Engage:

Learning Experience(s)

Review concepts from preview sheet. Take any questions. Students should review these concepts prior to this lesson.

Midpoint

Perpendicular Bisector

Distance Formula

Definition: Distance of a point to a line

Reflection about an axis.

Approx. Time___5__mins

Critical questions that will establish prior knowledge and create a need to know

1a. What is the definition of a midpoint?

1b.If no answer, draw two points, and label them A and B on the board and the segment in between. Wait for suggestions. Allow a volunteer to identify the midpoint. Ask the students what letter we-mathematicians usually designate midpoints

2a. Now we have a line segment AB. Can we find the perpendicular to segment S through M?

2b. Optional – if time is short, just sketch perpendicular. Explain that it can be sketched, but also a proof and method exists for finding a midpoint, and the perpendicular using a string for a compass and. Ask if anyone knows how to do this.

2c. What are the main features of the perpendicular bisector? What do we know about it?

3a. Distance Formula. Can someone state it?

3b. Given two points with coordinates (a,b) and (c,d) what is the distance between them?

4a. What is the distance from a point to a line? (Sketch a line and a point?)

4b. If needed continue with these questions: Is it on a diagonal? or along a squiggle? Draw some options.

5a. From this morning, what is a reflection?

Ask a student to draw a picture.

Also draw a curved surface.

Expected Student

Responses/Misconceptions

1a. The center point of a line segment.

1b. The center point on the line segment.

The midpoint is often labeled M. This is a convention, and it could be any convenient letter that distinguished it from other points in a diagram.

2a. Yes just draw it.

2b. A student may know how to draw the perpendicular bisector. If not, demonstrate.

2c. Any point on the bisector is equidistant from each endpoint.

3a. Student writes the distance formula.

3b. √[(a-c)^2 + (b-d)^2] = D

4a. The shortest distance, along the perpendicular to the line through the point.

4b. Students should see that it is the shortest distance, along a perpendicular.

5a. A mirror image, or light bouncing of a surface.

Hopefully a student will draw a reflective surface (line) and he path of a ray that hits the line at an angle and bounces back along a path that is at the same angle, but opposite side to a perpendicular drawn through the point of reflection.

Students should recall that a curved surface bends light with respect to the perpendicular, and that a curved surface can have a focus where exiting light rays converge.

Explore:

Learning Experience(s)

Students will be counted off as l, 3, 5.

Write the definition of a parabola on one flipboard:

A parabola is the set of points in the plane that are the same distance from a fixed point F as they are from a fixed line D. The fixed point is called the Focus, and the fixed line is called the Directorix.

Activity:

Give each student a sheet of wax paper, a ruler and a pencil.

Directions:

·      Place a line on the lower quarter of the paper -- demonstrate on a second flip chart.

·      Place about ten points on the line

·      Place a point on the paper low, medium or high above the line

·      Fold the paper so a point on the line is over the point you drew. Make the crease. Unfold the paper and, using the ruler, draw a line in the crease.

Demonstrate with second Flip Chart

Repeat for various parts of the line

Wait 5-10 minutes.

Ask students to compare their result with their neighbors.

Wait 2 minutes

Approx. Time____15_mins

Critical questions that will allow you to decide whether students understand or are able to carry out the assigned task (formative)

1. Ask students if they have seen this definition and terms before?

Ask students if they understand the directions

Ask students to raise their hand if they need help, but otherwise begin the activity.

2. Ask students what they made.

3. Ask students who were ls to hold up their result.

Next 3s, then 5s.

What do you notice?

Show some student samples from each category.

4. What about this method? Why are the directions so loose --- should they be more exact? Why or why not?

Optional.

Ask students to measure the distance from their line to the focus, or original point and write the number in cm on the upper right hand

Collect several papers and order by distance. Display in order. Have students come to table to observe.

Expected Student

Responses/Misconceptions

1. Students respond with whatever experience they have.

2. A bunch of lines that make a parabola.

3. The 1's are tall skinny parabolas and the 3's are wider and flatter. The 5's are the widest and flattest.

4. The directions do not have to be exact. All that matters is that the line is over the dot when the fold is made.

Explain:

Learning Experience(s)

Proof. Here we will relate the paper folding to the components of a geometric proof.

First, let’s discuss what we did with the paper.

Draw on flipchart 2.

Ask students to reread the definition of a parabola still displayed on board 1.

Now ask students to draw a vertical line from that same point.

Remind students that the parabola points are to be equidistant from a point, the focus, and a line, the Directorix

Be sure students notice that we go through the process of finding a perpendicular bisector and perpendicular to the Directrix uniquely for each point on the Directrix.

Approx. Time_15____mins

Critical questions that will allow you to help students clarify their understanding and introduce information related to concepts to be learned

1. Take a look at your wax paper model. Draw a line real or imagined from one of the points on the Directrix (line) to the focus. Redo that fold -- what do you observe?

2.     Do we have a name for a point halfway between two endpoints?

3.     What about the line we drew in the crease?

4. If the fold is the perpendicular bisector of the segment AF, what does that tell us about the points on the line in the crease?

4a. Does this mean that all the points on all the fold lines are part of the parabola? Why?/Why not?





4b. What do you notice?

5.     How does the requirement that the parabola points be equidistant from a point and a line affect which points meet the definition?

