Classic Math Problemswith Numbers

Today’s Learning Goal 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 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.

Establishing the Variables

There are two numbers whose sum is 72. One number is twice the other. What are the numbers?

The third step is to establish what the unknowns represent.

x --

y --

For this problem, the unknowns are two different numbers. So, what would you state x and y to represent for this problem?

If you have to find more than one quantity or unknown, x would represent the smallest unknown.

smaller numberlarger number

Establishing the Variables

There are two numbers whose sum is 72. One number is twice the other. What are the numbers?

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.

x --

y --

What equation could we write for the first statement?

smaller numberlarger number

x + y = 72

What could we write for the second statement?

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!

There are two numbers whose sum is 72. One number is twice the other. What are the numbers?

Establishing the Variables 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?

x + y = 72

What would you do to solve this equation?

y = 2x

What is the smaller number of the two?

x + 2x = 72 3x = 72

3 3

If we plugged x = 24 into either equation, we would get the y-value.

x = 24

y = 2(24)y = 48

Consecutive Integer Problems 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.

21, 22 -4, -3 102, 103

To represent consecutive integers with variables, we usually do the following:

x = 1st integerx + 1 = 2nd

integerx + 2 = 3rd integer

What would represent the third consecutive integer?

Consecutive Integer Problems 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 consecutive even integers.

2, 4 -68, -66 206, 208

To represent consecutive even integers with variables, we usually do the following:

x = 1st even integerx + 2 = 2nd even

integerx + 4 = 3rd even integer

What would represent the third consecutive even integer?

Consecutive Integer Problems 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 consecutive odd integers.

1, 3 -57, -55 405, 407

To represent consecutive odd integers with variables, we usually do the following:

x = 1st odd integerx + 2 = 2nd odd

integerx + 4 = 3rd odd integer

What would represent the third consecutive odd integer?

Another Problem Involving Numbers 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?

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.

Good…we are trying to find three consecutive even integers.

Another Problem Involving Numbers

Find three consecutive even integers such that the largest is three times the smallest.

How can we represent the consecutive even integers with variables?

Now, re-read the problem. What equations can we write given the information in the problem?

Perfect…let x = 1st even integer

x + 2 = 2nd even integer

x + 4 = 3rd even integer

Excellent… x + 4 = 3xlargest

3*smallest

Another Problem Involving Numbers Suppose we subtracted x

from both sides, what is the resulting equation?

x + 4 = 3x-x -x

4 = 2x What is the x-value to

the solution?

Find three consecutive even integers such that the largest is three times the smallest.

Great…x = 2.

2

22 = x

So, 2 is the first even integer. What would be the other two?

Fabulous… x + 2 x + 42 + 2 2 + 44 6 So, 2, 4, and 6 are three even consecutive integers

such that the largest is three times the smallest.

Partner Work You have 20 minutes to work on the following

questions with your partner.

For those that finish earlySolve 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 There are four steps to solving algebra word

problems.1) Read the problem through one time.2) Re-read the question to get the variables (unknowns).3) Establish what the unknowns represent.4) Re-read the other parts of the problem to get the

equations. Consecutive means one after another.

Homework Complete Homework Worksheet.