Why it matters on the HSPT
Negative numbers and the order of operations are the two rules that turn an easy arithmetic question into a wrong answer. The test uses them directly, in items like 3 minus negative 5, and it hides them in comparisons, where 2 plus 3 times 4 sits next to the same numbers in parentheses. A number series can run into negatives as well. None of this is hard math. It is a small set of rules that must be followed in order, every time, under a clock.
The rules
The number line. Negative numbers sit to the left of zero. Negative 7 is less than negative 3, because it is farther left, even though 7 is bigger than 3. Every comparison of negatives comes back to this picture.
Adding with signs. Same signs: add the sizes and keep the sign. Negative 4 plus negative 6 is negative 10. Different signs: subtract the smaller size from the larger and keep the sign of the larger. Negative 9 plus 4 is negative 5. Seven plus negative 3 is 4.
Subtracting. Change subtraction to adding the opposite, then use the adding rules. Three minus negative 5 becomes 3 plus 5, which is 8. Negative 2 minus 6 becomes negative 2 plus negative 6, which is negative 8. Two minus signs in a row become a plus.
Multiplying and dividing. Same signs give a positive; different signs give a negative. Negative 3 times negative 4 is 12. Negative 3 times 4 is negative 12. Negative 20 divided by 5 is negative 4. Count the negatives in a long product: an even count is positive, an odd count is negative.
Absolute value. The two bars around a number mean distance from zero, so the answer is never negative. The absolute value of negative 8 is 8. Do what is inside the bars first, then drop the sign: the absolute value of 3 minus 10 is the absolute value of negative 7, which is 7.
Order of operations. Parentheses, then exponents, then multiply and divide left to right, then add and subtract left to right. Two things trip students. First, multiply and divide are one level, so 12 divided by 3 times 2 is 4 times 2, which is 8, not 12 divided by 6. Second, a negative sign in front of a squared number is applied after the square unless parentheses say otherwise: negative 3 squared, written with the minus outside, is negative 9, while negative 3 in parentheses squared is 9.
Even and odd, and what they do. An even plus an even is even, an odd plus an odd is even, an even plus an odd is odd. An even times anything is even. The test uses these in questions that ask which choice must be odd without giving numbers.
Worked examples
Example 1
−6 + 10 =
- A.−16
- B.−4
- C.4
- D.16
Show answer
Answer: C: 4
Different signs, so subtract the sizes: 10 minus 6 is 4, and the sign belongs to the larger size, 10, which is positive. The answer is 4. Choice A adds the sizes, which is only right when the signs match.
Example 2
5 − (−8) =
- A.−13
- B.−3
- C.3
- D.13
Show answer
Answer: D: 13
Subtracting a negative is adding a positive: 5 plus 8 is 13. Choice B, negative 3, comes from treating the problem as 5 minus 8 and ignoring the second minus sign.
Example 3
(−4) × (−3) × (−2) =
- A.−24
- B.−9
- C.9
- D.24
Show answer
Answer: A: −24
Three negatives, an odd count, so the product is negative. The sizes multiply to 24, so the answer is negative 24. Choice D forgets to count the third negative.
Example 4
2 + 3 × 4 − 6 ÷ 2 =
- A.7
- B.11
- C.13
- D.17
Show answer
Answer: B: 11
Multiply and divide first: 3 times 4 is 12, and 6 divided by 2 is 3. Now add and subtract left to right: 2 plus 12 is 14, minus 3 is 11. Choice D, 17, comes from working straight left to right.
Example 5
|3 − 9| + |−4| =
- A.−10
- B.2
- C.10
- D.16
Show answer
Answer: C: 10
Inside the first bars, 3 minus 9 is negative 6, and its absolute value is 6. The absolute value of negative 4 is 4. Six plus 4 is 10. Choice B, 2, subtracts instead of adding the second value.
Example 6
Which of the following is the smallest?
- A.−2
- B.−7
- C.0
- D.3
Show answer
Answer: B: −7
On the number line, negative 7 is farthest to the left, so it is the smallest. It is tempting to think negative 2 is smaller because 2 is a smaller number, but a bigger negative is farther from zero and therefore less.
Example 7
If n is an odd number, which of the following must be even?
- A.n + 2
- B.3n
- C.n + 1
- D.n − 2
Show answer
Answer: C: n + 1
An odd plus an odd is even, and 1 is odd, so n plus 1 is even. Adding or subtracting 2 keeps a number odd, and 3 times an odd number is odd. Testing with n equals 5 confirms it: 6, 15, 6, 3, and only the n plus 1 choice is even for every odd n.
Common mistakes
- Working left to right through a mixed expression. Multiplication and division come before addition and subtraction no matter where they sit.
- Treating multiplication as coming before division. They are one level, worked left to right.
- Reading a bigger negative as a bigger number. Negative 10 is less than negative 1.
- Dropping the second minus sign. Five minus negative 3 is 8, not 2.
- Letting absolute value go negative. The bars always give zero or a positive number.
Practice
These rules sit inside arithmetic and pre-algebra items, appear in non-geometric comparison where order of operations is the whole trick, and show up in a number series that crosses below zero.