Activity #1 -Quadratic Functions Activity

The objective of this activity is for the students to discover the effects of a,b, and c in the quadratic function f(x) = a(x – b)2 + c. Students will also discover the vertex form of a quadratic equation. This activity takes approximately 45 minutes to complete.

In this activity students will construct 3 sliders, one for a, b, and c and plot the graphs f(x) = x2, g(x) = (x – b)2 + c , and

h(x) = a(x-b)2 + c. This activity take approximately 45 minutes to complete.

The students will move the sliders for a, b , and c, and compare the graph with the original graph of f(x) = x2 to determine the affects of each variable. This meets the NCTM Algebra Standard

( Understand patterns, relations, and functions & Analyze change in various contexts ).

The students develop conjectures about how a, b, and c affect the graph of the quadratic equation. Then students come to a conclusion concerning the vertex form of a quadratic equation. These discoveries that lead to conjectures meet the NCTM Reasoning & Proof Standard (Make and investigate mathematical conjectures).

The students are working in pairs discussing what they find and developing conjectures together. This collaborative effort meets the NCTM Communication Standard (Use the language of mathematics to express mathematical ideas precisely & Communicate clearly the mathematical ideas to peers, teachers, and others).

Quadratic Functions Activity

Before you begin the sketch go to the Edit Menu and select Preferences and under the Text Tab make sure For All New Points ischecked. Then click on OK.

In Steps 1 – 10 we will be constructing 3 sliders that we will use to observe relationships of the variables a, b, and c in quadratic functions.

Step 1Select the Straightedge Tool and hold down until it becomes a line. Then in the white space construct a line. Your line will have two points A and B on it.

Step 2From the Construct Menu choosePoint On Line. Point C will be constructed on your line. Drag C so that it is between A & B.

Step 3Select the Selection Arrow Tool and then select points A, B, and C (in that order) and from the Measure Menu choose Ratio.

Step 4Click in the white space so that the ratio is not highlighted, then click on the line and point B. Then press Ctrl + H to hide the line and the selected point.

Step 5Select point A, then point C and press Ctrl + L to construct a segment. Click in the white space so that no items are selected.

Step 6Using the Text Tool, double click the measured ratio and label a.

Step 7Using the Selection Arrow Tool, first click in the white space, then select the segment, endpoints, and ratio and press Ctrl + C to copy. Then click in the white space so that nothing is selected and press Ctrl + V to paste. Drag the copied objects so that they are below the original.

Step 8Using the Text Tool, double click on the copied measured ratio and label b.

Step 9Using the Selection Arrow Tool click in the white space so that nothing is selected, then press Ctrl + V to paste again. Drag the copied objects so that they are below the slider labeled b. Click in the white space.

Step 10Using the Text Tool, double click on the copied measured ratio and label c.

In steps 11 – 18 we will be graphing quadratic functions.

Step 11Press Ctrl + G and the New Function Calculator will appear.

Step 12To graph y = x2 we use the x key then the ^ key then select 2 on the key pad then OK.

Step 13Using the Selection Arrow Tool, click in the white space and then move the equation of the function next to the curve.

Step 14Select the graph and the equation and from the Display Menu choose Line Width then select Thick. Then from the Display Menu choose Color and select the royal blue color. Then click in the white space.

Step 15Press Ctrl + G and select the following keys in the order given.

(, x, -, the ratio in the white space labeled b, ), ^, 2, +, the ratio in the white space labeled c, then OK.

Step 16Using the Display Menu choose Line Width and select Thick. Using the Display Menu change the color to red. Then click in the white space.

Step 17Press Ctrl + G and select the following keys in the order given.

The ratio in the white space labeled a, *, (, x, the ratio in the white space labeled b, ), ^, 2, + , the ratio in the white space labeled c, then OK.

Step 18Using the Display Menu choose Line Width and select Thick.

Using the Display Menu change the Color to lime green.

Now we have 3 graphs to look at

f(x) = x2g(x) = (x – b)2 + ch(x) = a(x – b)2 + c

  1. Move the sliders labeled a and b so that they both measure 1. Then move the slider labeled c and notice the effects c has on the graph in red in comparison to the graph in blue ( f(x) = x2 ). What effect does c have on the graph (consider positive and negative values of c)?
  1. Move the slider labeled c so that it measures 0. Then move the slider labeled b and notice the effect that b has on the graph in red in comparison to the graph in blue ( f(x) = x2 ). What effect does b have on the graph (consider positive and negative values of b)?
  1. Move the slider labeled b so that it measures 0. Then move the slider labeled a and notice the effect that a has on the graph in green in comparison to f(x) = x2. What effect does a have on the graph (consider positive and negative values of a)?

For the following investigations, the graph of the function is

g(x) = (x-b)2 + c, therefore a = 1. You may hide f(x) = x2 by selecting the graph and the equation and press Ctrl + H.

  1. Now move the sliders so that b = 0, and c = 2. Write the equation of the quadratic with these given values. What is the vertex of the parabola?
  1. Now move the sliders so that b = 2 and c = 3. Write the equation of the quadratic with these given values. What is the vertex of the parabola?
  1. Now move the sliders so that b = -3 and c = 4. Write the equation of the quadratic with these given values. What is the vertex of the parabola?
  1. Now move the sliders so that b = -2, and c = -3. Write the equation of the quadratic with these given values. What is the vertex of the parabola?
  1. In general when the equation of a parabola is f(x) = a(x – b)2 + c what is the vertex of the parabola?

