October 2012 - Mathematics for Teaching

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.

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.

Math blogs

Math blog carnival

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This site will be hosting the 25th edition of Math and Multimedia Carnival which will go live at the end of this month, October 31.

A math blog carnival is a collection of articles from various math blogs and sites. So if you are a blogger, this is an opportunity for you to promote your favourite or latest posts and yes, your blogs for free. Below is a collection of blog carnivals I previously hosted.

If you have articles about math problems, puzzles and games, tips for teaching math and specific topics in math, videos, tutorials, lessons, curriculum materials and book reviews, math trivia especially about the the number 25, etc, you may email the permalinks to me or use the math and multimedia blog carnival submission form.

Please share, like, and tweet so more bloggers will know. Thank you.

Algebra, Number Sense

The many faces of multiplication

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The following table is not meant to be a complete list of ideas about the concept of multiplication. It is not meant to be definitive but it does include the basic concepts about multiplication for middle school learners. The inclusion of the last two columns about the definition of a prime number and whether or not 1 is considered a prime show that there are definitions adapted to teach school mathematics that teachers in the higher year levels need to revise. Note that branching and grouping which make 1 not a prime number can only model multiplication of whole numbers unlike the rest of the models. Multiplication as repeated addition has launched a math war. Formal mathematics, of course, has a definitive answer on whether 1 is prime or not. According to the Fundamental Theorem of Arithmetic, 1 must not be prime so that each number greater than 1 has a unique prime factorisation.

If multiplication is …	… then a product is:	… a factor is:	… a prime is:	Is 1 prime?
REPEATED ADDITION	a sum (e.g., 2×3=2+2+2 = 3+3)	either an addend or the count of addends	a product that is either a sum of 1’s or itself.	NO: 1 cannot be produced by repeatedly adding any whole number to itself.
GROUPING	a set of sets (e.g., 2×3 means either 2 sets of three items or 3 sets of 2)	either the number of items in a set, or the number of sets	a product that can only be made when one of the factor is 1	YES: 1 is one set of one.
BRANCHING	the number of end tips on a ‘tree’ produced by a sequence of branchings (think of fractals)	a branching (i.e., to multiply by n, each tip is branched n times)	a tree that can only be produced directly (i.e., not as a combination of branchings)	NO: 1 is a starting place/point … a pre-product as it were.
FOLDING	number of discrete regions produced by a series of folds (e.g., 2×3 means do a 2-fold, then a 3-fold, giving 6 regions)	a fold (i.e., to multiply by n, the object is folded in n equal-sized regions using n-1 creases)	a number of regions that can only be folded directly	NO: no folds are involved in generating 1 region
ARRAY-MAKING	cells in an m by n array	a dimension	a product that can only be constructed with a unit dimension.	YES: an array with one cell must have a unit dimension

The table is from the study of Brent Davis and Moshe Renert in their article Mathematics-for-Teaching as Shared Dynamic Participation published in For the Learning of Mathematics. Vol. 29, No. 3. The table was constructed by a group of teachers who were doing a concept analysis about multiplication. Concept analysis involves tracing the origins and applications of a concept, looking at the different ways in which it appears both within and outside mathematics, and examining the various representations and definitions used to describe it and their consequences, (Usiskin et. al, 2003, p.1)

The Multiplication Models (Natural Math: Multiplication) also provides good visual for explaining multiplication.

You may also want to read How should students understand the subtraction operation?

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