1 Sara offered to bring 200 cookies to a block party. She plans to take the very popular Nestle Toll House Chocolate Chip cookies. Sara has only 7 ½ cups of flour and no fresh eggs. She will be using powdered egg substitute for the eggs. Convert the cookie recipe to yield 200 cookies. Nestle Toll House Cookies—Yield 60 cookies 2 1/4 cups all-purpose flour 1 teaspoon baking soda 1 teaspoon salt 1 cup (2 sticks) butter, softened 3/4 cup granulated sugar 3/4 cup packed brown sugar 1 teaspoon vanilla extract 4 Tbsp egg powder 4 Tbsp warm water 2 cups (12-oz. pkg.) NESTLÉ® TOLL HOUSE® Semi-Sweet Chocolate Morsels 1 cup chopped nuts

2 To convert the recipe we must determine the scaling factor. Since the recipe is being scaled up (yield is greater than original recipe) this scaling factor should be a number greater than 1. The quantity of each ingredient in the recipe must be multiplied by the scaling factor. Scaling Factor: 200/60 = 10/3 Nestle Toll House Cookies (2 1/4 ∙ 10/3) OR 7 1/2 cups all-purpose flour (1 ∙ 10/3) OR 3 1/3 teaspoon baking soda (1 ∙ 10/3) OR 3 1/3 teaspoon salt (1 ∙ 10/3) OR 3 1/3 cup butter, softened (3/4 ∙ 10/3) OR 2 1/2 cup granulated sugar (3/4 ∙ 10/3) OR 2 1/2 cup packed brown sugar (1 ∙ 10/3) OR 3 1/3 teaspoon vanilla extract (4 ∙ 10/3) OR 13 1/3 Tbsp (3/4 cup and 1 1/3 Tbsp) egg powder (4 ∙ 10/3) OR 13 1/3 Tbsp (3/4 cup and 1 1/3 Tbsp) warm water (2 ∙ 10/3) OR 6 2/3 cups NESTLÉ® TOLL HOUSE® Semi-Sweet Chocolate MorselsNESTLÉ® TOLL HOUSE® Semi-Sweet Chocolate Morsels (1 ∙ 10/3) OR 3 1/3 cup chopped nuts

3 Peter's truck gets him 10 2/3 miles per gallon. Suppose Peter's tank is empty and he puts 5 1/2 gallons, how far can Peter go in the truck?

4 Since Peter put 5 1/2 gallons in his truck that gets 10 2/3 miles per gallon it is necessary to ________ (multiply) these values to find how far Peter can drive with that quantity of gas. (10 2/3 miles/gallon) (5 1/2 gallon) = (____ miles/gallon)∙(____ gallon) = (____)∙(____) miles = (____)/(____) miles = _____ miles Peter can travel _______ miles on ______ gallons of gas.

5 Solution: Since Peter put 5 1/2 gallons in his truck that gets 10 2/3 miles per gallon it is necessary to multiply these values to find how far Peter can drive with that quantity of gas. (10 2/3 miles/gallon) (5 1/2 gallon) = (32/3 miles/gallon) (11/2 gallon) = (16/3) (11/1) miles = (16∙11)/(3∙1) miles = 352/3 miles = 117 1/3 miles Peter can travel 117 1/3 miles on 10 2/3 gallons of gas.

6 Assignment 4.1: Identify and describe several additional instances of the problem or task that are of increasing complexity. Arrange these instances into a progression of increasing complexity. 1. Express an understanding of the process of multiplication of fractions. 2.Perform multiplication of proper fractions and express the product in the appropriate simplified form. 3.Perform multiplication of improper fractions and express the product in the appropriate simplified form. 4.Perform multiplication of mixed numbers and express the product in the appropriate simplified form. 5. Identify appropriate contexts for multiplication of fractions.

7 Assignment 4.2: Design and develop a prototype demonstration or application for at least 3 of these additional instances of the problem or task. Post your PPT to your wiki. After revision, post a notice to the discussion board so that other class members can review your prototype problem progression demonstration/applications. I will start by focusing on these tasks: 1. Express an understanding of the process of multiplication of fractions. 2.Perform multiplication of proper fractions and express the product in the appropriate simplified form. 3.Perform multiplication of improper fractions and express the product in the appropriate simplified form. Demonstration follows…

8 Multiplication of Fractions—Task 1 What fraction of the rectangle is shaded green? Express your answer as an unsimplified fraction. Click on the correct response. A. 1/2B. 2/1 C. 12/6D. 6/12 Note: Appropriate feedback will be provided based on the Learner’s response. The feedback will provide supplemental instruction, if needed.

9 What fraction of the rectangle is shaded brown? Express your answer as an unsimplified fraction. Click on the correct response. A. 3/12B. 4/1 C. 12/3D. 1/4 Note: Appropriate feedback will be provided based on the Learner’s response. The feedback will provide supplemental instruction, if needed. Multiplication of Fractions—Task 1

10 The standard algorithm for multiplication of fractions teaches us to multiply as illustrated: 2/3 ∙ 5/12 = (2∙5) / (3∙12) = 6 / 60 Why?

11 Multiplication of Fractions—Task 1 Visualize the process of multiplying 2/3 ∙ 5/12 by thinking or verbalizing the process as “Two of three equal parts of 5/12.” 1.Create a representation of 5/12. Click on “Original Fraction.” Original Fraction Note the rectangle consists of 12 parts of equal size. Five of those parts are shaded representing the fraction 5/12.

