Objective 6
The student will demonstrate an understanding of the mathematical processes
and tools used in problem solving.
For this objective you should be able to
●
apply mathematics to everyday problem situations;
●
communicate about mathematics using everyday language; and
●
use logical reasoning.
What Is Problem Solving?
Problem solving includes understanding a problem, making a plan to
solve it, carrying out that plan, and checking the answer to see whether
it is reasonable. A reasonable answer is an answer that makes sense. An
answer makes sense if it is not too big or too small for the question.
Louisa plans to serve fruit juice to her guests at lunch. Louisa, 1 of
her friends, and her 2 parents will all have juice. Each juice glass
holds 10 fluid ounces. If each person will have a full glass of juice,
how many cups of juice will Louisa need?
Understand the problem.
●
Identify the question you need to answer.
How many cups of juice will Louisa need?
●
Identify the information you know.
One glass holds 10 fluid ounces of juice.
Louisa, 1 friend, and her 2 parents will all have juice.
The Mathematics Chart has the information necessary to
convert fluid ounces to cups.
1 cup 5 8 fluid ounces
Make a plan.
●
Add to find the total number of glasses of juice needed.
●
Multiply the number of glasses by the number of fluid ounces
each glass holds.
●
Divide by 8 to convert fluid ounces to cups.
Carry out the plan.
●
Add: Louisa (1) 1 1 friend 1 2 parents 5 4 glasses of juice
●
Multiply: 4 glasses 3 10 fluid ounces per glass 5 40 fluid
ounces of juice
●
Divide: 40 fluid ounces 4 8 fluid ounces per cup 5 5 cups
Louisa will need 5 cups of juice.
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Objective 6
Check for reasonableness.
●
Is 5 cups of juice a reasonable answer? Is it enough juice for
4 people or too much juice for 4 people?
●
One glass holds 10 fluid ounces, which is a little more than
1 cup. So 4 people would need more than 4 cups of juice.
●
10 fluid ounces is not close to 2 cups. So 4 people 3 2 cups 5
8 cups of juice, which would be too much juice.
●
A reasonable answer is an amount between 4 and 8 cups. The
number 5 is between 4 and 8.
5 cups of juice is a reasonable answer.
Do you see
that . . .
Alexa studied for 1 hour 45 minutes on Tuesday, 1 hour
20 minutes on Wednesday, and 2 hours 30 minutes on Thursday.
How long did Alexa study during these three days?
Understand the problem.
●
Identify the question you need to answer.
How long did Alexa study during these three days?
●
Identify the information you know.
She studied 1 hour 45 minutes on Tuesday.
She studied 1 hour 20 minutes on Wednesday.
She studied 2 hours 30 minutes on Thursday.
The Mathematics Chart has the information needed to convert
minutes to hours.
1 hour 5 60 minutes
Make a plan.
●
Add to find the total number of hours and minutes Alexa
studied.
●
Convert minutes to hours and minutes.
Carry out the plan.
1 hour 45 minutes
1 hour 20 minutes
1 2 hours 30 minutes
4 hours 95 minutes
●
Since 95 minutes is greater than 1 hour, convert 95 minutes
to hours and minutes. Subtract 1 hour (60 minutes) from
95 minutes.
95 minutes 2 60 minutes 5 35 minutes
95 minutes 5 1 hour 35 minutes
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Objective 6
●
Now add this to 4 hours.
4 hours 1 1 hour 35 minutes 5 5 hours 35 minutes
Alexa studied a total of 5 hours 35 minutes during these three
days.
Check for reasonableness.
●
Alexa studied more than 1 hour on Tuesday, more than
1 hour on Wednesday, and more than 2 hours on Thursday.
1 hour 1 1 hour 1 2 hours 5 4 hours
Alexa studied more than 4 hours.
●
Alexa studied less than 2 hours on Tuesday, less than 2 hours
on Wednesday, and less than 3 hours on Thursday.
