Manipulatives are physical objects -- such as blocks or beads -- that students can handle and use during a math activity, and they are effective for linking abstract mathematical concepts to concrete applications. They are especially effective for operations such as multiplication and division, which focus on how groups are formed and separated. Furthermore, using manipulatives appeals to kinesthetic -- or hands-on -- learners, who might struggle with comprehending abstractions without a concrete model.

### Egg Carton Multiplication

Give each student the bottom half of an egg carton -- the part with all the separate compartments -- and a small cup full of beads. As you write a multiplication problem on the board, have students fill compartments of their cartons with the appropriate amount of beads. For example, if you write the problem "3 X 5" on the board, they could place three beads into each of five different compartments, and then count the total to give the answer of 15. Explain that they could also place groups of five beads into three different compartments and get the same answer.

### Cracker Multiplication and Division

Give each student 12 crackers. Then, write the problem, "12/ 4" on the chalkboard. To see how many times 4 goes into 12, have each student arrange their crackers into groups of 4, leading them to see that 4 goes into 12 three times. Then, write the problem "3 X 4" on the chalkboard. They will see that they already have 3 groups of 4, so the answer is 12. This will help them understand how multiplication and division are related. You can then have them arrange their crackers into a rectangle. Show them that if they count the number of crackers along the vertical and horizontal edges, they will see which numbers get multiplied to produce the total, which are again 3 and 4 in this example.

### Pizza Pie Division

Bring in a pizza that you've cut into 16 slices, and have the class gather around the box. Have the students count the slices. Ask them what half of 16 is. When they say eight, have one student count out eight slices so they see that eight slices make one half of the pie. Ask how many halves are in a whole. When they say two, explain that they've just done division, that 8 goes into 16 twice, or, 16/8 = 2. On the chalkboard write "16/4." Ask another student to count four slices, and then ask how many times that group of 4 slices will fit in the whole pie. You can also have four students hold their hands above each group of 4 slices to show that 16/4 = 4.

### Jelly Beans and Dice

Split the class into groups of three or four. Give each group a container full of jellybeans or another small object. Go to each group and roll a pair of dice. Whichever numbers are on the dice will determine how many piles of jellybeans they need to build, and how many jelly beans go in each pile. For example, if you roll a 5 and a 2, they have to build 5 piles of 2 jellybeans. They can then count the total beans to see that 5 X 2 = 10. You can also have them build 2 piles of 5 jellybeans, to illustrate the commutative property.

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