## Hopping on the Number Line

Pre-K-2,3-5
1

In this lesson, students generate sums using the number line model. This model highlights the measurement aspect of addition and is a distinctly different representation of the operation from the model presented in the previous lesson. The order (commutative) property is also introduced. At the end of the lesson, students are encouraged to predict sums and to answer puzzles involving addition.

Tell the students that they will find sums using the number line model. Then display a large number line and a 5+4 domino, that is, a domino with 5 spots on the left side and 4 spots on the right. Then demonstrate with a counter how a hop of 5 is taken on the number line. You may wish to encourage students to count aloud as the hop is made. Then make a hop of 4, starting at the place the counter landed. You might choose to have them record what happened using the equation notation 5 + 4 = 9, or to informally describe the moves this way: “If you take a hop of 5 spaces and then a hop of 4 spaces, you land on 9.” You may wish to highlight the fact that in this model, spaces are counted, not points on the number line.

After several trials, put the students in pairs and give each pair some dominoes, a counter, and individual number lines.

Ask the students to take turns moving the counter on the number line to find the sum shown on the domino and recording the hops in pictures and in equation form. Ask them to draw the first hop and write the first numeral in green and the second hop and numeral in red. Encourage the students to predict the sums and to verify their predictions by moving a counter on the number line.

After allowing time for exploration, ask the students to predict the answers to questions such as “If I take a hop of 3 and then a hop of 5, where will I land?” [8.] Now have students make up 2 similar problems on a piece of paper and trade them with a friend. Students should then solve their partners’ problems using the number line. When the pairs have finished, call them together to discuss what they did. Encourage them to use the number line in their explanation. Then ask “If I take a hop of 5 and then a hop of 4, where will I land?" [9.] "How about if I take a hop of 4 and then a hop of 5?" [9.] "Will this work every time?" [Yes.] Encourage them to explore the order property by writing each first addend in green and each second one in red.

Be sure to lead a discussion about the order (commutative) property. You may need to use other examples to illustrate this important property of addition.

As a concluding activity, pose puzzles such as “I am the number you land on when you take a hop of 5 and then a hop of 1. Who am I?” [6.] You may wish to encourage students to create and share similar problems. One or more of these puzzles could be added to their unit portfolios.

Assessment Options

1. The Questions for Students help students focus on the mathematics and aid you in understanding the students’ current level of knowledge and skill with the mathematical concepts of this lesson. You may want to add others that conversations with the students suggest.
2. A teacher’s resource, Class Notes, is provided to document your observations about student understanding and skills. You may find the information useful when planning additional learning experiences for individual students or for documenting progress for students with mandated instructional plans.

Extensions

1. Ask students, "How is using a number line like measuring? How is it different?"
2. Move on to the next lesson, Exploring Adding With Sets.

Questions for Students

1. What number did you land on when you made a 5-hop, then a 3-hop?

[8.]

2. Could you land on the same number if you took a 3-hop first, then a 5-hop? How do you know?

[Yes; 5 + 3 = 8, and 3 + 5 = 8.]

3. What sums did you model with hops? How did you record them?

[Student responses will depend upon the "hops" they performed.]

4. Were any of the sums the same? Why?

[Student responses will depend upon the "hops" they performed.]

5. How would you find the sum of 2 and 5?

[Make a hop of 2, and then a hop of 5, to reach 7.]

6. How would you tell a friend to add on the number line?

[Student responses may vary.]

Teacher Reflection

• Which students counted as they took hops and which moved directly to the number?
• What activities would be appropriate for students who met all the objectives?
• Which students had trouble using the number line? What instructional experiences do they need next?
• Did any children notice a connection with measurement?
• What adjustments would you make the next time that you teach this lesson?

### Counting to Find Sums

Pre-K-2
This lesson focuses on the counting model for addition and begins with reading a counting book. Students model the numbers with counters as the book is read. Then they count the spots on each side of a domino and write, in vertical and horizontal format, the sums suggested by dominoes. Finally, the students illustrate a domino and record a sum it represents.

Pre-K-2
This lesson builds on the previous two lessons and encourages students to explore another model for addition, the set model. This model is similar to the counting model in the first lesson, because it is based on counting. Reading a related counting and addition book sets the stage for this lesson in which students write story problems, find sums using sets, and present results in the form of a table. In the discussion of the table, the students focus on the order property and the effects of adding 0.

### Balancing Discoveries

Pre-K-2
This lesson encourages students to explore another model of addition, the balance model. The exploration also involves recording the modeled addition facts in equation form. Students begin to memorize the addition facts by playing the “seven-up game.”

### Seeing Doubles

Pre-K-2
In this lesson, the students focus on dominoes with the same number of spots on each side and on the related addition facts. They make triangle-shaped flash cards for the doubles facts.

### Finding Fact Families

Pre-K-2
In this lesson, the relationship of subtraction to addition is introduced with a book and with dominoes. Then students explore the concept of missing addends.

### Learning Objectives

Students will:
• See the number line model to find sums.
• Investigate the order (commutative) property of addition.
• Solve and create puzzles using the number line.

### NCTM Standards and Expectations

• Develop understanding of the relative position and magnitude of whole numbers and of ordinal and cardinal numbers and their connections.
• Understand the effects of adding and subtracting whole numbers.
• Understand various meanings of addition and subtraction of whole numbers and the relationship between the two operations.
• Develop and use strategies for whole-number computations, with a focus on addition and subtraction.
• Develop fluency with basic number combinations for addition and subtraction.
• Use a variety of methods and tools to compute, including objects, mental computation, estimation, paper and pencil, and calculators.

### Common Core State Standards – Mathematics

-Kindergarten, Counting & Cardinality

• CCSS.Math.Content.K.CC.C.6
Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group, e.g., by using matching and counting strategies.

-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.

• CCSS.Math.Content.1.OA.B.3
Apply properties of operations as strategies to add and subtract. Examples: If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known. (Commutative property of addition.) To add 2 + 6 + 4, the second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12. (Associative property of addition.)

• 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.

• 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).

• CCSS.Math.Content.1.OA.D.7
Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 - 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.

• CCSS.Math.Content.1.NBT.C.4
Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, 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 and explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten.

• 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.

### Common Core State Standards – Practice

• CCSS.Math.Practice.MP1
Make sense of problems and persevere in solving them.
• CCSS.Math.Practice.MP4
Model with mathematics.
• CCSS.Math.Practice.MP5
Use appropriate tools strategically.
• CCSS.Math.Practice.MP6
Attend to precision.