Why it matters on the HSPT
Two small topics share this lesson because each one is worth about one question and each is pure recognition. The test shows an equation such as 4 times the quantity 2 plus 5 equals 4 times 2 plus 4 times 5 and asks which property it shows. Or it shows a number with a small 2 after it and asks what it equals in ordinary numbers. Neither takes arithmetic skill. Both take one careful read and a name you have memorized. The properties also earn their keep on other questions, because they are the reason mental-math shortcuts work.
The rules
Commutative property. Order does not matter for addition or multiplication. 6 plus 9 equals 9 plus 6, and 4 times 7 equals 7 times 4. The numbers swap places. It does not hold for subtraction or division: 9 minus 6 is not 6 minus 9.
Associative property. Grouping does not matter for addition or multiplication. The quantity 2 plus 6, plus 7 equals 2 plus the quantity 6 plus 7. Only the parentheses move; the numbers stay in the same order. As with the commutative property, it fails for subtraction and division.
Telling them apart. Read the two sides and ask one question: did any number change position? If yes, commutative. If no, and only the parentheses moved, associative. An equation like 3 plus the quantity 7 minus 5 equals the quantity 7 minus 5, plus 3 is commutative, because the whole group in parentheses moved from second place to first, even though its inside did not change.
Identity properties. Adding 0 leaves a number alone: 8 plus 0 is 8. Multiplying by 1 leaves a number alone: 10 times 1 is 10. Zero is the additive identity and one is the multiplicative identity. A related fact the test likes: any number times 0 is 0, which is not an identity property, just the zero property of multiplication.
Distributive property. Multiplying a sum is the same as multiplying each part and adding: 3 times the quantity 2 plus 4 equals 3 times 2 plus 3 times 4, which is 6 plus 12, which is 18. It works with subtraction inside too: 4 times the quantity 6 minus 2 equals 24 minus 8, which is 16. Spot it by the multiplier that appears twice on the expanded side.
Using them for speed. These are not just names. To multiply 4 times 7 times 25, reorder to 4 times 25 times 7, which is 100 times 7. That is commutative and associative at work. To find 6 times 42, split it: 6 times 40 plus 6 times 2 is 240 plus 12, which is 252. That is distributive. To find half of 9 times 8, halve the 8 first, because 4 times 9 is easier than 4.5 times 8.
Base 10, said out loud. The number 4,372 means 4 thousands, 3 hundreds, 7 tens, and 2 ones. Each place is worth ten times the place to its right, which is why it is called base 10, and no digit is ever bigger than 9. Writing it as 4 times 1,000 plus 3 times 100 plus 7 times 10 plus 2 times 1 is called expanded form, and the test sometimes writes those place values as powers of 10.
Base 2. A number written in base 2 uses only the digits 0 and 1, and each place is worth twice the one to its right: from the right, 1, 2, 4, 8, 16, 32. A small 2 written after the number marks it as binary. To read one, add up the place values under each 1. The binary number 10011 is 16 plus 0 plus 0 plus 2 plus 1, which is 19.
Writing a number in base 2. Take the largest power of 2 that fits, subtract it, and repeat, writing a 1 for each power you used and a 0 for each you skipped. For 13: 8 fits, leaving 5; 4 fits, leaving 1; 2 does not fit, so write 0; 1 fits. From the 8s place down, that is 1, 1, 0, 1, so 13 is 1101 in base 2. Check by adding back: 8 plus 4 plus 1 is 13.
Worked examples
Example 1
Which property is shown by 5 + (8 + 3) = (5 + 8) + 3?
- A.commutative property of addition
- B.associative property of addition
- C.distributive property
- D.identity property of addition
Show answer
Answer: B: associative property of addition
The numbers 5, 8, and 3 stay in the same order on both sides. Only the parentheses moved, so this is the associative property. It would be commutative only if numbers had switched places, such as 5 plus 8 equals 8 plus 5.
Example 2
Which property is shown by 9 × (4 - 1) = (4 - 1) × 9?
- A.commutative property of multiplication
- B.associative property of multiplication
- C.distributive property
- D.identity property of multiplication
Show answer
Answer: A: commutative property of multiplication
The group in parentheses moved from the right of the multiplication sign to the left, so the two factors changed order. That is commutative. The inside of the parentheses did not change and nothing was multiplied out, so it is not distributive.
Example 3
Which equation shows the distributive property?
- A.7 × 1 = 7
- B.7 × (10 + 2) = 70 + 14
- C.7 + (10 + 2) = (7 + 10) + 2
- D.7 × 10 × 2 = 2 × 10 × 7
Show answer
Answer: B: 7 × (10 + 2) = 70 + 14
Only choice B multiplies a sum by multiplying each part: 7 times 10 plus 7 times 2. Choice A is the identity property of multiplication, choice C is associative, and choice D is commutative.
Example 4
Which is the fastest way to find 8 × 35 using a property?
- A.8 × 30 + 8 × 5
- B.8 × 3 × 5
- C.8 × 40 - 5
- D.8 + 30 + 5
Show answer
Answer: A: 8 × 30 + 8 × 5
Split 35 into 30 plus 5 and distribute: 8 times 30 is 240 and 8 times 5 is 40, so the product is 280. Choice B is wrong because 3 times 5 is 15, not 35; you cannot split a number into factors by breaking its digits apart. Choice C starts well with 8 times 40 but then subtracts 5 instead of 8 times 5.
Example 5
What is the base 2 number 1011 written in base 10?
- A.11
- B.13
- C.1,011
- D.22
Show answer
Answer: A: 11
From the right, the places are worth 1, 2, 4, and 8. The digits are 1, 0, 1, 1 from the left, so the 8s place and the 2s place and the 1s place are on and the 4s place is off. Add 8 plus 2 plus 1, which is 11. Choice C reads the binary digits as an ordinary number.
Example 6
Which is the number 15 written in base 2?
- A.1010
- B.1111
- C.1001
- D.11110
Show answer
Answer: B: 1111
Fifteen is 8 plus 4 plus 2 plus 1, every power of 2 up to 8, so every place from 8 down is on: 1111. Check choice D: it would be 16 plus 8 plus 4 plus 2, which is 30, exactly double, because adding a 0 on the end of a binary number doubles it just as adding a 0 on the end of a base 10 number multiplies by ten.
Common mistakes
Calling every parenthesis move associative. If the group in parentheses changed sides of the operation, the order changed, and that is commutative. Look at where the numbers are, not at whether parentheses exist.
Applying commutative to subtraction. 10 minus 4 is not 4 minus 10. The properties in this lesson hold for addition and multiplication only.
Distributing over the wrong operation. Multiplication distributes over addition and subtraction. It does not distribute over another multiplication: 2 times the quantity 3 times 4 is 24, not 6 times 8.
Reading a binary number as base 10. The digits 1011 with a small 2 after them mean eleven, not one thousand eleven. The subscript is the whole question.
Forgetting a zero place. When writing a number in base 2, every skipped power of 2 still gets a 0. Thirteen is 1101, four digits, because the 2s place is used as a placeholder even though it is off.
Practice
Property questions appear as arithmetic items, and the mental-math shortcuts they justify speed up every pre-algebra and non-geometric comparison question. Expanded form with powers of 10 is covered in the exponents lesson, and place value in base 10 is the opening rule of the decimals lesson.