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Balancing Equations

Number and Operations
Grace M. Burton
Location: unknown

This lesson encourages the students to explore another meaning for operations of subtraction, the balance. This meaning leads naturally into recording with equations. The students will imitate the action of a pan balance and record the modeled subtraction facts in equation form.

Teacher Note: In this lesson, subtraction will be demonstrated using a pan balance. If you have only a balance with hanging weights, please modify the directions accordingly. You may wish to use heavy weights such as sinkers for this lesson so that the balance will clearly register the differences in weight. The pieces should be identical so that groups of the same size weigh the same amount.

To review with the students how a pan balance operates, display one and call on a volunteer to "be" a pan balance. Ask the child to hold out his or her arms, then place a container in each hand. Tell the child to imagine that the container on the left side is very heavy, and to act out what the balance will look like. (You might also place a heavy object in the hand that represents that left side of the balance.) Then ask him or her to imagine the left side is much lighter than the right side and act that out. Tell students they will “act out” the behavior of a pan balance. Then ask the whole class to stand, stretch their arms out and show how a pan balance works. [You may wish to suggest which side of the pan balance is heavier, lighter.]

444 pan balance

Now ask a volunteer to place nine weights in the left pan of the balance and then put four weights on the other side. Ask: "How many more weights are on the left side than on the right side? What will we need to add to the left side so the pan balance is level?" Accept and model all the students' responses. When the response "five" is given, record this on the board using an equation [9 = 4 + 5] that models what was done with the weights to make the pan balance level. (Note that although the number sentence is an addition sentence, subtraction is used to find the missing addend.) Continue with other examples until the students are comfortable with the process. (Because this is an example of a missing addend problem, some students may have difficulty completing the equation correctly.) To continue the investigation, call on volunteers to pose problems and model them on the pan balance.

Now give each pair of students some crackers and a copy of the Pan Balance Activity Sheet.

pdficonPan Balance Activity Sheet 

Tell students they will take turns choosing two numbers and acting out how a balance would look if there were that many crackers on the left and right sides. Ask students to determine how many more crackers would be needed to balance the pan and record that information on the recording sheet. Then ask them to write the number sentence that they modeled. Invite the students to select one of the resulting equations to illustrate by drawing a pan balance that models the equation.

Assessment Option

Because a new and especially challenging meaning for subtraction has been added with this lesson, you may wish to make more entries on the Class Notes recording sheet begun earlier in this unit.


  1. Students may use the Pan Balance - Numbers Tool to model and solve additional subtraction equations, similar to those in this lesson. 
  2. Move on to the next lesson, Fact Family Fun.

Questions for Students 

1. When you modeled comparison subtraction on the balance, 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 followed in the lesson.]

2. Suppose you put seven fish-shaped crackers on the left side of the balance and three fish-shaped crackers on the right side. To balance the pan, which side should you add crackers to? How many more crackers should you add to balance the pan balance? What equation tells what you did?

[Right side; add 4 more; 7 = 3 + 4.]

3. How would you explain to a younger student how to make the sides balance?

[Student responses may vary, but they should be able to model the same steps used during the lesson.]

4. Choose one equation that you wrote when you acted like a pan balance. How does this equation show what you did?

[Student responses may vary.]

5. How could you use the balance to complete this number sentence: 3 + _ = 5? Which numbers are the addends?

[Put 3 on one side, 5 on the other, and balance; 3 and 2 are the addends.]

6. What does it mean if the pan balance is level before you add any fish-shaped crackers?

[Both sides have the same number of crackers.]

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?
  • When the class worked together in a whole class situation, did some students respond before they had thought the question through? Did others fail to respond even though they were sure of the answer? Did I call on all students equally often?
  • Which parts of the lesson helped students achieve their learning goals? Which parts will I change the next time that I teach this lesson?
Number and Operations

Counting Back

Students count back to compare plates of fish-shaped crackers, and then they record the comparison in vertical and horizontal format. They apply their skills of reasoning and problem solving during this lesson in several ways. [Because students have associated the word "more" with addition, the comparative approach to subtraction is typically more challenging for the students to understand.]
Number and Operations

How Many More?

Students write subtraction problems, model them with sets of fish-shaped crackers, and communicate their findings in words and pictures. They record differences in words and in symbols. The additive identity is reviewed in the context of comparing equal sets.
Number and Operations

Hopping Backward to Solve Problems

In this lesson, students determine differences using the number line to compare lengths. Because this meaning is based on linear measurement, it is a distinctly different representation from the meanings presented in Lessons One and Two. At the end of the lesson, the students use reasoning and problem solving to predict differences and to answer puzzles involving subtraction.
Number and Operations

Fact Family Fun

In this lesson, the relation of addition to subtraction is explored with fish-shaped crackers. The students search for related addition and subtraction facts for a given number and also investigate fact families when one addend or the difference is 0.
Number and Operations

Wrapping Up the Unit

During this final lesson in the unit, the students use the mathematical knowledge and skills developed in the previous lessons as they visit five stations to review comparative subtraction.

Learning Objectives

Students will: 

  • Model the balance meaning of subtraction.
  • Record the subtraction modeled on the balance.

NCTM Standards and Expectations

  • 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, 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.

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, 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 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.

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.