Why it matters on the HSPT
These three topics were not on the test a decade ago and still get only a question each, but each one is a free point for a student who has seen the idea once. Functions come up as "which table represents a function." Sets come up as a Venn diagram or a short list with a union or intersection symbol. Counting comes up as a word problem about combinations of shirts and pants or crusts and toppings. None of them needs arithmetic beyond multiplication, so the whole skill is recognizing what is being asked.
The rules
What a function is. A function is a rule that gives exactly one output for every input. In a table of inputs and outputs, that means no input may appear twice with different outputs. Outputs are allowed to repeat: the table with inputs 1, 2, 3 and outputs 5, 5, 5 is a function, because each input still has just one answer. The table with input 4 listed twice, once with output 7 and once with output 14, is not.
Checking a table fast. Scan the input column only. If every input is different, it is a function; stop there. If an input repeats, check whether its outputs match. Same input with the same output twice is still a function. Same input with two different outputs is not.
Ordered pairs. A list of pairs like (1, 3), (2, 5), (3, 5) is the same thing as a table, with the first number as input. The same rule applies: look for a repeated first number with different second numbers.
The vertical line test. For a graph, a function is any picture that no vertical line crosses more than once. A straight line that is not vertical passes. A circle fails, because a vertical line through it hits two points. The test rarely draws graphs for this, but the idea is the same as the table rule: one input, one output.
Evaluating a rule. If a question gives a rule in words, such as "the output is 3 times the input, minus 1," just substitute. An input of 5 gives 3 times 5 minus 1, which is 14. Read the rule twice to get the order of the operations right.
Sets. A set is a collection of items, written in curly brackets like the set of odd numbers under 10. A Venn diagram shows two sets as overlapping circles. Items in both sets go in the overlap. Items in only one set go in that circle's outer part.
Union and intersection. The union of A and B, written with a symbol shaped like a U, is everything that is in A or in B or in both. List every item once, even if it appears in both sets. The intersection, written with the same symbol upside down, is only the items in both. Memory hook: the union symbol looks like a U, and a union brings everyone together. The intersection symbol looks like an arch, and only what fits under the arch counts.
In the diagram, A is the set 1, 3, 5, 7, 9 and B is the set 2, 3, 5, 7. The union is 1, 2, 3, 5, 7, 9. The intersection, shaded, is 3, 5, 7. The items in A but not in B are 1 and 9.
Counting outcomes. When you make a choice in stages, the number of possible results is the product of the number of options at each stage. Three shirts and four pairs of pants make 3 times 4, which is 12 outfits. Two crusts, three sauces, and five toppings, one of each, make 2 times 3 times 5, which is 30 pizzas. The test only asks one-of-each problems, so multiply. The trap answer is the sum: 3 plus 4 is 7, and 7 is always one of the choices.
Listing when the numbers are tiny. For two coins or one coin and one die, listing the outcomes is fast and safe. A coin flipped twice gives heads-heads, heads-tails, tails-heads, tails-tails: 4 outcomes, which is also 2 times 2. The list and the product should always agree.
Worked examples
Example 1
Which set of input and output pairs represents a function?
(a) inputs 1, 2, 2 with outputs 4, 6, 8
(b) inputs 3, 4, 5 with outputs 9, 9, 9
(c) inputs 0, 1, 0 with outputs 2, 3, 5
(d) inputs 6, 6, 7 with outputs 1, 2, 3
- A.table (a)
- B.table (b)
- C.table (c)
- D.table (d)
Show answer
Answer: B: table (b)
Check the inputs only. In (a), input 2 appears twice with outputs 6 and 8, so it fails. In (c), input 0 appears twice with outputs 2 and 5. In (d), input 6 appears twice with different outputs. In (b), every input is different. The outputs are all 9, but repeated outputs are allowed. Choice B is a function.
Example 2
A rule says the output is 4 times the input, plus 3. What is the output when the input is 6?
- A.13
- B.24
- C.27
- D.42
Show answer
Answer: C: 27
Substitute 6 for the input: 4 times 6 is 24, plus 3 is 27. Choice B stops after multiplying. Choice D, 42, multiplies 6 by 7, which is what you get by adding the 3 before multiplying instead of after.
Example 3
Set P is the set 2, 4, 6, 8, 10 and set Q is the set 4, 8, 12. What is the intersection of P and Q?
- A.4, 8
- B.2, 4, 6, 8, 10, 12
- C.2, 6, 10
- D.12
Show answer
Answer: A: 4, 8
The intersection is only what the two sets share. The numbers 4 and 8 are in both, so the intersection is 4, 8. Choice B is the union, everything in either set with nothing repeated. Choice C is the part of P that is not in Q.
Example 4
Using the diagram, how many items are in the union of A and B?
- A.3
- B.5
- C.6
- D.9
Show answer
Answer: C: 6
The union is everything in either circle, counted once. A alone has 1 and 9, the overlap has 3, 5, and 7, and B alone has 2. That is 2 plus 3 plus 1, which is 6 items. Choice D, 9, comes from counting the overlap twice, once for each set. Choice A is the size of the intersection.
Example 5
A sandwich shop offers 3 kinds of bread, 4 kinds of meat, and 2 kinds of cheese. How many different sandwiches with one bread, one meat, and one cheese are possible?
- A.9
- B.12
- C.24
- D.36
Show answer
Answer: C: 24
Multiply the options at each stage: 3 times 4 times 2 is 24. Choice A, 9, is the sum of the three numbers, the trap for adding instead of multiplying. Choice B, 12, stops after bread and meat and forgets the cheese.
Example 6
A coin is flipped and then a number cube with faces 1 through 6 is rolled. How many different results are possible?
- A.8
- B.12
- C.36
- D.2
Show answer
Answer: B: 12
There are 2 results for the coin and 6 for the cube, so 2 times 6 is 12 results, from heads-1 through tails-6. Choice A adds the two numbers. Choice C, 36, is two number cubes, not a coin and a cube.
Common mistakes
Rejecting a table because outputs repeat. Repeated outputs are fine. Only a repeated input with different outputs breaks a function.
Counting the overlap twice. In a union, an item that is in both sets is listed once. Add the three regions of the diagram, not the two circles.
Mixing up the symbols. The U-shaped symbol is union, the bigger set. The upside-down one is intersection, the smaller set. If your answer for an intersection is larger than either original set, you took the union.
Adding instead of multiplying. When each stage of a choice is independent, multiply. The sum is always offered as a choice, and it is always wrong.
Misreading the rule. For a rule in words, do the operations in the order the sentence gives them. "Three times the input, minus one" is not the same as "three times the quantity input minus one."
Practice
Functions and counting questions appear in the pre-algebra and math problem solving formats, and sets are one of the rarer arithmetic items. Counting outcomes is the setup step for the probability questions in the data and probability lesson, and rules with inputs and outputs connect to lines in the coordinate plane lesson.