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Math Fundamentals for
Statistics I (Math 52)
Homework Unit 5:
Division
By Scott Fallstrom and Brent Pickett
“The ‘How’ and ‘Whys’ Guys”
This work is licensed under a Creative Commons AttributionNonCommercial-ShareAlike 4.0 International License
3rd Edition (Summer 2016)
Math 52 – Homework Unit 5 – Page 1
Table of Contents
This will show you where the homework problems for a particular section start.
5.1: Division .................................................................................................................................................2
5.2: Division as Subtraction (Algorithms) .................................................................................................3
5.3: Division of Integers ..............................................................................................................................5
5.4: Division of Fractions ............................................................................................................................7
5.5: Division of Decimals.............................................................................................................................9
5.6: Division with Exponents and Scientific Notation............................................................................11
5.7: Number Sense and Division ..............................................................................................................14
5.8: Division with Number Lines .............................................................................................................16
5.9: Mixed Numbers and Improper Fractions.........................................................................................17
5.10: Roots and Radicals ...........................................................................................................................20
5.11: 2-Dimension Geometry and Division/Roots...................................................................................21
5.12: 3-Dimension Geometry and Division/Roots...................................................................................24
5.13: Applications of Geometry, Division and Roots ..............................................................................26
5.14: Order of Operations .........................................................................................................................28
5.15: Division Summary.............................................................................................................................30
5.16: Division Wrap-Up (Practice) ...........................................................................................................30
5.1: Division
Vocabulary and symbols – write out what the following mean:




Dividend
Divisor
Quotient
Remainder



Exist
Unique
÷
Concept questions:
1. What does it mean for an answer to exist?
2. What does it mean for an answer to exist, but not be unique?
3. Give an example of a situation where an answer doesn’t exist.
4. Give an example of a situation where an answer exists, but is not unique.
5. Is 40 ÷ 10 defined? Explain why.
6. Is 38 ÷ 0 defined? Explain why.
7. Is 0 ÷ 32 defined? Explain why.
Math 52 – Homework Unit 5 – Page 2
8. Is 0 ÷ 0 defined? Explain why.
9. If a number is divided by 43, could we ever have a remainder of 50? Why or why not?
Exercises:
10. Try a few of these problems. Find the quotient and remainder, and explain why the value you find
meets the definition of division.
a. 63 ÷ 9
g. 53 ÷ 10
b. 45 ÷ 5
h. 93 ÷ 20
c. 72 ÷ 8
i. 100 ÷ 100
d. 35 ÷ 7
j. 1,000 ÷ 100
e. 38 ÷ 7
k. 100,000 ÷ 1,000
f. 41 ÷ 7
l. 50,000,000 ÷ 500
Wrap-up and look back:
11. If 57 = 2(25) + 7, can we say that 57 ÷ 2 = 25 R 7? Why or why not?
12. If 57 = 2(25) + 7, can we say that 57 ÷ 25 = 2 R 7? Why or why not?
13. Why does 38 ÷ 0 fail the definition of division?
14. Why does 0 ÷ 0 fail the definition of division?
15. Can we have a remainder that is negative? 16 = 5 × 3 + 1, but 16 = 6 × 3 – 2 also. Which is correct?
16. Did you have any questions remaining that weren’t covered in class? Write them out and bring them
back to class.
5.2: Division as Subtraction (Algorithms)
Vocabulary and symbols – write out what the following mean:

Scaffolding

Short Division
Concept questions:
1. What is the purpose of algorithms?
2. Why is short division not recommended for a divisor of 37, but is recommended for a divisor of 7?
Exercises:
3. For the following, determine the quotient and remainder if possible. If not possible, explain why it is
not possible.
a. 15 ÷ 0
c. 0 ÷ 52
b. 75 ÷ 30
d. 0 ÷ 0
Math 52 – Homework Unit 5 – Page 3
4. Use repeated subtraction to find:
a. 15 ÷ 6
b. 23 ÷ 7
c. 50 ÷ 20
e. 15 ÷ 0
d. 103 ÷ 25
f. 0 ÷ 0
5. Find the quotient and remainder. Write the result as a mixed number fraction, then check your work
using multiplication.
a. 28 ÷ 5
c. 59 ÷ 10
e. 912 ÷ 100
b. 14 ÷ 9
d. 243 ÷ 50
f. 50 ÷ 7
a. 315 ÷ 3
d. 757 ÷ 3
g. 42,683 ÷ 5
b. 912 ÷ 3
e. 8,637 ÷ 6
h. 593,284 ÷ 2
c. 912 ÷ 9
f. 8,637 ÷ 4
i. 40,392 ÷ 3
6. Use the Short Division algorithm.
7. Use the Traditional or Scaffolding (Partial Quotients).
8.
a. 5,943  11
c. 63,843  21
b. 4,023  13
d. 49,637  12
Here are some to try with the calculator; find the quotient and remainder. Use ÷R if possible.
a. 2,304 ÷ 9
c. 693,240 ÷ 5
e. 304,208 ÷ 15
b. 15,304 ÷ 7
d. 2,304,159 ÷ 11
f. 667,668 ÷ 37
9. Noel is checking out the problem 23 ÷ 5, and she sees that 5 × 4 = 20, so 23 ÷ 5 has quotient of 4 and
remainder of 3. Is Noel correct? Explain why or why not.
10. Carl is checking out the problem 23 ÷ 5, and he sees that 5 × 2 = 10, so 23 ÷ 5 has quotient of 2 and
remainder of 13. Is Carl correct? Explain why or why not.
11. Is division commutative? Explain why or give an example to show why not.
12. Is division associative? Explain why or give an example to show why not.
Wrap-up and look back:
13. [Intentionally left blank]
14. [Intentionally left blank]
15. How many times can you subtract 0 from 12 to get to 0? What does this indicate about 12 ÷ 0?
16. How many times can you subtract 2 from 12 in order to get 0? What does this indicate about 12 ÷ 2?
17. How many times can you subtract 0 from 0 in order to get 0? Is there more than one answer? What
does this indicate about 0 ÷ 0?
Math 52 – Homework Unit 5 – Page 4
18. Did you have any questions remaining that weren’t covered in class? Write them out and bring them
back to class.
5.3: Division of Integers
Vocabulary and symbols – write out what the following mean:

