Base 10 materials practice problems answers:

1. Give an example of a groupable (decomposable) base 10 material.

Possible answers include craft sticks, grouped with rubber bands or beans grouped in small cups.

2. What are the advantages of a groupable (decomposable) base 10 material? (What does it do a particularly good job of teaching)

Because children make and take apart the groups of 10 (and 10 groups of 10 to make 100) themselves, it makes visible the relationship between the numbers in the various place values.

3. At what grade(s) should students be working with groupable/decomposable base 10 materials?

Groupable/decomposable materials should be a big part of the first grade curriculum. They are also appropriate for kindergarten as an introduction to the base 10 structure of numbers, and for higher grades when extra reinforcement of the base 10 number structure is needed.

4. Are groupable/decomposable base 10 materials proportional materials? Why or why not?

Yes, they are, because a ten is exactly the same size as 10 ones (indeed, it is comprised of 10 ones).

5. Give an example of a pregrouped, proportional base 10 material.

Base 10 blocks, which are sometimes known as Diens blocks (or, if you are familiar with Montessori, you could cite the golden bead material). The Rekenrek style abacus can be considered a transitional material between groupable and pregrouped proportional materials.

6. What are the advantages of a pregrouped proportional base 10 material as compared to a groupable base 10 material?

• It encourages seeing 10 as a group that can be counted as a group rather than as a bunch of ones that have to be counted individually.
• It introduces the notion of trading or exchanging (trading and exchanging can be used interchangeably, and both are considered preferable to borrowing and carrying).
• It's easier to work with numbers into hundreds.

7. What are the advantages of a pregrouped proportional base 10 material as compared to a non-proportional base 10 material?

It shows the relative sizes of the place values.

8. At what grade(s) should students be working with pregrouped proportional base 10 materials?

Second and third grade.

9. How do base 10 blocks relate to Cuisenaire rods?

The sizes are compatible: ones have the same size in both sets of materials. Also, one of the standard colors for base 10 blocks is orange to match the color of the 10-rods in Cuisenaire rod sets.

10. Give an example of a non-proportional base 10 material.

Possible answers include: money, Montessori stamp game tiles, and chips (such as poker chips) with amounts written on them, and the traditional abacus (but not the Rekenrek style abacus).

11. What are the advantages of non-proportional base 10 materials?

• They take up less room, especially for working with numbers into the thousands
• In the case of money they also teach about money
• They provide a transitional step between proportional material thinking and numerical thinking.

12. At what grade(s) should students be working with non-proportional base 10 materials?

Grade 2 and up, but in grade 2 and probably 3 also, but children should generally be learning first with proportional materials, and then repeating the work with non-proportional material (for instance, proportional material work one week, and non-proportional material the following week) before moving on to abstract number work. Non-proportional materials should be used to extend and make more abstract ideas that children have become comfortable working with using proportional materials.

13. Describe two activities for learning the base 10 structure of our number system using manipulatives.

Options include:

• Race to 50 or 100: roll a die to see how many counters to take. Group 10 or trade for 10 when you have 10 counters. The first person to accumulate 50 (or 100) wins.
• The teacher names a number and the child builds that number using base 10 materials.
• The teacher provides a set of objects, which the child groups into 10s and 100s and finds the number of objects.
• Give away 50 or 100 (more difficult): each child starts with 50 (or 100). Roll a die to see how many to give away. Decompose 10 or trade a 10 for 10 ones when you don't have enough singles to give away.