EPEI - RUSD Mathrusdmath.weebly.com/uploads/1/1/1/5/11156667/g6_u4...6.EE.2.c 13. f3 = 2 • 2 • 2...
Transcript of EPEI - RUSD Mathrusdmath.weebly.com/uploads/1/1/1/5/11156667/g6_u4...6.EE.2.c 13. f3 = 2 • 2 • 2...
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ANSWERS FOR EXERCISES
MATH GRADE 6 UNIT 4
EXPRESSIONS
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Grade 6 Unit 4: Expressions
ANSWERSLESSON 1: NUMERICAL EXPRESSIONS
ANSWERS
6.EE.1 2. I will test the trick with both odd and even original numbers between 1 and 100.
Original number = 1 1 • 2 = 2 2 + 12 = 14 14 ÷ 2 = 7 7 – 1 = 6
Original number = 4 4 • 2 = 8 8 + 12 = 20 20 ÷ 2 = 10 10 – 4 = 6
Original number = 5 5 • 2 = 10 10 + 12 = 22 22 ÷ 2 = 11 11 – 5 = 6
Original number = 31 31 • 2 = 62 62 + 12 = 74 74 ÷ 2 = 37 37 – 31 = 6
Original number = 88 88 • 2 = 176 176 + 12 = 188 188 ÷ 2 = 94 94 – 88 = 6
Original number = 100 100 • 2 = 200 200 + 12 = 212 212 ÷ 2 = 106 106 – 100 = 6
Yes, this trick will always give the number 6.
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Grade 6 Unit 4: Expressions
ANSWERSLESSON 2: NUMERICAL EXPRESSIONS
ANSWERS
6.EE.1 1. C 6
6.EE.1 2. 7
6.EE.1 3. 35
6.EE.1 4. 2
6.EE.1 5. 176
6.EE.1 6. 125
6.EE.1 7. B Subtract 5 from 12.
6.EE.1 8. D 99 – (50 ÷ 5) + 5 = 94
E 99 – [50 ÷ (5 + 5)] = 94
6.EE.1 9. (22 + 3) • (9 – 5) = 100
6.EE.1 10. 16 • (3 + 3) • 5 = 480
6.EE.1 11. 82 – 16 ÷ 8 • (5 + 25) = 22 OR 82 – (16 ÷ 8) • (5 + 25) = 22
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Grade 6 Unit 4: Expressions
ANSWERSLESSON 2: NUMERICAL EXPRESSIONS
Challenge Problem
6.EE.1 12. The order of operations that you follow is important because it affects the value of an expression. Two people could get very different values for the same expression if they followed a different order in performing the operations.
An example of a real-life situation that uses the order of operations is converting a temperature in Fahrenheit to a temperature in Celsius. For example, to convert 50°F to Celsius, you use this equation:
(50 – 32) • 5 ÷ 9 = (18) • 5 ÷ 9 = 90 ÷ 9 = 10º C
If you didn’t follow the proper order of operations—if you didn’t complete the operations inside the parentheses first—you might say that 50°F converts to about 32.2°C.
50 32 5 9 50 160 9
50 17 7
32 2
−( ) ÷ = − ÷
= −
≈
•
.
. �C
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Grade 6 Unit 4: Expressions
ANSWERSLESSON 3: ALGEBRAIC EXPRESSIONS
ANSWERS
6.EE.2.a 1.A
x3
16+
6.EE.2.a 2. n + 12 or 12 + n
6.EE.2 3. Algebraic Expression
Numerical Expression
Algebraic Equation
Numerical Equation
7 1n +( )
n n− +( )3 105 2
18
3
•( ) ÷
6 36n = 4 2 4
14
2÷ ( ) =•
6.EE.2.a 4. D 12(3 + a)
6.EE.2.a 5. C 9a + 3b
6.EE.2.a 6. 253
2513
−−( )n
n or •
6.EE.2.a 7. 2(n + 8) or (n + 8) • 2
6.EE.2.a 8. 52
425
4n
n+ ÷
+ or
6.EE.2.a 9. n2 – 7 or n • n – 7
6.EE.2.a 10. nn
++( ) ÷
93
9 3 or
6.EE.2.a 11. 72 + 3n
6.EE.2.a 12. Multiply any number n by 3, and then add 12.
6.EE.2.a 13. Add 15 to any number b, and then divide by 3.
or
Add 15 to any number b, and then multiply by 13
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Grade 6 Unit 4: Expressions
ANSWERSLESSON 3: ALGEBRAIC EXPRESSIONS
Challenge Problem
6.EE.2.a 14. Here is one example.
