1. MATH 2160 2 nd Exam Review Problem Solving and Proportional Reasoning

2. Problem Solving +Polya’s 4 Steps Understand the problem What does this mean? How do you understand? Devise a plan What goes into this step? Why is it important? Carry out the plan What happens here? What belongs in this step? Look back What does this step imply? How do you show you did this?

3. Problem Solving + Strategies for Problem Solving Make a chart Make a table Draw a picture Draw a diagram Guess, test, and revise Form an algebraic model Look for a pattern Try a simpler version of the problem Work backward Restate the problem differently Eliminate impossible situations Use reasoning

4. Proportional Reasoning +Piaget Concrete A manipulative Something I can move around and investigate things with Semi-concrete A picture of the things I have been moving around If I have never moved it around before this becomes semi-abstract for me

5. Proportional Reasoning +Piaget Semi-abstract Symbols or alternate representations for my semi-concrete models Good time to use algebraic thinking with students Abstract Using numbers to represent all parts of the original concrete model

6. Proportional Reasoning +Equivalent Representations Percents Decimals Ratios Fractions 25% = 0.25 = 25/100 = ¼ Need to be able to fill in missing equivalent values on a table

7. Proportional Reasoning + Circle Graph Representations 5/3 12.5% 1.25 All circles contain exactly 100%

8. Proportional Reasoning +Integer Rods White1 cm x 1 cmW Red2 cm x 1 cmR Lime3 cm x 1 cmL Purple4 cm x 1 cmP Yellow5 cm x 1 cmY Green6 cm x 1 cmG Black7 cm x 1 cmK Brown8 cm x 1 cmN Blue9 cm x 1 cmB Orange 10 cm x 1 cmO

9. Proportional Reasoning +Integer Rods Equivalents / Combinations Number of Equivalents = 2 n-1

10. Proportional Reasoning +Integer Rods Equivalent Fractions ½ 1/3 ¼ Etc.

11. Proportional Reasoning +Scale Factors Enlargement >100% Reduction <100% Original = 100% Percent = per 100

12. Proportional Reasoning +Types of Percent Problems A% x B = C What is 27.37% of 123.4 to the nearest thousandth? 0.2737 x 123.4 = C C = 33.775 What percent of 785.46 is 800.03 to the nearest thousandth? A x 785.46 = 800.03 A = 1.019 A% = 101.9%

13. Proportional Reasoning +Types of Percent Problems A% x B = C 237.9% of what number is 48.37 to the nearest thousandth? 2.379 x B = 48.37 B = 20.332

14. Proportional Reasoning + Bar Graphs ALL ALL graphs must have the following Title Scales Labels

15. Proportional Reasoning + Bar Graphs A survey was taken and the following results were obtained from a group of 25 people: Hats = 5; Glasses = 10; Neither = ?

16. Proportional Reasoning + Circle Graphs A survey was taken and the following results were obtained from a group of 25 people: Hats = 5; Glasses = 10; Neither = ?

17. Proportional Reasoning +Additive Models A factory builds 500 widgets a week. They want to increase productivity by 25 widgets a week over the next 3 months. What is the factories production rate per week for any given week? 500 + 25W = P where P = Production rate and W = the number of weeks

18. Proportional Reasoning +Multiplicative Models A factory builds 500 widgets a week. They want to increase productivity by 2.5% widgets a week over the next 3 months. What is the factories production rate per week for any given week? 500∙1.025 W = P where P = Production rate and W = the number of weeks

19. Proportional Reasoning +Tangrams Recognizing pieces Whole to part Part to whole Conservation of area

20. Proportional Reasoning +Tangrams Recognizing pieces 1 whole square or large square composed of 7 pieces 2 large triangles 2 small triangles 1 medium triangle 1 small square 1 parallelogram

21. Proportional Reasoning +Tangrams Whole to part If the whole tangram has an area of 36 units 2, what is the area of the parallelogram?

22. Proportional Reasoning +Tangrams Part to whole If the parallelogram and a small triangle have an area of 3/8 units 2, what is the area of the whole large square?

23. Test Taking Tips Get a good nights rest before the exam Prepare materials for exam in advance (scratch paper, pencil, and calculator) Read questions carefully and ask if you have a question DURING the exam Remember: If you are prepared, you need not fear