ACCELERATED MATHEMATICS
CHAPTER 3
FRACTIONS
TOPICS COVERED:
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•
•
•
•
•
•
•
•
•
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Divisibility Rules
Primes & Prime Factorization
Greatest Common Factor
Least Common Multiple
Fraction sense
Adding and subtracting fractions and mixed numbers
Equations with adding and subtracting fractions
Hands-on multiplying fractions
Multiplying fractions and mixed numbers
Applications of multiplying fractions
Hands-on dividing fractions
Dividing fractions and mixed numbers
Applications of dividing fractions
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 3-1: Divisibility Rules
Name:
Divisibility Rules
2
3
A number is divisible by 2 if the ones digit is
A number is divisible by 3 if the sum of its digits
even.
is divisible by 3.
4
5
A number is divisible by 4 if its last two digits are
A number is divisible by 5 if it ends in 0 or 5.
divisible by 4.
6
9
A number is divisible by 6 if it is divisible by
A number is divisible by 9 if the sum of its digits
both 2 and 3.
is divisible by 9.
10
A number is divisible by 10 if its last digit is 0.
Complete the following in 4 minutes or less.
Circle the numbers that are divisible by 2.
34
58
324
243
196
825
67
432
4374
90
423
9701
241
234
65250
Circle the numbers that are divisible by 3.
48
75
761
762
46
51
76
763
913
77
764
834
78
765
7085
Circle the numbers that are divisible by 4.
934
924
732
742
944
752
954
762
964
772
Circle the numbers that are divisible by 5.
354
355
650
605
325
608
375
506
5280
380
560
8542
385
1056
49104
Circle the numbers that are divisible by 6.
78
62
69300
762
3054
765
5553
96
24718
104
Circle the numbers that are divisible by 9.
377
378
4876
5876
387
5976
837
9567
827
5796
Circle the numbers that are divisible by 10.
100
75
120
245
23
250
60
380
108
387
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 3-2: Divisibility Rules
Name:
2.
The Southlake Carroll Marching Band is getting ready to perform at halftime
of the football game. With 216 musicians, can the marching band form
equal rows of 3? of 4? of 5? of 6? of 9?
True or false: All numbers divisible by 5 are also divisible by 10.
3.
True or false: All numbers divisible by 10 are also divisible by 5.
4.
True or false: All numbers divisible by 9 are also divisible by 3.
1.
Determine whether the first number is divisible by the second.
5.
185; 5
6.
76,870; 10
8.
456; 3
9.
35,994; 2
11.
12,866; 9
12.
7,564; 4
7.
10.
13.
461; 1
6,791; 3
45,812; 9
A giant pizza is divided into 18 pieces. What are the different numbers of
people you can divide it among so that there are no pieces left over?
15. What is the smallest number you find that is divisible by 2, 3, 5, 6, 9, and 10?
14.
Complete the table. Answer Y (yes) or N (no) for each box.
Number
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
Divisible
by 2
Divisible
by 3
Divisible
by 4
Divisible
by 5
Divisible
by 6
Divisible
by 9
Divisible
by 10
324
475
525
600
1234
3951
4230
7803
9360
11,235
15,972
23,409
Marty said to Doc, “So we are going to travel back in time. What year did
you set the Delorean for?” Doc replied, “I can’t remember exactly, but I do
remember the following: If you divide the year by 2, you’ll get a remainder
28.
of 1. If you divide the year by 3, 4, 5, 6, 7, or 9, you’ll also get a remainder
of 1.” “What about 8? Do you also get a remainder of 1?” “No,” said Doc.
Marty then knew which year they were off to. Which year?
The page numbers of a book are numbered 1 to 300. How many page
numbers meet these conditions:
29. A. The page numbers have the digit 5 and are also divisible by 5.
B. The page numbers contain the digit 5 but are not divisible by 5.
C. The page numbers do not have a 5 but are divisible by 5.
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 3-3: Primes and Composites
Name:
1. Circle all the prime numbers in the first row.
2. Draw a line through the first column (except for 2) and through the third and fifth columns.
3. Draw a line through the second column (except for 3).
4. Draw a diagonal line from the 5 in the top row (not including the 5) to the 5 in the side column.
Repeat with diagonal lines between the pairs of 5’s in the side columns.
5. Draw a diagonal line down and to the right between the first 7 in the left side column to the first 7 in
the right side column. Do this again for the second 7’s.
6. Circle any number that does not have a line through it.
7. Explain why you are left with just the primes.
Side
Column
7A
5
7B
5A
5B
First
Second
Third
Fourth
Fifth
Sixth
2
8
14
20
3
9
15
21
4
10
16
22
5
11
17
23
6
12
18
24
7
13
19
25
26
32
38
44
27
33
39
45
28
34
40
46
29
35
41
47
30
36
42
48
31
37
43
49
50
56
62
68
74
51
57
63
69
75
52
58
64
70
76
53
59
65
71
77
54
60
66
72
78
55
61
67
73
79
80
86
92
81
87
93
82
88
94
83
89
95
84
90
96
85
91
97
98
99
100
101
102
103
Side
Column
5A
5B
7A
5C
7B
5C
Created by Lance Mangham, 6th grade math, Carroll ISD
Largest Known Prime Number Discovered;
Has 17,425,170 Digits
Feb. 13, 2013 — On January 25th at 23:30:26 UTC, the largest known prime number, 257,885,161-1, was
discovered on Great Internet Mersenne Prime Search (GIMPS) volunteer Curtis Cooper's computer. The
new prime number, 2 multiplied by itself 57,885,161 times, less one, has 17,425,170 digits. With
360,000 CPUs peaking at 150 trillion calculations per second, 17th-year GIMPS is the longest
continuously-running global "grassroots supercomputing"[1] project in Internet history.
Dr. Cooper is a professor at the University of Central Missouri. This is the third record prime for Dr.
Cooper and his University. Their first record prime was discovered in 2005, eclipsed by their second
record in 2006. Computers at UCLA broke that record in 2008 with a 12,978,189 digit prime number.
UCLA held the record until University of Central Missouri reclaimed the world record with this
discovery. The new primality proof took 39 days of non-stop computing on one of the university's PCs.
Dr. Cooper and the University of Central Missouri are the largest individual contributors to the project.
The discovery is eligible for a $3,000 GIMPS research discovery award.
The new prime number is a member of a special class of extremely rare prime numbers known as
Mersenne primes. It is only the 48th known Mersenne prime ever discovered, each increasingly difficult
to find. Mersenne primes were named for the French monk Marin Mersenne, who studied these numbers
more than 350 years ago. GIMPS, founded in 1996, has discovered all 14 of the largest known Mersenne
primes. Volunteers download a free program to search for these primes with a cash award offered to
anyone lucky enough to compute a new prime. Chris Caldwell maintains an authoritative web site on the
largest known primes as well as the history of Mersenne primes.
To prove there were no errors in the prime discovery process, the new prime was independently verified
using different programs running on different hardware. Serge Batalov ran Ernst Mayer's MLucas
software on a 32-core server in 6 days (resource donated by Novartis[2] IT group) to verify the new
prime. Jerry Hallett verified the prime using the CUDALucas software running on a NVidia GPU in 3.6
days. Finally, Dr. Jeff Gilchrist verified the find using the GIMPS software on an Intel i7 CPU in 4.5
days and the CUDALucas program on a NVidia GTX 560 Ti in 7.7 days.
GIMPS software was developed by founder, George Woltman, in Orlando, Florida. Scott Kurowski, in
San Diego, California, wrote and maintains the PrimeNet system that coordinates all the GIMPS clients.
Volunteers have a chance to earn research discovery awards of $3,000 or $50,000 if their computer
discovers a new Mersenne prime. GIMPS' next major goal is to win the $150,000 award administered by
the Electronic Frontier Foundation offered for finding a 100 million digit prime number.
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 3-4: Primes and Composites
Name:
A prime number is a number with only 2 factors. Composite numbers have more than 2 factors.
Using your 100 Board, answer the following questions.
1.
What is the smallest prime number that is greater than 30?
2.
What is the smallest prime number that is greater than 50?
3.
5 and 7 are called twin primes because they are both primes and they differ
by two. List all twin primes between 1 and 100.
4.
Find 5 composite numbers in a row.
5.
Why didn’t we have to keep going and cross out all multiples of 9?
6.
Which of the primes 2, 3, 5, and 7 divide into 84?
Cross out the boxes containing composite numbers to discover the hidden message.
D
7
Q
4
B
71
N
9
F
67
P
6
A
3
U
2
E
14
R
2
I
2
R
31
R
35
U
69
C
16
R
8
M
25
T
3
M
32
I
89
V
19
E
23
T
27
A
17
M
18
I
11
S
10
F
43
S
87
E
7
M
12
D
29
O
42
F
48
T
12
P
60
M
12
A
37
G
75
K
9
S
3
I
41
I
64
O
20
N
17
K
9
V
97
C
7
R
19
D
73
S
14
H
100
T
5
K
9
L
67
O
59
I
23
R
45
E
97
N
49
Z
35
N
83
O
13
Q
8
I
59
R
11
E
13
R
11
T
27
E
29
S
37
A
12
S
71
D
57
R
83
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 3-5: Monomial or not?
Name:
The following are all examples of monomials:
15
− 25
x
x2
12y 2 z
4 x3
7z2
11 • 14 • x • y • 5
−3 x
5 y4
The following are all examples of things that are NOT monomials:
x+4
−2 y − 2
2 x2 + 5x
y − 3x 2
3z 3 + 1
5
WHAT IS A MONOMIAL?
