[Note: Readers are strongly encouraged to read the extensive comments to this post, which clarify the ideas and provide more detail for introducing ratios to younger children.]
F R A C T I O N....
a:b ≡ a ÷ b
Ratio --
An expression showing how two quantities compare?
A fraction? A percent? A probability?
Other definitions?
Developing ratio concepts is generally considered to be a crucial part of a middle schooler's math development. The following activity is designed to help students develop an understanding of the equivalence of ratios, fractions and percents. Just as importantly, it provides an application of part:part and part:whole relationships.
OVERVIEW OF ACTIVITY
On a table in front of the room the instructor has placed a small empty cup, a large container filled with cashews, a large container filled with peanuts and four large empty containers. In this activity, students will make different mixtures of nuts that are respectively 50%, 20%, 10% and 40% cashews by volume. Students will need to describe the methods to accomplish this and provide mathematical details to support these methods.
INTRODUCTION
Let's get started, boys and girls. Notebooks open, dated ________ and title today's activity MIXED NUTS. Before we are all tempted to enjoy these delicious nuts, does anyone know which nuts are generally more expensive: cashews or peanuts? Other than for flavor, can anyone suggest why cashews are sometimes mixed with other nuts? Ok, suppose a company or grocer wants to sell a mixture that is 50% cashews by volume. Think about
(1) What you think this means
(2) How you would do this
Now write your ideas and methods in your notebook. Be very specific - like a recipe.
To be continued...
Friday, February 22, 2008
MIXED NUTS - A Middle School Activity to Promote 'Ratio Sense'
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13 comments:
I think a lot of teachers would have to use some other tasty treats, because so many schools (ours included) are nut-free due to allergy issues! ;-)
Duh--
What was I thinking, considering how allergic I am to everything - everything but nuts that is!
Ok, please change to organically grown, sugar-free fortified all-natural jelly beans (red and blue only to be politically correct!).
Beyond the poor choice of food items, I'm really interested in your reaction to the mathematical side of the activity. Perhaps we could even use virtual nuts in some simulation software!
The grocer isn't a scientists so
he probably expects us to fill the
bag in which he wants to sell the
nuts half with with cashew nuts.
A scientists would think that the
space occupied by the cashew nuts
(not counting the empty space
between the nuts) must make up 50%
of the overall volume available in
the bag.
Well, the jelly beans aren't as interesting since they all occupy the same volume.
I'm wondering how much attention to the details of finding the volume of an irregularly shaped solid you are expecting as part of this investigation.
If you're just thinking "fill half the space with cashews then the remaining half with peanuts" I think it's way to easy for middle schooler. Maybe with different ratios (which is where you appear to be headed) it would be more appropriate.
Mathmom and mathfr3d--
Hopefully we can get past the irregular shapes and volume considerations!
Originally the activity involved a 100% salt solution which was to be diluted. I chose to change that idea and I'm beginning to regret it! This would have avoided the volume issues but I thought the chemistry aspect required too much explanation. Oh well...
The 50% of course was intended as an opener for discussion purposes to invite all students. In the overview to the activity, the other percents are listed.
One could teach this procedurally:
(1) 20% = 1/5.
(2) Subtract: 5-1 = 4
(3) Combine 1 cup cashews with 4 cups of peanuts (or 1 cup of 100% salt solution with 4 cups water).
Guiding students toward enlightenment of the underlying ideas is far more challenging and is the raison d'etre for this blog.
Constructing a series of provocative leading questions (the Socratic method) takes training and practice.
Helping students understand part:part vs. part;whole relationships here is critical and transcends the particular vehicle one uses for demonstration purposes. N'est ce pas?
chocolate and raisins....
cheap chocolate and good stuff...
this year we did juices and juice drinks and water... far more discussion than I intended, but I am positive that some of my students read some juice labels as a result.
Jonathan
I like the fruit juice idea. Students can see and taste the effect of diluting say, pure OJ with water or other fruit juices. Chocolate and raisins would definitely be a hit -- reminds me of Raisinets or whatever they used to be called!
Jonathan -- the ideas that emerge from sharing our experiences make the whole so much greater than the parts of all of our blogs. I'm just planting seeds or germs of ideas. Since I'm not in the classroom at this point (other than SAT classes), I depend completely on the reality of those who are working with kids every day.
Also, do you think the 40% strength for example would be far too simple for most high school students or do you believe some would struggle with the ratio concept?
Just don't try the similar problem with water and ethanol, Dave. If you add 100 ml of ethanol to 100 ml of water, the mixture takes up only 191.2 ml.
Now, it would take a very tolerant school to let you do that experiment in a math class, but I present that to point out that one needs to show respect to other subjects when one puts forth examples in their domains.
