fractions

Beth, Sorry I can’t help here—I need Marilyn Burns’ book too. Does anyone know if it covers decimals in relation to fractions as well? Thanks

I don’t know if this will help but I use connecting Math and it teaches equivalent fractions like this:
2/4 x / = 4/8
You look at the numerators and the denominators as division problems and say 2x (what) = 4
Then,
You look at the denominators as a multiplication problem
If the answer for the numerator will fulfill the equation for the denominator tehn they are equal
This sounds hard but it really is easy
I teach division as a multiplication problem such as 5 x ____ = 25
so, doing the same thing with fractions is easy
Have him look at it as two seperate problems that must have the same answer or they are not equal

I posted a fairly long outline on things to do with fractions about six or seven months ago — perhaps you can pull it out of the archives and repost it.

Beth — it is a *good* thing that your son is relating fractions to concrete measures and seeing them. In the long run, this will lead to success in math a lot farther and in more depth than quick calculation tricks. Tell him if he wants to see the fractions, just sketch a bar in the margin of his page. A bar is better than a pizza because it is easier to divide in any number of equal parts evenly.

Example (this is easier to draw than write - draw along with me and you will see) draw a bar about an inch long (horizontally) and half an inch wide (vertically). Divide it the long way (by drawing two more vertical lines) into three equal parts. Colour two of them. Now you have 2/3, clearly represented. Now cut through it, if posible with another colour of pen, with a horizontal line dividing it in half. Now you have a total of six equal parts and four of them are coloured. So *exactly the same area* represents 2/3 and 4/6, and it makes sense to say they are *equal* because *they measure exactly the same space*.

For calculation you can note that every part in the original drawing became two parts in the new one; original total parts (denominator) 3, new total parts (denominator) 3 x 2 = 6; original coloured parts (numerator) 2, new coloured parts (numerator) 2 x 2 = 4. This leads to the rule 2/3 = 2/3 x 2/2 = 4/6

Here’s another one to do: draw an array of dots, four rows of three dots each. Single out the first three rows of dots by outlining them with a different colour or by putting x’s on them. We now have a diagram of 9 out of 12, or 9/12 coloured. Now circle each row of three. We see we have four rows, and three of them are coloured in, so we have 3/4. But it’s the *same* picture. So we see that 9/12 and 3/4 represent the *same measurement* and so are *equal*.

For calculation you can note that we grouped by three’s so each three objects in the original became one row. Original total (denominator) 12 dots, new total (denominator) 12 divided by 3 = 4 rows; original marked (numerator) 9 dots, new marked (numerator) 9 divided by 3 = 3 rows. So this leads to the rule
9/12 divided by 3/3 = 3/4

IMPORTANT: concrete measure is the basis of real math. Things are equal *because* they measure the same amount. Calculation shortcuts come *after* the measure. Nan’s calculation methods are absolutely correct, but need to have reasoning and solid number/measure sense behind them.

Note that you can teach your son to draw bars and dot arrays in the margins of his papers and he can do this for a year or two or longer until he is more comfortable with the calculations.

Another point: this comes up frequently with my algebra students. They have been taught calculation tricks without number sense, and they lose it completely when faced with a problem that looks any different.

The ONE rule for fractions is that fractions are *equal* (meaning *exactly the same size*) if you can get to one from the other by ** MULTIPLYING OR DIVIDING BOTH numerator AND denominator by exactly the SAME THING**

Vital stresses: multiplying or dividing. Say “times”, never “and.” Addition does not work in this field. In simplifying fractions, we use crossing out as a shorthand; well and good, as long as you stress that you are *dividing* when you cross out. A huge number of students get caught on (x + 3) / (x - 4) by trying to cross out the x’s — they have never had it stressed that it is division.

Vital stress: the same THING. It doesn’t have to be a whole number or even a number at all. In algebra you simplify 3ab/5abc by dividing out the ab numberator and denominator so 3ab/5abc = 3/5c. You rationalize denominators by multiplying both numerator and denominator by the square root of 2 or something even uglier. If you learn one *simple* logical rule now, you can save months and years of frustration in algebra later. Shortcuts in math, as in life, tend to lead to blank alleys and dead ends.

