My daughter has a problem in this week’s math homework:
The sum of two whole numbers is 72. Their difference is 48. What are the two numbers?
Now, this is supposed to be solvable by a nine year old that hasn’t had algebra, even the very simple stuff. In fact, on her math homework it always has a number of stars next to the problem that indicates the level of difficulty. Five stars is the toughest, one star is relatively simple.
Now, I know the answer, and even know how to solve for it. But I can’t think of how to get this answer with a simple, non-algebraic solution. Any suggestions?
Off of the top of my head:
Think of a number line. Pick any number and then add and subtract a second number to/from the first number. The first number is in the middle of the two answers. It makes more sense if you look at it visually.
x-a <=======|=======> x+a
x
So to solve this problem, what number is in the middle of 72 and 48? Go from there.
Well maybe, just maybe they were studying sets of twelves (dozens) that week.
You know like, I have 6 dozen and need a differance of 4 dozen between the two numbers. Let’s see 6 dozen plus one dozen satisfys the answer. (ie 60 +12 = 72).
Were ther pictures of egg cartons near the page they were studying?
I got 60 and 12, and I didn’t use algebra. I used trial and error.
The key for me was “their difference is 48.” So I started with 49 and 1. Their sum is 50. We’re looking for a sum of 72. So, (quite luckily), I jumped to 60 and 12 (whose difference I know is 48). Their sum is 72. Voila.
A more methodical way of doing this is to write down differences from 49 upward that total 48, along with their sums
Dooku nailed the intended method, if somewhat succinctly
Fourth Grade version:
Subtract the smaller number (the “difference”) from the larger one (the “sum”).
Divide the result in two. The result is the first answer.
Add the “difference” to the first answer (that’s what it’s all about, right?) to get the second answer.
Algebraic version: (AKA “why does this work?”)
x + y = A
x - y = B
Thus:
Subtracting 2) from 1) gives us:
A - B = (x+y) - (x-y) = 2*y
Dividing by 2:
(A - B)/2 = y (the “smaller” answer)
And then of course, from 2)
y + B = x (the “larger” answer)
My sister (in the States) home schools her kids and we were just talking about math education. She says that there’s a big push for kids to “discover” mathematical principles on their own, and this sounds suspiciously just like what she said.
Well, it seems that trial and error is the best approach for a 4th grader. However, the search can be significantly narrowed by noticing that both numbers (the sum 72 and the difference 48) are multiples of 12. So, one can surmise (or at least take a stab) that the problem can be restated equivalently as finding two numbers when added together is 6 (which is 72/12) and whose difference is 4 (which is 48/12). It’s easy to guess the answer is 5 and 1, which can then be translated back to 60 (from 512) and 12 (from 112), the answer to the original problem. I don’t think that the algebra-based solution that someone gave above will make any sense at all to a typical 4th grader.
Half both numbers, then add add half the diff to one and subtract the other half diff to the other.
72/2=36, 48/2=24
36-24=12, 36+24=60
60+12=72
60-12=48
a+b=72, a-b=48
Yes, of course that is the algebric solution, but the problem is how is this solved without algebra.
More importantly, why is this being taught this way? If the solution involves trial and error what can you learn out of this? How does that help a person learn math fundamentaly?
Judging by today’s public schools, this was a one-star problem, right? This is why more and more people are getting turned-on to home schooling. (By the way, Stanley his cup back…he’ll be needing it! )
Tutoring a Algera II student, he had to “solve” and graph the solution on a number line to some practical inequalities, AND something idiotic inequalities like:
2x + 3 < 2x - 10
What’s the point to reaching a statement that (a) eliminates the variable and (b) ends in a false statement? His text claims if it’s true, then all real numbers are solutions. If not true, then no real solutions. What’s the point of that? You might as well ask him to find the point of intersection where:
2x +3 = 2x - 10. (Do ya get it?)
So, are we now just teaching something to fill up the school days and school nights? Or, what?
It was a two star problem. Not the easiest, but not even near the hardest. Which is another reason I assumed they expected a non-algebraic solution. I’m old enough to have forgotten a lot of the simplest ways to do math, sometimes I just “know” the answer without the “why” part. But most of her homework is consistent and challenging enough without being ridiculous. And outside of “language arts” the school instruction is fairly good. Of course they only have budget to offer AGP (“gifted”) in math and science at her school.
Homeschooling might be a great option if I didn’t have to work for a living. I sort of consider it outside of the acceptable solution set.
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