Algebraic expression: 7(5n + 2)
Written expression: Add 2 to 5 times any number n, and then multiply by 7.
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Grade 6 Unit 4: Expressions
ANSWERSLESSON 4: WRITING EXPRESSIONS
ANSWERS
6.EE.3 1. B 4x + 17
6.EE.2.a 2. B 7a
6.EE.2.a 3. A 3a
6.EE.2.a 4. D 2a + 3b
6.EE.2.a 5. D a + b + 2c
6.EE.2.a 6. a + 2(b + c + d) and a + 2b + 2c + 2d
6.EE.2.a 7. Any two of these equations are acceptable. 2(a + b + c) + 2b + c + d + e 2a + 4b + 3c + d + e 2(a + 2b) + 3c + d + e
6.EE.2.a 8. a + 3(c + d) + e and a + 3c + 3d + e
6.EE.2.a 9. a + 4(b + c) + b and a + 5b + 4c
6.EE.3 10.
Addition
Distributive property of multiplication over addition
Commutative property of addition
Associative property of additiona c b c d c a c c c b d
a c c c b
+ + + + + = + + + + += + + +( ) +
2 2 2 2 2 2 2 2
2 2 2 ++= + + += + + +( )
2
5 2 2
5 2
d
a c b d
a c b d
Challenge Problem
6.EE.2.a 6.EE.3
11. Here is one example.
Total length of train: 2a + 2b + 2c + 2d + a + 2b + 2c + 2d + a + 2b + 2c + 2d + a + 2b + 2c + 2d + e
Combine like terms: 2a + 2b + 2c + 2d + a + 2b + 2c + 2d + a + 2b + 2c + 2d + a + 2b + 2c + 2d + e (2a + a + a + a) + (2b + 2b + 2b + 2b) + (2c + 2c + 2c + 2c) + (2d + 2d + 2d + 2d) + e 5a + 8b + 8c + 8d + e
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Grade 6 Unit 4: Expressions
ANSWERSLESSON 5: EVALUATING EXPRESSIONS
ANSWERS
6.EE.2.c 1. D 20
6.EE.2.c 2. 54
6.EE.2.c 3. 102.8
6.EE.2.c 4. 4
6.EE.2.c 5. 4
6.EE.2.c 6. 123.6
6.EE.2.c 7. 16.2
6.EE.2.c 8. 24
6.EE.2.c 9. 20.25
6.EE.2.c 10. 12f
6.EE.2.c 11. 12f = 12 • 2 = 24
The total length of the edges is 24 inches.
6.EE.2.c 12. f3 is the volume of the cube.
6.EE.2.c 13. f3 = 2 • 2 • 2 = 8
The volume of the cube is 8 cubic inches.
Challenge Problem
6.EE.2.c 14. Expression in words: Subtract 2 from the number of sides.
Algebraic expression: n – 2
n – 2 = 100 – 2 = 98 98 triangles can be formed if n = 100.
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Grade 6 Unit 4: Expressions
ANSWERSLESSON 6: VOCABULARY OF EXPRESSIONS
ANSWERS
6.EE.2.b 1. A 2 terms
6.EE.2.b 2. A 43
D 7(t + 4b2)
6.EE.2.b 3. The coefficients are 4, 5, 1, and 9.
6.EE.2.b 4. The exponents are 3, 1, 2, and 1.
6.EE.2.c 5. 125
6.EE.2.b 6. There is only one constant: 11.
6.EE.2.b 7.Expression Variables Coefficients Exponents Number of
Terms Constants
s, t 4, 6 1, 1 2 25
s, t 8, 12 2, 1 3 50
w, z –3, 4 1, 1 2 none
noneb, c, d 2, 3, 4 1, 1, 1 1
+ +t s2(4 6 ) 25
+ +t s8 12 502
−w z4 3
+ +d b c6(4 3 2 )
6.EE.2.a 6.EE.2.b
8. Here are three examples.
7a + b + 5
2(x + 42) + 6y + 19
8(m + 4) + 4(3y – 5) + 3
6.EE.2.a 6.EE.2.b
9. Here are three examples.
3(x + 9)
0.125(a + 9)
14
9m +( )
