## Geology Rocks Equations

• Lesson
6-8
1

In this lesson, students explore linear equations with manipulatives and discover various steps used in solving equation problems. Students use blocks and counters as tactile representations to help them solve for unknown values of x.

Students should work in groups or pairs. This will encourage discussion during the lesson, which will help with understanding the manipulative representation. Try to match students who are likely to understand the manipulatives with students who may have trouble initially.

Begin the lesson by distributing the Geology Rocks Equations Activity Sheet and the manipulatives. Have students read the opening paragraph and Question 1 on their own. Use the first question as a demonstration of what they will be doing with the questions that follow.

Tell students that the blocks represent the crates (which hold the same number of rocks), and each chip counter represents a 1-pound rock. For simplicity's sake, assume that the weight of each crate is also zero. Ask students to create a representation of the problem on their desks using the manipulatives. Then, ask them to discuss in their groups ways to figure out how many rocks are in the crates using only this information.

You may want to allow students time to explore and share their thoughts within their groups about how they are using the manipulatives to help them solve the problem. After you have given students time to explore, ask the groups to share their answers with the class. You could also allow students to create their own representation with the chips, which will help ensure that each student understands the problem. After 30 seconds or so, they can pair-share to correct any problems with their representations and discuss their solutions.

The correct answer to Question 1 is that there are 2 rocks in each crate. Have students who found the correct solution share their strategy with the class. If no students found the answer, lead the class through the process:

1. Mr. Anderson realizes that he can remove 2 rocks from each side of the pan balance and still have a balanced pan balance. This leaves him with the following:

2. Then, Mr. Anderson realizes that he can remove a crate from each side of the pan balance and still remain balanced. This leaves him with the following:

Students can represent these situations algebraically. Use boxes instead of x's:

+ + 2 = + 4

This could then be simplified to:

2 + 2 = + 4

Students can remove boxes and numbers just as they did on the pan balance. Teachers often give students manipulatives and expect them to complete the analogy on their own. However, quite often, students are not able to make the connection. The teaching is much more effective if you make the analogy explicit for students. That is, explain explicitly how the boxes and rocks represent variables and numbers.

Some students may have used a guess-and-check strategy until they found out the weight of the crate, while others may have simplified the equation. As an introduction to the problem, this is fine. As students begin to think-pair-share their strategies to balance the equation, encourage those who are using guess-and-check to collaborate with those using algebra. Guess and check is a valid strategy when beginning with these equations; it requires solving the problem mathematically using a step-by-step process. Students are performing the operations multiple times and finding solutions that do not work, but at the same time they are gaining practice until they find the solution that does work.

Allow students to work on their own to complete Questions 2 to 5. As you move through the class, ask students to explain to you how the equation they are building using the manipulatives represents the problem on the paper. Encourage students to use multiple approaches to solve each problem—this will help them build the connections between representations.

After students have finished Questions 2 to 5, have them share within their group how many rocks they think are in each crate. For each solution, ask the class if they agree. If all groups agree, move on to the next problem. If someone disagrees have the class re-create the problem using the manipulatives and walk through the solution steps together to agree on the solution. Use student volunteers where possible as you go through the answers.

2. x = 4

3. x = 12

4. x = 4

5. x = 7

Once the class has agreed on all the solutions, inform students that the solutions they created are actually linear equations and can be represented mathematically with equations, similar to the equations you showed earlier using boxes.

Ask students to look over each problem and write an equation they think mathematically represents the initial picture Mr. Anderson made. Some students may not immediately make the connection that the boxes stand for unknown values of x and the rocks represent constants. You should stress this connection as students are working. To help them with the connection, you can have students use x's or boxes or both. Some students may have an easier time making the connection using the instead of x.

### Equations

2. 4x + 2 = 18 or 4 + 2 = 18

3. 2x + 12 = 3x or 2 + 12 = 3

4. 2x + 8 = x + 12 or 2 + 8 = + 12

5. 5x + 3 = 2x + 24 or 5 + 3 = 2 + 24

As soon as students have created the equations, discuss them as a class so they are agreed upon. When students have finished writing their equations, allow them to complete Questions 6 to 9 on the activity sheet. Allow students to solve these equations without or without manipulatives, depending on their own preference.