None
Concept questions:
1. If A is to the left of 0 and B is to the right of 0 on the number line, where will A ÷ B be located? Why?
2. If A is to the left of 0 and B is to the left of 0 on the number line, where will A ÷ B be located? Why?
3. If A ÷ B is positive and B is positive, what do we know about A? Why?
4. If A ÷ B is positive and B is negative, what do we know about A? Why?
5. If A ÷ B is negative and B is positive, what do we know about A? Why?
6. If A ÷ B is negative, what do we know about A and B? Why?
7. If A ÷ B is positive, what do we know about A and B? Why?
8. If A and B are negative, what is the sign of A ÷ B – A? Why?
9. If A is negative and B is positive, what is the sign of A ÷ B + A? Why?
10. If A and B are negative, what is the sign of A ÷ B + A? Why?
Exercises:
11. Take a moment and fill out the appropriate sign:
a. neg ÷ neg = _____
c. neg ÷ pos = _____
b. pos ÷ neg = _____
d. pos ÷ pos = _____
Math 52 – Homework Unit 5 – Page 5
12. Concept Questions
When dealing with integers, the sign of the result depends on the sign of the starting numbers. The result
could be always positive (P), always negative (N), or sometimes positive and sometimes negative (S).
Label each of the following expressions as P, S, or N. If the answer is P or N, explain why. But if the
answer is S, give one example that shows the product could be positive and one example that shows the
product could be negative. Note: some questions review previous topics as well as division.
Expression
Sign (circle one)
a.
neg × neg
P
S
N
b.
pos ÷ neg
P
S
N
c.
pos – pos
P
S
N
d.
pos × pos
P
S
N
e.
pos + neg
P
S
N
f.
neg + pos
P
S
N
g.
neg × pos
P
S
N
h.
neg ÷ neg
P
S
N
i.
neg – pos
P
S
N
j.
pos × neg
P
S
N
k.
neg – neg
P
S
N
l.
pos ÷ pos
P
S
N
Examples or Explanation
13. Now try a few division problems with integers:
a. – 35 ÷ 7
b.  63   9
c. 81   9
d.
 55  5
e. 56 ÷ 7
f.  42   6
g.
h. 440   4
i.
j.
k.
l.
 2,627   37
 3,723  73
23,997   421
 1,625,576   317
 70  5
Math 52 – Homework Unit 5 – Page 6
Wrap-up and look back:
14. Are the division rules about signs the same as multiplication? Why?
15. Is the concept table above easier to determine with subtraction or division? Why?
16. Did you have any questions remaining that weren’t covered in class? Write them out and bring them
back to class.
5.4: Division of Fractions
Vocabulary and symbols – write out what the following mean:


Divide Straight Across
Multiplicative Inverse Property


Reciprocal
Multiplicative Inverses
Concept questions:
1. Albert says
2. Are
36 3
 because he “canceled” off the 6’s. Is this correct?
65 5
3
3
and  multiplicative inverses? Is this correct? If so, explain why. If not, explain why not.
5
5
3. Is the reciprocal of 0 still 0? Explain why or why not.
4. Rewrite using addition: A – B.
5. Rewrite using multiplication: A ÷ B.
6. Rewrite using subtraction: A + B.
7. Rewrite using division: A × B.
Exercises:
8. Divide straight across.
 36   12 
a.     
 25   5 
 36   9 
b.     
 55   5 
 35   5 
c.     
 49   7 
 28   7 
d. 


 189   189 
 36   2 
e.     
 21   7 
f.
 291   291 



 72   8 
Math 52 – Homework Unit 5 – Page 7
a c a
9. From the rule determined in class, we know         . Use this to determine the result of the
b b  c 
following division problems:
 36   12 
a.     
 25   25 
 36   7 
b.     
 55   55 
 35   63 
c.     
 49   49 
 28   53 
d. 


