Number Representations: Building Meaningful Math Connections

Representing numbers in different ways helps students connect what they can build, draw, say, and write. A collection of counters, a ten frame, and an equation can tell the same math story. Here are simple ways to use math manipulatives and number representations to build number sense, encourage math talk, and help students explain their thinking; from counting and place value to fractions.

Building Number Sense with Math Manipulatives

Before we ask students to represent a math concept, we want to give them the tools to do so. One of the easiest ways to build excitement about going to the teacher-led small group table is through the use of varied math tools and manipulatives. We’ve shared how you can quickly boost engagement each month and season by gathering the seasonal colors and trinkets that excite students in all grades.

Most of the manipulatives below are self-explanatory, but here are a couple of out-of-the-ordinary ideas.

The trays can be found on amazon. I undid the lid for these pictures, so it’s easy for students to grab and it doesn’t take up much space on the table, but It snaps right back on for safekeeping!

Representing Numbers in Different Ways

Let’s begin by having students show one way to represent a math concept. We are using our Make it Another Way Mats below. On the left, we see 16 made with a dime and six pennies. You can further explore coins and continue to ask for different combinations of coins that demonstrate the value 16. On the right, the student gives the bat a value of ten and the centimeter cubes the value of mosquito ones ;). When students make a representation, ask them to explain it. This can provide shared learning between students at the table as well as provide practice in communicating mathematical ideas.

There are a variety of mats to allow students to make different amounts of math representations. These representations can be done as partners, table groups, or individually. Below, we lovingly call this mat the round-about.

Math Talk Questions That Connect Representations

After students build a number, ask them to show the same value in another way placing the two models side by side so students can explain what changed and what stayed the same.

• How do you know both models show the same number?
• Where do you see a group of ten? Where are the extra ones?
• Can you find a group of five or ten to help you count?
• How could you draw what you built?
• What number or equation matches your model?

For example, a student might show 16 with a full ten frame and six more counters, then build it with one ten and six ones. Writing 16 = 10 + 6 beside both models helps connect the materials to the equation.

For students who need support, start with a smaller number and two familiar models. Extend the task by asking students to find another representation and explain why its value is still the same.

Connecting Number Representations to Place Value and Operations

Meaningful math connections can be made in all math strands. We typically think of number sense, computations, and place value when we think of using manipulatives, but meaningful math connections happen anytime we allow students to explain their thinking. The more this becomes a daily practice, the more students connect and creatively explore representations of different values across strands.

Model this practice in the classroom through daily calendar routines, having fun with the number of days in school, and representing all things math-related in your grade-appropriate way.

Upper-grade students can connect these models to 39 = 30 + 9, then explore how 3 × 13 = (3 × 10) + (3 × 3).

Keeping things hands-on vertically through the grade levels provides that deep understanding and the ability to connect math concepts.

Representing Fractions in Different Ways

Let’s discuss different ways to give students confidence in understanding fractions.

Free Number Representation Game: Creepy Crawly Cover Up

A few spiders and a web turn number representation practice into a hands-on math station! Creepy Crawly Cover Up includes picture cards with quantities shown through ten frames, tally marks, dice, dominoes, and collections of objects. Students flip a card, find the matching number on the web, cover it with a spider, and color the matching section on their recording sheet.

An addition-to-20 version is also included. Students solve the equation, cover the matching sum, and record the equation in the corresponding section.

Choose cards within the number range your students are practicing. Introduce the game at the small group table, then use familiar cards for partner or independent practice.

To keep the math connections going, pause for a quick question: “How did you see that number?” Students can point out groups, explain a counting strategy, or build the quantity another way before taking their next turn.

More Hands-On Math Activities

The following posts share many ways to keep up meaningful math connections through representations. Each image will take you to the post.