Partitioning Numbers into Tens and Ones

Introduction

Our previous lessons on multiples of ten focused on your learner's understanding of two-digit numbers in the context of cardinality, ordinality, and measures. Now, we will pick up again on the tens-and-ones structure by partitioning numbers into tens and ones with part-part-whole models. This paves the way for calculations such as 20 + 8 = 28 and 34 - 30 = 4.

Before beginning the lesson, your learner should be familiar with counting tens and ones for two-digit numbers.

Key Concepts for Partitioning Numbers into Tens and Ones

Here are a few concepts that are helpful to know for the lesson:

• Part-Part-Whole Models: Part-part-whole models, such as cherry diagrams and bar graphs, are visual representations that help learners understand how numbers can be broken down into parts and combined to form a whole. These models are essential for visualizing the relationship between the components of a number and the total value.
• Partitioning Two-Digit Numbers: A two-digit number can be represented with the "tens" as one part and the "ones" as the other part. For example, the number 57 can be broken down into 50 (tens) and 7 (ones). This representation helps learners understand the structure of numbers and the value of each digit.
• Using Part-Part-Whole to Add and Subtract: Understanding the part-part-whole relationships of numbers helps learners perform simple addition and subtraction calculations. For example, knowing that 57 can be represented as 50 and 7 allows learners to see that 57 = 50 + 7. Similarly, understanding that 57 is made up of 50 and 7 helps learners calculate that 57 - 7 = 50.

Teaching Plan

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

Examples and visuals to support the lesson:

1. Exploring Bundles of Tens and Ones

• Present a mixture of bundles of ten straws and individual straws, then ask your learner to collect a given number of straws.
• Ask your learner to verbalize how they collect, for example, 34 straws, and draw attention to whether they say you need three tens and four ones.
• Model the partitioning of the straws using a part-part-whole diagram. For example, show the tens and ones partitioned in the 'parts' circles. Using 34, the parts are 30 and 4.
• Then, move the tens and ones together into the 'whole' circle. Label the 'whole' with the digits. Move the tens and ones apart again into the 'parts' circles.

Skill Check
I can split groups of objects into tens and ones.

2. Solving Missing Whole Problems

• Progress to presenting missing whole problems on part-part-whole diagrams (addition).
• When using concrete representations of the tens and ones, move them together and apart as they are moved to/from the 'whole' circle.
• When using pictorial representations for the tens and ones 'parts', use numerals to represent the whole (showing an additional set of tens and ones in the 'whole' circle at the same time as showing them in the 'parts' circle is misleading as it shows double the given number of items).
• Include some part-part-whole models where the ones are represented on the left and the tens on the right to check that your learner is really thinking about the value of each digit in the number rather than just working from left to right.

Once your learner is confident with the tens and ones structure, encourage them to describe the structure in full sentences, using the stem sentences:

• "There are __ tens, which is __, and __ one(s), which is __. This makes __ altogether."
• "The __ represents __; it has a value of __ tens."
• "The __ represents __ ones; it has a value of __."

Skill Check
I can find the two-digit number that the whole represents in a part-part-whole model. I can describe what each digit of a two-digit number represents.

3. Solving Missing Part Problems

• Next, present your learner with a variety of missing 'part' problems.
• Until now, your learner may not have specifically learned how to identify missing parts in two-digit numbers. But if they have had lots of experience with partitioning in previous lessons, they have the building blocks in place to do so.
• Consider a whole two-digit number (for example, 57), show the tens part (for example, five tens or 50), and ask your learner to identify the missing part.
• Encourage your learner to reason: "The missing part must be seven because fifty-seven is five tens for fifty and seven ones."
• Then present a problem with the tens part missing (as in the example with 35). As before, spend time modeling and practicing clear, accurate explanations.

Skill Check
I can find the missing part of a two-digit number split into tens and ones.

4. Advanced Part-Part-Whole Models

• Once your learner can confidently work with the numbers represented pictorially, move on to part-part-whole models with all numbers represented as numerals.
• Provide your learner with practice completing missing 'parts' and 'wholes', varying the placement of the tens and ones 'parts', and the orientation of the diagrams. Continue to use the stem sentences that describe the value of each digit.
• Once your learner is confident, you can present problems based on partitioning two-digit numbers into three parts, building on your learner's knowledge of adding multiples of ten.
• For now, avoid extending to partitioning a two-digit number into a multiple of ten and another two-digit number (for example, 61 into 50 and 11). This skill is important but will be covered in depth in a future lesson.

Skill Check
I can solve missing number problems for two-digit numbers split into three parts.

Summary

By the end of this lesson plan, your learner should have a solid understanding of partitioning two-digit numbers into tens and ones, representing these numbers using part-part-whole diagrams, and solving addition and subtraction problems. Encourage your learner to continue practicing these skills and verbalizing their reasoning to reinforce their understanding.

Teaching Plan adapted from NCETM under OGL license v3.