Tag: teaching algebra

Posted in Number Sense

Teaching algebraic thinking without the x’s

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Students need not wait till they meet Mr. x to learn about algebra. In fact, the best way to learn about algebra is to learn it while there aren’t x’s yet; when all the learners need to deal with are concepts that still make sense to them. Here is a list of tips and ways for teaching algebraic thinking as pupils learn about numbers and number operations.

1. Vary the “orientations” of the way you write number sentences.

For example, 5 + 20 = 25 can be written as 25 = 5 + 20. The first expression is about ‘doing math’, the second engages students about ‘thinking about the math’, the different representations of the number 25. The thinking involved in the second one is ‘algebraic’.

2. Be mindful of the meaning of equal sign

If you want to ask your learners to find, for example, the sum of 15 plus 6, do not write 15 + 6 =___. It’s a recipe for misconception of the meaning of equal sign. I recommend: What numbers is the same as (or equal to) 15 + 6? Better, What number phrases are the same as (or equal to) 15 + 6? This last one promotes algebraic thinking.

3. Encourage learners to generalize. Continue reading “Teaching algebraic thinking without the x’s” →

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.

Posted in Algebra, High school mathematics

Teaching the concept of function

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Mathematics is not just about the study of numbers and shapes but also about the study of patterns and relationships. Function, which can define some of these relationships, is an indispensable tool in its study. Function is the central underlying concept in calculus. It is also one of the key concepts of mathematics that can model many quantitative relationships.

Textbooks and teachers usually introduce function via a situation with the related quantities already identified. What is required of the  students is to learn how to set up and represent the relationships in tables, graphs, and equation and analyze the properties. In the real world, when function is used as a model, the first thing that needs to be done is to identify the varying quantities. So, it is important to let students identify the quantities and let them determine which of these quantities may be related. This way they get a sense of what function really is and what it is for. The function is not the graph, not the table of values, and not the equation. The function is the relationship between the variables represented by these. The study of function is the study of these relationships and their properties, not finding y  or f(x) given x and vice versa, not reading graphs,  and not translations among the representations. These are important knowledge and skills, yes, but only in the context for investigating or learning more about the relationships between the quantities, that is, the function. Thus, for an introductory lesson for function, I find it useful to use a situation where students themselves will:

  1. identify the changing and unchanging quantities;
  2. determine the effect of the change of one quantity over the others;
  3. describe the properties of the relationship; and,
  4. think of ways for describing and representing these relationships.

These are the ‘big ideas’ students should learn about function. Of course, there are others like looking or dealing with function as a mathematical object and not only as a process or procedure for generating or predicting values. However, for an introductory lesson on function, teachers need not focus on this yet.

Sample introductory activity:

What are the quantifiable attributes or quantities can you see in the figure below? Which of these quantities will change and remain unchanged if GC is increased or decreased? Click the figure and move point C. Are there ways of predicting the values of these changing quantities?

teaching function
Identifying related quantities

Click here or the image above to go to dynamic window for the worksheet.

I like this particular activity because it gives students the opportunity to link geometry/measurement concepts to algebra and learn mathematics through solving problems.

Click link for a synthesis of the evolution of the definition of function and What are the big ideas in function?