Math teachers should grow *algebra eyes and ears. * To have algebra eyes and ears means to be always on the lookout for opportunities for students to engage in algebraic thinking which involves thinking in terms of generality and to reason in terms of relationships and structure, etc. In the post Teaching algebraic thinking without the x’s I described some tips on how to engage pupils in algebraic thinking as they learn about numbers. Likewise in Algebraic thinking and subtracting integers and Properties of Equality – do you need them to solve equation?

Here is another example. How will you use this number patterns in your algebra 1 class so students will also grow algebra eyes and ears?

Let me share how I teach this. I like to simply post this kind of patterns on the blackboard without any instruction. For a few seconds students would normally not do anything and wait for instruction but getting none would start scribbling on their notebooks. When asked what they’re doing they would tell me they are generating other examples to check if the the pattern they see works (yes, detecting patterns is a natural tendency of the mind). When I asked what’s the pattern and how they are generating the examples I sometimes get this reasoning: the first and the second columns increase by 1 so the next must be 5 and 6 respectively, the third and fourth columns increase first by 6, then by 8 so the next one must increase by 10 so the next numbers must be 30 and 31 respectively. That is, . Of course this is not what I want so I would ask them if there are other ways of generating examples that does not depend on any of the previous cases.

In generating examples, students usually start with the leftmost number. I would challenge them to start from any terms in the equation. After this, if no one thought of proving that the pattern will work for all cases, then I’ll ask them to prove it. It would be easier for me and for them if I will already write the following equation at the bottom of the pattern for students to fill up and prove but this method is for the lazy and lousy teacher. A good algebra teacher never gives in to this temptation of doing the thinking of representing an unknown by a letter symbol for their students.

In proving the identity, I have observed that students will automatically simplify everything so they end up with fourth degree expressions. This is another opportunity to challenge the students: show that the left hand side and right hand side simplifies to identical second degree expressions with only their knowledge of square of the sum .

The teaching sequence I just described is consistent with the levels of understanding of equation I described in Assessing understanding of function in equation form.

we can se at first that y=x+1 & z-1=x*y

x^2 + y^2 + z^2 = (z+1)^2

then

x^2 + y^2 + ( xy )^2 = ( x*y + 1 )^2

x^2 + y^2 + ( xy )^2 = ( xy )^2 + 2xy + 1

x^2 + y^2 = 2xy + 1

x^2 + y^2 – 2xy = 1

( x – y )^2 = 1

( x – ( x + 1 ) )^2 = 1

( – 1 )^2 = 1

1 = 1

I would approach this from a different point of view that relates this algebraic equation with simple geometry.

X^2 is the area of a square where each side is X.

The equation shows a pattern of X^2 + Y^2 + (Z-1)^2 = Z^2 since the left two numbers are consecutive and the last two numbers are consecutive.

So, X^2+Y^2 is the different in area between a larger square (Z^2) and a smaller suqare (Z-1)^2.

If you draw the larger and the smaller squares, you can easily see that the difference is two stripes of Z units with one unit overlap. So, we conclude that X^2 + Y^2 = 2Z-1.

Following the pattern that X and Y are consecutive numbers, the next logical pattern is X=5 and Y=6. In that case, X^2 + Y^2 = 61 and we get Z=31.

Thus, we have 5^2 + 6^2 + 30^2 = 31^2.

To generalize it in terms of the first variable: X^2+(X+1)^2+(Z-1)^2=Z^2. where Z = (X^2+(X+1)^2+1)/2.

This way the kids could visualize the equation and makes sense out of it.

Liang

Aurggh — the equation didn’t appear correctly – I thought I had the html tags right. Oh well; I didn’t expect it to be published immediately either.

Alright, Erlina – now you’re just being nasty! What do you mean by giving us a delicious problem without giving us the answer? How will we know if we’re right? And do you have 20 more of these problems so we can keep our students busy for an entire period?

ok, I’m kidding. Bravo, once again.

But, I do wonder if I missed something. I can show that the right and left sides are equivalent algebraically. But when simplified, it seems so abstract and removed from what I see the pattern as, descriptively. (“Add the squares of two consecutive numbers to the square of the product of those two numbers. The sum will be equal to the square of one more than that product.”

On the other hand, there is a certain elegance to: 1×4 + 2×3 + 3×2 + 2x + 1

Note: I understand you won’t want to publish this. 🙂