 189   189 
 36   51 
e.     
 21   21 
f.
 291   317 



 72   72 
10. Find a common denominator, then divide straight across. Be sure to check your results with
multiplication and simplify the quotients when possible.
3  1 
a.     
8  2
 11   5 
b.     
 3  9
2 3
c.     
5 4
11. Find the multiplicative inverse of the following numbers (if possible):
a.
8
17
b. 51
c. 
8
5
d.
1
858
g. 1
h. 
e. – 384
f. – 1
i.
242
938
0
12. Practice dividing fractions using one of the techniques shown here: (1) dividing straight across or (2)
rewriting as multiplying by the reciprocal. Be sure to simplify the quotient, if possible.
 5  5
a.     
 11   8 
 7  5
e.       
 3  9
 13   1 
b.     
 28   7 
f.
7  9
  
22  2 
g.
 876   33 

   
 91   70 
h.
 112   7 

   
 135   45 
c.
 3  3 
d.      
 8   10 
 19   10 
    
 22   33 
Math 52 – Homework Unit 5 – Page 8
13. Use the idea of FLOF to rewrite 6,000 ÷ 840 in at least 3 different ways that would produce the same
quotient.
14. Rewrite the division problem to make it easier. Show at least two different divisions for each.
a.
 200  60
e. 77,000 ÷ 3,300
b. 750 ÷ 150
f. 12,000  5,000
c. 15,000 ÷ 500
g. 3,200  20
d. 7,000 ÷ 40
h.
 7,200   60
Wrap-up and look back:
15. Arthur says he is thinking of a number that has no multiplicative inverse. What is this number and
why? Is there more than one possibility?
16. Sean rewrites 400 ÷ 20 to get 200 ÷ 40. He said he cut one in half and doubled the other. Does this
work?
17. Marti rewrites 6,000 ÷ 120 as 600 ÷ 12. She said that she divided out a 10 on each. Does this work?
18. If rewriting A ÷ B by cutting B in half, would you double A or take half of A? Why?
19. Did you have any questions remaining that weren’t covered in class? Write them out and bring them
back to class.
5.5: Division of Decimals
Vocabulary and symbols – write out what the following mean:

Repetend
Concept questions:
1. Can we rewrite 12.55 ÷ 2.3 as 12.55 ÷ 23? Why or why not?
2. Can we rewrite 0.055 ÷ 0.2 as 55 ÷ 2? Why or why not?
3. Would you recommend rewriting 0.005 ÷ 0.3 as 5 ÷ 300 or as 0.05 ÷ 3? Explain why?
4. Marcy says that
7
can’t be written as a decimal because it’s a fraction. How do you respond to help
8
her understand.
Math 52 – Homework Unit 5 – Page 9
Exercises:
5. Practice rewriting the following divisions with nicer (whole number) divisors. At this point, we won’t
perform the actual calculation.
a. 7.4 ÷ 0.004
f. 9 ÷ 0.003
b. 8.1 ÷ 0.02
g. 40 ÷ 0.03
c. 9.35 ÷ 6.1
h. 0.00041 ÷ 0.000618
d. 0.005 ÷ 0.03
i. 0.008 ÷ 0.0000044
e. 0.573 ÷ 0.2
j. 0.000007 ÷ 0.003
6. Perform the division by hand unless indicated with the calculator symbol. If the value is a fraction,
convert the fraction to division and then write as a decimal.
a. 17  0.004
b.
7
8
c. 0.08  0.9
d.
13
40
e. 0.0075 ÷ 0.00055
f. 18.4  0.022
g.
0.011  0.37
h.
16
18
i.
11
13
j.
19
27
7. Perform the division by hand; write as a decimal.
a.
5
9
b.
2
9
c.
7
9
8. Perform the division; write as a decimal.
a.
23
99
b.
17
99
c.
4
99
179
999
c.
74
999
9. Perform the division; write as a decimal.
a.
123
999
b.
Math 52 – Homework Unit 5 – Page 10
10. Based on the previous problems, you should see a nice pattern. Explain the pattern in words.
11. Use your pattern to find the value of the following (without a calculator). Then check your result using
the calculator.
76
99
1
b.
99
857
999
13
d.
999
a.
c.
12. Use your pattern to write as a decimal:
e.
f.
7,163
9,999
8,163
99,999
9
. Does the decimal match what you think?
9
Wrap-up and look back:
13. When rewriting a division problem A ÷ B, is it better to make A into a whole number or B? Explain.
14. Is 0.99999 the same as 1? Why or why not?
15. Is 0.9 the same as 1? If they are not the same, find a number that is between the two.
16. Did you have any questions remaining that weren’t covered in class? Write them out and bring them
back to class.
5.6: Division with Exponents and Scientific Notation
Vocabulary and symbols – write out what the following mean:

