## Visualizing Mathematics Related Posts

## The Progression of Division

Division From Fair Sharing to Long Division and Beyond Over the past school year, I have had an opportunity to work with a great number of K to 8 teachers in my district with a focus on number sense and numeration. As a secondary math teacher turned K-12 math consultant, I’ve had to spend a significant amount of time tearing apart key number sense topics including the operations. While I often he...

## Why Japanese Multiplication Works

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...

## Counting Principles – Counting and Cardinality

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...

## The Progression of Fractions

Exploring Fraction Constructs and Proportional Reasoning Fractions are a beast of a concept that causes struggles for many adults and students alike. While we all come to school with some intuition to help us with thinking fractionally and proportionally, the complexity quickly begins to increase as we move from concrete, to visual, to symbolic and from identifying, to comparing, to manipulatin...

## The Progression of Multiplication

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...

## The Progression of Proportional Reasoning From K-9

The Foundation and Development of Multiplicative Thinking Proportional Reasoning is a big idea that is connected in some way to all mathematical strands and stretches across many grades in the Ontario mathematics curriculum. While many focus on the importance of proportional reasoning for students in junior and intermediate grades, I believe that we can see the early development of proportional r...

## Connecting Relationships to Multiple Strands

Exploring the Relationship Between Diagonal Length and Side Length of a Square It's been a couple weeks since posting, so I thought I need to get back in the swing of things. I've been enjoying a really busy start to semester 2. I'm loving the implementation of the gamified assessment approach I've modified off of Jon Orr and Alice Keeler's work, the spiralled approach to teaching grade 9...

## Visualizing Proportional Reasoning: Working With Ratios and Proportions

Solving Proportions With The Toronto Maple Leafs Win:Loss Ratio As we began the Proportional Reasoning unit in my MFM1P Grade 9 Math Course, I was beginning to struggle with an idea to extend the idea of visualizing mathematics concepts into ratios, rates, and proportions. What I came up with was using a visual representation of Toronto Maple Leaf wins and losses in order to help scaffold stud...

## Visualizing Two Variable Linear and Non-Linear Relationships

Tables of Values, Graphing Scatter Plots, Describing Relationships and Making Predictions Although generally not the urgent student learning need in most classrooms, working with two-variable linear and non-linear relationships could also be made more visual to help students better understand some of the important pieces that will help with linear equations and beyond. As mentioned in posts pr...

## Visualizing the Maximum Area of a 3-Sided Rectangular Enclosure

Increasing Spatial Reasoning Skills: Optimization of Measurement Although maximizing the area of a 4-sided rectangular enclosure is usually considered quite basic for most students, understanding why maximizing a 3-sided rectangular enclosure does not yield a square can be more difficult. To help address this student learning need, here is the animation created for maximizing the area o...