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.

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?

Algebra

Equations, Equations, Equations

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Students deal with a different ‘types’ of equations: equations in one unknown, equations in two unknowns, and the equation representations of function. There are others like the parametric equations but let’s talk about the first three I just enumerated. Is there a connection among all these three apart from being equations?

Let’s take for example the equation 4x – 1 = 3x + 2. To solve the equation, students are taught to use the properties of equality. When the topic gets to equation in two unknowns, this equation is learned independently of the equation in one unknown especially in finding the solutions. When the topic gets to solving systems of equation say 3x+y = 4 and x–y = 5, the methods for solving the system of linear equation – substitution, elimination, graphing – are also learned without making the connection to the methods of solving equation they already know. Then, function comes in the scene; the y‘s disappeared and out of nowhere comes f(x). Most times we assume the students will make the connection themselves.

How can we help students make connection among these three? To solve equation in one unknown, I think we should not rush to teaching them how to solve it using the properties of equality. There are other ways of solving these equations one of which is generating values which I’m sure you use in introducing equations in two variables. Using the example earlier, students can generate the values of 4x – 1 and then 3x+2. This way, the question “What is x so that 4x-1 = 3x+2 is true?” will make sense to students. They will have to find the value of x that belongs to the group of numbers generated by 4x-1 as well as to the group of numbers generated by 3x+2.

Now, why go through all these? Two reasons: 1) to reinforce the notion that algebraic expressions is a generalized expression representing a group of numbers/values and 2) to plant the seed of the notion of function and equations in two unknowns which students will meet later. Of course this does not mean we should not teach how to solve equation using the properties of equality. I just mean we should teach them other solutions that will help students make the connection when they meet the other types of equations.

Another popular tool is the input-output machine which is the same really as the table of values. For some reason they are used mostly to introduce equations in two unknowns or to introduce function. Why not introduce it early with equations in one unknown? Of course you need a second machine for the other expression. The challenge for the students is to find what they need to input in both machine so they will have the same output. The outputs can be represented by the expressions on each side of the equal sign but later you get to the study of function you may introduce y provided that y = 4x-1. Students need to see that this equation does not just mean equality but that it also means the value of y depends on x according to the rule 4x-1. Since every x value generates a unique y value, y is said to be a function of x, in symbol, y=f(x). Since y = f(x), we can also write f(x) = 4x – 1.

In most curricula, the formal study of function comes after systems of linear equation so there’s no hurry with the f(x) thing. The use of the form y = 4x-1 would be enough. If students understand equations this way I think they can figure out the substitution method for solving systems of linear equations by themselves. Graphing would therefore also be a natural solution students can think of. Equation Solver is a simple GeoGebra applet I made to help students make the connection.

Wouldn’t it be nice if students see 4x-1=3x+2 not just a simple equation in one unknown where they need to find x but also as two functions who might share the same (x,y) pair? This will really come in handy later. Solutions #2 and #3 of solving problems by equations and graphs are examples of problems where this knowledge will be needed.

I recommend that you also read my post What Makes Algebra Difficult is the Equal Sign.

To understand is to make connections. This has become a mantra in this blog. Students will not make the connection unless you make it explicit in the design and implementation of the lesson.

Algebra, Math videos

Teaching Equations of Sequence with Mr Khan

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In the following video Mr. Khan’s gave an excellent task and solution on finding the equation of a sequence of blocks. I suggest you stop the video after the presentation of the problem. Let the students solve it first before you let Mr. Khan do the talking.

Mr Khan did give an excellent explanation especially the one about x – 1. The last solution involving the slope and equation of lines was not as clear. This is the part where your students need you. So I suggest that after viewing the video ask the students what part of the video made sense to them and which part was not very clear.

I think it would be best to ask students first about the rate at which the number of blocks is increasing rather than use the term slope. If you want to relate this to slope ask the students to plot the values in the table on a grid. You make then ask what the slope is of the line containing the points.

Additional solutions

Here are two more solutions to the problem. The first solution involve dividing then adding. This leads to a a different expression but will still simplify to 4x-3.

divide then add — Dividing and putting together

The second solution involve completing the figure into rectangles for easy counting then taking away what was added. This leads directly to the simplified equation. Don’t you love it:-) I do. So please share this post to FB and Google. Thanks.

algebraic expressions — Adding and taking away

This post is the second in my series of post on Teaching Math with Mr Salman Khan. The first is about Teaching Direct Variation with Mr Khan.

If students find Khan Academy’s math videos helpful and cool then by all means let’s use them in teaching mathematics. Just don’t let Mr Khan do all the teaching. Remember you are still the didactician.

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Additional solutions

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