Scientific Notation
Concept questions:
xA
1. How can we rewrite B ?
x
2. What is the base of  11 ? What is the value of  11 ?
0
0
3. What is the base of  220 ? What is the value of  220 ?
4. Peter said  220 = 1 because anything to the 0 power is 1. Is Peter correct? Explain.
5. Polly said that 00 = 1 because anything to the 0 power is 1. Is Polly correct? Explain.
6. x 0  1 is true Always (A), Sometimes (S), or Never (N)? Explain your result.
7. Would you say that x 2  2 x ? Explain.
8. Would you say that x 2   x 2 ? Explain.
Math 52 – Homework Unit 5 – Page 11
Exercises:
9. Write the result using a single exponent.
a.
415
47
d.
427 79
42715
g.
4413
44 20
b.
7 56
7 23
e.
2385
2385
h.
32 21
3232
c.
 1360
 1320
f.
 1553
 1554
i.
 1161
 1132
10. Find the value of the following expressions.
 350
a. 130
c.
b. 840
d. 6,7950
e.  710
f.
 230
11. Simplify and rewrite the following with positive exponents.
a.
x 23
e.  32
b. x  x 21
 3
c.   
 5
 2x 
d.  2 
y 
f.
2
 32
g. 10 3
h. 02
3
i.
x  x 
15 2
10
12. Are these big size numbers or small size numbers?
a. 5.7  1044
c. 8.5  1064
b.  3.8  1089
d.  4.55  10304
13. Where are these numbers on a number line?
Number
Sign
Size
a.
5.7  1044
Left of zero
Right of zero
Far from zero
Close to zero
b.
 3.8  1089
Left of zero
Right of zero
Far from zero
Close to zero
c.
8.5  1064
Left of zero
Right of zero
Far from zero
Close to zero
d.
 4.55  10304
Left of zero
Right of zero
Far from zero
Close to zero
Math 52 – Homework Unit 5 – Page 12
14. Fill in the table by writing the value in either standard notation or scientific notation.
Standard Notation
a.
– 48,000
b.
0.0000791
c.
68,100,000,000,000
Scientific Notation
d.
7.8  109
e.
8.2  1015
f.
 7.01  1011
15. Determine which numbers are written in scientific notation. If it is not written in scientific notation
properly, rewrite the number properly – but keep the same value!
a. 17  1043
e. 1.700  1020
b. 0.0427  1011
f.
c. 1.4  1045
g. 3.5  10773
d.  10.8  1084
h. 0.0007025  1069
0.000035  1051
16. Multiply or divide these numbers in scientific notation. Use your calculator to check your work.


3  10 2  10 
5  10 6  10 
7.2  10 2  10 
310 8 10 

Examples: 1.4  1045 2  1031  2.8  1076
a.
b.
c.
d.
9
31
19
31
11
9
40
31



and 6  10 4 2  10 3  12  10 7  1.2  108
e.
f.
g.
h.
6 10 9 10 
8  10   2  10 
4  10   8  10 
3  10   4  10 
9
9
60
4
31
31
31
31
Wrap-up and look back:
17. Does writing a number in scientific notation change the value of the number? Why or why not?
18. Did you have any questions remaining that weren’t covered in class? Write them out and bring them
back to class.
Math 52 – Homework Unit 5 – Page 13
5.7: Number Sense and Division
Vocabulary and symbols – write out what the following mean:

Scientific Notation
Concept questions:
1. Hector said that division makes things smaller. Is this true? Help clarify his statement so that it is
correct.
2. Helen said that multiplication makes things bigger. Is this true? Help clarify her statement so that it is
correct.
3. For positive numbers A and B. If A ÷ B is bigger than A, what do we know about B?
4. For positive numbers A and B. If A ÷ B is smaller than A, what do we know about B?
5. For positive numbers A and B. If A ÷ B is the same as A, what do we know about B?
Exercises:
6. Circle whether the quotient is greater or less than the values listed.
If the problem is…
Then the quotient will be …
a.
15  0.23
Greater
Less
than 1
Greater
Less
than 15
b.
15  7
Greater
Less
than 1
Greater
Less
than 15
c.
14  35
Greater
Less
than 1
Greater
Less
than 14
d.
14  0.1925
Greater
Less
than 1
Greater
Less
than 14
7. Circle the appropriate word that will complete the sentence involving A ÷ B:
If the size of B is…
a.
0
b.
Then the size of A ÷ B will be…
Larger than A Smaller than A
Equal to A
Undefined
Between 0 and 1
Larger than A
Smaller than A
Equal to A
Undefined
c.
1
Larger than A
Smaller than A
Equal to A
Undefined
d.
Greater than 1
Larger than A
Smaller than A
Equal to A
Undefined
8. Circle the appropriate word that will complete the sentence involving A ÷ B:
If the size of B is…
Then the size of A ÷ B will be…
a.
Greater than A
Larger than 1
Smaller than 1
Equal to 1
Undefined
b.
Less than A
Larger than 1
Smaller than 1
Equal to 1
Undefined
c.
Equal to A
Larger than 1
Smaller than 1
Equal to 1
Undefined
d.
0
Larger than 1
Smaller than 1
Equal to 1
Undefined
Math 52 – Homework Unit 5 – Page 14
9. Determine the size of the quotient without performing actual division.
Size is…
Quotient
a.
11  Number between 0 and 1
Greater than 11
Less than 11
b.
11  Number greater than 1
Greater than 11
Less than 11
c.
11  Number less than  1
Greater than 11
Less than 11
d.
11  Number between  1 and 0
Greater than 11
Less than 11
10. Determine the size and sign of the quotient without performing actual division.
Quotient
Size is…
Sign
a.
2
  0.5
3
Pos
Neg Zero
Greater than
2
3
Less than
2
3
b.
 2   17 
    
 3  9 
Pos
Neg Zero
Greater than
2
3
Less than
2
3
c.
 11  7
Pos
Neg Zero
Greater than 11
Less than 11
d.
 11   1.05
Pos
Neg Zero
Greater than 11
Less than 11
e.
6.92   0.32
Pos
Neg Zero
Greater than 6.92
Less than 6.92
f.
 0.2   5 
Pos
Neg Zero
Greater than 0.2
Less than 0.2
g.
 12.032  12.5
Pos
Neg Zero
Greater than 1
Less than 1
Equal to 1 Undefined
h.
 7.1  7.100
Pos
Neg Zero
Greater than 1
Less than 1
Equal to 1 Undefined
i.
25  0
Pos
Neg Zero
Greater than 1
Less than 1
Equal to 1 Undefined
j.
0   31,400
Pos
Neg Zero
Greater than 1
Less than 1
Equal to 1 Undefined
6
7
Math 52 – Homework Unit 5 – Page 15
Wrap-up and look back:
11. What do we divide by to make something bigger in size?
12. What do we divide by to make something smaller in size?
13. Did you have any questions remaining that weren’t covered in class? Write them out and bring them
back to class.
5.8: Division with Number Lines
Vocabulary and symbols – write out what the following mean:

