Session 2, Document A: The Big Ideas and Properties in Multiplication

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1 Session 2, Document A: The Big Ideas and Properties in Multiplication In groups, respond to one of the prompts below. Your group should: 1) Read the prompt you are assigned. 2) Read the sections of the handout that are related to the prompt. 3) Discuss and capture your best thinking on a piece of chart paper so that it can stand alone for others to interpret. Include a model in your response. PROMPT 1 Explain what unitizing is and why is it critical to students acquisition of multiplicative reasoning. Give some specific examples to illustrate unitizing. Where else in mathematics do we need unitizing? Include an illustration on how skip counting requires unitizing and is a higher- level strategy than repeated addition. PROMPT 2 Explain why arrays are a big idea in multiplication and across mathematics. Give specific examples. Include in your explanation some attention to the struggles students have as they move through stages of understanding arrays. PROMPT 3 Read the explanation about the big idea of compensation in multiplication. Compare and then illustrate the similarities and differences between compensation in addition and compensation in multiplication using specific examples. Include the potential misconceptions or difficulties that students may have moving from compensation with addition to compensation with multiplication. PROMPT 4 Use a 29 x 37 area to model the distributive property of multiplication and write a story problem that helps to illustrate your model. Write an equation that illustrates the distributive property and matches your model. PROMPT 5 Why is multiplying by 10 not just adding a 0? Explain what the underlying mathematics is to this trick. Apply this understanding to our place value system. Why is it important for students to understand the trick? PROMPT 6 Construct a visual model that could be used to help students understand the associative property of multiplication. Be sure to include mathematical notation that helps to make sense of your model. 1

2 Session 2, Document B: The Big Ideas Definitions Properties of Multiplication: There are four properties involving multiplication that will help make problems easier to solve. They are the commutative, associative, multiplicative identity and distributive properties. Commutative property: When two numbers are multiplied together, the product is the same regardless of the order of the multiplicands. For example 4 * 2 = 2 * 4 Associative Property: When three or more numbers are multiplied, the product is the same regardless of the grouping of the factors. For example (2 * 3) * 4 = 2 * (3 * 4) Multiplicative Identity Property: The product of any number and one is that number. For example 5 * 1 = 5. Distributive property: The sum of two numbers times a third number is equal to the sum of each addend times the third number. For example 4 * (6 + 3) = 4*6 + 4*3 Compensation: Multiplicative compensation involves adjusting the numbers in a problem without altering the solution e.g., 8 x 25 = 4 x 50; 6 x 4 = 6 x x 1 Compensation methods in multiplication use a product relates to the original. A compensation is made to the answer, or one factor is changed to compensate for a change in the other factor. (J. Van de Walle, Elementary and Middle School Mathematics, 8 th edition, 2012) Additive compensation is often initially introduced to students when considering the combinations to 10. Teachers will begin by showing that is the same as 9 + 1, 8 + 2, 7 + 3, etc and have students observe the pattern. This is an example of compensation by 1. Modeling this relationship can be very powerful for building understanding of compensation and increase the likelihood that the strategy will be accessible later on when students can use it to flexibly add numbers such as In these examples, the total has not been changed but the student has made a flexible adjustment of the numbers. (J. Van de Walle, Elementary and Middle School Mathematics, 8 th edition, 2012) Unitizing: Counting by equal groups e.g., Determining the number of wheels on 5 cars can be seen as counting by ones to 20: 1, 2, 3, 4, 5 20 OR by unitizing 5 cars each with 4 wheels: The student might say I have 5 cars with 4 wheels each and hold up 5 fingers and count 4, 8, 12, 16, 20 there are 20. Arrays Objects arranged in equal rows 1

3 Session 2, Document B: The Big Ideas Definitions Arrays can be describes as spatial structuring. When a number of second graders were interviewed it was found that children go through a series of four stages as they develop the ability to coordinate rows and columns. Initially, students structure arrays as one- dimensional paths. Asked to figure out how many squares would fill a 3 x 7 array, at the lowest stage a child fills the borders first, working along a unidirectional path. The second stage is characterized by structuring one of the dimensions (rows or columns), but not both. This might look like a student talking about the repeated addition in the rows but simultaneously not being able to consider the repeated addition in the columns or how many rows are there. In the third stage, students can use the square units as indicators of how many rows and how many columns but they struggle to understand how one square can belong to both a row and a column. Not until stage 4 can students consider what it truly means to have that square in a row and a column. This requires an understanding of the attributes of the square and its relationship to the array. This idea is a central organizing principal of mathematics. (Michael Battista et al. (1998). Students Spatial Structuring of Two- Dimensional Arrays of Squares. Journal for Research in Mathematics Education 29(5): ) Open area: A model to represent a solution for multiplication linked to the distributive property. This is a model that generalizes to many kinds of numbers and can be used to make sense of the relationship between two factors and the product in a multiplication equation as well as the relationship to division and the numbers in a division problem. When considering modeling 3 factors, as in the associative property, one might imagine stacking areas on top of each other, or using a cube to model the three dimensions (factors). Subitizing, coined in 1949 by E.L. Kaufman et al. [1] refers to the rapid, accurate, and confident judgments of number performed for small numbers of items. The term is derived from the Latin adjective subitus (meaning sudden) and captures a feeling of 2

