Ray's New Elementary Algebra · Arts. 172–173 · Unit 5: Chapter V: Supplement to simple equations
35. Negative answers and the discussion of a problem
Goal: Your student explains what a negative answer means and what happens to c/(a − b) when a − b or c is zero.
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
Negative solutions, with four short questions to reword so they make sense; then the discussion of x = c/(a − b) in five cases, including c/0 (infinity) and 0/0 (indeterminate).
The lesson, step by step
Warm-up
Ask: What number added to 5 gives 3? Let your student puzzle over it.
Negative solutions
Read Art. 172. The answer is −2: adding −2 is the same as subtracting 2.
1. A negative solution indicates some inconsistency or absurdity in the question from which the equation was derived.
2. When a negative solution is obtained, the question to which it is the answer may be so modified as to be consistent.
Guided practice
The book asks your student to reword these so the answer is positive:
2. What number must be subtracted from 20, that the remainder shall be 25 ? (x= — 5.)
3. What number must be added to 11, that the sum being multiplied by 5, the product shall be 40 ? (x= — 3.)
(What must be ADDED to 20 to give 25? What must be SUBTRACTED from 11?)
Discussion of a problem
Read Art. 173 and the five cases of x = c/(a − b). Try them with numbers: a = 8, b = 3, c = 20 (x = 4); a = 2, b = 5, c = 12 (x = −4).
Dividing by zero
Show c divided by 1, 0.1, 0.01, 0.001: the answers grow without limit. That is why the book writes c/0 as infinite. Then 0/0 can be any value at all, "the symbol of indetermination."
Wrap-up
Your student explains in two sentences why 0/0 has no single value.
Tip
Read p. 27 again (Observations on Addition and Subtraction) before this lesson, as the book itself suggests.