Linear, Quadratic and Cubic Function Explorer Teacher Notes

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1 Introduction LinQuadCubic Explorer Linear, Quadratic and Cubic Function Explorer Teacher Notes The aim of this.tns file is to provide an environment in which students can explore all aspects of linear, quadratic and cubic functions, their graphs and their equivalent algebraic representations. Students will be able to dynamically alter any of the functions by moving coordinate points that are the basis for each function, and see how the algebraic form of the function changes accordingly, whether it is in expanded form, factorised form or homogeneous form. In addition, they can plot distinct points on the axes to verify whether they lie on the function, or not. Resources The.tns file is not one designed for students to work through page by page without further direction. Instead, the teacher may base a whole lesson around the use of a single page to introduce and develop students understanding of functions, their graphs, the algebra and how they link to the coordinate system. See the lesson suggestions at the end of these notes. Skills required Using a Clickpad, students need to be move between pages, grab and release points and use CTRL+TAB to move between different applications on the same page. Using a Touchpad, students need to be move between pages, and grab and release points. Depending on the tasks set by the teacher, students may also need an interest to discover new situations on their own, checking their understanding as they go. The activity Linear Functions (pages 2.1 to 4.1) Every page of this.tns file has the same three part layout: (1) A graph, with multiple points which can be grabbed. (2) A list of (x,y) coordinates to be plotted (3) An algebraic expression of the graphed function. This is contained in an Interactive Notes Application at the foot of the screen. The scroll bar of this Notes Application allows the student to view different, equivalent algebraic expressions, one at a time.

2 The three screenshots below exemplify the range of different expressions available. Note: In OS2.1, the Notes Application does not allow for suppression of the quotes symbol, so students will need to ignore them. For example, y=3x + 7 needs to be read as y=3x+7. A future OS release is expected to address this matter. y=mx+c format factorised format ax+by+c=0 format Note: Advise your students not to click on the actual algebraic formula. Doing so will allow them to edit the expression, and possibly corrupt the code that s there. They should only click on the window s small scroll bar, on the right. If students do click on the algebra, then use / d (undo) to rectify the situation. You can also see that the points (3,15) and (-3,5) have been plotted, by simply entering in the coordinates in the lists. Page 3.1 is identical in functionality to page 2.1. The only difference is that the axes intercepts do not have the full coordinates displayed. So, instead of (-6,0), just 6 is shown. Page 2.1 & 3.1 also constrain the moveable points to finite sections of each axis. Furthermore, a vertical line on the y- axis cannot be formed. Page 4.1 allows the student to freely move two coordinates anywhere on the plane, and not be restricted to either axis. This leads to a less cluttered display, but at the price of not emphasising the values of both x and y coordinates at each point. The teacher or the student will need to decide which display format they prefer to use. This extra flexbility then permits vertical lines to be formed (see right) In this situation, whilst the Notes Application displays an undefined function, a new label will appear between the two points, displaying the correct equation of the vertical line.

3 Quadratic Functions (pages 5.1 to 8.1) Pages 5.1 (left) and 6.1 (right) have the same functionality, with the only difference being the visual labelling of the axes intercepts. Pages 5.1 & 6.1 allow the student to move the zeros of the function, and its y-intercept. Page 7.1 allows the student to move the turning point of the function, and its y-intercept. Page 8.1 is the same as 7.1, with the addition of the display of the value of the discriminant. On all pages, factorised and expanded algebraic expressions can be viewed using the Notes Application scroll bar. Cubic Functions (page 9.1) Cubic functions can be explored by controlling the locations of the zeros and their y-intercept. Both factorised and expanded algebraic expressions can be viewed using the Notes Application scroll bar. Author Nevil Hopley, Head of Mathematics, George Watson s College, Edinburgh. August to October 2010

