Section A - Lesson Plan Outline Lesson Title: Function

EMT525 Teaching the 7-12 Mathematics Curriculum Assessment Task 2 Jennifer Ralph

Section A - Lesson Plan Outline

Lesson Title: Function Transformations

Grade Level: 11/12 Duration: 90 minutes (or 2x50 min)

Objectives:

Students will

Be able to translate functions vertically and horizontally

Be able to dilate functions parallel to the x- and y-axes (equivalently, dilate

functions from the y- and x-axes)

Be able to reflect functions about the x- and y-axes

Know that order can be important when applying multiple transformations

This lesson relates to course content as described by the Tasmanian Qualifications

Authority. Specifically, it looks at “graphs derived from others, using translation,

reflection in x-axis, reflection in y-axis, dilation from axes and combinations of these”

(Tasmanian Qualifications Authority, 2014).

Introduction:

Remind students that we should be familiar with some standard functions. Suppose we

wanted to modify one though – for example, we want to model a bridge whose shape

looks like a parabola, but it is flipped upside-down and stretched. What if we are

modelling a real-life problem, like somebody's distance over time, but they ended up

leaving at a later time or travelling at half the speed? It would be helpful to know how to

modify functions in particular ways.

Main part of lesson:

Move to computer lab and perform teaching activity


Move back to classroom. Explain to the class how to modify equations of

functions. For example:

◦ To obtain f(x)+c from f(x), can just add “+c” to the function

◦ To obtain f(x-c) from f(x), need to replace every instance of x with (x-c)

◦ To obtain af(x) from f(x), put brackets around the original equation and

multiply the whole thing by a

Write up the worked example question.

◦ Discuss: Can the order of transformations matter? Does it always matter?

Work through the example

Give students exercises to complete

Conclusion:

Ask students to share something they've learned – make a list of these things on

the board. Ideally, suggested ideas will not just be “how to transform functions”, but

will include the teacher questions mentioned in section H. If not, some prompting may

be required. On scrap pieces of paper, get students to write something they want to

know more about, don't understand or need to work on. Collect these pieces of paper as

students leave, so that future lessons can clarify ideas and focus on areas in which

students seem to be struggling.

Section B – Teaching Activity

For this activity, students use computers with Geogebra, and will need the

“Transformations” ggb file. The teacher is at the front of the class with a whiteboard.

The general routine for the activity is as follows:

Write transformation on the board (e.g. f(x)+c)


Ensure the relevant graph is visible in Geogebra, and other graphs are hidden.

The visibility of graphs can be toggled on/off with the circular button to its left.

Ask students what f(x) would be for this particular example in Geogebra (when

c=0, n=1, a=1) – that is, what graph are we translating? Write this down. For

v(x) and w(x), ask how the graphs of x³ and x² have been altered to obtain f(x).

In Geogebra, make sure that students look carefully at the equations of functions

as they change, as well as the effect on the graph.

Ask students what the transformation does to the graph (e.g. “move graph c units

up”). Write on board.

Specific notes for transformations are below:

f(x)+c. Ask students what they think it might do before using Geogebra. Use

graph g(x) and slider 'c'.

f(x-c). Use h(x) and 'c' slider in Geogebra. In particular, note the direction it

moves – why does f(x-c) move c units right, not left? It has a minus sign, doesn't it? This

is an important teacher question. Consider f(x-2) as an example. Can write a table like to

help explain:

x -3 -2 -1 0 1 2 3

f(x) f(-3) f(-2) f(-1) f(0) f(1) f(2) f(3)

f(x-2) f(-5) f(-4) f(-3) f(-2) f(-1) f(0) f(1)

af(x), a>0. Consider what will happen at one point on the graph, and then what

will happen to the graph as a whole. Make sure 'n' is set to +1 in Geogebra. Use v(x) and

'a' slider. Does the x intercept change? The y intercept? x and y values of the stationary

point?


-f(x). As with af(x), but keep 'a' fixed and change 'n'. What is the axis of

symmetry?

f(ax), a>0 and f(-x). As with af(x) and -f(x) Why does f(ax) stretch the graph by

a factor of 1/a, rather than a? This is another important teacher question. Can write a

table similar to the one used for f(x-c).

Ask students to copy each of the transformations into their book. Expected

duration is 45 minutes.

Section C – Pedagogical Analysis of Teaching Activity

The teaching activity in section B was chosen for its interactivity and visual

effectiveness. Rather than just be told what each transformation does, students can use

the sliders to investigate for themselves. The activity accommodates different types of

learners by having auditory and written material, as well as a hand-on approach.

The key mathematical principles to be conveyed in the activity are the different

types of function transformation, and the effect that each of these has. These are

illustrated quite literally by Geogebra, which gives concrete examples for students that

may struggle in abstract thinking. Students' conceptual understanding of function

transformation is aided by the use of Geogebra, as they can see each graph moving

when they change the value of variables.

