Reasoning in Math
Connections
Have you ever . . .
• Had to calculate how much material a project required?
• Suddenly realized a simpler way to find an answer to
a problem?
• Thought through a better way to manage your budget?
Reasoning is a skill you use in every aspect of life, and reasoning is
the basis of math. Math gives you a way to reason about numbers.
How do the numbers relate to each other? What conclusions can
you come to based on what you know?
Some people compare doing math problems to solving puzzles. That’s because both are
about reasoning. Reasoning is your ability to think logically, and it’s the most valuable skill
you can have in math.
In math, you might use reasoning skills to:
• Find patterns and make comparisons.
• Understand what numbers and operations mean.
• Change the form of an equation.
• Think through a plan to solve a problem.
The key to reasoning in math is to ask why. If you don’t
understand a procedure in math, find out why it works. If
you don’t understand a concept, ask why it’s important.
Understanding the “why” behind math concepts is the key
to reasoning and solving problems.
45
Essential Math Skills
Learn
It!
Using Mathematical Reasoning
Every time you do a math problem, you use reasoning and your knowledge of
numbers and operations. The more you understand what numbers and operations mean, the more you can draw logical conclusions about them.
You have a small business selling watches with custom bands.
Currently, you have three boxes that originally had five dozen watch
faces each. You’ve opened two of the boxes, and used 3/5 of one and
8/15 of the other. Now, you need to calculate your inventory for your
records. How many watch faces do you have?
Un Understand
Think about the numbers involved and the types of operations you will need to use.
What do you need to find in this problem? What do you need to understand about
?1.
numbers and operations?
In this problem, you need to know how many watch faces you have. To find this, you must
find the number of boxes you have left. A box in the problem is a new kind of quantity.
You’re dealing with dozens and boxes, and each box is equal to five dozen.
12 = 1 dozen 5 dozen = 1 box
You also need to understand fractions. You used 3/5 of one box and 8/15 of another. That leaves
you with 2/5 of one box and 7/15 of another.
P Plan
Make a plan to solve the problem. You use reasoning to think of plans that will work and to
figure out the easiest or best plan.
Make a plan to calculate inventory. What reasoning did you use?
?2.
You might reason that you can find how many boxes are left and
convert it to watch faces. An alternate plan is to find how many watch
faces are in each box and find a sum total. If b = number of boxes, then:
2
7
b = 1 + 5 + 15
46
Review &
Practice
Learn more and
practice with
fractions on pages
305–312.
Reasoning in Math
You could plan to calculate the number of boxes by addition and then multiply the answer
by the number of watch faces in a box.
Number of watch faces = 5 dozen ◊ b
A Attack
You also use reasoning when you attack a problem. You think about how you can make
calculations and represent numbers in different ways.
How many watch faces are left? What reasoning did you use to make your calculations?
?3.
You use reasoning to add fractions and calculate the number of boxes. You can reason that
2/5 of a box is the same as 6/15 of a box, since one fifth is the same as three fifteenths.
2
7
6
7
13
b = 1 + 5 + 15 = 1 + 15 + 15 = 1 15 boxes
You can reason that five dozen watch faces is equal to five times 12, if you understand what
multiplication is. Multiplying five times 12 is the same as adding 12 + 12 + 12 + 12 + 12. Since
there are five sets of 12 watch faces, you have five times 12 watch faces, or 60.
13
watch faces = 1 15 # 60
You can reason though the multiplication problem. One box will contain 60 watch faces.
How much is 13/15 of 60? If you divide 60 into 15 parts, each part is four. So, a fifteenth of a box
will be four watch faces. Since 13/15 is 2/15 less than a whole box, it will be eight watch faces left.
13/15 of a box will contain 52 watch faces. The total is 112 watch faces.
C Check
When you check your solution, you’ll also use reasoning. Think through
the answer. Does it make sense? Does it fit with what you know?
Check your solution to the problem. What reasoning did you use?
?5.
You might reason that 2/5 of a box and 7/15 of a box are each almost half
a box. So, you’ll have a little less than two boxes worth of watch faces.
Since 112 is a little bit less than two boxes (10 dozen, or 120 watch
faces), the answer makes sense.
Build Your
Math Skills
Break up complex
problems into small
steps. It will help
you understand the
problem and think
about it clearly.
47
Essential Math Skills
e
ic
Pract
It!
Answer the following questions.
1.A scientist measures the width of tree rings from trees in a forested
area. The following table shows the measurements for outer rings of
two sample trees. Before the formation of the 2006 rings, the diameter
of Tree A was 0.256 meters and the diameter of Tree B was 0.408
meters. What was the difference in diameter at the end of 2010?
