1. CHAPTER 9 Meanings for Operations (Continued)

2. MULTIPLICATIVE STRUCTURES  There are four types of multiplicative structures  Equal Groups  Comparison  Combination  Area and Other Product-of-Measures

3. MULTIPLICATIVE STRUCTURES  Equal Groups: Whole Unknown (Multiplication)  Repeated addition multiplication should be thought of as m x n = n + n + … + n (m times). Note that this is a convention. Some other cultures reverse the order. This is the U.S. convention.  Ex: John puts 6 marbles into a bag. If he has 5 bags, how many marbles does John have in all?  Rate problems are similar to repeated addition except there is a ratio of units involved.  Ex: Sally ran 14 miles per hour for 3 hours. How far did she run?  Questions:  What is the computational equation here?  Draw a model.

4. MULTIPLICATIVE STRUCTURES  Equal Groups: Size of Groups Unknown (Partition Division)  These problems refer to situations where the size of the set is unknown, but the number of sets is known.  Note that sometimes people call this partitive division.  Fair sharing  Ex: Katie has 24 golden retrievers. She wants to share them equally among 8 friends. How many dogs does each friend receive?  Rate  Ex: Sally paid 42 cents for 6 bananas. How much did each banana cost?  Ex: If a car drives 180 miles in 3 hours. How many miles per hour did the car travel?  Questions:  What is the computational equation here?  Draw a model.

5. MULTIPLICATIVE STRUCTURES  Equal Groups: Number of Groups Unknown (Measurement Division)  These problems refer to situations where the number of sets is unknown, but the size of each set is known.  Sometimes this type is referred to as repeated subtraction.  Repeated Subtraction  Ex: Suppose that Owen swims 200 m in a 25 meter pool. How many laps did she swim?  Rate  Ex: Sarah bought some bananas that were 7 cents each. If the total cost of the bananas was 56 cents, how many bananas did she buy?  Ex: Suppose that Bobby walked 12 miles at a rate of 4 miles per hour. How many hours did it take Bobby to walk the 12 miles?  Questions:  What is the computational equation here?  Draw a model.

6. MULTIPLICATIVE STRUCTURES  Comparison: These problems refer to a situation where the comparison is based on one set being a particular multiple of the other.  Comparison: Product Unknown (Multiplication)  Ex: Bobby has 3 toy cars. Sue has 5 times as many cars. How many car does Sue have?  Ex: Anton ran 4 times as many miles this month than last. Last month, he ran 3 miles. How many did he run this month?  Comparison: Set Size Unknown (Partition Division)  Ex: Bobby has 15 toy cars and has 5 times as many as Sue. How many cars does Sue have?  Ex: Anton ran 12 miles this month. This month, he ran 4 times the amount of miles as he did last month. How many miles did he run last month?  Comparison: Multiplier Unknown (Measurement Division)  Ex: Bobby has 12 toy cars and Sue has 3. How many times as many cars does Bobby have than Sue?  Ex: Anton ran 12 miles this month and 3 miles last month. How many times as many miles did he run this month compared to last month?  Questions:  What is the computational equation here?  Draw a model.

7. HOMEWORK PRACTICE  In the following problems, draw a model, determine the type of multiplicative structure, write an appropriate computational equation and determine the answer. In your equations, leave the unknown as an empty box. For the division situations, also give the correct multiplication equation. Make sure your model demonstrates to me that you understand the relevant multiplicative structure. Note that you should be doing these without calculators.  The school district bought 84 kg of cheese. The cheese comes in boxes of 12 kg each. How many boxes did the school district buy?  Keeba collects stamps and has 28 stamps in her collection. Bronwyn is amazed since Keeba has 7 times as many stamps as herself. How many stamps does Bronwyn have?  Suppose a serving of chicken noodle soup is 3 cups. How many cups is 9 servings?  The school district used 81 kg of cheese in 3 weeks. What was the average number of kg of cheese used each week?  Anne types 88 words per minute and Billy types 22 words per minute. How many times as many words does Anne type than Billy?  In the following video, state the strategy the child uses to solve the problem and explain how she modeled her answer using physical modeling, i.e., manipulatives. You should be able to right click the link, select hyperlink and then select open hyperlink.  http://mediaplayer.pearsoncmg.com/_blue- top_640x360_ccv2/ab/streaming/myeducationlab/mathmethods/IMAP_Nicole3_23_iPad.mp4 http://mediaplayer.pearsoncmg.com/_blue- top_640x360_ccv2/ab/streaming/myeducationlab/mathmethods/IMAP_Nicole3_23_iPad.mp4

8. MULTIPLICATIVE STRUCTURES  Combinations  Combination problems involve counting the number of possible pairings that can be made between two or more sets.  Ex: Suppose Sally can choose a cup or a cone, three flavors of ice cream (chocolate, vanilla, strawberry), and three toppings (hot fudge, nuts, oreos). How many different ice cream treats can she make if she choose one container, one flavor of ice cream, and one topping?  Ex: Suppose you flip a fair coin and roll a six-sided die. How many different possible outcomes or results can this experiment have?  Area and Other Product of Measures Problems  This category involves problems where the unit represented by the product is literally a different unit from its factors.  Ex: Suppose you are carpeting a room that measures 8 feet in height and 12 feet in length. The product is the area of the carpet and measures 96 square feet. [Children will often assume that the area is 96 feet. Drawing models can help.]  Ex: Suppose that you have 7 employees each of which work 40 hours per week. Then your workers comprise 280 worker-hours.

9. REMAINDERS  In the real world, division does not often result in a whole number. The natural consideration is how does one deal with these extra numbers?  Left over/Discarded: There are situations where one simply has extra values.  Ex: You have 30 golden retrievers to share equally among 8 friends. How many golden retrievers does each friend get?  Ex: You have 25 feet of rope to make jump ropes. How many 7-foot jump ropes can you make?  Force to Next Highest/Rounding: There are situations where you have to consider the next highest whole number.  Ex: The roller coaster holds 12 people. How many times must it run to allow 30 people to ride?  Partitioned Into Fractions: There are situations where the left over forms a fraction.  Ex: Each bottle hold 8 ounces of juice. If there are 44 ounces of juice in a pitcher, how many bottles will that be?

10. OTHER PROPERTIES  Commutative Property  Associative Property  Zero and Identify Properties  Distributive Property  Division by Zero?