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What Balances?

Number and Operations
Location: Unknown

This lesson encourages students to explore another meaning of subtraction, the balance. They use subtraction facts to generate related addition facts and explore at the concrete level the idea of subtraction as the inverse of addition.

Note: In this lesson, subtraction will be demonstrated using a pan balance. If you only have a balance with hanging weights, please modify the directions for that type of balance. You may wish to use large pasta shapes 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.

686 pan balance

Call on a volunteer to play the role of a pan balance. Ask the child to hold out his or her arms while you 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 would look like. Then ask him or her to imagine that the left side is much lighter than the right side, and act that out. Then ask the class to stand and stretch their arms out and show how a balance works.

Now display a pan balance and review with the students how it operates. Ask a volunteer to place eight pasta shapes in the left pan of the balance beam and then to put two pasta shapes on the other side. Write on the board: 8 - __ = 2. Say: "How many pasta shapes would we need to take away from the left side so that the scale balances?” Accept and model all the students' responses. When the response "six" is given, select students to record this on the board using an equation [8 – 6 = 2]. Continue with other examples until the students are comfortable with the process, and then give them time to explore subtraction with a pan balance.

686 scale mac 

Give each pair of students a number cube, and assign each student to one side of the pan balance. Tell the students to roll the number cube and place that many pasta shapes on their side of the balance. Then have them work with their partner to write the subtraction example as an equation, and then to make the scale balance by taking some pasta shapes away from the heavy side. Ask them to record the difference and to repeat the activity several times. Balancing with a pan balance is concrete preparation for the algebraic procedure of balancing an equation.

Next, give the pairs of students a paper bag. Instruct students to not look in the bag. Assign one student to go first. Ask the student to take up to five pasta shapes from the bag and place it on the left side of the scale, and the same or a different number (between 5 and 10) of pasta shapes on the right side. Tell the other student to take away pasta shapes from the heavier side until the scale balances. Then have each student write a subtraction equation to describe the situation. Have them repeat the activity several times, switching roles each time. Then call the class together and give the students time to share one of the ways they balanced the scale. If you wish, ask them to record it in words or pictures.

  • Pan balances
  • Pasta shapes
  • Crayons
  • Paper bags
  • Number cubes

Assessment Options

  1. The Questions for Students will elicit information that will help you assess the students' current level of knowledge about using the balance to model subtraction.
  2. Because this new 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.
 Move on to the next lesson, Who's in the Fact Family?

Questions for Students 

1. When you modeled a subtraction equation on the balance, what did you do first? Then what did you do? How did you record this?
2. Suppose you put six pasta shapes on the left side of the balance and nine pasta shapes on the right side. What can you take away so the scale will balance? What equation tells what you did?

[3; 9 - 6 = 3.]

3. If you started with nine pasta shapes on the heavy side and the scale balanced after you took away four, how many pasta shapes were in the bag?


4. Choose one equation that you wrote. How does this equation show what you did? Can you write another subtraction equation with the same addends? Can you use those addends to write an addition equation?

[Student responses may vary.]

5. How would you explain to a friend how to use the balance beam to complete this number sentence: 6 - __ = 1?

[Put 6 on one side, 1 on the other, keep removing (5) until both sides balance.]

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?
  • Which students are still counting to find differences?
  • What parts of the lesson went smoothly? Which parts will I change the next time that I teach this lesson?
Number and Operations

Recording Two Ways

The students make sets of pasta shapes and count some away, then record the subtraction in vertical and horizontal formats. They draw a set and cross out some shapes, then write in both formats the subtraction that the drawing represents.
Number and Operations

How Many Left?

This lesson encourages the students to explore the familiar set model of subtraction. The students write story problems and find differences using sets, including subtracting all and subtracting zero. They record the differences in a chart.
Number and Operations

Where Will I Land?

In this lesson, the students find differences using the number line, a continuous model for subtraction. [Number can be considered in two ways: discrete and continuous. The counting and set models use the discrete form of number.] Students are encouraged to predict differences and to compose puzzles involving subtraction.
Number and Operations

Who's in the Fact Family?

In this lesson, the exploration of the relation of addition to subtraction is continued as the students use problem-solving skills to find fact families, including those in which one addend is zero or in which the addends are alike.
Number and Operations

What's the Difference?

During this lesson, students use reasoning to find differences from numbers up to 10, using real and virtual calculators and an addition chart as tools. They also play a concentration game.

Learning Objectives

Students will:

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

NCTM Standards and Expectations

  • Count with understanding and recognize "how many" in sets of objects.
  • 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.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.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.

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.