Read through the entire Chapter 3 about Ratios and Proportions
Reflect on the following questions and comment below:
1. Pick one or two tricks that relate to the grade level that you teach. Reflect on what is being said (you may agree or disagree) and offer further insight into how you currently teach or will teach this concept in the future and why.
2. Comment on a colleagues post to engage in a meaningful conversation.
** This is a pretty hefty chapter, so please feel free to offer additional comments about what you have read.
I use the formula triangle when I teach Ohms Law (Voltage = Current x Resistance) for basic electricity lessons in Electricity, Computer Technology, Auto Mechanics and machine shop classes. I think it does make it easier for students to see the relationship. When students calculate the problems I make them write down the formula, show all steps, and all units. The students always struggle with why they have to show units and work. No matter how much I hound them some students still fail to show formula, work, or units. Peter Jablonski ONBOCES
ReplyDeleteI am glad to hear it is not just me! With any math concept that involves a formula, I tell them, step one, write the formula. I take off points if they fail to write the formula. And some continue to not do it, but it is improving when they know I will deduct for something so easy.
DeleteIt's true students don't feel they need to show the work. When we did jeopardy the other day and the students were complaining that we wanted them to show their work. I'm not sure if cause they are lazy or they don't feel the need.
DeleteI picked 3.3 Mixed Numbers. Although I never heard of the “MAD” trick, I do just tell them to multiply the denominator and the whole number, then just add the numerator. That is your new numerator and you keep the old denominator. Well, that really isn’t teaching them what is really going on. The fix does make sense to me. I am proud to say that I do use fraction pieces to help the understand what is happening when they go from improper to mixed. I give them a some fraction pieces, say 17 one-third pieces. I have them count up the pieces and write the fraction as an improper fraction. Not so hard! But do they understand what is happening when we take those same pieces and make them a mixed number? They will when they start grouping the pieces! I tell them to make as many wholes as they can, and that is the new whole number. What pieces are left over is the new numerator, because there wasn’t enough to make another whole. The denominator is what the fraction pieces are, obviously. This can also work with 3.4 “Backflip and Cartwheel”. You can just use the pieces in the other direction. Take apart the wholes and count all the fraction pieces seperately. And keep the denominator. That never changes. It is really important to make sure students understand that fractions are like a division problem, the denominator is the divisor.
ReplyDeleteI like to use Money a lot when dealing with fractions. It also helps to show the connection to decimals. Fractions pieces work similar to money examples but sometimes it can be tough when you start getting in to pennies or even nickels because then you are entering the world of large denominators!
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DeleteAudrey, A few years ago I was showing my kids "why" we multiply the denominator by the whole number before we add the numerator. Kids knew what to do, but most had no idea why it worked. I had pictures and fraction pieces on the smart board. After showing a few examples, one of my students almost fell off his chair when the light bulb went on. He said, "That makes so much sense!" It was awesome. I think that once kids understand why something works, then that is the time to show the trick.
DeleteKaren~ Perhaps if children can really show that they are grasping the concepts, they might be able to come up with a trick to use for themselves. This is when I think of my own math skills... I certainly understand math conceptually (for the most part...HAHA), but I find myself using shortcuts and tricks when solving. Teaching tricks INITIALLY is clearly not best practice, but maybe after they have demonstrated conceptual understanding, a trick or two might not be so bad.
DeleteKaren, I agree with you. When kids are given a chance to repeatedly change mixed and improper fractions back and forth and use models first, they often come up with their own formula which happens to be the same steps at MAD as a shortcut.
DeleteSarah,
DeleteGood point on using money.. I too like to use that and pizza. They tend to have some understanding of the slices so I like to use that technique..
Cross multiplication is HOT item amongst students. They can whine all day long that they hate math but as soon as someone says cross multiply they start to cheer. I don't know what it is or when it happened but they fell in love with this phrase! But what they don't know is that it is a love/hate relationship because they want to use it every time they see two fractions next to each other, no matter the operation between them!
ReplyDeleteI completely agree that the wording must be chosen carefully when dealing with fraction operation or the students will always try to cross multiply the heck out of it! haha
I try to explain proportions and show them examples using scaled drawings and the "human" Barbie (who is not proportionate by any means). This starts to steer them away from thinking about ratios as just a fraction and the equal sign starts to have its own "worth." Then we see the connection between the cross multiplication and the scale factor or the ratio comparison etc.
I will definitely use reciprocal more often, I always try to remind myself to stick with that term more than I do but I fall victim to "old tricks" sometimes myself.
Sometimes we as teachers just teach the tricks we learned not even knowing the conceptual understanding behind it ourselves!
DeleteI like that idea with the "human" Barbie, I will have to remember to use it next year, or even this year with review. Thanks!
DeleteI see that all the time as well! I think we need to stress to students what they actually SEE....is it adding? Is it subtracting? Is it an equation? Based upon what type of problem, then they can draw pictures of use inverse properties to solve.
DeleteHi Sarah, I haven't run into the problem of my 6th graders trying to cross multiply everything. I introduce it only as a way to solve proportions. I am lucky we have a strong 5th grade math teacher who never introduces a cross multiply trick with fractions so they don't equate operations with cross multiply just proportions.
DeleteI can honestly say that I had never heard of many of these tricks. I suppose that’s a good thing because it means that I haven’t been using them in class!