None
Concept questions:
1. If C is positive, then where is 2 ÷ C located? Is it bigger than 0 or less than 0?
2. If C is positive, then where is 2 ÷ C located? Is it bigger than 1 or less than 1? Do we have enough
information to decide? Explain your answer.
Exercises:
3. Practice with the number lines and be cautious with the sign!
a. Find the region that best approximates B ÷ B.
b. Find the region that best approximates A ÷ C.
c. Find the region that best approximates D ÷ C.
d. Find the region that best approximates D ÷ B.
e. Find the region that best approximates B ÷ A.
f. Find the region that best approximates 1 ÷ A.
g. Find the region that best approximates (– 1) ÷ A.
A
R1
B
R2
R3
0
C
R4
1
D
R5
R6
Math 52 – Homework Unit 5 – Page 16
Wrap-up and look back:
4. If A is between – 1 and 0, will the reciprocal of A be to the left of 0 or right of 0?
5. If A is between – 1 and 0, will the reciprocal of A be to the left of A or right of A?
6. If A is between 0 and 1, will the reciprocal of A be to the left of A or right of A?
7. Did you have any questions remaining that weren’t covered in class? Write them out and bring them
back to class.
5.9: Mixed Numbers and Improper Fractions
Vocabulary and symbols – write out what the following mean:


Proper Fraction
Improper Fraction

Mixed Number
Concept questions:
a
b
1. If is a proper fraction, will
be proper or improper? Explain.
a
b
2. If
a
b
is an improper fraction, will
be proper or improper? Explain.
a
b
3. Manny said that an improper fraction is where the numerator is bigger than the denominator. That’s
why he says that
7
7
is improper… because 7  4 . Is
improper or proper? Explain your
4
4
answer.
4. Is 4
2
the correct notation for a mixed number? Explain. If not correct notation, correct the expression
3
but keep the same value.
5. Is 4
7
the correct notation for a mixed number? Explain. If not correct notation, correct the expression
3
but keep the same value.
6. Is 4
2
the correct notation for a mixed number? Explain. If not correct notation, correct the
3
expression but keep the same value.
7. Rewrite 4
2
using addition; be sure to keep the same value.
3
Math 52 – Homework Unit 5 – Page 17
8. Olivia converted  5
2
to an improper fraction. She multiplied  5  7  35 , then added 2 to this
7
result and got – 33. Her final response is  5
2
33
  . Is she correct? If not, help her understand the
7
7
error that she made.
9. Rewrite  5
2
using addition; be sure to keep the same value.
7
4
to an improper fraction. He multiplied 5  3  15 , then added 4 to this result and
5
4 19
got 19. His final response is 3  . Is he correct? If not, help him understand the error he made.
5 5
Exercises:
10. Patton converted 3
11. Identify the following fractions as proper or improper.
a.
Fraction
2
5
Label
Proper Improper
b.
Fraction
1
5
Label
Proper Improper
c.
41
3
Proper Improper
d.
 11
 33
Proper Improper
e.
52
 52
Proper Improper
f.
50
23
Proper Improper
12. Convert the fractions from improper to mixed number, or vice-versa.
a.
c.
Fraction
Notation
11
5

Mixed Number
Notation
b.
77
9
d.
Fraction
Notation
4
5
Mixed Number
Notation
41
37
3
11
f.
 11
1
3
h.
 333
e.
40
g.
81
2
7
1
3
Math 52 – Homework Unit 5 – Page 18
13. Convert the fractions from improper to mixed number, or vice-versa using your calculator .
Fraction
Notation
Mixed Number
Notation
11
5
a.

c.
77
9
Fraction
Notation
b.
4
5
d.
41
37
Mixed Number
Notation
3
11
f.
 11
1
3
h.
 333
e.
40
g.
81
2
7
1
3
14. Do we need common denominators when multiplying fractions? Why or why not?
15. Do we need common denominators when adding fractions? Why or why not?
16. Do we need common denominators when subtracting fractions? Why or why not?
17. Do we need common denominators when dividing fractions? Why or why not?
18. Perform the operations shown below (it is your choice as to the method you use, but be efficient).
a. 7
3
3
 19
4
4
d.  47
1
3
 29
6
4
5 
1

