Distinguish Repeated Amounts from a Fixed Extra

Supplied materials

• Paper or a browser scratch area; a calculator is optional unless rounding is requested.
• Supplied teaching case: A fictional kit needs 2 strips of length x plus one extra strip of length 3. Which expression gives total length? Candidate answers: (A) (2+3)x (B) 2(x+3) (C) 2x+3 (D) x+6.
• Independent check inputs: A fictional kit needs 3 strips of length x plus one extra strip of length 4. Which expression gives total length? Candidate answers: (A) (3+4)x (B) 3(x+4) (C) 3x+4 (D) x+12.

Learn the idea

An expression names a quantity without asserting equality to another quantity. Equivalent expressions give the same output for every allowed input, and their structures can reveal repeated groups, fixed extras, or shared factors. A context can describe a variable amount repeated several times and a fixed amount added once. Translating each part separately keeps the expression faithful to that structure.

Here is the method on the supplied teaching case. 2 variable strips contribute 2x; the one fixed extra strip contributes 3. The fixed amount is added once, not once per variable strip.

Parentheses mean the entire grouped amount is repeated. Decide whether the extra occurs once overall or once per repeated item before choosing the formula; draw the described strips if useful.

1. Write or sketch the supplied teaching case, labeling its quantities and restrictions.
2. Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
3. Solve the independent check with its changed inputs before revealing feedback.
4. Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.

Check your reasoning

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Take it further

Changed-input task: A fictional kit needs 9 strips of length x plus one extra strip of length 10. Which expression gives total length? Candidate answers: (A) (9+10)x (B) 9(x+10) (C) 9x+10 (D) x+90. Show the mathematical reason, not only the selected answer. Expected result: 9x+10. 9 variable strips contribute 9x; the one fixed extra strip contributes 10. The fixed amount is added once, not once per variable strip.

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