Solving Linear Equations Using Manipulative

 

Name:          Son Thieu

Title of lesson:  Solving Linear Equations Using Manipulative

Date of lesson:  5th Sixth-Weeks

Length of lesson:  2 Days

Description of the class: 9/10 Grade – Regular/Honor

TEKS addressed:

(a) Basic understandings

 (3) Function concepts

 (4) Relationship between equations and functions.

 

(b) Foundations for functions: knowledge and skills and performance descriptions.

(1) The student understands that a function represents a dependence of one quantity on another and can be described in a variety of ways. Following are performance descriptions.

(B) The student gathers and records data, or uses data sets, to determine functional (systematic) relationships between quantities.

(C) The student describes functional relationships for given problem situations and writes equations or inequalities to answer questions arising from the situations.

(D) The student represents relationships among quantities using concrete models, tables, graphs, diagrams, verbal descriptions, equations, and inequalities.

(E) The student interprets and makes inferences from functional relationships.

            I.  Overview

In this lesson, students use manipulative to represent visually the steps they take to obtain a solution to an algebraic equation. They develop an understanding of the connections between the solution involving manipulative and the symbolic solution. Integrate with the real world applications on the function.
II. Performance or learner outcomes

            Students will be able to:

  1. Use manipulative to solve linear equations of the form ax + b = c, where a, b, and c are constants.
  2. Use symbolic methods to solve linear equations.
  3. Verify and check the solutions to the equations by using substitution.
  4. Make representations of real world situations using linear function.
  5. Or integrate linear functions with everyday life.

 

   Teachers will need the following:

Students will need the following:

Lesson Resources:

            http://www.learner.org/channel/workshops/algebra/workshop1/lessonplan2.html

Five-E Organization

 

Teacher Does                                           Student Does

Engage: Day 1:

Learning Experience

Today, we will look at one form of linear equation.

  1. Who knows what the forms of linear equations are?
  2. By raising your hand, who can tell me what each form look like?

First, we are using the computer to look at a simulation applet through the web. Make sure you take notes of what happens as the values change. Also there is some important information under the applet that you need to look over. (Teacher will go over each step as he/she explains the procedures of using the simulation.)

http://id.mind.net/~zona/mmts/functionInstitute/linearFunctions/lsif.html

This applet is a program that will help you visualize how changing the values for the slope, m, and the y-intercept, b, will affect the graph of the equation. At first the program will be automatically cycling through several values for m and b. If you want to use the sliders to control it yourself, just press the 'You Control' button. Then change the values for m and b and observe what happens.

 

 

 

Hoped for student response

 Answer questions.

1. Slope-Intercept Form, Point-Slope Form, Standard Form, any equation with y, x + constant (i.e. 2y = x +6)

2. y = mx + b,

    y = m(x - x 1 ) + y 1 or y - y 1 = m(x - x 1 ),

   Ax + By + C = 0 or y = (-A/B)x + (-C/B)

 

 

 

                                                      Evaluate 

Teacher would walk around and monitor students if they are doing the activity correctly and ask questions to be certain that students understand what to do.

Examples:

1. Can you tell me what’s happening to the graph?

2. How can you tell?    

 

Explore:

Learning Experience(s)

For this lesson, you will be divided into group of 4. One member of each group will come up and get the materials quietly and orderly, while the rest sit quietly until further instructions.

Distribute a bag of chips, a set of cups, and a large sheet of paper or dry-erase board to each group of students.

You will be using a cups and chips activity to solve the equation 2x + 6 = 12.


Explain instructions for activities:

Ø      If the variable is positive, place the cup(s) facing up.

 

Ø      If the variable is negative, place the cup(s) facing down.

 

Ø      The coefficient of the variable indicates the number of cups to use.

 

1. How would you show the representation of 2x using the cups?

 

The chips represent the numbers.

 

Ø      If a number is positive, the chip should be yellow side up.

 

Ø      If a number is negative, the chip should be red side up.

 

2. How would you represent +6 using chips?

 

Now draw an equal sign to the right of the two cups and six yellow chips.

 

3. How can you represent +12 on the right of the equal sign?


4. What can be done to both sides of the equation to get rid of the six yellow chips (+6)?

On the overhead, add six red chips to both sides of the equation and have students repeat these actions in their groups.

