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

O VERVIEW : 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 ca lcula tor) Computer a nd Internet Overhead Computer projector TI Smartview Overhead Algebra Tiles Stud ent textbook Equation ma ts National Library of Virtua l Ma nipulatives Algebra ba la nce

DAY

1

Objectiv es: S tudents will m ultiply expressions us ing algebra tiles. Students w ill identify, us e and d escribe the Dis tributiv e Property Materials :

Algebra tiles Cornerpieces

Warm up: Stud ents will reca ll the work f rom the previous day by com pleting q ues tion 5 -10. T he students will work on the problems thems elves f or a couple minutes and then check their a nswers with their teams. The a ctivity is to check und ers tanding on writing the area as a product of the dimens ions a nd as a s um of tiles. Lesson/Activities: The q ues tions f rom today ’s activ ities will as k the students to m ultiply expressions by build ing recta ngles with tiles. It is the 12 th week of the year and the s tud ents have used the tiles numerous times so little time is need ed to go over how to us e them. It will be noted howev er tha t we will be using only the positiv e sides for this a ctiv ity. Stud ents will have the materials needed for this activity already on their tables. T hey w ill be asked to w ork on q ues tion 5 -11 as a group. T here are 8 parts to the q ues tion so ea ch team will be ass igned 4 pa rts to complete. The activ ity has them using the tiles to com plete an area model. For ques tion a, they will need two X tiles on one side a nd four X tiles on the bottom. Us ing the corner piece, they will find the tiles that complete the a rea mod el. Stud ents will then record the total am ount of tiles us ed to complete the area. As they finis h a n example, the

team’s recorder will go to the board to record their answer in the table provided (s ee Res ources). As a whole group dis cussion, students will s hare wha t patterns they may s ee between the two colum ns. Stud ents will be guided if needed to explain tha t the pattern is the Distributiv e Property . Students will then work on q ues tion 5 -13 w ithout using tiles and only the pattern. As the teacher circula tes the room, ques tions such as, how do you know y our a nswer is correct? Wha t happened to the parenthesis? What is ha ppening to the terms ins ide the parenthesis? To check for und ersta nding. Closure: students will record in their L ea rning Logs the pa ttern tha t they us ed to D istribute- ma ke up a n example a nd solve it w hile expla ining ea ch step a long the way. Homework:

Problems 5-15 through 5 -20

TABLE FOR DAY 1 ACTIVITY

AS PRODUCTS A B C D E F G H

AS SUMS

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 fo r t he 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

Objectiv es: Stud ents will continue m ultiplying express ions and w ill begin to us e generic recta ngles. Stud ents will find dimensions to generic rectangles given pieces of the area a nd visa v ers a. Materials :

Algebra Tiles Cornerpieces Overhead Overhead Algebra Tiles Stud ent textbook

Warm up: Stud ents will record their thoughts in their lea rning log a bout the following sta tem ent: When given a product is there a lways a sum of the terms? Lesson/Activities: The lesson will begin with the students individ ually answ ering 4 ques tions which they will need to perform the d istributive property, they may need to be rem ind ed to check their lea rning logs f rom yesterday. As they com plete the firs t task, the tea cher will need to rota te throughout the class a nd give assistance w here necessary . The students may s till use a lgebra tiles throughout the day, but it is recommend ed that they a re encoura ged to use the generic rectangles since the rectangles will be usef ul during the fa ctoring unit. S tud ents will then complete two questions that s how how the algebra tile method can be s implif ied into generic recta ngles. A tea cher lead dis cuss ion will ta ke place a fter the s tud ents complete the third question of the day. T he discussion will s how how the a rea us ing algebra tiles is the exa ct same when s implifying us ing the generic rectangle. T his will be done us ing the overhead and overhead algebra tiles. First, the problem w ill be done using the tiles. Then a recta ngle w ill be d rawn with the va lues in pla ce

using a different tra ns parency. The one trans parency will then be picked up over the first exam ple to show how the two m ethods are id entical. S tudents will then complete 2 q uestions in their groups. The first one has the rectangle draw n for them a nd the s econd q ues tion has examples in w hich they are only giv en the express ions. Closure: Stud ents will be given a n extension problem that has som e dimensions and s ome areas. T hey will be cha llenged to f ind the m issing values and lis t the answers as a prod uct and as a s um. Homework: Stud ents will complete q ues tions 5-27 through 5-32.

Classwork for day 2 1.

Use the D istributiv e Property to find each prod uct.

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 ca lled a generic rectangle. It a llows y ou multiply without using a lgebra tiles. FING T HE AREA OF EACH PART AND WRITE THE AREA OF THE WHOLE RECTANGLE AS A PROD UCT AND A SUM. a.

3 Y

+5

b. X 2X

c.

+5

X

2X

-3

d. -7 4y

6y

-1

4. Multiply a nd sim plif y the f ollowing expressions using either the Dis tributive Property or generic recta ngles. a. b.

(x + 5)(3X + 2) 3x(6y – 11)

c. d.

