Posted in Algebra

How to Teach the Equation of Linear Function

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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.

equation of linear functionActivity

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.)

Posted in Algebra

What is a Linear Function?

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What is a linear function?

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
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.

calculating the gradient
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 called rate 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

formula for slope
How to calculate the gradient
Can we consider all lines as representations of linear function?
vertical line
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.

Coming up next: How to teach the equation of a linear function.

 

Posted in Geometry

Convert a Boring Geometry Problem to Exploratory Version

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The following problem (or proving activity, if you like to call it that) is a typical textbook geometry problem. It is tough and guaranteed to scare the wits out of any Year 9 student.

proving triangle

When I used the given condition to construct the figure using GeoGebra, the only thing I can move is A or B and what it does is simply to reduce or enlarge the circle. Pretty boring. So I thought of making C dynamic. The way to do this is to construct point C along the circle and then construct a perpendicular line to AB. With C moving along the circle, the once static and close task is now a dynamic, exploratory and task.

kinds of triangles

Your students will observe that for triangle ABC to be an equilateral triangle, CD must be the perpendicular bisector of AB. You can now ask them the problem: Given that CD is the perpendicular bisector of radius AB, prove that ABC is an equilateral triangle, which is what the textbook is asking them to do.

In presenting the problem the way I’ve shown above, you did not only make the problem more interesting (hopefully) and accessible to the majority of the learners (I’m sure most of them can answer the questions), you have also given learners the chance to explore the problem first and be familiar with the situation.

Note that you will be doing a disservice to your students’ geometry life if you will stop at #5 and not give them the opportunity to prove. Proving is what makes mathematics different from other disciplines. It would be a shame if they will go through life only complaining about x and not of proving as well. I’m joking but you know what I mean. You may want to check some of my favorite post about teaching geometry through problem solving: Unpacking mathematics – a geometry example and Problem Solving Involving Quadrilaterals.

Posted in Algebra

Guest Post: Supporting Older Students Who Struggle with Maths

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Often as teachers we struggle to support those students for whom maths is challenging. As a teacher myself, I found many of these students experienced difficulties which continued from their primary years right through until adulthood. Working with these students is what inspired me to write ‘Real World Maths- building skills for diverse learners’ ISBN 9780987542724 – a new publication for Banksia Publishing, distributed by Admark Education. I decided to make this title available as both a print paperback book and a full colour PDF e-book so that teachers can access the information and student pages in the way which suits their classroom needs the best.

real world mathsSometimes students who struggle with maths might have additional needs such as an intellectual or learning disability, be learning English as an additional language, or have missed significant periods of their schooling due to reasons such as illness or travel. Some may have a condition known as dyscalculia, which makes learning and retaining maths and numeracy skills extremely difficult. Often these students in particular have trouble with skills such as memorizing times tables or performing calculations which work in a backwards direction, such as subtraction tasks. They also tend to have difficulty with using their working memory to keep facts or signs in their head whilst they read the rest of a question, or remember which steps to perform next in a process and which they have already done. They might experience difficulty visualizing objects such as three dimensional shapes and imagining what might happen if these objects are turned or transformed in some way. This makes answering questions related to space and transformation quite difficult to do.

As the rest of their class moves forwards, often these students continue to struggle and can becoming increasingly frustrated with their inability to retain skills and progress with their peers. This can lead to acting out in class, failure to attend, personal frustration and distress and a decrease in self esteem and confidence. As teachers, it is critical to identify which students are struggling to keep pace with their peers and the expected learning outcomes, and take steps to provide age appropriate work which is success focused and will ensure skills are able to continue to develop over time.

‘Real World Maths – building skills for diverse learners’ (Banksia Publishing, 2013) provides a wide range of practical, easy to use resources for upper primary and junior to middle secondary students. It is an innovative title which focuses on addressing areas of need for students who struggle with their maths skills. It includes:

o    Strategies for supporting visual and auditory working memory

o    Ideas for student goal setting and skill building in maths

o    Place value

o    Fractions, decimals and percentages

o    Working with maps, distances and directions

o    Time, dates and calendar skills

o    Managing money and building financial literacy skills

o    Working with shapes, patterns and designs

o    Collecting and using maths information through practical activities

o    Over 50 copiable student pages to use in the classroom

-Article Author: Anne Vize M.Ed (Special Education).

Feel free to email Anne Vize with any queries to banksiapublishing@gmail.com