b.   14     27 
6 
2

e. 52
3
5
 19
4
6
5
1
c. 32  19
6
2
f. 14
4
1
 31
5
3
19. Perform the operations shown below using either method.
3
 3 
a.   1     3 
7
 4 
b. 2
2
1
 15
23
3
c. 6
4 
3
  5 
5 
5
1
2
d. 10  5
8
5
Math 52 – Homework Unit 5 – Page 19
20. Estimating is a way to get close to the answer quickly. For estimating, we don’t care about what the
actual value is, but just about getting close to it. No calculator!
Estimating the value of …
Is closest to…
25,983  219
a.
1.3
13
2
20
130
1,300
c.
5
1
3 5
9
4
4,900  1.1
0.49
4.9
49
490
4,900
d.
49  0.12
0.49
4.9
49
490
4,900
e.
38
b.
45
f.
5
1
2
 14
2
3
g.
497,855  248
h.
239
30
20
1
2
 165
2
3
6
3
200
4
200
7
8
0.3
2,000
40
2,000
9
0.03
0.003
20,000
400
200,000
4,000
Wrap-up and look back:
21. When adding or subtracting mixed numbers, do you prefer to keep the numbers in mixed number form
or convert them to improper numbers? Explain your reasoning.
22. When multiplying or dividing mixed numbers, do you prefer to keep the numbers in mixed number
form or convert them to improper numbers? Explain your reasoning.
23. Could you find the sum of 9,145
8
15
by converting to improper fractions? What would you
 41,035
19
19
tell someone who was going to find the sum of 9,145
8
15
 41,035
by converting to improper
19
19
fractions? Explain.
24. Did you have any questions remaining that weren’t covered in class? Write them out and bring them
back to class.
5.10: Roots and Radicals
Vocabulary and symbols – write out what the following mean:



Square Root
Principal Square Root
Index



Radicand
Radical
Root
Math 52 – Homework Unit 5 – Page 20
Concept questions:
1. Karl claims that
36  6 because  6 6  36 . What parts is Karl correct about? Is he incorrect
about anything?
2. Since both  5 5  25 and 55  25 , what notation do we use to represent the negative square
root of 25?
3. What if we tried to find
 49 ; what real number squared would be – 49? Explain your answer.
Exercises:
4. Find the values given, if possible.
a. 
 16 
d.  144
b.
400
e.
c.
81
f.
g.  64
 25

  400

h.
 36
i.
 225
5. Use the table in the workbook to find the values, if possible.
Root
Form
Value
Root
Form
Value
Root
Form
a.
169
b.
 121
c.
225
d.
 324
e.
289
f.
900
Value
Wrap-up and look back:
6. Why is
 121 not a real number?
7. Did you have any questions remaining that weren’t covered in class? Write them out and bring them
back to class.
5.11: 2-Dimension Geometry and Division/Roots
Vocabulary and symbols – write out what the following mean:


Right Triangles
Legs


Hypotenuse
Pythagorean Theorem
Math 52 – Homework Unit 5 – Page 21
Concept questions:
1. When solving a problem with the Pythagorean Theorem, Thomas saw that c 2  100 and knows there
are two numbers that work: 10 and – 10. He asks why we don’t pick “– 10” as a possible solution?
How can you help him understand the reasoning?
2. Jamal saw that the Pythagorean Theorem confirmed that a triangle with sides of 3, 4, and 5 was a right
triangle. He wondered if he multiplied all the sides by 5 if the new triangle would still be a right
triangle?
3. We know that a 3-4-5 triangle is a right triangle, what about if we multiplied all the sides by 9. Would
the new triangle be a right triangle? Explain.
4. How do you know just by looking at a right triangle which side is the hypotenuse?
5. Is it possible to have a right triangle with legs of 12 and 15 that also has a hypotenuse of 10? Explain.
6. In a problem, Jessenia was trying to find the missing piece of a right triangle. The clues said that the
hypotenuse was 8 and one leg was 5. She plugged them into the formula to get: 52  82  x 2 . Has
she set up the equation correctly? Explain.
Exercises:
7. In the following right triangles, find the missing piece. If the result is not a whole number, then use
your calculator to find the length to 3 decimal places.
a.
c
7
c.
7
30
8
d.
b.
20
c
c
10
27
c
27
Math 52 – Homework Unit 5 – Page 22
8. In the following right triangles, find the missing piece. If the result is not a whole number, then use
your calculator to find the length to 3 decimal places. Remember, if a 2  b 2  c 2 , then a 2  c 2  b 2 .
a.
c.
9
7
15
m
11
x
b.
d.
31
x
50
30
n
21
9. Determine if the following sides will make a right triangle.
Side Lengths
Right Triangle?
Side Lengths
Right Triangle?
a.
12, 15, and 20
Yes
No
b.
13, 84, and 85
Yes
No
c.
11, 12, and 13
Yes
No
d.
14, 17, and 30
Yes
No
e.
15, 20, and 30
Yes
No
f.
140, 171, and 221
Yes
No
g.
39, 80, and 89
Yes
No
h.
11, 60, and 61
Yes
No
10. If a right triangle has legs of 6 and 8, what is the perimeter? Draw a triangle and show your work!
11. If a right triangle has legs of 10 and 21, what is the perimeter? Draw a triangle and show your work!
Wrap-up and look back:
12. Draw a right triangle with sides labeled so that k 2  w 2  p 2 .
13. Draw a right triangle with sides labeled so that b 2  c 2  a 2 .
14. Draw a right triangle with sides labeled so that a 2  c 2  b 2 .
15. Did you have any questions remaining that weren’t covered in class? Write them out and bring them
back to class.
Math 52 – Homework Unit 5 – Page 23
5.12: 3-Dimension Geometry and Division/Roots
Vocabulary and symbols – write out what the following mean:




