## Getting the (Decimal) Point with Blocks: Multiplying Two Decimals Using Base Ten Blocks

• Lesson
3-5
2

Students will use base ten blocks to model decimal multiplication. They will assign different values to the different base ten blocks to explore the consistent relationship between the types of blocks. They will also discover different factors for the same product. These activities will help students develop a conceptual understanding of decimal multiplication.

Before the beginning of class, make enough copies of the Multiplication Model Activity Sheet for each student. Also, a copy of the Decimal Multiplication Overheads will be necessary for the teacher.

Group students in pairs and distribute base ten blocks to each group. Tell students they will be using base ten blocks to model decimal multiplication. They will begin by modeling multiplication of two whole numbers: 34 × 23. Ask students to first make the number 34 using the traditional representation of a rod as "10" and a unit as "1" and to arrange their formation on the left of their Multiplication Model Activity Sheet. Be sure to tell students that all the rods should be together and all the units should be together (instead of alternating). Show Figure 1 Overhead to demonstrate this.

Next, have students construct the number 23 on the same array, placing the blocks above the top line this time. Tell students that if their rods representing 34 are at the top of the left side, then the rods representing 23 should be on the left side of the top row; if the rods for 34 are at the bottom of the left side, then the rods for 23 should be on the right side of the top row. Point to the spots on the overhead, so students are better able to visualize this. Ask students to display the rods first, and then, display the units (to be consistent with place value). As students complete this step, display OH- Figure 2 on the overhead projector.

The next step is to complete the area model by filling in the rest of the model, using blocks that match the length of the blocks that are on the left and at the top. After students complete this step, ask a student to share their model with the class by sketching what they did on OH-Figure 2.

After completing this model, tell students it is time to determine the product of this multiplication problem. Sample guidance: First, count the number of units. Because there are 12 of them, you are able to combine 10 of them together to make a rod. This leaves two units. Now count the number of rods. There are 18 rods, including the one you made by combining 10 units. Ten rods are equal to one flat. Therefore, combine 10 of the 18 rods, and exchange them for a flat. This leaves eight rods. Now count the number of flats. Including the one that you just made from the 10 rods, there are 7 flats. The 7 flats plus the 8 rods plus the 2 units is equal to 782, therefore, 34 × 23 = 782.

Depending on the students' work, you may choose to create additional examples.

Ask students to think about changing the value of the units, rods, and flats. Ask, "Suppose each flat was worth 1 whole, what would each rod be worth?" [1/10.] Using the same figures as before, show Figure 1 and ask, "What is the value of the blocks on the left now?" [3.4.] Show Figure 2 and ask, "What is the new value of the top row?" [2.3.] Then show Figure 3 again, and have students determine the value of the base ten blocks using the new values. As students are computing, ask, "What is the value of each single unit?" [1/100.] As students are working, remind them to think about the relationship between the initial value of a flat and the new value of a flat.

As students finish, ask them what the product is using the new values. [7.82.] Ask, "How does this value compare with the product of 34 times 23?" [It is 1/100 the value.] For advanced students, you may also ask, "Why is the value of 3.4 × 2.3 equal to 1/100 the value of 34 × 23?" [Because both numbers are equal to 1/10 the previous value, and 1/10 × 1/10 = 1/100.]

Begin the next class period with a review of the previous lesson. You can check for understanding using a similar problem, with different digits: "How can you find the value of 6.1 × 7.2?" [You know that 6.1 is 1/10 of 61 and 7.2 is 1/10 of 72. The product of these two decimals is 1/100 the product of the whole numbers. Because 61 × 72 is 4,392, the product of 6.1 × 7.2 is 4,392/100, or 43.92.]

Then show OH- Figure 3 on the overhead projector. Using the value of 1 flat = 1 whole, ask students to determine the following:

• The value of the far-left column (the first factor) [1.6.] State that this value is the first factor.
• The value of the top row (the second factor) [4.4.] State that this value is the second factor.
• The value of each "unit" [1/100.]
• The total value (the product of the two factors) of the base ten blocks [7.04.]

Now ask, "Can you model the base ten blocks to determine two different factors that also have a value of 7.04?" [Yes, 3.2 × 2.2 and 6.4 × 1.1.] Allow students to rearrange the blocks again to see if they can find more factors. [8.8 × 0.8, 17.6 × 0.4] Be sure to reinforce the concept that the product being modeled by the blocks must be a rectangle.

