Tag: solving equation

Posted in Algebra

How to derive the quadratic formula

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As I wrote in my  earlier post about solving quadratic equations, introducing the quadratic formula in solving for the roots of a quadratic equation is not advisable because it does not promote conceptual understanding. All the students learn in using the formula is to substitute the values and evaluate the resulting numerical expression. I have seen test questions like “In 2x^2-4x+4=0, what is the value of a, b and c?”  Where is the mathematics in this item?

Not teaching the quadratic formula in solving for the roots of a quadratic equation does not mean that the quadratic formula will not be part of the algebra lesson. It would be a good exercise at the end of the unit to ask students to derive a formula for finding the roots of ax^2+bx+c=0 because you will be talking about vertex and discriminant (if you think they need to know what a discriminant is as this will just add to the terms they need to memorize) in later lessons. However I suggest that you ask the students ‘to solve for x’than ‘derive the formula’.

Problem

Solve the equation ax^2+bx+c=0.

Solution

 

 

 

 

 

 

Express the left hand side as product: x(x+ \frac {b}{a}) = \frac {-c}{a}.

Complete the square:

The rest as they say is pure algebra:

As you can see, deriving the quadratic formula is a beauty. Using it is not. Completing the square and factoring will do for students solving quadratic equations for the first time (ninth grade, for most countries). What is needed at this point is exposure to different problem solving context requiring representations of and solving quadratic equations.

Coming up in the next post is the meaning of this in graphs.

 

Posted in Algebra

Solving quadratic equations by completing the square

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I’m not a fan of  teaching the quadratic formula for solving the roots of quadratic equations because the sight of the outrageous formula itself is enough to make students wish they are invisible in their algebra class. Indeed who wants to have to do withOf course not all quadratic equations can be solved by factoring. Here’s how I try to resolve the situation. Before quadratics, students have been solving linear equations. So if you ask them to solve x^2+4x-3 = 0, chances are, they will use the same technique they learned earlier and this is to put all the x‘s on one side of the equation and the constants on the other side. They will not think of factoring the expression on the left even if they have done hundreds of factoring exercises earlier. For them factoring is another way of representing an algebraic expression and indeed it is. Solving equation means to find the value of x and based on their earlier experience, the technique is to put the x on one side. So this is what they will do:

x^2+4x+3=0

   => x^2 +4x=-3

=> x(x+4)=-3

Students will try to guess and check until they find the values of x that will make the equation true. They will continue to use this technique until you give them something like x^2+4x-3=0 which will make the procedure very tedious. This will be the time to prompt them to think of how easy it would be if the one of the side where the x’s are is a perfect square like in x^2=10 where x = + \sqrt{10} or in (x+2)^2 = 10 so that they will have x+2= + \sqrt{10}. So the problem now is to make the side x^2+4x a perfect square. A visual representation of the equation will be handy. Students should have no problem thinking of a rectangle as visual representation of a product.

Clearly the left hand side is not a square. The way to make one is to cut-off half of the 4x area. But it makes an incomplete square!

Let’s complete it by adding a 2 by 2 square. To keep the balance we add the same amount on the right hand side.

It should be now easy solving for x by extracting the root and using the properties of equality.

I believe that this process will make sense more than using the quadratic formula. Students just memorize the formula without understanding. They also will not remember a piece of it the next day anyway. I’m not saying the quadratic fomula is not completely useful. One application of it is on using the Cosine Rule for ambiguous case.

Should the method of factoring be taught first? I believe it’s best to introduce the students to the method of completing the square first (with the visuals, of course). Once the students get the hang of this procedure, the first thing that they will drop is drawing the rectangle and square and just do it mentally.You can later ask them to investigate the structure of quadratic equations where it is  no longer necessary to transfer the constant on the other side. Solving quadratic equation by factoring therefore is a shortcut students should deduce from the procedure of completing the square.