6. x = 3

7. x = 6

8. x = 4

9. x = 3

To summarize the lesson, have students share with the class any strategies they came up with that make it quicker to solve the problems without creating a manipulative representation each time. Students may have realized that you are subtracting or adding a constant, and then dividing by the coefficient of x. Discuss with students the steps involved in solving these equations. You are isolating the numbers on one side of the equation and the unknown on the other. You are grouping like terms and solving. Use an example not on the activity sheet, such as the one below in your discussion:

Ask students to write down on their activity sheet a list of steps to solve the problems and compare their lists with one another. Most solutions will require at least 2 steps to solve. As students begin to more fully grasp the process of 2-step equations, ask them to solve the equation 5x – 8 = 1x + 2. In solving this equation, students will be faced with using negative integers and accepting a solution that is not a whole number (x = 2.5). Students may find it tough to create manipulative representations of these harder equations. Allow them to explore and work it out on their own. This is truly an exploration for students.

### Reference

Lappan, G., J. Fey, W. Fitzgerald, S. Friel, and E. Philips. 2005. Investigation 3.2: Exploring equality. In Moving straight ahead: Linear relationships. Connected Mathematics 2 Series, Pearson Prentice Hall.

Assessment Options

1. Create several model equations on the board and ask students to write the equations and solve them. Also, write up equations and ask students to create a model and solve them.

2. Read a sentence that describes a given situation and have students create equations and solution for the unknown. Example: Mikaela had 4 bags and 3 coins. Her money is worth the same amount as Sam's. He has 2 bags and 7 coins. How many coins are in each bag?

3. Have student write a paragraph explaining what it means if you are left with an equation that reads         4x = 14.

[In Mr. Anderson’s case, it would mean that there are parts of a rock in each crate. There would be 31/2 rocks inside each crate.]

Extensions

1. Present an altered form of Mr. Anderson's lab problems, in which he uses balloons that have exactly one pound of lift (–1). If a balloon is tied to a rock, together they weigh 0 pounds because they balance each other out. Each crate still represents x, the unknown number of rocks inside. Create similar pictures and equations for students, and have them find the number of rocks in each crate.
2. Extend this activity to include equations involving negative and non-integral unknowns. For example, 4x – 4 = x + 3 or –4x + 12 = x – 3.
3. Present this problem to students: A 40-pound rock is dropped on the ground, and it breaks into 4 pieces, each piece weighing an integral number of pounds. Using a pan balance, these 4 pieces can be used to measure any integer weight from 1 to 40 pounds. How much does each piece of the rock weigh?

Questions for Students

1. What are you actually doing to the equation when you remove a rock from both sides of the pan balance?

[Subtracting 1 from both sides of the equation, which creates an equivalent equation that has the same unknown value.]

2. What are you actually doing to the equation when you remove a crate from both sides of the pan balance?

[Subtracting 1 x from both sides of the equation, which creates an equivalent equation that has the same unknown value.]

Teacher Reflection

• Did you challenge the achievers? How?
• Will your students be able to come up with interesting ways to represent these equation models with negative integers?
• How did students illustrate that they were actively engaged in the learning process?
• What worked with classroom behavior management? What didn't work? How would you change what didn’t work?

### Learning Objectives

Students will:

• Explore equality using several equations and physical representation
• Combine like terms
• Solve 2-step equations involving 1 unknown

### NCTM Standards and Expectations

• Develop an initial conceptual understanding of different uses of variables.
• Recognize and generate equivalent forms for simple algebraic expressions and solve linear equations.
• Model and solve contextualized problems using various representations, such as graphs, tables, and equations.

### Common Core State Standards – Mathematics

• 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.

• CCSS.Math.Content.6.EE.B.6
Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.

• CCSS.Math.Content.7.EE.A.2
Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. For example, a + 0.05a = 1.05a means that ''increase by 5%'' is the same as ''multiply by 1.05.''

• CCSS.Math.Content.6.EE.B.5
Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.

• CCSS.Math.Content.7.EE.A.1
Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.

### Common Core State Standards – Practice

• CCSS.Math.Practice.MP1
Make sense of problems and persevere in solving them.
• CCSS.Math.Practice.MP4
Model with mathematics.
• CCSS.Math.Practice.MP5
Use appropriate tools strategically.
• CCSS.Math.Practice.MP7
Look for and make use of structure.