Category: High school mathematics

Posted in Algebra, Graphs and Functions, High school mathematics

What is an inverse function?

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In mathematics, the inverse function is a function that undoes another function. For example,  given the function f(x) = 2x. If you input a into the function f, the output is 2a. The inverse function of  f(x) is the function g(x) such that if you input 2a into g(x) its output is a. Now what is g(x) equal to? How does its graph look like? Is the inverse of a function also a function? These are the basic questions students need to answer about inverse function.

How to teach the inverse function
Functions and their inverses

The idea of inverse function can be taught deductively by starting with its definition then asking students to find the equation of the inverse function by switching the x and y in the original function then expressing the equation in the form y = f(x). This is an approach I will not do of course as I always like my students to discover things for themselves and see and express relationships in all three representations: numerical (ordered pairs or table of values), geometrical (graphs) and symbolic (equation) representations.

In teaching the inverse function it is important for students to realize that not all function have an inverse that is also a function, that the graph of the inverse of a function is a reflection along the line y = x, and that the inverse function does not necessarily belong to the same family as the given function.

The concept of inverse function is usually taught to introduce the logarithmic function as inverse of exponential function. Important ideas about inverse function such as those I mentioned are not usually given much attention. Perhaps teachers are too excited to do the logarithmic functions.

I suggest the following sequence for teaching inverse. I’m sure many teachers and textbooks also do it this way. What I may just be pointing out is the reason behind the sequence. I also developed three worksheets using GeoGebra. The worksheet is interactive so that students will be able to make sense of inverse of function on their own.

Start with linear function. Its inverse is also a function and it’s easy for students to figure out that all they need to do is to switch the x‘s and y‘s then solve for y to find the equation. You may need to see the inverse of linear function activity so you can make sense of what I am saying.

The next activity should now involve a quadratic function. The purpose of this activity is to create cognitive conflict as it’s inverse is not a function. The domain needs to be restricted in order to get an inverse that is also a function. Depending on your class, the algebraic part (finding equation of the inverse) can be done later but it’s important for the students at this point to see the graph of the inverse of a quadratic to convince them that indeed it is not a function. Click the link to open the activity inverse of quadratic functions.

The third activity will be the inverse of exponential function. By this time students will be more careful in assuming that the inverse of a function is always a function. Except this time it is! It is also one-to-one just like linear, but it’s equation in y belong to a new family of function – the logarithmic function. Click the link for the activity on inverse of  exponential functions.

Teaching principles

There are at least three math teaching principles illustrated in the suggested lesson sequencing for teaching the inverse function and introducing logarithmic function.

  1. Connecting with previously learned concepts. Start with something that students can already do but in a different context. In the above examples they are already familiar with linear function and they already know how to find its equation.
  2. Creating cognitive conflict. The purpose is to challenge possible assumptions and expose possible misconceptions.
  3. Making connections. Mathematics is only understood and hence powerful when there is a rich and strong connections among related concepts, representations, and procedures.

You may find the Precalculus: Functions and Graphs a good reference.

Posted in Algebra, High school mathematics

Using cognitive conflict to teach solving inequalities

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One way to teach and assess students understanding of math concepts and procedures is to create a cognitive conflict. Here is one way you can create cognitive conflict in solving inequalities:

To solve the inequality x – 7 > 5, the process usually involve adding 7 to both sides of the inequality.

solving_inequality

This process uses the principle a > b then a + c > b + c. There is no change in the inequality sign since the same number is added to both side.

Now, what if we add 7 to the left side of the inequality and 6 to the right side?

cognitive conflict

The process uses this principle: If a > b, c > d then a + c > b + d. Should this create a change in the inequality sign? Certainly not. There should be no change in the inequality sign when a bigger (smaller) number is added to the bigger (smaller) number side.  Both of these processes create a cognitive conflict and will be a good opportunity for your class to discuss what solving inequality means and, compare the processes of solving equations and inequalities. Comparing and contrasting procedures is also a good strategy to developing conceptual understanding.

For those interested to learn more about inequalities I recommend this book:Introduction to Inequalities (New Mathematical Library)

Posted in Algebra, High school mathematics

Free online calculator for problem solving and math investigation needs

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Meta-Calculator is  a free online calculator that should serve the needs of almost any high school student/college student for problem solving and math investigations tasks. It would also be useful for anyone who needs to analyze statistical data, do lots of calculations , graph equations or create images of equations—you can just hop on the internet browse to the webpage and download the graph!

Meta-Calculator is a multi purpose calculator that works both in your browser via the Flash Plugin and on your iphone/ ipad as an app–so pretty much every modern computer/phone  out there can use it.  It is really four calculators in one—a scientific one,  graphing calc, statistics calculator and a matrices/vector calculator.  Let’s look at each one in detail.

