1. Balancing Math and Science

2. Goals
To examine models that can be used to teach students to balance equations and to determine the equation for the slope of a line
To understand the difference between work and torque
To make connections among math, science and art

3. Procedural Memory Think About ...
... what happens to the body happens to the brain. This dual stimulus creates a more detailed “map” for the brain to use for storage and retrieval.
Eric Jensen, Teaching with the Brain in Mind, (Association for Supervision and Curriculum Development, 2005), 136.

4. Relevant Research
Physical science students with laboratory experiences demonstrated increased achievement and a more positive attitude toward science than those lacking such experiences.
M.P. Freedman, “Relationships among laboratory instruction, attitude toward science and achievement in science knowledge,” Journal of Research in Science Teaching 34:4 (1997): 343–357.

5. Solve the puzzle by determining the value of each piece of the mobile.
In the Balance
Solve the puzzle by determining the value of each piece of the mobile.
Note the rules

6. Need to Balance Our Levers
Set up a balance using the blue mat board provided. Make sure it is balanced and that you can identify this position again.
Find as many ways as you can to balance the following:
Five paper clips at 4 cm from the fulcrum

7. What Were Your Solutions?
It’s algebra! 5 at 4 = 20
Balance these with
4 at 5
5 at 4
10 at 2
2 at 10
Did you try: 2 at 5 and 1 at 10?
(2 x 5) + (1 x 10) = = 20

8. The Equations
m1 d1 = m2 d2
and
(m1 d1)+(m2 d2)= m3 d3

9. How does this differ from the original puzzle?
Distance effects the mass used to balance the mobile.

10. How do you balance mobiles?
Solve to find the mass of each object. 2g 1m 5m 2m b a

11. Balanced mobiles 2g 1m 5m 2m 2g x 1m = a x 1m 2 = a a = 2g
4g x 5m = b x 2m
20 = 2b
b = 10g

12. Now a Little More Difficult ... Solve for a, b and c5 3 1 d 2 10 a b c

13. Check Your Answers:
a=30
b=2
c=1.5 5 3 1 d 2 10 a b c

14. A Mobile Is ...
A type of moving sculpture developed by Alexander Calder in 1932
Often constructed of colored metal pieces connected by wires
Made of parts that move freely

15. Alexander Calder Mechanical engineer turned artist
Son and grandson of sculptors
Used his knowledge of physics to create mobiles
Interaction with space
Visually interesting: Use a tiny object to balance a large object (or many objects)

16. Six Dots Over a Mountain, Alexander Calder 1956
Smithsonian Hirshhorn Sculpture Garden, Washington D.C. (AD 08/07)

18. Untitled, Alexander Calder, 1976
National Gallery of Art, Washington D.C. (AD 08/07)

21. Balance the Calder Mobile Puzzle
This is not drawn to scale.
Note that at position D, the piece with a weight of 2 is located at a distance of 3.5

22. Where’s the science? Torque = lever arm distance x force
Balance = mechanical equilibrium
Rotational motion

23. How do we avoid confusing work and torque?
W = F d
T = F d
What is the difference between the two ds?

24. Work or Torque? Balance the large washer with three small washers.
Figure out how to measure the work done by adding the third small washer to balance the large one.

25. Record your understanding
In your interactive notebook, write your own explanation of:
Force
Torque
Work

26. Mobile Creations in the Classroom

29. Add the Math:
Have students create a puzzle that explains the math of their lever
Draw the mobile
Indicate the mass and distance for each section of the mobile
Prove that each lever is balanced using the equation for torque
F1 d1 = F2 d2

30. Mobile Creations in the Classroom: See the Math

31. T = F d

32. Build It Design and build a mobile Materials:
Rubber bands, cardboard (or matting)
Clothes hangers, string and anything
Jewelry wire and beads
Trash to treasure!

33. How Can You Balance a Very Small Object With a Very Large Object?
Think ... Write a solution

35. And now for something completely different (or not)
Reset your brain to use this model for a different purpose.

36. Models of Linear Equations
y = mx
y = mx + b
y – y1 = m(x – x1)

37. Set up and balance your blue lever (again)
Place one paper clip at 1, to the right of the center – this will not move
Then hang two clips from zero – these represent “x”
Mark one paper clip using a permanent marker, this will represent “y”
Move your “y” clip until the lever is balanced, and record this position in the data table
Continue to move the “x” and balance to find “y” (see directions)

38. In your equation for the line, how does the 2 in the equation relate to the model?

39. Create & draw the models for:
y = 3x – 2 (do not use x=0)
y = mx
y – y1 = m(x – x1)

40. Reflection
How can the models help students with different learning styles understand math and science?
Find a specific objective from your HAPG and give an example of how you can use one of the models in your classroom.

41. Questioning and Objectives