6.EE.2.a 6.EE.2.b
10. x 5
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Grade 6 Unit 4: Expressions
ANSWERSLESSON 6: VOCABULARY OF EXPRESSIONS
6.EE.2.a 6.EE.2.b
11. Here are three examples.
)
k
x
a
85
15
Challenge Problem
6.EE.2.a 6.EE.2.b
12. a. 2 4( )x + (number of terms: 1; coefficients: 1; exponents: 1) or 2 16x + (number of terms: 2; coefficients: 2; exponents: 1)
b. 2 4( )x + (number of terms: 1; coefficients: 1; exponents: 1) or 2 8x + (number of terms: 2; coefficients: 2; exponents: 1)
c. x x( )4 (number of terms: 1; coefficients: 1, 4; exponents: 1, 1) or 4 2x (number of terms: 1; coefficients: 4; exponents: 2)
d. 12
4x +( ) (number of terms: 1; coefficients: 1; exponents: 1)
or 12
2x + (number of terms: 2; coefficients: 12
exponents: 1)
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Grade 6 Unit 4: Expressions
ANSWERSLESSON 7: EQUIVALENT EXPRESSIONS
ANSWERS
6.EE.3 1. B 3a + 3b
6.EE.3 2. 7 28
6.EE.3 3. dx + 4d
6.EE.3 4.5
18
458
20x x−
= −
6.EE.3 5. Here are four examples.
4(x + 6 + x + 2.25) 4x + 24 + 4x + 9 8x + 33 4(x + 6) + 4(x + 2.25)
6.EE.2.c 6. 47
6.EE.3 7. D 8 a b+( )
6.EE.3 8. nx + ny = n(x + y)
6.EE.3 9. 12a + 4b = 4 • 3a + 4 • b = 4(3a + b)
6.EE.3 10. 19
23
19
69
19
6c d c d c d− = − = −( )
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Grade 6 Unit 4: Expressions
ANSWERSLESSON 7: EQUIVALENT EXPRESSIONS
Challenge Problem
6.EE.2.a 11. Baseball cards = x
Basketball cards = 13
x
Hockey cards = 713
73
• x x=
Football cards = 73
273
12
76
x x x÷ = =•
Total number of cards in Martin’s collection: x x x x+ + +13
73
76
Simplified expression: x x x x x x x+ + + = + + +
= + + +
=
13
73
76
113
73
76
66
26
146
76
296
The number of cards in Martin’s collection is 296
x
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Grade 6 Unit 4: Expressions
ANSWERSLESSON 8: GREATEST COMMON FACTOR
ANSWERS
6.NS.4 1. B 4
6.NS.4 2. D 15
6.NS.4 3. A 2
6.NS.4 4. C 15
6.NS.4 5. C 4
6.NS.4 6. Factors of 4 Only
Factors of 6 Only
Factors of Both 4 and 6
Not Factors of Either 4 or 6
43
6
1
25
6.NS.4 7. Factors of 12 Only
Factors of 15 Only
Factors of Both 12 and 15
Not Factors of Either 12 or 15
2 4
6 12
5
15
1
3
7 8 910 11 13 14
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Grade 6 Unit 4: Expressions
ANSWERSLESSON 8: GREATEST COMMON FACTOR
6.NS.4 8. Here is one example. I made a rectangle 16 units wide and 30 units long.
I tried different size squares to figure out which could fill the rectangle without any overlapping. I found the greatest common factor is 2. 1 is also a common factor for 16 and 30.
6.NS.4 9. Here is one example.
First, I found all of the factors of 12 : 1, 2, 3, 4, 6, 12 Then, starting with the greatest factor, I divided 16 by the factors of 12 to see which divided evenly without a remainder. When I got to a number that divided without a remainder, I knew that the number was a common factor. When I finished dividing 16 by all the factors of 12, I looked at the common factors and chose the greatest number as the greatest common factor.
16 ÷ 12 = 1 R4 16 ÷ 6 = 2 R2 16 ÷ 4 = 4 16 ÷ 3 = 5 R1 16 ÷ 2 = 8 16 ÷ 1 = 16
The greatest common factor is 4. 1 and 2 are also common factors of 12 and 16.
6.NS.4 10. Here is one example. I find the factors of each number.
14 = 1, 2, 7, 14
70 = 1, 2, 5, 7, 10, 14, 35, 70
I circle all of the common factors and look for the greatest one. The greatest common factor is 14.
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Grade 6 Unit 4: Expressions
ANSWERSLESSON 8: GREATEST COMMON FACTOR
6.NS.4 11. Here is one example. I made a table of the factors for each number. Then I highlighted the common factors.
19 95
1 1
19 5
19
95
The greatest common factor is 19.
Challenge Problem
6.NS.4 12. Multiply the common factors together using the lowest exponent that appears with each factor. For example, the greatest common factor of 72 and 126 is 2 × 32 = 18. This method can be used to find the greatest common factor of any set of numbers.
One method for determining the greatest common factor of a set of numbers inovlves finding the prime factorization of those numbers. This is done by finding the list of prime numbers whose product gives the original integer. For example, consider the numbers 126 and 72.
