Reverse Distribution with a Common Factor

Supplied materials

• Paper or a browser scratch area; a calculator is optional unless rounding is requested.
• Supplied teaching case: Factor the greatest common positive integer from 6x+10. Candidate answers: (A) 2(3x+10) (B) 2(3x+5) (C) 2(3x−5) (D) 2(8x).
• Independent check inputs: Factor the greatest common positive integer from 9x+15. Candidate answers: (A) 3(3x+15) (B) 3(3x+5) (C) 3(3x−5) (D) 3(8x).

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. Factoring rewrites a sum as a product. A greatest common integer factor divides each coefficient, making the grouped expression easier to interpret and verify by expansion.

Here is the method on the supplied teaching case. The coefficients 6 and 10 share greatest common factor 2. Dividing each by it gives 3 and 5; expanding recovers the original.

After taking out the common factor, divide every original term by it. The factor must multiply the entire remaining group; expanding is a direct check that no term changed.

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: Factor the greatest common positive integer from 27x+45. Candidate answers: (A) 9(3x+45) (B) 9(3x+5) (C) 9(3x−5) (D) 9(8x). Show the mathematical reason, not only the selected answer. Expected result: 9(3x+5). The coefficients 27 and 45 share greatest common factor 9. Dividing each by it gives 3 and 5; expanding recovers the original.

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