Now ask students to pair up. Provide them with base ten blocks. Ask them to configure three instances of decimal multiplication that yield the same decimal product. For each instance, they should do the following:

• Draw the configuration.
• Label the factors and product.
• Redraw using the same product to determine two different factors.

After the activity, conduct a class discussion by asking, "Are there certain types of factors you know will allow you to redraw them into two new factors?" "What types of factor pairs will not let you redraw them into two new factors?" [When both factors are prime numbers.] Students should be able to identify the characteristics that allow this to happen, namely that at least one of the factors of the first instance should be divisible by 2, 3, or some other reasonably small prime number.

Assessment Options

1. Have students determine which multiplication will yield a larger product:

2.5 × 4.2          or         2.7 × 3.9

Have students support their solution by using and tracing the base ten blocks.

[2.7 × 3.9, because 27 × 39 = 1053, which is greater than 1050, the product of 25 × 42.]

2. Have students write a real-world story about a decimal multiplication problem that they have created with the base ten blocks. The story should contain each of the different numbers that are involved with the rectangular construction. For example, stories could involve painting a rectangular wall in a room, or carpeting a rectangular floor in a room, and the dimensions of the two-dimensional space are the given decimals.

3. Ask students to draw their base ten blocks on figure paper to demonstrate student understanding. Note which students have mastered the skill.

Extensions

1. Have students explore what happens if the flat is equal to 1/10 and the rod is equal to 1/100.  What is the value of a unit? [1/1000.] How is a product using these values related to the product when a flat is equal to 1 and a rod is equal to 1/10? [The product, when a flat is equal to 1/10, is a hundredth of the product (when the flat is equal to 1).]
2. Have students work with factor trees to develop different models from the original model. For example, if they construct a 1.2 by 3.6 rectangle, the product will be 4.32. To make a model of a different shape, instead of moving the different shapes around, students should make a factor tree for the whole number 432. This moves the decimal two places, or multiplies 4.32 by 100. The prime factorization of 432 is 2 × 2 × 2 × 2 × 3 × 3 × 3. Now students should divide these prime factors into two groups. For example, they can place 2 × 2 in one group and the remaining factors 2 × 2 × 3 × 3 × 3 in the second group. The product of the first group is 4, and the product of the second group is 108. Returning to decimals, students can move the decimal place one place to the left in each factor (which equals two places total) to reverse what they did at the beginning. The 4 can be changed to 0.4, and the 108 can be changed to 10.8, and 0.4 ×10.8 = 4.32. In this example, a 0.4 by 10.8 rectangle can be created with the base ten blocks. Note that 4 × 1.08 = 4.32, also, but students may find that more difficult to represent with base ten blocks.

Questions for Students

1. Why is a product such as 0.3 × 0.1 is not equivalent to 3/10?

[Answers will vary, but students may note that one of the factors is 3/10, so the product could only be 3/10 if the other factor is 1. Also, the product will be 1/100 the product of the whole number product, or 1/100 of 3, which is equivalent to 3/100, or 0.03.]

2. Can you always write a decimal as a fraction? What denominators would you use for these fractions?

[10, 100, 1000.]

3. How could visualizing base ten blocks help you check a decimal product that you calculated using paper and pencil?

[Answers will vary, but students may note that they will be more likely to catch an error that is 10 times the correct value or 1/10 the correct value.]

Teacher Reflection

• When constructing decimals, how did students demonstrate that they understood a whole is equal to a flat, a tenth is equal to a rod, and a hundredth is equal to a unit?
• What difficulties did students have when constructing different-shaped models with the base ten blocks? How could these difficulties be addressed?
• Did students struggle making the various exchanges with the base ten blocks such as 10 units for 1 rod?
• What prerequisite knowledge do the students need to be successful in doing this lesson using base ten blocks?
• What are the misconceptions students have regarding the relationship between the three blocks?
• How can I demonstrate, using base ten blocks, that different sets of factors will have the same product?

### Learning Objectives

Students will:

• Use base ten blocks to find the product of two decimals.
• Demonstrate understanding of decimal multiplication by relating base ten blocks to area models.

### NCTM Standards and Expectations

• Understand the place-value structure of the base-ten number system and be able to represent and compare whole numbers and decimals.
• Select appropriate methods and tools for computing with whole numbers from among mental computation, estimation, calculators, and paper and pencil according to the context and nature of the computation and use the selected method or tools.
• Work flexibly with fractions, decimals, and percents to solve problems.

### Common Core State Standards – Mathematics

Grade 5, Num & Ops Base Ten

• CCSS.Math.Content.5.NBT.B.7
Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

### 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.MP6
Attend to precision.