Third Grade Math question

Factual Questions
81

With different wording, I give my third graders questions like that. “You’ve got four brownies, all the same size, and you want to share them equally among yourself and two friends. How much brownie does each person get?” Kids solve in one of two ways:

  1. Imagine dividing each brownie into thirds; each person gets a third of each of the four brownies, or four thirds brownie.
  2. Imagine giving one brownie to each person, then cutting the remaining brownie into thirds; each person gets one whole brownie and one third of a brownie, or 1 1/3 brownie.

The underlying math is totally appropriate for this age. My criticism is that the problem is so poorly worded that it’s unclear what’s expected.

edit: octopus, while common core encourages kids to think fractionally in second grade and to use words like “one half” and “one third” to describe what’s happening, fractional notation, with numerator and denominator, is introduced in third grade.

82

I’m an elementary math specialist; I teach math methods to “preservice” teachers at a local college; and I have been involved in educational and children’s publishing for many years and have written 1 1/3 metric tons of math content. So this is my beat, so to speak.

Most of what I would have said about this has already been said, and very clearly too, especially by Left Hand of Dorkness (but a bunch of others as well).

A few extra thoughts. First, it is increasingly common for districts and even states to provide lots and lots of problems like this one to schools, with the stated or implied directive “Use these.” As LHoD points out, these are often written on the fly, usually on difficult deadlines, with little editing or oversight. I’m not saying that standard textbooks never contain errors or poorly-thought-out questions and problems, because they certainly do; but they have many more layers of quality control, and so problematic problems like this one are much more likely to appear in “extra” materials like these. I hope I haven’t written problems that were this flawed, but it wouldn’t shock me if I have–that being said, I am pretty darn sure they didn’t make it into print.

Second, a couple of people have hinted at this: writing good problems in context (that is, word problems, or story problems, or problems set in situations) is difficult, and the brain turns to mush after a while. [It certainly has in my case.] The person who wrote the problem pretty clearly was casting around for a situation that kids could identify with in which 4 would reasonably be divided by 3. I bet the writer had already written a few of these: wolfpup’s four pints problem, and a variation of a pizza problem (4 kids are sharing 3 equal-sized personal pizzas; LHoD is correct, math writers can’t manage without pizzas), and a situation in which 4 kids are splitting up 3 same-sized granola bars equally, and needed something else, and had a half-formed idea about water bottles, and wrote it down without really thinking about whether it was a scenario that actually made sense or could be easily parsed by students. --And since there was no quality control and deadlines were super-tight, the question never got flagged, reworked, or removed. Anyway, that’s probably the genesis of it.

Third, just to address claims that mixed numbers are “obsolete” or “useless”–neither is the case.

*The Common Core (I know not all states are using it or planning to use it, but CC is driving textbook creation in this country) certainly expects kids to work with mixed numbers in both fourth grade and fifth grade (and possibly third, though I can’t remember for sure and I’m too lazy to look it up just now).

*Mixed numbers make a lot of sense even to young children: we’ve all heard kids give their age as “5 and a half” or “6 and three quarters,” and elementary teachers often hear kids estimate measurements as “3 and a half feet” (by which they often mean “somewhere between three inches and four inches,” but never mind). Anyway, once they understand fraction notation it is easy for them to understand how to write and read these in mixed number form.

*We also use mixed numbers in certain types of measurement. Several people have mentioned volumes in cooking; there’s also distance (I saw “5 1/2” today on a mileage sign) and length (6 3/8 inches, 5 1/2 feet); even baseball announcers will tell you that the starting pitcher went six and two thirds innings (never mind that the box score will write it as 6.2, which used to drive my father crazy).

In contrast: “improper fractions” (I always prefer the term “topheavy”) aren’t usually easy to gauge the value of just by looking (how big is 75 thirds, anyway?) and are seldom used in the wild. No child says “My age is nineteen quarters”; no one gives the distance to the museum as “seven halves blocks.” Decimals have a lot going for them, of course, but there are situations where we don’t use them, as in measuring sixteenths of an inch or thirds of cups; and for kids, decimals typically require a greater level of abstraction than fractions or mixed numbers. (Except in the case of money, which is sometimes linked directly to decimals, decimals are taught after fractions, at least in the US.)

Anyway, mixed numbers haven’t gone away, aren’t going away, and shouldn’t go away.

83

Yuck. Missed edit window. Wanted to point out

a) that kids don’t necessarily understand exactly what “6 3/4” means when they give it as their age, at least not in a formal sense along the lines of “six full years and then we cut the next year into four equal parts and I have lived through three of them”; but the basic concept of “six and most of the way through to the next birthday” does resonate;

b) “3 and a half feet” (by which they often mean “somewhere between three ***FEET ***and four FEET…”) Not inches.

Oh well.

84

Excellent posts, Ulf! My only addition is that in third grade, kids need to know improper fractions up to 2; it’s kind of a dip-your-toes in it exposure, with more coming in higher grades.

85

Thank you! Yes, I don’t mean to dis topheavy fractions, which certainly are important to learn about. If for no other reason than when adding fractions: 2/3 + 2/3 = 4/3, which is at the very least a logical step on the way to obtaining a mixed number answer of 1 1/3. And third grade is an excellent time for dipping those toes. I’m mostly responding to the idea expressed earlier in the thread that fractions of this type are somehow “better” or more “useful” than mixed numbers.

86

But the Unambiguous part is far more important than the others. If the question is ambiguous (like this one was) then it doesn’t matter if it’s concise, focused or relevant.

