Ray's New Elementary Algebra · Arts. 169–171 · Unit 5: Chapter V: Supplement to simple equations
34. General problems
Goal: Your student solves problems whose answers are formulas, and equations in which the square of the unknown cancels out.
You'll need
- The book, open to the page
- Pencil and plenty of paper (or a whiteboard)
- A card to cover printed answers
What's in the book
Three workers in pairs (a general solution), the pasture problem, ten general problems with formula answers, and equations where x² disappears. Answers are printed.
The lesson, step by step
Warm-up
Divide 20 into two parts so that one is 4 times the other. (16 and 4.)
A general problem
General problem 1: divide a into two parts so that one is n times the other. Let the parts be x and nx; x + nx = a, so x = a/(n + 1). Test with the warm-up: a = 20, n = 4 gives 4.
Guided practice
Your student does general problems 2 and 3 while you watch, then tests each formula with small numbers.
The pasture problem
Read Art. 170 together slowly. It is famous and tricky: the grass keeps growing. Say: The key step is finding that a week's growth on an acre is one twelfth of the first crop. The answer is 108 sheep.
When x² disappears
Read Art. 171, examples 1 and 2. In example 2, multiplying out gives x² on both sides, which cancel, leaving a simple equation (x = 8).
Independent practice
General problems 4 to 7, and Art. 171 exercises 3 to 5 and 8.
Wrap-up
Ask your student to check one formula answer with numbers they choose.
Tip
The pasture problem (Art. 170) is optional reading. If it confuses, skip it; nothing later depends on it.
Answers to the book's problems
The book prints the answer after each exercise.
Worked out for this site from the scan. If a number in your copy differs, trust the book. Every answer is in the Key ↗.