 

5. When you pair each red chip with a yellow chip, what happens?


Now remove each pairs of red and yellow chips, one pair at a time.

 

6. What happens? What is the equation?


7. If two cups equal six chips, what does that tell us about one cup?



Solve these four problems using chips and cups; each person within the group is responsible for one of the problems, and no two or more students will do the same problem. Be sure that everyone writes down on a clean nice piece of notebook paper the procedures and steps of how to solve his/her own problem with explanations.  

 

5x + 1 = -9

3x – 4 = 9

2x + 3 = 4

6x – 1 = - 13

 

 

Hoped for student response

Students come up to the front of the class and get materials quietly and orderly, while others sit quietly at desks.

Work on activities.

      Expected Student Response

1. Place two cups facing up on top of their paper or dry-erase board.

2. Place 6 chips yellow side up next to their two cups.

3. Place 12 yellow chips on the right side of the equal sign.

4. -6 should be added to each side (i.e., add six red chips to both sides); alternatively, +6 could be subtracted from each side (i.e., take away six yellow chips from each side).

5. Equal to 0.

6. 2 face up cups left = 6 yellow chips, 2x = 6.

7. Equals 3 chips.

      Evaluate 

Circulate as students are solving these problems. Allow a few minutes for students to complete both problems.

Example Questions:

Explain:

Learning Experience(s)

Ask each group to review the solutions to the problems to the class. For the second problem, ask the students to discuss the final step.(When students arrive at the equation 2x = 1).

1. Were you actually able to use the cups and chips to solve the problem?

2. When you had 2x = 1, what operation did we have to do?

In the problem -2x + 3 = -5?

3. What was the first step in solving this problem?


4. What is the next step to balance the equation and get x by itself?

Both sides need to be divided by -2, yielding x = 4, or turn over both the cups and the chips on both sides of the equation, which would represent multiplication by -1.

5. How can we check this to make sure it is the correct answer?

 

Hoped for student response

 Answer questions and explain solutions.

      Expected Student Response

1.      Yes

 

2.      Both sides divided by 2, or chip needed to be split in half.

 

 

3.      Subtract 3 from (or add -3 to) both sides of the equation, yielding -2x = -8.

 

4.      Both sides need to be divided by -2, yielding x = 4, or turn over both the cups and the chips on both sides of the equation, which would represent multiplication by -1.

 

5.      The value x = 4 can be substituted into the original equation to show that it works: -2(4) + 3 = -5.

      Evaluate

Give out quiz for the students (10-15 minutes) and go over it for the remainder of the class if time allows

Extend / Elaborate: Day 2:

You’ve solved linear equations using cups and chips and symbolic manipulation (or algebra), it's time to try solving similar equations with symbolic manipulation (algebra) only. Ok solve the following problems and you have 2 minutes for each.

  • 3x + 2 = 14
  • -3m - 1 = -10
  • -7x + 5 = 12
  • -w + 13 = 9
  • ½d + 7 = 10

Ok, lets us have 15 minutes of discussions to the questions and answers. (Call on students to review and explain their solutions, procedures, and any difficulty with the questions)

Learning Experience(s)

Ask students to identify other real life situations that use linear functions and how the functions are useful in each situation.

Now solve this problem as your test for this lesson.

Clem's balloon is 200 feet off the ground and rising at a rate of 5 feet per second. Mary's balloon is 100 feet off the ground and rising at a rate of 7 feet per second. Which of the two balloons will reach 500 feet first? Show how you found your answer.

       

 

Hoped for student response

 Solve problems using algebra.

Doing test problem.

 

      Evaluate:

For homework assignment, complete this problem:

Terry is going to the county fair. She has two choices for purchasing tickets, as shown in the table below.


Ticket Choices

Admission Price

Cost per Ride

A

$6.00

$0.50

B

$2.00

$0.75



· Write an equation for Terry's total cost (y) for ticket
Choice A. Then write an equation for Terry's total cost (y) for ticket Choice B. Let x represent the number of rides she plans to go on.

· How many rides would Terry have to go on for the total cost of ticket A and ticket B to be equal? Use mathematics to explain how you determined your answer. Use words, symbols, or both in your explanation.

· Terry plans to go on 14 rides. To spend the least amount of money, which ticket choice should Terry choose? Use mathematics to justify your answer.