Vital stress: equals means exactly the same size. Nothing more and nothing less. This is a great relief and clears up a lot of trash!
Avoid using the equals sign to mean “This is my next step”; don’t write things like 6 x 3 = 18 + 4 = 22 X 5 = 110 This, of read across the misused equals sign, say 6 x 3 is the same as 110. Sure, you’ve saved ten seconds of writing; and you’ve set yourself up to waste a whole year or more repeating algebra. Not a profit!

Also beware of rounding. 1/3 is NOT equal to .33 If you were supposed to get 1/3 of your grandmother’s million-dollar estate, you should get
$333 333.00 If you got .33 of it, you would get
$330 000.00 Unless you want to throw away $3333.00, think again about what “equals” means!

My son isn’t where Nan is so I appreciated your suggestions. He just doesn’t understand fractions enough to handle them mathematically. I am working with him along the lines you suggest but when you have never done this before it is good to know you are going the right way.

His book has a couple pages of concrete pictures and then jumps immed. to doing equivalent fractions without the pictures. It is just moving too fast for him. I can see when he tries to apply the “rules” he is just mixing things up. I started by having him color 1/2 of shapes I drew. He did that just fine. Then he had to color 1/4. I then tried 1/3 and discovered he didn’t seem to have a sense of it—kept drawing three lines instead of two. So this may take awhile!1!!

We’ve been doing stuff with pie–now I remember your post (which I will look for) being antifood but for my son it works. The reason is he loves pie and after we work with it, he gets to eat it. I start by asking if he wants 1/4 or 1/8 of the pie. He is motivated to know which one is bigger because it affects him!!!! Then I ask him if he would want 2/8 or 1/4–we use the pie for equivlent fractions and when he has done enough, he gets to eat a piece!!!
He now is bargaining for bigger pieces of the pie by using fractions. (No, I want a 1/4 of the pie not a 1/6.)

Beth

Thanks for your ideas but my son just isn’t this far yet. I do think though that I will have him practice multiplication as you suggested. I think that will help him computationally when we get that far!!!

Beth

My kids do the arrays too. I guess I thought you wanted to go beyone the concrete. Some of my students still need to draw it out and fill in to see the relationship. I like the computation along with it because it helps them see the relationship between equivalent fractions and multiplication.Hopefully, the connection will enable them to go beyond concrete examples.
Nan

Another good thing to use is centimeter cubes. They stick together and let you make all kinds of arrays. My kids use them and then write the fractions for each color. This is a fun way to play with parts of wholes and to help to make fractions real. I think kids really do not get the fact that a fraction is part of 1.

of relating fractions to other aspects of our lives- such as telling time and cooking. You can have a lot of fun with various fractions of a clock-1/4 hour etc, 1/12 hour, 1/6 hour- all of which you can figure out by looking at the five minute divisions. And the connections between cooking and fractions are legion…

I am a firm believer that kids need to stay with the concrete stuff until they are ready to move on- and they will signal this by beginning to develop strategies for representation because manipulatives just are not efficient. The stages are concrete, iconic representation (pictures, tallys etc) and abstract representation (numbers). There is litte benefit to pushing faster than they are ready to go- someone will just have to go back and reteach it. You will love the Burns book…
Robin

I think Robin has it absolutely right about not pushing.

I tutor math as a career, all levels from elementary to university, and 99% of the students I get have been pushed beyond their conceptual limits at some point, most often in fractions although often even earlier than that in multiplication and even addition. They have heads chock-full of formulas, procedures, and recipes, but unfortunately they keep pulling out the *wrong* procedure or applying it upside down and backwards, because to put it bluntly, they have not got a clue what they are doing or what a number is. Yes, this occurs right up through university calculus.

There is one caveat: sometimes students finally get something right after much struggle by using a certain concrete method and then they stick to it as a security blanket. There does come a point to say it’s time to leave that behind with the other childish things. But that point is after months or even years, not the first two days of introduction.

