Distributive Property: Solving Equations Involving

One Variable:

Developed for use in a first year Algebra class

Technology/ Manipulatives Used: TI-84 (or other graphing calculator)

Computer and Internet Overhead

Computer projector TI SmartView Algebra Tiles

The following set of plans is for the first half of Chapter 5 in the CPM algebra curriculum

Keith E. Falkowski I2T2 Final Project

Summer 2006

OVERVIEW:

New York State Standard A.A.22 Solution of all types of linear equations in one variable

NCTM Standards Algebra

Geometry Problem Solving Representation

Connections Communication

Day #1: Distributive Property ⇒ Students will multiply expressions using algebra tiles. Students will identify, use and describe the Distributive Property. Day #2: Distributive Property ⇒ Students will continue to use the algebra tiles to go further in depth with the distributive property and complete tasks involving binomials and trinomials. They will move from concrete to abstract at this point. Day #3: Solving Equations involving One Variable ⇒ Students will use the computer lab to work on the LVM website. They will be working with the algebra scale to balance the equation to solve for a variable. Day #4: Solving Equations involving One Variable ⇒ Students will use algebra tiles to balance the two sides of an equation to isolate the variable and solve the equation. Day #5: Solving Equations involving One Variable ⇒ Students will use the graphing calculator to solve linear equations

List of Materials

Algebra Tiles Cornerpieces

TI-84 (or other graphing calculator) Computer and Internet

Overhead Computer projector

TI Smartview Overhead Algebra Tiles

Student textbook Equation mats

National Library of Virtual Manipulatives Algebra balance

DAY 1 Objectives: Students will multiply expressions using algebra tiles. Students will identify, use and describe the Distributive Property Materials : Algebra tiles Cornerpieces Warm up: Students will recall the work from the previous day by completing question 5-10. The students will work on the problems themselves for a couple minutes and then check their answers with their teams. The activity is to check understanding on writing the area as a product of the dimensions and as a sum of tiles. Lesson/Activities: The questions from today’s activ ities will ask the students to multiply expressions by build ing rectangles with tiles. It is the 12th week of the year and the students have used the tiles numerous times so little time is needed to go over how to use them. It will be noted however that we will be using only the positive sides for this activ ity. Students will have the materials needed for this activity already on their tables. They will be asked to work on question 5-11 as a group. There are 8 parts to the question so each team will be assigned 4 parts to complete. The activ ity has them using the tiles to complete an area model. For question a, they will need two X tiles on one side and four X tiles on the bottom. Using the corner piece, they will find the tiles that complete the area model. Students will then record the total amount of tiles used to complete the area. As they finish an example, the

team’s recorder will go to the board to record their answer in the table provided (see Resources). As a whole group discussion, students will share what patterns they may see between the two columns. Students will be guided if needed to explain that the pattern is the Distributive Property. Students will then work on question 5-13 without using tiles and only the pattern. As the teacher circulates the room, questions such as, how do you know your answer is correct? What happened to the parenthesis? What is happening to the terms ins ide the parenthesis? To check for understanding. Closure: students will record in their Learning Logs the pattern that they used to Distribute- make up an example and solve it while explaining each step along the way. Homework: Problems 5-15 through 5-20

TABLE FOR DAY 1 ACTIVITY

AS PRODUCTS AS SUMS

A

B

C

D

E

F

G

H

Questions for Day 1 Warm up activity: For each of the following rectangles, find the dimensions and write the area as the product of the dimensions, then as the sum of the tiles.

b.

Questions for the Class Activity 5-11 Use your cornerpiece to build rectangles with the given

dimensions. Sketch each rectangle on your paper, label the dimensions and write an equivalence statement for its area as a product and a sum.

1. (2X)(4X) 2. 2X(X + 5) 3. X(2X + X) 4. 2(3X + 5) 5. (x + 3)(2X + 1) 6. (2X + 1)(2X + 1) 7. (2X + 5)(X + 2 + X) 8. X(4 + 3X) 5-12 Using the patterns that you identified, multiply the

following expressions without using your tiles 9. 2X(6X + 5) 10. 6(4X + 1) 11. 3Y(4X + 3) 12. 7Y(10X + 11Y)

