1 x < 1} Caryn Loves Math 14

15 KEY Graphing Piecewise Functions (3) Name: Graph the following by completing the tables below. x x 1 1. f(x) = { 2x x < 1 } 4 x < 2 2. f(x) = { x 2 2 x 2} 2x + 5 x > 2 3. f(x) = { (x 2)2 x 0 x 2 3 x < 0 } 1 x x 1 4. f(x) = { 3 x x < 1 } Caryn Loves Math 15

16 KEY Finding a Piecewise-defined Function given the Graph Name: Caryn Loves Math 16

17 Foldable: Writing Equations for Piecewise Functions Includes 2 version of the inside one completed one partially completed Print the foldable front to pack. Cut out and fold Caryn Loves Math 17

18 Finding Equations Foldable - Outside Caryn Loves Math 18

19 Finding Equations Foldable Inside - Completed 1 st Segment Middle Segment 3rd Segment Find the interval 4 x < 1 Since the graph is linear Find the slope & y-intercept Find the interval 1 x < 4 Since the graph is linear Find the slope & y-intercept Find the interval x 4 Since the graph is linear Find the slope & y-intercept Slope = 2 1 y = mx + b y = 2x 4 y-intercept (0,-4) Slope = 1 y-intercept (0,-1) 1 y = mx + b y = x 1 Slope = 0 y-intercept (0,4) 1 horizontal line y = 4 The equations for our functions is 2x 4 4 x < 1 f(x) = { x 1 1 x < 4} 4 x 4 Writing Equations for Piecewise Functions Caryn Loves Math 19

20 Finding Equations Foldable Inside - Blank 1 st Segment Middle Segment 3rd Segment Find the interval Find the interval Find the interval Since the graph is linear Find the slope & y-intercept Since the graph is linear Find the slope & y-intercept Since the graph is linear Find the slope & y-intercept Slope = y-intercept (0, ) y = mx + b Slope = y-intercept (0, ) y = mx + b Slope = y-intercept (0, ) horizontal l The equations for our functions is f(x) = { } Writing Equations for Piecewise Functions Caryn Loves Math 20

21 Note Pages Caryn Loves Math 21

22 Note Page: Finding the Piecewise Function Given the Graph This function has 3 sections. So we need to find the equation for each one 1st Segment 2nd section 3rd Section x starts at -4 stops at -1 (both have dots) so this equation is for -4 x -1 x starts at -1 stops at 2-1 < x < 2 x starts at 2 and doesn't stop x 2 Horizontal Line have slope= 0 b = 2 y = 0x+2 y = 2 b = -4 y = mx+b y = -2x-4 b = -1 y = mx+b y = x-1 Our Function is 2011/2014 Caryn White: Caryn Loves Math 22

23 Note Page: How to graph a piece-wise function by hand To graph a piece-wise function you need to identify each section, noting their restrictions, then graph each section on the same coordinate plane. Since our functions are linear, I will find 3 points for each section. Since x<4 we need to use x-values less than 4. One of the x-values must be 4. x y (x,y) 4 (4,-1) 0 (0,2) Since x > 4, pick x-value larger than 4, one x-value must be 4. x y (x,y) 4-1 (4,-1) 5-1 (5,-1) 6-1 (6,-1) -4 (-4,5) Graph the 3 points and connect the dots, since x<4 graph the point (4,-1) using a circle Graph the 3 point and connect the dots, since x > 4 graph the point (4,-1) using a dot. Combine the 2 graphs on the same coordinate plane. 2011/2014 Caryn White: Caryn Loves Math 23

