Depth · one concept from first principles · step 2 of 4
Recover each letter's sign from a quadrant
The point lies in Quadrant III. Which quadrant contains the point ?
Number system
Your answer
How to solve
Strategy Organize Information in More Ways — Turn the quadrant into two sign facts, then list the only two sign patterns a negative product allows and keep the one that fits the other fact. Once the signs of a and b are known, every coordinate of the new point has a definite sign.
1STEP 1
Translate the quadrant into signs
Quadrant III means the x-coordinate and the y-coordinate are both negative.
b-a ＜ 0, ab ＜ 0
A quadrant is just a pair of signs.
6.NS.C.6Organize Information In More Ways2STEP 2
List the sign patterns for a negative product
A product is negative exactly when its two factors have opposite signs, so either a is positive and b negative, or the other way round.
ab ＜ 0 → (a ＞ 0, b ＜ 0) or (a ＜ 0, b ＞ 0)
Only two cases are possible, so we can test both.
7.NS.A.2Make A Systematic List3STEP 3
Keep the case that fits b-a ＜ 0
b-a ＜ 0 means b ＜ a. If a were negative and b positive, b would be the larger one, which breaks b ＜ a. So a ＞ 0 and b ＜ 0.
b ＜ a → a ＞ 0, b ＜ 0
A positive number is always bigger than a negative one.
7.NS.A.2Make A Systematic Listis positive and is negative.
Why?
Like signs multiply to a positive and unlike signs to a negative, so means and have opposite signs.
🧱Signs of products and quotientsMultiplying by a negative number flips a number to the other side of zero, so two flips land back on the positive side: for two nonzero numbers, a product or quotient with like signs is positive and one with unlike signs is negative (zero itself has no sign).
Why?
negative with positive is ruled out, because it breaks .
Why?
says , because sliding and both up by keeps their order.
🧱Shift both by the same amount and the difference holdsSlide two points the same distance along the line and the gap between them never changes.
Why?
A negative sits below and a positive above it, so would be the larger one.
🧱Transitivity of orderIf the first is smaller than the second and the second smaller than the third, the first is smaller than the third, so pairwise comparisons chain into one full order.
Why?
Of the two sign patterns only , is left.
🧱Rule out the rest and the last one is forcedIf every possibility but one has been ruled out, the survivor must be the answer — no further checking is needed.
4STEP 4
Find the signs of the new coordinates
b ≠ 0, so b² ＞ 0 and -b² is negative. b ＜ 0 makes -3b positive, so a-3b is positive plus positive: positive.
-b² ＜ 0, a-3b = a+(-3b) ＞ 0
Squaring removes the sign; multiplying a negative by -3 makes it positive.
7.NS.A.2Organize Information In More Ways5STEP 5
Read off the quadrant
A negative x-coordinate with a positive y-coordinate is Quadrant II.
(-, +) → Quadrant II
Left of the y-axis and above the x-axis.
6.NS.C.6Organize Information In More WaysAnswer
Quadrant II
Try a=1, b=-1: the first point is (-2, -1), which is in Quadrant III, and the second is (-1, 4), which is in Quadrant II.
Takeaway
A quadrant is a pair of signs -- read the signs and the letters give themselves away.
- Translate the quadrant into signs
- List the sign patterns for a negative product
- Keep the case that fits b-a ＜ 0
- Find the signs of the new coordinates
- Read off the quadrant
Where next?
Level upsuggested
Another one like thissuggested
Step back and solidifysuggested
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