Integers UPCAT Reviewer: Signed Numbers, Context, and Mastery Practice

TEACHER ABI UPCAT MATHEMATICS

Integers and Signed-Number Reasoning

Interpret signs as direction and position, then solve multi-step integer problems without losing meaning in mechanical sign rules.

5-10 minute lesson27 original questionsAdaptive practiceSaves progress

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Integers

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A sign describes position or direction

Positive and negative values often describe opposite directions: above and below, deposit and withdrawal, gain and loss, rise and fall. Keep that meaning visible while calculating.

Change = final−initialDistance between a and b: |a−b|An even number of negative factors gives a positive product; an odd number gives a negative product.

Addition

Same signs: add magnitudes and keep the sign. Different signs: subtract magnitudes and keep the sign of the larger magnitude.

Subtraction

Rewrite a−b as a + (−b), then use addition reasoning.

Multiplication and division

Same signs give positive; different signs give negative.

Absolute value

|x| is distance from zero, so it is never negative.

DO IT FAST

Direction first, arithmetic second

1. Name zero or the reference level.

2. Translate each change: risedeposit/gain is positive; fallwithdrawal/loss is negative.

3. Combine in chronological order.

4. Interpret the final sign. A negative result may be a valid debt, depth, or below-zero temperature.

Why it works

Students often perform the arithmetic correctly but attach the wrong sign because they stopped tracking what the sign represents.

WORKED EXAMPLES

Five forms you should recognize

1. Subtract a negative

−7−(−12) = −7 + 12 = 5

Removing a negative has the same effect as adding its positive opposite.

2. Distance

|5−(−8)| = 13

Distance is nonnegative even when one location is negative.

3. Temperature change

4−(−6) = 10°C

Crossing zero means the change is larger than either endpoint’s magnitude alone.

4. Net movement

3−8 + 5−2 = −2

Each upward or downward movement is entered with its direction.

5. Product sign

Three negative factors give a negative product because the number of negative factors is odd.

COMMON TRAPS

Check before you commit

• Thinking the integer with the larger magnitude is always greater
• Treating subtraction of a negative as subtraction of its magnitude
• Forgetting parentheses around a negative base with an exponent
• Reporting a negative distance
• Dropping the initial signed value in a change problem
• Assuming every negative result is an error

FIVE-FORM SKILL CHECK

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One original question in each form recommends your next step. It does not yet verify mastery.

CHOOSE YOUR PRACTICE

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Foundations

Build the core procedure with immediate explanations.

Core Practice

Use mixed forms with less scaffolding.

UPCAT-Style Transfer

Apply the competency in unfamiliar representations.

FRESH MASTERY CHECK

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QUICK ANSWERS

Integers and Signed-Number Reasoning FAQ

Why is −2 greater than −9?

−2 lies farther right on the number line and is closer to zero.

Is (−3)² the same as −3²?

No. (−3)² = 9, while −3² means −(3²) = −9 under standard order of operations.

When should I use absolute value?

Use it for distance or magnitude when direction does not matter.

RELATED COMPETENCIES

Continue your mathematics review.

Number PropertiesAbsolute ValueLinear EquationsCoordinate Geometry

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