My kids nailed adding fractions w/unlike denominators

Michele,

I’m not familiar with math u see. Could you give me a step by step explanation (as you would to your kids) on how you use skipcounting to teach adding factions with unlike denominators. It think it was you who posted great ideas on rounding with mountains and telling time. I’ve used these strategies and my ld tudents have thrived. Please share more.

Thanks!
Ruth

Michelle,

I would also like to know more about mathusee.
Thanks Dorothea :wink:

I went to a workshop on MathUSee at the state IDA conference. It was just awesome. I maybe learned more about algebra in 12 minutes (it was an overview of the whole course) than in my 1 year I had it in school.
Take a look at their website: www.mathusee.com

I am using mathusee plus some ideas from on cloud nine with one of my students. Math is now her favorite subject. And this is from someone who is dxed dyscalculic (me).

I am getting no money from them or anything else.

—des

I would give some extra practice in the calculation part htan Math-U-See does — but I also really like their presentation. (I’ve never known any program to have enough practice, though, and I like using different resources anyway.)

Well, this is kind of tough. I’ll do my best.

You teach skip counting by 2’s, 3’s, 4’s etc…. Now, if they can’t remember they can add on by 3, then 3 more etc..

So you have the kid make equivalent fractions. They need to build say write the fractions. SAME AMOUNT DIFFERENT NUMBERS. They need this first.

So, they have the fraction ½, they then write 2/4, 3/6, 4/8, 5/10 out by about 5 fractions. It is easier on paper because the fraction bar is straight on not on the diagonal.

Then they do the same for the next fraction, let’s say 2/3

2/3, 4/6, 6/9, 8/12, 10/15

Next, the kids focus on the DENOMINATORS. Look at the 4 in your first fraction.

Again:

Give them this problem: ½ +2/3= ?
The kid will skip count all the numerators. In this example there is a one on top, so skip count by 1. Then there is a 2, so skip count by 2 In 2/3, skip count again by 2, and then 3.
The kid will skip count all the denominators.

½ 2/4 3/6 4/8 5/10

2/3 4/6 6/9 8/12 10/15

Now, take your first denominator which is 2 in the fraction ½. Look at your 2/3’s list of denominators. Is there a 2? Nope, cross off ½…look for a 4 in the 2/3’s row? Nope, cross of 2/4 then. Look for a 6, Yep, DING DING DING, we have a match, and it is the lowest. DING DING DING. Now circle both 3/6 and 4/6. WOW, now we have an easy problem just like we did before? Remember how easy adding like denominators was?

Now, add the tops, and keep the bottom the same. And Math u see has it written out with lines to plug in.

Hope this makes sense. We have not got into reducing them yet….but they can add/ w/unlike den. This is my first year, and I am not an expert. I’m just a little ahead of them. But, I’m impressed so far. I maynot be doing it exactly like the book but I’m trying, but it worked.

Michelle AZ

Michelle— good start, but a cautionary note:

It is one thing to get the “right answer” and get all those check marks correct.
It is unfortunately quite another thing to understand a concept, retain it for next year and all the years after that, and be able to transfer it to other work (say for example algebra) when you need to do so.

First, be very sure when students are working with equivalent fractions that they understand very very well that the numbers are the same *size*. Fractions are used for measurement; make sure they measure out 1/2 cup equals 2/4 cup etc.

I have to let another person use the computrer — bcdk on a serious concern about transfer later.

Point well taken. I do need more time with the manipulatives. And you are right. I do need to spend more time getting it in their head visually. So many interuptions. Field trip tomorrow, picnic on Friday with early release, spring break around the corner, quarterly testing of all IEP objectives this week, time for individualized goals, Oh, (sigh), I wish I had more time in the day to teach the way I’d really like to teach. I need to keep reviewing all the other things we’ve learned, give them time to explore the manipulatives plus keep up with math facts…and then the phone rings. Sorry to digress. I’rm not “there” but I feel like I’m doing better this year. I am especially pleased with their understanding of place value.

