A student thinks the sum of 4.3 and 8.4 is 12.7 because 4+8= 12 and 3+4=7. The student then adds 3.7 and 2.6 and gets 5.13 because 3+2=5 and 6+7=12 Identify the mistake in the students procedure and explain why the procedure wont always work
The mistake in the student's procedure is that they are incorrectly splitting the decimal numbers into their whole number and decimal parts and then adding them separately.
In the case of 4.3 and 8.4, the student splits the numbers into 4 and 3, and 8 and 4. However, this is not the correct way to add decimal numbers. When adding decimal numbers, the corresponding places must be added together.
In the correct procedure, we add the whole numbers first, and then add the decimal parts separately.
For 4.3 and 8.4:
(4 + 8) + (0.3 + 0.4) = 12 + 0.7 = 12.7
Now, let's examine the second addition of 3.7 and 2.6. In this case, the student incorrectly splits the numbers into 3 and 7, and 2 and 6. Again, this is not the correct way to add decimal numbers.
The correct procedure would be:
(3 + 2) + (0.7 + 0.6) = 5 + 1.3 = 6.3
The student's method of splitting the numbers and adding them separately only works when the decimal parts are within the range of 0-9. However, in the case of adding 0.3 and 0.4, or 0.7 and 0.6, the sum of the decimal parts may exceed 10. In such cases, splitting the decimal numbers and adding them separately does not give the correct result.
The mistake in the student's procedure is that they are adding the ones place and the tenths place separately, without considering the place value. This procedure does not work because when adding decimals, the place value is significant.
To add decimals correctly, we need to align the decimal points and add each place value together. In this case, we have:
4.3
+ 8.4
-------
12.7
When we line up the decimal points and add the numbers together, we get 12.7 as the correct sum. So, in this case, the student's procedure was successful.
However, let's now consider the second example:
3.7
+ 2.6
-------
5.13
Here, the student again applies their previous procedure, where they add each place value separately:
3.7
+ 2.6
-------
5.13
When we align the decimal points and add the numbers correctly, the sum is actually 6.3, not 5.13.
The reason the student's procedure doesn't always work is that decimals have specific place values, just like whole numbers. When we add decimals, we must consider the place value of each digit. In the student's procedure, they applied a method of adding only the ones place and the tenths place without considering the carryover or the place value rule. This can lead to incorrect results, as we have seen in the second example.