Hanover School Division (HSD) Math Workshop

HSD Math Professional Development Workshop

May 2017


Thanks for inviting me to learn with you at the Hanover School Division (HSD) for a day full of math professional development!


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 multiplication and the order of operations by writing expressions. There are extensions to this task that we did not explore today, but are applicable for grade 9 applied.

Counting Principles

We briefly discussed the importance of counting and quantity principles, but focused on the importance of 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 Having spent the majority of my professional life teaching secondary math and mentoring intermediate (grades 7 to 10) math teachers, 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 se...


Would You Rather?

After break, I shared Would You Rather?, a website created by John Stevens that provides all kinds of great warm-up problems for a range of grade levels including grade 9 applied:

Would You Rather - Popcorn Brand A B C or D

Here’s a visual animation of the second problem we did involving fractions:

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...


As an aside, you might consider looking at the “Japanese Multiplication” or “Stick Multiplication” which is derived from base ten block representation:

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...


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 didn’t have time to dive into division with base ten blocks, but I thought I’d put that here for you to reference:

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

Here are some other tasks we explored as well:

Stacking Paper [3 Act Math Task]

How many packages of paper will it take to reach the ceiling?

Stacking Paper - 3 Act Math Task - Real World Math

Real World Math in Proportional Reasoning and Linear Relations Here's a new Real World / 3 Act Math Task that is related to proportional reasoning and a direct variation linear relation with a partial variation sequel to boot! Please note that this is part one of three tasks in a series: Stacking Paper - Direct Variation (y-intercept equal to 0) Stacking Paper Sequel - Partial ...

Summary & Resources

Stacking Paper Sequel [3 Act Math Task]

What is the height of the table?

Finding the Equation of a Line Given Slope and a Point - 5 Stacks of Paper on a Table

Finding the Equation of a Line Given Slope and a Point This 3 Act Math Task is a sequel to the Stacking Paper Real World Math Problem I created a couple weeks back. In the previous task, students were asked "How many stacks of paper will it take to reach the ceiling?" That problem had a sequel that explored the scenario of stacking the paper on a table instead of on the floor in order to see i...

Summary & Resources

Thick Stacks [3 Act Math Task]

How tall is the table?

Thick Stacks 3 Act Math Task - Height of 12 Stacks and 5 Stacks

Finding the Equation of a Line Given Two Points This 3 Act Math Task is yet another sequel to the Stacking Paper and Stacking Paper - Sequel 3 Act Math Tasks where we look at linear relations with respect to a real world situation involving packages of paper stacked into a tower. While the original task begins with a direct variation requiring only basic proportional reasoning skills, th...

Summary & Resources

The Stacking Paper Tasks are also available on iOS as a multi-touch book for iBooks:

Stacking Paper Tasks Multitouch Book for iBooks

Grab It!

Learn more about what’s inside the multi-touch book here.

Or, consider accessing these tasks in an interactive way via Google Sites.

3 Act Math Tasks: Curious Math iTunes U Course

Access over 30 tasks with all resources downloaded directly to your iOS Device to avoid streaming media in your classroom through the Curious Math iTunes U Course:

3 Act Math Tasks - Curious Math iTunes U Course

Enrol in Course

Other Useful iTunes U Courses to Check Out:

Learn Pythagorean Theorem Through Exploration – Kyle Pearce
Stacking Paper Tasks – Kyle Pearce
Beautiful Functions – Jon Orr

OR, you don’t want to download Jon’s Beautiful Functions book? Check out all of the links here on his blog.


h2>Feedback Appreciated!



Hope you folks found this YCDSB math professional development experience useful.

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

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