----~-~-
rJlr:~~l
INVESiflGATION
Riding
Ferris Wheels
Common Factors
and Multiples
-.----.-----At a Glance
Thue are many e.hings in the world that
Grouping:
the same time. Knowing about facto,. and multiples can help you to solve such
bapptO 0Vtt and over again in SCI cycles.
Sometimes f t want to know when two things with diffuent cycles will happm at
Pairs 01' Small ~
problems.
[.(1'5 Start
by comparing me multiples of 20 and
~.
The multiples of 20 are 20, 40, 60, 50, 100, 120, .
Tht: multiples of 30 are 30, 60, 90, 120, ISO, 180, .
Launch
• With the elm, make a lis! of
factor$ for 24 and 36 and Sod
The numbers 60, 120, 180, 240, ... art multiples of both 20 and 30. We c.aJ.lth~
numbers common multiples of 20 and lO.
the common factors.
Now let's compart w factors of 12 and lO.
• With lhe cbss, make a liSt of
The factors of 12 are 1, 2,3,4, 6, and 12.
The factors of 30 are 1, 2, 3, S, 6, 10, IS, and 30.
multiples for 24 and 36 and
find the common multiples.
The num~rs 1, 2, 3, and 6 are factors of both 12 and 30. We ca1J thne numbtrs
&clan of 12 and 30.
• Ikad Problem 4. 1 aloud.
CommOD
Make SUit the srudenu
understand what is bring
asked.
ean you figurt ou[ how many people could ride In each car?
Riding Ferris Wheels
Explore
One of the most popular rides :u a camiva1 or amusement park is [he Ferris whetl.
• CirrulatC' as snKkms work 10
solve the problem.
SUlf'Jlf'Mrize
Did you know1
• Let studenu share their
The brgest Ferris Whed
wa:I buU, for the 'Ibrid's Columbian Exposition In
Chicago in 1893. Tht: whetl could wry 2160 people in Its 36 passenger C2I1.
answer and explain thdr
thinking.
• Point OUI that. in part A, t/l(:
answer is onc of the givtn
numbrrs, and. in part C the
answer is the product of the
twO
numbtrs.
Assign ....nt
Answer to Prablem 4.'
Choices
A. The multiples of 20 are 20, 40, 60, 80, 100, .... The mUltiples of 60 are 60, 120, 180,
240, 300, etc. The least common multiple (the smallest number that appears on both
lists) is 60. So both siblings will be on the ground again in 60 seconds.
ACE questions 1-4 and
unassigned choices from
earlier problems
B. 150 seconds
C. 70 seconds
)bu and your link: sister go to a carnrnJ that has both a largr and a small Frnis
wheel. )bu get on the large Ferris wiled at the same lime your sister gets on the
small Ferris whttl. The rides begin as klOn as you are both buckJed into your
sau. Determine the number of 5eCOrKb that will pass bem you and your sister
are both at the booom ag.Un
A. if th~ large wheel makes one morutlon in 60 srcoods and the small whetl
makes one revolution In 20 seconds.
B. if the brgt wberl rn.aka one molution in ~ !eCOnds and the small Wed
makes one moZution In 30 K'COOds.
C. if the brgt w~1 makes one revolution in 10 seconds and the small whetl
makes one revolution In 7 seconds.
•
Problem 4.1 Follow-Up
For partS A-C in Problem 4.1, determine the number of tlmes each Ferris wheel goes
around before you and your sisler are both on the ground again.
Investigation 4; Commoa Facton &JUt Uu!Upln
II
Answers to Problem 4.1 Follow·Up
A. The large Ferris wheel goes around one time, and the small Ferris wheel goes around
three times.
8. The large Ferris wheel goes around three times, and the small Ferris wheel goes
around five times.
C. The large Ferris wheel goes around seven times, and the small Ferris wheel goes
around ten times.
InvlHti ;IIt'on 4
Looking at Locust Cydes
Cicadas spend most of their liv~ underground. Some ctcadas--<:ommonly called 13·year
locust~ome above ground every n years, while othcC'S-<alled 17·year locusts-<omc
out every 17 years.
Looking at
Locust
Cycles
- At
- - a- Glance
---------
Sl:epban's grandfather told him about a lenible year when the cicadas wert so
numerous dw they ate aU the crops on his brm. Stephm conjectuttd thai both
B·year and 17·year ~ came out that year. Assume Stephan's con}ecttltt is
co=<
A. How OWly years pass between the years when both 13-yt2t and 17·yeat
locusts are OUI at the same: time~ Explain how you got your answer.
.. Suppose there wett 12·yeat, 14-year, and 16-year locusts, and they all came
out lhis year. How mmy yeatS will it ~ before they all come oUlIQ8Clher
again? Explain how you got your at\$WCl'.
Launch
• Remind students of the locust
problem posed :u the begin-
ning of the student edition.
• _Iem 4.2 Follow-Up
• Read Problem 4.2 with the
students and make sure they
understand the questiOI\$
being asked.