2 hours 1 2 hours 1 3 hours 5 7 hours
Alexa studied less than 7 hours.
●
Alexa studied more than 4 hours but less than 7 hours during
these three days. A reasonable answer is an amount of time
between 4 and 7 hours.
5 hours 35 minutes is a reasonable answer.
122
Objective 6
Try It
Shalika is buying hamburger buns for a picnic. She needs at least
60 buns. Each package of hamburger buns contains 8 buns and
costs $2. If Shalika buys enough packages to have at least 60 buns,
what will she pay for the buns?
Understand the problem.
What do you know?
Shalika needs to buy at least ______ buns.
Hamburger buns come in packages of ______ buns.
A package of buns costs $________.
Make a plan.
Use the operation of _________________ to find the number of
packages of hamburger buns she needs to buy.
Use the operation of _________________ to find the total cost of
the buns.
Carry out the plan.
______ buns 4 ______ per package 5 ______.
The remainder means that Shalika needs to buy an extra package.
Shalika needs to buy ______ packages of hamburger buns.
______ packages 3 $________ per package 5 $________
Shalika will pay $________ for the hamburger buns.
123
Objective 6
Check for reasonableness.
If Shalika buys 5 packages of buns, she will have ______ buns and
spend $________. Five packages of buns is
enough not enough .
(circle one)
If Shalika buys 10 packages of buns, she will have _____ buns and
spend $________. Ten packages of buns is too many not enough .
(circle one)
Shalika will spend more than $________ but less than $________ in
order to have 60 buns. Therefore, $16 is a _______________________
answer.
Shalika needs to buy at least 60 buns. Hamburger buns come in packages of
8 buns. A package of buns costs $2. Use the operation of division to find the
number of packages she needs to buy. Use the operation of multiplication to
find the total cost of the buns. 60 buns 4 8 per package 5 7 R4. Shalika
needs to buy 8 packages. 8 packages 3 $2 per package 5 $16. Shalika will
pay $16 for the hamburger buns. If Shalika buys 5 packages of buns, she
will have 40 buns and spend $10. Five packages of buns is not enough. If
Shalika buys 10 packages of buns, she will have 80 buns and spend $20.
Ten packages of buns is too many. Shalika will spend more than $10 but
less than $20 in order to have 60 buns. Therefore, $16 is a reasonable
answer.
What Is a Problem-Solving Strategy?
A problem-solving strategy is a plan for solving a problem.
The strategy you choose depends on the type of problem you
are solving. Sometimes you can use more than one strategy
to solve a problem. Some problem-solving strategies include:
●
drawing a picture;
●
looking for a pattern;
●
guessing and checking;
●
acting it out;
●
making a table;
●
working a simpler problem; and
●
working backward.
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Objective 6
Sometimes the numbers in a problem can make it seem difficult. You
can change the numbers in the problem so that you can solve a simpler
problem. Then use the same steps to solve the original problem.
An unopened box of cereal weighs a total of 1 pound 9 ounces, and
this includes 2 ounces of raisins. What fractional part of this total
weight is raisins?
The numbers in this problem represent two different units of weight.
This could make it complicated to solve. Change the numbers so
they are easier to work with.
●
Suppose the box of cereal weighed 10 ounces and the raisins
weighed 2 ounces.
●
Ask yourself: What fractional part of 10 is 2?
●
2 is part of the total weight and is the numerator of the
fraction.
10 is the total weight and is the denominator of the fraction.
2
10
The answer for this easier problem would be }}.
Use the same method to help you find the answer to the original
problem.
●
The box of cereal weighs 1 pound 9 ounces.
●
You need the weight of the box of cereal in ounces if you are
going to compare it to the weight of the raisins.
●
Use the Mathematics Chart to convert 1 pound 9 ounces to
ounces.
1 pound 5 16 ounces
●
1 pound 9 ounces 5 16 ounces 1 9 ounces 5 25 ounces
●
The box of cereal weighs 25 ounces.