4 Linear Functions A Teaching Suggestion Select from the following sequences of tasks and mini-investigations: Page 2.1 or Page Move y-axis intercept to (0,8) Look at the algebraic expression, y=mx+c Now move the x-axis intercept to any value. What stays the same about the algebra, and what changes? 2. Move y-axis intercept to (0,9) Look at the algebraic expression, y=k(px-q) Now move the x-axis intercept to any value. What stays the same about the algebra, and what changes? 3. Move y-axis intercept to (0,10) Look at the algebraic expression, ax+by+c=0 Now move the x-axis intercept to any value. What stays the same about the algebra, and what changes? 4. Now, instead of fixing the y-axis intercept and moving the x-intercept, we ll do the opposite. Fix the x-intercept and move the y-intercept. Look at each of the algebraic expressions in turn, and right down what stays the same about the algebra, and what changes? 5. Summary Conclusions What property of the line can be altered without affecting the c of y=mx+c? What property of the line can be altered without affecting the q of y=k(px-q)? Page 2.1 or Page Plot the point (4,9) Move the axes intercepts so that the line looks as though it goes through (4,9) How do you know that (4,9) is on the line? Give a reason, using all 3 forms of algebraic expression. 2. Plot the point (4,8) Move the axes intercepts so that the line looks as though it goes through (4,8) How do you know that (4,8) is on the line? Give a reason, using all 3 forms of algebraic expression. 3. Plot the point (4,7) Move the axes intercepts so that the line looks as though it goes through (4,7) How do you know that (4,7) is on the line? Give a reason, using all 3 forms of algebraic expression. 4. Summary Conclusions For a single point, you can always find a line that goes through it Is this statement true all of the time? Give a reason for your belief. 5. Generalising Describe all straight lines that go through the point (0,7) in one algebraic expression. Describe all straight lines that go through the origin in one algebraic expression. Describe all straight lines that go through the point (5,0) in one algebraic expression.

5 Page Move the points to read (-7,-16) and (2,11) The algebraic expressions should read y=3x+5 and 3x-y+5=0. Now, type in the coords (0,12) in the right panel, so that a point appears on the axes. Now, edit the y-coordinate of 12 to another number, so that the point lies on the line y=3x+5. How do you know your point is on the line? Write down a reason, using algebra. 2. Now plot another point (4,10). Again, this point is not on the line. Edit the y-coordinate of 10 so that the point is on the line. Give an algebraic reason to show that you know it is on the line. Remember that you have 3 different algebraic forms you can use. 3. Repeat this process for the following points you have to determine the values of a, b, c and d so that the points lie on the line. (-5,a) (b,8) (c,9.5) (d,0) Plot each point to visually check, and give an algebraic reason to prove that it lies on the line. 4. Now clear all the plotted points. Move the two points on the line to (-8,10) and (4,-8) Find the coordinates of the y-intercept, and plot it. Find the coordinates of the x-intercept, and plot it. 5. Now make a free choice of an x-value, between 8 and 8. With that value, find the corresponding y-value so that the point lies on the line. Which algebraic form is the best to use for this task? Plot your point to visually check that it works. 6. Now make a free choice of a y-value, between 15 and 14. With that value, find the corresponding x-value so that the point lies on the line. Which algebraic form is the best to use for this task? Plot your point to visually check that it works. 7. Now, clear all the points that are plotted. Move the two points so that they have the same x-coordinate, and watch what appears next to the line. Then create a few more vertical lines. Why can these lines not be expressed in the form y=...? A. Exploratory Task- work with a partner There were some lines that could be created on page 4.1 that could not be created on page 2.1 Can you find one such line and explain why it can t be created on page 2.1 B. Exploratory Task- work with a partner Choose whether to use page 2.1 or Your task will be to position a line that has equation y x 8 2 Can you determine where to locate the line's points before you actually move them there to check that you were right? C. Exploratory Task- work with a partner Given y=3x+2, what sequence of instructions would you give to someone about how to sketch the graph of it, plus marking on it the coordinates of a point that is somewhere on the line. Will your instructions always work, regardless of the equation you are given? D. Exploratory Task- work with a partner On page 4.1, you positioned two points and the Nspire generated the equation. Ultimately, you need to know how to generate the equation from two points yourself. Start trying to form a method to do this. You would do well to think about what information you need to extract from the points' coords.

6 Quadratic & Cubic Functions Teaching Suggestions LinQuadCubic Explorer Similar activities to the previously detailed Linear Functions Teaching Suggestion can be designed for Quadratic Functions and Cubic Functions. Such activities should focus on the following themes : - allow students to explore new situations on their own, before they are given any formal instruction. Allow them to first build their own framework of understanding, which you can then develop. - force students to link the graphical display with the various algebraic forms. Move from one to the other, constantly. - regularly require students to substitute in values to the algebraic forms, to plot points on the graphs. In particular, choose negative and fractional values for the x coordinates.