Module 12 of 12 Applying the Pythagorean Theorem and Applying ...

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Module 12 of 12
Applying the Pythagorean Theorem and
Applying Math in Everyday Life
8 th Grade Math
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Introduction to Applying the Pythagorean Theorem

More Introduction to Applying the Pythagorean Theorem
It is important to remember when working with the Pythagorean Theorem that the legs
are always labeled a and b. It doesn’t matter which you choose for a or b, but they
must be on the legs of the right triangle. The hypotenuse, the longest side of the
triangle is always going to be labeled as c.
I think that it is important that we also look at another method for applying
the Theorem. It still requires that you label the legs and hypotenuse correctly,
but allows you to usually solve the problem without making mistakes with the formula.
Essentially, what the Pythagorean Theorem is stating to us
is that when we add the squares of the two legs,
the sum will equal the square of the hypotenuse.
To prove it to ourselves, we will use the same image,
but we will give it some nice easy numbers, the first
Pythagorean Triple 3, 4, and 5.
Looking at the squares of the two
legs, we have areas of 9 and 16.
The sum of 9 and 16 is 25, which
is equal to the area of the square
of the hypotenuse.

Example 1
A contractor is building a basement. He pours the walls of
the foundation and then checks to make sure the walls are
perpendicular. The length of the basement is 60 feet. The
width of the basement is 45 feet, and the diagonal is 75 feet.
Do his measurements indicate the walls are perpendicular?
How do you know?

Example 2
A door has the dimensions shown at the right. Is the angle
opposite the diagonal a right angle? Round to the nearest tenth.
Question: I need to…
Information: The legs of the triangle are ____ ft. and ____ ft. The hypotenuse
(diagonal) is ____ ft.
Plan: Use the converse of the Pythagorean Theorem to prove that it is or isn’t a right
triangle that is formed by the diagonal.
Solve: Find the size of
all of the square on the
right triangle formed by
the diagonal of the door.
The areas of the two green
squares should add up to
the area of the purple
square. Remember to round
to the nearest tenth. If they are equal, then
it is a right triangle, but if they are not equal,
then it is not a right triangle.
Use the
converse of
the
Pythagorean
Theorem.

Time to Practice!!!

Practice Problem
An outline of a playground is shown at the right.
What is the perimeter of the playground?
Question: I need to…
Information: The lengths of the perimeter are
____ ft., ____ ft., ____ ft., and ____ ft. There is
one length missing.
Plan: Use the Pythagorean Theorem with the
right triangle formed at the top of the playground
to find the missing measurement.
Solve:
You know the
length of all of the
segments except
one. When you
know that length,
just add them all to
find the perimeter

Let’s see how much
you’ve learned…

Problem-Solving Strategies
There are several commonly used strategies that are used to solve problems:
• Draw a picture – this is one of the strategies that is most useful when dealing with temperature or
elevation problems. Sometimes we have difficulty “seeing” the problem and a simple picture will
allow us to visualize what is happening.
• Look for a pattern – this is very effective with problems that deal with algebraic reasoning. Find a
pattern and it can lead to the answer.
• Draw a diagram, chart, t-chart, or a graph – these were a couple of the ways that we attacked
the problems in this unit. They can be very effective tools.
• Working backwards – starting with the solution, you work backwards in the problem.
• Simplify the problem – you would recreate the problem, using simpler numbers in order to get an
understanding of how to solve the problem.
• Act out the problem – sometimes running through the problem in your mind, with you actually
dealing with the situation being described, can help you “see” the necessary steps to solve the
problem.
• Guess and check – this is effective on multiple-choice problems. You can take the answers and
begin plugging them into the problem until you find one that works. It is very effective when used
to prove/disprove all answers.

TAKS Practice 1
In his backyard, Jackson has a small vegetable garden, shaped like
an isosceles right triangle. The longest side of the garden is about
7 feet. To the nearest foot, what is the length of the two legs of
his garden?
A 3 feet
B 5 feet
C 7 feet
D 8 feet
Question: I need to…
Information:
Plan:
Solve:
An isosceles triangle
is one where two of
the legs are the
same length.

TAKS Practice 2
Which of the following measurements would form a right triangle?
A 3, 4, and 6
B 4, 5, and 7
C 6, 8, and 9
D 9, 12, and 15
Question: I need to…
Information:
Plan:
Solve:
Remember that
the two smaller
numbers will be
the legs of the
triangle since the
hypotenuse is
always the
longest side.

Now I want to see if
you can come up
with your own
problem!

Create Your Own
• You will now create your own real-life problem. It should be a problem that is complicated enough
that someone would have to come up with a good plan for solving it. In other words, don’t make it
too easy.
• You will have to show the solution and explain the steps for solving it, though – so make sure that
the problem makes sense and is logical.
• Create the problem so that it looks like the ones that you see on your TAKS test. Include answer
choices. Try to make the three wrong answers so that they will make sense to someone that tries
to solve the problem wrong.
• Show an explanation of what you did to solve it and explain why you chose those steps and/or
procedures.
• Finally, you will show what the correct solution is.
Let’s see if they can
solve this one!

