## Solve Simple One-Step Linear Equation

### Yunling ZhangPalm Beach County Schools

#### Description

Students learn how to solve one unknown number by using hands-on manipulatives after being introduced to the history of abstract mathematics through literature.

#### Objectives

The student uses pictures, models, manipulatives or other strategies to solve simple one-step linear equations with rational solutions.

#### Materials

-Books: [The Story of Mathematics] by Lloyd Motz and Jefferson Hane Weaver,
Plenum Press, New York.
-Paper cups
-Plastic chips
-Paper
-Pencils

#### Preparations

1. Prepare the lesson plan as stated in the procedure.
2. Have enough chips and cups for each student.
3. Obtain a copy of the book. (See Materials.)

#### Procedures

Note: The student should know basic addition and subtraction prior to this lesson and should understand the identity axiom for addition.

1. Read pages 57 to pages 77 in [The Story of Mathematics] by Lloyd Motz and Jefferson Hane Weaver, Plenum Press, New York. (This is for teacher's information so that he or she will be able to orally share with students.) This section of the book tells about the origins of abstract mathematics. Many people think algebra is difficult to comprehend because it uses letters instead of real numbers. This section says that the numbers are also abstract unless associated with real subjects. Numbers are abstract symbols. A letter is just a symbol. Algebra is compared with arithmetic in many ways. The letter is just to identify the relationship with any numbers in the equation.

2. Write on board: 3 + 0 = , and then ask students for the answer. Briefly review the identity axiom for addition. Adding zero to a number does not change the total value.

3. Write the objective on the board: Solve simple one-step linear equations. Explain to the students what a linear equation is. Ask: What is the meaning of the equal sign?

4. Pass out the plastic chips, papers, and pencils. Tell them to make two piles of chips. Each pile should have three chips. Let everybody write an equation for the two piles (3 = 3). Let them add two chips on one pile, and then ask them: What should you do to the other pile so that the two piles still have an equal amount of chips? Wait for the students' answers. After they all understand that they should add two more to the other pile, let them write the change done to the equation (3+2=3+2). Then ask the students to take away one chip from one pile, and ask them the following question. What should we do to the other pile so that two piles are still equal? Wait for the answer. (Yes, you should take away one chip from the other pile.)

Tell students: Let's write down what did we do to the equation on the paper. Help students to write this equation: (3+2)-1=(3+2)-1. Explain the purpose of parentheses in a math equation. Continue practicing in the same way. Ask students to generalize the rules of keeping two piles equal in quantity when making changes. Ask necessary questions to guide the students to draw the correct conclusion. Do the same to each pile. Then ask students the following question. What should we do to the equation when we make changes so that the equation is always true? Let the students discuss and guide them to the following conclusion. Do the same operation on both sides of the equation.

5. Pass out the cups. Let them make two piles. One pile should have one cup and two chips; the other should have only four chips. Help the students write an equation for the piles if the cup represents an unknown number and each chip represents 1. Use an X to represent the unknown number. The equation should be x + 2 = 4. Ask the students to guess how many chips the cup should contain to have an equal amount to the other pile. Some students may guess 2 as an answer. Ask them how did they guess it is 2? Did you use subtraction 4 minus 2 to get 2? Let's take a look at our pile. If we take away 2 chips from the cup pile, and 2 from the other pile, the two piles are still equal, right? Now the chips inside of the cup must be 2. Look at the equation on your paper. If you subtract 2 from both sides, what do you get? (Answer: x = 2, right?) So what is the value of X ? (Answer: 2. )

Ask students to do the following and write down equations when they play with manipulatives:

(1) 2 chips and 1 cup = 3 chips
(2) 3 chips and 1 cup = 5 chips
(3) 4 chips and 1 cup = 6 chips

Circulate and check their work, giving help if needed.
Go over the answers after they finish.
Answers: (1) 2+x = 3 x=1
(2) 3+x = 5 x=2
(3) 4+x = 6 x=2

6. Ask the students to summarize the rules to solve an unknown number in a linear equation. Do not give them the answer quickly. Guide them by asking many questions. Sample questions:
-What is the first step you do in solving linear equations?
-How do you know which number to eliminate?
-How do you know when to substract the number and when to add a number?

To sum up the correct rules:
Step 1- Look at the number on the same side of the equation as the unknown number.
Step 2- Make the number zero by adding an opposite of the number. Subtract the number if it is a positive number. Add the number if it is a negative number.

7. Give students the two real-world problems below and ask them to write equations on their papers.

Problem 1- I bought two shirts that cost \$10. I kept the price tag on one shirt which was \$3, and lost the other price tag. What is the price of the other shirt?

Problem 2- Bryan and Tom have four pets. Bryan has one pet. How many pets does Tom have?

Allow time to write equations and circulate, noting those who are having difficulty. Encourage children to use the manipulatives to help solve the problems. You may need to give them more chips.

8. Put students into groups of three to do the following:
-Come up with two real-life problems (written down).
-Write a linear equation with an unknown number.
-Solve the equation.

9. Give each group feedback about their problems as they share them aloud with the class. Once all groups have shared, and you feel as if students are ready, give them the assessment for this lesson. (See Assessment.)

10. Ask each student to create one real-life problem individually, to write a linear equation with an unknown number, and to solve the equation. This will serve as the assessment.

11. Collect the assignments.

#### Assessments

Assess students as you make the following observations:
1. Observe as students are questioned and note their answers when manipulating chips and cups.
2. Observe group work when creating the real-life problems and sharing them.
3. Assess individual problems for creating an equation and solving correctly for the unknown.

Since this is a formative assessment, students may need additional practice before they become proficient using manipulatives to solve simple, one-step linear equations.

#### Extensions

Another manipulative for the lesson would be a balance, using chips as weights. Ask the students to write equations for each weight balance. Then write changes made to the balance weight to connect the manipulatives to the process of solving the equations.