A new symbol is a tiny rule machine. Pin the current definition, send the left input to a, send the right input to b, and only then use ordinary arithmetic.
A symbol such as △, ★, or * does not bring its own arithmetic meaning. The current problem supplies that meaning.
The triangle means only this rule in this problem. Another problem may define the same-looking symbol differently.
The worked example introduces the factorial sign: 3! means 1 × 2 × 3 = 6, and 5! means 1 × 2 × 3 × 4 × 5 = 120. It is an instruction, not punctuation. For a positive whole number n, n! multiplies the whole numbers from 1 through n. A custom two-input symbol is also an instruction: read its current rule before calculating.
All three cards use the symbol ★. Click a card and notice that the same inputs, 4 and 7, produce different outputs because the rules differ.
Read the rule on each card. The symbol alone does not determine an output.
The left written input fills a and the right input fills b, even if a swapped calculation happens to agree.
Rebuild the worked example one line at a time. The custom symbol disappears as soon as the definition is substituted.
Replace the triangle first, then multiply before subtracting.
The expression is (2*3)*5. Parentheses require the first operation to finish before the second begins.
2*3=(3+2)×3
15*5=(5+15)×5
Finish 2*3. Use that entire result as the next left input.
| Rule card | Substitution | Output | What changed? |
|---|
Reset the definition on every row; do not reuse the previous output formula.
10*11=50, while 11*10=51.
Because a,b≥1, the least inputs are a=b=1:
Compare 3a+2b−2 with 3b+2a−2. For the minimum choose the least legal inputs.
Enter both inputs before testing a domain. An empty box is not zero. A natural-number convention can vary between books, so these activities explicitly distinguish positive integers from nonnegative whole numbers.
Zero and negative values are excluded.
Zero is allowed; negatives are excluded.
All whole-number values are allowed.
The sample rule is a*b=3a+2b−2. The operation is evaluated only after both inputs pass the domain gate.
Check whether zero and negative values belong to the named domain. A blank is not zero.
The test states that x*5 is 100 greater than 5*x. The operation is simple to decode, but the input order changes which value is added.
In ab+a, the extra term is the left input. Write both x*5 and 5*x before subtracting.
For this question only, a★b = 3a − b/2. Find 10★6. First multiply the left input by 3, halve the right input, then subtract.
Complete the fresh practice and all eight workshop questions to finish this checkpoint.
Answers: 7;28;27;24;27;45;3;yes. For Question3, 1*2=6, then 6*3=27. For the last question, ab−a−b=ba−b−a, so swapping inputs preserves the result for every allowed pair.
Answers: definition;32;100;not equal;105. Substitute each stated rule. For the last question, x*5=6x and5*x=5x+5, so their difference is x−5=100. Check x=105:630−530=100.
This certifies that the learner can read a custom rule, map left and right inputs, substitute in order, respect the domain, and test whether input order matters.