Tuesday, 15 December 2015

Rows of apples

I don't do word problems very often. But I liked what was happening over in this great 4th grade Learning Lab lesson described by Kristin. So I adapted the less-numbered prompt and asked my class what they noticed. 

They were indignant (maybe because they've been made wary now):
J's "It's a lie!" was vehement.

I disagreed: this situation, though it is made up, could be a true one. And we do get some information from the description. I asked if anyone could draw an apple display that might be possible:
That one at the top is my counter-example.

So now for some numbers. The class seemed relieved by their appearance! What questions could we ask?
I wanted to get them working individually at this point, so off we went straight away. "You can use cubes or Cuisenaire rods if that helps you. Show me your answer, and what you thought to get to that answer."

There was a great range of approaches. Here's a selection:
This student was the only one who just added the numbers together.
I suggested he got out 56 cubes and arranged them in four rows...
I was so pleased she chose to do this!
This student was so excited that he'd got this.
But when asked, he said his method was to estimate, and then tinker a bit to get the right answer. 
Halving 56 took this student a lot of time though!
I think someone told her in the end!
I only saw this at the end! He'd multiplied by four! I explained that there were meant to be 56 apples altogether. Maybe because his first language is not English, he'd not quite understood? I need to ask. I told him I thought what he had achieved was good even so!
I like the fractions added at the end!
There were quite a few of these halving and then halving again ones.
It wasn't a strategy I'd talked about, so it's great to see it coming up!

Again, I love K's link with fractions
We'd been writing a few fraction equations (whatever they wanted!) earlier in the lesson, and K had noticed a pattern ("Can it go on the claims board?"):
He's used that pattern
So, two people set off in the wrong direction. One student, whose work isn't scanned here, couldn't explain how she got to 14 and I suspect that maybe she'd simply copied the answer. I'll need to sit with her next time!

It definitely felt like starting with the less-numbered situation gave students a chance to understand it before any number grabbing happened. We got to talk about what information was actually there too. I loved seeing all the approaches, and scanning them means I can show them tomorrow and compare different strategies. We'll showcase K's claim too!


  1. Hi Simon,
    You mentioned you don't use word problems often. Have you replaced your problem solving with 3 Act Tasks or is there something else you do more often. I love your claims and the use of the rods. Is that what you do most of your problem solving with? I'm interested to hear what you use most often. In Ontario we are stuck still with EQAO which is a province wide test that still uses many word problems so many teachers that I work with as a coach are scared to move away from them. I have been trying to show them other alternatives. Any advice you have would be appreciated. Thanks
    Mark Stamp

    1. Hi Mark,
      I feel a bit embarrassed by your very good question - because I'm perhaps just not doing enough problem solving!

      I tend not to give this kind of task, where students have to make mathematical sense of a situation and decide which tools to use, and maybe need to find ways to up the number of lessons where students are having to manage part of the modelling process, in this case finding mathematics to model a real-world-ish question. I use estimation 180 on an almost weekly basis, but I'm thinking that some of these should be expanded from a 10-minute task to one that perhaps takes up 40 minutes. I like Graham Fletcher's 3-acts lots, for instance his recent

      I say perhaps and maybe, because it's only very recently that tasks like this, that show a way of giving a word problem without the heartache, have shown a way forward for me. I still need to think about it all...

      So sorry, I'm not much help!