A monomial is a….
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 3-6: Prime Factorization and Monomials
Name:
Consider the expression 5x. Since this notation means 5 • x , it follows that 5 and x are factors of 5x.
Determine whether each expression is a monomial.
1.
24
2.
− 3m
3.
4y + 2
4.
5xyz
5.
3c − 3
6.
6( xy )
7.
q+r
8.
166h
9.
−25 fgh
Why isn’t zero composite? For a number to be composite it has to be able to be written as a product of
two factors, neither of which is itself. Zero has an infinite number of factors; however they are all
multiplied by zero to equal zero.
Determine whether each number is composite, prime, or neither.
10.
57
11.
4,293
12.
13.
97
14.
434
1
15.
111,111
Create a factor tree for each number. Write your answer using exponents. Show all work/answers
on separate paper.
− 110
18.
280
16.
144
17.
19.
− 123
20.
− 200
21.
900
22.
108
23.
−1500
24.
324
Create a factor tree for each number. Write your answer without using exponents. Show all
work/answers on separate paper.
32 pq
26.
27.
25.
35xy 2
12x 2 z 2
28.
51e2 f
29.
−42mn3
30.
−64 jk
31.
143m2 p
32.
105ab 2
33.
525ac 2
34.
−150c 2 d 3
35.
600xy
36.
−450s 2t 3
37.
−860x2 y 4
38.
−3000x3 y3
39.
−1230a 3b
40. What is the maximum number of prime numbered dates in any two consecutive months?
41
On a prime date both the month and the date are prime numbers. For example, 2/3 is a prime
date. How many prime dates occur this year?
42.
I am a two-digit prime number. The number formed by reversing my digits is also prime. My
ones digit is 4 less than my tens digit. What number am I?
43.
Use the prime factorizations of these numbers to find the seventh number in the pattern: 8, 18,
32, 50, 72, …
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 3-7: Triple Venn Diagrams
Name:
You can use a triple Venn diagram to find the GCF and LCM of set of monomials.
Below is an example of how to complete a triple Venn diagram.
Find the GCF and LCM of 36, 90, and 120.
The prime factorization of 36 = 3 • 3 • 2 • 2
The prime factorization of 90 = 5 • 3 • 3 • 2
The prime factorization of 120 = 5 • 3 • 2 • 2 • 2
90
36
3
3 2
5
2
2
120
GCF = 6 because all three numbers have both a 2 and a 3 in common. 2 times 3 equals 6.
LCM = 5 • 3 • 3 • 2 • 2 • 2 = 360 because you multiply all terms in the diagram. Notice an easy way to do
this is to take the 120 circle and just multiply it by the only number, 3, sitting outside this circle.
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 3-8: Greatest Common Factor (GCF)
Name:
Find the GCF of each set of numbers or monomials.
28,32
1.
2.
48, 56
3.
25,30
4.
72,84,132
5.
20, 28, 36
6.
126,168, 210
7.
42,105,126
8.
14, 28, 42
9.
15ab, 10 ac
10.
14 xy , 28
11.
17 xy, 15x 2 z
12.
12am2 , 18a3m
13.
120 x 2 , 150 xy
14.
105x3 y 2 , 165x2 y 4
15.
280ac3 , 320a3c
16.
160 zw, 240w2
17.
21 pt , 49p 2t , 42pt 2
18.
5m2 , 10m, 15m3
19.
5a2 , 25b2 , 50ab
20.
9 x, 30xy , 42y
Find the GCF of each expression.
7 x + 14
9 x − 27
10 y − 25
8n − 28
3 x 2 − 21x
24 a 2 + 18a
15mn − 25n
−3 x − 27
−6 x + 30
−6 x 2 + 30 x
25 x + 15 xy + 10 y
−80 x 5 + 40 x 3
21.
A lady wants to buy plants for her garden and wants to plant them in rows with the same
number of plants in each row. Which of these would give her the most choices?
8 flats of 6 plants OR 7 flats of 8 plants OR 5 flats of 10 plants
22.
What numbers between 50 and 100 have the greatest number of factors?
23.
What calendar date(s) has the most factors? have exactly 3 factors?
24.
25.
26.
27.
28.
Rebecca’s little sister Tina has 48 yellow blocks and 40 green blocks. Tina builds some number
of towers using all 88 blocks. What is the greatest number of identical towers that Tina can
build? How many green and yellow blocks are in each tower?
Mr. Mangham has some cookies. They can be divided evenly among 9 students. They can also
be divided evenly among 6 students. What are two possibilities for the number of cookies?
A band of pirates divided 185 pieces of silver and 148 gold coins. Since pirates are known to be
fair about sharing equally, how many pirates were there?
Identify the number that satisfies all three conditions: a) It is a composite between 62 and 72. b)
The sum of the digits is a prime number. c) It has more than 4 factors.
The number 6 has exactly four whole number divisors: 1, 2, 3, and 6. What is the smallest
number with exactly five whole number divisors?
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 3-9: Least Common Multiple (LCM)
Name:
Multiple comes from “multi” meaning many and “pli” meaning fold. Fold a piece of paper in half,
in half again, and again. The resulting number of pieces is eight times the number of original
pieces. Remember: Every number has a multitude of multiples!
List the first six multiples of each number or algebraic expression.
1.
8
2.
25
3.
x
4.
y2
Find the LCM for each set of numbers or monomials.
5.
2, 4, 5
6.
5, 6, 18
7.
12, 16, 30
8.
12, 16
9.
5, 10, 20
10.
14 x3 , 22x2
11.
6 x,8 x,10 x
12.
18, 30, 50
13.
6, 8, 36
14.
24, 16, 30
15.
15 x, 20 xy ,30 xyz
16.
12, 25, 18
17.
36, 80, 75
18.
8, 52, 65
19.
14, 21, 35
20.
24, 36, 48
21.
5 x, 12 x
22.
15 x, 45y
23.
15k , 35k 2
24.
12h2 , 28
25.
6 p, 8 p , 12 p
26.
3x, 15x2 , 30
27.
8k , 20k , 24k 2
28.
3c, 5c2 , 7c
29.
24a3b, 36b2c
30.
40 x4 y 2 , 55 y 2 z 3
31.
48g 3 , 72h4
32.
8a 2bc2 ,12abc,16ab2c
33.
100 f 4 g 3 ,150 f 4 h2
34.
2w5 x3 , 4 x4 y,6 y5 z 2
Advanced Venn Diagram Problem
Every red car at an auto show was a sports car. Half of all the blue cars
35. were sports cars. Half of all sports cars were red. There were forty blue
cars and thirty red cars. How many sports cars were neither red nor blue?
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 3-10: GCF and LCM
Name:
How fast can you solve all the problems on this page? Can you solve them all mentally?
GCF
1st monomial
2nd monomial
x
x2
x2
x3
x5
x3
x100
x5
2x
2y
5x
5x
20x 2
10x
4xy
8xy
16xy
4x
2x 2 y
64x4 y 2
18x2 y 2 z 2
6xyz
36x5 y7
72x3 y9
15x4 y 6
60x6 y 4
x
y
1.
The GCF of two numbers is 850. Neither number is divisible by the other.
What is the smallest that these two numbers could be?
2.
The GCF of two numbers is 479. One number is even and the other number
is odd. Neither number is divisible by the other. What is the smallest that
these two numbers could be?
3.
Ms. Wurst and Mr. Pop have donated a total of 90 hot dogs and 126 small
cans of fruit juice for a math class picnic. Each student will receive the same
amount of refreshments. What is the greatest number of students that can
attend the picnic? How many cans of juice will each student receive? How
many hot dogs will each student receive?
LCM
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 3-11: Least Common Multiple Word Problems
Name:
Solve.
1.
At one store hot dogs come in packages of eight. Hot dog buns come in packages of twelve.
What is the least number of packages of each type that you can buy and have no hot dogs or buns
left over?
2.
Tongue Tickler Tooth Paste comes in two sizes:
9 oz. for $0.89
12 oz. for $1.19
A. What is the LCM of 9 and 12?
B. If you bought that much toothpaste in 9-oz. tubes, how much would it cost?
C. If you bought that much toothpaste in 12-oz. tubes, how much would it cost?
D. Which tube gives you more Tongue Tickler Toothpaste for the money?
3.
In the school kitchen during lunch, the timer for pizza buzzes every 14 minutes; the timer for
hamburger buns buzzes every 6 minutes. The two timers just buzzed together. In how many
minutes will they buzz together again?
4.
Two ships sail steadily between New York and London. One ship takes 12 days to make a round
trip; the other takes 15 days. If they are both in New York today, in how many days will they
both be in New York again?
5.
The high school lunch menu repeats every 20 days; the elementary school menu repeats every 15
days. Both schools are serving sloppy joes today. In how many days will they both serve sloppy
joes again?
6.
Two neon signs are turned on at the same time. One blinks every 4 seconds; the other blinks
every 6 seconds. How many times per minute do they blink on together?
7.
Gear B has 12 teeth. Gear C has 18 teeth. How many teeth should be on gear A if each turn of
gear A is to produce a whole number of turns of the shafts attached to B and C?
8.
A company ships in two different sized boxes. One box is 5 inches long and the other is 8 inches
long. What is the shortest length crate the company can use to ship its product in either sized box
without having extra space?
9.