Your Word Problem
Write your problem here
Show the work to solve the problem here Put the answer choices here. Circle
the answer choice that is correct.
A
B
C
D

Introduction to Applying Math in Everyday Life
In this lesson, you will learn how to recognize and apply math concepts in your everyday life. You actually
problem-solve naturally and may not realize you are thinking mathematically. you use math in all areas of
your life, in and outside of school, in other subjects, and in sports and extracurricular activities.
Sometimes the mathematical thinking does not involve calculations. Know how to find the answer may be
enough. Other times, you might have too much information and need to decide which information to use to
solve a problem.
Identifying the information you need to solve a problem is mathematical thinking. Learning how to develop a
plan with emphasis on the plan versus the solution is also mathematical thinking. Determining a process to
solve a problem and applying your methods to given situations are ways to use mathematical concepts in
your everyday life.
• Consider the following example.
Sandra is taking her two children and their grandmother to a spring concert.
Which of the four ticket prices can be found without any other information?
Can you determine the ticket price for each person? No, you need to know ages.
Use logic to determine the price of one of the tickets. Whose ticket can you find?
Sandra has two children so logically she is an adult. Sandra’s ticket is $12.
The price of Sandra’s ticket can be found without any further information.
Ticket Prices
Seniors (over 55) $8
Adults $12
Children $5
Children under 5 Free

Example 1
A baseball coach is purchasing new uniforms for his team. Each jersey costs $19.90 and each pair
of pants costs $22.50. The tax rate for the purchase is 6.25%. If he has $1,095 in the uniforms
budget, what other information is necessary to determine if he has enough funds to purchase the
new team uniforms?
Question: I need to figure out what other information is needed to answer the question.
Information: The price of the jersey is ________, the price of the pants is ________, the tax rate is
______, and the amount of money that the coach has to spend is ____________.
Plan: Determine the missing information. The coach knows all of the prices that are needed to purchase
the new uniforms and the tax rate. He knows how much money he has to spend, but wants to make sure
that his purchase does not exceed his budget. How much would it cost for 1 player? 5 players?
Solve: The information that is missing is how many players are on the team. Without knowing how
many uniforms he needs to buy, it is impossible for the coach to know whether or not he has enough money
to buy them.

Question: I need to…
Example 2
Isaac is planning to travel 2,600 miles. He plans to drive between 375 and 450 miles each day. At
this rate, what is a reasonable number of days it will take Isaac to complete his trip?
Information: He is planning to travel a total of _________ miles. Each day he will travel a minimum of
________ miles and a maximum of ________ miles.
Plan: Try to figure out how many days it will take him if he only travels the minimum of ______ per day.
How many days will it take him if he travels the maximum of _______ per day. A reasonable number would
be between those to number of days.
Solve:

Time to Practice!!!

Practice Problem
Jazmine wants to install new countertops in her kitchen.
The drawing to the right shows the dimensions of her
counter.
What is the least amount of material needed to cover the
top of the kitchen counter?
Question: I need to…
Information: The shape is a composite figure that can be divided into several
simple shapes.
Plan: Cut the figure into simple shapes that I can find the area of.
Solve:
A figure that can
be divided into
more than one of
the basic shapes
is called a
composite figure.

Let’s see how much
you’ve learned…

Problem-Solving Strategies
There are several commonly used strategies that are used to solve problems:
• Draw a picture – this is one of the strategies that is most useful when dealing with temperature or
elevation problems. Sometimes we have difficulty “seeing” the problem and a simple picture will
allow us to visualize what is happening.
• Look for a pattern – this is very effective with problems that deal with algebraic reasoning. Find a
pattern and it can lead to the answer.
• Draw a diagram, chart, t-chart, or a graph – these were a couple of the ways that we attacked
the problems in this unit. They can be very effective tools.
• Working backwards – starting with the solution, you work backwards in the problem.
• Simplify the problem – you would recreate the problem, using simpler numbers in order to get an
understanding of how to solve the problem.
• Act out the problem – sometimes running through the problem in your mind, with you actually
dealing with the situation being described, can help you “see” the necessary steps to solve the
problem.
• Guess and check – this is effective on multiple-choice problems. You can take the answers and
begin plugging them into the problem until you find one that works. It is very effective when used
to prove/disprove all answers.

TAKS Practice 1
Adrian purchased 3 sheet sets for $108.63. If each set cost $34 before tax, what tax rate did he pay
on the 3 sheet sets?
A 5.75%
B 6.5%
C 7.25%
D 8%
Question: I need to…
Information:
Plan:
Solve:

TAKS Practice 2
Jesse is practicing free throws. Out of 18 attempts, he missed 3 baskets. At this rate, how many free
throws should Jesse expect to complete if he makes a total of 72 attempts?
A 12
B 18
C 60
D 64
Question: I need to…
Information:
Plan:
Solve:
Read this one
carefully!
Maybe a T-
Chart will help
you.

Now I want to see if
you can come up
with your own
problem!

Create Your Own
• You will now create your own real-life problem. It should be a problem that is complicated enough
that someone would have to come up with a good plan for solving it. In other words, don’t make it
too easy.
• You will have to show the solution and explain the steps for solving it, though – so make sure that
the problem makes sense and is logical.
• Create the problem so that it looks like the ones that you see on your TAKS test. Include answer
choices. Try to make the three wrong answers so that they will make sense to someone that tries
to solve the problem wrong.
• Show an explanation of what you did to solve it and explain why you chose those steps and/or
procedures.
• Finally, you will show what the correct solution is.
Let’s see if they can
solve this one!

Your Word Problem
Write your problem here
Show the work to solve the problem here Put the answer choices here. Circle
the answer choice that is correct.
A
B
C
D

Congratulations!
You have completed this module. Turn this
module in to receive feedback on your work.