In the past, I've given diagnostics before a unit so as to be able to compare pre- and post-instruction scores. Now, in the spirit of differentiation, I'm going to go one step further.

The next unit is about adding and subtracting fractions and mixed numbers. On my diagnostic, I wanted to see what percent of the students are still "adders-across" (#25 down: snakes that are bad at math). That would be 68/80, or 85%. The remaining 12 students could all do the basic algorithms, but most stumbled on the more complicated mixed number subtraction problem.

So here's the plan. In each class, I will assign one of the non-adders-across (NAA) to an adder-across (AA), tasking the NAA to help the AA learn over the coming lessons. If I see that they remain on task during practice time, the NAA will not have to take the quizzes, earning an automatic 100% on them. This seems reasonable, since they have already shown me they know the skill. Additionally, if the AA passes the quizzes (i.e. becomes an NAA!) then the NAA helper will earn some oh-so-coveted **extra credit points**. This way, the NAA has strong incentive to help, but there is no penalty if the AA doesn't make enough improvement.

Since almost no students showed mastery of the mixed number subtraction problems, every one will need to take that quiz when we get to it.

Now, the only thing that remains is to pair up the NAAs with the AAs effectively. I need to factor in personality, motivation, and so forth. Also, this experiment really highlights the imbalance between classes, even though we try to avoid any tracking (a constant difficulty in a small school). Here are the numbers of NAAs by period... Period 1: 5, Period 2: 4, Period 4: 2, Period 6: 1.

What Do We Mean by ‘Understanding’?

1 day ago

## 4 comments:

So adding across, in its own way, is pretty cool. It doesn't give a sum, but it is a quick way of getting a number in between two numbers.

1/2 and 2/3 ? 3/5 is in between them.

2 and 3? That's 2/1 and 3/1. 5/2 is between them.

I don't know if that helps your kids at all (I kind of doubt it), but it's still pretty cool.

Good luck

Looking at the number of NAA's in each section, pairing will be interesting. Unless of course you have some really small sections. Good Luck!

jd - That's cool - I never thought about that before. But you're right, I think it would just confuse the issue for the students more than anything right now. Maybe I can go back to this toward the end of the unit and see if they can explain why it works.

Continuities - my Numeracy sections are at 20 students each (we keep the remedial classes as small as possible). So not all students can be paired up with someone who already has the adding skill. It will be interesting, because I don't want to make simple high-low pairs.. I want students who can effectively collaborate, which can be hard to predict sometimes. My 6th period has my weakest students in it - only 1 already can add fractions. But, since the start of the new semester, there have been a few kids added to that class, and they have been a breath of fresh air. They don't have the skills, but they have the motivation, and they are helping bring some of the others up. I love that feeling when you have a class that you used to dread going to, and now becomes one you look forward to. It's amazing what an impact switching a few students out of/into a room can have on the culture of a class.

This is one thing my students learned well this year. (The fact that i'd scheduled 3 months just for fractions might have something to do with it.)

We used fraction strip manipulatives to start. They had a lot of experience with these, so not a lot of prep work was needed. We started off just adding fractions with like denominators - they worked in pairs, each one finding their part of the sum and putting them together.

They quickly realized (during the recording part - making them write what they see is important) that they just added the numerators, and the denominators stayed the same. I asked them often enough that they soon had a mantra of "during adding, the denominator stays the same". Note that I didn't give them this as a mnemonic - they figured it out and I just asked them enough to cement it for the next step.

That was asking them to try the fraction strips with different denominators. Confusion! They couldn't figure out which denominator to use! Some tried adding the two numbers together, but quickly realized on their own that meant a new strip which obviously wasn't the sum of the two.

Fortunately, they had equivalent fractions down cold at this point, so when i rewrote one of them with a new denominator, they quickly got it, and were able to rewrite the second one. Everyone got the having to rewrite part after only a couple of tries - as long as I was providing them with the common denominator. They referred to this as "making the pieces the same size", since that's how it looked on their fraction strips.

It was at *this* point that I let them teach each other. Some were stuck with just multiplying the two denominators together, and having to simplify later. Others got the idea of using the least common multiple (why introduce a new term for something they already knew how to do?) and happily used that as their new denominator.

I had numbers like your 6th period for adders across, and now have 90% of my kids not only doing it right, but understanding why it has to work that way. None of them will use the phrase "lowest common denominator", but they can all do it.

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