Digging into the Common Core: Deep Division!

We are just wrapping up our unit on division, and I wanted to know if they really understood what it is, so I decided to have them create their own problems.  In order to get their best work, I told them we would be compiling our problems into a book to be given to our principal for Christmas.  K-5 Math Teaching Resources  is a fantastic site for problems and activities aligned to the Common Core Standards.  I found a problem aligned with 5.NBT6: Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.  The students were asked to:

1. Create a word problem that could be solved by dividing a three digit dividend by a two digit divisor.

2. Estimate the answer to your problem.  Explain your strategy.

3. Solve your problem.  Show all your work.

4. Use a different method of solving your problem to check that your answer is accurate.  Explain your strategy.  

I was really looking to see if they knew the difference between the operations.  We have discussed clue words here and there but not in depth.  Most students were on the right track; only a couple did an addition or subtraction problem.  I made them peer edit their problems before putting them on good paper.  The front showed the problem, while the back showed their estimate and solution.  We had a blast!  They are really proud of their problems, and most included me eating cookies....I wonder where they got that idea? :)

Here are some fantastic examples of our work!

Number Talks: Division

We've been doing a lot of division lately and I began to notice that the preferred method is the traditional algorithm, mainly because that’s what parents are more comfortable with at home when helping with homework.  Because of this, I was noticing that if students made a mistake in following the steps, they really didn't notice they were making a mistake, let alone that their answer was not reasonable.  So, I decided to encourage more mental math.  We did this through some number talks and the use of slates.  In previous Number Talks, I didn't allow my students to write anything down to force them to do work mentally, however, for the division number talks (at least at first) I allowed them to write down their thinking on slates so they wouldn't confuse themselves.  The only stipulation was that they weren't allowed to divide the “normal way.”  No “houses!”  

We started with smaller, simpler numbers: 25 ÷ 4.  Here is a picture of their thinking:

Molly began by saying, "I know that 4 x 5 = 20 , and that was too small, so I added another group of 4 to get to 24.  Then, I knew I just had to add one more to get to 25.  So I had 6 groups of 4 with one left over."

Once some students saw how others were using basic facts to help them, they began to catch on.  More and more students began volunteering their thinking.  This is something we will continue to build a knowledge base of.  My hope is that this will lead my students to a greater understanding of division and answers that are reasonable.

Blake is one of my students that has an amazing sense of numbers: he does almost all of his calculations mentally.  I always try to see what he is thinking to guide the other students.  

This was a great example of two students building from something they already knew: multiples of ten.  I love how one counted up from ten and the other counted down from 20!

Jillian wasn't afraid to say she started with a fact that was too low and built her way from there!  I always encourage them to admit when they make a mistake and tell us how they found it and fixed it!  Chances are, they aren't the only one making that mistake!

We are really improving and they are finally telling me how much they love math!