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A while back I stumbled on teacher's website that advocated a different way to teach factorization. Rather than jumping straight to factorization practice, the teacher first had their student's practice finding pairs of numbers with a given product and sum. That is, they would give them tons of problems of the form:

If $a \times b = 2$ and $a+b=3$, then what are $a$ and $b$?

Rather than teaching them a method for finding $a$ and $b$, the goal was to get students to 'see' how a given sum and product can be formed. The idea was that when the students were introduced to factorization, they would have extensive experience with one of the basic skills required, and they could focus on understanding the conceptual structure.

The thing is, I can't find the website again. Is there a standard name or reference for this kind of method?

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    $\begingroup$ I have heard this approach done with factor diamonds. In the link, see page two. Is this something like what you had in mind? $\endgroup$ – Benjamin Dickman Oct 26 '15 at 21:30
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    $\begingroup$ Yeah. It's not exactly the same format, but the basic idea is the same. A big difference is the above was taught without any reference to factorization. It was just presented as a kind of puzzle that's worth doing by itself. $\endgroup$ – Nathan Oct 26 '15 at 21:47
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    $\begingroup$ could you elaborate more on what you mean by "see how a given sum and product can be formed"? $\endgroup$ – celeriko Oct 26 '15 at 22:11
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    $\begingroup$ The contrast is with teaching a method to derive the terms. So rather than solving for b in the second equation and substituting the expression in the first, the goal is to get the student to spontaneously generate the answer "1 and 2", and likewise with other such problems. $\endgroup$ – Nathan Oct 26 '15 at 22:19
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One way this is sometimes done is with factor diamonds.

(Google or google image: factor diamond method. Here is a sample result.)

In your question, you gave the example of $a \times b = 2$ and $a + b = 3$.

One could present this as the following puzzle:

enter image description here

I made this image with the goal of including relevant vocab (product and sum) and I purposefully avoided writing $a = \square$ and $b = \square$ because I thought it might be a nice opportunity to vary the presentation of the equal sign (to help avoid some of the problems mentioned in MESE 7964).

(Note: These puzzles are sometimes used to scaffold towards factoring quadratic expressions.)

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    $\begingroup$ (Since there are two solutions, $a=1, b=2$ and $a=2, b=1$, this could present a nice opportunity to have students try and explain why the values of $a$ and $b$ are interchangeable. In particular, they are interchangeable in each of the given equations around the product and sum because multiplication and addition, respectively, are commutative.) $\endgroup$ – Benjamin Dickman Oct 26 '15 at 22:33

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