Michelle AZ

It’s been my experience that for a lot of kiddos if you get the procedures down and solid, then work on the concepts, they make all kinds of connections - and it’s sort of more interruptable than that first “getting it.”
OF course, what *usually* happens is they sort of almost get the procedure, with hardly any understanding, and so they forget it quickly — because most of the m *will* forget it too quickly if you don’t get the visuals and concepts in there, too. So … now that they “have” it — build on it, make sure they keep it. At *least* do some daily review of the process — as I’ve said, most of my college students (in the pre-algebra classes) don’t have that procedure down - and/ or they don’t know when to do what procedure. So, I always include “what are we doing? adding fractions… so what do we need to do?” and a fair amount of language abotu why & how.

Though most of the college students you have no doubt had the math teaching technique known as “why do you do it that way, the teacher said to”.

For example, I learned to divide fractions that way. The teacher said “reverse the divisor and multiply”. I don’t to this day know why that works. I suppose I’ll find out via MathUSee.

—des

Michelle — yes, getting the procedure and seeing it work is a good thing, and I’m not knocking it. Try to find time for more real math with manipulatives and measurements by leaving out some of the pointless drill sheets — better to do five in-depth questions where you really get a grip on the question than fifty fill-in-the-blanks by rote copying (put this number here and that one there and hurry up!).
The best calculus classes I ever saw assigned only five to ten problems a week to be handed in — but they were some problems! The *worst*, most ineffective calculus classes handed out and collected and graded fifty to a hundred problems a week, two hours a day of grub work. Quantity is not quality. Which is not to say to omit drill, not at all, just to make it meaningful.

Give some exercises of the sort where you draw/measue 1/2, 2/3, 2/4, 4/6, etc. on number lines, and match up the ones that are equal; why are they equal — because they are the *same size*.

I do worry about the skip counting method because I have seen it come to a screeching crash in algebra classes. OK, you can do 2/3, 4/6. 6/9, etc. etc.; what are you going to do with 1/x or a/b or (x+1)/(x-1) ???

It is very important to get the idea that fractions are *equal* if you *multiply* *both* parts (top and bottom, numerator and denominator) by the *same thing*; fractions are *equal* if you *divide* *both* parts (top and bottom, numerator and denominator) by the *same thing*.
BUT — the rule is the RESULT of concrete measurement; measurement and concrete grouping comes first, computational rule later!
As a rule of thumb, students who “explain” math by reciting a rule usually have real trouble in high school and cannot do advanced math at all.

Also please note I say the same thing, not the same number; you can multiply or divide both parts by a number, a letter, an expression in parentheses, another fraction, a complex or imaginary number, or a unit of measurement — don’t over-limit yourself.

Des: on dividing fractions

First, re-word your division problems to yourself as
“How many (blips) in a (blop)?”
Example: 12/2 asks how many two’s are in twelve? The answer is six two’s are in twelve, ie 6 x 2 = 12
Example: 8/3 asks how many threes are in eight? Well, there are two threes and then two units left over —
*** *** **
So the first answer to 8/3 is 8/3 = 2 remainder 2
But that remainder is a problem. Let’s divide the two units up into parts. If you had two pies to share among three people, you would slice each pie into three pieces and each person would take one piece of apple and one piece of mincemeat, for a total of 2/3 of a pie each. So 2 divide by 3 is 2/3.
Now 8 divide by three asks how many threes are in 8, and the answer is 2 2/3; We check (2 2/3) x 3 = 6 + 2 = 8

OK, now going to dividing by fractions. Draw yourself a bunch of bars twelve centimeters long and one centimeter wide. (This is an easy size for paper and twelve splits easily). Divide one bar into half, one into thirds, one into fourths, one into sixths, and one into twelfths.

Now ask yourself:
How many halves in one whole? Answer, two halves make one whole.
1 divided by 1/2 = 2 because 2 X 1/2 = 1.

How many thirds make one whole? Answer, three thirds make one whole.
1 divided by 1/3 = 3 because 3 X 1/3 = 1

Continue with 1/4, 1/6, 1/12. Make sure you *see* this work.

Next step:
what about 2 divided by 1/2? 3 divided by 1/2? and so on?
Well, each one whole unit is two halves. So 2 divided by one half must give us 2 x 2 halves, or 4 halves. 2 divided by 1/2 = 4 because 4 X 1/2 = 2.
________ ________
1/2 1/2 Here are three one-inch bars each split into
________ ________ halves. There are two half-length pieces
1/2 1/2 in each one inch line.
________ ________ We see there are six 1/2s in three
1/2 1/2 whole units, so that 3 divide by 1/2 = 6

Work out several whole numbers divided by 1/2 —draw them!