Key aspects requiring special attention are horizontal translation and horizontal

dilation. A graph of f(x-c), c>0 is shifted to the right. This can be confusing for students,

as the equation has a minus sign and negative numbers are on the left of the Cartesian

plane. Similarly, f(ax) can cause some confusion because it is stretched by a factor of

1/a (i.e. compressed by a factor of a) despite the fact that the pronumeral a is

multiplying (stretching) x by a factor of a.

Section E – Student Exercises

Stewart, J. (2009). Calculus (6th ed.). Belmont, CA: Brooks/Cole Cengage Learning

Section F – Model Solutions

Note: Students will be encouraged to use more space than this, and will be given graph paper

for questions 4 and 5. Language (e.g. 'dilated' vs 'stretched') is flexible, so long as it correctly

expresses the given transformation.


Section G – Pedagogical Analysis of Student Tasks

It is not expected that students finish these tasks in class – in fact, it is expected

that they won't. The exercises will be continued in the following lesson. These problems

provide a good opportunity to practise using the concepts that have just been explained.

The questions should be answered in the order they appear in the book, as this

represents a logical progression from simpler questions to more complex ones.

The first two questions can be answered using the definitions written by students

during the teaching activity. These questions allow students to become better acquainted

with which transformations are which, and to check that students can interpret their

definitions correctly – especially for the less intuitive transformations.

The rest of the questions build on the first two, as students will need to

understand how the graph of f(x) is changed to obtain other graphs. The questions also

involve skills in interpreting graphs and creating new equations.

The sequence of questions fosters learning by gradually introducing elements

which are more difficult. Students must not only decode the different types of

transformation, but must also consider how they would affect the shape and position of

the graph. Ultimately, they must consider how a series of transformations will change a

given equation.

Students are likely to experience difficulty in scaling parallel to the x-axis and

horizontal translation, as elaborated on in section H. It is also likely that they will find

the last two questions difficult. There is quite a difference between writing f(3x), for

example, and actually modifying a function.


Section H – Teacher Questions

Question 1: Why does f(x-c) translate the graph c units right?

This translation is often confusing. Rather than incorrect answers to the question

above, I would expect students to be unable to answer it. Supposing g(x)=f(x-c), correct

answers are something like “the value of g at x is the same as the value of f at x-c (c

units to the left of x). Therefore, the graph of y=f(x-c) is just the graph of y=f(x) shifted

c units to the right” (Stewart, 2009, p. 37).

Question 2: Why does f(ax) stretch by a factor of 1/a (a≠0)?

Again, students are more likely to be unable to answer this question rather than

answer incorrectly. Correct answers that I would hope to receive would be something

along the lines of noticing that the value of f(ax) at x is the same as the value of f(x) at

ax, and ax is x multiplied by a. The answers to this and question 1 are very difficult to

follow without simultaneously drawing and referring to a diagram.

Question 3: Does the order of transformations matter?

Students may not think that it matters. Alternatively, they may think that it

always matters. Although the latter will not affect the result students get when

transforming graphs, the former will. The correct answer is sometimes, but not always.

Section I – Personal Development

During this assignment, I've had to think about how to explain troublesome

concepts such as horizontal translation and scaling. The solution came from previous

experiences in the unit. When looking at multiplying a negative number by a negative

number, we were shown that writing a pattern can help it to make sense. In the same


way, patterns can help people to make sense of other things – horizontal translation and

dilation, in this case.

The unit also made me think about how to reach a wider audience. Although

some people may be able to read or hear an explanation and make sense of it, most

people won't. For these people, it is really helpful to have examples that they can see,

preferably ones that they can interact with. That knowledge was useful in planning for

this assignment, and prompted me to create a Geogebra file so that students can

transform graphs themselves, to see what happens to them.

Also, when planning this assignment I have had to think about timing and

content. I realise that the lesson content in this assignment is quite full on, and it may be

more appropriate for students to be given a whole lesson in the computer lab, with

exercises set the following lesson. The amount of time able to be dedicated to one

particular topic, of course, depends on the overall unit and how much needs to be

covered throughout the year.

In the unit, I have been made to realise a lot of problems with certain

explanations and students over-generalising concepts. I have never had a problem with

“fruit salad algebra” or irregular shapes, for example, but I can see now why they might

cause problems; I experienced a similar situation in professional experience. Students

were given two similar triangles with a scale factor that was not an integer. They were

asked if the triangles were similar, and most said no because the scale factor had been a

whole number in each of the examples they had been shown.

One last important lesson from the unit was to ensure that your message is clear.

For example, using 3² is a better example than 2², or calling -2 “negative two” rather

than “minus two” will help reduce confusion when talking about subtraction.


References:

Stewart, J. (2009). Calculus (6th ed.). Belmont, CA: Brooks/Cole Cengage Learning

Tasmanian Qualifications Authority. (2014). Tasmanian Qualifications Authority:

Mathematics Methods. Retrieved from

http://www.tqa.tas.gov.au/4DCGI/_WWW_doc/204703/RND01/MTM315114_

V1a.pdf

Documents

Section A - Lesson Plan Outline Lesson Title: Function