I'm new to teaching math and I am currently in our chapter on factors and fractions. We have completed the lessons on GCF and LCM and today I gave a lesson on comparing fractions by writing equivalent with an LCD.
A few students asked if they could just cross multiply bc it was a lot easier.
My take on it was that, the concepts I was teaching was different. I gave students strategies to use when one or both denominators were prime and when one denom. is a factor of the other.
What do you suggest? Should I teach to the easy/fast method or teach the LCM=LCD method instead? Or just offer it as another method they an use. But then, do I mark them off on a test/quiz for doing the fast method since it isn't what we took notes on?

First of all, what grade do you teach? (I teach 5th/6th Math & Science.) Second, I can't for the life of me figure out how you could cross multiply to compare fractions. Can you explain that a little more?

Personally, I hate teaching LCM and GCF. I tell my kids that they need to know what those mean and how to find them because they will be tested on it on standardized tests, but that in reality, it really doesn't matter if you find the LCM for a common denominator or the GCF to reduce a fraction as long as find *something* that works. The worst that can happen is that you end up with fractions with big numbers (using the LCD obviously gives you the smallest numbers to work with) or that you may have to simplify more than once at the end. So what? I tell my kids (maybe I'm wrong - maybe I'm not being a good math teacher here - ?) to just figure out any denominator that works for the two fraction and make equivalents, and then to just check their answers to make sure they are completely simplified - and if not, simplify again. (I'm thinking about adding fractions here - not just comparing.) I don't know - maybe my way sounds like more work, but who wants to sit there and write out all the factors/multiples of a pair of numbers just to find the "right" one? My kids sure don't - and neither do I.

Anyway, sorry for rambling - I've had a really long day and am running on very few hours of sleep. : )

Many years ago a more experienced teacher taught me this trick and I found that it worked great and we didn't end up with large denominators that had to be simplified.

Example: 3/4 x 4/12=
Children look at the larger of the two denominators (12). Ask yourself: Can I divide the smaller (4) into the larger (12) evenly. If yes, the the larger (12) is the LCM. They then changed 3/4 to 9/12 by multiplying the numerator and denominator by 3.

Example: 3/4 + 2/9=
Children look at larger denominator and ask themselves if the smaller can be divided into the larger evenly. In this case the answer is no. So we then multiply the larger numerator by 2 and get 18 in this case. Ask if 18 can be divide by 4 evenly. No. The next step is to multiply 9 by 3 to get 18. Can 18 be divided by 4 evenly. No. So we would go on to multiply 9 x 4 for 36. Can 36 be divided by 4 evenly? Yes, so 36 is the LCD and we would have to change both to an equivalent fraction.

My fifth graders caught on quickly. I introduced them to this after we had practice listing multiples to find the LCD several times for understanding.

As far cross multiplying, I only encouraged them to use it to determine if two fractions were equal or not on those higher level questions. Example: Is 2/3 and 4/10 equivalent. 2x10=20;3x4=12 so the answer would be no.

Hope some of this makes sense to you. Do you think the children have so much trouble with fractions because they rarely use them in their daily lives or is the concept introduced to them before they are developmentally ready? We've debated the issue at my school for years. It would be interesting to find out how others feel about it.

JudyW said: "Do you think the children have so much trouble with fractions because they rarely use them in their daily lives or is the concept introduced to them before they are developmentally ready?"

Wow, I think this statement is right on. I have vented plenty of times about this but no one I've talked to seems to feel it's anything to be concerned about. I teach 5th and 6th grade Math - which I really enjoy because I absolutely love math - and I have often felt that we are trying to cover way too much material. It often feels like we are scrambling every day to learn some new concept when a lot of the students have barely gotten ahold of the basics. Why do 6th graders need to know how to solve algebraic equations anyway? I've always felt that if those beginning years - yes, all the way up to 6th grade - were spent focusing more on getting a very solid foundation in basic number concepts and problem solving skills, that when the time came for them to do algebra, they'd pick it up with no problem - instead of struggling with it for years. I really believe that some of the algebra concepts that I teach are too abstract for some of these kids. (Although, I suppose, to be fair, it's possible that I don't know how to teach it "correctly".) I don't see the benefit to learning (if you can call it that) algebra at 11 years old. I just can't buy in to the belief that if they don't learn it starting in 5th or 6th grade, they'll end up way behind by high school. Many kids are *still* way behind in high school because they've never learned solid math/number concepts and yet are made to continue on to a new concept (or whatever you want to call it) with every new lesson.

