Blue-Ness ActivityEquivalence, Concentration and

Intensive Quantity

The University of Texas at Austin © 2008

The Generative Design Center at UTA © 2008

The National ScienceFoundation

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Table of Contents Subject Areas Topics Abstract/Summary Prerequisites Keywords Teacher Notes/Procedures

Introduction Activity I: Equivalent FractionsActivity II: Simulated Blue-nessActivity III: Making Blue SolutionsExtensions and Connections

Equipment/Materials Needed In ClassBasic Setup ChecklistActivity I: Equivalent Fractions PPT || Fractions Worksheet Acitivity II: Blue-ness with HubNet PPT || Blue-ness WorksheetActivity III: Making Solutions PPT || Make solutions Follow-Up Worksheet

Time Considerations Curriculum Standards Curriculum Objectives Support Resources and Links

Subject Areas

Physical Science and Mathematics

Topics

Concentration, measurement, ratio

Abstract/Summary

The purposes of this activity sequence is to introduce and explore intensive quantity using color. It is important that an intuitive, often perceptual, sense of comparative intensity -- where we might say this is more blue than that or this is the same blue-ness as that -- come to be integrated with the use of ratio. More broadly intensive quantities can be associated with the suffix “-ness” (e.g., blue-ness, fast-ness, warm-ness, sweet-ness, and likely-ness) and are increasingly important as the science curriculum progresses from late elementary through high school. This activity sequence has three major parts: (1) Building on and/or developing ideas related to creating equivalent fractions, (2) Using networked activity to move from fraction ideas to ratio ideas by having students create and share simulated solutions, (3) Making physical solutions with the same blue-ness.

PrerequisitesThis activity benefits, but may not require, that students have ways of discussing or illustrating why 1/2 can be considered “the same” (equivalent) to 2/4. Patterns associated with creating the same “fractions of a whole” are used to support creating equivalent ratios.

KeywordsConcentration, equivalence, fraction, ratio, extensive and intensive quantity, measurement

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Teacher Notes/Procedures

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Introduction

Elementary curricula will have students understand fractions as portions of one kind of quantity. So 1/3 is one out three equal parts of the whole of one kind of thing (a fraction of a candy bar is still the same sort of thing as the whole candy bar).

Science often uses ratios, like 1g / 3 cc, to represent intensive quantities. In this case the ratio represents something that is related to mass and volume but is distinct from either one of these extensive (how much) quantities. The ratio represents something important in itself. It represents the “dense-ness” or density of a material. Science has lots of these kinds of intensive quantities: density, concentration, velocity and pitch are just a few. This activity centers on blue-ness, as a ratio of drops of blue solution to milliliters of water, as an example of intensive quantity.

The activity has student use what they know about fractions and patterns that they observe related to creating equivalent fractions to begin to understand how to compare quantities that are expressed as a ratio.

Equivalence is a bridge between ideas related to fractions and ideas related to using ratio to represent intensive quantity.

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• We start this blue-ness sequence with students exploring mathematical patterns for creating equivalent fractions (Activity I) that can then be extended to ratio ideas used to work with the network-based blue-ness simulation (Activity II) and finally to the physical activity of making unique solutions with the same blue-ness (Activity III).

• Students will come to be articulate (I can explain in clear ways what it is I’m doing and why), adventurous (I want to make a solution that no one else has tried) and confident (in making solutions with the same blue-ness I come to believe in my ability be successful and innovative with challenging content).

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Activity I: Creating Equivalent Fractions

Activity II: Ratios & Blue-ness Simulation

Activity III: Solutions with the Same Blue-ness

Activity I: Equivalent Fractions Display the screen at left. Students can number and write their

answers in a notebook or on the worksheet (FractionsSAME.doc).

Students will progress through this activity at different paces. Some may need support to get started and working in “teams” can make sense. After a while ask the students (or teams) to put their favorite(s) on the board. (Or you can run this as a race where teams have to write ones that no one else has used on the board). Discuss strategies.

