Posted in Algebra

How to grow algebra eyes and ears

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, 5^2 + 6^2 +30^2 = 31^2. 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 (a+b)^2 = a^2+2ab+b^2.

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.

Posted in Algebra, Geogebra

Making connections: Square of a sum

One of the ways to connect concepts in mathematics is to make use of the same representation to teach mathematics. Let me take for an example the visual representation of the square of a sum, (a+b)^2 = a^2+2ab+b^2 . This concept is usually ‘concretized’ using algebra tiles. However, if you have facility for computer technology I would recommend using the GeoGebra applet below: [iframe https://math4teaching.com/wp-content/uploads/2011/09/square_of_a_sum.html 650 550]

Suggested tasks:

1. You can show the grid (click view to do that) to teach Grade 3 pupils about area. You can change the dimensions of the rectangles and squares by dragging D.

2. For older students you can give this task (don’t show grid but you can show lengths then drag D): The square AGHC is dissected into rectangles and squares.If the sides of square AEFD is 2 units and that of square DIJC is 3 units,

a) calculate the area of the other rectangles and square?

b) write two numerical expressions representing two ways of getting the area of the big square.

4. Having worked with numerical expressions, students will be ready to work with variables: If AD is x and DC is y, find two expressions for the area of the square AGHC. This of course leads to the identity (x+y)^2 = x^2 + 2xy + y^2 . The popular FOIL method should only come after this activity. For the record, I’m not a fan of this method.

5. This representation can also be used to teach how to calculate expressions such as 0.75 x 0.75 +0.25 x 0.75 x 2 + 0.25 x 0.25. This is one way to help students appreciate an application of the square of a binomial.

6. You can also use this figure to teach the idea of function: If you drag D along AC, how will a change in the distance of AD affect the area of the rectangles and squares?

a) Create a table comparing the area of the quadrilaterals as the side AD increases from 0 to 5 units.

b) Do as in a) but this time compare the perimeters.

c) Express the area/perimeter of each quadrilateral as a function of the length of AD.

I will write about square of a difference in the next post.