The Graphing Calculator

Meta Calc can graph up to 7 equations or inequalities,  find their intersections,  produce a table a values or trace a point along any equation. You can also zoom in/zoom out , set the x-scale or y-scale, x-min/max, y-min/max, pan around the graph with your mouse.  A distinctive feature is the ability to save any of your graphs as images to your computer ( .png files). Just hit the ‘save graph’ button, and you will download the graph. This is a feature that any student or teacher could appreciate—the next time you need a graph for a presentation or a worksheet for your math lesson, just type in the equation and hit ‘save graph.’  I actually know some teachers that have used this very feature to introduce slope. One teacher, for instance, graphed 7 equations, slightly changing the slope for each one, and then let her students explore the relationship between the slope of a line and its graph.

The Scientific Calculator

The scientific calculator provides a really intuitive user experience.  It has all of the basic functions and buttons you’d expect including sin, cos, sin-1, cosh, log and more. Plus it has some more advanced features including a button to calculate   least common multiples, permutations, combinations and—possibly most powerful of all—a linear equations solver that lets you input up to 6 equations with either two or three variables and the solver will calculate the solutions.

The Matrix Calculator

The matrices/vector calculator has a wide range of functions. You can calculate a matrix’s determinant, or its inverse. Also, you can add, subtract, multiply and transpose matrices. The same functions are available for vectors.

The Statistics Calculator

Last but definitely not least is the statistics calculator. This has the fundamentals that you’d expect: calculating quartiles, mean, median, mode as well as correlation coefficient and various types of regressions (linear, quadratic, exponential, cubic , Power, Logarithmic, Natural Logarithmic).    You can then plot the data to the graphing calculator! A stand-out features is the ability to compute  Student t-tests: either 1 or 2-Tailed T-Tests (paired and unpaired). I was unable to find any calculators online that let you enter raw data and calculate T-tests so this is quite a rare online find.

The calculator can be found here : http://www.meta-calculator.com/online/.

Posted in Combinatorics, High school mathematics

Linking combination and permutation with repetition

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Combinatorial problems are difficult because it’s hard to  know which formula to use in a particular problem and when you need to ‘tweak’ or totally abandon the formula. In this post I share two solutions to a problem which connects the multiplication  principle, the combination formula and the formula for counting the number of permutation with repetition. Knowledge of connections among concepts help in problem solving. The problem is generated from the rook puzzle presented in my post Connecting Pascal’s Triangle and Permutation with identical objects.

Find the number of different ways of arranging 14 letters 7 of which are E’s and 7 are N’s, in a row. Here is one arrangement:

The first solution shows how the formula for counting permutation with identical objects can be deduced from the solution involving the multiplication principle and the second connects permutation and combination.

Solution 1

The idea behind this solution is to initially treat each letter as distinct. There are 14 letters to be arranged in a row. If these letters are distinct from one another then by the Multiplication Principle, there are 14! different arrangements in all.

But the letters are not all distinct. In fact there are only two kinds – N’s and E’s. This means an arrangement, for example, consisting of 7 N’s followed by 7 E’s,

N E N E N E N E N E N E N E 

has been counted 7!7! times in all in 14!. The same is true for this arrangement:

N N N E E N N E E N N E E E.

Thus, to find the number of ways of arranging 14 letters where 7 of which are identical and the remaining 7 are also identical, 14! need to be divided by 7!7!

If the problem had been In how many ways can you arrange 10 N‘s and 4 E‘s?, the solution will be \frac {14!}{10!4!}.

Notice that this solution uses the technique of counting the number of permutations (arrangements) of n objects, r1 of which are identical, r2 are identical, . . . , and rn are identical, where r1 + r2 + . . . ri = n. The number of different permutations is denoted by

In the problem, n = 14, r1 = 7 and r2 = 7. Hence, the total number of arrangements is  \frac{14!}{7!7!} = 3432.

Solution 2

This solution simplifies the original problem to How many different ways can 7 N’s be arranged in a row of 14 spaces? Now, why is this problem equivalent to the original problem? What happened to the 7 E‘s? Why aren’t they not considered anymore? This is because for a particular arrangement of 7 N‘s in 14 spaces, there is one and only one way the 7E‘s can be arranged.

To solve, count the number of possible positions for the 7 N’s. You have 14 positions to choose from for the first N. For the next N you only have 13 positions to choose from, for the next N, 12 and so on until the 7th N. By the Multiplication Principle you have  14.13.12.11.10.9.8.7 different possible positions where you consider each N to be distinct.

But the N’s are identical. That is, in the arrangement for example

_ N _ N N N _ N N _ _ _ _ N

N has been counted 7! times in 14.13.12.11.10.9.8.7. So you have to divide this by 7!

Thus, there are  different positions for N (all N are identical). This can be written in as shorter way using factorial notation by multiplying it by \frac{7!}{7!}.

If there were 10N‘s and 4 E‘s, the problem would have been In how many ways can I arrange 10 N’s in 14 spaces?

In general, if there were n possible positions for arranging r objects, the formula is \frac {n!}{r!(n-r)}. Note that this looks like the combination formula which is used to solve problems for the number of combination and indeed it is. You just got used to applying nCr = \frac {n!}{r!(n-r)} to problems like In how many way can you arrange n different objects taken r objects at a time?