Classic Math Problems with 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 number larger 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 number larger 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 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, , 103 To represent consecutive integers with variables, we usually do the following: x = 1 st integer x + 1 = 2 nd integer x + 2 = 3 rd 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, , 208 To represent consecutive even integers with variables, we usually do the following: x = 1 st even integer x + 2 = 2 nd even integer x + 4 = 3 rd 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, , 407 To represent consecutive odd integers with variables, we usually do the following: x = 1 st odd integer x + 2 = 2 nd odd integer x + 4 = 3 rd 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 = 1 st even integer x + 2 = 2 nd even integer x + 4 = 3 rd even integer Excellent…x + 4 = 3x largest 3*smallest

Another Problem Involving Numbers Suppose we subtracted x from both sides, what is the resulting equation? x + 4 = 3x -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2 22 2 = x So, 2 is the first even integer. What would be the other two? Fabulous…x + 2x 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 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 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.