Math

Division

A Concrete Way to Show Division

Division can be the most difficult operation for students. The language we use can make a difference in their understanding. Getting students to see that it's really just repeated subtraction of equal groups can build their confidence.

For example, in the problem 38 ÷ 4 = ? instead of saying what is 38 divided by 4, we can ask how many fours are in 38? Or how many groups of 4 are in 38?
Another way could be giving it a context. I have 38 jelly beans. I want to share them with 4 friends. How many will each person get?
One more way is to say I have 38 jelly beans. I want to give my friends 4 jelly beans each. How many friends can get jelly beans?

A student used Cuisenaire Rods (each rod has a value of 1-10) to make a linear model of the division problem 38 ÷ 4 = ?. The top row of rods shows 38 (the whole which is made up of 3 ten rods and 1 eight rod). The student then found out how many groups of 4 are in 38 by placing them underneath. There are 9 groups of 4 and a remainder of 2. Another student drew the rods with their values.

Students had to convince me that 38 ÷ 4 ≠ 9. This student drew the rods with their values in a parts-whole model. The top row of rods shows the whole and the bottom row of rods shows how many groups of 4 are in 38. There are 9 fours in 38 but there is also a remainder or leftover group of 2.

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Watch this student use the rods to figure out 28 ÷ 7 = ? and then represent it using a number line.

Division As Equal Groups

For the problem 24 ÷ 3 = ?, this student drew an array of 24 circles. He then put a circle around groups of 3. He could easily see from his drawing that there are 8 groups of 3 in 24.

This student solved 24 ÷ 3 = ? by making groups of 3 green circles. She then circled them to show there are 8 groups of 3 in 24.

Number Lines for Division

This student solved 24 ÷ 4 = ? by creating a number line beginning with 24 and then repeatedly subtracting groups of 4. He could see there are 6 groups (or jumps) of 4 in 24.

This student figured out the answer to 19 ÷ 4 = ? by first using Cuisenaire Rods. He made the whole (19) and then found out how many 4s are in 19. He saw there is also a remainder of 3. He showed this on a number line as well.

Another example of the rods and a number line representing division.

This student used a number line more efficiently to find how many groups of 6 are in 56 (56 ÷ 6).

Flexible Division Algorithm

Student Created Story Problems and Solutions Using a Variety of Strategies

This student solved the problem of 46 ÷ 4 by efficiently using multiplication on a number line. From 46 he pulled out a group of 10 fours and that got him to 40 chicken wings. Then he pulled out one more group of fours and that got him to 44. There were no groups of 4 left in 46 so he saw the remainder of 2 chicken wings left over. He counted up his groups of fours (11) and said the answer was each friend gets 11 chicken wings and there are 2 left over.

This student made 4 circles to show the friends. She then shared or repeatedly subtracted chicken wings (5 for each friend, then another 5 for each friend, and then 1 for each friend. Notice how she organized them so it's easy to see and count how many chicken wings each student gets.

This student used multiplication on a number line and flexible division.

For the number line, she pulled out 5 groups of 4 and then another 5 groups of 4, then 1 group of 4. You can see how she shows the remainder of 2 chicken wings.

With her flexible division algorithm, she used it efficiently by pulling out 10 groups of 4 right away, then another group of 4. She even saw "there are 11 groups of 4 living in 46."

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