Year
Tree A
Tree B
2010
6.5 mm
7.3 mm
2009
2.8 mm
5.6 mm
2008
6.9 mm
7.2 mm
2007
12.1 mm
15.2 mm
2006
12.2 mm
15.8 mm
Try sketching
this problem.
a.
Solve the problem.
b.
What reasoning did you use to solve the problem?
2.Guadalupe scored 89%, 94%, and 98% on her first three chemistry tests. How much does
she have to score on her next chemistry test to have a 95% mean?
a.
Solve the problem.
Review &
Practice
b.
What reasoning did you use to solve the problem?
48
Find a table of units
of measure on page
304. Learn about
mean, median, and
mode on pages
253–260.
Reasoning in Math
3.Harold and Sadie entered a dance-a-thon to raise money for cancer research. They had
pledges of $87 per hour of dancing. The dance-a-thon started at 9:00 a.m. on Saturday,
and they danced until Sunday morning at 2:45 a.m. How much money did they raise?
a.
Solve the problem.
b.
What reasoning did you use to solve the problem?
4.On the first 200 miles of a trip, Alva’s car uses 7.89 gallons of gas. If the car continues to
use gas at the same rate, how much gas will Alva need for a total trip of 957 miles?
a.
Solve the problem. Round to the nearest hundredth.
b.
What reasoning did you use to solve the problem?
5.A concert venue sells floor tickets for $52.50, first balcony tickets
for $46.25, and second balcony tickets for $32.50. If the venue
sold 66 first balcony tickets, 54 second balcony tickets, and a total
of $9,217.50 worth of tickets, how many floor tickets were sold?
a.
Solve the problem.
b.
What reasoning did you use to solve the problem?
Build Your
Math Skills
Think about
what numbers
represent in real
life, and always
ask why if you
don’t understand
a procedure or
concept in math.
49
Essential Math Skills
6.Explain how you can use reasoning to multiply fractions.
7.Explain how you can use reasoning to create an equation to solve a word problem.
8.Explain how you can use reasoning to find the perimeter of an irregularly shaped object.
Review &
Practice
9.Explain how you can use reasoning to convert a decimal to
a fraction.
50
Get some review
and practice with
multiplying and
dividing fractions
on pages 305–308,
perimeter on
pages 327–328, and
converting decimals
and fractions on
pages 321–322.
Reasoning in Math
Check Your Skills
Use your reasoning skills to answer the following questions.
1.
In one week, it rained 4.29 inches. The next week, the rainfall was 15.31 inches.
On the third week, it rained 8.99 inches. What was the approximate mean daily
rainfall over the three week period?
a.
1.05 inches
b.
1.29 inches
c.
1.36 inches
d.
1.48 inches
2.Hollis plants 16.85 acres of corn. She expects a yield of 148 bushels per acre, and each
bushel will need 1.25 cubic feet of storage. Which storage area will hold all the corn with
the least empty space?
a.
31 feet by 25 feet by 6 feet
b.
15 feet by 21 feet by 10 feet
c.
9 feet by 9 feet by 16 feet
d.
13 feet by 8 feet by 19 feet
3.
Georgina takes out a loan of $2,381 to pay for her dream
vacation. The loan is for seven years at 3.9% simple interest
per year. What will be the total cost of the loan plus interest?
Math Tip
For simple interest,
remember the
formula I = prt.
I = Interest
a.
$2,892
p = principal
b.
$3,031
r = rate
c.
$3,290
t = time
d.
$3,408
4.Sammie is making four three-layer cakes for a reunion. The recipe for a single layer
of one three-layer cake calls for 3/8 cup cocoa, 17/8 cups flour, 21/6 cups of sugar, and
¼ teaspoon salt. How much cocoa, flour, sugar, and salt will he need?
CocoaFlour SugarSalt
51
Essential Math Skills
5.
Skin cells have an average area of 9◊10-10 square meters.
Approximately how many skin cells would it take to cover
the surface of a palm that measures 7.2 centimeters by
8.5 centimeters?
a.
6.2 × 106
b.
6.54 × 106
Review &
Practice
Learn about
exponents and
scientific notation
on pages 349–350.
c.
6.8 × 106
d.
7.18 × 106
6.Gale is interested in becoming a conservation scientist. He learns that there were
34,900 conservation scientist and forester jobs in 2010, and that 5% more jobs are
projected by 2020. How many conservation scientist and forester jobs are projected
by 2020?
a.
35,025
b.
36,645
c.
37,210
d.
38,595
7.
Tony is planning a wedding reception. He has $2,930 left in the wedding budget
and 64 guests. The cost of renting the hall is $391.50, and the cost of the
decorations and servers is $668. To the nearest cent, how much can Tony spend for
food per guest?
