## Balancing Shapes

Students will balance shapes on the pan balance applet to study equality, essential to understanding algebra. Equivalent relationships will be recognized when the pans balance, demonstrating the properties of equality.

This lesson will enhance algebraic understanding through an informal study of equality. A two arm balance pan, shown in the classroom (borrowed from the science department, or built with a meter stick balancing on a pencil) will help students see when the pans are balanced, the left side equals the right side. This important concept in algebra will be reinforced as students manipulate shapes in the pan balance with this applet. The Properties of Equality will be identified later in the lesson.

To demonstrate how this applet works, project the Pan Balance-Shapes Tool. Place a shape on the left side of the balance; the balance tilts to the left. This is unbalanced (or an inequality). Place the same item on the right side to demonstrate equality. Next place a red square on the left, and a blue circle on the right. Show students how to place shapes in either pan until they balance, adding squares and circles, red on left, blue on right, in a varied order, until equality is reached. When they balance, show students how equivalent relationships are recorded in the table on the screen, and will remain there until you click "New Problem", which creates new weights for each of the shapes. Encourage students to use Reset Balance to keep the same weights, and keeps the table, but clears the pans. Show Array counts the contents of the pans, leading to a transition to variables, such as, 9 diamonds = 9d.

Next, provide time for students to explore this applet in pairs or groups of four as you circulate and observe the student work, asking students to explain their findings. Look for examples of the Properties of Equality (explained below) to project later for the class.

After exploration time, facilitate a discussion with the class of the discoveries (25 minutes). Students may project their discoveries by bringing their lap tops to the projector. With each one, name the property, and have students record one example of each of the properties on the Pan Balance-Shapes Recording Sheet. The teacher may provide examples of any properties students did not discover.

Examples of discoveries may include:

- One red square = one red square, or 1
*s*= 1*s*, using variables. This demonstrates the*Reflexive Property of Equality*,*a*=*a.*Remind students to "Reset" the balance to show 2 pink triangles = 2 pink triangles, or one yellow diamond on each side.

This property may also be demonstrated with with a pan balance in the classroom, placing 3 blue blocks in the left pan, and 3 blue blocks in the right pan. To develop kinesthetic understanding of the Reflexive Property of Equality, have students hold 3 cubes in their left hand, and 3 cubes in their right hand.

When demonstrating to the class, you may use the Pan Balance—Shapes Interactive to keep a constant set of relationships, which may be helpful when leading the discussion.

- To demonstrate the
*Symmetric Property of Equality*, if*a = b*, then*b = a*, place 1 red square in the left pan and 2 blue circles in the right pan. Ask students, "What if I put 2 blue circles in the left pan? What must I put in the right pan?"[1 red square].

Reset, and demonstrate 1 pink triangle = 3 yellow diamonds. Using the Symmetric Property of Equality, ask students, "What will balance 3 yellow diamonds placed in the left pan?

[1 pink triangle in the right pan].

- To demonstrate the
*Multiplication Property of Equality*, if*a = b*, then*ca = cb*, place 6 blue circles in the left pan with 4 pink triangles in the right pan. Ask, "If I remove 3 blue circles in the left pan, how many pink triangles will balance in the right pan?"[2, dividing both sides of the equality in half]. (Note: Multiplying by a fraction, one half, is equivalent to dividing by a whole number, 2. This may eliminate the need for the Division Property of Equality).

Alternatively, you may place 1 pink triangle in one pan, and 3 yellow diamonds in the other. Ask, "If I place 2 pink triangles in one pan, how many yellow diamonds will balance?"

[6, doubling each side, with the Multiplication Property of Equality].

- To demonstrate the
*Addition Property of Equality*, if*a*=*b*, then*a*+*c*=*b*+*c*, place combinations of colors in the left and right pans, such as 2 blue = 1 red. Add a yellow to both sides. The pans remain balanced. (Note: Removing a tile may be considered adding a negative. This may eliminate the need for the Subtraction Property of Equality). - To demonstrate the
*Transitive Property of Equality*, if*a*=*b*and*b*=*c*, then*a*=*c*, use the pan balance to show if 1 red = 2 blue, and 3 blue = 2 pink, then 3 red = 4 pink.