4 Session 2, Document B: The Big Ideas Definitions immediately knowing how many items lie within the visual scene, when the number of items present falls within the subitizing range. [1] The accuracy, speed, and confidence with which observers make judgments of the number of items are critically dependent on the number of elements to be enumerated. Judgments made for displays composed of around one to four items are rapid [2], accurate [3] and confident. [4] However, as the number of items to be enumerated increases beyond this amount, judgments are made with decreasing accuracy and confidence. [1] In addition, response times rise in a dramatic fashion, with an extra 250 ms 350 ms added for each additional item within the display beyond about four. [5] Powers of Ten: The special role of 10 in the base- ten system is important in understanding multiplication of one- digit numbers with multiples of 10. For example, the product 3 x 50 can be represented as 3 groups of 5 tens, which is 15 tens, which is 150. This reasoning relies on the associative property of multiplication: 3 x 50 = 3 x (5 x 10 ) = (3 x 5) x 10 = 15 x 10 = 150. (CCSS progressions for grade 3 (NBT)). 3

5 Session 4, Document A: Problem Structures 1) The table below shows the cost of lollipops at Brian s Candy Store. Number of Lollypops Cost 3 $.15 4 $.20 5 $.25 6 $.30 7 $.35 8 $.40 A. What is the cost of 9 lollipops at Brian s Candy Store? Show your work. B. Write a rule to find out the cost of 10 lollipops at Brian s Candy Store. 2) Abby has 8 quarts of ginger ale. She plans to use it all to make the punch recipe below. How many gallons of juice does Abby need to buy? Fruit Punch Recipe 3 quarts juice 1 quart ginger ale 4 scoops sherbet 4 quarts = 1 gallon 1

6 Session 4, Document A: Problem Structures 3) Mr. Jones ordered office supplies. He ordered 7 cases of paper. There are 10 packages of paper per case. Each package contains 500 sheets of paper. How many sheets of paper did he order? Show your work. 4) Ms. Smith s class keeps track of the number of pages they read each week. Below is a chart with some missing information. Students Pages Read Mike 54 Tina Luke 39 Cara 27 Seth Seth read 4 times more pages than Cara. How many pages did Seth read? 2

7 Session 4, Document A: Problem Structures 5) It takes 14 inches of ribbon to make one bow. How many inches of ribbon will it take to make 7 bows? 6) The price of gas in 1960 was $.19 per gallon. Today the price is 14 times more than in What is the price of gas per gallon today? 3

8 Session 4, Document A: Problem Structures 7) How many millimeters are in centimeters? 10 millimeters = 1 centimeter 8) During migration, humpback whales swim about 6 miles per hour. After 25 hours, about how far have they traveled? 4

9 Session 4, Document A: Problem Structures 9) A. Mark bought 12 boxes of crayons. Each box contained 8 crayons. How many crayons were there all together? Show your work. B. John bought 12 boxes of crayons. Each box contained 64 crayons. How many crayons were there all together? Show your work. 5

10 Session 4, Document B: Problem Structures Step 1: With a partner sort the problems into three categories 1) Easiest; 2) Moderate difficulty; and, 3) Most challenging Step2: Make notes of features of the problems that would make them more or less challenging for students. Easiest Moderate Most Challenging List Problems: List Problems: List Problems: Features of problems: Features of problems: Features of problems: 1

11 Session 6, Document A: Algorithms and the Open Area Model 1) Solve the following problem using an open area model, partial products, and the traditional algorithm. 27 x 38 = 2) Explain how the partial products algorithm is related to the area model. 3) Explain how the traditional algorithm is related to the area model. 4) Explain how the open area model is related to multiplying algebraic expressions like (2x + 3)(3x + 4) =. 1

12 Session 12: Division Write two different word problems for the following: = Select one of your problems. Solve the problem 2 different ways without using the traditional algorithm. Consider the context when selecting your strategy for solving. How will you deal with the remainder? by NSF EHR and the US DOE (S366A20002) Marge Petit Consulting, E. Hulbert, & R. Laird 1

13 Session 12: Division Partitive/Quotative Sort E15 Ann and Billy baked 60 cookies for the bake sale. They put them on 5 plates. They wanted to know how many cookies to put on each plate. Ann used this number sentence: 5 x = 60 Billy used this number sentence: 60 5 = Explain why both these number sentences will work. 1

14 Session 12: Division Partitive/Quotative Sort EG4 There are 96 octopus legs in the aquarium. An octopus has 8 legs. How many octopuses are there in the aquarium? Show your work. 2

15 Session 12: Division Partitive/Quotative Sort EG38 There are 85 students in the cafeteria. Each table seats 12 students. How many tables are there? 3

16 Session 12: Division Partitive/Quotative Sort EG42 Mrs. Thompson ordered 144 pencils. How many dozen pencils did she order? Show your work. 4

17 Session 12: Division Partitive/Quotative Sort EG43 John had 30 stars. He put them in 6 rows. How many stars were in each row? 5

18 Session 12: Division Partitive/Quotative Sort EG45 There are 48 raisins and 8 children. How many raisins will each child get? 6

19 Session 12: Division Partitive/Quotative Sort EM1 You have 45 ounces of apple juice. The apple juice comes in bottles that contain 15 ounces. How many bottles of juice do you have? Show your work. 7

20 Session 12: Division Partitive/Quotative Sort EM2 3 children equally share 45 ounces of apple juice. How much juice will each child get? Show your work. 8