## Multiply and Conquer

• Lesson
3-5
2

Students decompose 2-digit numbers, model area representations using the distributive property and partial product arrays, and align paper-and-pencil calculations with the arrays. The lessons provide conceptual understanding of what occurs in a 2-digit multiplication problem. Partial product models serve as transitions to understanding the standard multiplication algorithm.

Prior to the lesson, students should understand and have practiced:

• Shading in multiplication arrays on 10 × 10 grids. If students struggle with this, you may want to use the Factorize Interactive as an opening activity.
Factorize Interactive
• Decomposing factors into addends, such as 47 = 40 + 7.
• Multiplying by powers of 10. For example:
•  40 × 300 = (4 × 10) × (3 × 100) = (4 × 3) × (10 × 100) = 12 × 1,000 = 12,000

2-digit × 1-digit Multiplication Arrays

Begin with 2-digit × 1-digit problems. Gauge your students' understanding in this activity to determine the pacing for the 2-digit × 2-digit activity.

Work through the example problem 14 × 6 to introduce the solution method. Before you begin showing students arrays, have them estimate the answer. This helps develop mental math skills and gives a benchmark to judge accuracy when they complete the problem. Since 14 is closer to 10 than 20, approximate the answer to be about 10 × 6 = 60. Since we rounded down, the actual answer will be a bit bigger than 60.

Model the partial product arrays for students using the 14 × 6 example. Model the following steps, using the illustration as a guide:

• Decompose the first factor into tens and ones to give (10 + 4) × 6. The problem can now be solved using the distributive property.
• Draw an array that is 14 × 6.
• Break the 14-unit horizontal line into a tens piece and ones piece. In this case, the pieces will be 10 and 4. Draw a vertical line through the rectangle at this point, creating two rectangles. Label the length and width of both rectangles. Label the 10 × 6 rectangle as 60, and the 4 × 6 rectangle as 24, as shown.
• The rectangles represent 2 partial products. Ask the students what to do with the 60 and 24. It is important that students know that partial means each array represents just part of the problem. They should also understand that to get the total product you need to add. The solution is 60 + 24 = 84.

As you model, have students participate by coming up to help build the model. Next, demonstrate solving another problem while students create their own base 10 block or paper grid models.

Have less capable students work with a partner. To transition to more individualized work, have individuals and pairs compare their work with that of others and describe their partial arrays aloud. Do as many problems as needed to support your variety of learners. Keep in mind that large 2-digit numbers get unwieldy to work with; numbers between 10 and 30 work the best for modeling.

2-digit × 2-digit Multiplication Arrays

Model the problem 13 × 16. Once students understand the process, they can work problems at your direction for continual practice. Important points for 2-digit by 2-digit multiplication include:

• The problem must be decomposed. It becomes (10 + 3) × (10 + 6).
• Draw an array that is 13 by 16.
• Decompose the factors by breaking the array into smaller pieces. Break the 13-unit horizontal line into 10- and 3-unit long pieces. Then, break the 16-unit vertical line into 10- and 6-unit long pieces.
• Draw lines to separate the tens from the ones (10 from 3 and 10 from 6). This creates 4 rectangles.
• Label the outside dimensions.
• Students should be able to see the 4 partial products of this problem: (10 × 10), (3 × 10), (6 × 10), and (3 × 6).
• Label the inside areas of the 4 partial products.
• Finally, add the partial products together: 100 + 30 + 60 + 18 = 208.

Pencil-and-Paper Calculating

After students show understanding and fluency with the models, show them how the models are reflected in a partial product method for recording multiplication. Showing the array model and the distributed recording method side by side helps students see the connection between the two.

Students use 2 × 1 Multiplication activity sheet and 2 × 2 Multiplication activity sheet. Each activity sheet has one example, workspace provided for additional problems, and an area to draw arrays. Teach students how to complete the tables using the examples already solved.

 2 × 1 Multiplication Activity Sheet 2 × 2 Multiplication Activity Sheet

Tips about using this multiplication method:

• Label the entire array model before you begin calculating.
• Perform your calculations beside the array model.
• While the decomposed factors can be multiplied in any order, direct students to use the order in the examples to begin. List the products in the order of the traditional algorithm.
• As you compute partial products, locate them on the array model to make the connection between the array and the calculations.

