A student who was experimenting with different methods of multiplying two whole numbers noticed that when she increased one number and decreased the other by the same amount and then multiplied, she did not get the correct answer. Explain how you would help this student.

Answer 1

To help the student understand the concept, I would explain the properties of multiplication and the reason why increasing one number while decreasing the other by the same amount doesn't yield the correct answer.

Firstly, I would remind the student that multiplication is a commutative operation, meaning the order of multiplication doesn't affect the product. In other words, multiplying A by B will yield the same result as multiplying B by A.

Next, I would clarify that increasing one number while decreasing the other by the same amount changes the values being multiplied but doesn't change their product. For example, let's consider the numbers 4 and 5. If we increase 4 to 6 and decrease 5 to 3 by the same amount (2), the resulting multiplication would be (6 * 3 = 18). However, if we revert to the original numbers and multiply (4 * 5), we get the correct product of 20.

The mistake lies in assuming that the increased number and decreased number are interchangeable. In reality, they are not equivalent, and this swap affects the outcome. Emphasizing the importance of understanding the actual values being multiplied is crucial.

To reinforce this concept, I would suggest using specific examples and guiding the student in solving them step by step. By comparing the correct products obtained using different approaches, the student will clearly see the difference in outcomes when applying the swapping method.

I would also encourage the student to think critically and question their initial assumption. Engaging in discussions and asking open-ended questions like, "If we increase one number and decrease the other, does it guarantee that the product remains the same?" can help the student reveal the flaws in their approach.

Finally, I would provide additional practice problems where the student can apply the correct method of multiplying two numbers to ensure they have a solid understanding of the concept.