## Balancing Act

- Lesson

Problems such as those in this activity help develop what students already know in preparation for writing equations and learning ways to solve for variables. Students use mathematical models to explore quantitative relationships. When presented with pictures of pan balances with one or more objects in each pan, they communicate relationships between the weights of the objects by comparing the balanced and unbalanced pans.

Prior to participating in this lesson, students should be proficient with the following:

- Identifying the left and right sides of a pan balance
- Identifying proportional relationships (halves, doubles)
- Identifying spheres, cylinders, and cubes
- Using a pan balance or a seesaw

Display the chart paper with pan balance (**Figure 1**). Ask
questions about the pan balances to help students understand the weight
relationships pictured. Point out that in each problem, blocks of the
same shape have the same weight. That is, all spheres weigh the same
number of pounds, and all cylinders weigh the same number of pounds.
However, the weights may be different in other problems.

**Figure 1**

Once students have discussed and answered the questions, model the relationship using the real pan balance and proportional weights so students can check their answers. You might use a 10 oz weight on the left side and two 5 oz weights on the right. Then place two 10 oz weights on the left side, and have students tell you what should go on the right side.

Display the chart paper with pan balance (**Figure 2**). Repeat the questioning and modeling procedures. Then do the same for **Figure 3**.

Figure 2 | Figure 3 |

Give the students the Balancing Act student Activity Sheet. The students can do the problems on their own or in pairs. Once they have completed the problems, be sure to allow time for the students to discuss and model their solutions as a whole group.

To conclude the lesson, ask any of the Questions for Students that you did not ask during a previous part of the lesson. You might also consider asking these questions as part of your end-of-class assessment.

### Reference

Adapted from Greenes, Carole, et al, Navigating Through Algebra in Pre-K - Grade 2, from the Navigations Series, NCTM (2001).

- One pan-balance scale
- Proportional weights, preferably of different colors or shapes (i.e. 1 oz, 5 oz, 10 oz, 25 oz)
- Balancing Act Activity Sheet
- Pan balance figures (draw each figure on a separate sheet of chart paper)

**Assessment Option**

Use the student activity sheet “Balancing Act” to assess whether students met the lesson objectives.

**Solutions to "Balancing Act"**

- Six spheres
- Remove one sphere from each pan in balance A. That leaves one cylinder equal in weight to three spheres. On balance B, six spheres will balance the two cylinders.
- Three cylinders
- On balance C, each cylinder balances three spheres. On balance D, three cylinders will balance the nine spheres.
- The cylinder
- On balance E, one cube balances two spheres. On balance F, one cylinder balances three spheres. Because the cylinder balances more spheres than the cube, the cylinder is the heaviest block.

**Extensions**

Show students the online interactive, Pan Balance— Shapes, and, as a class, use shapes to make the sides balance. For older students, ask students to assign numbers to the shapes once the sides are balanced. Give children the opportunity to work with this Web site during center time.

**Questions for Students**

**Figure 1**

1. What do you think would happen to balance A if I put one more sphere on each side? Why?

[It would still be balanced because both sides will still weigh the same amount.]

2. What would happen if I took away one sphere from the right side of balance A? Why?

[The pans would not balance. The left side would be heavier and would tilt downward; the right side would go up.]

3. Which block is heavier — the cylinder or the sphere? How do you know?

[The cylinder is heavier. It takes two spheres to balance one cylinder.]

4. What could you put on the right side of balance B to make the pans balance? How do you know?

[I could put two cylindersorfour spheresorone cylinder and two spheres. I know because one cylinder is the same weight as two spheres, so two cylinders would be the same weight as four spheres.]

**Figure 2**

1. Look at balance A in Figure 2. Which block is heavier — the sphere or the cylinder? How do you know?

[The cylinder is heavier. I know because it takes three spheres to balance one cylinder.]

2. What would happen to balance A if I took away one sphere from the right side? How do you know?

[The left side would tilt downward and the right side would go up. The pans would be out of balance. One cylinder weighs the same as three spheres, so it has to weigh more than two spheres.]

3. What would happen to balance A if I took away the cylinder from the left side of the scale? How do you know?

[The right side would tilt downward and the left side would go up. The pans would be out of balance. Three spheres weigh more than zero cylinders.]

4. How do you think balance A would look with one cylinder on the left and one sphere on the right?

[The cylinder side would tilt downward and the sphere side would be up because one cylinder weighs more than one sphere.]

5. If one cylinder weighs six pounds, how much would one sphere weigh?

[It weighs two pounds. If one cylinder is six pounds, then the three spheres weigh six pounds, and each sphere weighs two pounds since 2 + 2 + 2 = 6.]

6. Look at balance B. What can you put on the left side to make it balance? Can you think of another answer?

[I can put one cylinder and one sphereorfour spheres.]

**Figure 3**

1. Look at both pan balances. Think about what they show. Which block on balance A is heavier? On balance B?

[The cube is heavier on balance A. The sphere is heavier on balance B.]

2. Which is heaviest - the sphere, the cylinder, or the cube? How do you know?

[The cube is heaviest because balance A shows that the cube is heavier than the sphere, and balance B shows that the sphere is heavier than the cylinder. Together they show that the cube is heaviest, the sphere is next heaviest, and the cylinder is lightest.]

3. What would happen to balance A if I put one more sphere on each side?

[It would stay in the same position.]

4. If the cube weighs five pounds, could the sphere weigh six pounds? How do you know?

[No because balance A shows that the sphere must weigh less than the cube.]

5. What would a pan balance look like if the cube were on the left and the cylinder were on the right?

[The cube is heavier, so the left side would be tilted downward.]

**Teacher Reflection**

- Were the materials appropriate? Did the materials allow you to assess student understanding?
- Were students able to associate the pictorial pan balances with the real pan balances successfully to check their answers?
- Were students able to solve balance problems on the Balancing Act activity sheet independently? If not, what additional experiences will you plan to give them more practice? If they did, what additional algebra concepts are appropriate?
- Did students relate the mathematical term “equal” to an even pan balance?
- Did students explain their reasoning in ways that the rest of the class could follow and understand? How can you help students monitor this themselves?

### Learning Objectives

- Understand equality as a level (balanced) pan balance.
- Identify blocks of equivalent weights.
- Understand that adding or subtracting the same weights on both sides of a pan balance will not change the heavier-lighter relationship.

### NCTM Standards and Expectations

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

- Model situations that involve the addition and subtraction of whole numbers, using objects, pictures, and symbols.

### 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.