●
The raisins weigh 2 ounces.
●
What fractional part of 25 is 2?
●
2 is part of the total weight and is the numerator of the
fraction.
25 is the total weight and is the denominator of the fraction.
2
The weight of the raisins is 25 the weight of the box of cereal.
125
Objective 6
Another way to solve a problem is to work backward. When working
backward, you can start with the information that is given last.
Some fifth-grade students volunteered to clean up a park. They
worked one morning for 2 hours. Then they took a 30-minute lunch
break. After lunch they worked for 1 hour 35 minutes. The students
left the park at 2:00 P.M. At what time did the students start cleaning
the park?
To find the time the students started cleaning the park, use the
information that is given last.
●
The students left the park at 2:00 P.M.
11 12 1
10
2
3
9
8
4
7 6
●
5
The students worked for 1 hour 35 minutes after lunch.
What is 1 hour 35 minutes before 2:00? From 2:00 count back
1 hour. Then count back 35 minutes.
1 hour before 2:00 is 1:00 (Think: 2 2 1 5 1)
35 minutes before 1:00 is 12:25
(Think: 60 minutes 2 35 minutes 5 25 minutes)
The students finished lunch at 12:25 P.M.
●
The lunch break lasted 30 minutes. What time did the lunch
break start? From 12:25 count back 30 minutes.
12:25 2 25 minutes 5 12:00
12:00 2 5 minutes 5 11:55
(Think: 25 minutes 1 5 minutes 5 30 minutes)
The students started lunch at 11:55 A.M.
●
The students worked for 2 hours before lunch. What is 2 hours
before 11:55?
2 hours before 11:55 is 9:55 (Think: 11 2 2 5 9)
The students started cleaning the park at 9:55 A.M.
126
Objective 6
Try It
Two cans of fruit weigh 4 pounds. Four cans weigh 8 pounds. Six
cans weigh 12 pounds. If each can weighs the same, how much
would 7 cans of fruit weigh?
One way to solve this problem is to make a table to help you
organize the data. Then look for a pattern in the data. Write the
missing numbers in the table below.
Cans of Fruit
Number
of Cans
Weight
(pounds)
2
4
4
6
7
Look for a pattern in the table.
The pattern is to multiply the number of cans by _______ .
To find the weight of 7 cans, multiply _______ 3 _______ .
Seven cans of fruit weigh _______ pounds.
Cans of Fruit
Number
of Cans
Weight
(pounds)
2
4
4
8
6
12
7
14
The pattern is to multiply the number of cans by 2. To find the weight of
7 cans, multiply 7 3 2. Seven cans of fruit weigh 14 pounds.
127
Objective 6
Try It
What dimensions
do you need to
find the perimeter
of a rectangle?
The perimeter of a gym floor is 180 feet. The floor is shaped like
a rectangle. If the width of the gym floor is 40 feet, what is its
length?
One way to solve this problem is to draw a picture.
The floor is ________ feet wide.
The opposite sides of a rectangle are _______________ .
Therefore, the gym floor has two sides that are ________ feet wide.
The sum of the two widths of the gym floor is ________ feet.
Subtract ________ from ________ to find the sum of the two lengths
of the gym floor.
________ 2 ________ 5 ________
The sum of the two lengths of the gym floor is _______________ feet.
Divide ________ by ________ to find the floor’s length.
________ 4 ________ 5 ________
The length of the gym floor is ________ feet.
The floor is 40 feet wide. The opposite sides of a rectangle are congruent.
Therefore, the gym floor has two sides that are 40 feet wide. The sum of
the two widths of the gym floor is 80 feet.
Subtract 80 from 180 to find the sum
of the two lengths of the gym floor:
Gym floor
180 – 80 = 100. The sum of the two
40 ft
P
5 180 ft
lengths of the gym floor is 100 feet.
