15 Blue-ness Sequence Science By the Horns

1. 15 Blue-ness SequenceScience By the Horns Summer 2006

2. 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.

4. 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.

5. 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.

6. 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 inbetween say 3/6 and 4/8 for half (e.g. 3.5/7).

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.

8. 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.

9. 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.

10. 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.

11. 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.

12. Viola! • Wow! They really are “the same” in this very important way. • They have the same “concentration” of blue in the water.