To get back to your original statement - I think both things are true: They rarely use them (fractions or other) in their daily life AND they're not developmentally ready. But both make the point that maybe they don't need to be doing that kind of math yet. I think they can't see the relationship to real life BECAUSE they're not developmentally ready.

A story I read in one of my classes in college is a good illustration, I think. It's about physical development - not mental, but still makes a good point. It was a story about identical twins who were about three years old (I think). The researchers wanted to know if training and practice would help one twin learn to ride a tricycle sooner than the other one who had no training and practice. So they worked and worked for a long time with the one twin to teach him how to ride. The child struggled for a long time but finally learned and became proficient. About the time the first twin was finally able to ride, (I think it took a couple of years??), they put the second twin on a tricycle and almost immediately, he learned to ride it. It turned out there was no benefit to all the time spent teaching and struggling with the first twin, because they were both able to ride at about the same time (when they were developmentally ready) - regardless of how long the one had practiced beforehand.

Ok - some parts of the story don't exactly cross over to the math problem, but I would really be interested to see what would happen if we backed off on the level of difficulty and then presented those concepts when kids were more developmentally ready. My guess (maybe I'm wrong - I don't know everything) is that we'd have a lot less kids stressed out over math, and a lot more successful students farther on down the road.

Anyway, that's my opinion. Don't know how well I've articulated it, so I hope it makes sense.

I've used cross multiplication, myself. It has backfired on me at times, though, when the kids didn't do the "cross" part of the process. I have to explain that it's like making a sideways triangle.

Ok - apparently I was pretty ignorant about the cross multiplying to compare. My high school daughter told me "oh yeah - that's the way I do it all the time." So she showed me - boy, did I feel like a goof. Here I am a math teacher and didn't even know this one. : )

... cross multiplying should NOT be used. Have students find the LCM when comparing and ordering fractions so that they will have that practice when they use it to add and subtract fractions. Another reason for the LCM is the types of problems that involve buying the bag of hot dogs and hot dog buns so that there are none left over.

As far as reducing using the GCF, I always tell them that if the fraction can be reduced, its usually with 2, 3, 5, 7 or 11. I have them try those first. GCF is also used for the types of problems such as: Amy has 12 lollipops, 6 pieces of gum, and 9 jawbreakers. If she wants to place the same number of pieces of each kind of candy, what is the greatest number of baggies she can use?

Shortcuts are quick fixes. If the students doesn't understand why it works, they will suffer when it comes to learning new material in the later grades. Don't make it harder for them than it already is (with standardized testing and all).

As for those of you that say, when are we ever gonna have to use this? Math builds muscle in the brain. The more you think, the more you learn. Its all about critical thinking skills and processes.

Why do you think so many students HATE math. It makes them THINK!

so lets say you have 1/2 and 3/4. Write them down, not slanted, side by side. Start at the 4 and draw a line to the 1. Multiply them. write the answer above the 1. Now, Start at the 2 and draw a line to the 3. Multiply them. write the answer above the 3. Now compare the answers. that's the answer for the fractions. You must start at a denominator and then go diagonally to the other fractions numerator.

For reducing fractions, I find the GCD by collecting the factors of just the smaller denominator, starting from the largest factor (the smaller number itself) and working backward toward 1. Then the first number you come by that divides evenly by the larger denominator is your GCD. Then you divide both the numerator and denominator by that GCD to get the fraction in its reduced form.

This works when you think of it algebraically. Let's start with an example where we have an unknown. 4/n = 1/2. In order to solve, I would want to isolate n, but I would want it to be a numerator and not a denominator. This would mean that I would multiply both sides by n, which gives me 4 = 1n/2. Now to undo the division by 2 and isolate n, I would multiply by 2, which gives me 8=n. I can do these same steps without an unknown because I am doing the same thing to both sides of the equation which balances it out. Often when cross multiplication is taught, the algebra is left out and so we can become confused as to why it works. When comparing 2 numbers, we don't know if they are =, >, or <, but I can find that out after I get rid of the denominators. Here is an example:

1/2 and 3/4

I could make this look like a numerical equation but leave out the equal sign

1/2 ___ 3/4 (the ____ represents either =, >, or <
Multiplying both sides by 2, I get
1 ____ 6/4
Then multiply both sides by 4, and I get
4 ____ 6
Well 4<6 so 1/2 < 3/4

Hope this helps. I just taught this concept yesterday and didn't go into the algebra stuff, so I might do that today!

Mr. De Groot

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