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Hopping on the Number Line

Number and Operations
Grace M. Burton
Location: unknown

In this lesson, students generate products using the number line model. This model highlights the measurement aspect of multiplication and is a distinctly different representation of the operation. The order (commutative) property of multiplication is also introduced. Students are encouraged to predict products and to answer puzzles involving multiplication.

On the overhead projector or chalkboard, display a large number line and demonstrate with a counter how hops of 5 can be taken on the number line. You may wish to encourage students to count aloud as the hops are made. You might choose to introduce the equation notation 4 × 5 = 20, informally reading it as "Four hops of 5, and you land on 20." After several examples with 5 as a factor, ask the students to determine what size hop to use next. Encourage the students to predict the products and to verify their predictions by moving a counter on the large number line. You may wish to provide children with a counter and individual number lines at their desks.

pdficon Number Lines 

After allowing time of exploration, ask the students to predict the answers to questions such as "If I take 4 hops of 3, where will I land?"

1251 numline

Now give each student a piece of paper and ask them to make up 2 similar problems and trade them with a friend to solve 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.

Be sure students have the opportunity to explore different factors, such as:

2 × 3
4 × 4
3 × 6
7 × 2
and so on....

Then ask: "If I take 5 hops of 3, where will I land? How about if I take 3 hops of 5? Will this work every time?" Encourage them to explore the order property and state their findings. [In each case, the student should land on 15, because of the commutative property of multiplication.]

As a concluding activity, you may wish to pose puzzles such as "I am a number between 20 and 30. You say my name when you hop by 5's. Who am I?" and encourage students to create and share similar problems.

  • Counters for the number line (chips, markers, etc.)  
  • Number Lines  

Assessment Options 

  1. At this stage of the unit it is important for students to know how to:
    • use the number line model to find products
    • the order property of multiplication
    • solve and create puzzles using the number line
  2. The guiding questions suggest ways to help you determine if students have achieved these objectives. You may want to add others that the conversations with the students suggest. The Class Notes Recording Sheet provides a form on which to document your observations about student understanding and skills. You may find this information useful when discussing progress toward learning targets with individual students.
  3. If you wish to display a collection of the student puzzles in a public place, ask students to copy the puzzle, write the answer, and tape it under the puzzle. They might want to send a written challenge to students from other classes to solve the puzzles.


  1. Students can use the virtual number lines to model other multiplication problems on the number line. Please conduct a simple internet search for this.
  2. Move on to the next lesson, Exploring Equal Sets.

Questions for Students 

1. What numbers did you land on when you hopped by 5?

[5, 10, 15, 20, etc.]

2. What numbers did you land on when you hopped by 3?

[3, 6, 9, 12, 15, etc.]

3. Were any of the numbers the same?

[15, 30, etc.]

Teacher Reflection 

  • What extension activities would be appropriate for students?
  • Which students had trouble using the number line? What instructional experiences do they need next?
  • What adjustments would you make the next time you teach this lesson?

Learning Objectives

Students will:
  • Use the number line model to find products.
  • Solve and create puzzles using the number line.
  • Investigate the order property of multiplication.

NCTM Standards and Expectations

  • Develop understanding of fractions as parts of unit wholes, as parts of a collection, as locations on number lines, and as divisions of whole numbers.
  • Understand various meanings of multiplication and division.
  • Understand the effects of multiplying and dividing whole numbers.

Common Core State Standards – Mathematics

Grade 1, Algebraic Thinking

  • 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.)

Grade 1, Number & Operations

  • 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.

Grade 3, Algebraic Thinking

  • CCSS.Math.Content.3.OA.A.1
    Interpret products of whole numbers, e.g., interpret 5 x 7 as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as 5 x 7.

Grade 3, Algebraic Thinking

  • CCSS.Math.Content.3.OA.B.5
    Apply properties of operations as strategies to multiply and divide. Examples: If 6 x 4 = 24 is known, then 4 x 6 = 24 is also known. (Commutative property of multiplication.) 3 x 5 x 2 can be found by 3 x 5 = 15, then 15 x 2 = 30, or by 5 x 2 = 10, then 3 x 10 = 30. (Associative property of multiplication.) Knowing that 8 x 5 = 40 and 8 x 2 = 16, one can find 8 x 7 as 8 x (5 + 2) = (8 x 5) + (8 x 2) = 40 + 16 = 56. (Distributive property.)

Grade 4, Num & Ops Base Ten

  • CCSS.Math.Content.4.NBT.B.5
    Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

Grade 2, Algebraic Thinking

  • CCSS.Math.Content.2.OA.C.4
    Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.