A construction company uses 15-foot long concrete blocks for the width and 45-foot long blocks
for the length of any rectangular building. What is the shortest length square building the
company could construct?
On Southlake Highway there are rest stops every 50 miles. Lauren’s family stops at the rest stops
every 150 miles. Kyle’s family stops every 250 miles. The two families began their trips from
10.
the same place. What is the shortest distance the two families must drive before they stop at the
same rest stop?
11.
Write your own word problem that involves finding the least common multiple. Provide the
solution.
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 3-12: Least Common Multiple Word Problems
Name:
Solve.
1.
Jim is planning a party for his class at school. He is going to buy hot dogs and hot dog buns to
serve. The hot dogs come 10 to a package. The buns come 8 to a package. He doesn’t want any
hot dogs without buns and doesn’t want any buns or hot dogs left over. How many packages of
each does he need to buy?
2.
Hannah is in charge of organizing school supplies to sell in the school store. She has pencils that
come 10 to a package. She has pencil top erasers that come 12 to a package. Each pencil needs a
pencil top eraser. How many packages of each does she need so that every pencil has a pencil top
eraser and every eraser has a pencil?
3.
Stephanie has a problem. Her English teacher gives a test every 3 school days. Her mathematics
teacher gives a test every 4 school days and her social studies teacher gives a test every 5 school
days. She knows that she could have all 3 tests on the same day. How often will she have three
tests on one day?
4.
Lance is making cookies for the bake sale. He wants to use up the ingredients he is going to buy
to make cookies. The recipe calls for 1 cup of chocolate chips, 1 cup of pecans, and 1 cup of
coconut. The chocolate chip package contains 2 cups of chips. The pecan package contains 4
cups of pecans. The coconut package contains 7 cups of coconut. How many packages of each
will he need to buy to use up all of the ingredients? How many batches of cookies will that make?
5.
How many dates per year are multiples of 4?
6.
How many dates per year are multiples of 5?
7.
Danielle thinks of two numbers that are multiples of 9. The product of the two digits (neither a 9)
of either of her numbers is also a multiple of 9. What are her numbers?
8.
The after-school program at DIS includes a craft session. Danny is planning on having children
make ladybugs for their craft this week. Each ladybug needs 1 Styrofoam hemisphere, two fuzzy
stems for antenna, and six regular stems for legs. The Styrofoam hemispheres are 5 in a package,
the fuzzy stems are 6 in a package, and the regular stems are 12 in a package. How many
packages of each are needed to make ladybugs with no left over parts? How many ladybugs can
be made with these packages?
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 3-13: Juniper Green
Name:
Juniper Green – Round 1
Rules of the game:
1. Two players play at a time. The first player selects an even number.
2. On each turn, a player selects any remaining number that is a factor or a
multiple of the number just selected by his or her opponent.
3. The first player who cannot select a number loses.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
Juniper Green – Round 2
Rules of the game: The rules this time are very similar to the first game, except you and your partner are
now working together. Try to stay alive as long as possible by crossing out as many numbers as
possible. The game is over when a player cannot select another number.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
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Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 3-14: The Bubble Gum Factories
Name:
The Mangham and Underwood Bubble Gum Factories
(Taken from Lessons for Algebraic Thinking: Grades 6-8)
Show all work on a separate sheet of paper to receive full credit.
At the Mangham Bubble Gum Factory, lengths of gum are stretched to larger lengths by putting
them through stretching machines. There are 100 stretching machines, numbered 1 through 100.
Machine 1 does nothing to a piece of gum; machine 2 stretches pieces of gum to twice their
original length; machine 3 triples the length of the gum, and so forth. So, machine 23, for
example, will stretch a piece of gum to 23 times its original length. Gum can go through as
1. many machines as necessary and may travel through the same machine more than once.
An order has just come in for a piece of bubble gum 26 inches in length. The factory has pieces
of gum that are only 1 inch in length, and machine number 26 is broken. Mr. Mangham, a factory
worker, believes there is another way to create a piece of bubble gum 26 inches in length by using
other machines. Show or explain how this could be accomplished.
It appears some of the machines in the factory are unnecessary because combinations of other
2. machines could be used instead. Figure out which machines are actually unnecessary. List all
unnecessary numbers.
The Underwood Bubble Gum Factory is going to offer gum in the following lengths: 15, 28, 36,
65, 84. The Underwoods are about to order machines for their company and since they are
3. expensive, they want to order the fewest possible. What is the fewest number of necessary
machines needed to create all of these gum lengths? What is the number of each machine? Show
all work.
4.
For each of the lengths at the Underwood Bubble Gum Factory, what other machines could have
been used that were unnecessary? Show all work.
Back at the Mangham Bubble Gum Factory, which lengths between 1 and 100 would come out if
5. the bubble gum went through five machines and all five machines were necessary ones? It can
travel through the same number machine more than once. Show all work.
You know the gum can go through five machines and stay less than 100. Can it go through six
and stay less than 100? Seven and less than 100? Which lengths between 1 and 100 require the
6.
greatest number of runs through necessary machines? How did you figure out your answer? It
can travel through the same number machine more than once. Show all work.
Suppose you now have lengths of gum up to 200. Are you going to need to buy more necessary
7. machines? If so, do not list them all, but what do the machines have in common? Show all
work.
How many necessary machines are required for a 160-inch length? Which lengths between 100
8. and 200 inches require going through six necessary machines? Which lengths between 100 and
200 inches require going through seven necessary machines? Show all work.
SHOW ALL YOUR WORK AND
MAKE SURE I CAN UNDERSTAND WHERE ALL YOUR NUMBERS CAME FROM.
Created by Lance Mangham, 6th grade math, Carroll ISD
Bubble Gum FAQs
* How do I know if a machine is necessary or unnecessary?
Let’s say you need a piece of 6 in. gum. Can you make it without machine #6? Yes, you can use
machines #2 and #3. So machine #6 is unnecessary.
Let’s say you need a piece of 7 in. gum. Can you make it without machine #7? No, the only way to get
to 7 is with machine 7. So machine #7 is necessary.
* Can I put gum through the same machine more than once?
Yes, you can put it through as many times as you wish.
* Can I put gum through different machines?
Yes. For example, you can put it through machine #2 and then machine #5.
* What happens if I put gum through machine #2 three times?
Machine #2 multiplies the length of the gum times 2. Therefore the first time you put it through it will
become 2 inches. The second time it will become 4 inches and the third time 8 inches.
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 3-15: Basic Fractions
Name:
Find three fractions equivalent to each of the following.
2
1.
2.
3
1
3.
4.
7
Write each fraction in simplest form.
28
81
5.
6.
36
90
63
45
9.
10.
108
48
7.
11.
Write each improper fraction as a mixed number.
17
17
14.
15.
13.
5
6
20
15
17.
18.
19.
3
6
96
25
21.
22.
23.
10
2
5
4
4
12
8
21
12
27
21
10
30
8
36
15
Write each whole or mixed number as an improper fraction.
3
5
3
5
25.
26.
27.
4
4
6
3
7
3
2
4
2
29.
30.
31.
8
8
5
11
7
4
5
33. 3
34. 2
35.
12
12
15
Order each set of fractions from greatest to least.
3 5 1
, ,
37.
4 8 2
1 5 2
, ,
39.
2 12 3
1 3 3
, ,
41.
2 5 8
8 3 5 2
, , ,
43.
9 4 6 3
38.
40.
42.
44.
8.
12.
16.
20.
24
28.
14
35
17
51
25
4
100
75
22
12
1
6
32.
36.
9
10
2
7
15
5 5 2
, ,
8 6 3
3 2 7
, ,
5 3 12
7 3 13
, ,
8 4 16
4 13 7 31
, , ,
5 16 8 40
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 3-16: Basic Fractions/Addition & Subtraction
Name:
Use your fraction sense to solve each problem.
1
2
1. Name a fraction between and .
3
3
3
4
2. Name a fraction between and .
5
5
1
1
5. Name a fraction between and whose denominator is 16.
4
2
2
3
6. Name a fraction between and whose denominator is 10.
3
4
2
5
9. Name a fraction that is halfway between and .
9
9
2
4
10. Name a fraction that is halfway between and .
3
5
11. How many fractions are there between one-fourth and one-half?
Use the clues to discover the identity of the mystery fraction.
12. My numerator is 6 less than my denominator. I am equivalent to
3
.
4
4
.
6
My numerator and denominator are prime numbers. My numerator is one
less than my denominator.
My numerator and denominator are prime numbers. The sum of my
numerator and denominator is 24.
My numerator is divisible by 3. My denominator is divisible by 5. My
denominator is 4 less than twice my numerator.
My numerator is divisible by 3. My denominator is divisible by 5. My
denominator is 3 more than twice my numerator.
My numerator is a prime number. The GCF of my numerator and
denominator is 2. I am equivalent to one-fifth.
15. The GCF of my numerator and denominator is 5. I am equivalent to
16.
17.
18.
19.
20.
Add or Subtract. Write each answer in simplest form.
5
1
4 +2 =
21.
22.
8
4
2
1
5 +3 =
23.
24.
7
2
1
1
5 −2 =
25.
26.
8
3
4
9
10 − 3 =
27.
28.
9
10
3
7
8 +2 =
5
10
1
3
7 +8 =
4
8
6
8
7 −2 =
7
14
1
3
12 − 8 =
4
5
Created by Lance Mangham, 6th grade math, Carroll ISD
The Basics of Negative Fractions
−2
1
2
The number above represents negative two and one-half.
