Folding Activity 1

We’ll do this first activity as a class, and I will model the steps with the document camera.

Part 1

You’ll need: Patty paper Ruler Sharpie®

Colored paper Tape or glue stick

As you do the following, be careful not to smudge your work. For any step that includes the use of a Sharpie, wait about 30 seconds after marking before you do any folding.

1. With a ruler and a pencil, draw a straight line across the bottom of your wax paper, about a ruler’s width above the bottom of an edge.

2. Fold the paper in half so that one end of the line you drew lands on top of the other end, and crease it in that position.

3. Using a pencil, draw a point on this crease no more than 3 cm above the line (try and use a different value for each member in your group).

4. Fold the paper so that one end of the line (not the crease) lands on the point. 5. Crease the paper along this fold. 6. “Slide” the line along just a little bit so that a different place on the line is over the point. 7. Crease again. 8. Keep sliding and folding the paper so that different places along the line land on the

point. The closer together your creases are, the more refined your shape will be. 9. Keep doing this until you reach the other end of the line. 10. Unfold your wax paper. Do you see a definitive shape? 11. Carefully darken the outline of your shape with a Sharpie®. You’ve just drawn a

parabola! 12. Label the point “𝐹”†, and label the line “directrix”. 13. Tape or glue the edges of your patty paper to a piece of colored paper. 14. Write “The Parabola” and your name at the top of the colored paper.

† This point is called the focus.

Folding Activity 2

Part 2 You’ll need: Patty paper Protractor Sharpie®

Your colored paper Tape or glue stick

1. With a ruler and a pencil, trace the straight line and the point you made on your parabola layer.

2. Fold the paper so that the point lands somewhere on the line, but only once. Make sure your directrix lands on top of your focus. It should not be at either extreme end of the line or too near the middle. Aim to fold it about !

! or !

! of the way across the line from either

end. Try to arrange it so that everyone in your group has their fold in a different place. 3. With your paper still folded, set a protractor down on your paper so that the 90° mark

lines up with the directrix, and move it along the paper until its edge lines up with the focus 𝐹.

4. Using a pencil, draw a line segment on the back of your paper from the folded edge to the focus. Don’t draw beyond the focus or the directrix.

5. Leave your paper folded, and flip it over. 6. Trace the line segment you drew in step 3 (you should still use a straightedge), on the

other side of the back of the folded paper. 7. Unfold your paper, and draw a point where the two new line segments meet. 8. Label the new point “𝑇”. This is called the point of tangency. 9. Draw a point where your perpendicular line meets the directrix. 10. Label this point “𝐷”. 11. Using a Sharpie® and a ruler, trace over the line segments 𝐹𝑇 and 𝑇𝐷. 12. With the directrix and the focus 𝐹 carefully aligned, tape or glue this copy only on one

edge over the parabola on your colored paper.

Discovering the Parabola Parts 1 and 2 Name ______________________ When you have finished making and labeling your Parabolas, answer the following questions. Each person should fill in their own answers, and your answers should match your individual parabola where appropriate, but work together to compare answers on your different parabolas before answering: a) Using your two-layer wax

paper parabola, what do you observe about the distances between the points 𝐹 and 𝑇 and the points 𝑇 and 𝐷?

Answer for your parabola:

b) How could you explain your answer to part a) to someone who is not in this class?

Answer for your parabola:

c) Compare your results from part a) to your other group members’ results.

Do all your answers have the same values? How are they the same or different (What are the values?) Do all your answers have any of the same properties? How are they the same or different (What are the properties that are the same?)

Answer for your group:

d) Compare your parabola to those of your group members, and compare your answers to parts a) through c) to your group members’ answers. Use what you learn to decide as a group what the definition of a parabola is.

Answer for your group:

Folding Activity 3

Part 3 You’ll need: Patty paper Protractor Sharpie®

Your colored paper Tape or glue stick

1. With a ruler and a pencil, trace the straight line and the point you made on your parabola layer.

2. Using a protractor, draw a line segment in pencil that goes from the focus to the directrix at 90° to the directrix.

3. Draw a line segment in pencil that goes through the focus at 90° to the line you drew in step 1.

4. Lay this copy on top of your finished parabola and carefully align the focus and the directrix, but don’t tape it down yet.

5. Draw a point on the top layer where your vertical pencil line meets the parabola.

6. Label this point 𝑉. 7. Draw points on the parabola where your horizontal

line meets the parabola. If your pencil line is not long enough, use a straightedge to extend it before you draw your points. Remember that this line should be at 90° to the line that goes from 𝐹 to 𝑉.

8. Label these points 𝐴! and 𝐴!. 9. Darken the line segment from 𝐴! to 𝐴! with a

Sharpie, but do not draw beyond the parabola. 10. Darken the line segment from 𝐹 to 𝑉 with a

Sharpie, but do not draw beyond the parabola. 11. Measure the distance from 𝑉 to 𝐹 in cm and make

a note of your measurement in the right column. 12. Measure the distance from 𝐴! to 𝐴! in cm and

make a note of your measurement in the right column.

13. With the directrix and the focus 𝐹 carefully aligned, tape or glue this copy only on one edge over the two previous layers on your colored paper.

11. 𝑉𝐹 = cm 12. 𝐴!𝐴! = cm

Discovering the Parabola Part 3 Name ______________________

When you have finished making your third layer, answer the following questions. Each person should fill in their own answers, and your answers should match your individual parabola where appropriate, but work together to compare answers on your different parabolas before answering: e) If the point 𝑉 on your

parabola were on the origin ((0,0) on a graph), what would be the coordinates of the following points? Use the measurements you wrote down in steps 11 and 12 on the previous page. 𝐹? 𝐴!? 𝐴!?

