Ratio and Proportion ScriptMaterials:  Lesson 3(shells), calculators

The following script goes page by page on the slideshow called "Ratio and Proportion."  Many of the math concepts are studied in a pre-algebra class.  The connection with science helps students to appreciate mathematical reasoning and how it is used in science.

Slide 1.   “Searching how to Compare “the title of this presentation refers to the reasons why ratios were invented.   When you compare objects you need a way in which you can mathematically compare to either 1 or 100%.   This standardizes the comparison which will give you a valid interpretation of the data.

Slide 2. Many students are confused with the terms fraction and ratio.  A fraction in arithmetic could be:

a.      Part of a whole – so ¼ (one fourth) of a circle is a part of the entire circle.
b.
Division – the same ¼ is also a reference to divide the numerator by the denominator.  In this example 1 is divided by 4 or .25 or 25% which is equivalent to ¼
c.
Ratio – can also refer to 1:4  (one to four) or one part to four parts

Assure students that it just depends on the problem on how to handle the mathematical manipulation of the numbers.  It might not be obvious, but with practice it becomes second nature to compare things.

Slide 3.  In order to interpret data it is easier to have a standard in which to compare.   A fraction allows the user to compare one item with another.  The mechanics to compare differs depending on the arithmetic operation.  For example John has ½ cup of milk and Jane has 1/3 cup of milk.  Who has the most milk?  You can’t compare the numbers until you either have the same denominator  3/6 to 2/6  (John then would have 1/6th more milk) or by percentage  ½ = 50%  compared to 1/3 = 33.3% (John would have 16.7% more).  Please note to students that 1/6 = 16.7%.

Slide 4.  Fractions have rules depending on the arithmetic operation you are doing.  Make sure the students know the difference between the lowest or least common denominators (LCD) which is the smallest number that both numbers have in common.  (This helps to keep numbers small and manageable in our heads.)  You might want to remind them that knowing the different factors of numbers can help find the lowest number.

Addition:  Must find the “least or lowest common denominator” and covert the entire fraction.  Then you keep the denominator and just add the numerator.

Slide 5.  Subtraction is similar to addition in that you need to find the LCD and then subtract the numerator as if it was a regular subtraction problem.  The answer will have the same denominator as LCD, only the numerator is subtracted for the final answer.

Slide 6.Multiplication is easy.  All you do is multiply the numerator across and then the denominator.  Then you reduce the fraction.

Slide 7. Division is weird.  When you divide fractions you invert the second fraction and then multiply across and then reduce.

Slide 8. Collecting data and using fractions helps in interpretation.  It allows the observer to quickly compare the data. However, the fraction is more useful when converted to percentage, because the human brain can interpret one number quicker than several.

EXERCISE:

In this exercise students are given a sample of Mollusca from an area.  We are using San Francisco Bay area map, but these shells came from India and the Philippines.  First have the students sort shells that look similar.  Discuss with them how they should sort the shells.  Even though some look bigger or smaller they are the same species.  How do they know that?  They are proportional.  Some might have been older or younger when they died.

The mixture contains species from the Phylum Mollusca including 3 snails, 1 bivalve and 1 scaphopod.  Snails belong to a large group of mollusca.  Many have a spiral shell, but some do not have any shell.  These are marine snails and 2 represent carnivores (snail 2 and 3).  They have a small “lip” at the opening of the shell which is a clue that is a carnivore.  Snail 1 is an herbivore (no lip).

Bivalves have two shells that are a mirror image.  The living organisms would live inside and they can be marine or fresh water.  Scaphopods are also called “tusk shells.”  The shell is slightly curved, conical, and whitish in color.  They are usually found in marine offshore and live in the mud.

Then after they sort have them count the number of each species on a graph similar to that in the picture.  Count the total number and then make a fraction comparing the individual species (numerator) with the total number (denominator).    Then convert the individual species to a decimal and then percentage.

We suggest you use a calculator, so students don’t get confused with the arithmetic.

Slide 9. This is a map of San Francisco Bay area and it shows the distribution of snails #3 throughout the bay.  (Please remember this is a hypothetical data or not true.)  Go over with students that there is a trend in the data.  There is a higher percentage of this species in the south bay.  A scientist would continue this and then compare. The data could then be interpreted easily.  That is what fractions are all about, the ability to compare.

Slide 10.  A proportion is a ratio that is equivalent.  It is easy to determine if they are proportional.  Go over the terms mean and extreme, which refer to the numbers diagonally across from each other.  If you multiply the mean and then the extreme and they are equal, the ratios are proportional.

Slide 11. All cubes are proportional.   If we want to compare the length versus the width it will always be equivalent to 1/1

Slide 12. Proportions are used by many trades from construction to cooking.  There are many examples of proportions.  The body of humans is proportional to one another and sometimes the relationship of feet and hands; circumference of skull and hands, are also in proportion.  Also many of our body parts are proportional as we grow.  For example baby feet have a similar proportion as adult feet.

Artists also use proportions when they draw.  They can grid a person and look at the features in each grid and make them larger or smaller; these are all proportional.

EXERCISE (for students to do on their own):
You may want to use a ruler to see what is proportional on a student’s hand.   Is the palm as wide as the longest finger?  Is it proportional to other students?  Depending on how much time you have, comparing body parts is quite interesting.

Slide 13. Proportions are useful when you need to make something consistent.  For example, if you want to mix paint to get the same color every time you have to know the ratio of one color to another.

If you make a chart of the different ratios you are really setting up proportions.  Just for fun, if you graph them, they will always show a straight line or what is called “direct proportion.”

Slide 14.  This slide shows examples of how you can change parameters equally to a picture and how it changes proportionally.

Slide 15.  How music is recorded has its root in the ancient Greek Pythagoreans.  They observed that tones an octave apart are pleasing to the ear.  A ratio of 2:1 produces harmonious tones. Similarly, musical intervals involving tones in the ratios of 3:2 (a fifth) and 4:3 (a fourth) are also pleasing.

Out of these ratios, the Pythagoreans developed the so-called major scale, which consists of seven notes.  Today they are designated by the letters C, D, E, F, G, A, and B, or sounded out as do, re, mi, fa, sol, la, and ti. The chromatic scale includes five additional notes -- the sharps and flats (black keys of a piano).

Slide 16. Proportions can also be seen in chemical equations one side of the equation must be proportional to the other side

Slide 17.  This lecture is trying to get students to understand that learning to use fractions and proportions will help them solve problems in many situations.

Slide 18. Finding the height of something very tall can be easily determined by setting up proportions of similar right triangles.  For example, a building cast a 50 meter shadow and a 5 meter flagpole cast a shadow of 6 meters during the same time during the day.  Let’s think of them as 2 right triangles and look at the sides as proportional:  y/60 = 5/6.  You can solve the problem like any other proportion.  Multiply the means by extremes and you get: 300 = 6y 50 = yThe height of the building is 50 meters and you did not even have to measure it with a tape measure.  Understanding what is a proportion and what is not, helps to solve problems.