What is this student thinking? It’s a bit tricky, so extra points if you come up with a complete story that includes a list of things that we know that this kid knows.

The student seems to be comfortable with the idea that 1 can be represented multiple ways: 1/1, 6/6, 2/2.
The sentence, “It’s a number that’s the same as both to make it #2” is bit confusing, but at least shows that the student’s trying to put the symbols into English.
The phrase, “Has to go to 6” shows that he knows the denominators must be equal to add fractions. But, he doesn’t appear to know how to change the denominator when something foreign (the x) is in the numerator.

The work on the right looks to me like a guess and check method. Try x=1. 1/6 + 1/2 (accidentally wrote an =?) doesn’t equal 2. Try x=2. 2/6 + 2/2 doesn’t equal 2. Not sure why he didn’t proceed to x=3!

The work doesn’t seem to imply this is an algebra class: I might expect trying to add something to both sides or multiply through by something. I would guess pre-algebra?

For whatever reason I feel awfully tentative jumping into these analyses. Meaghan has blazed a trail though. I’d add my impression that our student thinks the only way to make two addends get two is if they’re both one. But that sucks because you can only get that if you have 2/2 and 6/6. (See bottom-right corner.)

The limb’s getting creaky out here but I’ll wager our student understands x can’t be two numbers at once, so she sets x/2 equal to x/6 and gestures wildly at the fact they have to be the same.

If I’m right about any of this, I’d ask the student to tell me two numbers that add to two, then two more numbers, then two more numbers, getting the student comfortable with other addends besides 1+ 1. That’s step one, I think.

Steve & I had a lot of fun with this. Here is what I noticed:
-the student is looking for fractions that will make one
-the student knows that 1 = 1/1 = 2/2 = 6/6
-the student knows that if there were a value of x such that x/2 = x/6 = 1 the problem would be solved
-the student knows that proportions can be used to try to find missing values to create equal fractions
-the student knows x can only have one value at a time

I wondered:
-if the student is stuck on 1 + 1 = 2 and has not entertained 1 1/4 + 3/4 = 2, 1 1/2 + 1/2 = 2, 4/3 + 2/3 = 2, etc.
-if the student realizes that x/2 and x/6 will only be equal if x = 0, and that for whole numbers, x/2 will always be bigger
-why the student rejected plugging 6 in for both x’s — did (s)he try it and get a value or give up seeing that 6/2 is not one.

Possible directions for next steps:
1) Asking about the values x/2 and x/6 will have in the final answer. Will they be the same? Different? Which will be bigger? How do you know?
2) Asking for the different ideas the student is working with in terms of assumptions, hypotheses, or wonderings: what were you wondering when you wrote 1/2 = 1/6? What assumptions are you making when you wrote x/2 = _/6?
3) Following up on the idea of making ones: what were you hoping for when you tried x = 6? Why did you try to make 1? If the answer is to get 1 + 1 = 2, then probing if that will always work/is that the only way? If the answer is to get rid of fractions, asking what happened when they tried x = 6. If the answer is anything else, then I need to do some more noticing and wondering.

Now that I posted, I went back and read Meaghan and Dan’s work. I like that there’s some places where we have the same interpretation of the work and others where we have different interpretations. For example, I didn’t notice the guess and check possibility in 1/2 = 1/6, 2/2 + 6/6, but now I can see where Meaghan sees it. One thing I’m really curious about is whether the student can comfortably add, say, 1/2 + 1/6. I’m wondering if there’s a simpler explanation for 1/2 = 1/6 than that it’s a “typo” and they meant +. Could it be the first attempt at what Dan refers to as “gesturing wildly that they have to be the same”?

I much prefer Max’s and Dan’s assessments of the right side of the student’s work to mine, for the record!

(I’m a bioengineering PhD student that day-dreams of teaching math – I’ve really enjoyed soaking in the collective knowledge on this site!)

Please do come teach math! It’s way more fun than bioengineering, I promise. Not that I’ve ever bioengineered anything…

I wonder what, if anything, the exposure Dan and I have had to student thinking & writing, might have contributed to what we saw that you appreciated? How does one learn to see student work maximally? (I’m not claiming I have learned that, just building on the comment, “I like what you saw better” and using it to wonder “What does it mean to get better? How does one get better?”)

In his earliest post Max says that he thinks that the student knows that x can have only one value at a time. I see the opposite where x = 2 on the x/2 fraction and x = 6 on the x/6 fraction. I suspect, as mentioned earlier, that this student thinks that 2 = 1 +1 (correct!!) and that this is the only way to get 2 as a sum. If I had seen a 0/2 + 12/6 proposal I would feel marginally better about this. The idea that 2 = 1/4 + 7/4 or that it is the sum of 2 fractions with like numerators and unlike denominators (as in this original equation) would probably fall under the category of things that this student might acknowledge when pressed a bit but not something s/he would have thought of on their own.

