Fractions Galore! The Week of January 26 - 30: Math Coaching at Doty

I had so much fun at Doty this past week working with Lisa (who teaches 6th grade math) and Sonya (who teaches 5th grade math) and their students.  Thanks to their careful questioning, I had a couple of "Eureka!" moments.

Right after Thanksgiving, we combined the 5th and 6th graders to teach the area of a rectangle model for multiplying fractions using the instructional plan:

• Outlined here.
• Illustrated in this blog post.

Since then, Lisa continued on to generalizing a conjecture for multiplying any two fractions and worked on developing contexts for multiplying fractions.  Then she moved on to teach multiplying mixed numbers, which I demonstrate in this video:

From there, she taught division of fractions and mixed numbers.

Goals

Lisa's goal for the week (and for Semester 2) is to solidify applications (contexts) for all operations on fractions.  Meanwhile, Sonya was hoping to focus on getting her 5th grade students to the Applications/Communications level for multiplying fractions.  Borrowing language from the UbD framework, the desired result is that we produce students who:

• Are able to perform any operations on any fractions efficiently with understanding;
• Can explain why the standard algorithms work using multiple representations (including concrete materials, pictorial symbolic models) as opposed to simply memorizing a process;
• Can explain why multiplication of a factor by a fraction less than one results in a product that is smaller than the factor and division by a fraction less than one results is a quotient that is larger than the dividend;
• Can generate equivalent fractions by multiplying (or dividing) the numerator and denominator by a common factor and can explain why the product is an equivalent fraction using the identity property of multiplication (any factor multiplied by one or a fraction that is equivalent to one has the same value as the original factor).
• Are able to apply operations on fractions flexibly in multiple contexts - both in mathematics and the real world - and can contextualize and decontextualize problems involving multiplication of fractions.

Monday 

We developed this formative assessment to gauge the students' understanding of operations on fractions: Linear Contexts for Operations on Fractions.  After Lisa's students completed the task, we took a look at their work and saw that students had a difficult time deciding which operation was which and using appropriate models (diagrams) to communicate their thinking.  So we worked up a plan for Lisa.

We chose two examples to combine for a "my favorite no" activity.

This piece had some solid reasoning, but the pictorial model was not particularly helpful and did not show a model that demonstrated an understanding of how the context involved multiplication.

This student used a more appropriate pictorial model, but confused division and multiplication.

Meanwhile in 5th grade, we realized that (although her students were using the standard algorithm to find the product of fractions) Sonya's students needed more work on understanding the models for multiplication and how they applied to fractions.

Tuesday and Wednesday

According to Lisa, her students did an "awesome job" talking about "My Favorite NO" and identifying that the pictorial model in the first example didn't match the problem, but that the pictorial model in the second example (while more appropriate) was more confusing to use to think about fractions.

Lisa gave each of her students a "track" - a strip of paper taped into an oval - and she walked them through using the model to visualize the problem.  According to Lisa, "we had a lot of 'aha' moments as we talked about what we were looking at."  Students decided it would be easier to find two thirds of the way around the track if it was in a straight line rather than an oval, so they "cut" it an represented it as a linear model.

Then, Lisa started to tie the multiplication problem back to the area model and students became confused.

With the 5th grade, Sonya spent time tool building using counting up by fractions on an open number line and on a grid (Tool Building Grid) and going back to look at "repeated addition," "equal groups" and the "area of a rectangle" models because understanding these models is key to being able to contextualize and de-contextualize multiplication.

Thursday

We pulled the 5th and 6th graders together.  In order to get students to understand multiplying fractions at the applications level, we need to tap into what they already understand about applying operations on whole numbers.  We started class with a tiered formative assessment, to find out what multiplicative problems students can contextualize with confidence.

We spent about 10 minutes on this activity, students chose a starting flavor and were counseled to try other flavors until they had done (at least) one they felt confident about and one that stretched them a bit.  We took note of who was picking what "flavor" (the colors made this easier to notice).   Sonya made note of the models her students were using to think about multiplying a whole number by a fraction.  Lisa and Sonya collected the student work for us to review in greater detail after class.