Any new procedure should be linked to previously learned procedure or it should be an improvement of the first. This is my reason why I think the process I described above is a natural sequence to the process of solving linear equation that students already learned. Another reason is that most of the problems students encounter involving quadratic equation is of the form x^2 +bx=c rather than x^2+bx+c=0. For example, “Two numbers differ by 4 and their product is 3. What are the two numbers?” The major reason of course is that it will always work for all quadratic equations. Check out the visuals for solving ax^2+bx+c=0.

I also developed a geogebra applets Completing the Square Solver and Quadratic Equation Solver that I posted in AgIMat. You can use them to solve quadratic equations and to investigate their roots.

 

Posted in Algebra, Mathematics education

When is a math problem a problem?

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One of the main objectives of mathematics education is for students to acquire mathematical habits of mind. One of the ways of achieving this objective is to engage students in problem solving tasks. What is a problem solving task? And when is a math problem a problem and not an exercise?

What  is a problem solving task?
A problem solving task refers to a task requiring a solution or answer, the strategy for finding such is still unknown to the solver. The solver still has to think of a strategy. For example, if the task,

If x^2 - 7 = 18, what is x^2 - 9 equal to?

is given before the lesson on solving equation, then clearly it is a problem to the students. However, if this is given after the lesson on solving equation and students have been exposed to a problem similar in structure, then it cease to be a problem for the students because they have been taught a procedure for solving it. All the students need to do is to practice the algorithm to get the answer.

What is a good math problem?

The ideal math problem for teaching mathematics through problem solving is one that can be solved using the students’ previously learned concepts/skills but can still be solved more efficiently using a new algorithm or new concept that they will be learning later in the lesson. If the example above is given before the lesson about the properties of equality, the students can still solve this by their knowledge of the concept of subtraction and the meaning of the equal sign even if they have not been taught the properties of equality or solving quadratic equation (Most teachers I give this question to will plunge right away to solving for x. They always have a good laugh when they realize as they solve the problem that they don’t even have to do it. They say, “ah, … habit”.)

Given enough time, a Year 7 student can solve this problem with this reasoning: If I take away 7 from x^2 and gives me 18 then if I take away a bigger number from x^2 it should give me something less than 18. Because 9 is 2 more than 7 then x^2 - 9 should be 2 less than 18. This is 16.

Why use problem solving as context to teach mathematics?

You may ask why let the students go through all these when we there is a shorter way. Why not teach them first the properties of equality so it would be easier for them to solve this problem? All they need to do is to subtract 2 from both sides of the equal sign and this will yield x^2 - 9 = 16. True. But teaching mathematics is not only about teaching students how to get an answer or find the shortest way of getting an answer. Teaching mathematics is about building a powerful form of mathematical knowledge. Mathematical knowledge is powerful when it is deeply understood, when concepts are connected with other concepts. In the example above, the problem has given the students the opportunity to use their understanding of the concept of subtraction and equality in a problem that one will later solve without even being conscious of the operation that is involved. Yet, it is precisely equations like these that they need to learn to construct in order to represent problems usually presented in words. These expressions should therefore be meaningful. Translating phrases to sentences will not be enough develop this skill. Every opportunity need to be taken to make algebraic expressions meaningful to students especially in beginning algebra course. More importantly, teaching mathematics is not also only about acquiring mathematical knowledge but more about acquiring the thinking skills and disposition for solving problems and problem posing. This can only happen when they are engage in these kind of activities. For sample lesson, read how to teach the properties of equality through problem solving.

Finally, and I know teachers already know this but I’m going to say it just the same. Not all ‘word problems’ are problems. If a teacher solves a problem in the class and then gives a similar ‘problem’ changing only the situation or the given ‘numbers’ but not the structure of the problem or some of the condition then the latter is no longer a problem but an exercise for practicing a particular solution to a ‘problem’. It may still be a problem of course to those students who did not understand the teacher’s solution. I’m not saying that this is not a good practice, I am just saying that this is not problem solving but an exercise.

You may also want to read How to Solve It: Modern Heuristicsto further develop your problem solving skills.

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.