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From a Grade 3 NAPLAN test:

"Question 28

Joel draws 24 rectangles. He separates them into 2 groups so that there are 6 more in one group than the other. How many rectangles in the larger group? Write your answer in the box."

I have a multitude of questions.

  • Critique this question.

  • How does this type of question fit in the primary curriculum?

  • How would you teach this?

  • In this context are any principles of linear algebra applicable to teaching kids without knowledge of algebra.

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    $\begingroup$ For the benefit of non-Australian readers, you might want to explain what NAPLAN is, and indeed tell people that you are in Australia. $\endgroup$ – DavidButlerUofA May 7 '15 at 9:29
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    $\begingroup$ Just one comment on how I would teach this: I would probably want to make sure that students can solve the problem even if the rectangles are not all of the same size. This might be broached by pulling from student work, or it might require a follow-up question from the instructor. (Some attention may have to be paid to the phrasing of the term "larger group" -- lest "larger" be thought to refer to, e.g., total area, rather than the number of discrete rectangles.) $\endgroup$ – Benjamin Dickman May 7 '15 at 18:55
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Some of the NAPLAN questions are supposed to be about "working mathematically" and so not be specifically in a particular content area, and be a little bit "out there" so as to test problem-solving. The minimum standard for Year 3 from the NAPLAN website is that the students can "recognise and respond to routine questions addressing known facts in familiar contexts". Based on that criterion, this question is a bit beyond the minimum standard. However, it's certainly not too difficult for most children.

Putting myself into a Year 3 child's mindset, I would solve this by a sort of trial and error method. I'd write down a whole lot of ways to split 24 into two groups and stop when I found some that were 6 apart. The first one I thought of would probably be splitting it equally: 12+12. Then I'd move one from one group into the other: 12+12, 11+13, 10+14, 9+15. Ah. There it is. So there are 15 in the bigger group.

In terms of teaching, I think you'd need to spend time helping children think about solving problems they haven't seen before. If this was a problem you were doing in class, I would ask them: "what does it mean to split them into two groups?" "What ways would there be to do that?" "What does it mean to have six more in one group than the other?". Perhaps they'll give examples for some of them. One of the cool things about this problem is that it requires them to think about how to add things to get a particular answer, which is helping them to think about addition backwards. This will reinforce a heap of number facts in ways you couldn't do just by telling them about it or drilling them on it.

For more problems in a similar vein with the express purpose of helping children learn both content and problem-solving skills, you could try the Mathematics Task Centre or the Back to Front Maths books and resources (both Australian projects).

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First, I think the title you have chosen for your post is misleading or somehow biased. Question 28 hasn't designed to be thought of as a system of linear equations. It can be seen as a problem that fits into the so-called early algebra movement (whether the designers had it in mind or not), the point of which is to help students recognize numerical patterns and structures. It also fits into The proficiency strands advocated by Australian National Curriculum: Understanding, Fluency, Problem Solving and Reasoning.

Critique the question. As it can be seen, I am in favor of the question. To solve it students need to decompose 24 into two addends, realize that the decomposition can be done in many different ways though it doesn't matter in which order they add the addends (commutativity of addition, saving time and effort), consider that they can make a new pair of addends from an old one (e.g., $7+17=(7+1)+(17-1)=8+16=24$), and so on.

How does this type of question fit in the primary curriculum? I hope you have already convinced that it fits and why. If not, I can also refer you to Progressions for the Common Core State Standards in Mathematics to find out that is not just about Australian Curriculum.

How would you teach this? First we are not teaching "this" as an isolated topic, let alone as a system of linear equations. As it can be seen in both curriculums mentioned above this kind of problem is just part of certain more general strands developing over the years.

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    $\begingroup$ completely agree with you about the misleading title and the rest of your answer :) $\endgroup$ – celeriko May 7 '15 at 14:58
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The Singapore curriculum has something I want to call bar diagrams, though I might have the name wrong. The child can draw a picture, show a longer train of blocks (because she thought of the rectangles as blocks) and a shorter one, show the 6 sticking out further on the longer one, then figure out that the shorter part (of one and the whole of the other) is the rest of the blocks, and split in two equal portions.

My only complaint with the problem is that using something concrete and all the same (like blocks) would have made visualizing (an important mathematical skill) easier.