(2y – 5)(5y + 7 ) (5w-2p)(3w+p-4)

CHALLENGE QUESTION -3xy

-4X

-10 -3 y

-2

DAY 3 Objectiv es: Stud ents will use a n Internet w ebsite to sta rt to work with solving linear equa tions? Materials :

Computer la b National Library of Virtua l Ma nipulatives Handout to record work

Warm up: Before logging into the la b, students will be as ked to com plete some one step equa tions to introd uce the concepts needed f or solv ing multiple s tep linear eq uations in one va riable. Lesson/Activities: students will log into the computer lab a nd go to the Na tiona l Libra ry of Virtual Manipulatives w ebs ite which will be sav ed in the favorites bar. http://nlvm.us u.ed u/en/nav /vlibrary.html This is the f irs t time using the algebra sca le so the tea cher will lead the example a s the students s uggest what s hould be done. B efore the tea cher lead q ues tion, the students w ill be req uired to read the directions on the right s ide of the s creen. The las t section of this unit has 5 pages of wha t the s tudents will s ee through the computer projector as the teacher introduces the lesson. T he s tud ents will complete 10 a ctivities us ing the algebra bala nce a nd record their w ork on a premade sheet. They will need to state the q uestion, s how the solution, how they a rriv ed a t the s olution and a check of their a nswer.

Closure: Students will make a n entry in their learning logs addressing the idea of bala ncing a n eq uation to solve for a va ria ble.

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

Objectiv es: Stud ents will solv e linear eq uations tha t involve m ultiplica tion; they will also solv e q uadratic equa tions tha t sim plif y to linear Materials :

Algebra tiles Equation ma ts for student use Equation ma t for the overhead

Warm up: Stud ents will sta rt the day by drawing a repres enta tion of how the eq ua tion would look if they where using a lgebra tiles. Each stud ent will be given a ques tion to repres ent. T here will be three different ques tions. Lesson/Activities: Stud ents will then volunteer to come up to the ov erhead to s how how they set up there equa tions with the ov erhead tiles. After the third one is represented, questions will be asked a bout how we could sim plify the ma t. T he s tudents hav e sim plif ied mats during cha pter 2 so the students s hould be able to identify elimination, f lipping a nd zeros. After the ques tion is s implified, the equa tion will be put back on the mat. Step by step it will be simplif ied a gain but this tim e each algebra s tep will be record ed. When finis hed, the algebra ic record of the problem s hould resem ble the work that they did in the computer lab on the prev ious day. Students will then be instructed to work on 5 -37 a nd 5-38 for the remainder of the period. Stud ents will be required to record ea ch s tep they take in obta ining the solution. As the students work on the problem, the tea cher will be circulating around the room to id entify a nd correct a ny mis conceptions. They will also encourage s tudents to us e generic rectangles on ques tion 5 -38.

Closure: To finish the lesson, the s tudents will be as ked to record the algebraic records of ea ch of the equations and the s olution. For the last five minutes a dis cuss ion will take pla ce to a nswer any f urther q ues tions. Homework:

Problems 5-39 through 5-44

WARM UP Q UESTIONS 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 wit hout 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

Objectiv es: Stud ents will use the calculator to solve linear eq uations that include the distributiv e property Materials :

TI Smartview Computer projector Gra phing calculators

Warm up: Stud ents will be given a linea r equa tion to gra ph using the gra phing calculator; they m ust then record the ta ble and record the window tha t they us ed. Lesson/Activities: Stud ents will be given a q ues tion from yes terday to start the les son off. The solution will a gain be dis play ed on the board. Q ues tions will then be as ked if we ca n obta in this solution by using the gra phing calculator. S tud ents will then be giv en time to come up with the solution without a ny a id from the tea cher. If a ny s tud ent comes up w ith the solution, they will be as ked to use the projector a nd Smartview to demonstrate how they did it. If not, the tea cher w ill then gather the class together and ask w hat things they tried. Af ter a dis cuss ion, the teacher will lead the students to com ing up w ith the process. In their Notebooks in the Technology section they will copy the ques tion a nd the key seq uence used. When the exam ple is finished they will hav e to jus tify w hy the process works. S tudents will then be given a s et of 10 questions tha t they need to a nswer firs t without the ca lcula tor and then with the calculator, showing all necessary work. T here w ill be a couple q uestion that end up as decima ls so the students w ill ha ve to adjus t the ta ble set in terms of the change in x values. Students will be instructed to put the lef t sid e of the eq ua tion into Y1 and the right sid e into Y2. Then obtain a table and f ind

where the two Y values a re the same, this will be their answer for X. Closure: Stud ents will be given a q ues tion that contains a mista ke in the work lead ing to the solution. F irst they will have to use the calculator to obtain the rea l answer, identif y where the m istake was mad e during the algebra and then correctly solv e the q ues tion algebra ica lly.

Question sheet fo r day 5 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

2(x-3) = 4 2x = 10-3x 5(4x+3) = 75 -6 = -6(3x-8) 2-3(2x-1) = 17 5+2(x+7) = 20 x(2x-1) = 2x 2 +5x-12 6(2x-5) = -(x-4) 2(3x-4) = 22 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 1a. 1b. 1c. 1d. 2. 3a. 3b. 3c. 3d. 4a. 4b. 4c. 4d.

2 -18x +12 4x2-2x 50t-15t2 -32 + 24k + 4y (4x +5)(2x +3) = 8x2 +22x + 15 3y+15 = 3(y+5) 2x(x) = 2x2 (2x-3)(x+5) = 2x2 + 7X – 15 (6y-1)(4y-7) = 24y2 – 46y + 7 3x2 + 17x +10 18xy – 33x 10y2 – 11y + 35 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