Long Division

Long division is always a struggle at the beginning of the year. Many students do not know the process and even more students do not understand what they are doing. I’m afraid that long division will join algebra soon and mathematics education will only require students to learn how to multiply.

Long division is a multi-step process that many students struggle with to the point of making up rote memorization mnemonics. Students that have been shown how to repeatedly subtract multiples of the divisor don’t understand how this connects to the more efficient method.

My sixth grade classes today looked at me stunned when I asked them to divide 32 into a ten digit dividend. Does it matter? Have they never been asked to do something that is not printed on a worksheet? Something that might be challenging?

Several students asked if they could just do short division. This is something that really bugs me! Am I the only one?

I think there is so much math in long division that short division loses. Remainders are so important. Which remainders are repeating? Can we have a remainder larger than our divisor? What are the possible remainders when we are dividing by 32? These are questions that sixth graders don’t know the answers to. These are questions that are important to their understanding of division.

When I divided 132 by 11 today students couldn’t tell me what it meant to say 11 goes into 13 once. Why didn’t I put the 1 over the two? Why don’t I just subtract the 11 from the 32? These students have all seen their fourth grade teacher repeatedly subtracting multiples of 11’s. What is missing? Is it repetition? How do we help students “really” understand what they are doing?

Partial Products

Every year I spend too long reviewing basic algorithms. This year I’m going to jump in feet first and begin by doing partial products to develop both the multiplication algorithm, and then the division algorithm. I know the majority of my students know the multiplication algorithm but only a handful of them can explain why it works.

Last year I spent some time doing partial products with base ten blocks and students really seemed to expand their knowledge of these basic algorithms. I found a great activity to expand on what I did here.

Once students understand this activity they can then begin to draw generalized rectangle models to lead into the distributive property.