Then, we launched into looking at Area Contexts for Whole Numbers, modeling each of them using Cuisenaire Rods (letting the length of the white cube equal 1 and defining "one whole" as "one brownie"), modeling each of them using pictorially two ways (using a linear model and using an area model), identifying the operation involved, and writing an equation to represent the problem.

Students worked at their own pace, needing to check in with a teacher before moving on to Linear Contexts for Whole Numbers.  All of the students needed more than one day to complete this work.

After class we took a closer look at the colored paper tiered formative assessment and discussed where to go next.  We decided that we would bring the activity back after students worked on Friday, to gauge student growth.  We made notes of what advice we have for students to move forward in their thinking.

Friday

We decided that on Friday we would have the 5th and 6th graders combined again, and we would focus on linking the area model to the linear model for multiplying fractions before I left Doty.

Here's how it went:

Intuitive Hook (Formative Assessment):

We gave each student two different colored squares of paper and asked them to fold them each to model the same problem: two thirds times one fourth.  We reminded the 6th graders that this was the problem they were modeling the other day using a linear model.  Initially, this was going to be a quick activity to activate their prior knowledge, but we saw that there was some confusion.  So, we had to do some concrete tool building.

Tool Building

I had every one hold up "square 1" and we read it as a multiplication problem (1 times 1 equals 1).  Using Lisa's "Council of Randomness" (that's what she calls her popsicle stick container - because Lisa is brilliantly creative with classroom management stuff like that), I selected students to give me one step at a time in the process of modeling the problem: two thirds times one fourth.  After each step, I asked them to read the square as a new multiplication problem.  Here's what that looked like:

Step 1:

Student 1: Fold it in half (horizontally)... 

I open it up and ask:  What is this as a multiplication problem now?

Student 2: As a multiplication problem, it's "two halves times one equals two halves"

I ask:  Is that equivalent to one whole? 

Student 3: Yes.

I ask:  How do you know?

Student 3:  Because it's still the same area, it's just been divided into two sections. 

Step 2:

Student 4: Fold it in half again (horizontally)... 

I open it up and ask, what is this as a multiplication problem now?

Student 5: As a multiplication problem, it's "four fourths times one equals four fourths"

I ask:  Is that equivalent to one whole? 

Student 6: Yes.

I ask:  How do you know?

Student 6:  Because it's still the same area, it's just been divided into four sections. 

Step 3:

Student 7: Fold it into three parts (vertically)... 

I ask: Three equal parts? (The student nods.)  Do you have a good way to do that?

Student 7: Yes.  I fold it so that the first part and the third part have the same width.

I open it up and ask, what is this as a multiplication problem now?

Student 8: As a multiplication problem, it's "four fourths times three thirds equals twelve twelfths"

I ask:  Is that equivalent to one whole? 

Student 9: Yes.

I ask:  How do you know?

Student 9:  Because it's still the same area, it's just been divided into twelve sections. 

Step 4:

Student 10: Flip it. (So, I flip it front to back.) No, turn it, rotate it 90 degrees.

Ask: What property allows me to do this?  (The Commutative Property of Multiplication.)

I hold it up and ask, Everyone, what is this as a multiplication problem now?

All: As a multiplication problem, it's "three thirds times four fourths equals twelve twelfths"

I ask:  Is that equivalent to one whole? 

All: Yes.

I ask:  Is that equivalent to four fourths time three thirds? 

All: Yes.

I ask:  How do you know?

Student 11:  Because it's still the same area, it's just been rotated. 

Step 5:

Student 12: Fold it back up so that the horizontal length is one fourth

I hold it up and ask, what is this as a multiplication problem now?

Student 13: As a multiplication problem, it's "three thirds times one fourth equals three twelfths"

I ask:  What is that equivalent to? 

Student 14: One fourth.

I ask:  How do you know?

Student 15:  Because it's still the same area as it was when it was one fourth times one. 

Step 6:

Student 16: Fold one third up and back up so that the vertical length is two thirds.

I hold it up and ask, what is this as a multiplication problem now?

