Friday, December 5, 2008

Math problems of the week: 6th grade Connected Math vs. Singapore Math

1. From the Connected Math unit "Data About Us," Investigation 5: "What Do We Mean by Mean?"

A store carries nine different brands of granola bars. What are possible prices for each of the nine brands of granola bars if the mean price is \$1.33? Explain how you determined the values for each of the nine brands. You may use pictures to help you.

2. From "Review 1" in 6th Grade Singapore Math, Primary Mathematics 6A.

The average price of three mugs is \$4. One of the mugs costs \$p and another mug coasts \$3. Express the price of the third mug in terms of p in the simplest form.

3. Extra Credit

There are two sixth grade math problems. One involves the open-ended pricing of nine granola bars, complete with optional pictures; another involves using the formula for averages to calculate an algebraic expression. Estimate the average level of difficulty of the two problems, and then determine which one is more likely to contribute to the Lake Wobegon effect.

Kathleen Jalalpour said...

Once the idea of what an average is has been established, the Connected Math question is quite easy. One simply lists pairs around the average of \$1.33, for example \$1.32 and \$1.34, and the ninth answer is \$1.33 itself. It is a good question for establishing the concept of what an average IS, but has little depth of thought beyond that.
The Singapore Math question on the other hand, is probably only accesible to 6th graders by using a bar diagram. Since 3 mugs @ an average of \$4 cost \$12 altogether, and we know the cost of 2 of them (\$3 and \$p), the model looks like this:

|-----12----|
___ ___ ___
|_3_|_p_|_?_|

Therefore, the 3rd mug costs 12 - (3+p) dollars. This visual approach to algebra is accesible to ALL students, which is the beauty of the Singapore math approach.

Karl Fisch said...

I actually like both problems, but for different reasons. I also have the same concern with both problems.

I like the Singapore Math problem because of the algebraic representation that it lends itself to. I think most 6th graders, with an adequate background of what average meant and the use of a variable like p, could have decent success with this problem. I would most likely have thought of it as 4 + 4 + 4 = 12, therefore 3 + p + the other mug also would have to be 12. Then it would’ve been a little bit of a struggle to get to 12 – 3 – p, but a good struggle.

But I think I disagree a little with Kathleen. While it’s true that the Connected Math problem is easy for a sophisticated mathematical thinker, I’m not so sure that it is for a typical sixth grader. And I like this problem precisely because it doesn’t have one correct answer. For Kathleen it’s relatively quick and easy to get to the answer she described, but in the hands of a good teacher this could take an entire class period (or more) to explore. Not only would I encourage the kids to talk about why they might’ve come up with different answers, but ask them how many possible answers there are. It would also lend itself nicely to explore the difference between mean, median and mode, and when and why it makes sense to use each. I also think encouraging them to use multiple representations (using pictures) is a terrific skill to learn.

But my concern with both problems is that, for any rational sixth grader, the correct answer is, “Who cares?” I know these problems are taken out of context, and both texts may or may not have used more interesting and realistic problems elsewhere in the unit. And I’m not totally against students practicing on problems with no relevance. But for a problem involving mean, and data, there are so many opportunities to make this meaningful, engaging and relevant, I think it’s a travesty not to.

Thanks for sharing these problems.

bky said...

I also like both problems. For one thing, I like the idea of introducing a little algebra as you go along. My 4-5th grade boys (homeschooled samle of 2) are starting to get very comfortable using variables and unknowns in equations -- it's not algebra yet but it is turning into it. The first problem is also good; if you can get kids to realize that you can make up a set of numbers with a given average by doing them in pairs, one above and one below the average by the same amount, whatever the amount is, well, that takes some fluency.

About Karl Fisch's comment about relevancy -- I can only splutter. I will assert that relevancy is not a useful concept in teaching kids math unless we are talking about whether something is relevant to the mathematical topic at hand, which is the case for both problems. Strictly speaking, data is irrelevant to kids if we are talking about relevancy to their lives. Why? Because they're sixth graders. That's why I have never been very enthusiastic about the proto-statistics that crops up in elementary school curricula.

If there is some situation in your class where real data really seems relevant to your kids, I would think you or anyone would promptly use that -- but it just doesn't seem necessary.

For example: problems about mixing paint are a great way to really understand fractions. But no one ever mixes paint. The problems are relevant because fractions are important and thinking about paint mixing forces the student to really use everything he knows about fractions.

Karl Fisch said...

@bky - Sorry, I didn't mean to make anyone splutter.

I'm curious, at what age do you think mathematics should be relevant for kids?

bky said...

The question about relevancy pertains to education in general, not just math. Education is supposed to make more of the world accessible to people than they would otherwise be able to grasp. The more the student learns the more can be relevant to him. The question is not, when to introduce math that is relevant to the student, but to develop the student so that more and more is relevant to him. Is arithmetic relevant to a 5- or 6-year old? Not in the normal sense of the word, but we start to teach it to them anyway; when they start to master adding, subtracting starts to seem relevant, and multiplication too. If a set of problems can help fifth graders learn about mean and median, I would not want to throw them out because they were not relevant to the quotidian lives of students. If you can find problems that are relevant to the quotidian lives of students, sure, those would be good.

lefty said...

In the hands of a good teacher this could take an entire class period (or more) to explore.

And I hope that a good teacher would recognize which students already grasp the underlying concepts (including how many solutions there are) and would allow these students to work quietly on problems like the more challenging Singapore problem while the rest of the class spends an entire class period or more discussing the Connected Math problem.

I suspect that there are more 6th graders who instantly grasp the concepts underlying Connected Math than many teachers realize. For their future, and the mathematical, technological, and scientific future of this country, let's not bore them.

Mrs. C said...

We cook at least once a week, and having a larger family we have to double or even triple recipes. The boys are just beginning to get proficient at doing things like doubling 3/4 cup.

But if you write it down as a math problem, I don't know that you would get that "connection" there. Sometimes you just do the math so that you can get it done. :]