DAY 2 Objectives: Students will continue multiplying expressions and will begin to use generic rectangles. Students will find dimensions to generic rectangles given pieces of the area and visa versa. Materials : Algebra Tiles Cornerpieces Overhead Overhead Algebra Tiles Student textbook Warm up: Students will record their thoughts in their learning log about the following statement: When given a product is there always a sum of the terms? Lesson/Activities: The lesson will begin with the students individually answering 4 questions which they will need to perform the distributive property, they may need to be reminded to check their learning logs from yesterday. As they complete the first task, the teacher will need to rotate throughout the class and give assistance where necessary. The students may still use algebra tiles throughout the day, but it is recommended that they are encouraged to use the generic rectangles since the rectangles will be useful during the factoring unit. Students will then complete two questions that show how the algebra tile method can be s implif ied into generic rectangles. A teacher lead discussion will take place after the students complete the third question of the day. The discussion will show how the area using algebra tiles is the exact same when simplifying using the generic rectangle. This will be done using the overhead and overhead algebra tiles. First, the problem will be done using the tiles. Then a rectangle will be drawn with the values in place

using a different transparency. The one transparency will then be picked up over the first example to show how the two methods are identical. Students will then complete 2 questions in their groups. The first one has the rectangle drawn for them and the second question has examples in which they are only given the expressions. Closure: Students will be given an extension problem that has some dimensions and some areas. They will be challenged to f ind the missing values and list the answers as a product and as a sum. Homework: Students will complete questions 5-27 through 5-32.

Classwork for day 2

1. Use the Distributive Property to find each product. a. 6(-3X + 2) b. X(4X – 2) c. 5t(10 – 3t) d. -4(8 - 6k + y)

2. Write the area as a product and as a sum.

12X

15

8X2

10X

3. The problem in question 3 uses a diagram called a

generic rectangle. It allows you multiply without using algebra tiles. FING THE AREA OF EACH PART AND WRITE THE AREA OF THE WHOLE RECTANGLE AS A PRODUCT AND A SUM.

a. 3

Y +5 b.

X

2X

c.

+5 X

2X -3 d.

-7 4y

6y -1

4. Multiply and simplify the following expressions

using either the Distributive Property or generic rectangles.

a. (x + 5)(3X + 2) c. (2y – 5)(5y + 7) b. 3x(6y – 11) d. (5w-2p)(3w+p-4) CHALLENGE QUESTION

-2

-3y

-3xy

-4X

-10

DAY 3 Objectives: Students will use an Internet website to start to work with solving linear equations? Materials : Computer lab National Library of Virtual Manipulatives Handout to record work Warm up: Before logging into the lab, students will be asked to complete some one step equations to introduce the concepts needed for solv ing multiple step linear equations in one variable. Lesson/Activities: students will log into the computer lab and go to the National Library of Virtual Manipulatives website which will be saved in the favorites bar. http://nlvm.usu.edu/en/nav/vlibrary.html This is the first time using the algebra scale so the teacher will lead the example as the students suggest what should be done. Before the teacher lead question, the students will be required to read the directions on the right s ide of the screen. The last section of this unit has 5 pages of what the students will see through the computer projector as the teacher introduces the lesson. The students will complete 10 activities using the algebra balance and record their work on a pre-made sheet. They will need to state the question, show the solution, how they arrived at the solution and a check of their answer. Closure: Students will make an entry in their learning logs addressing the idea of balancing an equation to solve for a variable.

Worksheet for day 3 To be copied on both sides for a total of 8 spaces

Question 1 Question 2

Question 3 Question 4

Question 5 Question 6

Question 7 Question 8

DAY 4 Objectives: Students will solve l inear equations that involve multiplication; they will also solve quadratic equations that simplify to linear Materials : Algebra tiles Equation mats for student use Equation mat for the overhead Warm up: Students will start the day by drawing a representation of how the equation would look if they where using algebra tiles. Each student will be given a question to represent. There will be three different questions. Lesson/Activities: Students will then volunteer to come up to the overhead to show how they set up there equations with the overhead tiles. After the third one is represented, questions will be asked about how we could simplify the mat. The students have simplif ied mats during chapter 2 so the students should be able to identify elimination, flipping and zeros. After the question is s implified, the equation will be put back on the mat. Step by step it will be simplif ied again but this time each algebra step will be recorded. When finished, the algebraic record of the problem should resemble the work that they did in the computer lab on the previous day. Students will then be instructed to work on 5-37 and 5-38 for the remainder of the period. Students will be required to record each step they take in obtaining the solution. As the students work on the problem, the teacher will be circulating around the room to identify and correct any misconceptions. They will also encourage students to use generic rectangles on question 5-38.

