Download Grade 8 - Newark Public Schools

THE NEWARK PUBLIC SCHOOLS
THE OFFICE OF MATHEMATICS
Grade 8
Linear Equations in One Variable
8.EE.7a-b
Answer Pages
2012 COMMON CORE STATE STANDARDS ALIGNED MODULES
Lesson 1-Introductory Task (pg. 3)
If the price of the calculator is “n”, then the question is “when is n – 20 = 20% of n”.
n – 20 = .2n
.8n = 20
n = 100
ans: When the calculator cost $100.00
Focus Question
You simplify an algebraic expression when terms are alike by adding (combining) the like term.
Guided Practice (pg. 4)
1.…..a. k
b. 12r – 20
g. –r
h. 11v
2. 27 feet below sea level
3. 0
4. a. 3 + 6x
b. 8y + 24
g. .2s + 1.4 h. -4p
c. 10n – 10
i. -19x
c. 7x – 14
i. 26x – 40
d. -14x
j 6p
e. –r -10
k. 16n
d. -7y + 56
j. -3x – 30
e. .2x + 1.25
k.
Collaborative Work (pg. 5)
1.…..a. positive
b. negative
c. positive
d. zero
2. – x – 26
3. a. 35.5m + 50 = 635.5
b.
35.5m = 585.5
m = 16.493
It will take Maria 17 months to reach her goal.
Journal Question
1…..x =
2.
– 16x + 12 = 52
3.
n=1
4.
f. 8x + 11
l. -4n -5
f. 3y – 2
l.
e. no solution
x= 12
Homework (pg. 6)
1.…..Anna made two mistakes. Correct answer: 15x – 50
2. a. 13 + 2x
b. 10.3 - .3x
c. 21x -7
d. 1.2 – n
e. 18x – 18
f. – 18 + 22x
g. 1.7y + 1
h. 10x + 6
i. 5d – 15
j. 60a + 30
3. – 16 + 4x = 21
Lesson 2-Introductory Task (pg. 7)
. 15d = 2025
.
.
.
d=
d = 135 days
If you travel 15 miles a day for 135 days, you will have traveled a total of 2,025 miles.
15(162) = total miles
2,430 miles
Focus Question
1.…..An equation has an equal sign, an expression does not! An equation is made up of two expressions
separated by and equal sign.
2. No. It can’t have a solution. It is not an equation.
Guided Practice (pg. 8)
1.….. . 0.8(200 – y) = 160 – 0.8y
.
.
2.
a.
b.
c.
d.
0.8(200 – 20) = 144 beats per minute
0.8(200 – 50) = 120 beats per minute
7(5n – 9) = 35n – 63
11(x + 2) = 11x + 22
4(5 + 2y) = 20 + 8y
3(4n – 2) = 12n – 6
(pg. 9)
3. $63.60
4. s = 1
5. 6m – 8 = - 5
6
–8=-5
3–8=-5
-5 = -5
Yes, m = is a solution to the equation
Collaborative Work (pg. 10)
1.….. Change 499 to 500. Multiply by 5 = 2500. Subtract 5 (due to the 1 added to 499 times 5 = 5).
Ans: 2495
2. – x – 13 = 9
- x = 22
x = - 22
3. 11 + 3x – 7 = 6x + 5 – 3x
3x + 4 = 5 + 3x
0  -1 no solution
4. h = 11.5 – 2.25
5. (a) x = 7; dimensions: l = 47, w = 21
(b)
x = 12; dimensions: 12, 29, 55, 38
Journal Question
1.…..Use the opposite mathematical operation to the one that is in the given equation.
2.
Homework (pg. 11)
1.…..520 + m = 105
m =- 415
2.
– 21 = p – 13
-8 =p
3.
16x = 100
x=6
4.
– 6x – 3 = -18
- 6x = - 15
x=2
5.
9.
6x + 5 – 2x = 4 + 4x + 1
4x + 5 = 4x + 5
Identity: all real numbers
5x + 150 = - 100
10.
5x = -250
x = - 50
13.
14.
.
.
6.
6x – (3x + 8) = 16
6x – 3x – 8 = 16
3x = 24
x= 8
10 = + 4
11.
7.
-0.5x = 6
x = - 12
9 = 3(5x – 2)
6=
30 = n
12.
8.