ReplyDeleteThe first trick I chose was 3.5…using cross multiplication to divide fractions. This is how I remember being taught division of fractions. It never made much sense. I just did it because that is what we were told to do. I am happy to say that I have never taught division of fractions this way. I have always taught students to multiply by the reciprocal. I do feel that I need to work on helping students understand where this algorithm comes from. If we had more time, I am sure that some of my students could figure this out independently.
I also chose 3.6…using cross multiplication to solve proportions. I am guilty of this one. I am going to work on changing the way that I teach this next year an use inverse operations to solve.
A lot of these tricks I have not heard of either, and I am hoping it is a good thing as well. 3.6 flip and multiply seems like it could be confusing to a student, flip what? I teach this "copy multiply reciprocal" and I show the students examples such as multiplying by 2 is the same as dividing by 1/2 to explain why we do this. I will continue to teach it this way because it entails math vocabulary and the students remember it. 3.7 Cross Multiplying. I teach both math 6 and math 7. In math 6 I show the students equivalent fractions, ratio tables, double number lines, tape diagrams. But, when they get to 7th I show them cross-multiplying. When they start to solve more complex problems and percent problems, cross-multiplying always has seemed the easiest way to go. Maybe this is a copeut on my part, but I think a lot of it has to do with the level of the students. Some students work better with the different models and some students have the abstract understanding.
ReplyDeleteI can honestly say that I have never heard of many of the tricks from this chapter. We have taught children tricks to change mixed numbers into improper fractions(3.3), although not quite the same way as stated in the chapter. I can see how teaching them to simply multiply down, add up, omits the conceptual steps that are happening.
ReplyDeleteIn the chapter, the author talks about how children have great difficulty with fractional concepts, however, students shouldn’t be robbed of the opportunity to develop that understanding. We need to challenge our students to think. I was at a conference yesterday and one key point that I took away was to try and change children’s mindset about dealing with difficult situations. Instead of saying “I can’t”, teach children to say, “I can do it, just not YET”.
We have heard many times Kelly...don't rob students of their struggle to perhaps understand. Having a trick is sometimes a pathway to support learning.
DeleteYES Kelly!! I agree. Im starting to really think the trick can come second once the concept is learned.
DeleteKelly I like that line at the end all too often my students like to put up the wall and say oh it's math I told you i wasn't good at that...I like to say I wasn't good at swinging a hammer either when I was your age but I also didn't let that stop me from trying to..
DeleteKaren~ I'm guilty of multiplying the reciprocal when dividing fractions as well. You made a good point that I hadn't thought of. If there were more time, would we be able to teach more concepts and less tricks?
ReplyDeleteI never knew there were so many tricks! The trick I feel most familiar with is 3.2 Man on the Horse, but I don’t call it Man on the Horse. When taking a fraction and deciding which number is the dividend and which is the divisor, I usually say “Bottom Out”, meaning the bottom number (denominator) goes outside, so then the numerator must be the dividend…inside.
ReplyDeleteWhen I look at all these tricks, I am still feeling on the fence about tricks. I feel that if the students understand the concept, the trick is just there to help further commit a process to memory.
Currently, I use the cross multiply method when I see 2 fractions set equal. After reading the chapter, I really like the inverse operation method to get x alone. This is something that I stress greatly with our students when solving multi-step and literal equations. I also used to use the trick for converting mixed numbers to improper fractions. I will be using more pictures for this in class. Earlier this year, I used a lot of tape diagrams. My students never experienced tape diagrams, but I was able to show them what I learned (from tutoring younger students) to help them visualize a scenario.
ReplyDeleteThats great Kyle!! Im sure over the next couple years you will see kids coming up with much more experience with Tape Diagrams
DeleteI have never seen the Formula Triangle Trick. It doesn't make sense. The fix they have is absolutely correct. Grade 6 students don't need a formula to know that how fast you are traveling times how long is how far you will go. It is just common sense. I use that common sense to write the formula d=rt so that when given the distance students know to use inverse operations to solve. It makes sense to them to divide.
ReplyDeleteOne trick I do use is to cross multiply when solving proportions. This is the last strategy I introduce however. First students solve proportions with tape diagrams, unit rates, double number lines, ratio tables, and equivalent fractions. I introduce the cross products for use when the numbers are not multiples or factors so the proportion is increased by a decimal. Then cross products is excellent.
The one technique that I used today to explain to my student was relating to 3.1.. The question was how many square feet are in 1450 sq"? Boy I could see the struggle was happening so after several attempts to have him tell me how to set up the problem I decided to show him ceramic tile. I have a 12" tile and a 12" with 1"x1" individual tiles for backsplash and showed him the proportion/ratio between the two.. Then the light became clear to use division after of course multiplication to find 144sqin...So I agree with the fix. One that I can't quite get is the 3.4 picture to represent 8/15 as a fraction I think the color got mixed up to represent that although I like the initial set-up to it. It makes it seem as though it is 12/15.
ReplyDeleteWhen showing the he kids how to figure it out do they get it the first time? It's seems pretty easy but also a need if they are going to be in the construction trade. How long does it take them to become good at it.
DeleteThe one technique I use in class is 3.1. Accurate measurement is important in the culinary profession, and as a result, we find that experienced chefs know the types of volumetric containers in the kitchen and have them committed to memory. A common kitchen measurement scheme that my students study. When learning the types of containers with the gallon the students can visualize fractions. Such as 1/2 gallon or 3/4 gallon.
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