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Lesson Plans
Relative Ratios
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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.
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.

Introductory Activities: Setting the Stage

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.


Learning Activities

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
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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.)


Cross-Curricilar Extensions

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.)


Cross-Curricilar Extensions

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?”


Community Connections
  • 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.