If you owed someone $2.50 you could write this number as −2
On a number line, the mixed number −2
1
dollars.
2
1
would be placed here:
2
It is located between the negative 2 and the negative 3. Other ways to think of this number is as
1
 1
−  2  or −2 + −
2
 2
because the negative sign applies to both the 2 and the
1
.
2
When converting a negative mixed number to a negative improper fraction, ignore the negative sign,
convert like normal, and then place the negative sign back in at the end.
Ex. 1
−5
1
3
Three times five equal fifteen plus one equals 16, so −
16
.
3
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 3-17: Improper Fractions/Mixed Numbers
Name:
To convert from an improper fraction to a mixed number:
1. Divide the denominator into the numerator.
2. Place the remainder over the divisor.
Example:
To convert from a mixed number to an improper fraction:
1. Multiply the whole number times the denominator.
2. Add the original numerator to your answer and this is
the new numerator.
3. Place the numerator over the same denominator.
3r1
13
1
= 4 13 = 3
4
4
Example: 2
1
9
→ 4 • 2 +1 →
4
4
Write each improper fraction as a mixed number. For the first three problems you complete
draw pictures that show how the improper fraction and mixed number are equivalent.
11
7
9
11
−
−
2.
3.
4.
1.
10
6
8
8
15
21
17
17
−
−
5.
6.
7.
8.
8
4
3
4
17
17
21
25
−
−
9.
10.
11.
12.
5
6
10
4
20
15
30
100
−
−
13.
14.
15.
16.
3
6
8
75
96
25
36
22
−
−
17.
18.
19.
20.
10
2
15
12
Write each whole or mixed number as an improper fraction.
3
5
3
−5
21.
22.
23.
4
4
6
3
7
3
−2
4
−2
25.
26.
27.
8
8
5
11
7
4
3
−2
5
29.
30.
31.
12
12
15
5
3
2
−1
9
−6
33.
34.
35.
6
5
3
2
1
1
2
−1
2
37.
38.
39.
5
3
2
41. Describe in writing and with a picture how
24.
1
9
10
28.
6
32.
−2
36.
40.
7
15
5
5
9
1
−7
2
7
1
compares with 2 .
3
3
7
13
or
? Explain it! Show it! Prove it!
6
12
43. Find five fractions between one-fourth and one-half.
42. Which is larger,
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 3-18: Fractions/Decimals
Name:
Converting fractions to decimals
Divide!
3
= 3 ÷ 5 = 0.6
5
If the decimal keeps repeating use bar notation.
2
Example 2:
= 2 ÷ 3 = 0.6
3
Example 1:
Fraction to a decimal
Write each repeating decimal using bar notation.
1.
0.22222…
2.
0.41666…
3.
0.54545…
4.
0.6363…
5.
0.2727…
6.
0.428572428..
Express each fraction or mixed number as a decimal. Use bar notation, if necessary.
4
7
5
1
7.
8.
9.
9
18
7
3
8
1
−6
2
10.
11.
12.
16
11
12
1
2
24
−9
−3
7
13.
14.
15.
18
5
25
1
8
6
−
4
5
16.
17.
18.
6
9
7
2
5
9
8
19.
20.
21.
3
16
11
17
11
2
−10
−2
−6
22.
23.
24.
20
18
7
5
9
3
−
14
7
25.
26.
27.
8
10
13
Order each set of rational numbers from least to greatest.
28.
30.
32.
34.
36.
1, 1.1, 1.01, 1.11
2 1 3
, ,
3 2 5
4
, 0.35, 0.36
11
1
−0.3, − , −0.33, −0.35
3
29.
31.
33.
35.
2, − 2, − 2.1, 2.1
3 2
0, −2.1, ,
2 3
3 9 7 2
, , ,
4 10 8 3
5 4
− , − , −0.83, −0.801
6 5
0.47, 0.474, 0.47, 47%, 47.4%
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 3-19: Fraction Bar Notation
Name:
Now that we have learned about bar notation with decimals here is a serious problem for you:
How do you write 0.999999999….
That would be 0.9 , right?
Well, 0.9999999….repeated forever equals what?
Would you say that number is equal to 1 or that it is less than 1??? Think about it.
In ordinary math, this number equals one. Does your head hurt yet?
So how can .9999…=1?
There are many different proofs of the fact that 0.9999... does indeed equal 1. So why does this question
keep coming up?
Do you agree that 0.3333… is equal to
1
?
3
Remember 0.9999... doesn't mean "0.9" or "0.99" or "0.9999" or "0.999 followed by some large but
finite (limited) number of 9's". 0.9999... never ends. There will always be another "9" to tack onto the
end of 0.9999.... So don't object to 0.9999... = 1 on the basis of "however far you go out, you still won't
be equal to 1", because there is no "however far" to "go out" to; you can always go further.
"But", some say, "there will always be a difference between 0.9999... and 1."
Well, sort of. Yes, at any given stop, at any given stage of the expansion, for any given finite number of
9s, there will be a difference between 0.999...9 and 1. That is, if you do the subtraction, 1 – 0.999...9 will
not equal zero. But the point of the "dot, dot, dot" is that there is no end; 0.9999... is inifinte. There is no
"last" digit. So the "there's always a difference" argument betrays a lack of understanding of the infinite.
We have learned that
1
= 0.333... in decimal form.
3
1 1 1
1
+ + = 3( ) = 1. Reasonably then, 0.333... + 0.333... + 0.333... = 3(0.333...) should also equal 1.
3 3 3
3
But 3(0.333...) = 0.999.... Then 0.999... must equal 1.
So
If two numbers are different, then you can fit another number between them, such as their average. But
what number could you possibly fit between 0.999... and 1.000...?
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 3-20: Comparing and Ordering Fractions
Name:
To compare and order fractions use the least common denominator (LCD). The LCD is the least
common multiple (LCM) of the original denominators.
Example
Order from least to greatest:
2 5 3
, ,
3 8 4
1. Find the LCD of the denominators: 3, 8, and 4. The LCD is 24.
2. Write equivalent fractions.
2 16 5 15
3 18
=
=
=
3 24 8 24 4 24
3. Order the fractions.
15 16 18
<
<
24 24 24
Compare the following fractions using <, >, or =.
2
1
5
−
1.
2.
9
3
6
3
4
2
−
−
4.
5.
6
11
3
3
5
1
−
−
7.
8.
7
8
3
5
9
2
−
10.
11.
4
7
3
Order each set of fractions from greatest to least.
3 5 1
, ,
13.
4 8 2
−
7
8
3.
4
6
3
9
6.
9.
7
10
12.
14.
5 5 2
− ,− ,−
8 6 3
−
15.
1
5
2
− ,− ,−
2 12 3
16.
3 2 7
, ,
5 3 12
17.
1 3 3
, ,
2 5 8
18.
7 3 13
− ,− ,−
8 4 16
19.
8 3 5 2
− ,− ,− ,−
9 4 6 3
20.
4 13 7 31
, , ,
5 16 8 40
22.
2 2 13 5
, , ,
3 7 14 6
Order each set of fractions from least to greatest.
9 17 1
7
− ,− ,− ,−
21.
14 35 2 10
23.
2 4 2 5
, , ,
3 5 10 7
24.
−
7
20
4
−
8
1
2
5
2
9
3
10
2
−
8
3
7
3
2
5
23 12
7
7
,− ,− ,−
52 13 26 8
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 3-21: Addition and Subtraction
Name:
2
1
Example: −6 − ( −8 )
First, change the problem to add the opposite. Then ask, “Are there more
3
2
positives or negatives?” There more positives, so the answer will be positive. How many more
positives?
1
8
2
2
−6
3
5
5
So, the answer is +1
=1
6
6
Solve each equation. Write the solution in simplest form.
5
3
k =7 −2
2.
1.
6
4
7
1
9 +2 =n
3.
4.
8
6
5 9
w= −
5.
6.
8 16
5 3
r=− +
7.
8.
12 8
3  2
− +−  = x
9.
10.
10  5 
3 9
−3 − = d
11.
12.
5 10
 4  1
13.
14.
 −4  −  −2  = w
 6  3
2
d = 3 −8
15.
16.
3
Evaluate each expression if a =
11 1
− =m
12 16
1 7
3 − =a
8 8
3
b = 3−
8
7
2 −6 = q
8
2  1
2 +  −4  = q
3  4
1
3
v = −2 − 3
5
4
2
b = 3 −8
3
2  1
p = −10 −  −3 
9  3
4
2
7
, b = − , and c = −3 . Write the solution in simplest form.
9
3
18
17.
a+c
18.
b−a
19.
c+b
20.
c−a+b
21.
a+b+c
22.
c + a −b
23.
a+b
24.
c −b
25.
c − a −b
3
7
5
Evaluate each expression if x = , y = 2 , and z = − . Write the solution in simplest form.
8
12
6
x+ y+z
y − (− z )
x−z
26.
27.
28.
29.
x+ z
30.
y−x
31.
z + (− y )
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity: Multiplying Fraction Levels
Name:
For each level, draw pictures to represent the problem. Look for the “rule” that works based on the answers in
the pictures.