Activity Objectives

The objective of this activity is for students to discover the effects of a, b, and c in

the vertex form of a quadratic equation. Students should be able to identify a

vertex of the graph of a parabola and its coordinates. Student should discover

that a affects the opening of the parabola, b affects how the vertex of the

parabola is shifted horizontally, and c affects how the vertex of the parabola is

shifted vertically. At the conclusion of this activity students should be able if

given the equation of a parabola in vertex form to identify its vertex without the

graph.

Activity Length

This activity is approximately 45 minutes in length. Students may need

extra assistance with steps 1 – 10 in the construction of the sliders.

NCTM Standards

Algebra Standard

  • Understand patterns, relations, and functions.
  • Analyze change in various contexts.

Reasoning & Proof Standard

  • Make and investigate mathematical conjectures.

Communication Standard

  • Use the language of mathematics to express mathematical ideas precisely.
  • Communicate their mathematical thinking coherently and clearly to peers, teachers, and others.

A Rectangle with Maximum Area

The objective of this activity is for the students to discover that

the maximum area of a rectangle with a given perimeter will always be a square. Students discover that the relation between the length of

therectangle and the area of the rectangle is a parabola with a

maximum value. This activity came from The Geometer’s Sketchpad

Teaching Notes and Sample Activities. This activity takes approximately 45 – 50 minutes to complete. It is designed for experienced sketchpad users.

Students will use the Sketchpad to construct a rectangle and take measurements for area and perimeter. The students will drag one vertex of the rectangle and investigate the measurements of the area to come to a conclusion as to what figure yields the largest area. This meets the NCTM Geometry Standard (Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships.) and the Reasoning & Proof Standard (Make and investigate mathematical conjectures).

Students verify that the maximum area for a rectangle is a square by graphing the relationship between the length of AC and the area of ACDF and examining the highest point of the parabola. This meets the NCTM Algebra Standard (Understands patterns, relations, and functions).

Students will be assigned to work in pairs so that they can discuss their conjectures with each other and support their reasoning to one another. This collaborative effort meets the NCTM Communication Standard (Organize and consolidate their mathematical thinking through communication & Communicate their mathematical thinking clearly to peers, teachers, and others).

Activity #2 -A Rectangle with Maximum Area

Before you begin this sketch go to the Edit Menu and choose Preferences. Under the Text Tab make sure that For All New Points is checked then click on OK.

Suppose you had a certain amount of fence and you wanted to use it to enclose the biggest possible rectangular field. What rectangle shape would you choose? In other words, what type of rectangle has the most area for a given perimeter? You’ll discover the answer in this investigation. Or, if you have an idea already the investigation below will help confirm your idea.

Step 1Select the Straightedge Tool hold down until it becomes a segment. Then construct segment AB in the white space.

Step 2From the Construct Menu choosePoint On Segment. You should have constructed point C on segment AB.

Step 3Using the Selection Arrow Tool click in the white space then select A and segment AB and from the Construct Menu choose Perpendicular Line. Then click in the white space so that no items are selected.

Step 4Select C and segment AB then from the Construct Menu choose Perpendicular Line. Then click in the white space so that no items are selected.

Step 5Select C and then B (in that order) and from the Construct Menu choose Circle By Center + Point.

Step 6Select the circle and the perpendicular line that intersects it and press Ctrl + I to construct the intersections. Click in the white space then select point E on the circle and press Ctrl + H to hide this point.

Step 7Select D on the circle and segment AB and from the Construct Menu choose Parallel Line.

Step 8Select the parallel line you just constructed in step 7 and the perpendicular line that intersects it (the perpendicular line that does not intersect the circle) and press Ctrl + I to construct the intersection. Click in the white space so that no items are selected.

Step 9Select the four vertices (A,C,D,and F) in consecutive order and press Ctrl + P to construct the interior of the rectangle.

Step 10From the Measure Menu choosePerimeter. Then click in the white space. Select the interior of the rectangle once more and from the Measure Menu choose Area. Then click in the white space so that no items are selected.

Step 11Drag point C back and forth and observe how this affects area and perimeter of the rectangle.

Step 12Select A and C and from the Measure Menu choose Distance.

Click in the white space. Select A and F and from the Measure Menu choose Distance. Click in the white space so that no items are selected.

Question 1

Without measuring, state how AB is related to the perimeter of the rectangle. Explain why this rectangle has a fixed perimeter.

Question 2

As you drag point C, observe what rectangular shape gives the greatest area. What shape do you think that is?

Step 13Select the measurement for AC and the measurement for the Area of ACDF and from the Graph Menu choose Plot As (x,y). This point should be labeled G.

Step 14Drag point C to see the plotted point move to correspond to different side lengths and areas.

Step 15Select G and C and then from the Construct Menu choose Locus. This will enable us to see a graph of all possible areas for this rectangle.

Step 16Drag C so that G is at the high point of the graph.

Question 3

Explain what the coordinates of the high point on the graph are and how they are

relatedto the side lengths and area of the rectangle.

Question 4

Explain what the coordinates of the two low points on the graph are and how they

arerelated to the side lengths and area of the rectangle.

Activity Objective

Students will discover that the rectangle with the most area for a given perimeter

is asquare. Students discover that the relation between the length of the

rectangle and the areaof the rectangle is a parabola with a maximum value.

Activity Length

This activity usually takes about 50 minutes. Students usually have questions on

Step 13. Make sure that when they are completing this step they actually click

on themeasurements that are in the white space and not the figure itself.

NCTM Standards

Algebra Standard

  • Understand patterns, relations, and functions.

Geometry Standard

  • Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships.

Reasoning & Proof Standard

  • Make and investigate mathematical conjectures.

Communication Standard

  • Organize and consolidate their mathematical thinking through communication.
  • Communicate their mathematical thinking coherently and clearly to peers, teachers, and others.