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Fraction multiplication task and tree diagrams

November 22, 2014

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

November 9, 2014

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

November 9, 2014

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:

estimation1.jpeg estimation2.jpeg estimation3.jpeg estimation4.jpegestimation5.jpegestimation6.jpeg

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.

Reflective writing in math

October 19, 2014

Somehow there’s a disconnect between my writing expectations in math and what’s being turned in. For the subtracting integers portfolio which I’ve written about here and here I provided ample time, included both the criteria for success and rubric, plus discussed the reflection in class on multiple occasions. Either I’m not making my expectations clear or my students are not used to doing reflective writing that includes an analysis. I’m guessing it’s both so I created a graphic organizer to help them be more reflective and analytical.

First of all here’s the rubric the students are using:


And here’s an example that would benefit from a graphic organizer:

“In my portfolio I use multiple strategies. For the first thing we had to We had to use visual models you things certain ways to solve it. Such as -x -y or x-(-y). Then we had to use five positive numbers and negative to do this I first made a bunch of positive and negitive fives and just put it together and solve it and then I just made of a problem of -20+ positive 25 equals five.”

The student below is beginning to meet my expectations, though a revision using a graphic organizer will help provide specific evidence that’s currently missing.

“I can solve problems without giving up. What I used to help me was examples from the past worksheets that we did a while ago. … I can use math tools and explain what I use them. I used a number line as a math tool and I use this to show in understand my work. Kind of like a visual for me. I can work carefully and check my work.  I work carefully keeping my work organize and neat. I also check my work by doing the opposite of the problem and see if it kind of matches. It’s hard to explain…”

This Written reflection subtracting integers graphic organizer isn’t perfect, but I think it will help.  I’ve clarified and provided some examples of insightful analysis:


Using the graphic organizer, students now have a work space to brainstorm which of the eight math practices were utilized and how.


Image edited for space. Practices 5-8 are also listed on the graphic organizer, just not shown.

On Monday, I’ll share student work samples and ask the students to both critique and evaluate the reflections based on the rubric and graphic organizer.

I’ll update and report progress in my next post.


Formative assessments using Explain Everything

October 12, 2014

A couple of weeks ago I wrote a piece about student portfolios, specifically a subtracting integers portfolio my students were about to create. Their media of choice is Explain Everything. Now that the projects are trickling in I’m discovering conceptual errors which I might have overlooked on a traditional paper-pencil assessment. Here’s an example:


While the screenshot does capture the number line error when subtracting a positive number, I’m drawn to the student’s voice and explanation, “Since 2 is negative I move to the right.” The portfolio is no longer a thing to grade, there’s a human being behind that work and I can see and hear her. Her portfolio submission has instead become a formative assessment.

This next problem demonstrates further confusion. Here 2 is negative, but she moves to the left.


The student below, on the other hand, has mastered subtracting integers. She begins by demonstrating the number line movements, explains why she moved to the right, then proceeds to show its equivalence by adding the opposite.


When solving this equation the student admitted to a lot of guess and check, however I’m not bothered by the extra practice she got from doing the problem.


After introducing this slide she explained how she turned it into an addition problem by adding the opposite.


Then she solved.


So far my favorite student created real world problem is:

integer real life problem

Only three projects have been submitted thus far, but I’m anticipating most of my time will be spent giving feedback rather than issuing grades. I don’t feel comfortable grading something that’s a work in progress. In terms of time, the portfolios run between 6 and 7 minutes each. I really need to front load the time now so it will be easier later when the students apply the integer rules to negative fractions and decimals.


Here’s another project I looked at this morning. How much conceptual understanding does this student have? He provides no explanation and relies on adding the opposite to demonstrate the number line. Also, he only provided a screenshot of the equivalence problem with no explanation of how it was solved.


Goal: student portfolios

September 28, 2014

I haven’t seen much talk lately on twitter or in the blogosphere about portfolios. I want to deepen students’ understanding of concepts so I thought I’d steal/modify a portfolio idea that was shared at last week’s math curriculum meeting. If you can help me figure out a way to add more choice while still addressing the concepts, I’d love your input. As it stands right now choice is limited to product, and in a small way–process.

The original idea shared was too open ended for me, “Prove you have mastered the learning objective.” While I’m a free spirit in many ways I think the students need some focus so I came up with specific criteria for both the artifacts and written reflections, along with a rubric for each.

Subtracting Integers Criteria for Success


Submit three artifacts demonstrating conceptual understanding of subtracting integers. Record your thinking using Explain Everything, iMovie, or other format.

  • create visual models. Include the following scenarios:
    • x– y
    • x– y
    • x– (– y)
    • x– (– y)
  • create and solve an equation using a series of five different positive and negative integers on BOTH sides of the equation to make the statement true.
    • Example: –a – (– b) + c – d – (–e) = f + (– g) – (– h)  –  i + (– j)
  • create and solve an original, real-world integer problem where absolute value is applied.

Here’s the artifact rubric. It’s generic so it can be applied to any collection of artifacts. I didn’t include point equivalents or percentages because I want the students to focus on the “feedback”. They’ll be able to resubmit.



Submit a written reflection of your artifacts. Include:

  • an analysis of the artifacts
  • the math practices applied and how you have applied them
  • the modes of representation used
  • proper grammar, spelling, conventions

Here’s the reflection rubric. It’s also generic.


When I’m ready to grade I’ll translate the levels into percent equivalents. For example:

  • Level 4 = 100%
  • Level 3 = 90%
  • Level 2 = 70%
  • Level 1 = 60%
  • Level 0 = 50%

As the portfolios are turned in I’ll share student work and let you know how it goes.

Other bloggers sharing their goals can be found by clicking the link below.

Good, bad, ineffective feedback using Canvas

September 20, 2014

Last week I introduced integers using a James Tanton Math Without Words visual puzzle.  Below are examples of the student feedback I gave. I used our new Canvas LMS system as the work flow, but what’s important is the varying degrees of feedback I found myself giving students.

Since some students don’t bother to read feedback–especially when everyone, including me, is getting used to Canvas, I started class by asking the students to pull up the assignment and read the comments.

The student below read my feedback and asked very politely, “What am I supposed to do?” This is yet another reason why I will never earn National Board Certification.


Had I thought about it ahead of time, my feedback should have read, “If every bump is a hill and every dip is a valley, what remains?” That’s something I did for another student. See last image.

I gave this kid nothing. At best I offer hope and a growth mindset but in terms of informative and constructive feedback, it’s terrible–absolutely terrible.

Here’s another student. I acknowledge he’s made the connection to integers, but…

canvas feedback3

…I should have added this:


For students who clearly made the connection to integers, I should have been prepared to offer feedback that extended their thinking, but I wasn’t.


Here’s another example of awful feedback. I acknowledge the problems are correct, but I offer nothing.

canvas feedback2

I should have extended this student’s thinking. I could have added, “If these were numbers what would they be?”

Finally, here’s some feedback that is potentially helpful.


When I gave the feedback I didn’t have a plan. I just viewed the student work one at a time and made comments. That made my feedback inconsistent and not effective. What I should have done was examine the student work collectively, and prepare feedback based on common misunderstandings and extending student thinking.



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