More of the same...
In a certain group:
The ratio of males to females is 4:5.
The ratio of left-handed people to right-handed is 1:11 (assume no one is ambidextrous!).
64% of of the left-handed people are males.
(a) What % of the males are left-handed?
(b) What % of the females are left-handed?
Comments;
- If students or any of us see enough variations of these, will they become almost mechanical or does one have to decide which method/model/representation is needed for each problem?
- How many models should students be shown for these? Usually students or our readers will find a method or model no one else imagined!
- Is algebra the most powerful method? The most efficient? How many variables? Is it usually best to use one variable and let it represent the total number of people in the group?
- Anyone ever use a matrix/spreadsheet/table/Punnett square model to represent the data for these kinds of relationship problems? Specifically, problems in which the entire group (the universe) is divided into either groups A and B or groups C and D. This will become less cryptic as the discussion unfolds.
- Do you think most students today would feel more comfortable working in %, decimal or fraction form? What about rest of us out there?
- Too challenging for middle schoolers or not? Math contest problem or just a challenge to develop facility with ratio thinking? How would most algebra students fare with this?
Posts
6 comments:
This is not too challenging for them. In fact, an understanding of Bayesian analysis is vital to any citizen.
Agreed, Eric.
So our next poll will be to determine what % of our citizenry or readership is as comfortable with Bayesian principles as you are! I would like to see conditional probability and Bayes formula developed before students graduate high school, however, I'm not sure how much time is spent on discrete math, probability in particular, in most secondary curricula. I'm hoping others who visit here will be able to enlighten us about their local district or state standards.
Since economic and other decisions that affect all of us are based on statistical principles, I believe that students need to have a somewhat deeper understanding of fundamental principles here than is currently the case.
Yes, I'd like to see conditional probability and problems like this, but NOT Bayes formula -- that formula tends to lead to mindless memorization and bizarre attempts to plug in.
If you give them a tool or two or three -- I particularly like the "table" or "punnet square" approach for these problems -- then they can reason through them quite easily.
Simpson's paradox is another good one to show, and to explain what kinds of situations will cause it.
In response to your questions:
1) I think they can become almost mechanical depending on how you ask them. But if what becomes mechanical is the idea of organizing the information in a table, that's fine. If it gets to be a formula they're plugging into, not so good. You have to be careful to mix some Bayes-type ones with some where all you do is multiply the ratios, so they don't get the idea that all such problems are complicated.
2) I show them several models, emphasizing organizing the information in a table as the way to come back to. I show some algebraic approaches, some with tree diagrams or Venn diagrams, and some with "common sense". What I would suggest here for the "common sense" method is that seeing 4:5 and 1:11 means that there are groups of 9 and groups of 12. So, imagine that there are 9*12 = 108 people. Then (probably) all your calculations can be with integers instead of fractions/decimals/percents. It gets to be a lot more concrete for people that way. If the 108 guess is too hard, I just suggest that they start with 100 people and say how many people are in each category (not being afraid of fractions, since we know it's really a ratio, not an actual number of people). So doing it as a percent seems to help at least a little, because numbers between 0 and 100 seem to hit people's intuition better than numbers between 0 and 1 or fractional expressions.
3) I don't think algebra is all that good. Organizing in a table, usually you can figure everything out without needing to use a variable. If you do need a variable, I think using just one is a good idea, and then chasing it around the table using the given ratios. I might see 1:11 and suggest to people that it should mean 1x and 11x, and then progress from there, figuring out everything in terms of x. I don't like the one variable representing the total: that gets into too many fractions.
4) YES, the matrix/Punnet square approach is the main one I use. Look in an AP Statistics textbook (such as Bock/Velleman/Deveaux) for lots of examples like this.
5) I think percent is better than decimal or fraction with today's students, but integers are even better. I personally prefer fractions.
6) Too challenging for middle schoolers if you do it "cold" but if you teach them the idea of organizing information in a two-way table first, and give them problems that are not Bayesian but just multiplication of ratios first, then I think they can make the transition to this kind of problem.
Joshua,
Thank you for that extensive and thoughtful reply! I agree with almost everything you've said. This question is hardly entry level for middle schoolers. Clearly there needs to be a developmental progression. You and I are usually on the same page when it comes to formulaic approaches vs. understanding.
I do believe that the table is very powerful, although I plan on posting a problem in which the tree model is actually more useful. I strongly agree that choosing 'smart' numbers like 100 or 108 is a great approach. That needs to be explored further in this discussion. I also need to show the Punnett square approach and that should probably also be a separate posting.
"I do believe that the table is very powerful, although I plan on posting a problem in which the tree model is actually more useful."
I agree -- and I do encourage that students should be exposed to the tree approach. But I find that trying to get them fluent with too many approaches usually ends up with them just being confused. So I encourage them to choose one, and I recommend the table, as their main approach; and as we go along, I'll usually do each problem with the table plus one other method for comparison, and some few students will latch on to one of those other methods that makes sense to them.
I'm working with older students for this -- mostly AP Stats students, 11th and 12th grade -- so my decisions might be different with actual middle schoolers. But I doubt it; I think the approach would be pretty much the same.
Our school teaches a unit on counting and probability at the beginning of honors geometry. I'm pretty sure they don't get into any conditional probability though, and I bet the non-honors kids don't get even that.
I didn't get conditional probability until my 2nd year of college (math and CS degree)
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