For parts A and Bof Problem 4.2, teU whether the answer is less than, greater than, or
equaJ to the product of tht locust cycles.
Explore
• Circulate as students work on
the problem.
• If srudtnl5 get Stuck, ask
them to think about how this
problem is similar to the Last
problem.
Summarize
• Havc: students present their
strlltegies and solutions.
CH991 by Sidney 112ITiI. From ItIU II/l", f'rooP /'U Glvt Ibu hooft
• Point OU I WI, in pan A, the
WH. F~iU\, Nn. 'tNt.
solution is the product of the
two numbers. and in pan S,
the solution is smaller than the
prtxlUC1 of the IWO number5.
............
Assignment
Answers to Problem 4.2
Choices
A. 221 years; Possible explanation: The multiples of 13 are 13, 26, 39, 52, 65, 78, 91, 104,
117,130,143,156,169,182,195,208,221, .... The multiples of 17 are 17, 34, 51,
68,85, 102, 119, 136, 153, 170, 187, 204, 221, .... The least common multiple is 221.
B. 336 years; Possible explanation: The multiples of 12 are 12, 24, 36, 48, 60, 72, 84, 96,
108,120, 132, 144, 156, 168, 180, 192,204,216,228,240,252,264,276,288,300,
312,324,336, . ... The multiples of 14 are 14, 28, 42, 56, 70, 84, 98, 112, 126, 140,
154,168,182,196,210,224,238,252, 266,280,294,308,322, 336, .... The multiples of 16 are 16, 32, 48, 64, 80, 96, 112, 128, 144, 160, 176, 192, 208, 224, 240, 256,
272,288,304,320,336, .... The least common multiple of 12, 14, and 16 is 336.
ACE questions 5-7,
14-16, 18, and unassigned choices from
earlier problems
Answers to Problem 4.2 Follow.Up
A.
Investi .. tion 4
221 "" 13 )( 17
B. 336 < 12 )( 14 )( 16
Planning a Picnic
Common factors and common multiples can be used to figure out how many people can
share things equally.
Planning
a Picnic
- At
- - a- Glance
--------Grvuping: Individuals,
then Pairs
Launch
r
• Reld tbe problem to the
class, and as k studentS to sug·
gest how they might solve it
Miriam's uncle donated 120 cans of juke :and 90 packs of cbccse crackers for the
school picnic. Each student is to rttcive the milt number of cans of jui« and the
same number of pacb of ctaeken.
• Help srudents see that, 10
so!\·e the problem, they need
10 find faCtors.
What is the largest numbcr of students that can come to the picnic and shate the
food equally? How many cam of juice and how many packs of crackers will each
student receive? F.xpbin how you got your answers.
Explore
• Circulate as students work on
the problem.
• Problem 4.3 Follow-Up
If Miriam·s uncle eatS two packs of crackers before he sends the supplies to the school,
what is the largest number of stu<icnts thai can come to the picnic and share the food
equally? How many cans of jui~ and how many packs of crackers will each re~ivd
Summarize
• Have srudenlS present their
SU1Itcgies and solutions
• Hold a discussion about
how students might Slart
Iheir responses 10 the
Mathematical Reflections.
Investigation 4: Common FactoR aDd IIultJplel
_II
Answer 10 Problem 4.3
Assignment
The factors of 120 are " 2, 3, 4, 5, 6, 8,10,12,15,20,24,30,40,60, and 120. The factors
of 90 are " 2, 3, 5, 6, 9, 10, 15, 18, 30, 45. and 90. The greatest common factor is 30. So,
30 students can attend the picnic. Each student will receive 4 cans of juice and 3 packs of
crackers.
Choices
Answer to Problem 4.3 Foll_·Up
Now there are only 88 packs of crackers. The factors of 88 are " 2, 4, 8, 1" 22. 44, and 88.
The greatest common factor of 120 and 88 is 8. So, 8 students can attend. Each student
will receive 15 cans of juice and 11 packs of crackers.
ACE questions 17, 21. 22.
and unassigned choices
from earlier problems
(Assign 19, 20, and 23 as
an extra challenge.)
Investl ation 4
• Riding Ferris Wheels
[n this problem, students usc common multiples to solve a problem about riding ferris wheels.
Launch
It is important to discuss the differences and connections between f1ctors and multiples. Ask stu·
dents for the fallors of 24 and the factors of 36. As students find the factors, write them on the
board. When you bave a list of factors for each number, ask which factors appear in born lists,
and discuss what it means for a number [0 be in both lists. Explain that these numbers art the
common factors of 24 and 36.
Next, ask students to find the first ten multiples of 24 and the first len muJtiples of 36, and list the
multiples of each number on the board. Srudents may not include the numbers themselves in the
list. This gives you an opportunil)' to emphasize that the smallest multiple of any number is the
number itself When the lists art complete, ask students to find the numbers mat appear on both
lists. ExpWn that these numbers are the common multiples of 24 and 36. Ask what the numbers
that appear on both listS have in common. Somt students may see that evay number that appears
on both lists is divisible by 72. When you feel students arc: ready, move on to the problem.
