Mental Math

Fraction Kits – beginnings

So far in summer school, we’ve been working on sociomathematical goals: getting students accustomed to routines and procedures for sharing their thinking in pairs, in written work, and in whole-group dicussions.  Mathematically, the activities and lessons we’ve used to develop these goals have been focused on multiplication and the relationship between skip counting, grouping, and array models of multiplication. This has helped us identify students who need extra support (since multiplication isn’t ‘new’ content for 3rd grade) while also being a slightly more approachable area to start number talks with.

Last week in our classroom, we began work on the content standards that gave our 3rd graders the most trouble on the end-of-year assessments administered by the state and by the school: fractions. Specifically, these students need additional support in understanding what a fraction represents, how fractions can be represented on linear models, and how fractions can be different and yet equivalent. To start this work last week, we began by creating Fraction Kits based on the insight and advice of Marilyn Burns, whose book has been an excellent tool for lesson planning and mapping the mathematical terrain we need to help students explore this summer.

Monday and Tuesday we spent time counting collections, and observing how students were grouping objects.

Wednesday we began math class with a mental math number talk, solving and discussing strategies for 6×7. I was glad to see that students were willing to persist even if they didn’t “just know it” from memory – they were willing to think and find a means of figuring out the problem.

After our number talk, students returned to their seats where each received a set of pre-cut strips of construction paper and a large envelope cut from similar construction paper. They wrote their names on the envelopes and tucked them inside their desks.

Next we went through the process of folding our strips. For each strip, we:

-folded (I first modeled how to fold them)
-predicted how many sections there would be before opening to see
-decided as a group how we might then label each section
-waited for each student to fold, unfold, label each section, and label the back of each section with initials.

Once every student had their fraction pieces folded, labeled, and affixed with initials, we passed out scissors and got them cutting.

Why do, We do, You do?

I want to note that ordinarily, I steer away from “I do, we do, you do” type teaching. However, for creating fraction kits, I modeled first, together we decided how we might name/label, and then asked students to do it on their own. The reason I thought modeling and duplication was fitting in this instance is that our goal was for students to create tools that they could then use to explore fraction relationships. If their kits weren’t created correctly (and some weren’t: a couple students mis-folded and needed help to tape or re-cut later), they might develop some sneaky misconceptions about how much bigger 1/4 was than 1/8, etc.

So why not pre-cut the strips?

Giving the students a chance to fold and cut did elicit some students’ initial ideas about what those weird hyroglyphs called fractions might mean: many students were catching on to the pattern that “8 pieces mean each is called 1/8” and so on. Folding their own strips was also a useful opportunity for them to consider what happens when you take a half and fold it in half. For instance, when we folded the strip into 1/4ths, many students predicted that we would unfold the strip to find 6, 8, or 10 segments, rather than 4. Similar guesses of 10, 15, and 20 came out when we folded the strip into 1/8ths by folding it in  half, half again, and half again.

Wrapping up

Because we opened with a number talk, we only had time to get the strips cut, labeled, and put away on Wednesday. But we did close by asking students to see if they saw any “matches” between strips of different size. This got a few initial ideas out on the table (or in our case, the smart board): two 1/4th sized pieces was the same as one 1/2. Similarly, two 1/8 sized pieces was the same size as 1/4.

8+9: four strategies for counting and adding

Yesterday was our first mental math as a 3rd grade summer school class. This mental math number talk was designed to be a ‘warm up’ to the instructional routine; in terms of the standards, this problem would be well suited for late first grade. However, I wanted a problem that would be slightly easier than the 3rd grade standards, so that the students could focus on developing productive norms for number talks.

I started by discussing with the students what a number talk was, what mental math was, and what their jobs would be: to think about the problem and multiple ways of solving it, to talk at a conversation level to their partner and listen to their strategies, and to be ready to share with the class AND listen to their peers’ ideas during whole-group discussion.

We rehearsed these norms and actions several times, first with questions designed only to get them used to holding their ideas privately until asked to share with a neighbor: “Think of as many animals as you can that are green” and then “Think of something you have exactly 3 of in your house.” Students are generally pretty eager to share responses to these kinds of questions, so they are helpful for giving them a chance to practice holding their ideas private until asked to talk to a partner or share with the class.

We then did one more ‘practice round’ with the problem 2+2 to again reinforce, review, and label productive student actions, like showing on their fist with fingers how many strategies they had, holding their ideas privately until asked, then sharing at a low conversation level with a partner once asked.

When it was finally time to do the ‘real’ problem for the day (8+9), these norms were pretty firmly in place. I had a few kids whisper “ooh i know it!” to themselves, but using a gesture of “quiet” and modeling my own fist to my chest was sufficient reminder for them to be thinking about strategies, holding their ideas private until called on.

After I saw that nearly every student had at least 1 finger raised (probably 45 seconds), I asked them to turn and talk to a neighbor about how they thought about the problem. The students did so well, and I heard many of them trying to explain their thinking in addition to sharing their answers.

After about 30 seconds of partner time, I called their attention back to me (I use “class class” –> “yes yes” call and response for this) and asked students to share answers. I called on just about every student and got 17 for all but 1, who said 14.

I then asked for volunteers to share their strategies.

The predominant strategy students shared was counting on, either from 8 or 9, using fingers. Two students adjusted and compensated – one by solving 9 + 9 and removing 1, another by taking 1 from the 8 and adding it to the 9 to make the problem 10 + 7. The last strategy (bottom left) was contributed by my coteacher.

Overall, this number talk was a great start to our summer school theme of using number talks in regular instruction to work on the OA strand. I’m looking forward to Monday!