Multiply Every Term in a Group

Supplied materials

• Paper or a browser scratch area; a calculator is optional unless rounding is requested.
• Supplied teaching case: Expand 2(x+3). Candidate answers: (A) 2x+3 (B) 2x+6 (C) 5x (D) x+6.
• Independent check inputs: Expand 3(x+4). Candidate answers: (A) 3x+12 (B) 3x+4 (C) 7x (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. The distributive property rewrites a product of a factor and a sum as a sum of products. Every term inside the group receives the outside factor.

Here is the method on the supplied teaching case. Multiply both terms inside the parentheses by 2: 2·x+2·3=2x+6.

A factor outside parentheses cannot be applied to just the first term. Expand term by term, preserving signs; then substitute a simple value to compare the original and rewritten expressions.

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: Expand 9(x+10). Candidate answers: (A) 19x (B) 9x+10 (C) 9x+90 (D) x+90. Show the mathematical reason, not only the selected answer. Expected result: 9x+90. Multiply both terms inside the parentheses by 9: 9·x+9·10=9x+90.

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