# Fraction multiplication task and tree diagrams

My students are rising to the occasion. They never encountered the following type of problem before but by the second day they were becoming masters of the universe.

A little elephant has 12 peanuts. In the spirit of giving each day he gave away a fraction of his peanuts, kept the rest, and then ate one. These are the fractions he decided to give away:

$\large \frac{2}{3}, \frac{1}{4}, \frac{1}{8}$

In which order did he use the fractions so that he was left with just one peanut at the end?

This task is the second in a series of three. I wanted to give the students a challenge while practicing multiplying fractions, but I also wanted the problem to be approachable. Frankly I think it would have been a disaster had I not introduced the tree diagram the previous day. While in previous years students had used tree diagrams for an occasional probability problem that approach was the farthest thing from their minds. My goals for the lesson were to practice multiplication of fractions while using the diagram to organize their work.

As they became more comfortable with the tree diagram student work became more systematic. That freed up their brains to concentrate on what is happening with the fractional amount. Several groups needed to be reminded that the elephant gives away that amount, which means he keeps the rest, then eats one.

Here’s an example of how one group organized their work:

One student, Jeff, had missed the previous day’s lesson where I introduced the students to a simpler version of the problem involving only two fractions.

He admitted he was a bit confused, “There’s a lot to keep track of.”

At the start of day two’s lesson several students were able to determine which fractions to rule out as starting points. After I reviewed the first peanut task one student said, “We have to start with
$\large \frac{1}{2}$ because 3 doesn’t go into 14 evenly.”

On Monday, groups will tackle the third peanut task with minimal oversight. Those who finish early will work to create their own peanut task.

I want you to know that the inspiration for these problems was provided by the Peaches Today, Peaches Tomorrow task on the NRICH website. Part 2 of the series captivated me, yet I thought dealing with 6 fractions in a 41 minute period would be too much for my seventh graders. Designing these problems was a collaborative effort and I want to thank my colleagues for their help in creating a meaningful twist to the problem (give away instead of keep).

# Effective effort, reflections, and P-T conferences

One of my colleagues created a student reflection sheet for students to complete at the end of each quarter. He then meets briefly with each one to discuss how they’re progressing on student skills. I’ve refined it just a bit, but I’m wondering what you would add or modify?

While I’ve met with only a few students this reflection has been incredibly valuable as talking points during last week’s parent-teacher conferences. I hate discussing grades. Our grade books are transparent so parents can see their student’s grade at any time. What I can give parents however is a sense of how their child sees himself as a learner.

This quarter the biggest talking point for several students was effective effort in terms of:

• 1) following the rubric or criteria of success, and
• 2) implementing feedback

for the integer portfolio project.

The portfolio was extremely time consuming to assess, but the project gave me incredible insight into a student’s understanding. The two biggest drawbacks to the project were some students didn’t submit all the artifacts and/or they’re implementation of the feedback I provided was done only after I got parents involved. These were the two elements I added to the reflection and they became the major talking point I had with parents. They will also become a major talking point when I meet with my students.

One talking point that we’re quite familiar with is class participation. Students know themselves pretty well and parents do too. But asking students to self assess on active class participation invites a conversation on why one would try not to participate or volunteer answers. Not meaning to dwell on the negative, I do want to say it also gives me the opportunity to provide positive attention and acknowledge those who do participate, collaborate, and actively listen.

Other areas on the reflection sheet include class time use, assignment completion, quality of work, and attitude. For next quarter I want to add something about assessment preparedness and retakes, but I haven’t found the words yet.

I’m interested in making this better so if you have something that works for you, please share.

# Frayer model for estimation

Before we jump into adding and subtracting negative decimals I thought we should spend some time defining what estimation is and is not. That way students would be able to determine if their answer was reasonable. It took about a period and a half to clear up misconceptions but I think it was time well spent.

The basic structure of the lesson was 15 minutes of group work using giant whiteboards, followed by a 10 minute a gallery walk for feedback, then the remainder of the period to refine their definitions and examples.

I posted the following directions: Create your “bestest” definition of estimation and provide one example and non-example.

Here are the first drafts of student work from one of my classes:

Some groups were on the right track, but their definitions and examples could be more explicit. When it came time for the gallery walk the groups examined each others’ work, commenting on two things: one thing they like about the work and one thing that needs clarification. Groups struggling with the definition or example/non-example really benefited from seeing others’ work. The post it feedback also helped them to refine thinking.

At the end of the period I asked students to use their iPads to take a photo of their work and the next day we debriefed. The definition of estimation being an educated guess was lacking for me, so I added the phrase, “by using a strategy.” I think  that condition to the definition will allow us to continue our estimation conversation when we talk about scaling up or down with proportional reasoning.

Having students create their own version helped them participate in the final draft. We talked about where rounding occurs in estimation so we could identify the characteristics. One student thought it helpful to include arrows to clarify what is being rounded, and I threw in my caveat about using a strategy as part of the definition.

While it took a lot of time the students ended up owning the definition more than if I had told them what to write in their notes. The collaboration and post it feedback also helped those students who may have had some trouble getting started on their own.