= 2.5
x=5
7(5x – 2) = 6(6x – 1)
9 = 15x – 6
15 = 15x
1=x
35x – 14 = 36x – 6
-x=8
x=-8
100d = 29887
d = 298.87 days
Yes! If you substitute 5 for x, You’d multiply 5 times 3 and get 15, then subtract 3 and get 12.
Lesson 3-Introductory Task (pg. 12)
1.…..On the third line, Hoover multiplied two negatives and got a negative – incorrect!
Line three should be: 6 – 4 = 2
Therefore, x = - 2 is a solution !
2. The distributive property was used incorrectly. There should be a “plus sign” between (3 . y) and
(3 . 6). The answer should be 3y + 18.
Focus Question
Equivalent equations are equations that are equal in value but look differently.
Guided Practice (pg. 13)
1.…..4n = 21 – 3n
2.
n = 21
5.
2(x + 3) = 3(x – 3)
2x + 6 = 3x – 9
-x = - 15
x = 15
8a – 5 = 3a + 25
8a – 3a = 25 + 5
5a = 30
a = 6
6.
3.
2(x – 5) = x
2x – 10 = x
x = 10
4(m + 3) – 2m = 3(m – 3)
4m + 12 – 2m = 3m – 9
6m + 12 = 3m – 9
6m – 3m = -9 – 12
3m = - 21
4.
3(2x – 1) = 9
6x – 3 = 9
6x = 12
x=2
7.
2a + 3 = 3 + 2a
2a – 2a = 3 - 3
0=0
Identity: All real numbers
m=-7
8.
a.
3y – 5 = y + 2y – 9
3y – 5 = 3y – 9
Contradiction: no solution
2x + 4 = 2(x + 2)
c.
2x + 4 = 2x + 4
Identity: infinitely many solutions
d. 2(x + 5) = 3x + 4 – x
2x + 10 = 3x + 4 – x
2x + 10 = 2x + 4
Contradiction: no solution
9.
b.
e.
7(x – 5) = 2(x + 5)
7x – 35 = 2x + 10
7x – 2x = 10 + 35
5x = 45
x= 9
one solution
f.
2y – 4 = 3y – 5
1=y
one solution
8–x=x–4
-x – x = - 4 – 8
- 2x = - 12
x=6
one solution
Let p = number of pages in the book
P = 360
There are 360 pages in the book.
10. If you have $16.00 and a $5.00 coupon, you can spend $21.00. A bottle of olive oil costs $7.00,
therefore, you can buy 3 bottles of olive oil.
Collaborative Work (pg. 14)
1.…..3x = 48 – x
2. 6(a – 3) = 12
4x = 48
6a – 18 = 12
x = 12
6a = 30
a=5
5.
46 = 5x + x + 16
46 = 6x + 16
30 = 6x
5=x
7.
8.
9.
10.
8
lost 14; won 21
Chip has $26; Al has $52.
60 boys
3. 4(m – 3) = m + 15
4m – 12 = m + 15
3m = 27
m=9
6.
4. 8x + 4 – 3x = 24
5x + 4 = 24
5x = 20
x=4
9a + 2 – 6a = 35
3a + 2 = 35
3a = 33
a = 11
Journal Question
2x
x
1.
2.
x=9
32 = 2x + 12
32 = 2(6) + 12
32 = 12 + 12
32  24
x = 6 is not a solution
2x + 3
Homework (pg. 15)
1.…..a. –x + 7 – 6x = 19 – 7x
- 7x + 7 = 19 – 7x
-7x + 7x = 12
0 12
no solution
b. 2b + 3 = 2b – 1
2b – 2b = -4
0  -4
no solution
d. 2(3x – 4) = 2x + 4x + 8
6x – 8 = 6x + 8
6x – 6x = 8 + 8
x 16
no solution
c. -2x + 3 = 5x – 1 – 7x
-2x + 3 = -2x – 1
-2x + 2x = - 4
0  -4
no solution
e. 9m – 4 + 2m = 8m + 3m + 5
11m – 4 = 11m + 5
11m – 11m = 5 + 4
m9
no solution
Each of the equation in #1 is a contradiction: i.e. the variable term cancels leaving an untrue
statement, therefore, there is not solution to any of these equations.
2.
f. 4x – 2 = 2(2x – 1)
g.
4x – 2 = 4x – 2
4x – 4x = -2 + 2
0 = 0
identity: All real numbers
i.…..
