1. April 7, 2009 While we teach, we learn. ~Seneca

2. April 7, 2009 Bring Class Notes on Thursday, 4/9 Test 3  Thursday, 4/16 Covers: Text sections 4.1, 4.2, 4.3, 5.2, 5.3 Class Notes (fractions) Explorations 4.2, 5.8, 5.9, 5.12, 5.13, …

3. April 7, 2009 Section 5.2 continued Section 5.3 Exploration 5.12 Exploration 5.13 Assign Homework

4. Sec 5.2 – (cont’d) 2.Is 10/13 closer to 1/2 or 1? Use a diagram to explain how you know. Are there certain diagrams that are more effective? Discuss this with your group.

5. Sec 5.2 – (cont’d) 3.If a/b = 3/4, will the value of (a + x)/(b + x) be less than, equal to, or greater than ¾? (Assume that x is a positive number). Use a diagram to explain how you know. Are there certain diagrams that are more effective? Discuss this with your group.

6. Sec 5.2 – (cont’d) 4.A teacher asks for examples of fractions that are equivalent to 3/4. One student replied: How would you respond to this answer? Use a diagram to explain. Are there certain diagrams that are more effective? Discuss this with your group.

7. 5.2 – (cont’d) Terminology: A rational number is a number that can be expressed as a/b where a and b are integers (with b  0). Examples: 1/7  5 0.25 0.3333333…

8. 5.2 – (cont’d) Terminology: A fraction is a number that can be expressed as a/b where a and b are any numbers (with b  0). Example: (note that “fractions” include all of the rational numbers)

9. 5.2 – (cont’d) Terminology: In a fraction a/b, we call a the numerator of the fraction and we call b the denominator of the fraction.

10. Sec 5.3 – Understanding Operations With Fractions You need to know HOW to do the operations on fractions (see me if you need review). You need to understand the rules – why do we use these algorithms.

11. 5.3 – (cont’d) Addition: When we add 3 cats + 4 cats, we get 7 cats. But when we add 3 cats and 4 dogs, we do not end up with 7 cats or 7 dogs. Likewise, when we add fractions, we can only add when we are talking about the same denominator. So, 3/12 + 4/12 = 7/12, but 3/5 + 4/7 ≠ 7/5 or 7/7. We need a common denominator.

12. 5.3 – (cont’d) If I have 2/5 shown with the part-whole model, and I have 1/3 shown with a part- whole model, can I add them? Note that the whole is the same for both. But I can only really compare them if I look at 15ths.

13. 5.3 – (cont’d) Subtraction: Ex: Show 5 1/3 – 3 1/2. First, show 5 1/3 (in blue) Next, subtract 3 (in green), and then find ½ (shown in red). When we take away 3 (green) and ½ (red), we are left with 1 5/6

14. 5.3 – (cont’d) Multiplication: Look at this in a different way. Let the black square = 1 whole. Can you see the area 3/4 5/7? The product is 15/28.

15. 5.3 – (cont’d) 3/4 5/7: Can you explain why this is less than 3/4? Can you explain why this is less than 5/7? 3/4 of a number is less than the original number, so 3/4 of 5/7 is less than 5/7. 5/7 of a number is less than the original number, so 5/7 of 3/4 is less than 3/4.

16. 5.3 – (cont’d) Division: In division of fractions, the ideas of partitioning and repeated subtraction can help us determine the meaning of a situation. Ex: Suppose I have 3 3/4 yards of material. If I want ribbons of length 5/8 yards, how many ribbons can I make?

17. 5.3 – (cont’d) 0 1 2 3

18. 5.3 – (cont’d) Ex: Suppose I order 6 pizzas for my pizza party, but the delivery guy eats part of my pizza order – he provides only 5 1/4 pizzas. I had planned for each person to eat 3/8 of a pizza. There are 15 people at the party. Do I have enough pizza?

19. 5.3 – (cont’d) Is 5 1/4 ÷ 3/8 ≥ 15 ?

20. 5.3 – (cont’d) Each different color of the dots represents one person. Continue on your own. (Is 5 1/4 ÷ 3/8 ≥ 15 ?)

21. Explorations Exploration 5.12 –Do Part 1 #1. Write a reason for each decision in a, b, c, and d. –Do Part 3. Exploration 5.13 –For each of the 7 “Problems” in Part 1, do the following (1) Write a mathematical expression for the solution. (2) Then, show one (or more) pictures that can help you find the answer. (3) Identify the answer for each problem based on your picture.

22. Homework Due Thursday, 4/9 Link to online homework list: http://math.arizona.edu/~varecka/302AhomeworkS09. htm *Bring Class Notes again on Thursday* Exam 3: Thursday, 4/16.