24 Note Page: Graphing Piecewise Functions with 3 sections We need to graph each section separately then put them together. Pick any x you want that fits your requirements (x < 0; -3 x 0; x > 0). Be sure to include the endpoints f(x) = 2x when x < -3 f(x) = -2 when -3 x 0 f(x) = 4-x when x > 0 x y (x,y) -3-6 (-3,-6) -4-8 (-4,-8) (-5,-10) x y (x,y) -3-2 (-3,-2) -2-2 (-2,-2) 0-2 (0,-2) x y (x,y) (0,4) (0,3) (2,2) Be sure when graphing to use OPEN circle for < or > and CLOSED circle for or > Now put all three graphs together to get the graph of 2011/2014 Caryn White: Caryn Loves Math 24

25 How to graph a piece-wise function using a TI83/84 Graphing Calculator Press [y=] and enter each function separately. Place parenthesis around each equation If I graph the equation now, I will see the entire graph of each equation. Not the piece-wise graph We need to restrict the domain for each piece of our graph. Our first equation y1= -3/4x+2 should only show when x < 4. Move your curse to the end of the first equation. But where is the < you might ask? [2nd][Math] Then add / (x < 4 ) [ / ][ ( ] [x] [<] [4] [ ) ] If I graph it now. We see only the portion of the 1 st equation when x < /2014 Caryn White: Caryn Loves Math 25

26 Now add the domain to the other equation Y2= (-1) /(x 4) Keystrokes [/] [2nd] [Math] [4] [4] [)] Graph the following piece-wise defined functions using your graphing calculator. Skech the resulting graph /2014 Caryn White: Caryn Loves Math 26

27 KEY /2014 Caryn White: Caryn Loves Math 27

28 Walk Around Activity: Level 1 Graphing Piecewise-defined Function By Caryn White Instructions This activity is designed to help students with graphing piecewise-defined functions (or with writing them). This activity also gets students up and about. Place the 10 cards on the wall around your room. Students pick any card to begin with. They should graph the function on the bottom of the page. After graphing on their own paper, students look around the room for a graph that matches theirs. Then they graph the next problem. They should continue until they return to the card they started with. Option 2: Have the student start with the graph and write a piecewise-defined function to match the graph. Then continue until they return to the card they started with. Included are 2 version of this activity. One with 10 cards for use as a walk-around activity A shorter version with only 5 cards great for table work, bell work or use as a review. To help with grading, I have included 2 student answer sheets: One that includes graph paper, and another with just boxes. Hints and suggestions: I have a set of laminated graph paper for student to use with this activity. The lamination allows students to write on it with dry erase marker. I give student 3 markers, and have them graph each section of the graph with a different color. They walk around with their graph to find the match. Then they come back to their seat and write the matching number and sketch the graph on the answer sheet. Mistakes on the dry erase boards are easier to correct then pencil and paper. When making copies of the shorter version, I use different colored paper. This allows for easier identification when students misplace a piece. 2011/2014 Caryn White: Caryn Loves Math 28

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39 Absent Student Sheet Graph the function on each card. Then look for a matching graph on another card. Name: 2011/2014 Caryn White: Caryn Loves Math 39

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41 Student Response Sheet Version A1 Student Answer Sheet Name Period Pick a card to start with. Write the number in the 1 st box. Graph the piecewise-defined function, then locate the card with the correct graph. Write the number in the 2 nd box, and then continue until you reach your first card again. Show work below Card # Card # Matches Card # Matches Card # Card # Card # Matches Card # Matches Card # 2011/2014 Caryn White: Caryn Loves Math 41

42 Card # Card # Matches Card # Matches Card # Card # Card # Matches Card # Matches Card # Card # Card # Matches Card # Matches Card # 2011/2014 Caryn White: Caryn Loves Math 42

