Balancing

Pre-K-2
1

This lesson encourages students to explore another model of subtraction, the balance. This model leads naturally to recording with equations. Students use actual and virtual pan balances in their explorations and record the modeled subtraction facts and the related addition facts in equation form.

Note that in this lesson, the balance model for subtraction will be demonstrated using both an actual and a virtual pan balance.

Display a pan balance and review with the students how it operates. Explain that when the scale balances, both sides have the same value.

Then ask a volunteer to place 7 connecting cubes in the left pan of the balance beam and 4 connecting cubes on the other side. Ask the students how many connecting cubes are needed on the lighter side so that the scale balances. Accept and model all student responses. When the response “3” is given, ask the children what question “3” answers. [How many more than 4 is 7?] Then ask them to record the answer using the equation form 7 – 4 = 3. Ask if they know what addition sentence would also fit the situation. [3 + 4 = 7 or 4 + 3 = 7.] Continue with other weights until the students are comfortable with the process.

Divide the students into pairs and give each pair a die. Assign each child one side of the pan balance. Tell individuals to roll the die, read the number of dots on the upward face, and place that number of connecting cubes on their side of the balance. Then have partners work together to balance the scale by adding connecting cubes in a different color. They should record the subtraction equation that shows what they did. Have them repeat the activity several times.

Now call the class together and demonstrate how to use the Pan Balance— Shapes Tool.

Assign some students to work with this site. Because the shapes represent different values, advise the students to use only one shape as they explore with the online balance. While some children are using the online balance, others can play “What’s in the Bag?” To start the game, provide pairs of children with a pan balance, a paper bag, and some connecting cubes. Assign one child to go first, placing up to 10 connecting cubes in the bag and placing the bag on the left side of the scale. Then that child will place up to 10 loose connecting cubes on the right side. The other child is to add connecting cubes to the lighter side until the scale balances. Then each child writes subtraction and addition equations to describe the situation. Have them repeat the activity several times, switching roles each time.

• Pan Balance with weights
• Connecting cubes in two or more colors
• Spinners or number cubes
• Paper bag
• Pan Balance— Shapes

Assessment Option

Because a new model for subtraction has been added today, you may wish to make more entries on the Class Notes sheet begun earlier in this unit.

Extension

Move on to the next lesson, Fact Families.

Questions for Students

1. When you modeled comparison subtraction on the balance, what did you do first? Then what? How did you record this?
2. Suppose you put 7 connecting cubes on the left hand side of the balance and 3 connecting cubes on the right hand side. How would you balance the scale by adding cubes? What equation tells what you did?
3. How did you find out how many connecting cubes were in the bag?
4. How would you explain to a younger child how to make the sides balance?
5. Choose one equation that you wrote when you played the game. 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?
6. How could you use the balance to complete this number sentence: 3 + _ = 5?
7. What does it mean when the scale balances before you add any cubes?
8. If I have 5 red cubes and 7 blue cubes, what addition and what subtraction equations can I write about the cubes?

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? What, if any, misconceptions that need cleared up?
• Which parts of the lesson helped the students achieve their learning goals? Which parts would you change the next time that you teach this lesson?

Counting Back and Counting On

Pre-K-2
In this lesson, students model subtraction with connecting cubes while the teacher reads to them from counting books. Then children make a train of connecting cubes and write in vertical and horizontal format the differences suggested by adding to and subtracting from the train one cube at a time. Finally, they record, in written form, a train showing one cube being taken away and record the difference in vertical and horizontal format.

Comparing Sets

Pre-K-2
A children’s book sets the stage for this lesson which encourages students to review counting back. In this lesson, children write subtraction problems and model them with cubes. They compare sets and record differences in the form of a table. The additive identity is reviewed in the context of comparing equal sets.

Using the Number Line to Compare

Pre-K-2
In this lesson, students determine differences using the number line to compare lengths. Because this model is based on linear measurement, it is a distinctly different representation from the models presented in the previous two lessons. At the end of this lesson, children are encouraged to predict differences and answer puzzles involving subtraction.

Fact Families

Pre-K-2
In this lesson, the relationship of addition to subtraction is explored with books and with connecting cubes. Students search for related addition and subtraction facts for a given number using a virtual or actual calculator to find differences. They also investigate fact families when one addend is 0 as well as when the addends are the same.

Looking Back and Moving Forward

Pre-K-2
This final lesson of the unit reviews the work of the previous lessons and suggests a framework for summative assessment. During this lesson, students use the mathematical knowledge and skills developed in the previous lessons to demonstrate understanding and the ability to apply that knowledge to playing subtraction games.

Learning Objectives

Students will:
• Explore the balance model of subtraction.
• Write the subtraction modeled on the balance in equation form.
• Write the addition sentence related to a given subtraction sentence

NCTM Standards and Expectations

• Count with understanding and recognize "how many" in sets of objects.
• Develop a sense of whole numbers and represent and use them in flexible ways, including relating, composing, and decomposing numbers.
• Develop understanding of the relative position and magnitude of whole numbers and of ordinal and cardinal numbers and their connections.
• Understand the effects of adding and subtracting whole numbers.
• Develop and use strategies for whole-number computations, with a focus on 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.

-Kindergarten, Algebraic Thinking

• CCSS.Math.Content.K.OA.A.5
Fluently add and subtract within 5.

• 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.

• CCSS.Math.Content.1.OA.C.5
Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).

• 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).

• 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.

• 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.