Fraction Adventure #1: Post-Homecoming… Coming Home to Fractions

Homecoming is over.

The bands have played. The celebrations have ended. The alumni have gone home. And now, it is time for my preservice teachers to come home—to class!

Well…at least five of them did. 😂

Today was our first class after Homecoming, and 5 of my 14 preservice teachers made it to our 8:00 a.m. class. Knowing that Homecoming festivities can take a toll, I gave them a little grace and started class at 8:35. At first, there were only four students, but eventually we made it to five.

And because we are on our journey of reasoning with fractions, attendance immediately became our first mathematical task of the day.

So…What Fraction of the Class Is Here?

I asked a simple question:

“What fraction of our class is here today?”

Five students out of 14 gave us:

5/14

But simply naming the fraction wasn't enough. I wanted my preservice teachers to see it, represent it, and reason about it.

We placed 5/14 on a number line, giving us a linear model of the fraction. Then we created a 2 × 7 array to represent the 14 students as an area/array model.

From there, we ventured into fraction equivalence by creating another number line and lining up equivalent fractions.

And this was awesome!

Rather than beginning with rules, procedures, or “multiply the numerator and denominator by the same number,” we were reasoning about equivalence through representations.

But then I gave them another challenge.

Homecoming Football Meets Fraction Reasoning

Why leave Homecoming completely behind when we could bring it into mathematics?

Alabama A&M scored 61 points, while the opponent scored 31 points.

That gave us a total of:

61 + 31 = 92 points

I gave my preservice teachers square tiles and challenged them:

Can you create an area model representing the total 92 points while preserving each team's contribution to the score?

Then I got out of the way.

No step-by-step directions. No demonstration to copy. No “Here is how you do it.”

I provided only a few scaffolding questions and allowed them to wrestle with the mathematics.

What happened next was pure mathematical brilliance.

Let the Productive Struggle Begin

First, the students represented the opponent's 31 points. They counted 31 tiles and arranged them as five rows of six, with one tile remaining.

Next came AAMU's 61 points, represented with red—or, for our purposes, maroon—tiles. They began arranging those tiles in groups that corresponded with their first representation: five rows of 12, again with one remaining.

Then came my next challenge:

“Can you put the two groups together so that no tile is left out?”

Now the real mathematics began.

They moved tiles.

They rearranged rows.

They questioned one another.

They calculated.

They disagreed.

And, most importantly, they reasoned.

One student recognized that the combined score, 92, was even and suggested making two groups of 46. Another student grabbed her calculator to verify the thinking.

Then came another realization:

92 is divisible by 4, too!

That discovery opened the door.

The students began offering possible arrangements, accepting some and rejecting others based on their mathematical reasoning. Eventually, all 92 tiles fit beautifully into a:

4 × 23 array.

There it was—a visual model of the Homecoming score composed of maroon and gold tiles.

And Here Come the Fractions!

Once the array was complete, we returned to our fraction adventure.

Of the 92 total points:

AAMU contributed 61/92 of the points.

The opponent contributed 31/92 of the points.

And, of course,

61/92 + 31/92 = 92/92 = 1 whole.

Now the fractions were not just numbers written on a board. The whole had meaning. The parts had meaning. The denominator represented the total points scored, while each numerator represented one team's contribution to that whole.

That is exactly the type of fraction reasoning I want my preservice teachers to experience before they teach fractions to children.

Sometimes Five Is Enough

I started the morning thinking about the nine students who were missing after Homecoming.

By the end of class, I was thinking about the mathematical conversations happening among the five who were there.

A lecture would never have produced what I witnessed.

The students had to make sense of a problem, choose representations, test ideas, revise their thinking, justify their decisions, and listen to the reasoning of their peers.

They weren't simply doing fractions.

They were reasoning with fractions.

And I was impressed.

As for creating a linear model for 61/92 and 31/92?

Absolutely not.

Who wants to partition a number line into 92 iterations after Homecoming? 😂

We had already done enough.

Fraction Adventure #1 complete.

Next stop: more fraction reasoning—and hopefully more than 5/14 of my class!