# Thread: Number systems "method" problem: 67+86; step 1: 7x2=14 and 14-1=13

1. ## Number systems "method" problem: 67+86; step 1: 7x2=14 and 14-1=13

I'm stuck on a practice problem that seems so easy but I'm drawing a blank. I need to explain what the "student" was doing and why in steps 1&2. I need to figure out a computation to complete the problem using the method the "student" is using.

67+86

step 1: 7x2=14
14-1=13

step 2: 6+8=4+10= 14 so 60+80=140

2. Step 2 should be fairly straightforward as to what's going on and why the step was done. But step 1 is a little less clear. Essentially, what the "student" is doing is breaking apart the problem into several smaller, much easier problems. A good hint that might enable to you better see what's going on is to note that 67 = 60 + 7 and 86 = 80 + 6. So 67 + 86 = (60 + 7) + (80 + 6).

3. Originally Posted by ksdhart2
Step 2 should be fairly straightforward as to what's going on and why the step was done.
6+8=14 is clear, to me, but why state also that 4+10=14?

4. Originally Posted by mmm4444bot
6+8=14 is clear, to me, but why state also that 4+10=14?
You can never be sure from what someone writes, what they are thinking! But my guess is that they are seeing that the 8 needs 2 more to make 10, so they take two away from the 6 to add to the 8:

6 + 8
o o o o o o + o o o o o o o o
o o o o|o o + o o o o o o o o
o o o o + o o|o o o o o o o o
4 + 10

"Making ten" is a common way to do mental math (if you haven't just memorized the whole addition table), and is taught in some curricula.

5. Originally Posted by Jojo18
I'm stuck on a practice problem that seems so easy but I'm drawing a blank. I need to explain what the "student" was doing and why in steps 1&2. I need to figure out a computation to complete the problem using the method the "student" is using.

67+86

step 1: 7x2=14
14-1=13

step 2: 6+8=4+10= 14 so 60+80=140
$67 + 86 = (60 + 80) + (6 + 7) = 10(6 + 8) + \{(7 - 1) + 7\} =$

$10(6 - 2 + 2 + 8) + (7 + 7 - 1) = 10(4 + 10) + (2 * 7 - 1) =$

$10 * 14 + 14 - 1 = 140 + 13 = 153.$

Inane.

6. Originally Posted by JeffM
$67 + 86 = (60 + 80) + (6 + 7) = 10(6 + 8) + \{(7 - 1) + 7\} =$

$10(6 - 2 + 2 + 8) + (7 + 7 - 1) = 10(4 + 10) + (2 * 7 - 1) =$

$10 * 14 + 14 - 1 = 140 + 13 = 153.$

Inane.
The ideas here are commonly taught today as ways to figure out additions without having to memorize everything. It can look overly complicated when all written out, and one can question the value of teaching all this as specific methods; but it may be just what a skilled person might actually think. And it makes a lot of sense as a thinking method that is never written out.

I may not recall immediately that 7+6 = 13, but remember that 7+7 = 14; knowing how numbers work, I can just subtract 1 and see the 7 + 6 = 13. (In real life, I tend to think 7 + (3 + 3) = 13, using the make-ten strategy. This is very much real-life stuff.)

Teaching these ideas (if it is done right) should help students really understand numbers, rather than follow memorized rules and lists.

7. Originally Posted by Dr.Peterson
The ideas here are commonly taught today as ways to figure out additions without having to memorize everything. It can look overly complicated when all written out, and one can question the value of teaching all this as specific methods; but it may be just what a skilled person might actually think. And it makes a lot of sense as a thinking method that is never written out.

I may not recall immediately that 7+6 = 13, but remember that 7+7 = 14; knowing how numbers work, I can just subtract 1 and see the 7 + 6 = 13. (In real life, I tend to think 7 + (3 + 3) = 13, using the make-ten strategy. This is very much real-life stuff.)

Teaching these ideas (if it is done right) should help students really understand numbers, rather than follow memorized rules and lists.
Based on having volunteered for years to teach math at the local public library to kids who are failing at math, I find these ideas to be actively harmful. Children have difficulty with abstract concepts (as is shown in the transition from arithmetic to algebra or the difficulty in having students grasp the concept of a function). Indeed, many adults have a great deal of trouble with abstraction and generalization. Children of 7 years can more easily memorize the very simple and relatively few SPECIFIC facts necessary to add and multiply whole numbers than they can memorize and comprehend a somewhat smaller number of SPECIFIC facts and a number of generalized rules. It may look cool to trained mathematicians (and it is very profitable for authors and publishers of text books) to issue new texts in the series Bourbaki for Dummies, but the education establishment wants kids to be badly educated so that it can wring more money out of the taxpayers. The caveat "if it is done right" is truly depressing: have you dealt with the people who teach in primary school? I took my course in Abstract Algebra along with a whole slew of students who wanted to get certified to teach high school math: it was the slaughter of the innocents. Does anyone truly believe that the average graduate of a public high school today has higher comprehension of the math that they will actually use than the average graduate of a public high school 60 years ago did?

8. Originally Posted by JeffM
The caveat "if it is done right" is truly depressing: have you dealt with the people who teach in primary school?
That's exactly why I gave the caveat. I've taught some of the future teachers.

My point was only that the ideas in themselves are not inane; the way it is taught, including perhaps the very idea of formally teaching it, may be. But if we could help those teachers really understand how to think about math, things could get better than if all we do is put them down.

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