6.     Which points meet the definition of parabola?

Expected Student

Responses/Misconceptions

1. Students should notice that the line also lies on top of itself.

2. Students will hopefully note that the intersection of the line with the fold is the midpoint of the segment between the focus and the point.

3. The fold is perpendicular since the line is a reflection around it. So the fold is the perpendicular bisector of the segment.

4.     The points on the crease are all equidistant from the endpoints A and F.

4a. Students may think either is true. Encourage some discussion. ( Point out if a student does not that the midpoints of each folding proceedure actually lie on a line.)

4b. Hopefully some students can trace the line back to the fold and see that the parabola points determined by the folds are in fact directly above the points on the line that were used to make the fold.

5. Students will hopefully say that because the distance to a line is always on the perpendicular, that for a given point on the line, the parabola point must lie on the perpendicular to the line at that point.

6. Just the points that are equidistant from the Focus and Directrix - eg. For a point X on the Directrix, the parabola point is located on the perpendicular to the Directrix and also on the Perpendicular bisector to the segment FX.

Extend / Elaborate:

Learning Experience(s)

Now we can take the informal geometric proof and use it to generate the formula for a parabola.

Pass out the worksheet. Take a look at the sheet.

The parabola is already drawn -- it could be a copy of one of the ones we generated with the paper folding.

[Rather than talk students through this portion, we will for the sake of time give them the set up of the coordinate axes with the vertex of the parabola at (0,0).]

Preview Worksheet:

Give a brief overview of the sheet. In particular, ask students if they understand the assignment of the height "a" to the focus, and how that affects the coordinates of the focus.

Also mention that D here is a point even though we used it in the distance formula to represent Distance.

Approx. Time_____mins

Critical questions that will allow you to decide whether students can extend conceptual connections in new situations

1. Now we want to place our parabola on the coordinate plane and describe it with an equation? What are we trying to do?

2. One way we can do this is to let the bottom or vertex of the parabola be at (0,0), and so the focus will fall on the y-axis. Could we do it another way?

Note - numbering starts over to match worksheet.

1.                    We already have the vertical line through D on the Directrix. Draw a line from D to the Focus, and sketch the perpendicular bisector. It should go through P -- Does it?

2.                    The axes will form a coordinate grid. Assume that the vertex or bottom of the parabola is at the point (0,0).

Let the coordinates of P be (x,y).
Let the distance from the vertex to the focus be an unknown, called a.

Find representations for the following:

q      What is the coordinate of the Focus? ________________

q      What is the coordinate of D? ______________________

q      What is the equation of the Directorix? ____________________

3.                    What do we know about the distance from the Focus to P as it relates to the distance from P to D?___________________________________________________

4.                    Recall the distance formula: Given two points with coordinates (a,b) and (c,d) the distance between them is √[(a-c)^2 + (b-d)^2] )

Use the coordinates you wrote above to find:

Distance from P to the Focus:________________________________


Distance from P to D:_______________________________________

7)    Now set your answers equal to each other and solve for y:

Distance from P to the Focus = Distance from P to D

This result is the basic equation of a parabola. We will explore this equation further with a sketchpad activity in the next half hour.

Expected Student

Responses/Misconceptions

1. Find an equations so that if you have a value on the x- axis, you can use the function to find the y value, and draw or generate the parabola.
   
   
2. Students may say no it has to be this way. Explain that the location of the vertex at (0,0) is arbitrary-explain arbitrary if necessary--, and that it could be assigned another location. This representation makes computations of the other points which are in positions relative to the focus easier to work with.

1. Students should recognize the set up of the diagram as analogous to the paper folding. Students should see that the perpendicular bisector of D to the Focus goes through P.
2. Students should write (0,0) at the origin. And they should lable P (x,y)

(circulate to verify)

The focus is at (0,a)

D is at (-a,x)

x=-a

The distances are the same

P: (x,y)

Focus: (0,a)

Distance =

(x^2 +(y-a)^2)^.5

P: (x,y)

D: (x,-a)

Distance =

((y+a)^2)^.5

x^2+y^2-2ay+a^2=y^2+-2ay+a^2

x^2 - 2ay = 2ay

y = x^2/(4a)

Evaluate:

Lesson Objective(s)

Learned (WRAP –UP at end) -> Summarize

Do this activity if there is time. The mirrors we use are spherical, but we will see in the sketchpad activity that parabolic and spherical mirrors behave similarly close to the vertex.

Students will be given a chart to find all four parabola equations on their own as homework. They should be able to do this.

Approx. Time__N/A___mins

Critical questions that will allow you to decide whether students understood main lesson objectives

Now that we have the equation of a parabola, if we have a parabolic mirror that has height ______ -measure height of telescope mirror, and diameter ______ calculate the location of the focus - in other words, find a.

Be sure to center your parabola on the coordinate grid to find the appropriate x and y that go into your parabolic equation.

Students are asked to make some comparative statements about the four parabola equations.

Expected Student

Responses/Misconceptions

Students may not be ready to make this attempt without guidance. Depending on time do as a group activity, or allow them to attempt in pairs.

The answer is to halve the diameter of the mirror, to get an x value, and use the height from bottom to top of curve for y so:

Y = x^2/4a -- plug in radius for x and height for y and solve for a. a will be distance above (0,0) on the y axis.

Not all students will be successful. Computation errors can be a problem, as well as making a mistake in the assignment of coordinates.

Vertex

Coordinates. Vertex

Coordinates

Focus

Equation

Directrix

Equation of Parabola