To read past book study posts, visit our Number Talks Book Study Archive!
Chapter 7: How Do I Develop Specific Multiplication and Division Strategies in the 3-5 Classroom?
As in the beginning of chapter 5, Parrish presents the overreaching goals for number talks at this level: number sense, place value, fluency, properties, and connecting mathematical ideas. For a discussion of the importance of each at the 3-5 level, visit my previous post of chapter 5. Accuracy, efficiency, and flexibility are always encouraged with number talks as well.
Parrish goes on to stress the importance of using array models to anchor student strategies with multiplication and division. She compares the importance of the array model with multiplication and division to the importance of the number line to addition and subtraction. If you are not currently using arrays to help students understand the concepts/strategies associated with multiplication and division, it is important to make this shift. With an array, students can apply their understanding of the factors/dimensions of an array to find the product/area. See array for 6 x 22 below.
From an array with boxes delineated within the area, students can move to the use of an open array. Like an open number line, and open array can be customized to model thinking/strategies. Below is an example of 6 x 22 and 132/6 using an open array.
The remainder of chapter 7 illustrated five common strategies for multiplication and four common strategies for division. I will give examples of many of these strategies as I discuss chapter 8.
Chapter 8: How Do I Design Purposeful Multiplication and Division Number Talks in the 3-5 Classroom?
The number talks presented in this chapter are organized by operations and strategies. A rationale for helping students develop each strategy is presented and specific instructions are given for their implementation.
Even if you are not using number talks, this chapter does an exceptional job of illustrating essential strategies for multiplication and division.
First, Parrish stresses the importance of using number talks that focus on fluency with small numbers BEFORE moving on to using those that focus on computation with greater numbers. Number talks with small numbers help students focus on strategies rather than the magnitude of numbers and foster confidence. In this chapter, specific number talks are presented to bring about the use of specific strategies, but it is also understood that students will share other methods. "The ultimate goal of number talks is for students to compute accurately, efficiently, and flexibly."
Multiplication Number Talks
Repeated Addition or Skip Counting: Specific number talks are not presented for this strategy because the goal of number talks is to move students beyond additive thinking to multiplicative thinking. Praise students for using this type of thinking, but don't forget to make a connection to multiplication.
Making Landmark or "Friendly " Numbers: It is important to remember here that if an adjustment is made to one of the factors, an adjustment must also be made to the product.
Examples:
6 x 21
6 x 20 = 120
120 + 6 = 126
3 x 19
3 x 20 = 60
60 - 3 = 57
Partial Products: This strategy requires the breaking up of one or both factors into addends using expanded notation and the distributive property. This strategy can be used with any multiplication problem.
Examples:
6 x 23
6 x (20 + 3)
6 x 20 = 120
6 x 3 = 18
120 + 18 = 138
6 x 31
(3 + 3) x 31
(3 x 31) + (3 x 31)
93 + 93 = 186
Doubling and Halving: This strategy can be used to make problems with multiple digits easier to solve.
Example:
8 x 35
8/2 = 4
35 x 2 = 70
4 x 70 = 280
Breaking Factors Into Smaller Factors: It is important to expose students to number talks that lead to the use of this strategy to help students understand the associative property.
Examples:
6 x 21
3 x 2 x 7 x 3
32 x 8
4 x 8 x 2 x 4
Division Number Talks
Repeated Subtraction or Sharing/Dealing Out: Specific number talks are not presented for this strategy because the goal of number talks is to move students beyond removal to multiplicative thinking. Praise students for using this type of thinking, but don't forget to make a connection to multiplication.
Partial Quotients: When the partial quotient strategy is used, students are able to understand the value of each digit in a number being divided. No longer is there the "goes intos" thinking based on single digits without an understanding of each digit's value. When I taught fifth grade, this strategy along with the use of base ten tools helped students understand the concept of addition and set aside a series of memorized steps they had previously used.
Here is a great post by Tara, the Elementary Math Maniac, all about teaching the partial quotient strategy--Teaching Division with Partial Quotients: Moving from Concrete to Abstract Models.
Multiplying Up: Students build upon multiplication they know until they reach the dividend.
Example:
12 x 35
12 x 10 = 120
12 x 10 = 120
12 x 10 = 120
12 x 2 = 24
12 x 2 = 24
12 x 1 = 12
12 x 35 = 420
Proportional Reasoning: Division is considered from a fractional perspective. Halving and halving or thirding and thirding can be explored with the number talks included. Some number talks include: 800/40 and 144/6.
I hope you find the overview of each strategy helpful, if you have not purchased a copy of the book. I highly recommend you do!
Please feel free to share your comments, ideas, or experiences related to chapters 6 and 7. We would love to hear your thoughts!
AND don't forget to stop by this coming Sunday for our Makin' It Math mid month linky!
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