Transcript Classic Math Problems With Age

Classic Math Problems
with Numbers
Today’s Learning Goal
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We will learn how to read algebra word problems to
help us solve them.
We will apply the steps to reading an algebra word
problem to solving problems involving numbers.
We will continue to solve systems of equations and
begin to determine which strategy would be the
easiest to use.
Reading Algebra Word Problems
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Consider the following classic algebra problem:
There are two numbers whose sum is 72. One
number is twice the other. What are the numbers?
The first step to solving an algebra word problem is
to read the problem all the way through to see what
type of problem it is and what it is about.
The second step is to re-read the question at the end
of the problem!
Often times, the question lets you know what you are
solving for and what the unknowns (variables) are.
Sometimes two or three things need to be found.
There are two numbers whose
sum is 72. One number is twice
the other. What are the numbers?
Establishing the Variables
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The third step is to establish what the unknowns
represent.
If you have to find more than one quantity or
unknown, x would represent the smallest unknown.
For this problem, the unknowns are two different
numbers. So, what would you state x and y to
represent for this problem?
x -- smaller number
y -- larger number
x -- smaller number
y -- larger number
There are two numbers whose
sum is 72. One number is twice
the other. What are the numbers?
Establishing the Variables
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The fourth step is to read the problem again a piece
at a time. The other parts to the problem will give
you equation information.
What equation could we write
for the first statement?
What could we write for the
second statement?
x + y = 72
y = 2x
So, we have a system of equations. Which method
do you think would be the easiest to use to solve
this problem? Great…the substitution method!
x + y = 72
y = 2x
There are two numbers whose
sum is 72. One number is twice
the other. What are the numbers?
Establishing the Variables
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If we used the method of substitution
on the equations above, what would
be the resulting equation?
When we simplify the left-hand-side,
what is the resulting equation?
What would you do to solve
this equation?
What is the smaller number of the two?
If we plugged x = 24 into either
equation, we would get the y-value.
x + 2x = 72
3x = 72
3
3
x = 24
y = 2(24)
y = 48
Consecutive Integer Problems
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Another classic algebra problem involves consecutive
integers. Consecutive means one after another.
What is the difference between any two consecutive
integers?
Nice…1 is the difference between
consecutive integers.
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To represent consecutive
integers with variables, we
usually do the following:
x = 1st integer
x + 1 = 2nd integer
x + 2 = 3rd integer
What would represent the third consecutive integer?
Consecutive Integer Problems
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Consecutive even integers means one
even number after another.
What is the difference between any two consecutive
even integers?
Nice…2 is the difference between
st even integer
x
=
1
consecutive even integers.
nd even integer
x
+
2
=
2
To represent consecutive
even integers with variables, x + 4 = 3rd even integer
we usually do the following:
What would represent the third consecutive even integer?
Consecutive Integer Problems
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Consecutive odd integers means one odd
number after another.
What is the difference between any two consecutive
odd integers?
Nice…2 is the difference between
st odd integer
x
=
1
consecutive odd integers.
nd odd integer
x
+
2
=
2
To represent consecutive
odd integers with variables, x + 4 = 3rd odd integer
we usually do the following:
What would represent the third consecutive odd integer?
Another Problem
Involving Numbers
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Consider the following classic algebra problem:
Find three consecutive even integers such that the
largest is three times the smallest.
After reading the problem once, what are we trying
to figure out?
Good…we are trying to find three consecutive even integers.
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Some of you might be able to figure out what those
numbers are with a guess-and-check strategy. But,
we are trying to learn how to use algebra to solve
problems.
Another Problem
Involving Numbers
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Find three consecutive
even integers such that
the largest is three
times the smallest.
How can we represent the consecutive even integers
with variables?
Perfect…let
x = 1st even integer
x + 2 = 2nd even integer
x + 4 = 3rd even integer
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Now, re-read the problem. What equations can we
write given the information in the problem?
Excellent…
x + 4 = 3x
largest
3*smallest
Another Problem
Involving Numbers
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Suppose we subtracted x
from both sides, what is the
resulting equation?
What is the x-value to the
solution?
Find three consecutive
even integers such that
the largest is three
times the smallest.
x + 4 = 3x
-x
-x
4 = 2x
2 2
2=x
Great…x = 2.
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So, 2 is the first even integer. What would be
the other two?
Fabulous…
x+
2
42
2
x+
6 4
So, 2, 4, and 6 are three even consecutive integers
such that the largest is three times the smallest.
Partner Work
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You have 20 minutes to work on the following
questions with your partner.
For those that finish early
Solve the following problem:
1. In a 3-digit number, the hundreds digit is four more
than the units digit and the tens digit is twice the
hundreds digit. If the sum of the digits is 12, find
the three digits. Write the number.
Big Ideas from Today’s Lesson
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There are four steps to solving algebra word
problems.
1)
2)
3)
4)
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Read the problem through one time.
Re-read the question to get the variables (unknowns).
Establish what the unknowns represent.
Re-read the other parts of the problem to get the
equations.
Consecutive means one after another.
Homework
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Complete Homework Worksheet.