4•3
2•6
Multiplication
Level 0
2•
Multiplication
Level 1
Multiplication
Level 2
Level 3
Multiplication
Level 4
Multiplication
Multiplication
Level 5
Mixed Numbers
1 2
=
3 3
3•
1
3 1
=
=
6 6
2
2•
2 4 2
=
=
6 6 3
3•
2 6
1
=
=1
5 5
5
1
Multiplication
4 groups of 3
2 groups of 6
4
• 8 = 2 (or
8
)
4
2
20
• 10 = 4 (or
)
5
5
1 4 2 4
• = =
2 5 5 10
2 1 1 2
• = =
3 4 6 12
Sample of how to read:
Five groups of one-sixth
Sample of how to read:
Three groups of two-fifths
Sample of how to read:
One-fifth of a group of ten
Sample of how to read:
One-half of a group of four-fifths
2 3 3
6
• =
=
5 4 10 20
1
1
1 •2
2
2
1
1
1 •4
3
2
You can not simply multiply the whole numbers and
then the fractions!
Show distributive property = drawing with pictures!
Multiplication
Include negative values
Level 6
Negatives
Multiplication
Level 7
Simplify before you
multiply
4 5
•
5 8
The commutative property allows simplifying before
multiplying.
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 3-22: Multiplying Fractions
Name:
1. Multiplication is the same as repeated addition when you add the same number again and again.
2. Times means “groups of.”
3. A multiplication problem can be shown as a rectangle.
4. You can reverse the order of the factors and the product stays the same.
5. When you multiply two whole numbers, the product is larger than the factors unless one of the
factors is zero or one.
6. When you multiply two positive numbers, the product is larger than the factors unless one of the
factors is zero or a fraction smaller than one.
1. Four times three means four groups of three. Three times four means three groups of four.
2. The problem three times four can be shown by a three by four rectangle.
3. The problem one-half times one-half can be shown by a two-by-two rectangle representing one
whole.
Draw a picture for
3 2
•
4 5
1
3
4 • 2 . Estimate the answer. Use the distributive property.
2
4
1 2
3 • . Estimate the answer.
2 3
1 1
1
2 • 5 = 10 . Why not?
2 4
8
Example #1: You can model
Show
1
.
4
2
1
of .
3
4
Example #2: Model
Show 5.
Divide into thirds.
Find
Shade
2
1
of the .
3
4
2
1 2 1
of =
=
3
4 12 6
1
of 5.
2
1
of each.
2
Combine the 5 halves.
1
1 1 1 1 1 5
1
of 5= + + + + = = 2
2
2 2 2 2 2 2
2
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 3-23: Multiplying Fractions
Name:
What multiplication expression is represented by each model?
1.
2.
Draw a model to represent each product.
1
3
of
4.
6
4
3.
2
1
of
5
2
5.
Find each product. You do not have to simplify improper fractions into mixed numbers.
1
2
2 1
5
of
•
•6
6.
7.
8.
9
3
7 2
8
3
4
7
1
3 5
of
of
•
9.
10.
11.
4
7
10
3
4 6
3
7
3 1
2
of
•
of 8
12.
13.
14.
8
10
4 9
9
1
5
3
2
of 2
of 4
of
15.
16.
17.
3
9
4
5
1
1
5 1
4 3
of
•
•
18.
19.
20.
3
5
8 2
9 4
2
3
5
6•
• 10
12 •
21.
22.
23.
3
5
6
6
8
5
8 9
e = of
• 10 = t
h= •
24.
25.
26.
7
15
12
9 10
4 15
3 5
3
d= •
• =k
h = • 35
27.
28.
29.
9 16
10 8
7
3 5
9 7
5 4
n=
•
• =z
f = •
31.
32.
30.
20 6
14 12
12 15
3 2
3 8
3
• =c
• =t
y = 15 •
33.
34.
35.
14 9
4 9
10
Write true or false.
3 2 1
36.
• <
4 3 2
3 4 3
38.
• >
8 5 8
37.
39.
13 1 13
• >
14 2 14
5 5
<
7 7
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 3-24: Multiplying Mixed Numbers
Name:
To multiply mixed numbers, first convert each to an improper fraction. Then multiply the fractions.
Express each mixed number as an improper fraction.
2
5
5
9
1.
2.
9
6
3.
For all problems on this page, show all work on a separate sheet of paper.
You may leave your answer as an improper fraction for all problems below.
Find each product.
1 3
2
3 •
9•4
4.
5.
6.
5 4
3
4 1
3 2
•3
1 •2
7.
8.
9.
7 9
8 7
3
4
5
3
3 •2
•4
10.
11.
12.
4
5
7
8
Solve each equation.
4 6
2 • =s
13.
9 11
3
d = 14 • −1
16.
4
3
k = −2 • 16
19.
8
22.
25.
28.
1 2
−5  −  = a
3 3
1 1 1
j= + •
4 3 2
1 2
t = •1
6 5
14.
17.
20.
23.
26.
29.
1 3
p = −1 • 3
8 7
2
8
−5 • − = t
5
9
2
1
n = • −5
3
6
15.
18.
21.
5
−4   = q
8
1 1
g = −2  − 
4 3
1  6 
p = 6 •− 
6  37 
24.
27.
30.
3
1
12
5
1
2 •4
6
2
1 9
4 •
6 10
2
20 • 1
5
6
5
− •2 = x
7 12
3
2
3 •2 = a
5
9
5
7
−1 • − = y
14
8
1 6
s=7 •
3 11
 2
6  −5  = r
 3
 1
(−6)  3  = k
 3
Solve.
31.
1 2
1
−4 •   + 3
3 3
6
32.
The weight of an object on Venus is about
34.
15
−4 − 3  
28
33.
1
1 6
−5 − 7 •
2
3 11
9
of its weight on Earth. The weight
10
2
of its weight on Earth. What is the difference in
5
the weight of a 170 pound person on Venus compared to Mars?
of an object on Mars is about
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 3-25: Multiplying Mixed Numbers
Name:
You may leave your answer as an improper fraction for all problems below.
Solve each equation.
6  33 
5 14
a=− •
1.
2.
3.
−  = b
11  34 
7 15
 1   10 
 3  8 
m =  −9   
4.
5.
6.
 −  −  = n
 5   23 
 4  9 
1 4
 8  9 
−5 • 1 = r
7.
8.
9.
−   = q
3 5
 9  8 
 1
a = − 
 2
10.
2
Evaluate each expression if b =
4
b = 3 
5
11.
 2   15 
c =  −6   − 
 3   16 
1
p = 4 •8
2
 1
9  −3  = s
 3
2
 3
c = −1  − 
 5
12.
2
2
3
1
1
, c = −2 , t = , and w = −1 .
3
4
5
2
13.
w2
14.
t 2 ( −c )
15.
2b
16.
cw
17.
bc
18.
tw
19.
ctw
20.
bcw
21.
t 2 w2
1
5
1
1
Evaluate each expression if a = − , b = , c = −1 , and d = 2 .
4
6
2
3
22.
4c
23.
bd
24.
3b − 4a
25.
18b − 6c
26.
a + cd
27.
9d +
28.
a (c + 4)
29.
b( a + 8)
30.
d (b + 6)
7
8
A bulletin board measures 45 feet wide and 18 feet high. Eight posters measuring
1
1
2 feet wide and 3 feet high are placed on the bulletin board as well as a border
31.
2
4
covering a total area of 25 square feet. What fraction of the bulletin board is still
available for other items?
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 3-26: Fraction Word Problems
Name:
CHOCOLATE BANANA RAISIN COOKIES
Semi-Sweet Chocolate
Flour
Cocoa Powder
Baking Powder
Salt
Mashed Bananas
White Chocolate
(Makes 24 cookies)
3
of a pound
Butter
8
1
2 cups
Sugar
8
1
cup
Light-Brown Sugar
4
3
of a cup
4
1
of a cup
2
1
of a cup
2
2 teaspoons
Chocolate Extract
1 teaspoon
Eggs
2 large
1
of a teaspoon
4
1
3 medium size
3
5
of a pound
8
Pecans
Raisins
1
1
cups
2
1 cup
You gather the ingredients above. Assume each problem uses the ingredients as they start above (do not
use the remaining amount for the next problem.) Determine whether the following problems require
subtraction, multiplication, or both. Then solve. Show all work on a separate sheet of paper.
Correct
Answer
Operation(s)
1
Nicole ate of a pound of semi-sweet chocolate for
1.
5
breakfast. How much semi-sweet chocolate is left?
1
Carter ate of the semi-sweet chocolate with his broccoli
2.
5
soup. How much semi-sweet chocolate did he eat?
1
Brandon ate of the semi-sweet chocolate while standing
3.
5
on his head. How much semi-sweet chocolate was left?
3
Grant accidentally threw
of the flour in the trash. How
4.
4
much flour did he throw in the trash?
3
Brooke gave
of a cup of flour to her friend Eric. How
5.
4
much flour was left?
3
Jared put
of the flour on Connor’s face. How much
6.
4
flour is not on Connor’s face?
Created by Lance Mangham, 6th grade math, Carroll ISD
1
times the normal recipe, so
2
that her sisters could have some also. How many mashed
bananas did she use?
2
Emily does not like pecans so she uses only
of the
7
normal amount. How many cups of pecans does she use?
One-third of a pound of white chocolate magically
disappears, but some people say Crissy took it. How
much white chocolate was left?
Three-fourths of the butter somehow turns up sitting on
Andrew’s head. How much butter is left for the recipe?
Chris has determined that eating small amount of cocoa
powder can help you calculate pi more accurately. If
1
Chris ate
of a cup of the powder, how much was left to
16
use in the recipe?
2
Chad loves pi also. He accidentally takes
of the light9
brown sugar instead. How much light-brown sugar is left?
