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Balance Beam Discoveries

Number and Operations
Grace M. Burton
Location: unknown

This lesson encourages students to explore another model of multiplication, the balance beam, and another relationship, the inverse of multiplication. This exploration leads naturally into representing multiplication facts in equation form. In addition to extending their understandings of the concept of multiplication, students begin to practice the multiplication facts by playing the Product Game.

In this lesson, the balance beam model for multiplication will be demonstrated using an actual balance beam. The best type to use is one with hanging weights, but you can modify the directions if you have only pan balances available by enclosing sets of weights in baggies.

Display a balance beam and review with children how it operates. Ask a volunteer to place 3 weights at position 2 of the left arm of the balance beam. Then ask, "Where would we need to place just 1 weight on the other side so that the beam balances?" Accept and try all student responses. When the correct response of 6 is given, ask students to record this using the equation 3 × 2 = 6. (If a balance scale is being used instead, this same demonstration can be completed by placing 3 baggies with 2 weights in each baggie on the left side of the pan balance; then, 6 baggies with 1 weight each can be used on the right side to create balance.)

Repeat these steps with other multiplication facts as necessary.

Students can also use the Pan Balance—Numbers Applet to create multiplication numbers for a partner to solve.

appicon Pan Balance—Numbers Applet 

For example, on one side of the pan balance, a student could enter 7 * 3, and the other student would type 21 on the other side. (Note: The applet requires the use of an asterisk (*) instead of × to indicate multiplication, because the standard keyboard does not include the times symbol.)

Next show children the Product Game. Then, play a sample game with them, using volunteers from the class.

appicon The Product Game Applet

As students work in pairs, they should find other multiplication equations with the balance beams or play the Product Game. Students can also use number facts (4 × 3 = 12, for example) to play a "paper version" of the Product Game.

Allow the students to use both electronic tools until the class period is nearly over. Then call them together to discuss the experiences they have had that day using the Questions For Students section.

  • Balance beam and hanging weights  
  • Computers or tablets with internet access

Assessment Options 

  1. At this stage of the unit, it is important for students to know how to:
    • Find products using a balance beam
    • Write multiplication sentences in equation form
    • Use the inverse property of multiplication to complete multiplication equations
  2. The guiding questions provided may help you elicit information which will help you assess the students’ current level of knowledge about multiplication. As a new concept has been added today, you may wish to add to your entries on the Class Notes recording sheet begun earlier in this unit.

Questions for Students 

1. When you modeled an equation on the balance beam, what did you do first? Then what? How did you record this?

[Student responses may vary, but they should be able to model the steps covered in the first part of the lesson.]

2. Suppose you put a weight on the 12 on the right hand side of the beam, and you wanted to put weights of 3 on the left hand side. How many would you need?

[You would need 4 weights of 3; 4 × 3 = 12.]

3. How could you use the balance beam to complete this number sentence: 3 × _ = 15?

[Use three weights of 5 to get 15.]  

Teacher Reflection 
  • Which students met all the objectives of this lesson? What extension activities are appropriate for these students?
  • Which students did not meet the objectives of this lesson? What instructional experiences do they need next?
  • What adjustments would you make the next time you teach this lesson?
Number and Operations

Hopping on the Number Line

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.
Number and Operations

Exploring Equal Sets

This lesson builds on the previous lesson and encourages students to explore another model for multiplication, the familiar set model. Students find products using equal sets and present results in the form of a table. The students apply their knowledge about multiplication in the creation of pictographs.
Number and Operations

Modeling Multiplication With Streets and Avenues

This lesson encourages students to explore the array model of multiplication, a model that lays an important foundation for the later study of area. The lesson focuses on the factors 4 and 6. Students will also explore products with 0 or 1 as a factors. First students make arrays with counters, and then they create a second concrete example of the array model using toothpicks. They also write problems which involve multiplication.

Learning Objectives

Students will:
  • Explore the balance beam model of multiplication.
  • Write multiplication sentences in equation form.
  • Use the inverse property of multiplication to complete multiplication equations.

NCTM Standards and Expectations

  • Identify and use relationships between operations, such as division as the inverse of multiplication, to solve problems.
  • Understand and use properties of operations, such as the distributivity of multiplication over addition.

Common Core State Standards – Mathematics

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.C.7
    Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 x 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.

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 4, Algebraic Thinking

  • CCSS.Math.Content.4.OA.A.1
    Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 x 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations.

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.