Common Mistake: Remainders
Often, students are confused by the idea that the remainder means "how many things are left over," because there are no "left overs" in multiplication. Here is an example, where the student is solving the problem 176 / 6:
Incorrect Example
![Picture](/uploads/7/4/3/8/7438442/1305151566.jpg)
In the above example, the student is using groups of 60 (6 x 10) to get close to the dividend, 176. However, they go past the dividend to 180 (6 x 30 = 180). Since 180 is 4 away from 176, the student believes the answer to be 30 remainder 4.
Correct Example
![Picture](/uploads/7/4/3/8/7438442/1305151976.jpg)
In the above example, the student uses the same strategy of adding groups of 60 (6x10), but this time takes notice that they must land on a number that is less than or equal to the dividend.
In strategy A, the student sees that after multiplying 60 x 20, which equals 120, they only needed 54 more to get to 176. So instead of adding 60 more, they chose to try multiplying 6 x 9 (6 x 9 = 54). They then added 60 + 60 + 54 = 174. They knew that adding 6 more to 174 would be too big, so they subtracted 176-174 = 2. Finally, they added 10 + 10 + 9 = 29 (6 x 29 = 174), which has a remainder of 2 (174 + 2 = 176).
In strategy B, the student uses the same strategy of adding 60 (6 x 10, three times), up to 180. They saw that they went past 176, so they knew they needed to subtract a group of 6 (180 - 6 = 174). They subtracted 176 - 174 = 2, to find the difference up to the dividend.
This error indicates:
that the student is not thinking about what the division problem actually means. Because they are using multiplication (which is the inverse of division, and is a good problem-solving strategy), they are adding the items up instead of taking them away. This can lead to confusion because instead of having x items left over at the end, they need to remember that they should have x left over under the dividend (i.e. multiply to 174, not 180 when the dividend is 176).
How to Remediate the Error:
If your child is encountering this issue, it may be easier to start using smaller numbers. For instance, you could pose the problem 32 / 6. The student may think 6 x 6 = 36, and
36 - 32 = 4, so the answer would be 6 R4. However, a correct strategy would be 6 x 5 = 30, and 32 - 30 = 2, so the answer is 5 R2. This concept could be demonstrated using manipulatives, such as pennies, bottle caps, or cubes. Give your child 32 cubes, and ask them to divide them into 6 groups. They will find that there are 5 cubes in each group, with 2 left over. Ask, "Could you make 6 groups of 6 with 4 left over, with the 32 cubes we have here?" The students will see that they cannot make 6 groups of 6, because they do not have enough cubes. Once they have mastered this skill with smaller numbers, try using more difficult problems, such as the one above.
Another way to allow your child to check and see if their answer is correct, especially if they prefer to use equations to problem solve, is to have them check their multiplication. With our incorrect example above (176 / 6 = 30 R4), the student would multiply out 6 x 30 (the divisor times the quotient), and then add on the remainder, since the remainder is how much is left over at the end (it is not included in the multiplication). The student would do 6 x 30 = 180, and 180 + 4 = 184 (which does not equal 176, our dividend). Ask, "Is that number too big or too small?" Be sure to explain that the remainder is always added on when checking our work because it is extra; the leftovers. Emphasize once again that when you add up all the units in all the groups, PLUS the remainder, we can never have more units than we started with (as can be seen using the cubes).
36 - 32 = 4, so the answer would be 6 R4. However, a correct strategy would be 6 x 5 = 30, and 32 - 30 = 2, so the answer is 5 R2. This concept could be demonstrated using manipulatives, such as pennies, bottle caps, or cubes. Give your child 32 cubes, and ask them to divide them into 6 groups. They will find that there are 5 cubes in each group, with 2 left over. Ask, "Could you make 6 groups of 6 with 4 left over, with the 32 cubes we have here?" The students will see that they cannot make 6 groups of 6, because they do not have enough cubes. Once they have mastered this skill with smaller numbers, try using more difficult problems, such as the one above.
Another way to allow your child to check and see if their answer is correct, especially if they prefer to use equations to problem solve, is to have them check their multiplication. With our incorrect example above (176 / 6 = 30 R4), the student would multiply out 6 x 30 (the divisor times the quotient), and then add on the remainder, since the remainder is how much is left over at the end (it is not included in the multiplication). The student would do 6 x 30 = 180, and 180 + 4 = 184 (which does not equal 176, our dividend). Ask, "Is that number too big or too small?" Be sure to explain that the remainder is always added on when checking our work because it is extra; the leftovers. Emphasize once again that when you add up all the units in all the groups, PLUS the remainder, we can never have more units than we started with (as can be seen using the cubes).
Why it Works:
By using only equations to problem solve, it is easy for students to lose sight of the big picture. By using physical units to figure out a problem, students can visualize the division, helping them to actually see what the remainder looks like. While remembering that multiplication is the opposite function of division can be very helpful in problem solving, it is important to remind the student that division is the breaking down of larger numbers into small groups, so as you take units away, you may have some left over which can not fit evenly into a group. These extras are the remainder.