Lesson 5: In Sync
Introduction
In the last lesson you saw that the hand timelines are periodic, with the same period, and phase shifted. Similarly, the ball timelines are periodic, but shifted. The timelines for juggling three balls have more than phase shifts in common. In this lesson you will look for a connection between the ball and hand juggling timelines.
In order for the juggler to catch the ball, the ball and the hand must work in synchronization. That is, if you think of these actions as drum beats, the ball and the hand must beat at the same moment. We will examine several different periodic patterns and determine when they beat at the same time. You will develop a way to predict this mathematically.
This orchestration of beats or actions shall lead to orchestration of production outputs in a manufacturing plant.
Learning Objectives
• Determine when two different timelines beat in unison.• Develop a method of predicting when two timelines will beat together.
Materials
• graph paper• ruler
In Sync
One strategy in problem solving is
to organize the information. Frequently we use tables. In
this activity you will gather data. It will probably be
beneficial to organize the data into a table. After you have
gathered the data, you will be asked to examine it and make
conjectures.
Activity 1-Finding
Phase Shifts
Determine the period of
each of the beat tempos A-G, given in Figure 2. That is,
determine how often each of the beat tempos A-G beats. For
example, A beats every 2 seconds, and the period for A is 2
seconds. Then, use this information to determine the
frequency of each of the beat tempos A-G. For example, A
beats 1 time every 2 seconds and 30 times every minute. The
frequency of A is 30 beats per minute.
Period A 2 sec Frequency A 30 beats per minute
Period B _____ Frequency B _________
Period C _____ Frequency C _________
Period D _____ Frequency D _________
Period E _____ Frequency E _________
Period F _____ Frequency F _________
Period G _____ Frequency G _________
Determine when each combination of 2 tempos will beat in sync. For example, tempos A and B beat in sync on the 6th second (and every multiple of six thereafter). This number that you are finding is the common period for the two beat tempos.
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
As a class, compare matrix entries and come to agreement.
Develop a conjecture on how to determine when two different beat tempos will beat in unison. Be prepared to share your conjecture with the class.
Activity
2-Orchestrating Factory Outputs
Suppose that to construct a
certain computer board, equal numbers of X-chips and Y-chips
are needed. The X-chip machine makes 4 X-chips per minute
and the Y-chip machine makes 6 Y-chips per minute. Answer
the following questions about factory production.
What is the period for each of these machines?
What is the common period for the machines?
In this length of time (the common period), how many of each kind of chips have been made?
To balance production (get the same number of each kind of chips) we are going to need to put some more X-chip and Y-chip machines on-line. How many of each will we need?
The machine that constructs the computer board needs an X-chip and a Y-chip at the exact same time. Use the timelines that follow and the concept of phase shifting to synchronize the machines. Be sure that everything is clearly labeled and that the X- and Y-chips are produced simultaneously.
Closure
Restate your conjecture here, as a complete English sentence. Define any new words.
Homework
1. For each of the following pairs of beat tempos, determine when they will beat in sync. That is, find their common period.
a) 6 seconds and 14 secondsb) 4 seconds and 15 seconds
c) 12 seconds and 42 seconds
d) 20 seconds and 15 seconds
2. Determine when the three beat tempos given below will beat in sync. That is, find their common period.
a) 3 seconds, 5 seconds and 6 secondsb) 2 seconds, 7 seconds and 8 seconds
c) Explain how you determined the answers to parts a) and b).
3. Look up the definition of least common multiple in a math book, at the library or on the web. See how it compares to the definition developed in class.
4. Look back at your juggling timelines. When are they in sync? Draw a dotted vertical line to connect when a ball is in a hand. These are the moments of syncopation. Do your timelines and the mathematics agree? If not, why don't they?
5. Machine A produces an X-chip every three seconds and machine B produces a Y-chip every seven seconds.
a) When will the machines produce both chips in sync?b) At this time, how many of each type of chip will there be?
6. Machine A produces an X-chip every six seconds and machine B produces a Y-chip every eight seconds.
a) When will the machines produce both chips in sync?b) At this time, how many of each type of chip will there be?
7. Machine A produces an X-chip every three seconds and machine B produces a Y-chip every nine seconds.
a) When will the machines produce both chips in sync?b) At this time, how many of each type of chip will there be?
8. Explain how you got your answers to part b of the preceding three problems. Develop a mathematical procedure (algorithm) that enables you to find the answer to this type of question.
9. Explore the numbers in the following tables. Try to determine a pattern and express that pattern in words or mathematical symbols.
|
|
10. Another use of least common (LCM) multiples is in adding fractions.
Example: 
I. Find the least common multiple of the denominators.
6 • 15 = 90
II. Find a common factor of the denominators.
A common factor of 6 and 15 is "3"
III. Divide the least common multiple of the denominators by the common factor of the denominators.
=
30
IV. Use the quotient "30" and the denominator of each fraction to find the multiplier for each fraction.
= 5 
= 2
V. Multiply each fraction by the
number one in the form of 
.
= 
VI. Add the fractions in their new form.
a) 
b) 
c) 
d) 
e) 
f) 