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3-3 Combining Equal GroupsofLarger Blocks3-3

When students can easily represent 3 6 as 3 groups of 6 single blocksand find that the total number of blocks is 18, they can readily workwith larger blocks in the same fashion. That is, since a block-of-10 is one

block, it is also possible to make 3 groups of 6 of these larger blocks. This is

similarly true for blocks-of-100. It is clear that there are 18 blocks, no matter

which size is involved.This physical reality helps students use their basic

fact knowledge to multiply tens and hundreds by a single-digit number.

Discovery of this idea develops mental computation abilities and supports

paper-and-pencil techniques.

Applying Basic Facts

If your class made the large multiplication chart (up to 15 26) to correspondto the Array mat, discuss the idea that they could expand the chart indefinitely,but that it would take a tremendous amount of work. Indicate the area of thechart that corresponds to the basic facts up to 9 9. Tell the students thatthey will now discover a way to use just this small area of the chart to multiplyany two numbers!

They begin by applying the 9 9 portion of the chart to blocks-of-10 andblocks-of-100, just as they did with the single blocks. Have students work inpairs to make 2 groups of 4 single blocks and find the total number of blocks.

Keeping the models in view, have studentsrepeat the task twice more, first making 2groups of 4 blocks-of-10, and then 2 groupsof 4 blocks-of-100. Ask,

How many single blocks are there? (8) how many

blocks-of-10?(8) how many blocks-of-100? (8)

These answers are quite obvious to studentsbecause the images are the same except forsize. To emphasize the relationship, talkabout these using parallel language:

2 4 = 8

2 4 tens = 8 tens

2 4 hundreds = 8 hundreds

Focus Extending basic fact knowledge to

larger numbers

Its easy for students to extend a basic fact such as 2 4 tofind the answer to 2 40 and 2 400.

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Repeat, having students make 4 groups of 6 using the single blocks, blocks-of-10, and blocks-of-100. Again ask,

How many single blocks are there? how many blocks-of-10? how many blocks-of-100?

Next ask students to set out 5 blocks-of-10. Ask,

What number do these blocks show? (50) How can you show 3 50?

Have students make the 3 equal groups and then ask,

How many blocks-of-10 are there? (15) What number do these blocks show? (150)

Have the students summarize this model in a multiplication sentence.(3 50 = 150)

Repeat with blocks-of-100, having the students first set out 5 blocks-of-100

and tell the number represented. Then have the students represent 3 500and write the corresponding multiplication sentence. (3 500 = 1500)

Finally, present the examples 5 40 and 4 200. Have students predict theanswers and explain their thinking. They can use blocks to check their predictions.

Comparing Ways to Multiply

To illustrate the mathematical power of our base ten system, have students

find 3

20, first by placing the single blocks on an Array mat, and then bysetting out 3 groups of 2 blocks-of-10. Have students talk about why it waseasier to show the number in its base ten representation.

This is also a good time for students to think about the usefulness of the factthat the order of the factors doesnt change the product. To demonstrate, havestudents work together to make 20 groups of 4 single blocks and pack them tofind the answer. Then have students make 4 groups of 20 (2 blocks-of-10)and find the answer. Ask,

Which groups were easier to make? Why? What would be an easy way to find the total

number of blocks in 200 groups of 3?

Practicing Key Ideas

Matching Groups

Students work in pairs.One student makes equal groups with single blocks.The

other student makes the same number of equal groups, but with blocks-of-10 or

blocks-of-100.For example, if the first student makes 3 groups of 7 singles, the

partner makes 3 groups of 7 blocks-of-10 or 3 groups of 7 blocks-of-100. Together

students record their work with pictures, words, or multiplication sentences. Students

can then repeat the activity, reversing roles.

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Assessing Learning

1. Ask the student to show you 4 groups of 6 blocks-of-10. Ask,

How many blocks-of-10 are there? What number do these blocks show?

Does the student identify the correct number of blocks-of-10? name the number correctly?

2. Present the example 6 200 and have the student find the answer.Does the student predict the answer or use the blocks? answer correctly?

3. Ask,

How many blocks are there in 30 groups of 2?Does the student predict the answer without representing the example with blocks? use blocks and make 30 groups of 2 or make 2 groups of 30? answer correctly?

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