Divide 100 by 2 to find the floor’s length:
100 ÷ 2 = 50. The length of the gym floor
is 50 feet.
?
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Objective 6
Try It
Matthew is 5 years older than his sister Janet. The sum of their ages is 37.
How old are Matthew and Janet?
Use the guess-and-check strategy.
1st guess: Suppose Janet is 12 years old.
Then Matthew is 12 1 _______ years old. Matthew is _______ years old.
The sum of their ages can be represented by: 12 1 _______ 5 _______
This guess is too large
small .
(circle one)
Try a
larger
smaller number for Janet’s age.
(circle one)
2nd guess: Suppose Janet is 20 years old.
Then Matthew is 20 1 _______ years old. Matthew is _______ years old.
The sum of their ages can be represented by: ______ 1 ______ 5 ______
This guess is too large
small .
(circle one)
Try a number for Janet’s age that is between _______ and _______ .
3rd guess: Suppose Janet is 16 years old.
Then Matthew is 16 1 _______ years old. Matthew is _______ years old.
The sum of their ages can be represented by: ______ 1 ______ 5 ______
This guess is ____________________________ .
Janet is _______ years old, and Matthew is _______ years old.
If Janet is 12, then Matthew is 12 1 5, or 17. The sum of their ages is 12 1 17 5 29.
This guess is too small. Try a larger number for Janet’s age. If Janet is 20, then
Matthew is 20 1 5, or 25. The sum of their ages is 20 1 25 5 45. This guess is too
large. Try a number for Janet’s age that is between 12 and 20. If Janet is 16, then
Matthew is 16 1 5, or 21. The sum of their ages is 16 1 21 5 37. This guess is
correct. Janet is 16 years old, and Matthew is 21 years old.
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Objective 6
How Do You Change Words into Math Language and Symbols?
The sum is the answer to
an addition problem. The
difference is the answer
to a subtraction problem.
The product is the answer
to a multiplication
problem. The quotient is
the answer to a division
problem.
An important part of solving story problems is to rewrite the problem
using math language and symbols. The words in the problem will give
you clues about what operations to use.
Sometimes you will not be asked to solve a problem. Instead, you will
be asked what math needs to be done to solve the problem.
Jim bought 6 sandwiches for $5 each. He gave the cashier two
$20 bills. What steps will show how much change Jim should
receive from the cashier?
The question doesn’t ask for the amount of change Jim should
receive. It asks about the steps to find the solution. Think about the
math that needs to be done to find the solution.
●
First use multiplication to find the total cost of the sandwiches.
Find the product of 6 and $5.
●
Then use multiplication to find the total amount he gave the
cashier. Find the product of 2 and $20.
●
Then use subtraction to find the difference between the amount
Jim gave the cashier and the total cost of the sandwiches.
In order to know how much change Jim should receive from the
cashier, subtract the product of 6 and 5 from the product of 2 and 20.
130
Objective 6
Try It
A store had a sale on backpacks. On Monday the store sold
65 backpacks. For the next 4 days, the store sold 25 backpacks a
day. How can you find how many backpacks the store sold in these
5 days?
Think about the math that needs to be done to find a solution.
On Monday the store sold _________ backpacks.
The store sold _________ backpacks a day for the next 4 days.
Use the operation of __________________ to find how many
backpacks were sold during the 4 days.
To find how many backpacks were sold during these 5 days,
_________ 65 to the _________ of 25 and 4.
On Monday the store sold 65 backpacks. The store sold 25 backpacks a day
for the next 4 days. Use the operation of multiplication to find how many
backpacks were sold during the 4 days. To find how many backpacks were
sold during these 5 days, add 65 to the product of 25 and 4.
131
Objective 6
What Is Logical Reasoning?
Logical reasoning is thinking about something in a way that makes
sense. Being logical can mean looking for a pattern or looking for what
a group has in common.