87

Assuming it requires all 4 bottles to fill three cups
4/3 or 1 1/3 answer D

Now on the other hand if one bottle fills one cup, and we have 4 bottles and 3 cups
we have use 3/4 of the water to fill the three cups

And what english failure wrote the question?
Shouldnt the teaching establishment be able to write better than the student?
The problem is worded in a very awkward way that one might take it either way.

me i would have written the question like so

Dick has 4 7.5 fluid ounce bottles of water
Jane has 3 10 ounce cups

In a fraction, how much of the total combined water will john need to use per cup to fill all three?

A) 1 1/3
B) 3/4
C) 3.1411
D) 2 1/8
Yes i know, it’s fun with Dick and Jane, but at least you know 100% what i am asking

88

C) i turn around and quickly eat one brownie, eliminating the need for fractions :smiley:

89

That’s just including additional points of confusion if all you’re trying to do is figure out if the kid knows how fractions work. You really don’t need the 7.5 oz and 10 oz info there–it’ll just muddy the waters for a third grader, I would think. It should be something along the lines of “the entire contents of 4 bottles of water completely fills 3 glasses” or something in that vein (and you can clarify that the bottles are all identical in size and that the glasses are all identical in size). Don’t introduce unnecessary numbers. Or simply rewrite the question to something more like Susie needs to evenly divide 4 (identically sized) slices of pizza among 3 people, yadda yadda yadda.

90

Irrespective of who feels they are correct in this discussion, think of how inane all of this would sound to a third-grader.:smack:

It is pathetic that grade schooled children should be faced with such a dilemma in garden variety math classes.

91

I agree with all that 's been said, except to observe that the person writing the questions is not /always/ someone with relevant formal qualifications and experience.

And I’ll add that sometimes questions like this slip through even with the best of intentions and completely adequate budgets.

92

I’ve got an engineering degree with minors in both math and physics and this question is confusing simply because of the ambiguity of the purpose of the exercise.

Look at the original question:

my bolding

And the explanation

my bolding

It’s confusing because the phrase “to fill” is distracting and irrelevant to the idea of sharing.

Once you start to go down a rabbit hole of “filling” cups, you need to know the sizes of the cups compared to the bottles or other information, all of which doesn’t matter.

The problem should have been something like this:

OK, putting “Coke” in would probably be a distraction, so milk or juice.

93

Nah. My kids tease me because I make up so many problems about M&Ms or brownies or whatever. “It’s making me hungry!” they complain. My thinking is that kids like to think about candy and brownies; making problems about these subjects orients their brains toward the work and makes it a little more fun.

So make it about Coke!

If I had to choose the single most important one, it’d be “focused”: if your math problem includes concepts that kids haven’t learned about yet, it’ll torpedo the problem faster than anything else. But yeah, “unambiguous” comes in a close second.

94

I nominate this as a new member of standard responses to :smack: questions.

95

What’s the solution to this, assuming this is meant as a legitimate “fix” for the original question (with the exception of answer option e))? Since you say “What fraction of EACH Coke bottle,” the answer can’t be b or d, since the fraction must be less than 1 (due to the inclusion of EACH).

She also can’t put 3/4 of each Coke bottle into 3 cups, because that would mean she’s putting 9/4 of each Coke bottle into all the cups combined, which is again not possible.

So I guess you intend that she put 1/3 of each Coke bottle into each cup, but you never say she’s emptying each Coke bottle entirely, so the answer could really be any positive fraction less than or equal to 1/3. I guess 1/3 is the intended answer, but only by default, in which case the question should be:

“What is a POSSIBLE fraction of each Coke bottle that she put into each cup?”

(And all this is without noting the issue I have of putting “bottles” into cups rather than “Coke” into cups.)

I’d have written this, and this is what I do - write test questions - for a living.

John makes 4 cupcakes. He and 2 friends eat all the cupcakes John made. Each of them eats the same amount of cupcakes. How many cupcakes does John eat?

(The answer choices are fine.)

Forget about volume - the item is assessing sharing and improper fractions, so a context dealing with volume is not necessary. Also, forget about John’s mom - just have John do it. Why is his mother necessary in this problem?

If the original context must be followed:

John has 4 cups of juice. He pours all the juice into 3 empty glasses. Each glass ends up with the same amount of juice in it. What fraction of the juice John started with is in each of the 3 glasses?

(You could change the question to be “What fraction of the juice John started with is in each glass” but I feel like with 3rd grade, spelling out “each of the 3 glasses” to emphasize the EACH part is a good idea. There is some ambiguity in that “cups” is a measurement and an object, but it doesn’t affect the answer, regardless of which way it’s interpreted.)

96

I had the same reaction to the word “each,” there, as well.

97

Missed my edit window (by a fair margin), but in my second example, the question should be:

What fraction of a cup of juice is in each of the 3 glasses?

(to maintain the intended answer of 1 and 1/3).

(Writing these is harder than it looks, as others have pointed out.)

98

I think the question is written incorrectly. the only unit possible is the cup because each on represents 100% of something. The answer should be A and it should say what percentage of a cup is removed from each bottle.

With the cup as the unit of measure then 3/4 of a cup is removed from each water bottle.

It should read as follows to make any sense:
John’s mom has 4 water bottles. She needs to fill three cups with water. What fraction of cups of water from each bottle will she put into each cup?

99

Is she using all the water? Is she using an equal amount of water from each of the 4 bottles?

100

They could have added “equal amounts are poured from each bottle” but I suppose it’s assumed for the purposes of doing math. It appears the purpose of the question is to think logically about fractions.

I truly think the problem is worded wrong.