Comparing Numbers with Concrete Manipulatives

Early experiences with numbers in kindergarten should include plenty of opportunities to use manipulatives. Hands-on experiences help students build mental images of numbers, develop number sense, and make mathematical ideas visible. One important concept students develop is comparing quantities. Let’s explore how using counters to compare numbers helps students build a deeper understanding of greater than, less than, and equal to.

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Compare Numbers with Manipulatives

In her book How Children Learn Number Concepts: A Guide to the Critical Learning Phases, Kathy Richardson explains that comparing quantities develops in stages. When two sets are lined up so they can be compared visually, it’s relatively easy for children to determine which group has more and which has less. A more challenging task is determining how many more one group has than the other. More challenging still is determining how many fewer, or how many less, one group has.

This activity is ideal for introducing comparing numbers in a small-group lesson before moving it to a workstation for independent practice. Students use counters as concrete manipulatives while the five-frames organize the quantities, making the comparison easier to see. Print the mat on cardstock and laminate it so students can write the numerals with dry-erase markers and use it again and again. For the playing cards, use only the Aces (as 1) and the 2–5 cards so students compare numbers within 5.

Working with a partner, students each draw a playing card to determine the two numbers they will compare. They build each number on the five-frames using counters and write the corresponding numerals on the recording mat.

If the two quantities are equal, students simply state the comparison (for example, “Five is equal to five.”). If the quantities are not equal, they spin the spinner to determine the comparison they will make: greater than, less than, more than, or fewer than. In the example shown, the partners drew a 4 and a 1. After building the two sets and writing the numerals, they spun fewer than, so they circled the numeral 1 and explained their thinking by saying, “One is fewer than four.”

How Many More? How Many Fewer?

Once students are comfortable identifying which quantity is greater or less, the activity can be extended to asking how many more or how many fewer (or less). Students first need to recognize that the two sets have a certain number of counters in common. The remaining counters in the larger set are the extras, which tells us how many more one quantity has and, at the same time, how many fewer the other quantity has.

Kathy Richardson points out that the question “How many more?” is often misunderstood. Many children interpret it as “How many are in the larger set?” and simply answer with the greater number. Instead, she recommends asking, “How many extras are in the larger set?” That language helps students focus on the unmatched counters rather than the total quantity.

When introducing this extension, begin with pairs of numbers that are only 1 or 2 apart, making it easier for students to identify the extras before progressing to larger differences.

Extend the Activity to Larger Numbers

As students become ready to compare larger numbers, continue using a linear model before transitioning to ten-frames. Linear arrangements make it easier for students to match the quantities and identify the extras, reinforcing the understanding of how many more or how many fewer. Trains of linking cubes are a great option because students can physically line them up and compare the unmatched cubes. You can continue to randomize the numbers students compare by using playing cards, ten-sided dice (for numbers within 9), or two standard dice (for numbers within 12).

If you prefer to continue using counters as concrete manipulatives, simply glue two five-frames together to create a strip of 10, as shown below. This simple modification extends the original activity while allowing students to compare quantities within 10 using the same playing cards and spinner.

Comparing numbers is much more than identifying which number is greater or less. As students use concrete manipulatives to build, compare, and discuss quantities, they develop the number sense needed to reason about how many more and how many fewer. Those early experiences provide an important foundation for subtraction, missing addends, and other comparison situations they’ll encounter in the primary grades.

If you’d like to use these activities with your own students, download the free comparing numbers activity and start building deeper number sense through hands-on learning.