Sometimes you will need to apply what you know about certain math
concepts to determine what characteristics a set of numbers have in
common.
Which statement about this group of numbers is true?
3, 4, 6, 12, 24
1. All the numbers are multiples of 2.
2. All the numbers are factors of 24.
3. All the numbers are divisible by 3.
●
The first statement is about multiples. The multiples of 2 are
2, 4, 6, 8, 10, … The number 3 is not a multiple of 2. The first
statement is not true.
●
The second statement is about factors. The factors of 24 are
1, 2, 3, 4, 6, 8, 12, 24. All the numbers listed in the problem
are factors of 24. This statement is true.
●
The third statement is about the numbers being divisible by 3.
A number is divisible by another number if there is no
remainder when you divide.
34351
There is no remainder, so 3 is divisible by 3.
4 4 3 5 1 R1 There is a remainder, so 4 is not divisible by 3.
The third statement is not true.
The second statement is true. All the numbers in the group are
factors of 24.
132
Objective 6
Another way to use logical reasoning is to look for a common
characteristic in a set of figures or objects.
Mary drew the figures below and named them quisks.
A
Special made-up words,
such as quisks, may be
used to name a set of
examples.
B
C
Notice that each of Mary’s quisks is made up of two shapes: one on
the outside and one on the inside. Count the number of sides on
each shape.
●
Figure A is a square with a triangle inside of it. The square has
4 sides, and the triangle has 3 sides.
●
Figure B is a pentagon with a rectangle inside of it. The
pentagon has 5 sides, and the rectangle has 4 sides.
●
Figure C is an octagon with a 7-sided figure inside of it. The
octagon has 8 sides, and the inner figure has 7 sides.
In Mary’s quisks, the shape on the outside has one more side than
the shape on the inside.
Now look at Figures D and E. Which of these could be a quisk?
D
E
●
Figure D is a trapezoid with a pentagon inside of it. The
trapezoid has 4 sides, and the pentagon has 5 sides. The shape
on the outside has one fewer side than the shape on the inside.
Figure D is not a quisk.
●
Figure E is a hexagon with a pentagon inside of it. The
hexagon has 6 sides, and the pentagon has 5 sides. The shape
on the outside has one more side than the shape on the inside.
Figure E is a quisk.
133
Objective 6
Try It
Mrs. Cole sells eggs from her farm. She separates her eggs into three
groups: small, medium, and large. The size of the egg is determined
by its mass as measured in grams.
This list shows the masses of some medium-size eggs.
Medium-size eggs:
52.3 grams
49.6 grams
56.2 grams
This list shows the masses of some eggs that were not put into the
medium-size group.
Not medium-size eggs:
47.3 grams
58.1 grams
48.0 grams
Put the masses of the medium-size eggs in order from least to
greatest.
__________
__________
__________
A medium-size egg has a mass between __________ grams and
__________ grams.
The eggs that were not put into the medium-size group were either
__________ or __________ than the medium-size eggs.
Which eggs shown below could be medium-size eggs?
58.3 grams
45.9 grams
54.2 grams
Egg X
Egg Y
Egg Z
Look at the pictures of the eggs. According to the rule, which egg
could be a medium-size egg?
Egg X
is is not a medium-size egg. Its mass is too __________.
(circle one)
Egg Y
is is not a medium-size egg. Its mass is too __________.
(circle one)
Egg Z
is is not a medium-size egg. Its mass is between
(circle one)
__________ and __________ grams.
The masses of medium-size eggs from least to greatest are: 49.6, 52.3, and
56.2. A medium-size egg has a mass between 49.6 grams and 56.2 grams.
The eggs that were not put into the medium-size group were either smaller
or larger than the medium-size eggs. Egg X is not a medium-size egg. Its
mass is too large. Egg Y is not a medium-size egg. Its mass is too small.
Egg Z is a medium-size egg. Its mass is between 49.6 and 56.2 grams.
Now practice what you’ve learned.
134