43 Student Response Sheet Version A2 Name Period Pick a card to start with. Write the number in the 1 st box. Graph the piecewise-defined function, then locate the card with the correct graph. Write the number in the 2 nd box, then continue until you reach your first card again. Student Response Sheet Name Period Pick a card to start with. Write the number in the 1 st box. Graph the piecewise-defined function, then locate the card with the correct graph. Write the number in the 2 nd box, then continue until you reach your first card again. Student Response Sheet Name Period Pick a card to start with. Write the number in the 1 st box. Graph the piecewise-defined function, then locate the card with the correct graph. Write the number in the 2 nd box, then continue until you reach your first card again. Student Response Sheet Name Period Pick a card to start with. Write the number in the 1 st box. Graph the piecewise-defined function, then locate the card with the correct graph. Write the number in the 2 nd box, then continue until you reach your first card again. 2011/2014 Caryn White: Caryn Loves Math 43

44 Version B /2014 Caryn White: Caryn Loves Math 44

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46 Student Response Sheet Version B1 Name Period Pick a card to start with. Write the number in the 1 st box. Graph the piecewise-defined function, then locate the card with the correct graph. Write the number in the 2 nd box, and then continue until you reach your first card again. Card # Card # Matches Card # Card # Matches Card # Card # Matches Card # Matches Card # Card # Matches Card # 2011/2014 Caryn White: Caryn Loves Math 46

47 Student Response Sheet Version B2 Name Period Pick a card to start with. Write the number in the 1 st box. Graph the piecewise-defined function, then locate the card with the correct graph. Write the number in the 2 nd box, then continue until you reach your first card again. Student Response Sheet Name Period Pick a card to start with. Write the number in the 1 st box. Graph the piecewise-defined function, then locate the card with the correct graph. Write the number in the 2 nd box, then continue until you reach your first card again. Student Response Sheet Name Period Pick a card to start with. Write the number in the 1 st box. Graph the piecewise-defined function, then locate the card with the correct graph. Write the number in the 2 nd box, then continue until you reach your first card again. Student Response Sheet Name Period Pick a card to start with. Write the number in the 1 st box. Graph the piecewise-defined function, then locate the card with the correct graph. Write the number in the 2 nd box, then continue until you reach your first card again. 2011/2014 Caryn White: Caryn Loves Math 47

48 KEYS Version A - Set of 10 Cards Note: Students may start with any cards; shift the answer to match the starting point. Version B: Set of 5 Cards /2014 Caryn White: Caryn Loves Math 48

49 Walk -Around Activity: Level 2 Graphing Piecewise-defined Function By Caryn White Instructions This activity is designed to help students with graphing piecewise-defined functions (or with writing them). This activity also gets students up and about. Place the 10 cards on the wall around your room. Students pick any card to begin with. They should graph the function on the bottom of the page. After graphing on their own paper, students look around the room for a graph that matches theirs. Then they graph the next problem. They should continue until they return to the card they started with. Option 2: Have the student start with the graph and write a piecewise-defined function to match the graph. Then continue until they return to the card they started with. Included are 2 version of this activity. One with 10 cards for use as a walk-around activity A shorter version with only 5 cards great for table work, bell work or use as a review. To help with grading, I have included 2 student answer sheets: One that includes graph paper, and another with just boxes. Hints and suggestions: I have a set of laminated graph paper for student to use with this activity. The lamination allows students to write on it with dry erase marker. I give student 3 markers, and have them graph each section of the graph with a different color. They walk around with their graph to find the match. Then they come back to their seat and write the matching number and sketch the graph on the answer sheet. Mistakes on the dry erase boards are easier to correct then pencil and paper. When making copies of the shorter version, I use different colored paper. This allows for easier identification when students misplace a piece. 2011/2014 Caryn White: Caryn Loves Math 49

50 Version A 1 f(x) = { x2 x 1 1 x < 1 } 2011/2014 Caryn White: Caryn Loves Math 50

51 2 f(x) = { 2x + 1 x 2 1 x x > 2} 2011/2014 Caryn White: Caryn Loves Math 51

52 3 f(x) = { x x 0 x x < 0 } 2011/2014 Caryn White: Caryn Loves Math 52

53 4 f(x) = { x2 + 2x 3 x 2 2x + 1 x > 2 } 2011/2014 Caryn White: Caryn Loves Math 53

