HSPT: Made-up symbols: follow the rule exactly
Learn to evaluate a defined operation by putting each number into the right letter of the rule.
How do you work out a question that invents its own symbol, like ★ or ◆?
Short answer
Copy the rule, replace each letter with its number in the right order, and compute with the usual order of operations.
Know first: Exponents such as 5², Order of operations
Step 1 of 5
A new symbol means only what its rule says
Some questions invent a symbol and give it a rule, such as a ◆ b means 3a − b². The symbol is not multiplication or any operation you already know. It means exactly what the rule says, for this question only.
The letters are placeholders. Whatever number sits in the first spot replaces a everywhere, and the number in the second spot replaces b.
| Expression | Substitute | Value |
|---|---|---|
| 4 ▲ 1 | 2(4) + (1)² | 9 |
| 1 ▲ 4 | 2(1) + (4)² | 18 |
Swapping the numbers changes which one is doubled and which one is squared.
Step 2 of 5
Substitute in the right spots
Worked example: A rule with a square
For this question, a ◆ b means 3a − b². What is 6 ◆ 2?
- A14
- B16
- C−30
- D12
- 1
Match the spots
The first number is a = 6. The second is b = 2.
- 2
Substitute
3(6) − (2)².
- 3
Compute
Powers first: 2² = 4. Then 18 − 4 = 14.
- 4
See why the other choices are there
16 forgets to square b: 18 − 2. −30 swaps the numbers: 3(2) − 6². 12 treats ◆ as ordinary multiplication.
Answer
14
Wrong: 6 ◆ 2 = 3(2) − 6² = −30
Right: 6 ◆ 2 = 3(6) − 2² = 14
The first number always replaces a, the letter that comes first in the rule. Swapping them changes the answer.
Step 3 of 5
Follow the rule exactly
Check yourself · Question 1
For this question, a ▲ b means 2a + b². What is 5 ▲ 4?
Answer: B
Substitute a = 5 and b = 4: 2(5) + (4)² = 10 + 16 = 26.
Trap answer: 18
Doubling instead of squaring
- Why it looks right
- Doubling and squaring both turn 4 into a bigger number, and 2b looks a lot like b².
- Why it's wrong
- b² means b × b, so 4² = 16, not 8.
- The right answer
- 26, because 2(5) + 4² = 10 + 16 = 26.
Check yourself · Question 2
For this question, m ⊕ n means m² − 2n. What is (4 ⊕ 3) ⊕ 1?
Answer: D
Work inside the parentheses first: 4 ⊕ 3 = 4² − 2(3) = 16 − 6 = 10. Then 10 ⊕ 1 = 10² − 2(1) = 100 − 2 = 98.
Common mistake
"The symbol looks like an operation I know, so I use that."
- Why it's tempting
- A strange symbol between two numbers looks like a multiplication sign or plus sign.
- Do this instead
- Ignore what the symbol looks like. Copy the rule and substitute each number into its own letter.
Step 4 of 5
What to remember
Remember
- A made-up symbol means only what its rule says.
- The first number replaces the first letter and the second replaces the second; order matters.
- Use parentheses when you substitute, and do powers before multiplying and adding.
Step 5 of 5
Practice on a real question
Use what you just learned on this question, then check the explanation.
In your own words
Which number replaced each letter in the practice question, and what would you get if you used ordinary multiplication?