Preparing for Success in Algebra Demonstration Center

1. Preparing for Success in AlgebraDemonstration Center A Collaboration among: Los Angeles USD University of California, San Diego San Diego State University University of California, Irvine

2. Why is Proportional Reasoning Important? • It is primary goal of the CCSS in grades 5 - 8 • Scaling a problem allows you to solve real life problems such as recipe modification, scale drawings or photo enlargements. • An ability to reason proportionally indicates a facility with rational numbers and their multiplicative concepts

3. Different views of fractions • Fractions as measures • Fractions as part/whole representations • Fractions as operators i.e. scaling • Fractions as ratios/percents

4. Different views of fractions • Fractions as measures • Fractions as part/whole representations • Fractions as operators i.e. scaling • Fractions as ratios/percents

5. Fractions as measures The number line is the primary tool for this view. A group of Ms. Guzman’s students was following a set of directions to move a paper frog along a number line. Their last direction took them to 1/2. The next direction says: Go 1/3 of the way to 3/4. What number will the frog land on?

6. Different views of fractions • Fractions as measures • Fractions as part/whole representations • Fractions as operators i.e. scaling • Fractions as ratios/percents

7. Fractions as part/whole representations For most students, this is the most common and perhaps their only interpretation. Jose bought 8 pizzas for the 28 students in his class. Normally each pizza is cut into 8 pieces but the pizzeria was willing to cut them into a different number (the same for each pizza however). Jose wanted everyone to receive the same number of pieces. What is the smallest number he could choose for each pizza?

8. Different views of fractions • Fractions as measures • Fractions as part/whole representations • Fractions as operators i.e. scaling • Fractions as ratios/percents

9. Fractions as operators Scale factors. This is a functional view of fractions where there is an input and a resulting output after performing a fixed operation. Scaling up or down is a natural interpretation e. g. scaling up or down a recipe depending on the number of guests and recognizing when multiple objects are related by the same scale factor. Producing an enlargement of a photograph.

10. Different views of fractions • Fractions as measures • Fractions as part/whole representations • Fractions as operators i.e. scaling • Fractions as ratios/percents

11. Fractions as ratios and percents Ratios are ordered pairs (as are fractions) but care must be used in interpretation. (2:3) is usually interpreted as 2/3 but may also be represented as 2/5 or 40%. This latter view indicates the first entry is 40% of the two quantities combined. Rates may be interpreted as ratios e.g. 35 mph could be interpreted as (35:1) or (70:2) or (87.5:2.5). Thirty five percent may be represented as (35:100).

12. Theater Seats There are 100 seats in a theater, with 30 in the balcony and 70 on the main floor. 80 tickets were sold including all the seats on the main floor. What is the ratio of empty seats to occupied seats? What is the ratio of empty seats to occupied seats in the balcony?

13. What is the meaning of 24x? The camera has a zoom lens ranging from a wide angle of 25mm to a full telephoto of 600mm. (Both 35mm equivalents) Thesetwo numbers produce a ratio of 600:25or the equivalent ratio of 24:1 The ratio of 24:1 is represented in cameras as 24x

14. Estimating populations One method of estimating the number of fish in a lake is to catch and tag a collection of fish and then release them. At a later time, we return and catch a sample of fish and determine the proportion of tagged fish. We can then use that proportion to estimate the total fish population in the lake.

15. Estimating Populations (Continued) For example, suppose that we initially caught, tagged, and released a total of 50 fish. If we return later and catch a total of 60 fish, 18 of which were tagged, we can then estimate the number of fish in the lake? Fish population estimate

16. Using a Tape Model ? 50 Tag All fish in the lake 18 60 Tag sample Fish population simulation

17. All Proportional Reasoning Problems 1 2 There are 4 quantities. If you know any 3 of them, you can calculate the 4th. 3 1/2 = 3/4 or 1/3 = 2/4 4

18. Ratings and Shares • A rating is defined as the percentage of all TVs in the market that are tuned in to a particular program. • The share is defined as the percentage of the TVs tuned in to a particular program among all TVs that are actually ON.

19. Consider a Market consisting of 5 televisions. What rating and share would the Dodger game receive? Create a tape diagram to illustrate your solution. 5 Set Market

20. The Dodger Game in a 5 Set Market 4 TVs on during game 5 TVs on during game All TVs

21. In This Market • The Dodger game rating would be 40(%) because 2 out of 5 are watching the game. • The Dodger game share would be 50(%) since 2 out of 4 sets which are ON during the gameareactuallytuned to the game.

22. July 22nd , 2009 was Manny Ramirez’ bobblehead night at Dodger stadium. Since he had injured his hand the previous night, he was not expected to play and in fact, he came into the game as a pinch-hitter and faced only one pitch. He did hit a grand slam with that one pitch and that was the difference in the game – just Manny being Manny.

23. Manny Continued MLB.tv had a 15 rating (15% of all TVs in the Los Angeles area were tuned in) and a 25 share (25% of all TVs in the Los Angeles area that were actually ON were tuned in to the game). What fraction of TV's (in the Los Angeles area) were ON during the game, that is, tuned into some program during the time slot of the game?

24. An informal Solution Solve the problem by considering a sample of 100 televisions in the LA area. Note that percent means the number per one hundred so 100 is a natural choice to use for a sample size for computational purposes. A Dodger game TV ratings

25. ALL TVS TVs on During the Game

26. The Dodger Game with a Tape Diagram 25 rating :15 share TVs on during the game TVs on during game All TVs

27. The Dodger Game with a Tape Diagram 25 rating :15 share TVs on during the game 25% 25 25 25 Dodger Game` 15% 15 15 15 60% of all TVs on during game All TVs

28. The students in Ms. Baca’s art class were mixing yellow and blue paint. She told them that two mixtures will be the same shade of green if the blue and yellow paint are in the same ratio. The table below shows the different mixtures of paint that the students made. How many different shades of paint did the students make? Some of the shades of paint were bluer than others. Which mixture(s) were the bluest? Show your work or explain how you know.

29. Carefully plot a point for each mixture on a coordinate plane like the one that is shown in the figure. (Graph paper might help.) Carefully plot a point for each mixture on a coordinate plane like the one that is shown in the figure. (Graph paper might help.) Yellow-blue paint

30. To make A and C, you add 2 parts blue to 1 part yellow. To make mixtures B, D, and E, you add 3/2 parts blue to 1 part yellow. Mixtures A and C are the bluest because you add more blue paint to the same amount of yellow paint. The students made two different shades: mixtures A and C are the same, and mixtures B, D, and E are the same because they represent equivalent ratios. If two mixtures produce the same shade, they lie on the same line through the point (0,0).

31. And Finally Even if solving problems algebraically is not appropriate for your students – true for nearly all of us, you are laying the foundation for them to do so in the future. That number to symbol sense transition is critical and we want to help you help them make that transition.