At first students will tend to count by the same amount (1’s on top, 2’s on the bottom)

OR make the bottom two times the top (or the top 1/2 the bottom)Attend to these and other strategies as part of getting them ready to

extend them to other examples of equivalent fractions.Eventually students may begin to see that multiplying the top and

bottom number by the same thing works in getting ones “in between” (5/10 and 8/16) or maybe not (yet).

The decimal example will cause a stir because eventually they want something like .5/.10 to work and get 5 instead of 0.5 … hmm… a little decimal review (what’s 10 10ths?)

The Pepsi example is designed to push the students to develop strategies to deal with “big numbers”.

Continue the conversation by encouraging them to talk strategies and understand the strategies of others.

7/8 is seen as more challenging and will encourage the students to push their strategies.

You might ask them if there are any ones in between say 3/6 and 4/8 for half (e.g. 3.5/7).

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Activity II: Simulated Blue-ness

The Blue-ness WorksheetCompare to 2/10Compare to other = Compare others not equalFive in order not equal16/24

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1 10

2 10

3 10

=><

What the Blue-ness Activity Does

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1. You are a Brown Cactus. 2. Edit your fractions.

3. Test your fractions. 4. Send your fractions to show your class.

What it does

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5. When do solutions have the same blueness?

Equipment/Materials Needed

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Time Considerations

As part of developing shared meaning …

Curriculum Standards

Physical science typically encounter at least three times… Benchmarks

Curriculum Objectives

Student will be able to

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Support Resources and Links

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Blue-ness Sequence

OVERVIEW:

• Start with equivalent fractions on calculator• Use the blue-ness computer game to match the blue-ness of 2

drops in 10 mls of water. Post with Post-It notes equivalent blue-nesses.

• Use the blue-ness game to match a blue-ness students make up. Post your blue-ness in a squence using Post-It notes.

• Make at least one blue-ness solution, put it in the window to see if it matches the others.

[1] Using the calculator, what does it mean for 1 to be the 2 ?

2 SAME as 4

What do you see on the calculator? [2] Find ten more fractions that are the same as the

ones above. Be sure to write them down.-- put your best or most interesting fraction(s) on the board --

[3] Explain your strategy for getting fractions that were the SAME (How could someone get another fraction using your approach?):

[4] Can you create some examples of fractions that are the SAME as 1/2 that use decimals (hint: an example would be 0.1/0.2) Try for 10:

[5] If you are the boss of a marketing company and you want to show that half of the people who go to a certain Mall like Pepsi better than Coke, how many people would have to say "Pepsi" if you interviewed a total of 1050 people? How could you check your answer?

[6] Can you come up with 10 fractions that are the same 7/8? Write them down.

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To turn ON(2nd then ON to Turn OFF)

Press ENTER toGet Answer

Divide

Press 1 ÷ 2 ENTER Press 2 ÷ 4 ENTER

Fractions - What does it meanTo be the SAME?

[1] Using the calculator, what does it mean

for

13

62

to be the

SAME as

?

[2] Find 10 more fractions that are the

SAME as the ones above? Write

them down.

[3] Explain your strategy for getting

more of the SAME.

[4] Can you do some examples of the

the SAME with decimals on the

calculator?

[5] If you are boss of a marketing

company and you want to show

that half of the people that

go to a certain Mall like Pepsi

better than Coke, how many

people would have to say

"Pepsi" if you interviewed a

total of 1050 people?

On the back of this paper write out your

answers:

To turn ON

divide

Press ENTER

to get Answer

Press 1 ÷ 2 ENTER Press 3 ÷ 6 ENTER

Responses[1] What does it mean to be the same? (1÷2 and 3÷6)[2] List your ten fractions that are the SAME as 1/2 and 3/6:-- put your group number and your ten examples up on the board -- (you should start to see patterns in how we got our 10)[3] Explain your strategy for getting fractions that were the SAME. If you had to explain your strategy to someone else, what would you say?[4] Can you do some examples of fractions that are the SAME as 1/2 that use decimals (an example would be 0.1/0.2) Try for 10 examples:[5] How many people would have to say “Pepsi”? (See [5] on

previous page).[6] Can you come up with 10 fractions that are the same 7/8?