We are going to use the ideas of common factors and common multiples to solve the problems in this investigation.
Read Problem 4.1 aloud. Make sure srudents understand what is being asked.
Explore
Have students Vt'Ork in pairs or small groups to solve the problem. Remind them to record their
rtsults in their journals. If a group finishes early; have them work on Problem 4.1 Follow·Up.
Summarize
Let students share their answers and expWn their thinking. You may W7Jlt to spend some time
having them mentally conven each answtr to minutes. For example, in part a, students should
realize that the Siblings will both be at the bottom every (wo and a half minutes.
Students may notice that in part A the answer is one of the two numbers, and in part C the
answer is the product of the two numbers. If students do not make these obserntioru, prompt
them:
I noticed that in part A the answer is one of the given numbers. In part
C the answer is the product of the two numbers. (You may want to
write these observations on the board.) I wonder what is special about
these numbers that made this happen. As we work on the rest of this
unit, try to notice other number pairs that produce these results.
This is an important question to pose before moving on to the next problem. Although you art nO(
ready to answer the question, if you ask it now, students may notice this pattern in other problems.
This will make your 'WOrk easier later on.
- ---'P'Ii':w,.
In",ntigiltion 4
• Looking at Locust Cycles
In this problem, students find common multiples to predict when B -year and 17-year locusts
will be out at the same time.
Launch
Remind students of the locust problem posed at the beginning of the student edition.
Problem 4.2 involves the 13-year and 17-year locusts described at the
beginning of your book. Like the Ferris wheel problem, the locust problem involves determining when two or more things will happen at the
same time.
Read Problem 4.2 'Nith the students, and discuss the situation until you feel your students understand the questions being asked.
Explore
Have srudents work in pairs or small groups to explore the problem. Help them stay focused on
the problem, and remind them to be ready to share their strategies as weU as their solutions. If
students are sruck, ask them to explore how this problem is similar to the Ferris wheel problem
and to revicw the s[r.llcgics they used to solve that problcm. If you have time for presentations,
have students write their responses on the board.
Summarize
Have students present their strategies and solutions. After the fust group of students has shared
its strategy, ask if any group has a different strategy for solving the problem. You can ask these
additional questions.
I noticed that in part A the solution is the product of the two numbers,
and in part B the so lution is smaller than the product of the three num·
bers. (Show this if it was not shown by a student.) Why do you suppose
this happens?
How can we tell whether the least common mUltiple will be less than or
equal to the product of the numbers?
Your students may not be able to answer this. Don't push them; it is a subtle idea. If the numbers have no common factors, then the least common multiple will be the product of the numbers (for example, the least common multiple of 13 and 17 is 13 x 17, or 221). If the numbers
have a common factor, the least common multiple will be less than the product of the numbers.
Irwestlg_tJon 4
n-~.
ili'~---
°
4.3 • Planning a Picnic
This problem is different from the first two problems bea.use it involves factors instead of multi·
pies. We want students to wrestle with determining what signals a situation in which finding
common multiples is useful and what signals a situation in which finding common factors is use·
ful. After this problem, you 'Will want to ask your students to make some notes about this for
their Mathematical Reflections.
Launch
Read Problem 4.3 aloud. Ask students to suggest how they might solve the problem. Students
may look for multiples, because this is how they solved the earlier problems. If this occurs, help
students find multiples up to the least common multiple, 360.
Is 360 a reasonable response to the question?
If students think ;60 is a reasonable answer, ask:
360 what?
They should set that 360 students is not a reasonable answer. Remind students that part of pro!>
lem solving is asking whether the answer seems reasonable. Labeling the answer and restating it
in the COnlext of the problem helps to determine whether an answer is ceasolUble.
Continue to ask questions that help students see that they need to find factOC5 of the numbeC5.
Explore
When you feel your students have come up with a strategy, let them work in pairs on Problem
4.3 and Problem 4.3 Follow·Up.
Summarize
Part of your strategy for this summary should be to prepare students for the Mathematical
Reflections. After students have shared their solutions and strategies for Problem 4.3, discuss
how {hey might Slart their responses to the Mathematical ReflectiOns. Don't push them to write if
they ace not ready. Some students may need more practice with problems of this sort-pnctice
provided in the ACE questions--before they ace ready to complete the Mathematical Reflections.
_
Additional Answers
Mathematical Refledl.".
2. (continued) Solution: The multiples of 45 are 45, 90, 135, 180,225,270• .... The multi-
ples of 75 are 75. 150. 225. 300•.... The least (ommon multiple of 45 and 90 is 225. The
t rains would be at the station at the same time every 225 seconds.
3. List the factors for each number and then find the factors that are in both lists.
4. List the multiples for each number and then look for the numbers that are in both lists.
_ ___ a
.,.~
.
InvKti,nion 4