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    $\begingroup$ +1 for My only complaint [...] would have made visualizing [...] easier. I was going to say essentially the same thing until I saw that you beat me to it. Are the rectangles on different pieces of paper so they can be sorted, or does Joel have to cut them out with sissors before sorting? And while I understand that the use of "rectangle" is probably an attempt to kill two birds with one stone (get them familiar with certain math words), I think one bird here would have been enough. Otherwise, I like the problem and think it is grade appropriate (even for the U.S.). $\endgroup$ – Dave L Renfro May 7 '15 at 14:46
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Another (quite elementary) way to solve this problem is:

First, if we divide the $24$ rectangles into two equal groups, there will be $12$ in each group. Now, we actually want one group to have $6$ more than the other group, so we move $3$ from one group into the other. (Why $3$ instead of $6$? Because each one that we move results in the difference between the two groups growing by two.) So we have $12-3=9$ in one group and $12+3=15$ in the other group.

It seems like quite a reasonable 3rd grade problem. It only requires knowledge of two number facts ($12+12=24$ and $3+3=6$) and one important idea (that moving something from one group to another decreases the size of the first group by $1$ and at the same time increases the size of the second group by $1$, so the difference in sizes changes by $2$.)

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Joel draws 24 rectangles. He separates them into 2 groups so that there are 6 more in one group than the other. How many rectangles in the larger group? Write your answer in the box."

Critique this question.

It's a good question, and doesn't need to involve systems of linear equations per your question title. The question is clear and simple to understand with no complex language or strange/confusing sentence structures.

How does this type of question fit in the primary curriculum?

Grouping and counting of objects is well within this age range's comprehension and ability, and this is the grade level where formal division is approached.

How would you teach this?

Two methods:

The larger group has six more than the smaller group, so go ahead and take six and put them in one group. Then split the rest equally between the two groups.

Alternately, a brute force approach:

Group them however you like. Are there six more in one than the other? If so, you're done, count the larger group and answer. If not, move one from one group to the other, and check again.

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  • $\begingroup$ +1 for the multiple methods; I would definitely want to see the first method you suggested reached by the end of the lesson. (But wouldn't expect it to appear at the start [for everyone]...) $\endgroup$ – Benjamin Dickman May 7 '15 at 18:56
  • $\begingroup$ I like these two methods. These could be taken as model answers. $\endgroup$ – pdmclean May 9 '15 at 10:20
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Because you can solve this problem with a system of linear equations does not mean you should.

So I'm with all the respondents who have found the title of the question a little misleading.

However, it is interesting that this question should make you think of a system of linear equations and thereby make you question its relevance to grade 3 students (which I gather must be a young age), because, in my humble and uninformed opinion, this question should make the student think of down-to-earth and practical ways of approaching the problem, rather than mechanically reach for a rigid tool.

This question, it seems to me, should teach students to 1) see that the problem is not about rectangles, but about any kind of discrete objects, like marbles or apples and 2) since 24 is a very manageable number, the student should solve it by trial and error playing with small objects and/or dots on a sheet of paper. After finding a solution, the student should think about whether there might be more than 1 solution. They may also think about numbers other than 24, think about numbers other than 6, groupings other than 2, etc..

I doubt very much that the student is expected to compute the discriminant of a system of linear equations.

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    $\begingroup$ I have this problem, too, sometimes. Knowing too much. A problem that I would solve using algebra, but assigned to kids not knowing algebra. Should I tell them it's too hard for them? Or maybe admit it's too easy for me... $\endgroup$ – Gerald Edgar May 8 '15 at 19:33
  • $\begingroup$ @GeraldEdgar, good point! I've had on a couple of occasions had the opportunity to sit next to mathematical wizards and observe their thought processes, and more than once I noticed that they often map a difficult problem to something simple, by example, by visualization, by analogy, and very quickly too. :-) $\endgroup$ – PatrickT May 9 '15 at 9:47
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I think the wording of the question is a bit abstract. Why rectangles? How does Joel separate them after drawing them? Perhaps specifying objects or using manipulatives may be better.

I suppose I know that it doesn't require linear equations, but, surely, that's the underlying maths involved and recognizing that can only aid the teaching of this question. As long as it doesn't go over their heads one could have a go at treating it as an algebra question. How about writing down _ + ? = 24 and _ - ? = 6 (or larger + smaller = 24 and larger - smaller = 6) and solving by, say, trial and error. It could be generalized (ie replace 24 and 6 by other numbers, or that the larger is twice as large of the smaller). In any case this type of question be used in later years too.

So if it's not an algebra question, is doing it on paper the best way, involving lots of erasing and mess. Otherwise, it's a part-part-whole question with part unknown, according to http://www.math.niu.edu/courses/math402/packet/packet-2.pdf Or is a compare question. Though, this classification just confuses me.

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