Student 17: As a multiplication problem, it's "two thirds times one fourth equals two twelfths"

I ask:  How do you know?

Student 18:  Because there are twelve sections in the whole square and the part we're looking at is two out of twelve.

I ask:  What is that equivalent to? 

Student 19: One sixth.

By the end of Tool Building, students had two identical area models with two twelfths shaded.

Concept Building

I told the students to set aside one of the squares to keep it pristine because we were going to refer back to it.  Then, we revisited the track problem from Monday: How far did I run if I ran two thirds of the way around a track that is one fourth of a mile long?  I asked the students, what does "Square 1" (one whole) represent in that problem and what does the shaded area represent?  They thought about it independently, then talked in pairs.  While we walked around listening to their reasoning.  

A common misconception was that "square one" represents one whole lap.  But they were able to see that the shaded area represents the distance I ran.  So we were able to cut off one strip of the whole square to represent one lap which is one fourth of a mile.

Then we cut off a strip to represent each of the four laps needed to run one whole mile.

I asked students to think about whether or not cutting the square into strips changed the area.  Do the strips still represent one whole mile?  While I was walking around, I saw one student placing her strips to cover her "pristine" square one like this:

I got really excited and asked everyone to come over to see.  I asked another student why I was so excited.  What does this model demonstrate?  He was able to see that she was showing that the two models have the same area.  This demonstrates the idea of "conservation of area" which is crucial for mastering multiplicative reasoning and fractions.

Randomly, I asked the students what kind of model this was.  It was an area of a rectangle model.  I asked them how they know it's an area model.  It has two dimensions.  Then I asked students if it was possible to organize their model in a line like a number line or line plot.  Would that change the area?  Would it still represent the same problem?

Students were able to arrange their strips in a horizontal line, but many did this (having the shaded area somewhere in the middle):

I had the students do a "museum walk" around the room for 30 seconds, I told them I wanted them to notice how people arranged the shaded pieces.  When they returned to their seats, I randomly picked students to share out what they noticed and they saw that the most popular arrangement was like this (with the shaded area to the extreme left):

We decided that this makes sense because the first strip represents the first lap, and I ran two thirds of one lap (what would have been my first lap).  Rolling the first strip to represent a lap (as Lisa did on Tuesday) helps to illustrate this:

Summarizing

Finally, I asked students to think about these questions:  How does this linear model represent the problem?  What is the part?  What is the whole?  Students thought for one minute independently, then shared in pairs.  Meanwhile, I taped the linear model to Lisa's white board and listened for a good summary.  While I was listening, I found a student to share out her ideas.  She walked up to the board and explained:

 Each strip represents one fourth of a mile and the shaded part is two thirds of one fourth.  The whole line represents one whole mile, which has been split up into twelve sections.  The shaded part is two of the sections, so it represents two twelfths of the whole mile.

Other students built on her idea, explaining that the shaded area is equivalent to one sixth.

Then we emphasized that this was a linear (one dimensional) model for the multiplication problem that can be recorded pictorially using a line plot.  This is a different way of modeling the very same problem that we modeled using area (two dimensions).

Exit Card

At the end of class we had students take another stab at their color coded formative assessment from Thursday giving them time to thing about how to apply multiplicative reasoning on fractions to new contexts.  Lisa and Sonya collected these at the end of class.

Next Steps

This week, Lisa and Sonya can revisit the work on contexts for operations that we started on Thursday letting students work at their own pace.  Once students have finished Contexts for Operations on Whole Numbers, they need to check in with their teacher before moving on to Linear Contexts for Operations Whole Numbers,  Area Contexts for Fractions (not expecting the 5th graders to be able to do the division), and Linear Contexts for Operations on Fractions.

It would be a good idea to give students more feedback on their color coded formative assessment sometime early in the week, and perhaps create a new color coded formative assessment for the end of the week.

FeedBack

Teachers:  With this communication I am trying to give a wider audience the benefit of the collaborative work that is happening at the building level.  Please let me know if this kind of post is useful to you.  It is very time consuming for me, but is entirely worth it if you find it valuable.  So please let me know. 

Thanks!