A long way to compute 7·14…
Place value? We ain’t got no place value. We don’t need no STEENKING PLACE VALUE!”
The student above attempted a strategy that gave she/he the wrong answer. I am curious to know why is it the student chose to multiply the two digits on the left by the two digits on the right. I noticed that there’s a side note saying something about standard algorithm. I wonder why the student did not use standard algorithm and what led them to want to multiply 3 and 6 together. As a future teacher, I would probably ask the student to explain their strategy to me to understand where they are coming from. For myself, I was never encouraged to use standard algorithm. I would solve this problem by breaking the problem into chunks like 34 X 30 and 34 X 38.
At first, I was really surprised that the student thought in this way. Even it is wrong way to solve it, I had not thought about this kind of problem in this way. My first reaction on this post was “Wow~!” Anyway, I think that the student does not understand place value and standard algorithm as well. For 4 x 8 = 32 it is correct since it is product of ones (the place value). However, the other three multiplication got wrong because the student did not understand the place value enough. If the student’s way can be correct, the student may can do 30x 8 = 240 instead of 3 x 8 = 24, 4x 60 = 240 instead of 4 x 6 = 24, and 30 x 60 =1800 instead of 3 x 6 = 18 and add everything up to get right answer “2312”. To do this easier, the student also can draw the area models make 34 into chunk like 30 and 4 and 86 into chunk like 60 and 8.
The way this student approached this problem reminds me of the FOIL method we use for binomials, but maybe they just learned the distributive property and thought it could apply for a multiplication problem like this when it doesn’t actually work this way. We know that the easiest way for adults to do this problem is to do the standard algorithm, but for a student, they probably could’ve still used the distributive property to solve this multiplication. Imagine a table that’s 34 by 68 inches long. To break it down into units, we can have 30+4 on one side and 60+8 on the other side. In order to account for all areas of this table, we need to multiple the numbers with each other. The equation would look like this A=(30+4)(60+8). If this was the given problem, the student probably would’ve solved it correctly with his thinking. To get the correct answer, you have (30×60)+(30×8)+(4×60)+(4×8)=1800+240+240+32, which gives you 2312. Very creative mistake!
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I think this child has a fantastic way of looking at this problem! Although there was some confusion with the place value of the numbers being multiplied together. The student did understand a system that could work very well, and I think with a little help in understanding the place value in this equation and others, they might have a way that works for them. This creativity, if possible to get the correct answer, should be encouraged. Although, down the line it maybe easier to do the standard algorithm the student is well on there way to the standard algorithm.
I would be curious what grade this child is in? Have they been introduced to a FOIL type method? Is this a method the teacher has shown before, and if so does it seem to be more successful than the common algorithm? Also, would the teacher except the answer if the place value and answer was correct, but the way of doing it was as creative as this child displayed?
There’s no such thing as “a FOIL method” and I strongly recommend that teachers eschew teaching the mnemonic at all, let alone as a method.
We have in the real numbers and all its major subsets (rational and irrational numbers, integers, whole numbers, and natural (or counting) numbers, something called “the distributive property of multiplication over addition.” That’s all you need to explain how 34 x 68 = (30 + 4)(60+8) = 30*60 + 30*8 + 4*60 + 4*8. And this extends to any number of terms in any factor. Teaching a mechanical mnemonic that ONLY works for binomials times binomials (or two-digit numbers multiplied by two-digit numbers) has to be one of the worst notions brought into the K-12 curriculum ever. It is now routine, in fact, to hear both teachers and students in American high schools and middle schools call multiplication “foiling.” Curses, foiled again, mathematicians! Your silly generalizations can easily be ruined by teachers who can’t seem to grasp even minor abstraction.
This student clearly understands how to multiply numbers; it’s just that the student does not fully understand the use of place value. The 3 represents 3 tens, aka 30. However, the student understood it as the value of 3, because the student was not fully aware of place value: ones, tens, hundreds, etc. Same for the latter number, 68; The number represents 6 tens and 8 ones, aka 60 and 8. One way to help the student understand better is to show that, for example, 68= 60+8. Likewise, 34 equals 30+4, 3 tens and 4 ones. Of course, letting the student know about place value and how that plays such an important role in multiplying numbers.
One amazing thing to point out is that this student knows the ‘FOIL Method’. I can’t say for sure that the student has learned how to use the FOIL method, but the student knows how to process the FOIL method. It’s important for the student to learn when it’s appropriate to use the method and how it can done in those situations.
Clearly, as others have mentioned, this student does not have a firm grasp on place value but they do know how to multiply well, at least for single digit numbers. However, I was thinking that this mistake may have something to do with how the problem was written with the numbers side-by-side instead of on top of each other. For the standard algorithm I was taught about how to multiply double digit numbers like this, the numbers had to be written one on top of the other and then you multiply the numbers one by one like that and add them together. Perhaps that is what this student was trying to do horizontally and he lost track of place value in the process of writing it out like this.