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Standards Addressed: M8A1 M8A4 M8N1 Zak’s Wealthy Uncle Question: What are the differences in plans that Zak’s uncle has proposed to donate money to the school for new computers? Launch: What experiences do you have with fundraisers? How was money raised? What is the difference between linear and exponential relationships? What is an example of a linear relationship? What is an example of an exponential relationship? Investigation: The differentiated investigation activity is attached below. There are three levels for this activity – below level, on level, and above level. They are labeled in red, so make sure to delete this before reproducing for students. Conclusions: Have groups write their tables on the board for each plan. Compare the equations of the graphs. Discuss if the relationships are linear or exponential. Also discuss which plan will raise the most money for the school. Have the students display their graphs on the overhead or on chart paper on the board. In Class Problems: Write each expression in exponential form: 1. 3 x 3 x 3 x 3 2. 11 x 11 x 11 x 11 x 11 x 11 x 11 3. 1.5 x 1.5 x 1.5 x 1.5 x 1.5 Write each expression in standard form: 1. 210 2. 102 3. 39 Standards Addressed: M8A1 M8A4 M8N1 Closure: On a note-card have students write 3 things they learned today, 2 real life problems mentioned in class, and 1 thing they are confused about as a ticket out the door. (3-2- 1 strategy) Homework: The Relay for Life is suggesting three plans to raise money for a fundraiser. Plan 1 was proposed by a teacher at the school. Plan 2 was proposed by a parent. Plan three was proposed by a student. Copy and extend the table to show the number of pennies donated for each miles walked for Relay for Life. Number of Miles Number of Pennies Donated Plan 1 Plan 2 Plan 3 1 1 1 1 2 2 3 4 3 4 4 1. How are the patterns of change in the number of pennies donated under plans 2 and 3 similar to and different from the pattern of change for Plan 1? 2. Are the growth patterns for Plans 2 and 3 exponential relationships? If so, what is the growth factor for each? 3. Write an equation for the relationship between the number of miles walked m and number of pennies donated p for Plan 2. 4. Make a graph of Plan 3 for p = 1 to 10. How does your graph compare to the graphs for Plans 1 and 2? 5. For each plan, how many pennies would be raised if the maximum miles to walk is 16 (per person)? Standards Addressed: M8A1 M8A4 M8N1 Zak’s Wealthy Uncle (Below Level) Standards M8A1a. Represent a given situation using algebraic equations. M8A1i. Interpret solutions in problem contexts. M8A4c. Graph equations in the form of y = mx + b. M8A4f. Determine the equation of a line given a graph, numerical information that defines a line, or a context involving a linear relationship. M8A4g. Solve problems involving linear relationships. M8N1i. Simplify expressions containing integer exponents. Problem: Zak’s wealthy uncle wants to donate money to Zak’s school for new computers. He suggests three possible plans for his donations. Plan 1: He will continue the pattern in this table until day 12. Day (x) 1 2 3 4 Donation (y) $1 $2 $4 $8 Plan 2: He will continue this pattern until day 10. Day (x) 1 2 3 4 Donation (y) $1 $3 $9 $27 Plan 3: He will continue this pattern until day 10. Day (x) 1 2 3 4 Donation (y) $28 $31 $34 $37 Your Task: a. Complete each table to show how much money the school will receive until the end of each plan. Plan 1: Day (x) 0 1 2 3 4 5 6 Donation (y) $1 $2 $4 $8 Plan 2: Day (x) 0 1 2 3 4 5 6 Donation (y) $1 $3 $9 $27 Standards Addressed: M8A1 M8A4 M8N1 Plan 3: Day (x) 0 1 2 3 4 Donation (y) $28 $31 $34 $37 b. Graph each plan on the same coordinate plane (use a different color for each plan). Remember: Independent variable goes on the x-axis and dependent variable goes on the y-axis c. Match the equations below to the proper plan: i. y = 3x-1 goes with Plan ____ ii. y = 3x + 25 goes with Plan ____ iii. y = 2x-1 goes with Plan ____ d. Using your graph, explain what the y-intercept means for each plan. (Why do the different plans touch the y-axis at different places?) e. Which plan gives the highest donation at day 4? At day 10? How come the same plan isn’t always the best for the school? (How come Plan 2 ends up being the best at day 10 when Plan 3 was best at Day 4?) f. Zak believes that y = ½ (2x-1) is an equivalent equation to one of the equations in part c. Is he correct and why? (Substitute values for x into Plan 1 to see if the equations are the same.) Standards Addressed: M8A1 M8A4 M8N1 Zak’s Wealthy Uncle (On Level) Standards M8A1a. Represent a given situation using algebraic equations. M8A1i. Interpret solutions in problem contexts. M8A4c. Graph equations in the form of y = mx + b. M8A4f. Determine the equation of a line given a graph, numerical information that defines a line, or a context involving a linear relationship. M8A4g. Solve problems involving linear relationships. M8N1i. Simplify expressions containing integer exponents. Problem: Zak’s wealthy uncle wants to donate money to Zak’s school for new computers. He suggests three possible plans for his donations. Plan 1: He will continue the pattern in this table until day 12. Day 1 2 3 4 Donation $1 $2 $4 $8 Plan 2: He will continue this pattern until day 10. Day 1 2 3 4 Donation $1 $3 $9 $27 Plan 3: He will continue this pattern until day 10. Day 1 2 3 4 Donation $28 $31 $34 $37 Your Task: g. Copy and extend each table to show how much money the school will receive until the end of each plan. h. Graph each plan on the same coordinate plane (use a different color for each plan). i. Match the equations below to the proper plan above: i. y = 3x-1 ii. y = 3x + 25 iii. y = 2x-1 j. Using your graph, explain what the y-intercept means for each plan. k. Which plan gives the highest donation at day 4? At day 10? How come the same plan isn’t always the best for the school? l. Zak believes that y = ½ (2x-1) is an equivalent equation to one of the equations in part c. Is he correct and why? Standards Addressed: M8A1 M8A4 M8N1 Zak’s Wealthy Uncle (Above Level) Standards M8A1a. Represent a given situation using algebraic equations. M8A1i. Interpret solutions in problem contexts. M8A4c. Graph equations in the form of y = mx + b. M8A4f. Determine the equation of a line given a graph, numerical information that defines a line, or a context involving a linear relationship. M8A4g. Solve problems involving linear relationships. M8N1i. Simplify expressions containing integer exponents. Problem: Zak’s wealthy uncle wants to donate money to Zak’s school for new computers. He suggests three possible plans for his donations. Plan 1: He will continue the pattern in this table until day 12. Day 1 2 3 4 Donation $1 $2 $4 $8 Plan 2: He will continue this pattern until day 10. Day 1 2 3 4 Donation $1 $3 $9 $27 Plan 3: He will continue this pattern until day 10. Day 1 2 3 4 Donation $28 $31 $34 $37 Your Task: m. Copy and extend each table to show how much money the school will receive until the end of each plan. n. Graph each plan on the same coordinate plane (use a different color for each plan). o. Write an algebraic equation for each donation plan p. Using your graph, explain what the y-intercept means for each plan. q. Which plan gives the highest donation at day 4? At day 10? How come the same plan isn’t always the best for the school? r. Can you write a second equation representing each of the donation plans? Why are why not?