There is a prejudice, strongly fostered by a lot of elementary teachers, that symbolic math is “real”: math and concrete approaches are somehow second-rate baby stuff. Exactly the opposite is the truth!! Up to and including university second-year differential equations, ALL math is is applied math! Newton invented calculus to explain why the moon doesn’t fall down and hit the Earth, not because he liked filling up pages of parchment with squiggles. It is an extreme frustration as a college instructor to work for hours trying to teach a physical concept (the slope of a graph — how steep is it? Uphill or downhill?) and have some student tell the others they know a “better” way to do it, you just put this number here and that number there … In high school and university we fight and fight to get students to draw graphs and diagrams, to undo the damage done by elementary teachers who told them not to draw pictures.

Remember, you can always sketch the equivalent fraction in the margin. And the same goes for more advanced work.

I guess he just isn’t there then. Sigh. I will try the clock and other ideas you have. I can see very clearly that he is not going to go back to school next week knowing how to do equivalent fractions.

He was in resource room for math for first and second grade and had been doing OK in the classroom this year until fractions. Do you think I should have them put him back in resource room?

Beth

He is in third grade? Sheesh…I might wait then- and just make the homework help as concrete as I could. It is relatively easy to create models for equivalent fractions. I would do it-or better- help him do it and scaffold him off to being able to do that part on his own. It may very well be that his teacher is not really expecting mastery on this you know- equivalent fractions at the number level requires a lot of math understanding that third graders typically do not have- and shouldn’t becuse their brains are not wored for it yet in most cases. I know that Everyday Math- which is the program of choice in my children’s district- does not expect mastery of about 2/3’s of what is presented. exposure is the the thought behind their spiral- with mastery expected a couple of years later. I think that many programs are structured this way now- which can be a mind boggling concept for the folks who help the kids with their homework… We just were not taught that way. Lots of teachers have a hard time getting over this too. Be calm…
Robin

Thanks Robin. I appreciate your good sense. I really hate the way math is taught today. Now I did have his teacher tell me that he isn’t “getting fractions” and supplied me with all sorts of “pizzas” when I offerred to work with him over break.
I saw her notes on his papers and she showing how to do equivalent fractions using division. There is no way he is going to get that this year. He barely has division down (pretty good at multiplication however). It is just too abstract for him.

BTW, what is the logic of introducing ideas several years ahead of when most children can master them? It escapes me. I see this approach backfiring for my son who needs lots of repetition for mastery as well as my nonLD daughter who does catch it the first time. She ends up complaining that she is doing the same thing as she did last year.

Beth

You are welcome…

Anytime you design a program it is going to be a poor fit for kids at both ends of the spectrum- it is inevitable. The philosophy -as I understand it anyway- is to develop recognition- so that procedures and concepts are not totally unfamiliar when they are REALLY taught for mastery and they are integrated into a more complete understanding of the way numbers work. Preteaching vocabulary follows the same thought- as do politicians running for nationwide office.There is also a desire to raise the instructional level of the students- and there is some logic to that BTW- if high level work is not asked for then it will never be received. However, well meaning teachers do not want their kids to fail- really and truly-so they do one of two things until they are comfortably knowledgeable. They either expect mastery far too soon- as in the case of your son’s teacher I guess- or they retread the previously taught material until every single child passes the test. Your daughter’s teacher is somewhere in between I guess:) The book drives the program in both cases. Sigh…

Robin

I guess I do the same thing when I make my college students read the textbook before class but don’t expect mastery!!

I pulled out the Singapore math books I was using last summer. I have been doing the fractions stuff in there. Very concrete and where he is at. I am not even going to try equivalent fractions. He is finally getting something about fractions!! I will use Marilyn Burns book this summer to try and preteach fourth grade since it is obvious this is not going to be easy for him.

Besides FL school seem to slow down in May. I figure it is only a month left to really struggle here!!!

Thanks for all your help again.

Beth