The prime factorization of 126 is 2 × 3 × 3 × 7.
2
3
126
63
37
21
126 = 2 × 3 × 3 × 7
The prime factorization of 72 is 2 × 3 × 3 × 2 × 2.
2
3
72
36
3 12
224
72 = 2 × 3 × 3 × 2 × 2
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Grade 6 Unit 4: Expressions
ANSWERSLESSON 8: GREATEST COMMON FACTOR
Find the common factors and multiply all of the common factors together. For example, the common factors in the prime factorization of 72 and 126 are 2, 3, and 3.
126 = 2 × 3 × 3 × 772 = 2 × 3 × 3 × 2 × 2
Multiplying these together, the greatest common factor of 72 and 126 is 2 × 3 × 3 = 18. This method can be used to find the greatest common factor of any set of numbers.
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Grade 6 Unit 4: Expressions
ANSWERSLESSON 9: LEAST COMMON MULTIPLE
ANSWERS
6.NS.4 1. B 4
6.NS.4 2. C 10
6.NS.4 3. D 12
6.NS.4 4. D 21
6.NS.4 5. D 20
6.NS.4 6. Multiples of 4 Only
Multiples of 8 Only
Multiples of Both 4 and 8
Not Multiples of Either 4 or 8
4
12
20
8
16
24
6 10 14
18 22
6.NS.4 7. Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40 … Multiples of 7: 7, 14, 21, 28, 35, 42 … Least common multiple: 35
Multiples of 5 Only
Multiples of 7 Only
Multiples of Both 5 and 7
Not Multiples of Either 5 or 7
5 10 15
20 25
30 40
7 14 21
28 4235
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Grade 6 Unit 4: Expressions
ANSWERSLESSON 9: LEAST COMMON MULTIPLE
6.NS.4 8. Here is one example.
0
8
16
24
32
12 24 36
On graph paper, I made squares using 8-unit-by-12-unit rectangles. The smallest square I was able to make was 24 units by 24 units. So, the least common multiple is 24.
6.NS.4 9. Here is one example.
Multiples of 10:
10 × 1 = 1010 × 2 = 2010 × 3 = 3010 × 4 = 4010 × 5 = 5010 × 6 = 6010 × 7 = 7010 × 8 = 8010 × 9 = 90
10 × 10 = 10010 × 11 = 11010 × 12 = 120
Multiples of 11:
11 × 1 = 1111 × 2 = 2211 × 3 = 3311 × 4 = 4411 × 5 = 5511 × 6 = 6611 × 7 = 7711 × 8 = 8811 × 9 = 99
11 × 10 = 110
The least common multiple is 110.
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Grade 6 Unit 4: Expressions
ANSWERSLESSON 9: LEAST COMMON MULTIPLE
6.NS.4 10. Here is one example. I made a table and listed the first 10 multiples of each number.
Multiples of 6 6 12 18 24 30 36 42 48 54 60
Multiples of 10 10 20 30 40 50 60 70 80 90 100
I highlighted the multiples that the two numbers have in common and found the least one. So, the least common multiple is 30.
6.NS.4 11. Here is one example.
Multiples of 7 Multiples of 9
7 9
14 18
21 27
28 36
35 45
42 54
49 63
56
63
The least common multiple is 63.
Challenge Problem
6.NS.4 12. The least common multiple is the product of each prime number raised to the highest power that appears in each factorization.
Prime factorization of 12 and 21: 12: 22 × 3 21: 3 × 7
Prime numbers in the factorizations: 2, 3, 7 Highest power of 2: 22 Highest power of 3: 3 Highest power of 7: 7
The least common multiple of 12 and 21 is 22 × 3 × 7 = 84.
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Grade 6 Unit 4: Expressions
ANSWERSLESSON 10: USING EXPRESSIONS
ANSWERS
6.EE.1 6.EE.2.a
1.Number of Hours 0.5 1 1.5 2 t
Expression for Total Distance Jason Runs (5 miles per hour)
5 • 0.5 5 • 1 5 • 1.5 5 • 2 5t
6.NS.4 2. 8
6.EE.2.a 3. A 2 4n −
6.EE.2.a 4. The wheels on the bicycles is represented by 2n. The wheels on the tricycles is represented by 3m. The total number of wheels is represented by 2n + 3m.
6.EE.2.a 5. Martin walked 2w + 5.3 miles.