12 Multiplication of Fractions—Task 1 Visualize the process of multiplying 2/3 ∙ 5/12 by thinking of process as “Two of three equal parts of 5/12.” 2. Find 2 of 3 equal parts of 5/12. This requires dividing each of the 12ths in the original fraction into 3 equal parts creating 36 parts of equal size. (Click New Representation). New Representation

13 Multiplication of Fractions—Task 1 Visualize the process of multiplying 2/3 ∙ 5/12 by thinking of process as “Two of three equal parts of 5/12.” 3. Now that each of the original parts has been divided into three parts of equal size, select only two of the parts. The two of the three equal parts must be located in the region that was originally shaded. Click on Product Fraction to view this. Click on Answer to view how the answer is obtained. Original Fraction Product Fraction Answer Answer: 10/36 There are 10 parts shaded green and a total of 36 parts.

14 Multiplication of Fractions—Task 1 Now It’s Your Turn: Which of the following is an appropriate interpretation of multiplying 3/5 ∙ 9/14? a) Three of five equal parts of 9/14. b) Five of three equal parts of 9/14. c) Nine of fourteen equal parts of 3/5. d) Fourteen of nine equal parts of 3/5. Appropriate feedback to be inserted based on Learners response.

15 Multiplication of Fractions—Task 1 Now It’s Your Turn: 1.How would you verbalize the process of multiplying 3/4 ∙ 7/10 ? __________________________________ 2. Submit a representation (picture) of the fraction 7/ Submit an explanation and illustration of how to find the product 3/4 ∙ 7/10.

16 It is a standard practice to express fractions in their simplified form—meaning the same shaded area is represented using a fewest possible number of parts; thus, the simplified fraction is equivalent to the original fraction. The shaded region represents the fraction 20/32. Which is 20/32 in simplified form? Click on answer. Multiplication of Fractions—Task 2 Appropriate feedback to be inserted based on Learners response.

17 The fraction 20/32 is equivalent to 5/8. It is not very efficient to make sketches. It is more efficient to devise a system for simplifying fractions. Compare the fractions 20/32 and 5/8. Each part in the fraction 5/8 can be formed by combining 4 parts in the fraction 20/32. To simplify 20/32 divide both the numerator (shaded parts) and denominator (total parts) by 4. This creates the 1/8 th size parts in the simplified fraction. (20÷4)/ (32÷4) = 5/8 Multiplication of Fractions—Task 2

18 When simplifying the fraction 20/32 The 20 shaded parts in the fraction formed groups of 4. The 32 total parts in the fraction formed groups of 4. Examine the fraction represented below and determine the largest size groups that both the shaded parts can form and the total parts can form. Click on your answer. Multiplication of Fractions—Task 2 Appropriate feedback to be inserted based on Learners response. a) 2 b) 3 c) 4

19 The figure below represents the fraction 9/12. The parts in the shaded region can form groups of 3. The parts of entire figure can form groups of 3. The simplified form of this fraction is found by dividing both numerator and denominator by 3. A very important observation at this point is that 3 is the Greatest Common Factor of 9 and 12. The GCF(9,12) is the number used to divide both the numerator and denominator. What is the simplified form of 9/12? _______ (Answer: 3/4) Multiplication of Fractions—Task 2 Appropriate feedback to be inserted based on Learners response.

20 The figure below represents the fraction 16/24. What is the GCF(16, 24)? Multiplication of Fractions—Task 2 Appropriate feedback to be inserted based on Learners response. a) 2b) 4 c) 6d) 8

21 The figure below represents the fraction 16/24. The GCF(16, 24) = 8. To simplify 16/24 divide both the numerator and denominator by 8. Click on your answer. Multiplication of Fractions—Task 2 Appropriate feedback to be inserted based on Learners response. a) 2/3 b) 3/2 c) 4/6d) 8/12

22 Which of the following fractions simplify to 3/8? Click on your answer(s). Multiplication of Fractions—Task 2 Appropriate feedback to be inserted based on Learners response. ○ 12/32 ○ 15/40 ○ 9/24 ○ 6/18

23 Find the product of 2/3 and 6/11. (Hint: The product of two numbers is the answer to a multiplication problem. That is, (2 ∙ 6)/(3 ∙ 11).) Click on your answer. Multiplication of Fractions—Task 2 Appropriate feedback to be inserted based on Learners response. ○ 12/33 ○ 22/18 ○ 4/11 ○ 11/9

24 Which of the following products simplify to 1/4? Click on your answer. Multiplication of Fractions—Task 2 Appropriate feedback to be inserted based on Learners response. ○ 3/6 ∙ 4/8 ○ 1/2 ∙ 3/6 ○ 4/8 ∙ 1/2 ○ 1/4 ∙ 1/4

25 Recall that improper fractions are fraction in which the numerator as greater than or equal to the denominator. The process for multiplying improper fractions is the same as for proper fractions. The product should be written in simplified form. ______________________________________________________ Find 3/7 ∙ 11/6. Click on your answer. Multiplication of Fractions—Task 3 Appropriate feedback to be inserted based on Learners response. a) 33/42 b) 18/77 c) 11/14d) 11/21

26 Which of the following products simplify to the fraction 2/3? Click on your answer(s). Multiplication of Fractions—Task 3 Appropriate feedback to be inserted based on Learners response. a) 12/6 ∙ 1/3 b) 6/4 ∙ 2/3 c) 6/3 ∙ 1/3 d) 5/6 ∙ 4/5