Quantitative comparison and number reasoning
| English | 中文 | Pinyin |
|---|---|---|
| constraint/kənˈstreɪnt/ | 约束条件 | yuē shù tiáo jiàn |
| indeterminate/ˌɪndɪˈtɜːmɪnət/ | 不能确定的 | bù néng què dìng de |
A decision before an answer
- When x²=9, comparing x with zero has two possible answers. One convenient substitution can hide the other.
- Your goal: Compare quantities across all allowed values.
Read the relationship
- Quantitative Comparison has four fixed relationships: A greater, B greater, equal, or cannot determine.
- Use arithmetic, ratios, divisibility and powers.
x²=4; compare x and 1.
x=2 makes A greater; x=−2 makes B greater.
Use the defining rule
- Check all constraints. A variable can be negative, zero or fractional unless the stem restricts it.
- Recognise when the relationship cannot be determined.
x>0; compare x² and 0.
The square of a positive nonzero number is positive.
Check the conditions
- Use strategic substitutions to disprove a fixed relation. They can show indeterminacy but a few examples do not prove universality.
- Recognise when the relationship cannot be determined.
Given x²=9, x can be 3 or −3. Quantity A=x; B=0. A is greater for 3 and smaller for −3, so the relationship cannot be determined. If x is positive, only 3 is allowed and A is greater.
The greatest common divisor of 18 and 24 is ____.
Both divide by 6; no greater integer divides both.
Apply the task format
- Simplify both quantities without silently dividing by a value that could be zero or negative.
- Recognise when the relationship cannot be determined.
Figures are not necessarily drawn to scale. Given labels and constraints carry the evidence.
Which answer fits this case?
Compare quantities across all allowed values
A real variable is always a positive integer unless otherwise stated.
Real values can be negative, zero or fractional.
Keep the distinctions
- constraint 约束条件 — A condition limiting allowed values.
- indeterminate 不能确定的 — Not fixed by the given information.
- Compare quantities across all allowed values.
- Use arithmetic, ratios, divisibility and powers.
- Recognise when the relationship cannot be determined.
Match each term with its precise meaning in this lesson.
Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
The order follows the stated process; check each stage before the next.