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Prep for Teachers
Prepare 12 to 15 beanbags or envelopes by placing 30 white and 20 black
beans in each.
Make sure all organizers are copied, Web sites are bookmarked, and the
video is CUED to the start of the first learning activity.
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When using media, provide students with a FOCUS
FOR MEDIA INTERACTION, a specific task to complete and/or information
to identify during or after viewing of video segments, Web sites, or other
multimedia elements.
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In this activity, students will learn about equal ratios and the nature
of proportions using manipulatives. The activity will prompt students
to calculate ratios in different combinations to develop an understanding
of how ratio and proportion are related and how to convert the fractions
to decimals.
Step 1:
Review ratios with the class. Stress that their main purpose is to compare
two or more amounts. Written in fraction form, ratios can be reduced.
Improper ratios are not simplified.
Break the class up into groups of two or three. Distribute the bags of
beans, one white and one black to each group. Distribute the Exploring
Proportions Activity student organizer. Tell students that they will explore
equal ratios and the nature of proportions. They will also answer the
questions on the organizer as they complete the activity.
After completing the activity, groups will report their findings. Discuss
with the students what they discovered. (Students will separate 30 white
and 20 black beans into 10 piles each with 3 white and 2 black. Each pile
represents the simplest ratio – 3/2 – derived from the whole
– 30/20. By combining piles, students will discover that ratios are
equal if they are built with the same simplest ratio. Their parts have
a relationship to the whole. Equal ratios are in proportion.)
Ask the group what tests they would do to see if two ratios are in proportion.
(See if each ratio has the same simplest form – building block –
or whether they have the same decimal fraction.)
Ask the group if 6/4 and 4/6 are in proportion? (No, they are made up
of different building blocks: 3/2 and 2/3.) Remind the class that ratios
are used to compare amounts and that besides having the same building
blocks, equal ratios are written in the same comparative order.
Tell the class that they will now examine how problems are solved using
proportional reasoning.
Step 1:
Students will understand that proportions are direct relationships.
Insert the video Mathvantage #2: Proportional Reasoning into the
VCR and CUE the tape to the blue screen showing “Mathvantage.”
Distribute the “Proportional Reasoning” organizer. This sheet
will be used during the video for answering questions and taking notes.
Provide your students with a FOCUS FOR MEDIA INTERACTION, asking
them to explain what is meant by a direct relationship between quantities.
START the tape and STOP when the girl on the escalator says,
"...I go up." Ask students what direct relationships are between
quantities. (As one amount goes up, the other goes up too; as one amount
goes down, the other goes down, too.)
Elicit from the class other direct relationships they may know. Have students
complete question #1 on the organizer. CHECK answers for accuracy.
Step 2:
Students will learn that the key to proportional relationships is multiplication.
Say to the class that some direct relationships are proportional, or direct
proportions. Provide your students with a FOCUS FOR MEDIA INTERACTION,
asking them to listen for the key to proportional relationships. START
the tape and STOP when the girl says, "...get great results."
Ask students what the key to proportional relationships is. (Multiplication.)
Explain to students that when direct relationships go up or down by a
multiplier, the direct relationship is proportional and called a direct
proportion.
Ask your students how the girl showed that multiplication is the key to
proportional relationships. (She showed a recipe. If one cup of flour
needs two eggs, then two cups of flour will need four eggs.) Write on
the board 1/2 = 2/4. Point out that there is a multiplier of two.
When the girl put the flour and egg ratios on a graph, what did the graph
look like? (A straight line.)
Have students complete question #2 on the organizer. CHECK answers
for accuracy.
Step 3:
Students will see applications of proportional reasoning.
Tell students that they will see how proportional reasoning is used to
estimate attendance at sports stadiums. Provide your students with a FOCUS
FOR MEDIA INTERACTION, asking them what fact was most important in
estimating the attendance at the football game. START the tape
and STOP when the girl says, "...in the entire stadium."
Ask students what the key fact was for estimating attendance. (One section
has about 1,000 seats.)
Have students complete the first part of question #3 on the organizer.
Question #3 provides different examples so they may apply the estimation
technique in various situations. Check answers for accuracy.
Tell students that they will now see how proportional reasoning is used
to estimate deer population growth in a woodland habitat. Provide your
students with a FOCUS FOR MEDIA INTERACTION, asking what was the
number of fawns born during the winter.
START the tape and STOP when the park official says, "...1,070
white tail deer." Ask students how many fawns were born. (570.) Ask
why it's important to track the population growth of deer. (Too many deer
will destroy the woodland habitat because the deer will eat all the foliage.)
What was the total population before the winter and after the winter?
(500, 1,070.)
Of the 500 deer in the habitat before the winter, how many were does?
(300.)
Did all the does have the same number of fawns? (No, some had one, two,
or three.)
What proportion did he use to figure out the number of does that had twins?
(20/100 = 60/300) If students do not remember the proportions used, REWIND
the tape until the doe proportions appear and then PAUSE the tape.
How many fawns were twins? (60 x 2, or 120 fawns.)
Have students complete the second part of question #3 on the organizer.
CHECK answers for accuracy.
Tell students that they will now see how proportional reasoning is used
to compare objects of different size. Provide your students with a FOCUS
FOR MEDIA INTERACTION, asking what multiplier is used in both the
state population and bacteria examples.