## 7 replies on “Fractions and Equations”

The student seems to be comfortable with the idea that 1 can be represented multiple ways: 1/1, 6/6, 2/2.

The sentence, “It’s a number that’s the same as both to make it #2” is bit confusing, but at least shows that the student’s trying to put the symbols into English.

The phrase, “Has to go to 6” shows that he knows the denominators must be equal to add fractions. But, he doesn’t appear to know how to change the denominator when something foreign (the x) is in the numerator.

The work on the right looks to me like a guess and check method. Try x=1. 1/6 + 1/2 (accidentally wrote an =?) doesn’t equal 2. Try x=2. 2/6 + 2/2 doesn’t equal 2. Not sure why he didn’t proceed to x=3!

The work doesn’t seem to imply this is an algebra class: I might expect trying to add something to both sides or multiply through by something. I would guess pre-algebra?

For whatever reason I feel awfully tentative jumping into these analyses. Meaghan has blazed a trail though. I’d add my impression that our student thinks the only way to make two addends get two is if they’re both one. But that sucks because you can only get that if you have 2/2 and 6/6. (See bottom-right corner.)

The limb’s getting creaky out here but I’ll wager our student understands x can’t be two numbers at once, so she sets x/2 equal to x/6 and gestures wildly at the fact they have to be the same.

If I’m right about any of this, I’d ask the student to tell me two numbers that add to two, then two more numbers, then two more numbers, getting the student comfortable with other addends besides 1+ 1. That’s step one, I think.

Steve & I had a lot of fun with this. Here is what I noticed:

-the student is looking for fractions that will make one

-the student knows that 1 = 1/1 = 2/2 = 6/6

-the student knows that if there were a value of x such that x/2 = x/6 = 1 the problem would be solved

-the student knows that proportions can be used to try to find missing values to create equal fractions

-the student knows x can only have one value at a time

I wondered:

-if the student is stuck on 1 + 1 = 2 and has not entertained 1 1/4 + 3/4 = 2, 1 1/2 + 1/2 = 2, 4/3 + 2/3 = 2, etc.

-if the student realizes that x/2 and x/6 will only be equal if x = 0, and that for whole numbers, x/2 will always be bigger

-why the student rejected plugging 6 in for both x’s — did (s)he try it and get a value or give up seeing that 6/2 is not one.

Possible directions for next steps:

1) Asking about the values x/2 and x/6 will have in the final answer. Will they be the same? Different? Which will be bigger? How do you know?

2) Asking for the different ideas the student is working with in terms of assumptions, hypotheses, or wonderings: what were you wondering when you wrote 1/2 = 1/6? What assumptions are you making when you wrote x/2 = _/6?

3) Following up on the idea of making ones: what were you hoping for when you tried x = 6? Why did you try to make 1? If the answer is to get 1 + 1 = 2, then probing if that will always work/is that the only way? If the answer is to get rid of fractions, asking what happened when they tried x = 6. If the answer is anything else, then I need to do some more noticing and wondering.

Now that I posted, I went back and read Meaghan and Dan’s work. I like that there’s some places where we have the same interpretation of the work and others where we have different interpretations. For example, I didn’t notice the guess and check possibility in 1/2 = 1/6, 2/2 + 6/6, but now I can see where Meaghan sees it. One thing I’m really curious about is whether the student can comfortably add, say, 1/2 + 1/6. I’m wondering if there’s a simpler explanation for 1/2 = 1/6 than that it’s a “typo” and they meant +. Could it be the first attempt at what Dan refers to as “gesturing wildly that they have to be the same”?

I much prefer Max’s and Dan’s assessments of the right side of the student’s work to mine, for the record!

(I’m a bioengineering PhD student that day-dreams of teaching math – I’ve really enjoyed soaking in the collective knowledge on this site!)

Please do come teach math! It’s way more fun than bioengineering, I promise. Not that I’ve ever bioengineered anything…

I wonder what, if anything, the exposure Dan and I have had to student thinking & writing, might have contributed to what we saw that you appreciated? How does one learn to see student work maximally? (I’m not claiming I have learned that, just building on the comment, “I like what you saw better” and using it to wonder “What does it mean to get better? How does one get better?”)

In his earliest post Max says that he thinks that the student knows that x can have only one value at a time. I see the opposite where x = 2 on the x/2 fraction and x = 6 on the x/6 fraction. I suspect, as mentioned earlier, that this student thinks that 2 = 1 +1 (correct!!) and that this is the only way to get 2 as a sum. If I had seen a 0/2 + 12/6 proposal I would feel marginally better about this. The idea that 2 = 1/4 + 7/4 or that it is the sum of 2 fractions with like numerators and unlike denominators (as in this original equation) would probably fall under the category of things that this student might acknowledge when pressed a bit but not something s/he would have thought of on their own.