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January 09
 Inside the Classroom - Ooooh! Groups OF!

This is the start of a series of posts that show  a glimpse into some of the learning inside our classrooms at Belfast.  The following documentation of an interview with  Mrs.  Ayer one of our Grade 3/4 teachers.

I wanted to have students visually see what happens when you multiply a whole number and a decimal, so I gave students a series of 10 x 10 grids. We had a discussion about what one tiny square represented (hundredth), a row (tenth) and the entire square (one whole). Then I posed the question:  what is 6 groups of 1.3?

We talked about that we know what 6 groups of 1 is (6) and we set that aside.  But what is 6 groups of 0.3? I asked the students to color one group of 3 tenths.  Then I told them to color another group a different color, then another…

This is where we had a confused moment.  Some students were coloring in three hundredths, some were confused by what I meant by how many groups of tenths do we have (many said 3 instead of 1)...

After struggling for a bit I then changed the equation to one without decimals (6x3) and drew visual groupings of three on the board. “Oh!!! Groups OF!!!” ( the true languaging of multiplication; 6 x 3 means 6 groups of 3). Then they were able to understand that the three tenths were one group we were counting and then able to draw 6 groups of 3 tenths.

Some kids then asked, “What if I need the next grid because I filled in my first square? Is that allowed?”
Yes! The connection is that the one filled in  is a whole and then the remaining tenths left over became the decimal number (1.8).

That’s when a student said  “I just put my decimal between the two squares…” and everyone was like  - brilliant!!

We then tried a few more examples so they could practice and visualize. After this point is when they were let in on the trick of removing the decimal point from the equation and adding it back in afterwards.  Some students commented “Oh. That’s way easier.”

It’s important to understand what’s actually happening to the numbers. Students  need to know how it has grown or changed. Multiplying a decimal needs to be visualized so they can see why the number or place value is changing. If they only learn the trick they know the trick, not the concept.
What I found more often was that the visual learners liked using the visual over and over, whereas other students were ready to make the leap to using only the numbers; however, both groups can successfully  do the multiplication.

What would you do differently?
I will work with the grids before introducing multiplication, so the students  really understand how decimals build to wholes.

We used this learning to EXPAND to applying sales tax in a ”build your own aquarium project.”  Several students used the  model to calculate sales taxon what they had purchased for their aquarium.  We will also use this same model to extend to fractions  (to solidify parts of a whole understanding) and to visually prove that fractions and decimals are different ways of representing equivalent values.

Reflecting on the School Development Plan:
There is so much to celebrate here about mathematical learning, for both our students and our staff.  As our School Development Plan outlines, it is clear that students and staff alike are shifting their mathematical mindsets to one that encourages visualizing math, embracing mistakes as an opportunity for more learning, and working together collaboratively to discover new understandings in math. Actively designing and engaging students in rich mathematical tasks that go far beyond simple computation is allowing students to make deeper connections to mathematical concepts, and to view themselves as competent, confident mathematicians. And the staff as competent, confident math teachers!​