1. ALGEBRA IN ACTION BY: THE WHIZ KIDS Joel Bradshaw Tracey Guida Nina Mun Nancy Scott Steve Weitlich

2. How we see math word problems: If you have 4 pencils and I have 7 apples, how many pancakes will fit on the roof? Purple, because aliens don’t wear hats. WHY ALGEBRA IN ACTION?

3. BRIDGE DESIGN PROBLEM 1. Design a truss bridge (Warren, Howe, or Pratt) with the following specifications: Span = 102 cm Width = 11.5 cm (# 3) Vertical and Horizontal Beams = 17 cm (# 4) Diagonal Beams = 24 cm (#5) 2. How many joints, vertical and horizontal beams, and diagonal beams will be used in your design? 3. Sketch your design. 4. Build your bridge.

4. BRIDGE DESIGN: PRATT A B C D E F G H I J K L Externally Applied Force, F Using PASCO Bridge Set: L = 17 cm = 0.17 m h = 17 cm = 0.17 m a = 24 cm = 0.24 m a h L

5. FREE-BODY DIAGRAM (FBD) A B C D E F G H I J K L Externally Applied Force, F RARA RLRL Find the reaction forces at A and L, R A and R L, respectively by applying the equilibrium equations:  F = 0 and  M = 0

6. METHOD OF JOINTS  Use the Method of Joints to analyze each beam in terms of:  The magnitude of internal force  Whether the force is in compression or tension  Use symmetry and analyze only ½ of the truss.

7. ANALYSIS OF JOINT A A RARA FBD of Joint A F AB F AC a h L x y  F y = 0 R A – F AC = 0 F AC = R A ahah haha  F x = 0 - F AB + F AC = 0 LaLa F AB = R A For the internal force in member AC resolved into rectangular component forces:

8. METHOD OF JOINTS: ANALYSIS RESULTS A B C D E F G H I J K L Externally Applied Force, F RARA RLRL C C C C CCC CCC T TTT T T Tension: T Compression: C

9. BUCKLING FORCE, F BUCKLING F EF = R A ahah = ahah F4F4 ahah F buckling 4 = F buckling 4h F EF a 4h (  2 EI) a (a 2 ) 4h  2 EI a3a3 = = = F buckling = 769 N 0.01016 m 0.00254 m E = 2.29 x 10 9 N/m 2 I = 6.92 x 10 -10 m 4 Taking a safety factor of 2, the resulting maximum load is 384.5 N or a load mass of 39 kg. 0.006756 m

10. TESTING THE BRIDGE 5. Place your load in the middle of the bridge. Analyze the joint at the support.  Why should you begin your analysis with this joint?  Analyze the load distribution throughout the truss including the magnitude of an internal force and whether it is in tension or compression. 6. Now, change the location of your load. What happens to the load distribution? Does it change?

11. HOW CAN WE USE THIS IN OUR CLASSROOMS?  Discussion Questions on Bridges  Intro: The following diagram is a picture of a Warren Truss Bridge. The bridge is comprised of horizontal, vertical, and diagonal beams. Each of the beams are connected using a joint. Note: the horizontal and vertical beams are the same length.

12. CLASSROOM USE CONTINUED.  1. Use the picture above:  A. How many horizontal deck beams on the bottom chord are there?  2. Determine how many horizontal beams will be required to construct the bottom chord of the bridge in the following situations.  A. Span of the bridge is 170 cm. The length of the beam is 17 cm.  B. Span of the bridge is 24 m. The length of the beam is 4 m.  C. Span of the bridge is 4km. The length of the beam is 5m.  D. How are your answers in problems 1a and 2a related?  E. Span of the bridge is S. The length of the horizontal beam is L.

13. RESEARCH BASED APPROACH Based on the idea of “Pattern Tasks” by Margaret Smith Develops students’ algebraic reasoning Begins with observing a pattern from a picture The pattern is used for concrete problems Students connect patterns to algebraic equations Allows multiple representations

14. MAZE DESIGN PROBLEM  Total Area for the Shipping Terminal: 20,000 ft 2  Container Storage Area: 75% of the total terminal area.  Loading and Unloading Area: 10% of the total terminal area.  Administration Area: equivalent to 50% of the loading and unloading area.  Rail and Trucking Space: 7% of the terminal area.  Repair and Maintenance Area: (to be determined by you!) Design a shipping terminal with the following specifications. Your goal is to maximize the number of storage containers in the container storage area.

15.  Specifications for the Container Storage Area:  The container storage area is comprised of containers and pathways around the container.  The dimensions of a standard storage container are 8 ft x 20 ft.  Each container must have a pathway on at least 2 sides.  Pathways must be 8 ft wide.

16. THE MATH Total Area for the Shipping Terminal: 20,000 ft 2 Container Storage Area: 15,000 ft 2 Loading and Unloading Area: 2,000 ft 2 Administration Area: 1,000 ft 2 Rail and Trucking Space: 1,400 ft 2 Repair and Maintenance Area: 600 ft 2 Maximum Capacity: 33.48 storage units. Interpret this and incorporate it into design. Pertinent Variables: Amount of pathway around container, design, and layout

17. HOW CAN WE USE THIS IN OUR CLASSROOMS?  Apply system analysis method of problem solving  Introduce in early education  Generic approach to setting up problems by  Define system  Draw a picture  Identify variables  Identify independent equations  Identify given information  Solve literal equations  Plug in numbers

18. HOW CAN WE USE THIS IN OUR CLASSROOMS? You are going to design a rectangular garden. The area of the garden is 100 ft 2. The perimeter of the garden is 80ft. The length of the garden 2 times more than the width. Find the dimensions of the garden.  System: Rectangular Garden  Variables: Area (A), Perimeter (P), Length (l), Width (w)  Independent Equations: A = l*w P = 2l + 2w  Given Information: A = 100 ft 2, P = 80 ft. l = 2w  Solve Literal Equations: l 2 – (lp)/2 + A = 0  Plug in your numbers!

19. CONCLUSION  Problems are easily adaptable for any age  More scaffolding for lower levels  More variables and difficult math for higher levels  If we introduce system analysis at a young age, they will be better prepared for their STEM careers in the future