Write them below.

Note

• At first students will count by the same amount (1’s on top, 2’s on the bottom)

• 1/2, 2/4, 3/6, 4/8 etc.• OR make the bottom two times the top (or

the top 1/2 the bottom)• 1/2, 3/6, 4/8, etc.• Attend to these and other strategies as part of

getting them ready to extend them to other examples of equivalent fractions.

Notes Continued

• Eventually students may begin to see that multiplying the top and bottom number by the same thing works in getting ones “in between” (5/10 and 8/16) or maybe not (yet).

• The decimal example will cause a stir because eventually they want something like .5/.10 to work and get 5 instead of 0.5 … hmm… a little decimal review (what’s 10 10ths?)

• The pepsi example is designed to push them to try and develop strategies that allow them to deal with “big numbers”.

• Continue the conversation by encouraging them to talk strategies and understanding the strategies of others.

• 7/8 is seen as more challenging and will encourage the students to push their strategies.

• You might ask them if there are any ones in between say 3/6 and 4/8 for half (e.g. 3.5/7).

Blue-ness Computer Game

• Keep exploring equivalent fractions as much as you want.

• Transition to the computer game by saying we want equivalent mixtures (ratios) to 2 drops of blue in 10 mls of water. They can use the calculator to “get” the target number if they want (or not).

• Use the computer game to enter the 2 drops of blue in 10 mls on the left. On the right they are to find a mixture that matches. The edges of the container will turn pink when they get a match.

• They should find more examples (keeping the left the same).

• Each time they find one they should post it using a Post It note or by writing it on the board.

Make up your own

• Students should now make up a blue-ness to go on the left (e.g., 8 drops in 40 mls).

• Then they should try and find as many matches as they would like. To make the sliders go to higher values, select the slider [by clicking and stretching a rectangle around the slider] and click on edit in the menu bar of NetLogo. Don’t worry if they go “off the screen” the sides will still turn pink if they get a match.

• They should post the original and each match as they get them. This can get quite animated as students see which group is coming up with the most matches.

Discuss the Strategies

• It is important that the students articulate their strageties for coming up with equivalent ratios (blue-nesses). Connections to how to how they found equivalent fractions are good. They may also begin to articulate new strategies.

Blue-ness Solution

• Prepare in advance a blue solution using 10 drops of food coloring in 5 mls of water (or, equivalently, 20 drops in 10 mls, etc.). You won’t need large amounts of this solution as drops are very small by comparison to these amounts.

For Students

• Ask them to make a solution matching the blueness of putting 6 drops of your prepared solution in 30 mls of water but they can use this combination.

• A lot of procedural issues will come up about using the droppers and measuring water using a graduated cylinder.

• Students measure the water using a graduated cylinder and then pour this water into a see-through bottle. Then they should carefully add drops of the prepared solution.

• Put your original mixture (6 in 30mls) in the window and ask them to place their solutions alongside as they get them done. View from straight to the side. They should be the same “blue-ness”. Same blue-ness means they found equivalent ratios!

• If they finish one they should try and make another. They should try and come up with ones that are “interesting” (not like the others or an “easy” one) up to the limits of what you have available in terms of water, drops, etc.

Viola!

• Wow! They really are “the same” in this very important way.

• They have the same “concentration” of blue in the water.

Table of ContentsDISEASESubject AreasTopicsAbstract/SummaryPrerequisitesTeacher Notes/Procedures• Activity I: Dice and Disease• Activity II: HubNet Disease Simulation• Getting Simulation Data into the Calculators and Discussion• Re-Running the Activity• Using/Extending/Authoring NetLogo Models and ActivitiesTime ConsiderationsCurriculum StandardsCurriculum ObjectivesSupport Resources and LinksQuickStart InstructionsInformation TabEquipment/Materials Needed

Activity I: PresentationAcitivity II: PresentationDisease HandoutsDisease DataFollow-Up Worksheet

Keywords

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