## Hopping Backward on the Number Line

In this lesson, students generate differences using a number line model. Because this model highlights the measurement aspect of subtraction, it is a distinctly different representation from the models presented in the previous lessons of this unit. The order property for subtraction is investigated. At the end of the lesson, children are encouraged to predict differences and solve puzzles involving subtraction.

Inform the students that they will find differences using the number line model. Make a number line on the floor with numbered carpet squares or by drawing on a shower curtain. Display a subtraction problem, such as 9 – 5 = ___. Ask one student to stand on the number line at 9. Ask the rest of the students which way the student would hop to subtract. When students answer that the student should hop to smaller numbers, have the volunteer hop back 5 spaces. Encourage children to count aloud as each backward hop is made. Describe the action this way:

If you start at 9 and take 5 backward hops, you land on 4.

Allow other students to demonstrate additional subtraction sentences.

After several examples, model how to record counting back on a number line. Display a number line on the board or overhead projector. Write a subtraction sentence, such as 10 – 3. Use a counter to act out this problem, asking students where to place the counter and how many backward hops you should take. After acting this out, record it by circling the number 10 on your number line and drawing 3 backward hops.

Put the children into pairs, and give each pair a counter and an individual number line (or a 12‑inch ruler that they can use as a number line). Have students work in pairs to solve subtraction problems on their number lines and share their answers. Optionally, have students record their action using the Number Line Hopping Activity Sheet before sharing the difference. After some practice, encourage the students to predict the differences and verify their predictions by moving a counter on the number line.

Number Line Hopping Activity Sheet

Discuss the order of numbers in subtraction by asking questions
such as, "If I start at 9 and hop backward 5 spaces, will I get the
same answer as if I start at 5 and hop backward 9 spaces?" Encourage
the students to conclude that the *order property*, which works for addition, does not work for subtraction.

- Large Physical Number Line
- Counters
- Number Lines (Positive Numbers Only) or 12-inch Rulers
- Number Line Hopping Activity Sheet (optional)

**Assessment Option**

Have students answer the following prompt in their math journals: "How would you tell a friend to find the difference between 2 and 5 using a number line?"

**Extensions**

- Pose such puzzles as, "I am the number you land on when you a take a hop of 5 and then hop back 1. Who am I?" Have children create and share similar problems. They can write their puzzles on file cards and give the cards to students in other classes to solve.
- Move on to the next lesson,
*Finding the Balance*.

**Questions for Students**

1. What number will you land on if you start at 5, then hop back 3?

[I landed on 2.]

2. When you hop backwards on a number line, how do you know where to start? How do you know when to stop hopping?

[The first number in the subtraction problem tells me where to start. The second number tells me how many hops to take.]

3. If I start at 9 and hop backward 5 spaces, will I get the same answer as if I start at 5 and hop backward 9 spaces?

[No; if I start at 5, I cannot hop backward 9 spaces.]

4. If I want to find the difference between two numbers, how will I know which number to start with?

[I will start with the biggest number.]

**Teacher Reflection**

- Which children counted as they took hops, and which moved directly to the number?
- What activities would be appropriate for children who have met all the objectives?
- Which children had trouble using the number line? What instructional experiences do they need next?
- Do students need additional experiences to understand the order property?
- Were the numbers used appropriate for the age of your students?
- What adjustments would you make the next time that you teach this lesson?

### Counting Back and Counting On

### Taking Away Sets

### Finding the Balance

### Finding Fact Families

### Practice Makes Perfect

### Looking Back and Moving Forward

### Learning Objectives

- Use the number line model to find differences.
- Investigate whether the order property holds for subtraction.

### NCTM Standards and Expectations

- Use concrete, pictorial, and verbal representations to develop an understanding of invented and conventional symbolic notations.

### Common Core State Standards – Mathematics

-Kindergarten, Counting & Cardinality

- CCSS.Math.Content.K.CC.A.2

Count forward beginning from a given number within the known sequence (instead of having to begin at 1).

-Kindergarten, Counting & Cardinality

- CCSS.Math.Content.K.CC.A.3

Write numbers from 0 to 20. Represent a number of objects with a written numeral 0-20 (with 0 representing a count of no objects).

-Kindergarten, Algebraic Thinking

- CCSS.Math.Content.K.OA.A.1

Represent addition and subtraction with objects, fingers, mental images, drawings1, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.

-Kindergarten, Algebraic Thinking

- CCSS.Math.Content.K.OA.A.2

Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.

-Kindergarten, Algebraic Thinking

- CCSS.Math.Content.K.OA.A.5

Fluently add and subtract within 5.

Grade 1, Algebraic Thinking

- CCSS.Math.Content.1.OA.B.4

Understand subtraction as an unknown-addend problem. For example, subtract 10 - 8 by finding the number that makes 10 when added to 8.

Grade 1, Algebraic Thinking

- CCSS.Math.Content.1.OA.C.5

Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).

Grade 1, Algebraic Thinking

- CCSS.Math.Content.1.OA.C.6

Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 - 4 = 13 - 3 - 1 = 10 - 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 - 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).

Grade 2, Algebraic Thinking

- CCSS.Math.Content.2.OA.B.2

Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers.

Grade 2, Number & Operations

- CCSS.Math.Content.2.NBT.B.7

Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.

Grade 2, Measurement & Data

- CCSS.Math.Content.2.MD.B.6

Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, ..., and represent whole-number sums and differences within 100 on a number line diagram.

### Common Core State Standards – Practice

- CCSS.Math.Practice.MP4

Model with mathematics.

- CCSS.Math.Practice.MP5

Use appropriate tools strategically.

- CCSS.Math.Practice.MP6

Attend to precision.