1
Justin took
of a teaspoon of salt and threw it over his
32
shoulder, accidentally hitting Valerie in the face. How
much of the salt did not hit Valerie?
8
Haley loves raisins. She took of the raisins to create a
9
picture of Raisin Guy. How much of a cup of raisins did
she use?
Michael loves to mix things up so he put the pecans,
4
butter, and both sugars together. He then threw of this
5
mixture directly at Danielle’s face. How much of the
mixture was still left for Michael to throw at Christen?
Brittany, like, loves to put extra butter on everything. So,
1
like, she uses 5 times the normal amount of butter in the
2
recipe. How much, like, butter did she use?
2
Heather lost
of the mashed bananas while she was
3
helping Crissy and Haley learn their dance. How many
mashed bananas did she lose?
8
Leslie found ants in of a cup of flour. How many cups
9
of flour did not contain ants?
Bailey made 60 cookies, 2
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
Addition PLUS
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 3-27: Multiplying Fractions Visually & Mentally
Name:
In order to demonstrate your understanding of the concept of multiplication of fractions, show how to
find each of the following products visually, rather than with rules. Make a diagram or sketch to
illustrate your visual methods for solving each problem. Use graph paper.
1.
4•
2
3
2.
4.
1 6
•
2 7
5.
7.
1
of 22
4
8.
3
4
of
4
7
3
of 8
4
2
3
of
3
5
3.
6.
9.
1
• 12
4
1 3
•
4 4
1
1
of
2
5
12.-13. Write two interesting word problems that require finding the product of fractions. Solve each
of your word problems using visual models.
14.
I am a fraction in simplest form. One-sixth of me is the same as one-half of one-fourth. What
fraction am I?
Solve the following problems mentally.
2
• 12
1.
3
3
20 •
3.
4
2
2 • • 15
5.
5
2
8 •7
7.
7
3
3 •8
9.
4
1
2•6•4
17.
2
2.
4.
6.
8.
10.
18.
1 1
3• •
4 3
1
• 11 • 12
4
1
5 •6
3
1
12 • 5
2
1
12 • 3
2
1
4 • 20
5
Show how to use a sketch to solve the following problems.
After Kevin spent half of his week’s pay on food, then spent one-third of what was left on rent,
19. and then spent one-half of what was left on fun, he had $20 left of his paycheck. How much
money was his week’s check?
Mr. Mangham has a class of 32 students. Three-fourths of the students are weird and, of those,
seven-eighths are clueless. What fraction of all the students are weird and clueless and how
20.
many students are neither weird nor clueless? (This assumes that if you are not weird, you are
not clueless.)
21.
In an Algebra class, half of the students are boys. One-third of the students are wearing glasses.
Half of the boys are wearing glasses. What fraction of the girls is wearing glasses?
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 3-38: Muppets
Name:
In Drew’s class, one-fifth of the boys are absent and two-fifths of the girls are absent. Bert believes that
you multiply, thus two twenty-fifths of the class is absent. Elmo believes that means that a total of
three-fifths of the class is absent. Ernie believes three-tenths of the class is absent. Kermit can’t figure
out how much of the class is absent, but he believes it is somewhere between one-fifth and two-fifths.
Analyze each character’s answer to determine if they are correct or not. From this make a conclusion on
which character knows the most math.
A lot of students believe that you just add the two numbers together and get three-fifths. Look at the
problem carefully. Do you add those two numbers together? One-fifth relates to the boys and two-fifths
relates to the girls. The key missing part to this question is that it does not tell you how many boys are
in the class or how many girls are in the class. Might those numbers make a difference in what the true
answer is?
To test this theory, one good method is to guess how many boys and girls might be in the class and then
determine your answer. Based on these answers you can then see which character knows the most math.
Total students
Boys
Girls
20
10
10
50
25
25
100
50
50
20
15
5
40
10
30
60
25
35
80
50
30
25
15
10
25
10
15
Boys absent
Girls absent
Fraction of
class absent
Based on your results above, which character does know the most math? Why?
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity: Dividing Fractions with Pictures
Name:
Division
Level 0
(draw pictures)
12 ÷ 3 = 4
15 ÷ 5 = 3
How many groups of 3 are there in 12?
How many groups of 5 are there in 15?
Division
1
=4
2
1
4÷ = 8
2
2 1
÷ =2
3 3
3 1
÷ =3
4 4
5 1
÷ =5
6 6
NOTE: When divide by positive fraction, the
answer gets bigger.
Sample of how to read:
How many groups of one-half are in two?
Level 1
(draw pictures)
Division
Level 2
(draw pictures)
Division
Level 3
(draw pictures)
Division
Level 4
(draw pictures)
Division
Level 5
(draw pictures)
2÷
Sample of how to read:
How many groups of one-eighths are in foureighths?
4 2
÷ =2
6 6
6 2
÷ =3
8 8
4 1
÷ =2=
6 3
8 1
÷ =4=
12 6
3÷
2
1
=4
3
2
3 3
2
2 ÷ =3
4 4
3
Sample of how to read:
How many groups of three-tenths are in nine-tenths?
4
2
8
2
Sample of how to read:
How many groups of one-forth are in nine-twelfths?
Sample of how to read:
How many groups of two-thirds are in three?
Level 6: Introduce reciprocal rule with mixed numbers.
2
1
3 ÷4
3
4
Level 7: Introduce reciprocal rule with negative numbers.
1
1
−2 ÷ −6
4
5
3 rules for dividing fractions:
1. Divide straight across.
2. Find a common denominator
3. Use the reciprocal to make the denominator one
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 3-29: Dividing Fractions
Name:
(Taken from TEXTEAMS Rethinking Middle School Mathematics: Numerical Reasoning)
1. You can solve a division problem by subtracting.
2. To divide two numbers, a ÷ b , you can think, “How many b’s are in a?”
3. You can check a division problem by multiplying.
4. The division sign means “into groups of.”
5. The quotient tells “how many groups of” there are.
6. You can break the dividend apart to make dividing easier.
7. Remainders can be represented as whole numbers or fractions.
8. If you divide a number by itself, the answer is one.
9. If you divide a number by one, the answer is the number itself.
10. If you reverse the order of the dividend and the divisor, the quotient will be different unless the
dividend and divisor are the same number.
Mr. Mangham plans to make small cheese pizzas to sell during the weekends to make some extra
money. He has 9 bars of cheddar cheese. How many pizzas can he make if each takes the amount listed
in the table? On a piece of notebook paper or computer paper create the five columns shown below.
Answer the following questions by completing the row for each question.
1.
2.
3.
Situation
Think
about it
(what is the
question
asking)
1
bar of
3
cheese
How many
1
’s are in
3
9?
Picture It
Write it
in
symbols
9÷
1
3
Process
it (What
do you
actually
do?)
Solve it
9•3
1
bar of
4
cheese
2
bar of
3
cheese
Mr. Mangham still needs more money so he is going to sell small bags of coffee. He buys a large
twelve-pound bag. How many small bags can he make based on the following situations?
Think about it
Write it
Process it (What
Situation
(what is the
Picture It
in
do you actually
Solve it
question asking)
symbols
do?)
1
pound
4.
5
1
pound
5.
6
3
pound
6.
4
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 3-30: Dividing Fractions
Name:
(Taken from TEXTEAMS Rethinking Middle School Mathematics: Numerical Reasoning)
Mr. Mangham brought jars of M&M®’s to be shared by members of the student groups winning a class
game. How much of a pound of candy will each student get based on the following?
On a piece of notebook paper or computer paper create the six columns shown below. You will answer
the following questions by completing the row for each question.
1.
2.
3.
4.
Situation
Think about it
(what is the
question asking)
1
pound and 4
2
students
1
pound
2
divided by 4
Picture It
Write it in
symbols
Process it
(What do you
actually do?)
Solve it
1
÷4
2
Divide whole
in half and
the half into
4 equal parts
1
8
1
pound and 3
4
students
1
pound and 3
3
students
1
pound and 2
5
students
Next, Mr. Mangham brought very long chocolate bars to give away as prizes in another team
competition. What fraction of a bar did each team member get assuming it was shared equally?
Situation
5.
6.
7.
Think about it
(what is the
question asking)
Picture It
Write it in
symbols
Process it
(What do you
actually do?)
Solve it
a two person team
3
won
of a bar
4
a four person team
7
won
of a bar
8
a four person team
1
won 1 bars
2
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 3-31: Dividing Fractions
Name:
(Taken from TEXTEAMS Rethinking Middle School Mathematics: Numerical Reasoning)
Mr. Mangham is making ribbons for all of his students to wear in honor of his former student, Marci
1
Holden. It takes of a yard to make a ribbon for each student. How many badge ribbons can he make
6
from the lengths listed below? For each answer that has a remainder, some ribbon left over, tell what
fractional part of another badge ribbon you could make with the amount left over.
On a piece of notebook paper or computer paper create the six columns shown below. You will answer
the following questions by completing the row for each question. The first one has been done for you.
1.
2.
3.
4.
Situation
Think about it
(what is the
question asking)
1
yard
2
1
yd. divided into
2
1
pieces of yd.
6
Picture It
1 2
6 6
3
6
6
6
Write it in
symbols
Process it (What
do you actually
do?)
Solve
it
1 1
÷
2 6
Divide yard in
half, whole into
sixths. How many
sixths in a half?
