Peel Elementary Teachers Local (PETL)


PETL Math Professional Development Workshop

Saturday March 25th, 2017

Thanks for inviting me to learn with you on a Saturday in Peel! The group was energetic and excited to learn about meaningful manipulative use, counting principles, the progression of multiplication/division and the importance of concreteness fading!

Here’s a summary of what we explored today. Looking forward to connecting again soon!


After a brief presentation about avoiding the rush to the algorithm, we took some time to explore a 3 act math task with a very low-floor and high-ceiling, called the Airplane Problem:

Airplane Problem – Trip to Toronto [3 Act Math Task]

How many seats are there?



Airplane Task - Trip to Toronto - From Subitizing and Unitizing to Multiplication and Algebra

From Subitizing and Unitizing to Multiplicative and Algebraic Thinking This 3 act math task was designed specifically to have a very low floor in order to be useful from primary grades and a high ceiling with an opportunity for many extensions so the task can be used in junior and intermediate classrooms. While I believe this task can touch on many different specific expectations at man...


Summary & Resources

In the airplane problem, we made predictions and made connections to subitizing and unitizing.


Counting Principles

We briefly discussed the importance of counting and quantity principles like subitizing and unitizing. A full summary is below:

Counting Principles – Counting and Cardinality


Counting Principles - Principles of Counting and Quantity Featured Image

A Progression of Counting and Quantity As a former secondary math teacher and intermediate math coach, my new role as K-12 math consultant has led to a wealth of knowledge that I wish I had during my years spent in the classroom. My conversations about student learning needs with intermediate and senior math teachers always seems to come down to gaps in student understanding, however rarely wer...


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Tiny Polka Dots

After break, I shared Daniel Finkel‘s game, Tiny Polka Dots. We made some connections between the counting and quantity principles and the different card decks in this game.

Tiny Polka Dots - Math Game - Daniel Finkel

Also be sure to check out his TED Talk here.


Exploring the Progression of Multiplication

We made a leap from counting principles and unitizing to multiplication and specifically, using arrays and area models.

The Progression of Multiplication


Progression of Multiplication - Area Models and Standard Algorithm Featured Image

Arrays and Area Models to The Standard Algorithm Did you know that the words "array" and "area model" appear in the Grade 1-8 Math Curriculum a combined 22 times? Not only do arrays and area models help to support the development of proportional reasoning when we formally introduce multiplication in primary, but they also help us understand how to develop strategies that lead to building numbe...


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We also made a connection to “Japanese Multiplication” or “Stick Multiplication”:

Why Japanese Multiplication Works


Why Japanese Multiplication Works - Using Lines to Multiply Is Not a Math Trick

Japanese Multiplication? Chinese Multiplication? Line Multiplication? Whatever it's called, it's only a trick if you simply memorize without meaning Have you ever wondered why Japanese multiplication works? I've heard some call it Chinese multiplication, multiplication from India, Vedic multiplication, stick multiplication, line multiplication and many more. While many might argue as to the...


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3 Act Math Task: Donut Delight

After lunch, we explored the Donut Delight 3 act math task where we played with multiplication to predict how many donuts there were in the “double hundred dozen” box and then used repeated subtraction to lead to a flexible division algorithm for division when we tried to find how many layers there were.

Donut Delight [3 Act Math Task]

How many doughnuts are there?



Krispy Kreme Donut Delight - 3 Act Math Task Screenshot

Sparking Curiosity to Fuel Sense Making of Multiplication and Division When one of our district math leads, Brennan Jones asked me to brainstorm some ways we could help his staff engage in some professional development around division and incorporate a 3 act math task into the learning, I immediately thought of some contexts where arrays, base ten blocks and area models could be used to h...


Summary & Resources


Progression of Division

We spent some time working with the progression of division in most groups as well. Here are some screenshots of what we looked at:

Quotative and Partitive Division

Division With Base Ten Blocks 112 Divided by 8

Division With Base Ten Blocks 120 Divided by 12

Division With Base Ten Blocks 189 Divided by 9

Division With Base Ten Blocks 221 Divided by 13


Concreteness Fading

We attempted to summarize the use of manipulatives on a continuum called “Concreteness Fading”. While the name suggests that concrete manipulatives fade away over time, it is important to remember that they fade away with one layer of abstraction and then reappear as a new layer is “piled” on.

Applying Concreteness Fading to Progression of Multiplication

Hope you folks found this professional learning experience useful.

I’d be delighted to come back and learn alongside you all again sometime soon!

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