Here’s the breakdown of student thinking about double-digit multiplication that I’m seeing as we begin our unit in my 4th Grade class.

Direct Modeling:

Directly Modeling

 

Direct Modeling With Composition Into Groups:

Composing

 

Breaking The Numbers Apart With Addition:

Breaking The Numbers Apart

 

Breaking The Numbers Apart 2

 

Breaking The Numbers Apart With Arrays:

Arrays Arrays - Breaking Numbers Apart

 

Use of Standard Algorithm:

Standard Algorithm Standard Algorithm - Wrong

 

No Real Strategy, But Knowledge Of Multiplication by Multiples of 10:

Not Sure

  • Finally got to my computer and had a chance to really look at these. We’ve talked about #1 on twitter. I have more questions about her boxes, but sounds like you went there already.

    Here are my thoughts for the rest:

    #2 There’s a lot that’s great here. Really good math thinking. Clearly some organizational issues, and I think I’d work with this student on strategies for keeping track. Would particularly help with second error (first was drawing 36 dots instead of 38). Missing the third group of 10 was a big deal in the ultimate answer this student gave. How could he/she keep track of which chunks he/she has dealt with? Also, it’s not quite clear where the 96 came from, although I think it’s the 8 remaining dots times 12. Calculated correctly, but wrong number of dots. It’s clear the student’s not sure about the 96, so it’s surprising he/she kept it as a mental strategy and didn’t write anything down. I’d really push showing the work in an organized way so someone else can follow it, and so errors would be less likely.

    #3 Keep one number the same, break up the other by place value. Efficient, accurate, effective strategy. Nice use of distributive property.

    #4 Holy fascinating! What it looks like to me–although I’m not positive because so much of the work wasn’t written down–is this student might have thought of 12 as 3 x 4. Multiplied 38 x 4 to get 152 (not shown). Then multiplied 152 x 3 = 456 (but used addition to do it). So, thinking of 38 x 12 as 38 x 3 x 4. Great example of associative property. Not the easiest numbers to work with though. Would 6 x 2 have been easier?

    (I wouldn’t put 3 and 4 in the same category of “breaking the numbers apart with addition.” Yes, they both do that, but one is using distributive and the other associative (I think), so different approaches.)

    #5 Area model using place value. Another great strategy.

    #6 Area model using place value. Well done. Next logical step is to break only one of those numbers all the way into 10, 10, 10…. Could they do 35 x 28 = (10 + 10 + 10 + 5) x (20 + 8)? What happens in the 20 x 10 boxes?

    #7 Standard Algorithm. I might use Kazemi and Hintz’s “Compare and Connect” structure from Intentional Talk (you can preview it free online here: http://www.stenhouse.com/html/intentional-talk.htm, look at 39-54) with #3, #5, and #7. Did you get any full partial products strategies? 38 x 12 = (30 x 10) + (30 x 2) + (8 x 10) + (8 x 2)? If you did, I’d use that one instead of #3, and compare it to area model in #5 first. Then look at #7. Do students see the relationships?

    #8 Standard Algorithm with error. Total focus is on reasonableness here. Why should this answer jump out as unreasonable?

    #9 Hmm. Lots of ideas here. Drawing the 22 pencils, but then not using that. Some kind of version of standard algorithm on 20 x 20? The 12 must have come from multiplying 2 x 6. That leads me to believe we’re seeing this common error: 22 x 26 = (20 x 20) + (2 x 6). Cure for that one is the area model, shading in the boxes that were accounted for (the tens x tens and the ones x ones, but not the tens x ones combinations). Glad this student gave the second problem more of a try than the first one, because then you have more to go on.

    Those thoughts are from the work samples only; you obviously know these kids and have more to go on. Anyway, that was fun, thank you! I miss fourth grade and looking at student work!

  • Great feedback Tracy! I don’t know if there is much to add, but I will give my thoughts…I did miss the full convo on Twitter so sorry if I repeat…

    #1 – I am assuming the work on the left was trying to make arrays that had 12 oranges and the three extra ones at the bottom are the three to add to the 35 to make 38? I am quite confused with the 2_3 but think the 38 _ underneath is looking for the operation to use (as they have them listed underneath) to solve for the answer. I can’t see the bottom but it looked like the student started adding 12’s…cannot tell how many 12s but that would show at least a start of the repeated addition strategy. I guess we cannot tell if the student has seen a problem similar and thinks that it is an “array” question so feels the need to go with the “boxes of oranges”?

    #2 – I do love this one (a personal favorite strategy). I think my first question for the student would be “Could you solve this problem without the dots?” It is a great use of the distributive property to make “friendly numbers” to work with. I agree with Tracy that missing that extra 120 was the critical error. My guess is the student thought that the top 120 was their first group and they needed two more (a repeated addition mix-up?). I think having he/she explain the work aloud may either unveil the error to him/her?

    #3 – It would be interesting to have student 2 and 3 explain their strategies to each other and come up with the similarities and differences. Being similar, it may push student 2 to a bit of a more efficient strategy or maybe even begin looking at more numerical modeling over pictures?

    #4 – I love the mental math piece of this. Not one “carried one”, nothing. I LOVE that! I do see it as a distributive property though… (38×4)+(38×4)+(38×4). Maybe putting this student with 2 and 3 in a small group could come up with some really great similarities/differences in their strategies!

    #5 – Great strategy, the work I see with my students who use this method is pushing them to not get in the habit of always breaking every two digit number into tens/ones. Could they have kept the 12 whole and just broken the 38?

    #6 – Wow…did he/she add all of those boxes in their head?? Tracy mirrored my thoughts, work multiples of ten to not have to do all of those 10 boxes, definitely movable from this point. It reminds me of the younger students moving from jumping 3 tens on the number line to jumping 30 all in one jump with adding/subtracting.

    #7 – I think this student and the area model in #5 could chat. Where are their similar products?

    #8 – Yes, having students estimate before solving would help this student. What makes sense?

    #9 – This one actually interests me…typically my “number grabbers” who just grab the numbers in a problem and add/sub/mult/divide them use both numbers in the problem. This student completely ignored the 12. Did they think it was a 2?

    All of that said, that is just purely from looking at work. I think I would put them in groups based on similar strategies the conversations of similarities/differences would be interesting. That would free you to work in a small group with students like #9 and #1 who seem to struggle with a starting point or possibly just comprehension of what is happening in the problem.

    Good Luck!
    Kristin

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