- 18a + 3 = 3 – 18a
- 18a + 18a = 3 – 3
0 = 0
identity: All real numbers
+x
4x – 8 = 3x – 8 + x
4x – 8 = 4x – 8
4x – 4x = - 8 + 8
0 = 0
identity: All real numbers
j.
h. 2(3x – 4) = 2x + 4x – 8
6x – 8 = 6x – 8
6x – 6x = -8 + 8
0 = 0
identity: All real numbers
9m – 4 + 2m = 8m + 3m – 4
11m – 4 = 11m – 4
11m – 11m = - 4 + 4
0 = 0
identity: All real numbers
Each of the equations in # 2 is an identity: i.e. the result will be 0 = 0 and any real number can be
substituted in for the variable and the equation will be true. Ans. All Real Numbers.
Lesson 4-Introductory Task (pg. 16)
. 3(x + 5) = 4x
.
.
3x + 15 = 4x
15 = x
A piece of pizza has 65 calories
Focus Question
If, at any point in the procedure for solving and equation, both sides of the equation are obviously the
same, then there are infinitely many solutions (Identity). If both sides are obviously NOT the same,
then there is no solution (Contradiction). Otherwise, there is just one specific real number that will
solve the equation.
Guided Practice (pg. 17)
A.…..1.
2.
4(x + 1) = 4x + 1
4x + 4 = 4x + 1
No number is a solution
3. 6(x + 3) = 2(3x + 1)
6x + 18 = 6x + 2
No number is a solution
4.
x + 3x – 2x = 12
2x = 12
x=6
One number is a solution.
5. 2(8x + 6) = 4(4x + 3)
16x + 12 = 16x + 12
There are infinitely many solutions.
6.
2x + 5 = 5 + 2x
There are infinitely many solutions.
B.
.
.
.
.
9a + 2 – 6a = 35
3a + 2 = 35
3a = 33
a = 11
One number is a solution.
m + 3m = 40
4m = 40
m = 10
$30.00
Yes. If Al has $30, then Peter has $10 and together they have $40.
Collaborative Work (pg. 18)
1.…..2x + 5 = - 25
-3m – 6 = 48
2x = - 30
-3m = 54
x = - 15
m = -18
x . m = (-15)(-18) = 270
2.
Let n = the number Mrs. Reno is thinking of
7n – 6 = 85
7n = 91
n = 13
3.
Let r = number of rides that Robin went on
5 + 3n = 17
3n = 12
n=4
Robin went on 4 rides.
4.
3(2x – 5) = 2(3x + 5) – 20
6x – 15 = 6x + 10 – 20
6x – 15 = 6x – 10
Contradiction: no solution
5.
4p – 6p + 12 = 52
-2p + 12 = 52
-2p = 40
p = - 20
6.
Let y = age of youngest brother
y + 1 = age of the middle brother
y + 2 = age of the oldest brother
y + y + 1 + y + 2 = 63
3y + 3 = 63
3y = 60
y = 20
The middle brother is 21 years old.
7.
2(3p – 4) = 10
6p – 8 = 10
6p = 18
p=3
One, unique, real number solution.
8.
3x = 2x
x=0
One, unique, real number solution.
Journal Question
Let x = the cost of one granola bar
1250x + 25 = 800
1250x = 775
x = .62
Each Granola bar cost $0.62.
Homework (pg. 19)
1.….Sherry has $14, Sherlene has $25, and Lou has $50
2. Rich is 9, Ruth is 18, and Carla is 6.
3. 3x = 48 – x
4x = 48
x = 12
4.
6(a – 3) = 12
6a – 18 = 12
6a = 30
a=5
5.
4(m – 3) = m + 15
4m – 12 = m + 15
3m = 27
m=9
6.
3x – 8 + 4x = 10x + 8 – 3x
7x – 8 = 7x + 8
Contradiction: no solution
Golden Problem (pg. 20)
.
Let d = number of miles your car is towed
34 + 2d = 70
2d = 36
d = 18
Your car was towed 18 miles.
.
Company B: 28 + 3(18) = 82
We do not agree. Company B would have charged $82 to tow a car 18 miles.
. Set their fees equal to each other to determine when to use which of the companies for the lower
rate.
34 + 2d = 28 + 3d
-d = - 6
d=6
The companies would change the same amount of money to tow a car six (6) miles. Therefore,
If the car needs to be towed less than 6 miles, it would be cheaper to use Company A. If the car needs
to be towed more than 6 miles, it would be cheaper to us Company B.