You may discuss substitution at this time, and begin to write
equations for the relationships. To do so, transition to "Count Items."
For example, count it shows 4 × red squares = 3 × blue circles. Write 4*r* = 3*b*. Have students practice writing these equivalent arrays, as they are displayed on the computer.

Below is an example of one of the new problems that can result, with the equation shown using the "Count Items" feature.

Students should complete the Shape Pan Balance Recording Sheet after completing this lesson.

Shape Pan Balance Recording Sheet

The recording sheet can be reviewed the next class day, or it can be turned in for assessment.

### In the Classroom

Read about how Pamela Olmstead uses this lesson in her Success Story.

- Shape Pan Balance Recording Sheet
- Computers or tablets with internet access
- Optional: Pan Balance (from science department or homemade version)
- Optional: Square tiles or cubes to place on balance

**Assessment Option**

The Shape Pan Balance Recording Sheet may be collected to assess student engagement and success with using this applet to learn the properties of equality. Students’ answers to key questions will help the teacher determine student understanding of equivalence.

**Extensions**

- Replace shapes with variables. Have students write variable expressions for the relationships found. For example, 1
*r*= 2*b*. Therefore, 2*b*= 1*r*. 4*b*= ___ .[2

*r.*] - Apply the properties of equality to solve traditional algebraic equations with algebra tiles using the handout, Balancing Equations.
- Move on to the next lesson,
*Balancing Algebraic Understanding*.

**Questions for Students**

1. If one red cube plus one pink triangle plus one blue circle equals two blue circles plus two yellow triangles, ask students, what do you think will happen when I remove the same thing from both sides? Will it stay balanced?

[Remove one blue circle from each side to see. This is the main idea in solving equations, to get the variable alone by removing items (or adding items) to both sides.]

2. 6 red = 3 blue. If I remove one blue, how many reds must I remove to stay balanced?

[2.]

3. Now I have 4 reds = 2 blue. Is this the same relationship? What if I remove another blue? How many reds must I remove?

[2 reds.]

4. Explain why this happens.

[In this relationship, 1 blue = 2 reds. You see it in 6/3, 4/2, 2/1.]

5. Show what happens if you take the balanced shapes from the right pan and place them in the left pan, and take the balanced shapes from the left pan and place them in the right pan. Write a conclusion.

[If 2r = 3b, then 3b = 2r.]

6. What happens when you double the left pan, and also double the right pan?

[If r = 3b, then 2r = 6b].

7. If r + p + y = 2b + y, what happens when I remove y from both sides?

[The pans stay balanced, showing the subtraction property of equality].

**Teacher Reflection**

The following questions should be used by the teacher to reflect upon the lesson:

- What evidence did you see that students were able to transition from the concrete to the abstract? How could you help other students to make connections with the shapes to algebraic symbols?
- How did the balance pan applet enhance understanding? How did it confuse students? How can it be better used in the future?
- How did the balance pan applet enhance understanding? How did it confuse students? How can it be better used in the future?
- What adjustments did you make while teaching the lesson? How will you do it next time you teach this lesson?

### Everything Balances Out in the End

### Algebra in Balance

### Balancing Algebraic Understanding

### Other Related Resources

- Success Story

Read about how Pamela Olmstead uses this lesson in her classroom.

### Learning Objectives

Students will:

- Develop an understanding of equality using shapes and a pan balance.
- Apply the properties of equality to show that the balance is maintained.
- Informally explore the Reflexive, Symmetric, Multiplication/Division, Transitive, and Addition/Subtraction Properties of Equality.
- Recognize the relationship between a pan balance and an equation.
- Draw conclusions based upon patterns in a table.

### NCTM Standards and Expectations

- Represent, analyze, and generalize a variety of patterns with tables, graphs, words, and, when possible, symbolic rules.

- Relate and compare different forms of representation for a relationship.

- Develop an initial conceptual understanding of different uses of variables.

- Model and solve contextualized problems using various representations, such as graphs, tables, and equations.

### Common Core State Standards – Mathematics

Grade 6, Expression/Equation

- CCSS.Math.Content.6.EE.A.3

Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y.