Adding and Subtracting with Tens and Ones

Introduction

In this lesson plan, we progress from representing two-digit numbers as part-part-whole diagrams to solving equations. Your learner will explore adding and subtracting with tens and ones as a strategy for solving equations and story problems. This will provide your learner with a deeper understanding of number composition and the ability to solve related word problems.

Before beginning the lesson, your learner should be comfortable partitioning numbers into tens and ones.

Part-Part-Whole Diagrams

Part-part-whole diagrams, such as cherry diagrams and bar models, are visual representations that help learners understand how numbers can be broken down into parts and combined to form a whole. For example, the two-digit number 37 can be decomposed into 30 (tens) and 7 (ones).

• Cherry Model: In a cherry model, the whole number (37) is at the top, with two branches leading to the two parts, 30 and 7.
• Bar Model: In a bar model, the whole number (37) is represented by a bar, with sections of the bar labeled 30 and 7 to show the parts.

Linking Diagrams to Equations

Part-part-whole diagrams can be represented as addition and subtraction equations.

• Addition Equations: Using the example of 37, the part-part-whole diagrams can be translated into these addition equations: 7 + 30 = 37; 30 + 7 = 37; 37 = 7 + 30; 37 = 30 + 7.
• Subtraction Equations: The same diagrams can also be used to form subtraction equations: 37 - 7 = 30; 37 - 30 = 7; 30 = 37 - 7; 7 = 37 - 30.
• Different Arrangements: Notice that some of the equations represent the same calculation but with the numbers arranged on different sides of the equal sign.

Common Mistakes for Adding and Subtracting with Tens and Ones

Here are some common mistakes that your learner may encounter during the lesson:

• 32 = 3 + 20 (Incorrect): Learners may generalize a pattern of taking the digits in the order they appear and copying them down, ignoring the 0. Explain that 32 is composed of 3 tens and 2 ones. The equation 3 + 20 is incorrect because it does not correctly represent the tens and ones. Ask your learner to describe what each digit represents: "The 3 represents three tens (30) and the 2 represents two ones (2)."
• 17 = 3 + 20 (Incorrect): Learners may overgeneralize from seeing equations of the form a + b = c, assuming the sum always comes at the end. Emphasize that both sides of the "=" symbol must represent the same value. Ask your learner to evaluate the value on each side of the equation. In this example, 17 is not equal to 3 + 20. Explain that 3 + 20 should be 23, not 17.

Teaching Plan

The following activities will introduce your learner to adding and subtracting with tens and ones. Be sure to work at a pace that is comfortable for your learner.

Examples and visuals to support the lesson:

1. Linking Part-Part-Whole Diagrams to Equations

• Look at a part-part-whole diagram (cherry or bar model) representing the decomposition of a two-digit number into tens and ones. Together, write all of the equations that can be represented by the model.
• If necessary, reinforce the link between the part-part-whole diagram and the equations by using physical resources, such as sticks.
• Present missing 'part' or 'whole' equations, supported by the corresponding missing 'part' or missing 'whole' diagrams.

Skill Check
I can write equations that represent part-part-whole models.

2. Solving Equations Without Diagrams

Provide varied practice for your learner without the aid of part-part-whole diagrams. Look out for particular mistakes that indicate underlying misconceptions or lack of understanding. Here are a few examples:

• 32 = 3 + 20 (incorrect): Learners may generalize a pattern of taking the digits in the order they appear, ignoring the 0. Address this by asking learners to describe what each digit represents in their given answer (e.g., "The 3 represents three tens and the 2 represents two ones"); then ask how many tens and ones are represented in 3 + 20.
• 17 = 3 + 20 (incorrect): Learners may overgeneralize after seeing equations of the form a + b = c, assuming the sum always comes at the end. Question learners about the value of each side of the equation and whether they are the same. Emphasize that the expression on each side of the '=' symbol should represent the same value.

Encourage deeper thinking by providing a challenge task that links addition and subtraction back to comparison of two-digit numbers (e.g., 50 + 6 < 65 or 45 - 40 > 46 - 40).

Skill Check
I can solve equations that have two-digit numbers broken down into tens and ones.

3. Solving Word Problems with Diagrams

• Work towards your learner being able to solve word problems involving adding tens and ones, including drawing their own part-part-whole models for support.
• Begin by telling a story with the calculation answer included, and ask your learner to draw or complete a blank part-part-whole diagram to represent the story.
• Focus on the link between the story and the diagram rather than trying to encourage proportional drawings of the two parts in a bar model. Ask your learner to describe what each number in the diagram represents.
• Repeat using a missing "whole" story and then a missing "part" story. Encourage your learner to fill in the two parts that the story tells us, then use their knowledge of two-digit number composition to find the missing number. Continue to use full sentences to relate each diagram to the story.

Skill Check
I can solve story problems that have two-digit numbers broken down into tens and ones.

4. Additional Practice with Story-Based Problems

Provide your learner with practice solving a range of story-based problems. Initially, provide an empty part-part-whole diagram for each question. As your learner gains confidence, encourage them to draw their own models. Examples include:

• I had twenty conkers and then my friend gave me nine more. How many do I have now?
• The teacher had thirty-seven pencils. He gave out thirty of them. How many did he have left?
• There are sixty-three children sharing fruit. Sixty of them like apples. How many children do not like apples?
• I find a two-cent coin and a fifty-cent coin on the ground. How much money have I found altogether?
• I need to save forty-five pounds to buy a new bike. I already have five pounds. How much more do I need?

Summary

By the end of this lesson plan, your learner should have a deeper understanding of adding and subtracting with tens and ones. This will give them a strong foundation for performing calculations with two-digit numbers. Encourage them to continue practicing these skills and applying their knowledge to solve a variety of problems, reinforcing their mathematical reasoning and problem-solving abilities.

Teaching Plan adapted from NCETM under OGL license v3.