Tag: properties of equality

Posted in Algebra

FOIL Method is Distributive Law

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The FOIL method is not that bad really for teaching multiplication of two binomials as long as it is derived from applying the distributive law or more officially known as the Distributive Property (over addition or subtraction).

distributive property
Distributive Law

The FOIL method is a mnemonic for First term, Outer term, Inner Term, Last term. It means multiply the first terms of the factors, then the outer terms, then the inner terms and the last terms. I would suggest the following sequence of examples before the teacher introduces product of two binomials: Click here for the complete description of how to teach this with meaning.

Simplify the following expressions
  1. 3(2x-1)
  2. 3x(2x-1)
  3. -3x(2x-1)
  4. (x+3)(2x-1)
FOIL Method
The Distributive Law and the FOIL Method

The FOIL method is also related to Line Multiplication. However, while the latter is applicable to any number of terms in the factors like the Distributive Law, the FOIL method is not. It only works for getting the factors of binomials. This is why it is not a powerful tool. The most powerful knowledge is still the distributive property of equality. Before the teacher should introduce any fancy way of calculating, he or she should make sure this knowledge is in place. Sample lesson on how to do this is presented in Sequencing Examples.

Want a more challenging problem for distributive law? Click Application of Distributive Law.

Posted in Algebra

Teaching the properties of equality through problem solving

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problem solvingI like to teach mathematical concepts via problem solving. It right away engages students mind. It creates a need for learning a more systematic way of doing things and hence a reason for learning the concept. It provides a context for making connections. Most important of all, it gives students opportunity to learn before they are taught.

One of the ways of creating a problem solving task for this kind of lesson is to start by giving the solution/answer.  For example, a standard textbook task is to solve for x in an equation.  Why not do it the other way and ask students to find the equation given the solution?

Here is my favorite problem for introducing the properties of equality:

Write three equations in x with solution x = 5.

Fresh from their elementary school math experience of solving equation of the type  15 – ___ = 20 or x + 12.3 = 20, students will generate equations by trial and adjust. That is, they think of an equation then check if it gives x = 5 by substituting the value of x to the equation.  It will not take long for them to realize that this is something very tedious.

Other students will verify their equations by expressing the terms into equivalent structure. I explained this method in my previous post. This method is not also very efficient for some equations. But students have to experience these so that they will be able to appreciate the efficiency of solving equations using the properties of equality.

Usually when they already have at least two equations I will extend the task to:

Two of the equations should have x on both sides of the equal sign .

Challenge them further to think of a more systematic way for making the equations. You will not hear the bell ring before students will realize that all they need to do to have an x on both sides is to add or subtract x on both sides of the equal sign! (Multiplying or dividing by x is a different matter as there is a possibility that x can be zero). Now they know that those ubiquitous equations in their algebra textbooks did not just drop from they sky. They can even create one themselves. The students are applying the properties of equality before they even know what they are! Speaking of constructivist teaching here.

Teachers I shared this with were also very happy because they no longer have to do trial and adjust when they make exercises for solving equations.

Posted in Algebra, High school mathematics

Properties of equality – do you need them to solve equations?

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Should students learn the properties of equality before we ask them to solve equations in one variable? Would it be too hard for them to solve the equation 2(x + 7) = 4x without knowing the properties of equality?

In the grades, pupils learn to find equivalent ways of expressing a number. For example 8 can be written as 4 + 4, 3 + 5, 4 x 2, 10 – 2. Now, what has the pupils previous experience of expressing numbers in different ways got to do with solving equations in one variable?

Let us take this problem. What value of x will make the statement 2(x-5) = 20 true?. The strategy is to express the terms in equivalent forms.

2(x-5) = 20 can be expressed as 2(x-5) = 2(10).

2(x-5) = 2(10) implies (x – 5) = 10

x-5 = 10 can be expresses as x-5 = 15 – 5. Thus x = 15.

This way of thinking can be used to solve the equation 2(x + 7) = 4x.

2(x+7) = 2(2x)

=>    (x+7) = 2x

=>    x + 7 = x + x

=>    x = 7.

Of course not all equations can be solved by this method efficiently.   So you may asked ‘why not teach them the properties of equality first before asking them to solve equations like these?’  Here are some benefits of asking students to solve equations first before teaching the properties of equality:

1.  It makes students focus on the structure of the equation. Noticing equivalent structure is very useful in doing mathematics.

2.  It makes the equations like 3x = 18, x + 15 = 5, which are used to introduce how the properties are applied, problems for babies.

3. It is easier to do mentally. Try solving equations using the properties of equality mentally so you’ll know what I mean.

4. I hope you also notice that the technique has similarities for proving identities.

So when do we teach the properties of equality? In my opinion, after the students have been exposed to this way of solving and thinking.

Here’s on how to introduce the properties of equality via problem solving.