STATION 1 – THE PRODUCT OF INTEGERS What are two integers that differ by three? 5 and 8 21 and 24 98 and 101 x and ……? a) Write the product of two unknown integers that differ by three  Did you notice that this is a quadratic expression? b) Can this product ever be 810? If so, what are the integers? c) What is the minimum possible product of two integers that differ by three? And what are the integers that yield this product? STATION 2 – FINDING THE EQUATION OF A CURVE The graph of a quadratic function has x‐intercepts of 18 and ‐22. It has a maximum value of 40 Does the point (‐12, 28) also fall on this curve? Provide evidence. STATION 3 – THE GOLDEN MEAN Consider this line segment. Let’s divide this segment up into two chunks, one of length A and one of length B: A B And the length of the original segment is A + B Now let’s compare longs and shorts: Original segment (A+B) to longer (A) AND Longer (A) to shorter (B) If we choose our division carefully, then these two ratios will be equal. A B A
 A
B
This ratio, A , is called the Golden Mean. What is the exact value of the B
Golden Mean? STATION 4 – THREE CONSECUTIVE ODD INTEGERS Consider three consecutive odd integers: 7 , 9, 11 15, 17, 19 99, 101, 103 2x + 1 , ………? What are the next two odd integers after 2x + 1? OK… Stephen and Cooke are playing a game. Stephen takes the first two consecutive odd integers and Cooke takes the third.  Stephen squares an odd integer. He then squares the next consecutive odd integer, and he adds these two squares together.  Cooke squares the third of the consecutive odd integers.  When the boys compare their results, Stephen’s sum of squares is 209 larger than Cooke’s single square. What were their numbers? STATION 5 – A PLAY PEN FOR ELLIE Woods wants to build an outdoor play pen for his dog Ellie. It is to be rectangular in shape. To save materials, he plans on using one side of his house as a wall. So he needs to only fence the three remaining walls: x y Pen House x Notice that the dimensions of this pen are x and y. Woods wants to buy 50m of fencing for the perimeter. So, for example, Woods could make a pen that has dimensions of x = 10 and y = 30 (because 10 + 10 + 30 = 50m of fence). In general: 50 = 2x + y (50m is how much fence we available) And the area of this pen is Area = xy (the area of a rectangle). And we can use the first equation to make the second one simpler: 50 = 2x + y 50 ‐ 2x = y A = xy Sub in 50 – 2x for y
in the area eq A = x(50 – 2x) Use A = x(50 – 2x) to answer these questions: a) Is it possible to make a pen with an area of 400 square metres? b) What is the maximum possible area of the pen that Woods can build? c) Woods decides that he only wants Ellie to have 168 m2 of play area (any more and she’ll get into trouble). What are the dimensions that create this size a pen? STATION 6 – DIVING! A diver stands on a platform that is 10m above the water. At time = 0s, the diver begins her jump. 0.35 seconds later, the diver has reached her maximum height, 0.6m above the platform. How long, from the start of the jump, does it take for the diver to hit the water? Assume a parabolic dive path. STATION 7 – DONUTS! Do you remember what the area of a circle is? r2 A donut shop wants to put a strawberry glaze on top of its donuts. The baker wants the area of the glaze to be the same as the area that is not glazed. The whole donut has a radius of 6cm. Let the radius of the glaze be x cm. Therefore, the area of he glaze is Aglaze = x2 Notice that the for the area of the unglazed portion is the area of the whole donut less the area of the glaze. So: Aunglazed = Awhole – Aglaze = 2 – x2 Use these relationships to find the radius of the glazed portion if these two areas are equal. STATION 8 – FINDING THE EQUATION OF A QUADRATIC FROM A MODEL The following table of values represents (x,y) solutions to a quadratic function. Can you find the equation matching this set of values? x
1
2
3
4
5
6
…
15
16
17
18
19
y
‐117.5
‐86
‐57.5
‐32
‐9.5
10
…
50.5
40
26.5
10
‐9.5