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.
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.
To help the student understand why increasing one number and decreasing the other by the same amount does not produce the correct answer, you can follow these step-by-step explanations:
1. Begin by explaining the concept of multiplication to the student. Multiplication is the process of repeated addition, where you add a number to itself a certain number of times. For example, multiplying 3 by 4 is the same as adding 3 four times: 3 + 3 + 3 + 3 = 12.
2. Next, explain that when you increase one number and decrease the other by the same amount, the overall value remains the same. For example, if you increase 5 by 2 to get 7 and decrease 3 by 2 to get 1, the sum of those numbers remains the same (7 + 1 = 8). So, adding these changed values together cannot result in a correct answer for multiplication.
3. Illustrate this concept with examples. Take a simple multiplication problem like 2 * 3. If the student increases 2 by 1 (to make it 3) and decreases 3 by 1 (to make it 2), the resulting calculation is 3 * 2 = 6. However, the correct answer for 2 * 3 is 6 as well. This shows that changing the numbers equally does not affect the outcome of the multiplication.
4. Reinforce the concept by trying this method with different numbers. Encourage the student to choose different pairs of numbers and experiment with the technique of increasing one number and decreasing the other by the same amount. Show them that no matter which numbers they choose, the incorrect result will still be obtained.
5. Suggest alternative methods to multiply two whole numbers correctly. Remind the student that when multiplying two whole numbers, they should simply multiply the two numbers without any additions or subtractions. Emphasize that changing the value of one number and the other equally doesn't affect the multiplication process.
By following these step-by-step explanations, you can help the student understand why increasing one number and decreasing the other by the same amount does not give the correct result when multiplying two whole numbers.
To help the student understand what went wrong when she increased one number and decreased the other by the same amount and then multiplied, you can use an example and walk her through the mathematical process.
1. Begin by explaining the correct method of multiplication.
- Multiplication is the process of adding a number to itself a certain number of times.
- For example, multiplying 3 by 4 means adding 3, four times: 3 + 3 + 3 + 3 = 12.
- This is the basis of multiplication, where the two numbers being multiplied are added repeatedly.
2. Now, demonstrate the issue with increasing one number and decreasing the other by the same amount.
- Let's consider an example: multiplying 4 by 5.
- Instead of multiplying 4 by 5 directly, the student is increasing one number (4) and decreasing the other (5) by the same amount, let's say by 2.
- This results in multiplying (4 + 2) by (5 - 2) = 6 by 3 = 18.
- However, we know that the correct answer should be 4 multiplied by 5, which is 20.
- So, the incorrect method didn't yield the expected result.
3. Explain the reason behind the incorrect result.
- The issue with increasing one number and decreasing the other by the same amount is that the numbers being multiplied are not the original numbers.
- When we manipulate the numbers before multiplication, we end up with different values, leading to an incorrect result.
- In our example, by increasing 4 to 6 and decreasing 5 to 3, we changed the numbers we started with, which gives us the wrong answer.
4. Emphasize the correct approach to multiplication.
- Multiplication should be done directly on the original numbers without altering them beforehand.
- Encourage the student to focus on multiplying the original values as they are to obtain the correct result.
By explaining the correct method of multiplication and highlighting the issue with altering the numbers before multiplication, the student should understand why her approach didn't yield the expected answer. Encourage her to practice using the correct method to reinforce the concept.