1. Metro RESA...Building leaders of teaching and learning Digging Deeper Into Mathematics Clayton County Summer Math Academy Algebra I – Linear & Exponential Functions Sarah Ledford June 10, 2015 1

2. Metro RESA...Building leaders of teaching and learning Pet Shop 2 Sixty percent of the animals at the neighborhood pet store were dogs. If there were a total of 40 animals at the pet store, how many of the animals were dogs? The answer: There were _____ dogs at the pet store. From Step-by-Step Model Drawing (pg 79).

3. Metro RESA...Building leaders of teaching and learning Tips & Taxes? How do you determine how much tip you will leave at a restaurant? 3

4. Metro RESA...Building leaders of teaching and learning Pet Shop 4 Sixty percent of the animals at the neighborhood pet store were dogs. If there were a total of 40 animals at the pet store, how many of the animals were dogs? The answer: There were _____ dogs at the pet store. From Step-by-Step Model Drawing (pg 79).

5. Metro RESA...Building leaders of teaching and learning % Model 5 Draw the unit bar.

6. Metro RESA...Building leaders of teaching and learning % Model 6 Chunk the bar.

7. Metro RESA...Building leaders of teaching and learning % Model 7 I’m chunking it into 10 equal chunks so that each chunk represents 10%. Why do you think that I chose 10? 10%20%30%40%50%60%70%80%90%100%

8. Metro RESA...Building leaders of teaching and learning % Model 8 Show where the desired % lies. 10%20%30%40%50%60%70%80%90%100%

9. Metro RESA...Building leaders of teaching and learning % Model 9 Now let’s insert the total number of pets at the pet shop. 10%20%30%40%50%60%70%80%90%100% 40

10. Metro RESA...Building leaders of teaching and learning % Model 10 One reason we broke the bar into 10 equal chunks is because division by 10 is easy. Each chunk of the bar represents 10% AND 1/10 of 40. 10%20%30%40%50%60%70%80%90%100% 40

11. Metro RESA...Building leaders of teaching and learning % Model 11 One-tenth of 40 is 4. So, each chunk of the bar represents 4 animals. Put the 4 in each chunk of the bar. 10%20%30%40%50%60%70%80%90%100% 4444444444 40

12. Metro RESA...Building leaders of teaching and learning % Model 12 Now, along the bottom of our bar, I want to show a running total. 10%20%30%40%50%60%70%80%90%100% 4444444444 481216202428323640

13. Metro RESA...Building leaders of teaching and learning % Model 13 We were trying to find 60% of 40. By our model, we can see that 60% of 40 is the same as 6 4’s or 24. 10%20%30%40%50%60%70%80%90%100% 4444444444 481216202428323640

14. Metro RESA...Building leaders of teaching and learning Pet Shop 14 Let’s make sure we answer the question! There were 24 dogs at the pet store. 10%20%30%40%50%60%70%80%90%100% 4444444444 40

15. Metro RESA...Building leaders of teaching and learning % Model 15 What other questions could we ask? What if 65% of the pets were dogs? 10%20%30%40%50%60%70%80%90%100% 4444444444 481216202428323640

16. Metro RESA...Building leaders of teaching and learning Kohl’s Coupons 16 I have a Kohl’s coupon for an additional 15% off to be taken at the register. The item I want to buy is $50 and is on sale for 20% off that price. How much is the item?

17. Metro RESA...Building leaders of teaching and learning Kohl’s Coupons 17 10%20%30%40%50%60%70%80%90%100% 5555555555 5101520253035404550 First we need to find 20% off of $50. Note that 20% of $50 is $10 so that 20% off is $40. Where else do you see $40 in our model? 20% off is the same as paying 80% of the price!!

18. Metro RESA...Building leaders of teaching and learning Kohl’s Coupons Now, we need to find 15% off of $40 with a new model. 15% is not shown in our model. 15% is exactly halfway between 10% and 20%. The value we need is halfway between the corresponding 4 and 8, which is 6! 10%20%30%40%50%60%70%80%90%100% 4444444444 481216202428323640

19. Metro RESA...Building leaders of teaching and learning Kohl’s Coupons At the register, we get an extra $6 off (15% off of sale price). Therefore, the item that I want to buy that was originally $50 can be purchased for $34 (+ tax!).

20. Metro RESA...Building leaders of teaching and learning What if % isn’t nice? What if our coupon was for 23% off? Let’s say the item is $80. 10% off is $_____. 20% off is $_____. 30% off is $_____. If each 10% is $8, then 1% is (1/10 of $8) $____. I have 3 1%, which is $______. 20

21. Metro RESA...Building leaders of teaching and learning What if % isn’t nice? 23% off is $_____. The item costs $______ after using the coupon. Which is the same as paying ____% for the item. 21

22. Metro RESA...Building leaders of teaching and learning Task Review What GSE (content) was addressed? – Domain, cluster, and standards What SMPs were addressed? 22

23. Metro RESA...Building leaders of teaching and learning 6-8 GSE 6.RP.3 Use ratio and rate reasoning to solve real-world and mathematical problems utilizing strategies such as tables of equivalent ratios, tape diagrams (bar models), double number line diagrams, and/or equations. 6.RP.3c Find a percent of a quantity as a rate per 100 (e.g. 30% of a quantity means 30/100 times the quantity); given a percent, solve problems involving finding the whole given a part and the part given the whole. 7.RP.3 Use proportional relationships to solve multistep ratio and percent problems. 23

24. Metro RESA...Building leaders of teaching and learning Mathematics Assessment Project http://map.mathshell.org Tools for formative and summative assessment that make knowledge and reasoning visible, and help teachers to guide students in how to improve, and monitor their progress. – Formative Assessment Lessons (FALs  grades 6-11)