3
3
yard
4
5
yard
8
2
2 yard
3
Next, Mr. Mangham is making bows for the math students that are not in his classes so that he can easily
1
recognize them. It takes
yard of ribbon to make one bow. How many bows can Mr. Mangham make
2
from each of the following amounts of ribbon?
Situation
5.
6.
7.
8.
Think about it
(what is the
question asking)
Picture It
Write it in
symbols
Process it
(What do you
actually do?)
Solve it
4
yard
5
8
yard
9
3
1 yard
4
1
2 yard
3
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 3-32: Painting
Name:
Avery’s father donated 6 gallons of paint to her school. The teachers at Avery’s school have decided to
1
paint tabletops with the paint. It will take of a gallon of paint to cover a small-sized table. It will
5
1
1
take
of a gallon of paint to cover a medium-sized table, and of a gallon of paint for a large-sized
4
3
table.
Solve. Show all work on a separate sheet of paper.
1.
How many small-sized tables can the teachers paint if they use all of the paint? Show a model and
write an expression that relates to your model.
2.
How many medium-sized tables can the teachers paint if they use all of the paint? Show a model
and write an expression that relates to your model.
3.
How many large-sized tables can the teachers paint if they use all of the paint? Show a model and
write an expression that relates to your model.
4.
Based on your observations, write a rule that makes sense for dividing any whole number by a unit
fraction.
Avery found some green tables and blue tables that needed painting in the cafeteria. Her father agreed
2
4
to donate 6 more gallons of paint. It will take
of a gallon to repaint the green tables and
of a
3
5
gallon to repaint the blue tables.
Solve. Show all work on a separate sheet of paper.
5.
If the teachers use all the paint, how many green tables can they paint? Use a model and write an
expression that relates to the model.
6.
If the teachers use all the paint, how many blue tables can they paint? Use a model and write an
expression that relates to the model.
7.
In question #4 you described a method for dividing fractions. Does your method work for
questions #5 and #6? If needed, revise your rule so that it does work.
8. Use your rule to solve the following problems. What do you notice about your answers?
4÷
9.
1
12
4÷
2
12
4÷
Based on your answers above can you predict the answer to 4 ÷
4
12
8
? Solve the problem. Was
12
your prediction correct?
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 3-33: Dividing Fractions
Name:
Any number times its reciprocal
4
5
equals 1. The reciprocal of
is .
5
4
So we need to multiply both the
5
numerator and denominator by .
4
To divide fractions, multiply the
first fraction by the reciprocal of
the second number.
The Multiplicative Inverse Property
Any number times its multiplicative inverse or reciprocal is equal to 1.
Reciprocal comes from “re” meaning backward and “pro” meaning forward. In writing the
reciprocal you have gone “back and forth” and returned to the identity for multiplication.
Find the reciprocal of each number.
1.
1
4
2.
5
6
3.
7
4.
8
15
Find each quotient. Show all work on a separate sheet of paper. Simplify, but improper fractions are
okay.
5 1
3 5
1 3
÷
÷
÷
5.
6.
7.
6 3
4 8
2 5
8.
8÷
4
5
9.
1 2
÷
6 9
10.
9 1
÷
10 4
11.
3
÷9
8
12.
8 2
÷
9 3
13.
2 4
÷
5 7
5
9
15.
7 7
÷
8 10
16.
1 5
÷
9 12
14.
15 ÷
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 3-34: Dividing Mixed Numbers
Name:
To divide mixed numbers, convert them to improper fractions and follow the same rule as dividing
fractions.
Find the reciprocal of each number.
1.
8
3
4
2.
9
6
7
3.
7
5
6
4.
3
5.
14
6.
1
5
12
7
16
7.
6
7
8
8.
10
1
10
Solve each equation. Show all work on a separate sheet of paper.
9.
12.
j=
6 3
÷
7 14
5
1
÷2 = p
12
2
10.
4 14
÷ =b
9 15
11.
9 3
÷ =s
16 4
13.
2 5
s = 2 ÷1
3 6
14.
4
a =1 ÷6
5
You may leave your answer as an improper fraction for all problems below.
Find each quotient. Show all work on a separate sheet of paper.
2
1 3
4÷2
3 ÷1
15.
16.
17.
5
4 8
8
1
÷ −5
9
3
18.
1
2
−2 ÷ 4
2
7
19.
1
−3 ÷ −7
9
20.
2
4
6 ÷4
3
5
21.
2
5
1 ÷ −1
9
6
22.
7
5
÷2
10
8
23.
1 7
−3 ÷ 1
5 9
24.
3
−1 ÷ −14
4
25.
2
2
5
÷3
15
9
26.
27.
3
7
6 ÷1
4 20
28.
18 ÷ −1
1
8
29.
−2
1
7
÷−
10
8
1 3
−4 ÷ 1
6 7
Evaluate each expression. Show all work on a separate sheet of paper.
30.
y ÷ z , if y =
32.
f ÷ g , if f = 5
4
2
and z =
5
3
31.
c ÷ d , if c = 14 and d =
7
8
1
5
and g = 1
4
9
33.
w ÷ z , if w = 9 and z = 2
1
7
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 3-35: Dividing Mixed Numbers
Name:
Name the reciprocal (also called the multiplicative inverse) for each rational number.
1.
−3
5.
1
3
3
4
1
5
4.
0.6
1
12
8.
1.5
2.
−10
3.
3
6.
−
8
7
7.
−
You may leave your answer as a simplified improper fraction for all problems below.
Solve each equation. Show all work on a separate sheet of paper.
1
f = 18 ÷ −3
5 ÷ −6 = l
9.
10.
3
11.
1
3
2 ÷3 = k
2
5
12.
4
÷ −6 = b
9
13.
1 
7
c = 2 ÷  −1 
5  10 
14.
1 17
−1 ÷ 1 = d
9 63
15.
 3
−10 ÷  −5  = k
 4
16.
1 7
g=− ÷
5 8
17.
1
2
−12 ÷ 4 = x
4
3
18.
2  3
−5 ÷  −  = r
7  8
19.
3 
9
−7 ÷  −1  = y
5  10 
20.
2  10 
6 ÷−  = p
3  3
21.
1  7
−12 ÷  −  = k
4  8
22.
2 
4
h = −5 ÷  −2 
3  15 
1
miles on a measured
2
23.
1
1
course 1 miles long. To set the record, he crawled without stopping for 9 hours. How many
2
2
laps did Morris complete? What was his crawling speed in mi/h?
Reg Morris holds the world’s crawling record (true story). He crawled 28
24. A turtle walked
25.
1
1
mile at the rate of mile per hour. How long did it take?
2
5
A certain math book is
3
of an inch thick. How many of these books will fit on a shelf that is 3
4
feet wide?
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 3-36: Dividing Mixed Numbers
Name:
Write an expression for each word problem and solve on a separate sheet of paper.
3
Draw a diagram to show how many -ft. pieces of string can be cut from a piece of string four
1.
4
and a half feet long.
3
2. How many -c servings are there in a 6 c package of rice?
4
3.
George cut 5 oranges into quarters. How many pieces of orange did he have?
1
Anna bought a package of ribbon 10 yd long. She needs 1 -yd pieces for a bulletin board. How
4.
3
many pieces can Anna cut from the ribbon?
2
5. Using #4, what if Anna decided to use -yd pieces? How many pieces can she now cut?
3
1
6. A bulletin board is 56 in. wide and 36 in. high. How many 3 -in columns can be created?
2
1
There are 3 boys and 2 girls in the Krunch family. Mr. Krunch bought 3 pounds of candy to
7.
2
divide equally among them. How much candy did each child get?
1
It takes 1 cup of liquid fertilizer to make 7 gallons of spray. How much liquid fertilizer is
8.
2
needed to make 80 gallons of spray?
1
Darlene has 2 hours to complete three household chores. If she divides her time evenly, how
9.
4
many hours can she give to each?
1
1
10. Elizabeth bought 3 pounds of tomatoes for 2 dollars. How much did she pay per pound?
3
2
3
pound box of candy. How much is that per pound?
11. Dad paid $2.00 for a
4
1
3
The runner ran 2 miles in
hour. At that rate, how many miles could he run in 1 hour? That
12.
2
4
is, what is his speed in miles per hour?
13.
Farmer Brown measured his remaining insecticide and found that he had two and a quarter
gallons. It takes three-fourths gallon to make a tank of mix. How many tankfuls can he make?
Linda has 4
14.
2
yards of material. She is making baby clothes for her niece. If each pattern
3
1
yards of material, how many patterns will she be able to make?
6
2
On Sunday, Mr. Underwood swam 15 miles in 2.1 hours. As a mixed number, what was Mr.
5
Underwood’s swimming pace in miles per hour?
requires 1
15.
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 3-37: Rectangles
Name:
Determine the missing dimension of each of the following rectangles. All area units are in square
inches.
1.
3.
2.
A = 32
1
2
A = 38
1
2
A = 42
3
4
4.
A = 72
1
4
5.
A = 117
7
8
6.
A = 120
3
8
7.
A = 63
15
16
8.
A = 54
1. Length = 6
3. Width = 9
1
inches
2
1
inches
2
5. Length = 10
7. Width = 8
1
inches
4
1
inches
4
27
32
1
inches
2
Width =
2.
Length = 5
Length =
4.
Width = 8
Width =
6.
Length = 4
1
inches
2
Width =
Length =
8.