Then do the same for several whole numbers divided by 1/3, several whole numbers divided by 1/4.

By now you should see a pattern that whole number divided by 1/2 = same number multiplied by 2; divided by 1/3 = multiplied by 3, etc.

Next step, divide by a fraction with something other than 1 in the numerator. What about 4 divided by 2/3?
For a real example, you are making cookies for the holidays and you have four cups of sugar left. Your favourite recipe requires 2/3 cup of sugar. How many times can you make this recipe?

Draw four cups of sugar. Divide them into thirds.

[____] [____] [____] [____]
[____] [____] [____] [____]
[____] [____] [____] [____]

Now, we see that 4 divide by 1/3 = 4 x 3 = 12
But we need 2/3. Shade in 2/3 at a time

[oooo] [oooo] [yyyy] [yyyy]
[xxxx] [eeee] [aaaa] [nnnn]
[xxxx] [eeee] [aaaa] [nnnn]

I see 6 groups of 2/3 here (note that some groups are shared over two cups, but you can always pour the sugar back together.) So you can make six times the recipe.

Let’s look at the numbers:

4 divided by 1/3 = 4 x 3 = 12
4 divided by 2/3 = half of the above = 6 (use twice as much each time, get half as many groups, makes sense)
4 divided by 2/3 = (4 x 3) divide by 2 = 6
= 4 x 3/2

check:
4 divided by 1/3 = 12 because 12 x 1/3 = 4
4 divided by 2/3 = 6 because 6 X 2/3 = 4

Do several concrete problems like this and you will see that yes, you multiply by the denominator and divide by the numerator or in shortand, multiply by the reciprocal of the divisor.

Dividing a fraction by a fraction is the same process, just that the answer often will come out to a fraction or a whole number with a fraction remainder.

Example: 2/3 divide by 1/4
real problem: have 2/3 yard of ribbon, doll’s dress asks for 1/4 yard, how many dresses can we make?
Ask: How many 1/4s go into 2/3?
Think ahead and estimate in reality: two times 1/4 is 1/2, so we can get at least two. three times 1/4 is 3/4 and that is just a bit bigger than 2/3, so we will be a bit short of three times.

2/3 divide by 1/4 = 2/3 x 4/1 = (2x4)/(3x1) = 8/3 = 2 and 2/3
Answer agrees with estimate; we can make two dresses and either get more ribbon or leave a bit off the third dress.
Check: 2/3 divide by 1/4 = 8/3 or 2 2/3 b3cause 8/3 x 1/4 = 2/3

This works, and it is nice to see how and feel safe with it.

Thanks for the post. It was informative. I will try it.

Michelle AZ

Though… not sure I’d waste my time making 2/3 of a doll’s dress —one of the things that aggravated me about some math books is that you were supposed to creatively interpret that and truncate, except of course, for the times when you weren’t.
I’m happy to say that our current text would not have a problem like that without a whole number answer.

Sue — generally I agree with you — that’s the beginning and end of the problem-solving process, understanding and setting up the problem in the beginning, and re-interpreting the answer at the end, which the general push for speed-speed-speed omits. I call this approach telling the whole shaggy-dog story and leaving off the punchline.

On that topic, I have actually seen people praise a set of texts that I could hardly believe when my student brought his homework, texts where the “problems” are presented in minimal note form to reduce the reading load. Sure, you get an “answer” of some sort much faster and with less strain, but what’s the point? You think people are going to pre-digest work for you? What have you actually learned?
In fact, my student, who was language gifted (this is the one who was passed through in reading for three years on a primer level) found the pre-digested notes confusing.

As far as the 2/3 of a doll’s dress, well this is the time to analyze and discuss. What would you do in real life? Buy a bit more ribbon? Leave the trim off the sleeves? Make two and save the rest of the ribbon for something else? It’s in this kind of discussion that you learn to really *use* numbers as opposed to playing magical mystery tour in school.