Assessments

1. If students work individually or in small groups with base  blocks or grids, watch how they create their models. This can help you determine if a student is having trouble with decomposing, drawing arrays, or making the connection between the conceptual model and the calculation.
2. Give students a real-life problem to solve using 2 × 1 or 2 × 2 multiplication.
3. Consider showing a labeled array and asking students to identify what multiplication problem it models. Ask students to create a word problem that is modeled by the array.

Extensions

1. This model of multiplication can be extended to 3-digit numbers although it becomes less efficient. Some students may enjoy the challenge of being given larger numbers to distribute and multiply. Three-digit numbers should be decomposed into hundreds, tens, and ones. Since three dimensions are required, they must be modeled with base ten blocks rather than paper grids.
2. Ask students to write journal entries comparing array models with the traditional algorithm. Ask them to determine what is happening when they use a regrouping method and how it works.
3. To practice the array models, play a game whose object is to color in as many squares as possible on a sheet of graph paper. Roll 3 dice, 2 of one color and 1 of another color. The same-color dice represent a 2-digit number and the other die represents a single-digit number. Students color in the partial product arrays. The first to correctly color in as much of their paper as possible is the winner. For an extra challenge, have students play in pairs, coloring the same grid. This adds strategy to the game.
4. Instead of giving students the factors to use in a model, have them discover the best strategy on their own. Students need to think flexibly about the factors as they develop strategies for multiplicative thinking. For example, ask students to compute 14 × 6, but let them discover other factors that work, such as 6 × (10 + 4), 6 × (9 + 5), 6 × (2 + 12), etc.Use questioning, discussions, and mathematical communication throughout the process.

Questions for Students

[In the case of a 2 × 1 problem, the single-digit factor must multiply both the tens and the ones digit of the 2-digit factor. That's why the model shows 2 arrays, which together represent the entire product. The explanation of 2 × 2 products is similar. Answers should show understanding of what happens when a number is decomposed.]

2. Why do we decompose a 2-digit number?

[It helps us arrive at a correct answer by simplifying the multiplication components of the problem into pieces that are easy to multiply. Reinforce for students that decomposing a number doesn't change its value.]

3. Is this method easier than the traditional way or harder?

[Answers will vary. Students generally find that multiplying by multiples of ten and single-digit numbers is easier to grasp because they know the basic multiplication facts. They are often able to complete the problems using mental math.]

4. What do the partial products show?

[They show the result of distributing over decomposed factors.]

Teacher Reflection

• Were students able to transfer previous knowledge of making arrays and the distributive property to this activity? Was that a benefit to new learning, or did problems arise as students worked?
• Do you think this way of recording a multiplication problem makes better sense to some or all of your students?
• Was it a challenge that some students had previously learned a different way to record a multi-digit multiplication problem?
• What are the benefits for students in using a partial product model?

### Factorize

3-5, 6-8
Dividing Numbers into Two Factors and Building Arrays to Represent Each Factorization

### Learning Objectives

Students will:

• Create and solve multiplication stories.
• Practice selected multiplication facts.

### Common Core State Standards – Mathematics

• CCSS.Math.Content.3.OA.A.1
Interpret products of whole numbers, e.g., interpret 5 x 7 as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as 5 x 7.

• CCSS.Math.Content.3.OA.B.5
Apply properties of operations as strategies to multiply and divide. Examples: If 6 x 4 = 24 is known, then 4 x 6 = 24 is also known. (Commutative property of multiplication.) 3 x 5 x 2 can be found by 3 x 5 = 15, then 15 x 2 = 30, or by 5 x 2 = 10, then 3 x 10 = 30. (Associative property of multiplication.) Knowing that 8 x 5 = 40 and 8 x 2 = 16, one can find 8 x 7 as 8 x (5 + 2) = (8 x 5) + (8 x 2) = 40 + 16 = 56. (Distributive property.)

• CCSS.Math.Content.3.OA.C.7
Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 x 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.

Grade 4, Num & Ops Base Ten

• CCSS.Math.Content.4.NBT.B.5
Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

• CCSS.Math.Content.5.OA.A.1
Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.

Grade 5, Num & Ops Base Ten

• CCSS.Math.Content.5.NBT.B.5
Fluently multiply multi-digit whole numbers using the standard algorithm.

### Common Core State Standards – Practice

• CCSS.Math.Practice.MP1
Make sense of problems and persevere in solving them.
• CCSS.Math.Practice.MP4
Model with mathematics.
• CCSS.Math.Practice.MP5
Use appropriate tools strategically.
• CCSS.Math.Practice.MP7
Look for and make use of structure.