8.At the local frozen yogurt shop, customers get a 35% discount
after 3:00 p.m. Janice buys two yogurts at 4:00 p.m. for $6.27.
How much would Janice have paid before three o’clock?
a.
$8.90
b.
$9.17
c.
$9.65
d.
$10.02
Remember
the Concept
Use your understanding
of numbers and
operations to think
through math concepts
and problems.
Try more than one
approach to build your
math reasoning skills!
52
Answers and Explanations
Reasoning in Math
page 45
Using Mathematical Reasoning
Practice It!
pages 48–50
4b.You might solve this problem in different ways, but
you need reasoning to find the gallons used per mile
or the miles per gallon. Then you need to understand
how to find the total gallons used in the trip from one
of those figures.
1a.0.1732 meters
5a.84 floor tickets
Tree A
52.5x + 46.25(66) + 32.5(54) = 9217.5
6.5 + 2.8 + 6.9 + 12.1 + 12.2 = 40.5 mm
52.5x + 3052.5 + 1755 = 9217.5
0.0405 m × 2 = 0.081 m
52.5x + 4807.5 = 9217.5
0.081 + 0.256 = 0.337 m
52.5x = 4410
Tree B
x = 84
7.3 + 5.6 + 7.2 + 15.2 + 15.8 = 51.1 mm
0.0511 m × 2 = 0.1022 m
0.1022 + 0.408 = 0.5102 m
Difference
0.5102 - 0.337 = 0.1732 m
1b.You could give the difference in either meters or millimeters. You need reasoning to plan how to calculate
the present diameter of each tree in either meters or
millimeters, especially to realize that the width of the
tree ring will add twice that to the diameter. You also
need to realize that you need to subtract to find the
difference in diameter.
2a.Guadalupe needs 99%.
(0.89 + 0.94 + 0.98 + x) ÷ 4 = 0.95
0.89 + 0.94 + 0.98 + x = 3.8
2.81 + x = 3.8
x = 3.8 - 2.81 = 0.99
2b.You need reasoning skills to write an equation to find
the missing test score. You also need reasoning skills
to use algebra to solve the equation.
3a.They raised $1,544.25.
9:00 a.m. to 2:45 a.m. the next morning is 17.75 hours.
17.75 × 87 = 1544.25
5b.You need reasoning skills to write and solve an equation or to work through the problem step-by-step
without an equation.
6.
You can use reasoning to multiply fractions by recognizing that the result must be smaller. Half times a
quarter is half of a quarter, so it will be smaller than
a quarter (by half). By multiplying the denominators,
you’re finding the smaller units of the answer. By
multiplying the numerators, you’re finding the correct
number of those smaller units.
7.
You can use reasoning to create an equation to solve
a word problem by finding the relationships between
quantities in the word problem. Identify known and
unknown quantities, and identify operations that
describe their relationships. Translate the relationships that you find into a mathematical equation.
8.
You can use reasoning to find the perimeter of an
irregularly shaped object by finding the values of each
side of that object and adding those values together.
Even if you don’t know the length of a portion of the
perimeter, you can either find that length or use a
variable to represent it.
9.
You can use reasoning to convert a decimal to a fraction if you realize that decimals are based on fractions
with a denominator of 10, 100, 1,000, etc. Take the
decimal number and place it over the implied denominator, so that 0.7 becomes 7/10 or 0.94 becomes 94/100.
3b.You need reasoning skills to find the number of hours
danced and to understand that you need to multiply
the hours danced by the hourly pledge.
Check Your Skills
4a.37.75 gallons
(4.29 + 15.31 + 8.99) ÷ 21 = 1.36
2.
b. 15 feet by 21 feet by 10 feet
7.89 ÷ 200 = 0.03945
0.03945 × 957 = 37.75365 c 37.75
pages 51–52
1.
c. 1.36 inches
16.85 × 148 = 2493.8
i
Essential Math Skills
2493.8 × 1.25 = 3117.25
15 × 21 × 10 = 3150
3.
b. $3,031
2381+ 2381 × 0.039 × 7 = 3031
3
1
=
8 4 2 cups
7
1
Flour: 12 # 1 = 22 cups
8
2
4.Cocoa: 12 #
Sugar: 12 # 2
Salt: 12 #
1
=
6 26 cups
1
=
4 3 teaspoons
5.
c. 6.8 × 106
0.072 × 0.085 = 0.00612 square meters
0.60012 ÷ (9 × 10-10) = 6,800,000
6.
b. 36,645
34,900 × 1.05 = 36,645
7.
$29.23
(2930 - 391.5 - 668) ÷ 64 = 29.226 . . .
8.
c. $9.65
0.65x = 6.27
x = 9.646 . . .
ii