## Block Pounds

- Lesson

Students explore the use of variables as they solve for the weights of objects using information presented in pictures. They model situations that involve adding and subtracting whole numbers, using objects, pictures, and symbols.

**Prior Knowledge**

- Adding basic facts.
- Subtracting basic facts.
- Naming cubes, spheres, and cylinders.

**Introducing the Activity**

Draw the picture of the two weight scales shown in Weight Problem 1 on the board. Label the scales A and B, as shown.

**Weight Problem 1**

Tell the students that these are scales and that they show the weights of the blocks that have been placed on them. Ask the following questions:

- (Point to scale B.) What is on scale B? [A sphere.] How much does it weigh? [Six pounds.]
- (Point to scale A.) What is on scale A? [A cube and a sphere.] How much do the objects weigh all together? [Nine pounds.]
- Figure out the weight of the cube. How did you do it? [The sphere is 6 pounds, so the cube weighs 9 - 6, or three, pounds.]

Next, draw the two scales for weight problem 2 on the chalkboard or on poster board.

**Weight Problem 2**

As in problem 1, the weight of one block is given in this problem. But unlike in problem 1, to find the weight of a cube, two operations must be performed. First the total weight of the cubes must be determined. Then the weight of each cube must be found. Ask the following questions:

- What block is on scale A? [A sphere.]
- How much does the sphere weigh? [Four pounds.] How do you know? [The scale shows four pounds.]
- (Point to scale B.) What is on scale B? [One sphere and two cubes.]
- How much do the blocks weigh all together? [Fourteen pounds.]
- (Point to the sphere on scale B.) How much does this sphere weigh? [Four pounds.]
- How can you figure out how much each cube weighs? [The sphere is four pounds. So the two cubes are 14 - 4, or ten, pounds. So each cube weighs five pounds.]

Point out to the students that the blocks of the same shape have the same weight. So since the sphere on scale A weighs four pounds, the sphere on scale B must also weigh four pounds. In like manner, the cubes weigh the same number of pounds.

Present weight problem 3 to the students. Unlike in the first two weight problems, in this problem the weight of one of the blocks is not given directly. The students have to decide which scale to consider first. The scale with two identical blocks is the best place to begin because, through guess and check or the recall of the addition of doubles, the students can find the weight of one sphere.

**Weight Problem 3**

Ask the following questions:

- Which blocks are on scale A? [A sphere and a cube.] How much do they weigh all together? [Eleven pounds.]
- Do you know how much the sphere weighs? [No, we can't tell.]
- Do you know how much the cube weighs? [No, we can't tell.]
- Which blocks are on scale B? [Two spheres.]
- How much do they weigh all together? [Twelve pounds.]
- Do you know how much each sphere weighs? [Yes, each weighs six pounds.]
- How did you figure it out? [6 + 6 = 12, so half of 12 is 6.]
- Can you figure out the weight of the cube? [Yes.]
- How will you do that? [The sphere weighs 6 pounds, so the cube is 11 - 6, or five, pounds.]

Distribute copies of the Block Pounds Activity Sheet for students to complete individually or in pairs. Encourage the students to record the weights on the blocks as they are determined. Once the students have completed the problems, have them talk about how they solved them.

Note that the problems on "Block Pounds" are ordered by difficulty. Problem A gives the weight of one of the blocks directly. Problem B requires a knowledge of a doubles addition fact to find the weight of one block before the weight of the other can be computed. Problems C and D show three scales with three different types of blocks. In problem C, the weight of the sphere is given directly and the students have to replace the sphere on each of the other two scales with its weight to find the weights of the cylinder and cube. In problem D, no weight of a block is given directly.

**Reference**

Greenes, Carole, et al., Navigating Through Algebra in Pre‑K–2. Reston, VA: NCTM. 2001.

- A pan balance
- Small wooden cubes
- Block Pounds Activity Sheet
- Pan Balance - Shapes Tool (Extension)

**Assessment Option**

Use the student Block Pounds Activity Sheet to assess whether students met the lesson objectives.

**Solutions**

A. The cube is 5 pounds; the cylinder is 9 pounds.

B. The sphere is 10 pounds; the cylinder is 8 pounds.

C. The cube is 9 pounds; the cylinder is 4 pounds; the sphere is 3 pounds.

D. The cylinder is 5 pounds; the cube is 4 pounds; the sphere is 8 pounds.

**Extensions**

- Show
the children problem A from the student activity sheet and how it can
be represented using addition equations, as shown below.
Say, "The equation for scale A says that the weight of the cube plus the weight of the cylinder is fourteen pounds. The equation for scale B says that the weight of the cube is five pounds. Let's talk about how you solved the problem. I will write the steps."

Call on the students to tell how they solved the problem. Encourage them to use the term replace in their discussion. Show them how to record the solution steps. For example:

Step 1: Scale B shows that the cube is 5 (pounds).

Continue with "Block Pounds" problem B and record and discuss the steps. Have the students record the steps to the remaining two problems, C and D, on their own or in pairs.

Step 2: Replace the cube with 5 in equation A. Then 5 + cylinder = 14.

Step 3: Find the weight of the cylinder by subtracting 5 from 14: 14 - 5 = 9.So the cube weighs nine pounds.

Some students may count up from 5 to 14 to find the weight of the cylinder. Others may recall the addition fact 5 + 9 = 14. Still others may use subtraction or guess and check to find the missing addend.

- Alternatively, students may explore similar types of problems using the Pan Balance - Shapes Tool.

Pan Balance - Shapes Tool

Questions for Students

Refer to the instructional guide.

Teacher Reflection

- Were the materials appropriate? Did the materials allow you to assess student understanding?
- Were students able to solve balance problems on the Block Pounds 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

Students will:

- Solve problems involving variables as unknowns.
- Replace variables with numbers to solve problems.
- Recognize pictorial and equation representations of the same relationship.
- Write addition equations for relationships presented in pictures.

### Common Core State Standards – Mathematics

-Kindergarten, Geometry

- CCSS.Math.Content.K.G.A.2

Correctly name shapes regardless of their orientations or overall size.

-Kindergarten, Geometry

- CCSS.Math.Content.K.G.A.3

Identify shapes as two-dimensional (lying in a plane, ''flat'') or three-dimensional (''solid'').

-Kindergarten, Geometry

- CCSS.Math.Content.K.G.B.4

Analyze and compare two- and three-dimensional shapes, in different sizes and orientations, using informal language to describe their similarities, differences, parts (e.g., number of sides and vertices/''corners'') and other attributes (e.g., having sides of equal length).

Grade 1, Algebraic Thinking

- CCSS.Math.Content.1.OA.D.8

Determine the unknown whole number in an addition or subtraction equation relating to three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 + ? = 11, 5 = _ - 3, 6 + 6 = _.

### Common Core State Standards – Practice

- CCSS.Math.Practice.MP5

Use appropriate tools strategically.