I'm interested to see where you go with this... especially what questions you use to get at the differences between part-part and part-whole relationships. Because, you're right, figuring out exactly what questions to ask when is key to getting students to understand - and it is the most difficult part of lesson planning.
By the way, you've got to leave out the red and blue jelly beans - those are gang colors!
Dan--
The questions would vary of course according to the age, background and overall ability level of the group. There is no methods text that will guide this process. As educators, we all need to experiment and learn from experience (i.e., our mistakes!).
For me, the questions needed to develop the ideas for 6th-8th grade would be very different from questions asked of older students who ostensibly know how to do this (review mode). For the younger child, I would probably begin by mixing equal amounts of each ingredient, say, one cup of A and one cup of B (Thanks to all of the previous comments, I'm now gun-shy to name these!).
"Girls and boys, what part of the mixture is 'A'?" Hopefully, most would say one-half. I might then say, "Another way to show this relationship is to write 1:2. This is called the ratio of A to the whole mixture. Now, how would we write the ratio of A to B?"
To encourage more participation, I've become enamored of 'think-pair-share' since it lessens the anxiety of the shy or unsure student. I used this technique frequently when I did not want chorus response (which meant only the confident ones replied).
In other words, at an introductory level, I would do the demonstration myself and ask the students to give the ratio. Also, note that I would NOT bring in percents at this stage, if the primary goal is to develop pure ratio concepts. That would naturally occur at a later stage, although, depending on the group, someone might suggest 50% immediately. I would still hold off.
An advantage of using discrete objects like jelly beans would be that we can choose random samples of the mixture and see how close we come to a 1:1 ratio between A and B or 1:2 ratio between A and the total sample. No harm in showing connections to relative frequencies and probabilities. That's how children develop the ability to make connections, but this is a judgment call for the instructor (time factor).
Now we would proceed to one part A and 2 parts B. I'm changing this from the original problem, but it's a natural progression when teaching. Again, I would ask BOTH questions, one at a time, and use think-pair-share: "What is the ratio of A to B now? Of A to the total mixture?" Some children will already see how these two ratios are connected.
[If time permits]
After thoroughly mixing the ingredients, we would take a random sample: "Elvira, come up, take this small plastic cup (3 oz) and fill it with some of the new mixture." We would then count the total number of objects, as well as the number of A's and B's. After recording this, we would form the two ratios A:Total and A:B to see how close we come to the theoretical answers from above, namely 1:3 and 1:2.
Yup, this lesson would require the entire 6 1/2 hour school day! Not quite, but it takes time for children or anyone else to acquire concepts. Oh, and yes, the students would need to practice several more of these (a few minutes of guided practice in class and at least 15 minutes of written homework, heaven forbid!).
Dan, when publishing a post, I'm only suggesting a starting point for the instructor. This comment shows just how much thought and planning goes into a lesson, and I'm certain you or others can significantly improve upon these suggestions. This type of extensive 'lesson sharing' rarely occurs in my opinion in the US, whereas, in Japan, teachers meet regularly to evaluate lessons and discuss ways to improve the process. This is not a criticism of educators, rather it is a comment about organization and structure.
Further, my original question involving percents was not intended as an introduction, since the child needs to feel comfortable with 1:3 before tackling 33 1/3%!
Of course, I still like the 40% question for the 8th grader, since all of the other percents involved mixing just ONE part of ingredient A with more parts of B.
Is this what you were looking for? I feel like I just published another post and this may just get buried...
A similar problem appeared yesterday on the Kitchen Table Math blog. It was an SAT question involving ratio. I posted there under a signed Anonymous listing because my google account didn't seem to be working at first. So the more recent 'Anonymous' replies are mine.
I'm not certain of the 'proper' etiquette here. Is it okay to refer to another blog? It just seemed easier than typing in all the same replies. Sorry, Dave, if I'm doing anything wrong.
For the hands on thing, at least one of the ratios should reduce to n:1, so including 40% is important (3:2).
Controlling the ratio and the total amount might be much harder. That's where algebra really kicks in.
Jonathan
Hypatia--
We're one big 'blog-o-family' so of course it's ok to refer to others' blogs, in fact, it's common for links to other blogs to appear in the comments section. BTW, it was coincidental that I chose to do a post on ratios as KTM was doing something similar. Usually, I do the SAT problems!
Jonathan--
The algebraic approach is part of the natural progression from middle school prealgebra to the next level of working with ratios. My question for you was: Do you believe high school students would regard all of this as trivial or have you seen conceptual issues with ratios in any of your students in the past or currently? Better yet, how would you raise the bar with ratios for secondary or stronger students (no pun intended with the 'bar models' used in Singapore math!)?
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