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Finding the Balance

Links Away
Grace M. Burton
Location: unknown

This lesson encourages students to explore another model of subtraction, the balance. Students will use real and virtual balances. Students also explore recording the modeled subtraction facts in equation form.

In this lesson, the balance model for subtraction will be demonstrated using both an actual and a virtual pan balance. If you have only a balance with hanging weights, please modify the directions given as needed.

Display a pan balance and review with the students how it operates. Place 5 weights on one side of the balance and 3 weights on the other side. Discuss with students how to make the scale balance using take away. Relate this action to the equation 5 – ___ = 3. Explain that the equals sign is like the middle point of a balance scale — both sides must be equal. The Pan Balance Overhead can be displayed if you do not have access to a pan balance.

overhead Pan Balance Overhead 

Ask how many links need to be taken away so that the scale balances. Accept and model all student responses. When the correct response [2.] is given, complete the equation so that it reads 5 – 2 = 3. Repeat by having students place different weights on the balance, write an equation, and solve. Continue until the students are comfortable with the process.

537  scale blocks

Now show students the online Pan Balance- Shapes, and assign some children to work with this tool. Because the different colored on-screen weights have different values, tell the children that they should use only one color of weight in a problem. Have students place an unequal number of weights on each side (for example, 5 on one side and 3 on the other), then remove some from the heavier side until the beam balances. Have them write the equation suggested by their action — in this case, 5 – 2 = 3.

appicon Pan Balance— Shapes 

Split the class into two or more groups for center activities. While some children are using the online balance, others can play "What’s in the Bag?" To start the game, give pairs of children a pan balance, a paper bag, and some links. Assign one child to go first, placing 10 links (or connecting cubes) on the right side of the balance. The same student then places up to 10 links (or connecting cubes) in the bag and places the bag on the left side of the scale. The other child is to take away from the right side until the scale balances. Then the students write a subtraction equation to describe the situation. Have them repeat the activity several times, switching roles each time.

  • Pan Balance, or Balance Beam and Hanging Weights  
  • Pan Balance Overhead 
  • Computer with Internet connection  
  • Links or Connecting Cubes (in two or more colors)  
  • Paper bags 

Assessment Options 

  1. Have students respond to the following prompt in their math journals: "How can a balance help you find differences?"
  2. Have students write equations for balance problems. Display an unbalanced scale. For example, place 6 weights on one side and 2 weights on the other side. Ask students to write a subtraction equation for this balance. Repeat with other problems.


  1. Give dice or spinners numbered 1‑6 and a pan balance to pairs of students. Assign each child one side of the pan balance and tell them to roll the die (or spin the spinner) and place that many links on their side of the balance. Then have them work together to make the scale balance by taking some links away, and to record the subtraction equation that shows what they did. Have them repeat the activity several times.
  2. For advanced students, have them use the online Number Balance instead of the shape balance.
  3. Move on to the next lesson, Finding Fact Families.

Questions for Students 

1. Place 5 weights on one side and 3 weights on the other side of a balance. Which side of the scale is heavier? How do you know?

[The side with 5 weights is heavier because it is lower than the side with 3 weights.]

2. If we want the sides to balance, which side will we have to take weights from? How many weights will we have to take away?

[We have to take 2 weights away from the side with 5, since it is heavier.]

3. Suppose you put 6 links on the left side of the balance and 9 links on the right side. How would you balance the scale? What equation tells what you did?

[Take 3 links away from the side with 9. The equation is 9 – 3 = 6.]

4. How did you find out how many links were in the bag?

[Take weights away until the sides balance.]

Teacher Reflection 

  • Which students met all the objectives of this lesson? What extension activities are appropriate for these children?
  • Which students did not meet the objectives of this lesson? What instructional experiences do they need next?
  • When the students worked in pairs, did both children contribute equally?
  • Which students are counting to find differences? Which students use known facts to solve balance problems?
  • What adaptations did you make for special needs students? Were these appropriate?

Counting Back and Counting On

This lesson, which focuses on the counting model for subtraction, begins with reading a counting book. The students model the numbers as the book is read. Then they make a chain of links and write in vertical and horizontal format the differences suggested by adding and subtracting one link at a time from their chains. Finally, they draw a chain showing one link being taken away and write in two formats the difference it represents.

Taking Away Sets

This lesson encourages students to explore another model for subtraction, the familiar set model. Reading one of the many books that feature subtraction set the stage for this lesson in which the students write story problems, find differences using sets, and present results in a table. In the discussion of the table, they focus on the effects of subtracting all and subtracting 0.

Hopping Backward on the Number Line

In this lesson, students generate differences using a number line model. Because this model highlights the measurement aspect of subtraction, it is a distinctly different representation from the models presented in the previous lessons of this unit. The order property for subtraction is investigated. At the end of the lesson, children are encouraged to predict differences and solve puzzles involving subtraction.

Finding Fact Families

In this lesson, students explore the relation of addition to subtraction with books and links. Then the children search for related addition and subtraction facts for a given number. They also investigate fact families, including those where one addend is 0 and where the addends are alike.

Practice Makes Perfect

During this lesson, students use what they know about fact families to play a concentration game. They will also identify subtraction facts they need to learn.

Looking Back and Moving Forward

This final lesson reviews the work of the previous lessons and suggests a framework for summative assessment. Students will self-select a solution strategy for subtraction from the models introduced in this unit. An extension activity is suggested in which students use the mathematical knowledge and skills developed in the previous lessons to demonstrate understanding and ability to apply that knowledge to playing a new game.

Learning Objectives

Students will be able to:

  • Explore the balance model of subtraction.
  • Write the subtraction modeled on the balance in equation form.
  • Practice subtraction facts from 10.
  • Demonstrate understanding of the equals sign (=).

NCTM Standards and Expectations

  • Use concrete, pictorial, and verbal representations to develop an understanding of invented and conventional symbolic notations.

Common Core State Standards – Mathematics

-Kindergarten, Measurement & Data

  • CCSS.Math.Content.K.MD.A.1
    Describe measurable attributes of objects, such as length or weight. Describe several measurable attributes of a single object.

-Kindergarten, Measurement & Data

  • CCSS.Math.Content.K.MD.A.2
    Directly compare two objects with a measurable attribute in common, to see which object has ''more of''/''less of'' the attribute, and describe the difference. For example, directly compare the heights of two children and describe one child as taller/shorter.

Common Core State Standards – Practice

  • CCSS.Math.Practice.MP5
    Use appropriate tools strategically.
  • CCSS.Math.Practice.MP6
    Attend to precision.