# Back and Forth in Time (lesson)

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Students explore the graphs of a sine function by creating a “seismograph” type instrument. Students develop an understanding of amplitude, period, and phase shift. (see attachment for full lesson w/handouts)

Main Standards
T 2.0 Students know the definition of sine and cosine as y – and x -coordinates of points on the unit circle and are familiar with the graphs of the sine and cosine functions.
T 4.0 Students graph functions of the form f(t) = A sin (Bt + C) or f(t) = A cos (Bt + C) and interpret A, B, and C in terms of amplitude, frequency, period, and phase shift.

Materials

Strips of butcher paper, cut ( 4 strips per team)
Marker (1 per team)
Stopwatch (1 per team)
1 can ( 1 per team)
Copies, Graphs of Sine and Cosine Functions
Graphing Calculators, 1 or 2 per group (notes for modification included if graphing calculator not available)
Meterstick, 1 per team
Ruler, 2 per team
Scissors, 1 per team (optional, but helpful)
Index cards, 2 per group

Teacher Notes

***May take more than 1 traditional class period****

Demonstration
Demonstrate the measuring instrument (much like a seismograph). Have 3 student volunteers help you. One should be the marker, the puller, and the axle.

Wrap the butcher paper around the can. Set the can down with the paper wrapped around it and one end sticking out to pull. Have the timer say “go” and then another student is to pull the end of the butcher paper, unraveling it at a steady pace for 4 seconds. Practice this one more time.

Next, have the marker stand in place, ready to move the marker directly back and forth on the paper. (like a needle on a seismograph). Let the marker know that he/she is to keep a steady pace. (May help to think “tick/tock” in his/her head to keep the same pace.)

When the timer says go, the puller pulls and the marker marks simultaneously for 4 seconds. The result should be what looks like a sine curve. Cut off the sheet of paper at the beginning and the end of the marks so that the sheet of paper only represents 4 seconds.

Group Work
Now, hand out activity sheet Back and Forth in Time to each person and break up the class into groups of at least 4. Pass out the necessary materials to each team. Remind the students that even though it is a team activity, each person is to have the response to the questions noted on their own activity sheet.

Questions 1-8 has student teams set up their curve as a coordinate system directly on the butcher paper. With the x-axis representing time in seconds and the y-axis representing distance from the center in centimeters. Make sure as you walk around that the students are measuring from the x-axis up to the curve along a perpendicular line (using their index card as an aid). Also, encourage them to work as a team such that more than one person is taking measurements at the same time.

Question 9 asks student teams to find the equation that models the curve. Hopefully they chose a sine or cosine curve, and not a polynomial. Some hints to give them as they find the sine or cosine curve is to find the period, amplitude, and phase shift.

Discuss Questions 9-12 prior to going to the back of the sheet of paper.

The back of the sheet requires the use of a graphing calculator. If graphing calculators are not available, then modify the activity and have the student teams skip Question 8 & 12 and do only Questions 9-11 without the graphing calculator.

Discussion: Application in Science

A Seismograph records the movements within Earth. It shows amplitude/time relationship. Waves leave the focus in groups and travel at different speeds depending on their type of motion and the substances they travel through. Seismographs show this separation of travelling rates and the arrival times of seismic waves to seismic stations (lag time). From this data epicenters can be located.

The difference between what we have simulated and a seismograph is that the “marker”/ needle moves according to the waves that leave the focus and travel through the ground. The waves are not consistent, and the amplitude and frequency of the waves change depending on the distance from the epicenter and the strength of the earth’s movements (earthquakes).