Elementary School Science:Formal Level Skills:

Proportional Reasoning

Presented byFrank H. Osborne, Ph. D.

© 2015

EMSE 3123Math and Science in Education

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Proportional Reasoning• Proportional Reasoning is a very important

ability in both math and science.

• The formal operational student can imagine all possibilities, not just the situation being presented.

• This student can see not only ‘what is’ but also ‘what could be’.

• The student can apply this to hypothetical situations or those that cannot immediately be seen.

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Proportional Reasoning• Consider the Shadow Problem.

• A boy 4 feet tall sees that he casts a shadow that is 2 feet long. If the shadow of a tree is 10 feet long, how tall is the tree?

• This problem tests the ability to reason with proportions.

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Proportional Reasoning• The ratio of the boy’s height to the length of

his shadow is a constant that can be appied to hypothetical problems or those he cannot see.

• Since his height is twice the length of his shadow, this ratio is also applied to the tree.

• So the tree will have a height of twice its own shadow or 20 feet.

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Proportional Reasoning• Proportional thinking is one of the hardest

skills for children at the elementary level to learn because they have to pay simultaneous attention to two variables, adjusting one as the other changes, while at the same time applying this thinking to hypothetical situations that the student cannot see.

• The student must understand that the ratio applies to all possibilities.

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Proportional Reasoning• In the shadow problem, this means that no

matter how tall the tree is, the ratio of the shadows is the same.

• This is what Piaget means when he says that the formal operational student can imagine all possibilities, even those he cannot see concretely.

• The rods problem is another example.

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Proportional ReasoningA ratio problem with rods.•The distance between two points is measured to be the equivalent of 10 dark green rods. By looking at the size relationship of the rods, figure out how long this distance would be in terms of the light green, red, violet, and yellow rods.

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Proportional ReasoningYou should not need to line up rods for this one.

•The length of a dark green is 6 so 10 of them total a length of 60.

•The numbers 2, 3, 4, and 5 are all factors of 60. How many of each do you need to make a total length of 60?

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Proportional Reasoning• The length of red is 2 so 30 are needed.• The length of light green is 3 so 20 are needed.• The length of purple is 4 so 15 are needed.• Finally, the length of yellow is 5 so 12 are needed.

• This can be thought of as a balance problem where different numbers of rods are required to keep the balance (total length) even.

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Proportional ReasoningThe Triple-Beam Laboratory Balance.

•When you adjust the weights, the combination of them and their distances restores the balance of the object and the weights.

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Proportional Reasoning• Science and mathematics are full of situations

and concepts that require students to reason with proportions. One such concept is density.

• How to teach Density?• Begin by thinking about what you know about

the concept of density. Outline how you would go about teaching it. After doing the Lab Activity, return and reflect on what you wrote.

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DensityDensity, proportional reasoning, and why we can ice skate.

•We can think of density as the lightness or heaviness of objects.

•We compare heaviness by comparing identical amounts of materials in the form of a unit cube.

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DensityDensity is also the ratio of mas to volume for a particular material.

•For example, the density of Copper is about

9 gm/cm3

•This means that the ratio of mass to volume for copper is 9.

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DensityWe can use proportional reasoning to calculate how much any number of copper cubes will weigh.

•For example, 10 cm3 of copper will weigh 90 grams because the ratio between mass and volume for copper is 9 to 1 for copper.

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DensityHow many cubes would we need to have 99 grams of copper?

•As each cube is 9 grams, we need to determine how many 9s there are in 99. Since there are 11 9s in 99, and each represents 1 cube, we will need 11 cubes of copper.

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Density Table for Various Materials

We can compare relative heaviness of materials because the table lists the masses of exactly the same amount of materials, 1 cc (1 cm3). Density of water is 1 cc.

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DensityWill something sink or float?•Sinking or floating depends on the density of the material and the liquid. Most metals will float on mercury with density of 13.59 g/cc.•Ice (0.92 g/cc) will float on water (1.00 g/cc) but it will sink in rubbing alcohol (0.80 g/cc).•Most matter can exist in both liquid and solid form where the solid form of the material usually sinks in the liquid form (paraffin wax is like this).

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DensityDensity compared to water.

•With water being the standard, materials with a density less than 1.00 g/cc will float on water, while materials with a density greater than 1.00 g/cc will sink.

•When water freezes and ice forms, it floats on the liquid so you can ice skate.

•Which is denser, milk or heavy cream? Ans: milk is denser because cream floats on milk.

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Density and UnitsWhat the density formula means.

•When you learned density, you probably started with the formula:

Density = Mass/Volume

•Because of this, you probably did not realize that you were finding the mass of one unit cube of the material you were measuring.

•This is because you were taught the formula and not the concept.

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Density and UnitsThe Learning Cycle approach.

•We learned about density using the learning cycle approach. Only later did we state that the formula is just a way to find the mass of one cube without having one cube of the material available.

•Just take any number of cubes, find their mass and divide by the number of cubes that you have.

•It is simply a logical consequence of simple math.

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Density and Units• Example: If we have 50 cubes and they have a mass

of 150 grams, then the mass of one cube is found as

Density = Mass/Volume

= 150 grams/50 cubes

= 3 grams/cube

• We say that the density is 3 grams per cube.

Density = 3 grams/cube

• This is generally how units work in both science and mathematics.

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Density and Units• We can multiply and divide units of the same type.

Back when we were studying multiplication we had a figure like this.

• If these measurements were in cm, then the area would be 3 cm x 7 cm = 21 cm2.

• Units are of utmost importance in science. It is really important to keep track of them.

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Density and Units• We can also divide units. For example, we can

divide 15 cm3 by 3 cm.

• The method is to cancel just as in regular division. Both the numbers cancel and the units cancel.

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Density and Units• First, we divide cm3 by cm to get cm2.• Then, we divide 15 by 3 to get 5.

• Keeping this method in mind will make understanding science (especially physics) a whole lot easier for you and also make it easier to teach.

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Density and UnitsWhat would happen if the units were different?•Imagine we have a basket of 9 cherries and 3 apples.

•What do we get when we divide the number of cherries by the number of apples?

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Density and UnitsWhat would happen if the units were different?

•We know 3 into 9 is 3.

•But, we cannot divide a

cherry into an apple.

•So our answer is 3 cherries

per apple or 9 cherries = 3 cherries/apple 3 apples

•This means that there are 3 cherries per apple or a ratio of cherries to apples of 3:1.

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Density and UnitsRemember the definition of density.•The definition of density has much in common with the cherry example. Both are simple ratios that remain constant.•A density of 3 grams per cc means that there are 3 grams of mass for every cube in exactly the same way as there were 3 cherries for every apple. We can use this fact to answer questions•“How many cherries correspond to 10 apples?”•“How many apples correspond to 12 cherries?”

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The End

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