1 of 10 · rationals-c01
−7 + 12 = ?
- Different signs mean compare the distances from zero: 7 and 12.
- The positive distance is larger, so subtract 12 − 7 = 5 and keep the positive sign.
- So −7 + 12 = 5.
- Check: 5 − 12 = −7.
Answer: 5
Solution key: The numbers between, and the ones below zero
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Camp · The numbers between, and the ones below zero
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Start with the main idea.
2 of 10 · rationals-c02
4 − 9 = ?
- Subtracting 9 from 4 moves 9 steps left on a number line.
- Four steps reach 0, leaving 9 − 4 = 5 steps to the left of 0.
- That lands on −5, so 4 − 9 = −5.
- Check: −5 + 9 = 4.
Answer: -5
3 of 10 · rationals-c03
2 − (−6) = ?
- The question subtracts a NEGATIVE: 2 − (−6).
- Subtracting −6 undoes a move left by 6, so it is the same as adding 6.
- 2 + 6 = 8.
- The answer is 8. Check: 8 + (−6) = 2.
Answer: 8
4 of 10 · rationals-c04
(−3)(−5) = ?
- The touching parentheses mean multiplication: (−3) × (−5).
- For MULTIPLICATION, two negative factors give a positive product.
- Multiply the sizes: 3 × 5 = 15.
- The answer is positive 15. This sign rule does not mean adding two negatives is positive.
Answer: 15
5 of 10 · rationals-c05
(−3)(5) = ?
- The parentheses mean (−3) × 5.
- For multiplication, one negative and one positive factor give a negative product.
- Multiply the sizes: 3 × 5 = 15.
- Apply the sign: −15. Check: −15 ÷ 5 = −3.
Answer: -15
6 of 10 · rationals-c06
−18 ÷ 3 = ?
- Divide −18 into 3 equal groups.
- A negative divided by a positive gives a negative quotient.
- Divide the sizes: 18 ÷ 3 = 6.
- The answer is −6. Check: −6 × 3 = −18.
Answer: -6
7 of 10 · rationals-c07
1/2 + 1/4 = ?
- To add fractions, use the same-size pieces: denominator 4.
- 1/2 = 2/4, because multiplying the top and bottom by 2 keeps the value the same.
- Add numerators and keep the denominator: 2/4 + 1/4 = 3/4.
- 3 ÷ 4 = 0.75, so 3/4 or 0.75 is the answer.
Answer: 0.75
8 of 10 · rationals-c08
0.7 + 0.45 = ?
- Line up decimal places so tenths add to tenths and hundredths to hundredths.
- Write 0.7 as 0.70; the extra zero does not change its value.
- 70 hundredths + 45 hundredths = 115 hundredths.
- 115 hundredths = 1.15. Check: 1.15 − 0.45 = 0.70.
Answer: 1.15
9 of 10 · rationals-c09
−2.5 + 1 = ?
- Start at −2.5. Adding positive 1 moves one unit right.
- That is not enough to reach 0, so the answer is still negative.
- Subtract the sizes: 2.5 − 1 = 1.5.
- The answer is −1.5. Check: −1.5 − 1 = −2.5.
Answer: -1.5
10 of 10 · rationals-c10
3/5 − 1/5 = ?
- Both fractions already have fifths as their pieces.
- Subtract numerators, not denominators: (3 − 1)/5 = 2/5.
- 2 and 5 share no factor greater than 1, so 2/5 is simplified.
- 2 ÷ 5 = 0.4. Either 2/5 or 0.4 names the answer.
Answer: 0.4
The numbers between, and the ones below zero · Camp
1 of 10 · rationals-c01
−7 + 12 = ?
- Different signs mean compare the distances from zero: 7 and 12.
- The positive distance is larger, so subtract 12 − 7 = 5 and keep the positive sign.
- So −7 + 12 = 5.
- Check: 5 − 12 = −7.
Answer: 5
2 of 10 · rationals-c02
4 − 9 = ?
- Subtracting 9 from 4 moves 9 steps left on a number line.
- Four steps reach 0, leaving 9 − 4 = 5 steps to the left of 0.
- That lands on −5, so 4 − 9 = −5.
- Check: −5 + 9 = 4.
Answer: -5
3 of 10 · rationals-c03
2 − (−6) = ?
- The question subtracts a NEGATIVE: 2 − (−6).
- Subtracting −6 undoes a move left by 6, so it is the same as adding 6.
- 2 + 6 = 8.
- The answer is 8. Check: 8 + (−6) = 2.
Answer: 8
4 of 10 · rationals-c04
(−3)(−5) = ?
- The touching parentheses mean multiplication: (−3) × (−5).
- For MULTIPLICATION, two negative factors give a positive product.
- Multiply the sizes: 3 × 5 = 15.
- The answer is positive 15. This sign rule does not mean adding two negatives is positive.
Answer: 15
5 of 10 · rationals-c05
(−3)(5) = ?
- The parentheses mean (−3) × 5.
- For multiplication, one negative and one positive factor give a negative product.
- Multiply the sizes: 3 × 5 = 15.
- Apply the sign: −15. Check: −15 ÷ 5 = −3.
Answer: -15
6 of 10 · rationals-c06
−18 ÷ 3 = ?
- Divide −18 into 3 equal groups.
- A negative divided by a positive gives a negative quotient.
- Divide the sizes: 18 ÷ 3 = 6.
- The answer is −6. Check: −6 × 3 = −18.
Answer: -6
7 of 10 · rationals-c07
1/2 + 1/4 = ?
- To add fractions, use the same-size pieces: denominator 4.
- 1/2 = 2/4, because multiplying the top and bottom by 2 keeps the value the same.
- Add numerators and keep the denominator: 2/4 + 1/4 = 3/4.
- 3 ÷ 4 = 0.75, so 3/4 or 0.75 is the answer.
Answer: 0.75
8 of 10 · rationals-c08
0.7 + 0.45 = ?
- Line up decimal places so tenths add to tenths and hundredths to hundredths.
- Write 0.7 as 0.70; the extra zero does not change its value.
- 70 hundredths + 45 hundredths = 115 hundredths.
- 115 hundredths = 1.15. Check: 1.15 − 0.45 = 0.70.
Answer: 1.15
9 of 10 · rationals-c09
−2.5 + 1 = ?
- Start at −2.5. Adding positive 1 moves one unit right.
- That is not enough to reach 0, so the answer is still negative.
- Subtract the sizes: 2.5 − 1 = 1.5.
- The answer is −1.5. Check: −1.5 − 1 = −2.5.
Answer: -1.5
10 of 10 · rationals-c10
3/5 − 1/5 = ?
- Both fractions already have fifths as their pieces.
- Subtract numerators, not denominators: (3 − 1)/5 = 2/5.
- 2 and 5 share no factor greater than 1, so 2/5 is simplified.
- 2 ÷ 5 = 0.4. Either 2/5 or 0.4 names the answer.
Answer: 0.4