Length = 6
3
inches
4
Width =
1
inches
2
Width =
Length =
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 3-38: A Camping Trip
Name:
You and your friends are planning a camping trip. You are in charge of making and bringing enough
trail mix. The trail mix needs to provide each person with one serving for each day of the trip. Here is
the recipe for trail mix:
1
cups raisins
4
3
cup peanuts
4
2
1
cup walnuts
3
1
cup cashew nuts
8
2
cup dried pineapple pieces
3
1
cups coconut pieces
8
1
cup chocolate chips
4
Combine all ingredients and mix them.
1. How many cups in all does the recipe make?
1
cup, how many servings does the recipe make?
2
Suppose 12 people go camping for 3 days. How many cups in all
3.
should you make?
2. If one serving is
For #3, how much of each ingredient should
you use to make enough trail mix for each
4.
1
person to have a -cup serving on each day?
2
Raisins
Cashews
Peanuts
Coconuts
Pineapple
Chocolate
Walnuts
Suppose 3 people are going on the camping trip for 3 days and each
1
5.
serving of trail mix is
cup. How many cups in all should you make?
4
For #5, how much of each ingredient should
6.
you use?
Raisins
Cashews
Peanuts
Coconuts
Pineapple
Chocolate
Walnuts
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 3-39: Fraction Word Problems
Name:
Write an expression for each word problem and solve on a separate sheet of paper.
1
Liz Ann and Bryan are running a race. Bryan only ran 1 miles before he dropped out. Liz Ann
4
1.
1
ran 3 times farther than Bryan. How far did Liz Ann run and how many more miles did Liz
2
Ann run than Bryan?
2
George Bush was eating his daily jelly beans. He decided he would share his jar of 5 pounds of
5
2.
jelly beans amongst himself and seven other world leaders. How many pounds did each leader
get?
1
3
Trent bought 6 pounds of tomatoes. An average tomato weighs of a pound. Approximately
3.
2
8
how many tomatoes did Trent buy?
The Honey Baked Ham Store is busy during the holiday season. The average ham they sell
1
1
4. weighs 9 pounds. If they say each ham will make 23 servings, how much does one serving
2
4
weigh?
Bob is very interested in knowing exactly how fast he drives. On his last trip, he figured he drove
1
6
5.
42 miles in 1 hours. How fast was he driving?
4
7
1
Bill owns a bakery that makes really good sugar cookies. It takes 12 cups of sugar to bake 5
5
6.
1
complete batches and of another batch. How many cups of sugar is used for each batch?
3
1
3
Garrett can make 3 pies in an hour. If he works for 7 hours, how many pies can Garrett
7.
2
4
make?
1
In a recipe you need 4 cups of milk and you want to use all 15 cups of milk in the refrigerator.
8.
2
How many batches of the recipe can you make?
1
Larry, who lives in London, won the equivalent of 2 million dollars in the lottery. If one dollar
2
9.
1
is equal to 1 pounds (money in England), how many pounds did he win?
2
1
Becky has 3 pizzas left over from a party. The next day each person who visits Becky’s house
5
10.
2
eats
of a pizza. How many people visit before the pizza is all gone?
5
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 3-40: Fraction Word Problems
Name:
Write an expression for each word problem and solve on a separate sheet of paper.
1.
2.
3.
3
of a pumpkin pie. She decided to split it between herself, Emily Sue, Billy Bob,
4
Joe Jill, and Danny Dude. How much of a whole pie will each person get?
Lizzy Lou had
1
chocolate bars for Christmas. If there are 6 munchkins, how
5
many chocolate bars does each one get?
Dorothy gives the munchkins 9
There were 10 cakes at Johnny’s birthday party. If each person was given
2
of a cake and all 10
5
cakes were eaten, how many people attended the party?
There was
4.
1
of a pizza at Joe’s Pizza, Pasta, & Subs. If each person at the restaurant was given
2
1
of a whole pizza for free, how many people were at Joe’s?
16
5.
The Griswold’s sat down for Christmas Eve dinner. They bought a 24 pound turkey, which
amazingly enough they cooked correctly this year. There are 6 people in their family so how
many pounds does each person get? If Rusty eats only one-half of a pound and they split up the
rest for the other five people, how much extra turkey will each get?
6.
If Emily practices band for 3
2
hours over 4 days, how many hours did she average each day?
5
1
of a cookie because he is on a diet. The rest of the cookie was divided up amongst
4
all of the seven reindeer. Then Comet stole Blitzen’s serving. How much of a cookie does Comet
get?
Santa eats
7.
8.
9.
5
1
of a pound of turkey and she has to make equal groups of of a pound each. How
7
3
many groups are there?
Taylor has
You are on a 9
1
2
mile hike with 6 gallons of water. How many gallons of water can you drink
3
3
every mile?
1
2
cups of dog food and needs to divide it up into serving sizes of
of a cup each.
10.
3
3
How many servings can Scooby make?
Scooby has 4
Created by Lance Mangham, 6th grade math, Carroll ISD
Activity 3-41: Delivering Packages
Name:
Jordan has a job delivering packages to schools in CISD. This morning her first stop was at Johnson
where she picked up some packages. Her instructions were to drop off three-fifths of those packages at
a second building and pick up six more packages. At her third stop, Rockenbaugh, she dropped off
three-sevenths of her packages and picked up four. At her fourth stop, Durham, she dropped off threefourths of her packages and picked up six. The fifth delivery was to Eubanks Intermediate where she
left one-third of her packages and picked up two. On her sixth stop Jordan left one-half of her packages
and picked up two. She then made her last delivery of six packages, all that she had left. Then she took
a lunch break. How many packages did Jordan deliver before lunch?
One way to go about solving this problem would be to guess and check. You could guess the amount of
packages Jordan starts with and work through the problem to see if she has six at her last stop. You
know enough math now to not have to make numerous guesses. You can combine what you know about
variables, fractions, and equations to solve each step of the way working backwards through the
problem.
For example, at Jordan’s fifth delivery she left one-half, picked up two, and left with six. How would
1
you turn this into an equation? How about: • x + 2 = 6 That would translate into one-half of some
2
number (because she still is carrying one-half) plus the two she picked up equals the six she is taking to
the last stop.
Continue this process of writing and solving equations all the way back to the first stop. After solving
the last equation, you can then use your information to determine how many total packages she dropped
off. Show all equations and all work.
Created by Lance Mangham, 6th grade math, Carroll ISD
THINK YOU ARE HAVING A BAD DAY?
Dear Sir:
I am writing in response to your request for additional information. In Block #3 of the accident reporting
form, I put "trying to do the job alone" as the cause of my accident. You said in your letter that I should
explain more fully and I trust that the following details will be sufficient.
I am a bricklayer by trade. On the day of the accident, I was working alone on the roof of a new six story
building. When I completed my work, I discovered that I had about 500 pounds of bricks left over.
Rather than carry the bricks down by hand, I decided to lower them in a barrel by using a pulley which,
fortunately, was attached to the side of the building on the top floor.
Securing the rope at ground level, I went up to the roof, swung the barrel over the side and loaded bricks
into it. Then I went back to the ground and untied the rope holding it tightly to insure a slow descent of
the 500 pounds of bricks. You will note in Block #11 of the accident report form that my weight is 185
pounds. Do to my surprise at being jerked off the ground so suddenly, I lost presence of mind and forgot
to let go of the rope. Needless to say, I proceeded at a rather rapid rate up the side of the building.
In the vicinity of the third floor, I met the barrel coming down. This explains my fractured skull, minor
abrasions, and broken collarbone.
Slowed only slightly, I continued my rapid ascent, not stopping until the fingers of my right hand were
two knuckles deep into the pulley. Fortunately by this time I had regained my presence of mind and was
able to hold tightly to the rope despite the excruciating pain I was beginning to experience.
At that time however, the barrel of bricks reached the ground - and the bottom fell out of the barrel when
it hit. Now devoid of the 500 pounds of bricks the barrel now weighed only 50 pounds.
As you might imagine, I began a rapid descent down the side of the building. In the vicinity of the third
floor, I met the barrel coming up. This accounts for the two fractured ankles, broken tooth, and the
severe lacerations on my legs and lower body.
Here my luck began to change slightly. The encounter with the barrel seemed to slow it up enough to
lessen my injuries when I fell into the pile of bricks and fortunately only three vertebrae were cracked.
I am sorry to report, however, that as I lay there on the pile of bricks in pain, unable to move and
watching the empty barrel six stories above me, I lost hold of the rope. The empty barrel weighed more
than the rope and came back down on me and broke both my legs.
I hope I have furnished the information you required as to how the accident occurred.
Created by Lance Mangham, 6th grade math, Carroll ISD
Additional problems
1
cups of flour for a recipe that
3
makes 12 cookies. She needs to make 120 cookies. How many batches of cookies will she have to
make to get 120 cookies? How many cups of flour will she need in order to make 120 cookies?
1. Joni is making cookies for the school picnic. The recipe calls for 2
2
1
pound of walnuts which cost $4.25 per pound, pound of pecans which cost $5.69
3
3
3
1
per pound,
pound of peanuts at $2.99 per pound and
pound of almonds at $3.98 per pound. How
4
2
many pounds of nuts did Haley buy?
2. Haley bought
3. For the problem above mow much was the total cost of the nuts she bought?
4. If each loaf of banana bread weighs 1
3
pounds, what is the total weight of 6 loaves of bread?
4
5. Brad is making cookies for his Boy Scout troop. He needs 116 cookies. He can bake a dozen
1
cookies every 12 minutes. How long will it take him to bake 116 cookies?
2
Created by Lance Mangham, 6th grade math, Carroll ISD