# Digging Deeper Into Mathematics Clayton County Summer Math

• Slides: 45

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

Pet Shop 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). Metro RESA. . . Building leaders of teaching and learning 2

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

Pet Shop 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). Metro RESA. . . Building leaders of teaching and learning 4

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

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

% Model 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% Metro RESA. . . Building leaders of teaching and learning 7

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

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

% Model 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 Metro RESA. . . Building leaders of teaching and learning 10

% Model 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% 4 4 40 Metro RESA. . . Building leaders of teaching and learning 11

% Model Now, along the bottom of our bar, I want to show a running total. 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% 4 4 4 8 12 16 20 24 28 32 36 40 Metro RESA. . . Building leaders of teaching and learning 12

% Model 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% 4 4 4 8 12 16 20 24 28 32 36 40 Metro RESA. . . Building leaders of teaching and learning 13

Pet Shop 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% 4 4 40 Metro RESA. . . Building leaders of teaching and learning 14

% Model What other questions could we ask? 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% 4 4 4 8 4 12 4 16 4 20 4 24 4 28 4 32 4 36 4 40 What if 65% of the pets were dogs? Metro RESA. . . Building leaders of teaching and learning 15

Kohl’s Coupons 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? Metro RESA. . . Building leaders of teaching and learning 16

Kohl’s Coupons First we need to find 20% off of \$50. 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% 5 5 5 10 15 20 25 30 35 40 45 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!! Metro RESA. . . Building leaders of teaching and learning 17

Kohl’s Coupons Now, we need to find 15% off of \$40 with a new model. 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% 4 4 4 8 12 16 20 24 28 32 36 40 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! 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!). 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 \$______. Metro RESA. . . Building leaders of teaching and learning 20

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

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

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. 3 c 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. Metro RESA. . . Building leaders of teaching and learning 23

Mathematics Assessment Project • http: //map. mathshell. org • Tools 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) 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 Metro RESA. . . Building leaders of teaching and learning 25

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

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) Metro RESA. . . Building leaders of teaching and learning 27

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

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. Metro RESA. . . Building leaders of teaching and learning 29

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 Metro RESA. . . Building leaders of teaching and learning 30

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 Metro RESA. . . Building leaders of teaching and learning 31

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)2 Metro RESA. . . Building leaders of teaching and learning 32

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)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)3 Metro RESA. . . Building leaders of teaching and learning 33

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

Odd One Out? Investment 1 A = 500 × 1. 064 Investment 2 A = 250 × 1. 062 Investment 3 A = 500 × 1. 032 Metro RESA. . . Building leaders of teaching and learning P-35

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. Metro RESA. . . Building leaders of teaching and learning 36

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

Comparing Investments Solutions P 1 F 6 P 2 F 3 P 3 F 2 P 4 F 5 P 5 F 1 P 6 F 4 Metro RESA. . . Building leaders of teaching and learning 38

Comparing Investments Solutions P 1 F 6 G 6 T 6 P 2 F 3 G 4 T 4 P 3 F 2 G 3 T 5 P 4 F 5 G 5 T 2 P 5 F 1 G 1 T 1 P 6 F 4 G 2 T 3 Metro RESA. . . Building leaders of teaching and learning 39

Comparing Investments Solutions P 1 F 6 G 6 T 6 S 4 P 2 F 3 G 4 T 4 S 1 P 3 F 2 G 3 T 5 S 2 P 4 F 5 G 5 T 2 S 1 P 5 F 1 G 1 T 1 S 5 P 6 F 4 G 2 T 3 S 3 Metro RESA. . . Building leaders of teaching and learning 40

Double Your Money Investment 1 A = 500 × 1. 06 n Investment 2 A = 250 × 1. 06 n Investment 3 A = 500 × 1. 03 n Metro RESA. . . Building leaders of teaching and learning P-41 Which two investments will take exactly the same time to double the money?

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

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. Metro RESA. . . Building leaders of teaching and learning 43

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 inputoutput 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. Metro RESA. . . Building leaders of teaching and learning 44

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. Metro RESA. . . Building leaders of teaching and learning 45