Answer for your parabola:

𝑉 (0,0) 𝐹( , ) 𝐴!( , ) 𝐴!( , )

f) Using what you know

about graphing lines, your group’s definition of a parabola, your answer from part a) and your measurements from steps 11 and 12 in part 3, find the equation of the line that is the directrix of your parabola from part 3.

Answer for your parabola:

g) What do you observe about the measurements of 𝐴!𝐴! and 𝐹𝑉?

Note: the segment 𝐴!𝐴! is called the latus rectum.

Answer for your parabola:

h) Is this observation consistent across your team? (Do you all get approximately the same result from this comparison with potentially different measurements for your parabolas?)

Answer for your group:

i) Using the coordinates of either point 𝐴! or 𝐴!, find the generalized formula (equation) for your parabola, and write it in the form 𝑦 = 𝑎𝑥!. Note that you are stull assuming that your parabola is sitting so that its vertex is at the point (0,0).

Answer for your parabola:

j) Solve the equation in part i) for 𝑥! and divide the coefficient of the right hand side by 4, so that your equation looks like this: 𝑥! = 4𝑐𝑦 Your final answer should have a 4 in it, as above, but your value for 𝑐 may be different from those of your other group members.

Answerforyourparabola:

k) Compare your results to those of your group members. Each person should look at their answer to part j), and then look back at their answers to parts g) and h). Could you have written the equations for your parabolas without first having to write it as you did in part i)?

Answerforyourgroup:

l) Use your answers to these questions to decide as a group what the parabola looks like which has the equation 𝑥! = 20𝑦. Accurately describe its features (vertex, latus rectum, directrix, focus) or draw a labeled sketch.

Answerforyourgroup:

Folding Activity 4

Part 4 You’ll need: Cartesian grid (see back page of this packet) Patty paper (1 piece) pencil

1. LabeltheaxesofaCartesiangridwithvaluesfrom−10to10inboththe𝑥and𝑦directions.

2. Layapieceofwaxpaperoverthegrid.

3. Onthewaxpaper,notonthegraphpaperplotthesepoints,à and“connect”themtoformthegraphofaparabola:

4. Determinetheequationofthisparabolaintheform𝑦 = 𝑎𝑥!

5. Determinetheequationofthesameparabolaintheform𝑥! = 4𝑐𝑦 Youdidthisforadifferentparabolain“DiscoveringtheParabolaPart3”,questionj)

6. Holdoutyourlefthandsothatyou’relookingatthebackofit,likethis:

Nowrotateyourhandsothatyou’relookingatthepalm,likethis:

7. Pickupthewaxpaper,andwiththesametypeofmotion,flipandrotateit,andsetit

downonthesamesetofaxessothattheparabola“hugs”the𝑥-axisinsteadofthe𝑦-axis.

8. Whatistherelationshipthatthisparabolahaswiththeoneyouhadbeforeyouflipped?

The rotated parabola is the __________________________ of the original one. 9. Findtheequationoftherotatedparabolainthesameformasyoudidinquestion5.10. Howarethesetwoequationsthesame?Howaretheydifferent?11. Findthecoordinatesofthefocusofthisparabola.

12. Findtheequationofitsdirectrix

𝒙 𝒚 -4 2-2 !

!

0 02 !

!

4 2

Discovering the Parabola Part 4 Name ______________________

13.

Nowthatyouknowsomethingaboutparabolaswhoseverticesareattheorigin(thepoint(0,0)),discussthefollowinginyourgrouptodeterminewhattheequationofaparabolawouldbeifitwerecenteredatadifferentpoint.

m) Iftheequationofaparabolais𝑥! = 4𝑐𝑦,whatwouldtheequationbeoftheparabolawhere𝑐 = 3?

Answersforyourgroup:

n) Whatwouldtheequationbeofthesameparabolaifitwereshiftedoversothatinsteadofhavingitscenteratthepoint(0,0),itscenterwasatthepoint(1,−4)?

Writethisequationsothattheshiftinthe𝑥directionisonthesamesideoftheequationasthevariable𝑥,andtheshiftinthe𝑦directionisonthesamesideoftheequationasthevariable𝑦.

o) Graphthisparabolaononeofthegridsatthebackofthispacket,andlabelallofitsfeatures,including:

i) Vertexii) Coordinatesofvertexiii) Equationofdirectrixiv) Latusrectumv) Coordinatesofbothendpointsofthe

latusrectumvi) Coordinatesofthefocus

p) Usingalgebra,writetheequationofthisparabolainpolynomialform

(𝑎𝑥! + 𝑏𝑥 + 𝑐𝑦 + 𝑘 = 0).

q) Rewritetheequation

𝑦! − 6𝑦 − 8𝑥 − 7 = 0intheform

𝑥 − ℎ ! = 4𝑐 𝑦 − 𝑘

or

𝑦 − 𝑘 ! = 4𝑐 𝑥 − ℎ .Nowthatit’sinthisform,howdoesthisequationforaparabolacomparetotheoneinquestionn)?Whathaschangedintheequation?Howdoyouthinkthiswillmanifestinthegraphofthisparabola?

r) Onanothergridatthebackofthispacket,graphtheparabolafrompartq).

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