In my previous post about examples, I described different uses of examples in teaching mathematics. In this post I’ll give a series of examples for us to be conscious about sequencing examples in our lesson. What are the things do you consider when you think of an example in a math lesson? And how do you sequence them? I’ll give an example to answer the questions I posed.
Let us consider a lesson on multiplying polynomials or more specifically getting the product of two binomials. In my previous post FOIL is Distributive Law, I gave a sequence of three examples before the main problem (x+3)(2x-1).
My first example was 3(2x-1). There are at least two ways learners can do it. One is by applying the distributive law. This would be 3(2x)-3(1) and this simplifies to 6x-3. The other solution is by applying the meaning of multiplication by 3. This would be (2x-1)+(2x-1)+(2x-1). Removing the brackets and collecting the terms will yield 6x+3. Both should be discuss in the lesson. The second solution will remind the students why the distributive law works. I could have chosen x(2x+1) but with this you cannot have the second solution.
My second example was 3x(2x-1). This too, has a purpose. After doing example 1, most if not all students will be able to apply the distributive law. This is also an opportunity for reviewing multiplying two x’s. The purpose of this example is not only to check if learners can do distributive law but also for them to think about what is the same and what is different about the expression 3(2x-1) and 3x(2x-1). The second example is actually the first example multiplied by x. So if 3(2x-1)=6x-3, then 3x(2x-1)=x(6x-3)=. Here you did not only make the learners be conscious of the structure of algebraic expressions, you have also created actually another example, x(6x-3), where they can apply the distributive law.
My third examples was -3x(2x-1). It is obvious what dimension I have added to this example. I will leave to you as challenge to think of the different ways learners can work on this based on what they learned in the previous two examples. Because your main problem will need the distributive law, you still need to give emphasis to it among the different method your learners will provide.
The last example I gave was (x+3)(2x-1). This last example is of course the main objective of the lesson. The previous example did not only review students’ knowledge about the distributive property. It also reviewed them about the product of two x’s, of the meaning of multiplying by a number (that this do not generalize to multiplying by a variable), and of course multiplying by a negative number. More than this, you have also created an awareness and communicated the importance of looking at the structure of the symbols they work with. This is important since it will only be by looking at the structure that they will realize that (x+3)(2x-1) is actually the same structure as 3(2x-1) to which they can apply the distributive law. Click here to see the solution.
So what needs to be considered when selecting and sequencing examples? The mathematics. More specifically, the mathematical structure that we want to highlight or the property of the concept that we want the students to focus on. This can be achieved by carefully selecting that what we want to change or vary. This idea is based on variation theory of learning.