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!

  

Division!

Oh the joys of division!  For whatever reason, my students are afraid, deathly afraid of division.  From their pre-assessments, I could tell they weren't really thinking of the actual numbers being divided, but more concerned with the PROCESS, and because of this were making silly, careless errors and their answers made no sense!  So, I decided to start at the basics: the concept of division and how it relates to multiplication.  We soon moved on to estimation and using compatible numbers (multiples of ten) to help us solve.  On Friday, we explored three different ways to divide.

We began by using base-ten blocks.  We are pretty good at using them after using them for modeling decimals as well as multiplication.  They knew that if they didn't have enough flats (100's) to fill in the groups evenly, they would have to break one into ten 10's and go from there.  I was extremely happy that they were so fluent in this from adding and subtracting decimals!  They had no idea they could divide this way.  I chose some of my struggling students to model this in front of the class.  They were extremely successful and were proud to show the class even they could divide large numbers.  Here are some pics of Sam and Bobby dividing using models:

Sam had to break a long into 10 ones! He did a great job!

 Next, we moved on to using the distributive property and expanded notation to divide.  The kids are pretty proficient in writing numbers in expanded notation, so we really just had to take it a step further to create the area model.  Here is Allison using her whiteboard to solve.  You can see she broke 248 into 200+40+8, then divided each part by two, and finally added her quotients.  Her area model is below, except I caught her before she had completed it, but you get the idea!

Below: Max bypassed drawing the model and opted instead to just divide each part by three.  Then, he added up the quotients.

Lastly, we learned the partial quotients method for division.  Personally, this is my favorite.  It allows kids who aren't as proficient in their math facts to still be successful in division.  It also really shows the value of the numbers, something my students clearly struggle with.  Here is a shot of Keegan flying through a problem with ease.  I think he's found his favorite way to divide!

Today, after a long, warm weekend of fun, we spent some time going back over each method for division.  I split the kids up into partners: a student who "gets it" with a student who doesn't.  They really worked well together and as I circulated I heard the helpers saying things like "how many times does 3 go into 400-some?  Does it go in at least 10 times?  Do you think we can go higher?"  Sometimes, all it takes is some guidance from a peer rather than teacher lady.  I'm OK with that!

Lastly, we created anchor charts showing the ways for division.  We added a fourth way, the traditional way.  I found that my "high" kids like the traditional way, and I'll let them use it as long as they can explain each step using place value! (Muahahahaha!)  Don't forget to check out our division math rap on YouTube!

Common Core Connection: Standard: 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.

Mathematical Practices: Make sense of problems and persevere in solving them.