Thursday, January 17, 2013

Math problems of the week: 5th grade Investigations vs. Singapore Math

I. The final problem set from the "Measuring Polygons" unit of the 5th grade TERC/Investigations Student Activity Workbook (Unit 5, p. 63) [click to enlarge]:

II. The final exercise from the "Perimeter, Area and Surface Area" unit of the 5th grade Singapore Math Primary Mathematics 5A Workbook (Unit 5, p. 122) [click to enlarge]:

III. Extra Credit:

What do you think is meant by "size" in the Investigations problem above? Does doubling the sides of a swimming pool actually double its "size"?

Anonymous said...

Well, it's not all bad. I may take this problem, show it to my nieces, and use it to demonstrate the perils of sloppy thinking and bad math.

Rivka said...

How foolish of me. As I started to read the problem, I assumed that it would lead the student towards discovering by what proportion doubling the side lengths increases the area.

Hahaha, as if 5th grade Investigations were something like 3rd grade Beast Academy! How unrealistic is that?

Anonymous said...

I need some brainsoap stat. I made the mistake of actually clicking on the TERC page and reading the instructions. Nothing will clean the feeling of "you have got to be s4!++ing me" out of my mind now.

There is no degree to which I could eff up homeschooling that would leave my son as behind as he would be if he'd stayed in our public schools.

KimS said...

Among so many other things, this example highlights one of the primary shortcomings of Investigations. This problem, where the sides of a rectangle are doubled, could be used as a gateway for so concepts, like ratios, proportion, and scaling or perimeter, area, and surface area. The questions students could be asked and scenarios they could explore, to use Investigations type language, are endless. Instead you get an overly simplistic example with nothing more than "double the sides". It's the lack of depth in Investigations that constantly shocks me!

R. Craigen (WISE Math co-founder) said...

Hi Katharine. In this exercise I think we have a slight advantage to investigations. It's not that their page has much to commend it. There is an interesting question that is completely missed by it: what happens to area when lengths are scaled by a constant factor? And it is far too childish for grade 5, and it does not cover sufficiently advanced ideas. It is, frankly, a waste of student time.

Standing by itself, however, the Singapore page is problematic. It is at a good level for grade 5 and the content is quite good. But the presentation is horribly flawed. For one, what do those hand-drawn numbers in the shaded regions represent? Not the surface area of the face, evidently. And not the total surface area, in the examples shown. And if so, one expects correct units. They are also not correct volumes -- they just look random to me, though numerically they are closely-enough related to the dimensions given that they must have SOME meaning.

Awkward wording "...and width and height BOTH of 4 in EACH" ? And why is it "a" box in #2 and #3 but "the" box in #4? Why is it "the surface area" in #2 and #4 but "the area" in #3? Getting pickier still, why is "in." given a period in both text and diagram, indicating an abbreviation, but "cm" is not, even though it is also an abbreviation, though in #3 it is followed by a period, presumably because it ends a sentence?

I'm not sure whether to say that it is commendable or problematic that the exercises randomly use imperial or metric measures. Are both prescribed for use in Singapore? It's better than NWCP, however, where we are using numerical conventions that have been specified by U.N. and government mandate but are not in use in the general public.

In #3 we can forgive that only one side length is specified because it is said to portray a cube. But in #4 we have a "box" with only two side lengths specified, 4 and 6. Okay -- so presumably we are to use extra information to get the other side length or the surface area. But again we're given the unexplained shaded region value "48". Is this the area of the face? No, given the values shown it should be 24 cm^2. Abandon that. We also have the information "32 cm^2 floating above the figure. If the 48 is supposed to be an "area" then one wonders why it is given in large handwriting and the 32 in small typesetting, the former without units and the latter with? Perhaps we can guess that it is the surface area of the top surface (it surely isn't the whole surface area, which is the required value!). But this would require the box to be 16 cm long. This would defy geometric intuition given the figure, in which it looks to be about 6 - 8 cm long.

Auntie Ann said...

Uh...you do realize that this is a photocopy from a *used* workbook page? The numbers on the ends of the boxes represent the student working on the solution. In the first one, the 32 represents a student note of the sum of the surface area of that end plus the paired end on the other side of the box.

Ditto for the others. Any mistakes in the handwriting are by the student, not the curriculum.

Anonymous said...

Craigen, you're pretty funny. I hope you're having us on, because the alternative is frightening.

As far as the hand-lettered numbers, is it really not apparent to you that they were written by a child given these problems, and not printed in the book? Surely you jest.

As to the wording, it's all the way it should be. Are you really having difficulty understanding plain English, or are you imagining what a particularly obtuse child might say? A description like "both of four inches each" might be overly careful, but this should not be a barrier to comprehension for a child of average intelligence.

If you don't know why it's "a box" one place and "the box" the other, that's your deficiency, not the book's. It seems fair to assume that a normal fifth grader knows what a definite article is for, if not why it is required in one case and not the other.

The question of why cm doesn't have a period after it might have been resolved with a bit of investigation on your part. You see, cm is not an abbreviation but a symbol. It is correct to use it without a period, much as km or m. Seriously. Look it up.

As for the last problem, again, most fifth graders of average intelligence would be able to note not only that the 32 cm ^2 refers to the surface area of the top, but also to correctly calculate the width of the top surface as 8, given the depth of the solid printed as 4.

That is because most fifth graders should know that 32 / 4 = 8. With the three dimensions now 4, 6, and 8, a fifth grader should be able to do all the necessary multiplication and addition to arrive at the total surface area. (Hint: there are two points where 48 will be an important number for the student).

Anonymous said...

I did do this with my niece. She estimated that a page that asks for no more math than multiply 4 and 8 by 2 would be first grade work, because first graders are still learning their basic arithmetic.