Surface Area
Volume
Circular Cylinder
Circular Cone



Sphere
Pyramid
Rectangular Solid
Concept questions:
1. Rebekah is working on solving for the volume of a shape that looks like this. She used the formula
4
V    r 3 . Is this formula going to result in the right answer? Explain.
3
2. Sarah is working on solving for the surface area of the shape below. She used the formula
4
V    r 3 . Is this formula going to result in the right answer? Explain.
3
3. What’s the difference between surface area and volume?
4. If you were going to paint the outside of a box, would you want to calculate the volume or surface
area? Explain.
5. If you wanted to fill up a cone with water, would you be finding the volume or surface area? Explain.
6. Wendel has a rectangular box that is 5 inches by 7 inches by 10 inches. He measures the wrapping
paper and finds that he has 300 square inches. Will this be enough paper to cover the whole box… or
will Wendel have to use the Sunday comics again?
7. When calculating the surface area, would you be measuring square inches or cubic inches? Explain.
8. When calculating the volume, would you be measuring square inches or cubic inches? Explain.
Math 52 – Homework Unit 5 – Page 24
Exercises:
9. Use the formulas from the book to find the surface area or volume of the 3-dimensional shapes.
a. Find the surface area and volume of this shape.
10 in.
60 in.
b. Find the volume and surface area of this shape.
15 cm
8 cm
c. Find the surface area and volume of this shape.
9 yards
d. Find the surface area and volume of this shape.
4m
10 m
13 m
Wrap-up and look back:
10. Which formula looks the most confusing to use? Explain why.
11. Create a problem that forces you to use that formula – then swap with a classmate and solve their
problem.
12. Did you have any questions remaining that weren’t covered in class? Write them out and bring them
back to class.
Math 52 – Homework Unit 5 – Page 25
5.13: Applications of Geometry, Division and Roots
Vocabulary and symbols – write out what the following mean:

None
Concept questions:
1. Which type of application problem is your favorite to work with? Why?
2. Where in your life could you use some geometry?
Exercises:
Geometry Applications:
3. An enormous can of cheese dip called “Que Bueno” is sold at Costco. The can has a diameter of
6.1875 inches and a height of 7 inches.
a. What is the volume of cheese in the can in cubic inches?
b. I wanted to heat this up and grabbed a rectangular pan that was 9 inches by 13 inches. The pan is 2
inches high… will all the cheese sauce fit in the pan?
4. One cubic inch of gold weighs about 0.697 pounds. In a movie, a titanium briefcase was filled with
gold and the bad-guy ran away with it in one hand. The briefcase is 13 inches long by 18 inches wide,
and was 4.5 inches deep.
a. What is the volume of the briefcase (in cubic inches)?
b. If the briefcase was full of gold, how much would the gold weigh (in pounds)?
c. The briefcase itself weighs about 10 pounds. Use your analytical skills to determine if you think it
is possible for anyone to run around with this briefcase in one hand. Be sure to include the total
weight of everything.
5. A pile of dirt is to be moved from a quarry to a local home in order to create an enormous worm farm.
The pile of dirt is shaped like a circular cone with a height of 10 feet and a radius of 12 feet.
a. What is the volume of the pile of dirt (in cubic feet)?
b. If you use a standard pick-up truck, the bed of the truck is 6.5 feet by 4.5 feet, with a depth of 22
inches. Find the volume of the truck bed in cubic feet (make sure all dimensions are in feet!).
c. Using the previous parts, how many trips will you need to take in order to move the entire pile of
dirt?
Math 52 – Homework Unit 5 – Page 26
6. A friend has a fish tank that is a rectangular box shape, with dimensions of 13 in long by 48 in wide by
20 in high. She fills the tank so that the water level is 2 inches below the top. Then she puts in a sphere
for decoration, and the sphere has a radius of 8 inches.
a.
b.
c.
d.
What is the volume of the water that the tank will hold?
What is the volume of the water that she put in the tank?
What is the volume of the sphere?
Will the water overflow the tank?
Time/Date Applications:
7. It is currently Thursday. In 90 days, what day of the week will it be?
8. It is currently Monday. 40 days ago, what day of the week was it?
9. If a friend has a birthday on Wednesday, October 21, 2015, what day of the week will their birthday be
on in 2016?
10. For the same friend in the previous part, what day of the week was their birthday on in 2014?
11. If a friend has a birthday on Monday, July 27, 2013, what day of the week will their birthday be on in
2014?
Money Applications:
12. The President/Superintendent of MiraCosta College earns an annual salary of $250,000.
a. What is her monthly salary?
b. What is her daily pay?
c. MiraCosta uses an 8 hour day and 5 days per week as a typical work week, for 52 weeks in the
year. Determine how many hours the President works per year.
d. Using the previous part, determine the President’s hourly rate.
13. In a will, a parent leaves $813,219 for each of 3 kids to split. About how much would each child
expect to receive?
14. If Angelica makes $2,500 per month, about how much is she earning per week?
15. If Angelica makes $753 per week, and works 40 hours per week, how much is she earning per hour?
16. If you are earning $21 per hour and work 40 hours per week for 52 weeks per year,
a. what would your annual salary be?
b. how much would your annual salary change if you received a raise of 50 cents per hour?
Wrap-up and look back:
17. Did you have any questions remaining that weren’t covered in class? Write them out and bring them
back to class.
Math 52 – Homework Unit 5 – Page 27
5.14: Order of Operations
Vocabulary and symbols – write out what the following mean:



PEMA
PEMDAS
Order of Operations
Concept questions:
1. Why is PEMDAS confusing to memorize?
2. If you were memorizing PEMDAS, which operation would you do first: addition or subtraction?
3. Where does absolute value come into the PEMDAS?
4. What does PEMA stand for, and why does it not include division or subtraction?
5. If you remove parenthesis from a problem, will you change the value? Give examples to back up your
answer.
6. Which operation would you perform first in these expressions (just determine the first operation – you
don’t need to simplify further):
a. 15  5  6  3  1
d. 7   5  8
b. 28  19  5  3
e. 8  16  4  2
c. 28  19  5  3
f.
3  8 2  12  3  2
Exercises:
7. Does changing the parenthesis actually change the value of the expression?
First Expression
Second Expression
a.
(17 – 5) – 3
17 – (5 – 3)
Same
Different
b.
 x  5
 x5
Same
Different
c.
52
52
Same
Different
d.
 23
 23
Same
Different
e.
 52
 52
Same
Different
f.
 24
 24
Same
Different
g.
19 – 5
19 5
Same
Different
h.
1
14 
2
Same
Different
i.
4  6  2
Same
Different
14
1
2
4  6  2
Same or different (circle one)
Math 52 – Homework Unit 5 – Page 28
j.
4  6  2
46  2
Same
Different
k.
30  36  3  2
30  36  3  2
Same
Different
l.
0.35
0  35
Same
Different
m.
812  5
812  5
Same
Different
n.
812  5
812  5
Same
Different
8. Simplify the following expressions using the order of operations.
a. 15  5  6  3  1
b. 28  19  5  3
c. 28  19  5  3
d. 7   5  8
e. 8  16  4  2
f.
3  82  12  3  2
g.
3  82  12   3  2
h.
3  82  12   3  2
i.
 3  8  12   3  2
2
9. Fill in the missing boxes with the correct numbers.
×3
–4
a.
11
b.
–8
c.
d.
e.
f.
g.
–4
–9
×3
–5
+4
+3
÷ (–3)
÷5
×3
÷ 11
2.1
7
–5
×3
+ 29
+ 94
x2
(find 2 solutions for this one)
361
Math 52 – Homework Unit 5 – Page 29
5.15: Division Summary
When dealing with division, the key is to think of it in terms of multiplication. Here are some wrap up
questions to tie all the concepts together.
If you recall from multiplication, there were a number of properties that held true. But, do those same
properties work with division?
1. Is division commutative? Write a few equations that would be true if division was commutative,
then explain your conclusion.
2. Is division associative? Write a few equations that would be true if division was associative, then
explain your conclusion.
3. Is there a distributive property with division? Is either of these true:
o a  b  c  a  c  b  c
o c  a  b  c  a  c  b
o Explain your conclusion.
4. What is the Multiplicative Inverse Property?
5.16: Division Wrap-Up (Practice)
1. For the following, determine the quotient and remainder if possible. If not possible, explain why it is
not possible. Try to do some of these in your head, and use the algorithms when that is not possible.
a. 728  7
f. 10,000 ÷ 100
b. 61  0
g. 789  7
c. 0  0
h. 7,843  11
d. 75  11
i.
98,431  15
e. 8,200 ÷ 200
j.
529,157  5,341
2. Now try a few division problems with integers:
a.
 42   7
b. 32   8
c.
d.
 48  8
 3,591   63
Math 52 – Homework Unit 5 – Page 30
3. Practice dividing fractions using one of the techniques: (1) dividing straight across or (2) rewriting as
multiplying by the reciprocal. Be sure to simplify the quotient, if possible.
 6   3
a.      
 11   8 
 18   9 
c.       
 24   7 
 88   11 
b.     
 21   7 
d.
 176   88 

   
 95   45 
4. Perform the division by hand unless indicated with the calculator symbol. If the value is a fraction,
convert the fraction to division and then write as a decimal. Rewrite the division if necessary.
a. 0.46  0.0005
b.
3
8
c.
7
37
5. Determine which numbers are written in scientific notation. If it is not written in scientific notation
properly, rewrite the number properly – but keep the same value!
a. 1,700  10 50
e. 15,700  10 50
c. 1.57  10 254
d. 0.0035  10 61
b.  0.00083  10 61
6. Simplify and rewrite the following with positive exponents.
a.
x 41
 x7
e.   2
 3y
b. x 55  x 63
c.
d.
x 
x 
5 2
5 2
 x 35
f.
x
2




2x 7
 8 x 25
7. Determine the size of the quotient without performing actual division.
Quotient
Size is…
a.
23  Number between 0 and 1
Greater than 23
Less than 23
b.
 73  Number less than  1
Greater than 73
Less than 73
c.
 73  Number between  1 and 0
Greater than 73
Less than 73
d.
23  Number greater than 1
Greater than 23
Less than 23
Math 52 – Homework Unit 5 – Page 31
8. Perform the operations shown below with mixed numbers.
3
7
a. 50  26
5
10
b.  324
d. 10
3
2
 19
4
3
1
7
4
2
8
e.  324
1
2
c. 3  10
9
7
f.
1
2
 19
4
3
3
7
72  19
5
10
9. Find the values given, if possible.
c.  1
a.  64
b.
d.   81
1,600
10. In the following right triangles, find the missing piece. If the result is not a whole number, then use
your calculator to find the length to 3 decimal places.
a.
c
5
8
b.
25
15
x
Math 52 – Homework Unit 5 – Page 32