FORWARD the tape to the scene of bicycles on a highway with the
bicycler on the left raising his hand. PLAY the tape and PAUSE
when the screen shows the proportion a/b = c/d. (1,000,000.)
Have students complete the third and fourth parts of question #3 on the
organizer. CHECK answers for accuracy.
Step 4:
Students will use cross products to find missing information in proportions
.
Tell students that they will see how proportions can be used to find missing
information. Provide your students with a FOCUS FOR MEDIA INTERACTION,
asking them to calculate the number of people wearing sunglasses in a
crowd of 12. PLAY the tape and PAUSE when you see the proportion
2/3 = n/12. Ask students to provide their calculation. (n=8)
Have students do the calculation in the first part of question #4 on the
organizer. Students may use any method or reasoning. Elicit answers and
methods from the class.
Let's see how the tape finds the missing information. Provide your students
with a FOCUS FOR MEDIA INTERACTION, asking what operations are
used to calculate the answer. PLAY the tape and PAUSE when
the screen shows 8 = n. Ask students what operations are used.
(Multiplication and division.) Tell the class that in a proportion, cross
products are equal. This fact can be used to test if a proportion is a
true proportion and to find missing information.
Have students complete the second and third parts of question #4 on the
organizer. (Students will use cross products to find the missing number:
n = 4; they will also conclude that two ratios are equal if they
have equal cross products.)
STOP the tape.
Step 5:
Students will apply proportional reasoning to solving problems in planetary
science.
Create a discussion by asking the following questions:
If people want to know how much they weigh what should they
do? (Step on a scale.)
What makes the scale work? (Objects push down on the scale. The scale
measures how much the objects push down.)
What force makes people and objects push down on the scale? (Gravity.)
Scientists tell us that weight is a measure of the pull of gravity on
the mass of an object. Double the gravity and you double the weight.
What type of relationship exists between gravity and weight? (Direct
proportion.)
Where do people and objects have no weight? (Space.)
Why? (There is little or no gravity.)
Would the Earth's Moon have more or less gravity than the Earth? (Less.)
Why? (The Moon is smaller than the Earth.)
We will now examine how to calculate the weight of a 150-pound person
on the Earth's Moon. Distribute the “Proportions and Planetary Science”
organizer and the class set of calculators. Have the class log on to NASA's
National Space Science Data Center fact sheet at http://nssdc.gsfc.nasa.gov/planetary/factsheet/index.html.
Provide your students with a FOCUS FOR MEDIA INTERACTION, telling
them to find the gravity of the Earth and Moon and to write down that
information in question #1 on the organizer sheet. Elicit data. (Earth
gravity is 9.8 m/s2; the Moon is 1.6 m/s2.)
Ask the class to write a proportion using the gravity facts that would
help them calculate the weight of a 150-pound person on the Moon. (150/9.8
= x/1.6, where x is the weight of the person on the moon.)
Reinforce that equal ratios make up a proportion and that those ratios
must be written in the same comparative order. Once the proportion is
written, have students calculate the weight to the nearest tenth of a
pound. (24.5 pounds.)
Have the students calculate on the organizer the weight of the 150-pound
person on Mars and on Jupiter to the nearest tenth of a pound. (Mars:
56.6; Jupiter: 353.6 pounds.)
Ask students to draw a circle. Tell them to draw a line through its widest
part. What do they call this line? (Diameter.)
On the organizer, have students write which planets have a diameter smaller
and larger than the Earth. (Smaller: Mercury, Venus, Mars, and Pluto.
Larger: Jupiter, Saturn, Uranus, and Neptune.)
Ask students to use the diameter information on the NSSDC fact sheet to
determine if gravity and planet diameter have a direct relationship. (No,
not all gravities go up as planet size increases, i.e. Saturn, Uranus.)
Students will engage in a hands-on activity using proportions to solve
problems in physical anthropology.
Physical anthropologists study the physical characteristics of humans.
They have long known that there is a relationship between bone size and
height. We will examine if bone size can be used to predict the height
of a person.
Distribute the class set of calculators and the Proportions and Physical
Anthropology organizer. Break the class up into groups of three or four.
Tell students to complete the steps on the organizer and draw a conclusion
about whether bones are good predictors of a person's height. Students
will take measurements of one member of the group and determine the proportion
from bone length to height. They will then predict the height of other
group members based on the bone measurement.
When groups are finished, have them come together to report their findings.
(Students will discover that although individual predictions may differ,
bone size is a good predictor of height. Forensic scientists and physical
anthropologists often use bone size to reconstruct the height of a person
when there are few remains.)
SOCIAL STUDIES
The United States Congress consists of the Senate and the House of Representatives.
Look up the number of members in each. How are the numbers of members
determined for each? How are proportions used to determine the number
of Representatives from a state? Name two states that lost Representatives
and two states that gained. A good Web site for this information is the
U.S. Census Bureau, Census 2000.
http://www.census.gov/population/www/censusdata/apportionment.html
This site explains how the number of representatives to the House of Representatives
is determined by population statistics.
ART
Explore how artists use the average size of a human head to draw a body
in proportion.
ENGLISH/LANGUAGE ARTS
What is meant by the statement, “A person’s ability to achieve
is directly proportional to their motivation for success?”
- Invite an architect to explain how proportions are used in designing
a building or building complex.
- Invite a zoologist to speak about how proportions are used to create
viable living conditions in modern zoos.
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