25. Metro RESA...Building leaders of teaching and learning MAP FALs Lesson Plans – Concept development vs. Problem Solving Process – think independently & jot down ideas/work, work within a group to come up with a better/more efficient solution, discuss student solutions, & discuss provided student solutions PPT slides Worksheets 25

26. Metro RESA...Building leaders of teaching and learning Comparing Investments Making Money - pre-assessment Odd One Out? - powerpoint slides Matching Game Double Your Money - powerpoint slides Making Money (revisited) 26

27. Metro RESA...Building leaders of teaching and learning Simple Interest What is simple interest? Do you remember the formula? How do you compute the “future value,” A? I = Prt A = P + I  A = P + Prt  A = P(1 + rt) 27

28. Metro RESA...Building leaders of teaching and learning Odd One Out? P-28 Investment 1 $100 Simple Interest Rate: 5% Investment 2 $400 Simple Interest Rate: 5% Investment 3 $200 Simple Interest Rate: 10%

29. Metro RESA...Building leaders of teaching and learning Compound Interest What is compound interest? Do you remember a formula? How do you compute the “future value,” A? What is the difference between 5% simple interest over 3 years and 5% interest over 3 years compounded annually? For simplicity, let’s say you invested $100. 29

30. Metro RESA...Building leaders of teaching and learning Simple Interest Let P = 100; r = 0.05; and t = 3 (for each year) 5% simple interest over 3 years A = 100(1 + 0.05(3)) = 100(1 + 0.15) = 100(1.15) = 115 30

31. Metro RESA...Building leaders of teaching and learning Compound Interest Let P = 100; r = 0.05; and t = 1 (for each year) 5% interest over 3 years compounded annually Year 1: A 1 = 100(1 + 0.05) = 100(1.05) = 105 Year 2: A 2 = 105(1 + 0.05) = 105(1.05) = 110.25 Year 3: A 3 = 110.25(1 + 0.05) = 110.25(1.05) = 115.76 31

32. Metro RESA...Building leaders of teaching and learning Compound Interest Let’s generalize for any P or r where we have r% interest over 3 years compounded annually Year 1: A 1 = 100(1 + 0.05) = 100(1.05) = 105 Year 1: A 1 = P(1 + r) Year 2: A 2 = 105(1 + 0.05) = 105(1.05) = 110.25 Year 2: A 2 = A 1 (1 + r) = P(1 + r)(1 + r) = P(1 + r) 2 32

33. Metro RESA...Building leaders of teaching and learning Compound Interest Year 2: A 2 = 105(1 + 0.05) = 105(1.05) = 110.25 Year 2: A 2 = A 1 (1 + r) = P(1 + r)(1 + r) = P(1 + r) 2 Year 3: A 3 = 110.25(1 + 0.05) = 110.25(1.05) = 115.76 Year 3: A 3 = A 2 (1 + r) = P(1 + r) 2 (1 + r) = P(1 + r) 3 33

34. Metro RESA...Building leaders of teaching and learning Compound Interest When compounding annually, A t = P(1 + r) t where t = # of years If interest is not compounded annually, our formula is more complex: A t = P(1 +r/n) nt where n = # of times interest is compounded per year 34

35. Metro RESA...Building leaders of teaching and learning Odd One Out? P-35 Investment 1 A = 500 × 1.06 4 Investment 2 A = 250 × 1.06 2 Investment 3 A = 500 × 1.03 2

36. Metro RESA...Building leaders of teaching and learning Comparing Investments Look at the first pair of cards – the plans & formulas. What do you notice about the cards? Match the plan with the formula. There are 2 blank cards for the formulas that you should write in. Leave your cards as they are and match the tables & graphs to them. If any information is missing, fill it in. 36

37. Metro RESA...Building leaders of teaching and learning Comparing Investments Once your group feels really good about your matched groups, match the last set of cards (statements) to your sets. 37

38. Metro RESA...Building leaders of teaching and learning Comparing Investments Solutions P1 F6 P2 F3 P3 F2 P4 F5 P5 F1 P6 F4 38

39. Metro RESA...Building leaders of teaching and learning Comparing Investments Solutions P1 F6 G6 T6 P2 F3 G4 T4 P3 F2 G3 T5 P4 F5 G5 T2 P5 F1 G1 T1 P6 F4 G2 T3 39

40. Metro RESA...Building leaders of teaching and learning Comparing Investments Solutions P1 F6 G6 T6 S4 P2 F3 G4 T4 S1 P3 F2 G3 T5 S2 P4 F5 G5 T2 S1 P5 F1 G1 T1 S5 P6 F4 G2 T3 S3 40

41. Metro RESA...Building leaders of teaching and learning Double Your Money P-41 Investment 1 A = 500 × 1.06 n Investment 2 A = 250 × 1.06 n Investment 3 A = 500 × 1.03 n Which two investments will take exactly the same time to double the money?

42. Metro RESA...Building leaders of teaching and learning Task Review What GSE (content) was addressed? – Domain, cluster, and standards What SMPs were addressed? 42

43. Metro RESA...Building leaders of teaching and learning Alg I GSE A.CED.2 Create linear, quadratic, and exponential equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. F.BF.1 Write a function that describes a relationship between two quantities. F.LE.1 Distinguish between situations that can be modeled with linear functions and with exponential functions. 43

44. Metro RESA...Building leaders of teaching and learning Alg I GSE F.LE.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input- output pairs (include reading these from a table). F.LE.3 Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. 44

45. Metro RESA...Building leaders of teaching and learning 6-8 GSE 6.EE.1 Write and evaluate numerical expressions involving whole-number exponents 7.RP.3 Use proportional relationships to solve multistep ratio and percent problems. 8.EE.1 Know and apply the properties of integer exponents to generate equivalent numerical expressions. 45