Methods for Comparing Two-Digit Numbers

Introduction

Welcome to our lesson plan on comparing two-digit numbers. First, we will review how to identify and represent two-digit numbers using various tools and methods. We will explore different ways to compare these numbers and learn how to use digits to determine which number is larger or smaller. By the end of this lesson, your learner will be able to confidently compare two-digit numbers using both visual aids and numerical strategies.

Before beginning the lesson, your learner should be able to identify the tens and ones values of two-digit numbers. They should also be comfortable using number lines up to 100.

Key Concepts for Comparing Two-Digit Numbers

The following concepts are helpful to know for this lesson:

• Cardinality: Cardinality refers to the quantity of items in a set or group. Dienes blocks, hundred squares, and other visual aids represent the quantity of tens and ones in two-digit numbers.
• Ordinality: Ordinality refers to the position of numbers in a sequence. Number lines and Gattegno charts can be used to visually demonstrate the order of numbers.
• Structure of Two-Digit Numbers: Two-digit numbers consist of a tens place and a ones place. For example, he number 52 has 5 in the tens place and 2 in the ones place.
• Cardinal Method for Comparing Numbers: Compare numbers based on the quantity of tens and ones. For example, 52 has five tens (50) and 2 ones, while 28 has two tens (20) and 8 ones. Since 50 is greater than 20, 52 is greater than 28.
• Ordinal Method for Comparing Numbers: Compare numbers based on their position in the counting sequence. For example, when counting, 28 comes before 52. Therefore, 52 is greater because it comes later in the sequence.
• Symbols for Comparison: Greater Than (>) indicates that the first number is larger than the other (52 > 28). Less Than (<) indicates that the first number is smaller than the other (28 < 52). Equal To (=) indicates that two numbers are the same (30 = 30).

Teaching Plan

The following activities will help your learner become confident with comparing two-digit numbers. Be sure to work at a pace that is comfortable for your learner.

Examples and visuals to support the lesson:

1. Review of Two-Digit Numbers

Begin with a review of identifying and representing two-digit numbers. Discuss cardinality using Dienes blocks, hundred squares, and other visual aids. Explain ordinality with number lines and Gattegno charts. These models will be used in the lesson to help your learner compare numbers.

Here are some examples that reinforce cardinality:

• Represent the number 36 with three tens blocks and six ones blocks. Show how to count the blocks to understand the value of the number.
• Highlight the number 52 on a hundred square by showing five rows of ten squares and two additional squares.

These examples can be used to review ordinality:

• Provide your learner with a number line and have them label the location of two-digit numbers such as 19, 29, and 48.
• Practice using a Gattegno chart to show the order of numbers 1 to 10 and multiples of ten. For example, ask your learner to point to 3 on the chart, then locate 30.

Skill Check
I can use different math tools to represent two-digit numbers.

2. Comparing Numbers with Different Methods

Present examples of two methods for comparing numbers. Here's an example of two students comparing 52 and 28 in different ways.

• Javed says, "I know that fifty-two is greater than twenty-eight because fifty-two has five tens sticks, and twenty-eight only has two tens sticks."
• Nancy says, "I know fifty-two is greater than twenty-eight because it comes later in the counting sequence. When I count to fifty-two, I have to count past twenty-eight."
• Discuss with your learner the reasons given by Javed and Nancy, highlighting their experiences with cardinality and ordinality. Encourage your learner to make connections to the tools the used earlier (Dienes blocks, hundred squares, number lines, Gattegno charts, etc.).

Skill Check
I can use math tools to compare two-digit numbers.

3. Comparing Two-Digit Numbers with Different Tens Digits

• Explore how to compare two-digit numbers by examining the digits without building the numbers with Dienes blocks or counting through them.
• Demonstrate that when the tens digits are different, you can compare two-digit numbers by looking only at the tens.
• Compare two numbers with different tens digits such as 37 and 14. Have your learner compare the numbers using the method of their choice.
• Show that changing the ones digits does not affect the comparison. Just as 37 > 14, you can demonstrate that 32 > 18 or 35 > 12. In other words, any number with 3 tens is greater than another number with 1 ten.
• You can visually represent this concept by removing or adding ones to the Dienes representation or sliding the numbers along on the number line.

Skill Check
I can compare two-digit numbers by looking at digits in the tens place.

4. Comparing Two-Digit Numbers with the Same Tens Digits

• Investigate what happens when the tens digits are the same such as 63 and 68.
• Hide the ones digits and ask your learner to identify which number is greater. They won't be able to determine the larger number.
• Reveal the ones digits and ask them to compare the numbers. Your learner should be able to see that 63 is less than 68.
• If needed, you can incorporate visual aids to demonstrate why it is important to compare the ones digits when the tens are the same.

Skill Check
When the digits in the tens place are the same, I can compare the digits in the ones place.

5. Working Towards a General Rule

• Provide your learner with varied practice. Initially, they can use Dienes blocks or a number line as a scaffold. Encourage them to progress to working only with numbers written as numerals.
• Work towards the simplified generalized statement: "To compare two-digit numbers, we need to compare the tens digits. If the tens digits are the same, we need to compare the ones digits."
• Include comparisons with both equal and unequal tens digits to solidify the concepts. Continue to use >, <, and = symbols for comparing quantities and measures.

Skill Check
I know that when I compare two-digit numbers, I have to compare the digits in the tens place first. If they are the same, I have to compere the digits in the ones place.

Summary

In this lesson plan, your learner has reviewed how to identify and represent two-digit numbers using various tools. They have explored different methods for comparing numbers and learned how to compare two-digit numbers by examining the digits. By practicing these skills, your learner will be able to confidently compare two-digit numbers using both visual aids and numerical strategies. Keep encouraging your learner to practice these concepts to reinforce their understanding and proficiency in comparing numbers.

Teaching Plan adapted from NCETM under OGL license v3.