Algebra – 2.15

Building Equations

GUESS-CHECK-GENERALIZE

• This is a method that we use for solving word problems.

• GUESS: make guesses (they don’t have to be good guesses)

• CHECK: use math to check if your guess is correct

• GENERALIZE: use the same steps to check if some guess ‘x’ is correct, and use this to solve

Remember this problem?

• Vanessa works 32 hours per week at her job. Her boss offers her a full-time position (40 hours a week) and a $2-per-hour raise. She says, “Super! Now I’ll make $200 more a week!” What was her hourly wage before the raise?

GUESS-CHECK• First, guess what the hourly wage was before

the raise.• *You’re not trying to get the right answer right

now, you’re just trying to learn how to check a guess.

• Check your guess. Keep track of the steps that you take.

• Repeat with another guess. Keep guessing until the checking becomes automatic.

GENERALIZE• Make your guess a variable.• Take the same steps that you used before.

Some practice• Ancient Egyptian mathematicians used the concept of

“false position” to solve equations. You guess a convenient answer and then adjust it to find the correct answer.

• Example: A number plus one fourth of itself equals 210. What is the number?

• A convenient guess is 4. Why is this convenient?• Suppose you guess 4. What is the result of adding

one fourth of 4 to itself?• Guess 12. How does the result of the same

calculation compare to the result with 4?• Use the GCG method to solve the problem.

Last summer, Katie mowed lawns to earn money. She mowed 35 lawns per week and charged $6 per lawn. This summer, Katie wants to earn an additional $150 per week.

She will raise her price to $8 and find more customers.

• If Katie finds 4 new customers and still keeps her former customers, will she earn an additional $150 a week?

• If Katie finds 13 new customers, will she earn an additional $150 a week?

• If Katie finds 8 new customers, will she earn an additional $150 a week?

• Use GCG to build an equation for finding the number of new customers Katie needs to earn $150 extra.

• Solve the equation.

Suppose Katie charges $9 per lawn. How many new customers must

she find in order to make an extra $150 a week?

In 36 years, Anna will be five times as old as she is today.

How old is Anna?