Visualizing Structure

The underlying structure behind many mathematical ideas is illustrated for students when teachers intentionally highlight these ideas using pictures and symbols. Students who see the structure behind ideas are able to think flexibly about numbers, shapes, and relationships and begin to build a foundation for algebraic thinking and reasoning. The structure of odd numbers will be used to illustrate this focus.

Odd Numbers

Think about your answers to the questions below for two minutes.

What is an odd number?
How do you visualize odd numbers?

Discuss your responses in your PLC

The Structure of Odd Numbers

Some typical definitions and visualizations are described below.

1. What is an odd number?

Numbers that end with a 1, 3, 5, 7, or 9.

Rex: Numbers that have pairs plus one.

2. How do you visualize odd numbers?

Otis: The numerals 17, 145, or 73.

Rex: Pairs of dots in a rectangle plus one more, 10+10+1, or 2(27)+1.

Why Visualizing Numbers is Important

Even and odd numbers provide a way for students to investigate patterns, make conjectures, and provide justification about the results. One conjecture that students make when adding even and odd numbers is that the sum of two odd numbers is always even.

Rex’s definition and matching visualization supports his conjecture as well as the justification for why it is always true. Imagine an odd number visualized as pairs of dots combined with one more dot, with another odd number visualized as pairs and one more. This new collection of dots will be made up of pairs or dots with two leftovers that make one more pair.

If Rex's definition and visualization of an even number is a collection of pairs then his visualization of an odd number plus another odd number would always make an even number.

Otis’ definition and visualization of adding two numbers that have an odd number in the ones digit does not support the making of a similar conjecture and furthermore does not support a convincing justification.

One of the eight Standards for Mathematical Practice (SMP) is: Look for and make use of structure (CCSS.MATH.PRACTICE.MP7). These practices highlight what it means for students at all grade levels to think and act like mathematicians. When teachers are intentional about writing symbols and highlighting pictures to bring out structure, they support students as they make sense of patterns and develop algebraic reasoning skills.

The Structure of Algebra

The emphasis on structure supports students as they learn algebra.

Look at the task below and discuss it with your PLC members.

Print out the Mod 3 Act 1 What is Writing for Structure Handout

Write on the pictures and symbols. It helps if you and your group members have pens or pencils with different colors available so you can highlight what you see and the connections you are making between symbols and pictures.

Task 1: Connecting Representations

Discuss with your PLC

How are the dots shown in each picture on the handout, connected to each symbolic expression.

Why Emphasizing Structure is Fundamental to Algebraic Thinking

A very important skill in algebra is the ability to notice how things are changing in a pattern. Symbolic representations of patterns take on new life when used to highlight things that are changing and things that stay the same. Writing for structure can be thought of as a tool that teachers can use to show how things change and how they stay the same. It gives teachers the power to highlight ways to compose and decompose numbers or shapes and gives students tools to represent their thinking as they investigate mathematical ideas.