Monthly Archives: July 2010

Teaching the properties of equality through problem solving

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.

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

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.

Click here on how to introduce the properties of equality via problem solving.

Teach for conceptual and practical understanding

Whit Ford left this comment on my post Curriculum Change and Understanding by Design: What are they solving? He makes a lot of sense. I just have to share it.

I believe the method of planning lessons is less important than WHAT you are asking the students to think about. Most Algebra I and II texts I have come across suffer severely from “elementitis” (see “Making Learning Whole” by David Perkins), which makes it very challenging for teacher to convey “the whole game” to students while still following the text. For example…

A teacher who is talking about how to “collect like terms” is not going to motivate the students as much as one who succeeds in relating this to a more interesting and complex problem which is related to student’s daily lives in some way. This is a HUGE challenge when teaching mathematical abstractions, one I am struggling with as I prepare to teach the first semester of Algebra I using a traditional text. However, it does lead to some interesting potential exercises:

– Ask students to give you examples of two objects in their lives (or in the room). Chances are you will get answers like two apples, two desks, two eyes, etc. Note these on the board as students mention them, then ask… so do you ever come across “two” all by itself? The answer is NO – “two” is an abstract concept, one which we apply constantly in our daily lives, but an abstraction nevertheless.

– So how do we come up with “two” of something? By finding them, collecting them, putting them together, etc. The abstraction of this process is what we have called “addition”. But what kinds of things can you add together and have it make sense? A foot and another foot – certainly. An apple and a pear – only if you recast each as “a fruit” – then you have “two pieces of fruit”. A meter and three centimeters – only if you recast each in the same units – then you have “103 centimeters”. So what is to be learned from this process? We can only add “like” things together, or quantities that are measured in the same units, if the answer is to make any sense. Addition certainly lets us add the quantities of one apple and one pear… 1+1=2, but 2 of what? The answer must make sense in the real world, and the abstract process of adding abstract quantities does not always result in a useful answer.

– So what about 2x+3y? We have two of “x” and 3 of “y”. Can we simplify this abstract expression? Until we know what “x” and “y” represent, until we have been given values for each of them (with units), we don’t even know if adding them together will produce an answer that makes any sense (see apples and pears above). Furthermore, since we have differing quantities of each, we will have to postpone combining them until we know values for each variable (since one value must be doubled, while the other must be tripled). On the other hand, if the problem were 2x+3x, we are being asked to assemble two and three of the same quantity “x”… intuitively, this MUST be 5 of the same quantity “x” – no matter what quantity and units “x” represents, since the units of both terms will always be the same.

I am hoping that such approach (extended considerably with more examples and practice) will begin to build both a conceptual and a practical understanding of the mathematical abstraction “like terms”, along with how to combine them when they occur… yet, this is just ONE of the many topics covered at a very procedural level by most Algebra I texts. Our challenge is to get students to understand the forest, when the textbook spends most of its time talking about trees.

His post Learning the Game of Learning is a good read, too.

You may also want to check-out my post on combining algebraic expressions . It links conceptual and procedural understanding and engages students in problem posing and problem solving tasks.

Patterns in the tables of integers

Mathematics is said to be the science of patterns. Activities that involve pattern searching is a great way to engage students in mathematical thinking. Here are some of my favorites for teaching positive and negative integers. If you are wondering why most of my posts are about integers it’s because I’m doing a Lesson Study with a group of Mathematics I (Year 7) teachers about this topic. Last week we concluded the first cycle of our research lessons on teaching subtraction of integers.

The first task students need to do with the tables is to list 3-5 observations. From there you can start asking the ‘whys’ for each observation. Sample questions are provided for each table below:

1. Adding integers

Sample questions for discussion:

a) Under what conditions will the sum be positive? negative? zero?

b) Why are there the same numbers in a diagonal?

c) How come that the sum is increasing from left to right, from bottom to top?

2. Taking away integers

Sample questions for discussion:

a) Under what conditions will the difference be positive? negative? zero?

b) Why are there the same number in a diagonal?

c) How come that for each row/column, the difference is decreasing?

3. Multiplying integers

Sample question for discussion:

a) Under what conditions will the sum be positive? negative? zero?

4. Dividing Integers

Sample question for discussion:

a) Under what conditions will the difference be positive? negative? zero?

Feel free to share your ideas/questions for discussion.

You may also want to share other  math concepts that students can learn with these tables.

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