54 5 2x 2 x < 1 f(x) = { x x 3} 2x + 10 x > /2014 Caryn White: Caryn Loves Math 54

55 6 f(x) = { 2x 2 x 1 3x x > 1 } 2011/2014 Caryn White: Caryn Loves Math 55

56 7 f(x) = { 2x + 2 x < 1 x x 1 } 2011/2014 Caryn White: Caryn Loves Math 56

57 8 f(x) = { (x 3)2 x 0 (x + 3) 2 x < 0 } 2011/2014 Caryn White: Caryn Loves Math 57

58 9 x x 3 f(x) = { 3 3 < x < 3} x x /2014 Caryn White: Caryn Loves Math 58

59 10 x 2 x 0 f(x) = { x x > 0 } 2011/2014 Caryn White: Caryn Loves Math 59

60 Absent Student Sheet Graph the function on each card. Then look for a matching graph on another card. Name: 2011/2014 Caryn White: Caryn Loves Math 60

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62 Student Response Sheet Version A1 Student Answer Sheet Name Period Pick a card to start with. Write the number in the 1 st box. Graph the piecewise-defined function, then locate the card with the correct graph. Write the number in the 2 nd box, and then continue until you reach your first card again. Show work below Card # Card # Matches Card # Matches Card # Card # Card # Matches Card # Matches Card # Caryn Loves Math 62

63 Card # Card # Matches Card # Matches Card # Card # Card # Matches Card # Matches Card # Card # Card # Matches Card # Matches Card # Caryn Loves Math 63

64 Student Response Sheet Version A2 Name Period Pick a card to start with. Write the number in the 1 st box. Graph the piecewise-defined function, then locate the card with the correct graph. Write the number in the 2 nd box, then continue until you reach your first card again. Student Response Sheet Name Period Pick a card to start with. Write the number in the 1 st box. Graph the piecewise-defined function, then locate the card with the correct graph. Write the number in the 2 nd box, then continue until you reach your first card again. Student Response Sheet Name Period Pick a card to start with. Write the number in the 1 st box. Graph the piecewise-defined function, then locate the card with the correct graph. Write the number in the 2 nd box, then continue until you reach your first card again. Student Response Sheet Name Period Pick a card to start with. Write the number in the 1 st box. Graph the piecewise-defined function, then locate the card with the correct graph. Write the number in the 2 nd box, then continue until you reach your first card again Caryn Loves Math 64

65 Version B Caryn Loves Math 65

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67 Student Response Sheet Version B1 Name Period Pick a card to start with. Write the number in the 1 st box. Graph the piecewise-defined function, then locate the card with the correct graph. Write the number in the 2 nd box, and then continue until you reach your first card again. Card # Card # Matches Card # Matches Card # Card # Card # Matches Card # Matches Card # Card # Matches Card # Caryn Loves Math 67

68 Student Response Sheet Version B2 Name Period Pick a card to start with. Write the number in the 1 st box. Graph the piecewise-defined function, then locate the card with the correct graph. Write the number in the 2 nd box, then continue until you reach your first card again. Student Response Sheet Name Period Pick a card to start with. Write the number in the 1 st box. Graph the piecewise-defined function, then locate the card with the correct graph. Write the number in the 2 nd box, then continue until you reach your first card again. Student Response Sheet Name Period Pick a card to start with. Write the number in the 1 st box. Graph the piecewise-defined function, then locate the card with the correct graph. Write the number in the 2 nd box, then continue until you reach your first card again. Student Response Sheet Name Period Pick a card to start with. Write the number in the 1 st box. Graph the piecewise-defined function, then locate the card with the correct graph. Write the number in the 2 nd box, then continue until you reach your first card again Caryn Loves Math 68

69 KEYS Version A - Set of 10 Cards Note: Students may start with any cards; shift the answer to match the starting point. Version B: Set of 5 Cards Caryn Loves Math 69