6.EE.3 6.EE.4
6. C x(3x + 2)
D 3x2 + 2x
F 2x(x + 1) + x2
6.EE.3 6.EE.4
7. A 2(3x + 2) + 2x
B 2(4x + 2)
C 6(x + 1) + 2(x – 1)
F 8x + 4
6.EE.3 6.EE.4
8. B 50 20x y+
D 5 10 4x y+( )
6.EE.2.c 9. 335 or 335.00
6.NS.4 10. 24
![Page 21: EPEI - RUSD Mathrusdmath.weebly.com/uploads/1/1/1/5/11156667/g6_u4...6.EE.2.c 13. f3 = 2 • 2 • 2 = 8 The volume of the cube is 8 cubic inches. Challenge Problem 6.EE.2.c 14. Expression](https://reader035.fdocuments.us/reader035/viewer/2022071401/60eb6011eecc3d3fa248219a/html5/thumbnails/21.jpg)
Copyright © 2015 Pearson Education, Inc. 67
Grade 6 Unit 4: Expressions
ANSWERSLESSON 10: USING EXPRESSIONS
Challenge Problem
6.EE.2 11. Here is one example for x – 3: Emma took out x books from the library. Mia took out 3 fewer books than Emma. How many books did Mia take out?
Here is one example for 5n + 10d: Denzel planted 5 rows of tomato plants and 10 rows of cucumbers. There are n tomato plants in each row and d cucumber plants in each row. Write an expression for the total number of plants (tomato and cucumber) Denzel planted.
![Page 22: EPEI - RUSD Mathrusdmath.weebly.com/uploads/1/1/1/5/11156667/g6_u4...6.EE.2.c 13. f3 = 2 • 2 • 2 = 8 The volume of the cube is 8 cubic inches. Challenge Problem 6.EE.2.c 14. Expression](https://reader035.fdocuments.us/reader035/viewer/2022071401/60eb6011eecc3d3fa248219a/html5/thumbnails/22.jpg)
Copyright © 2015 Pearson Education, Inc. 68
Grade 6 Unit 4: Expressions
ANSWERSLESSON 11: PUTTING IT TOGETHER
ANSWERS
6.EE.2.b 6.EE.4
3. Here are two examples.
Word or Phrase Definition Examples
expression consists of numbers, letters, or both connected by one or more of the arithmetic operations (addition, subtraction, multiplication, division, and raising to a power)
Algebraic expressions have one or more variables and numerical expressions have no variables.
Algebraic Expressions
x2
3(y + 2)
16
4 4 53m n g− + ( ) +
Numerical Expressions
5 – 2
412
10 ÷ •
2 4 6
3
2 −( )
Expression Vocabulary
1
64 4 3 5m n g− + ( ) +
coefficientvariable
exponent
term
constant
} } }Not Expressions (equations):
5 – 2 = 3 3(y + 2) = 24
numerical expression
consists of numbers linked by operation signs—addition, subtraction, multiplication, division, and raising to a power
has no variables
5 – 2
412
10 ÷ •
2 4 6
3
2 −( )
Not examples (these are numerical equations):
5 – 2 = 3
412
10 9 ÷ • =
![Page 23: EPEI - RUSD Mathrusdmath.weebly.com/uploads/1/1/1/5/11156667/g6_u4...6.EE.2.c 13. f3 = 2 • 2 • 2 = 8 The volume of the cube is 8 cubic inches. Challenge Problem 6.EE.2.c 14. Expression](https://reader035.fdocuments.us/reader035/viewer/2022071401/60eb6011eecc3d3fa248219a/html5/thumbnails/23.jpg)
Copyright © 2015 Pearson Education, Inc. 69
Grade 6 Unit 4: Expressions
ANSWERSLESSON 11: PUTTING IT TOGETHER
ANSWERS
algebraic expression
consists of letters (or letters and numbers) connected by the operation signs—addition, subtraction, multiplication, division, and raising to a power
has variables
x2
3(y + 2)
16
4 4 53m n g− + ( ) +
Not examples (these are algebraic equations):
x2 3=
3(y + 2) = 24coefficient a number that appears
before a variable and multiplies it
1
64 4 3 5m n g− + ( ) +
constant terms in an algebraic expression that contain only numbers
1
64 4 3 5m n g− + ( ) +
exponent the number of copies of a number or variable that are multiplied together
1
64 4 3 5m n g− + ( ) +
x2
term number, variable, or combination of numbers and variables separated by operation signs in an expression
x2 There is one term: x2
3(y + 2) There is one term: 3(y + 2)
16
4 4 53m n g− + ( ) +
There are four terms: 16
m and 4n and 4 3g( ) and 5
variable a symbol (usually a letter) that stands for an unknown value in an expression or equation
( )1
64 4 3 5m n g− + +
x2
3 2y +( )