I just want to promote in this post James Tanton’s latest pamphlet on fractions. It’s FREE for download. Just click Pamphlet on Fractions. Tanton writes:
If fractions are pieces of pie, then what does the multiplication of fractions mean? (You can’t multiply pie!)
If fractions are proportions, then what are their units? Amount of pie per student (and not just pie)?
If fractions are points on the number line, then what does half a pie mean?
Fractions are slippery and tricky and, in the end, abstract. It is actually unfair to expect students to have a good grasp of fractions during their middle-school and high-school years. This pamphlet explains why, and offers the means to have an honest conversation with students as to why this is the case. Their confusion and haziness about them is well founded!
What do you say about the statement I highlighted above? What a relief to know that? 🙂
I highly recommend that you also checkout his collection of past MATH ESSAYS.
You may also want to read a couple of my posts on fractions:
The FOIL method is not that bad really for teaching multiplication of two binomials as long as it is derived from applying the distributive law or more officially known as the Distributive Property (over addition or subtraction).
Distributive Law
The FOIL method is a mnemonic for First term, Outer term, Inner Term, Last term. It means multiply the first terms of the factors, then the outer terms, then the inner terms and the last terms. I would suggest the following sequence of examples before the teacher introduces product of two binomials: Click here for the complete description of how to teach this with meaning.
Simplify the following expressions
3(2x-1)
3x(2x-1)
-3x(2x-1)
(x+3)(2x-1)
The Distributive Law and the FOIL Method
The FOIL method is also related to Line Multiplication. However, while the latter is applicable to any number of terms in the factors like the Distributive Law, the FOIL method is not. It only works for getting the factors of binomials. This is why it is not a powerful tool. The most powerful knowledge is still the distributive property of equality. Before the teacher should introduce any fancy way of calculating, he or she should make sure this knowledge is in place. Sample lesson on how to do this is presented in Sequencing Examples.
In my earlier post on Linear Function, I described how a linear function can be recognized based on equation, graphs, and tables. In this article, let us talk about how to derive the equation of linear function (as they are called in calculus) or the equation of a line (as they called in analytic geometry). I will be using linear function and line interchangeably in the discussion as most teachers would do. It is important to note and to make sure that students are aware that when the topic is on linear function and the teacher says “line” as in “What is the equation of this line?” it actually mean, “What is the equation of the linear function represented by this line?”
What is the equation of linear function?
Textbooks define a linear function as a function defined by the equation y=ax+b (or y=mx+b as m is commonly used for this form) or the equation ax+by+c=0. The latter equation is called the standard form of the equation of a line. In my opinion, this form should not be used when talking about equation of linear function because it does not show clearly the relationship between x and y. The equation y=ax+b shows the relationships between the independent variable x and the dependent variable y, where the value of the variable y is determined by the rule ax+b. But as I said earlier, in school mathematics, this is used interchangeably since we can transform one to the other by algebraic manipulation and we would be getting the same set of points from both. For example the equation 2x-y-3=0 could be transformed to y=2x-3.
Where did y=ax+b come from?
In my previous post about linear function I introduced linear function as a polynomial function of at most degree 1 so ax+b defines the linear function x?ax+b (also expressed as y=ax+b), where a and b are constant. In this case, linear function is defined based on the structure of the equation defining it. Do you think this would make sense to students? Telling is never a good way of teaching mathematics. I propose here a simple activity that would lead to the derivation of of the linear equation of a linear function from graphs.
Activity
The Cartesian plane is made up of points and each point is named by the ordered pair (x,y). In the figure on the right, there is something special about the points E, D, C, F, B, G, and A. They all belong to one line. They are collinear. What does it take for a point to be a member of this elite group? Can you just think of any point and say that it belongs to that group? Observe the x and y coordinates of the points on the line. What do you think would the coordinates of the middle point between D and C? How did you get that? What about the middle point between E and D? What would be their coordinates? If you think you have discovered the condition for membership on this line, try more points. Find the coordinates of the midpoints between C and F, F and B, B and G and G and A.
Challenge: Does the point P=(-12,-20) belong to the line where points A, B, C, D, E, F and G are?
Question to the teacher-reader: How would you proceed from here to derive the equation y=mx+b?
Next: How to derive the equation of linear function from its graph (This is the continuation of the Activity presented in this post.)
In mathematics, a linear function is used to name two different but related notions. In calculus and analytic geometry, a linear function is a polynomial with a highest degree of one. In linear algebra, the linear function is the linear map. This article is about the linear function in calculus and analytic geometry. This is the one we study in high school.
What does a polynomial with highest of degree 1 mean?
Will that include zero? Yes. That is why it would include what is also referred to as a constant function or the zero polynomial. Will that include algebraic expression with negative exponents or fractional exponents (they are also less than zero)? No. Because these expressions are not polynomials.
Will any polynomial of degree 1 qualify as a linear function?
Yes. For example if the polynomial of degree 1 has only one variable say 2x+3, then that defines a function x?2x+3. In symbol we can write this as f(x) = 2x+3 of if we let y=f(x) then we write y=2x+3. If the polynomial has several independent variable, say the polynomial 2x+3y+z, then it is the linear function defined by f(x,y,z)=2x+3y+z.
What does the graph of a linear function look like?
For the linear function in one variable, it is a line not parallel to the x-axis (inclined). For the linear function of degree zero, it is a line parallel to the x- axis. For the linear function with several independent variables, the graph is a hyperplane. In this post we will stick with the linear function in one variable. Examples of their graphs are shown below.
Graphs of linear function
What is common about the two lines?
They are both lines, that’s for sure. However for both graphs, the change in y is the same for every unit of increase in x. If the coordinates are tabulated as shown below, we can see the increase/decrease in y stays the same or constant for every increase in x. The top table is for the red line and the bottom table is for the blue line. This is also how you can tell from the table of representation whether the relationship between x’s and y’s is linear or not. The change in y should be constant for per unit change in x.
The change in the value of y is constant per unit change in x
What do you call the ratio between the change in y vs the change in x?
If you look at the line as a representation of a function, we say that it is the rate of increase or decrease (also calledrate of change). If you look at the line simply as a geometric figure, we say that it is the gradient or the measure of the slope of the line. Sometimes textbooks and teachers use this interchangeably. Since the slope refers to the change in y for every unit of increase in x, its formula is
How to calculate the gradient
Can we consider all lines as representations of linear function?
This is not a linear function
Take a look at the line on the right? Does it have the same slope? If you calculate it using any two points, you will get k/0. The number is undefined. You could argue that the value of the slope is still the same anywhere only that it is undefined. Alright.
Is it a function? No. Remember that a function is a relationship between the x and the y values such that for every x, there is one unique y value.
