Wednesday, November 13, 2013

'Half as many as' vs 'Twice as many as' Revisited

OVERVIEW
Hundreds of math posts over the last 6+ years and my most popular post is still by far "There are twice as many girls as boys...".
http://mathnotations.blogspot.com/2008/08/there-are-twice-as-many-girls-as-boys.html

Still hard to do word problems without coping with the vagaries of language and grammar in particular.

Contrast phrases such as

'Half OF the girls' and 'Half AS MANY Girls as' OR

'Twice as many X's as Y's'

These are confusing enough for native English speakers. Imagine the terror if you're not! Phrases like these don't respect anyone!

THE PROBLEM

Attendance at a stadium broke down as follows...
Of the children, half as many girls as boys
Of the adults, half as many women as men
The number of females was what fractional part of the total attendance?

(A) 1/2  (B) 1/3  (C) 1/4  (D) 1/6  (E) cannot be determined from info given

Answer: B

REFLECTIONS

1. Although the title of the post suggests the focus was on interpreting the phrase "as many as", there are some significant underlying ratio concepts here.

2.  As I discussed in my earlier post, I've observed that students do better with the semantics if they first decide which is the larger of two quantities. Thus "half as many girls as boys" hopefully suggests that there are fewer girls but my experience is that some are simply blocked by the sentence structure and will guess randomly or say nothing. Remember we can always go back to concrete numerical relationships:

"Ok say there are 12 boys.
If there are half as many girls, then how many girls?"

Note how I not only used numbers but I also inverted the problem by giving the number of boys first. Yes we are also teachers of reading! Will students do this on their own? If trained!!

3. From this point there are many solution paths and I would definitely allow upper  elementary or middle school students to play with this for a few minutes in their groups. There's no rushing this process.

We can simply explain our method to them but this is only a part of their learning. Of course that's my opinion and there are many out there who would cringe at this. The eternal battle between The Direct Instructionists and the Constructivists! Those are just labels about which I care little. Whatever works... Since I can speak from 40 yrs of experience I know what worked for my students...
Besides I've already debated this ad infinitum and ad nauseam with some of the best. No one ever changes their mind!

4.  There are some subtle part:whole ratio concepts embedded here. Isn't it tempting to pick choice (E) here because we're not given the Adult:Child ratio. But the question asked for the ratio of the combined female to the total. It is very instructive to see this algebraically.

5. 'Plugging in' convenient numbers for the subgroups in this problem makes it accessible to 4th-5th graders. Organization is very helpful. I use a tree model to represent this kind of data but most do not do this.  Say there are 2 girls, 4 boys; 5 women, 10 men. Then the number of females (7) is still one-third of the total (21)!

6. Hopefully my readers will suggest more efficient ratio methods, Singapore models, other algebraic representations, etc... OR no comments at all!

2 comments:

letsplaymath.net said...

I love ratio problems, because they are so confusing at first and then a little bit of work can make it all clear. In bar models, this is a two-unit puzzle that simplifies to a single unit.
First unit = girls:
[---]
Boys are double that:
[---][---]
Second unit = women:
[-]
Men are double that:
[-][-]
What we are really interested in, however, is females, so we can tie these units together:
[---][-]
And males are double that:
[---][-][---][-]
Of course, the big trick is to get students to see that "half as many girls as boys" is the same as a 1:2 ratio, one unit of girls for every two units of boys. As you said, it often helps to ask students first which group is largest, and then encourage them to try simple numbers until they can see the relationship. In Singapore math, kids get lots of experience with ratio-like word problems in the elementary years working up to the study of ratios in middle school.

Dave Marain said...

Beautiful bar model explanation! I've missed this type of interaction. Thank you!