Closure: To finish the lesson, the students will be asked to record the algebraic records of each of the equations and the solution. For the last five minutes a discussion will take place to answer any further questions. Homework: Problems 5-39 through 5-44 WARM UP QUESTIONS 3(X+4)= 18 4(-4+X)= 4 2X + 3(X-1) = 7

Classwork questions for Day 4

5-37 Work with a partner to solve the equations using algebra tiles and the algebra mat. Record each of your steps as you solve the problem. Check your solution by substituting your answer back into the original equation.

3(X-4)=15 1-2(3X-5)=11 5(y-4)=10 -2(X-2)=11

6(X+4)=3(5X+2) 5-X(X+3)=-(X+5)(X+1)

5-38 Now work with your team to solve each of these equations without using tiles. You may want to draw generic rectangles to help you rewrite the products.

2(y-2)= -6

43= 4(X+6)-1 (X+3)(X+4) = (X+1)(X+2)

2(X+1)+3 = 3(X-1)

DAY 5 Objectives: Students will use the calculator to solve linear equations that include the distributive property Materials : TI Smartview Computer projector Graphing calculators Warm up: Students will be given a linear equation to graph using the graphing calculator; they must then record the table and record the window that they used. Lesson/Activities: Students will be given a question from yesterday to start the lesson off. The solution will again be displayed on the board. Questions will then be asked if we can obtain this solution by using the graphing calculator. Students will then be given time to come up with the solution without any aid from the teacher. If any student comes up with the solution, they will be asked to use the projector and Smartview to demonstrate how they did it. If not, the teacher will then gather the class together and ask what things they tried. After a discussion, the teacher will lead the students to coming up with the process. In their Notebooks in the Technology section they will copy the question and the key sequence used. When the example is finished they will have to justify why the process works. Students will then be given a set of 10 questions that they need to answer first without the calculator and then with the calculator, showing all necessary work. There will be a couple question that end up as decimals so the students will have to adjust the table set in terms of the change in x values. Students will be instructed to put the left side of the equation into Y1 and the right side into Y2. Then obtain a table and find

where the two Y values are the same, this will be their answer for X. Closure: Students will be given a question that contains a mistake in the work leading to the solution. First they will have to use the calculator to obtain the real answer, identify where the mistake was made during the algebra and then correctly solve the question algebraically.

Question sheet for day 5

1. 2(x-3) = 4 2. 2x = 10-3x 3. 5(4x+3) = 75 4. -6 = -6(3x-8) 5. 2-3(2x-1) = 17 6. 5+2(x+7) = 20 7. x(2x-1) = 2x2+5x-12 8. 6(2x-5) = -(x-4) 9. 2(3x-4) = 22 10. 2x-5(x+4) = -2(x+3)

The following are the 5 screen shots to aid in the presentation of NLVM 1. This is the first screen

Students will click on the appropriate boxes to represent the situation. They will obtain the next screen

After they click continue, they will be asked to + - x or / from both sides and the value When they chose subtract 1 the next screen will appear

Next they will choose to subtract an x to obtain the next screen

Now they will divide by 2 to obtain the next screen

They will then get the correct answer of 2

ANSWER KEY

Day 1 Warm ups (x+2)(2x+3) = 2x2 +7x=6 (x+3)(x+4) = x2 + 7x +12 activities

8x2

2x2 +10x 3x2

6x+10 2x2 +7x+3 4x2 + 4x +1 4x2 + 14x +10 4x + 3x2 12x2 + 10x 24x + 6 12xy + 9y 70xy +77y2

day 2 1a. -18x +12 1b. 4x2-2x 1c. 50t-15t2 1d. -32 + 24k + 4y 2. (4x +5)(2x +3) = 8x2 +22x + 15 3a. 3y+15 = 3(y+5) 3b. 2x(x) = 2x2 3c. (2x-3)(x+5) = 2x2 + 7X – 15 3d. (6y-1)(4y-7) = 24y2 – 46y + 7 4a. 3x2 + 17x +10 4b. 18xy – 33x 4c. 10y2 – 11y + 35 4d. 15w2 – 2p2 – wp – 20w + 8p day 3 students answers will vary depending on the questions the program gives them to solve. I would recommend correcting a couple to check for understanding.

Day 4 Warm up

1. 1 2. 5 3. 2

activities 1. 9 2. 0 3. 6 4. -3.5 5. 2 6. -3 1/3 7. -1 8. 5 9. -2.5 10. 8

day 5

1. 5 2. 2 3. 3 4. 3 5. -2 6. .5 7. 2 8. 34/13 9. 5 10. -14