Using your imagination, show or describe 10 ways to connect dots A and B. 1)    What information is needed for you to calculate the area of the square?

2)    How can the two triangle measurements help you figure out the area of the square?

3)    Figure ADE is a right triangle. Where is the right angle? Where is the hypotenuse?

4)    How can you use the Pythagorean Theorem to figure out the length of line segment AD?

5)    What is the length of line segment AD? What is the length of line segment AB?

6)    What is the area of the square?

Figure ABCD is a square. Figure ADE is a right triangle. What is the area of the square? 1) What information is needed to calculate the perimeter of the triangle? What information do you already have?

2) How can you figure out the length of line segment AD? You are given the area of square ABCD; how can that help you discover the length of AD?

3) How is the area of a square or rectangle usually calculated? How can you calculate this area, if you don’t know the length of any of the sides?

4) But the area has already been calculated. Can you think backwards to figure out what the length of the sides must be?  Are the sides all the same length? How do you know for sure? What number times itself equals 4? What is the square root of 4 square meters? What is the length of AD?

5) How can you figure out the length of line segment AE? You know the length of AD and DE, which are two sides of the triangle. What kind of triangle is it? Which side of the right triangle is the hypotenuse? If this is a right triangle, can you use the Pythagorean Theorem to calculate the length of the hypotenuse? What is the Pythagorean Theorem: Is it a + b = c? Is it a² + b² = c²? Does the Pythagorean Theorem tell you the length of the hypotenuse, or the length of the hypotenuse squared?

6) Extra hint for students who already know about 3-4-5 right triangles: double the measurements of AD and DE; what do you notice?

Figure ABCD is a square with an area of 4 square meters. Figure ADE is a right triangle. One side measures 1.5 meters as indicated. What is the perimeter of the triangle?  What prior concepts or procedures do students need in order to solve this problem? 1) What information is needed for you to calculate the circumference of the circle? Is that information presented on the drawing? Can you figure out the needed information from the information that is given to you in the picture? How will you do it?

2) What does “isosceles” mean? What is the length of side ED of the triangle ADE? How can you use the Pythagorean Theorem (a2 + b2 = c2) to figure out the length of side AD? What is the length of it? Is Line AD part of the triangle? Is it also part of the square?

3)  Do we know the exact value of the square root of 2? What do you get if you multiply the square root of 2 times the square root of 2? How do we calculate the area of square ABCD? What is the area?

4) Draw a diagonal from A to C. How can you use the Pythagorean Theorem to figure out the length of Line AC? What is the length of it? Is Line AD part of a triangle? Is it also part of the circle? Is the diagonal Line AC the same as the diameter of the circle? How do you know? If you know the length of the diameter, how  can you calculate the circumference of the circle? What is it?

Figure ABCD is a square. Points A, B, C, and D are all located on the circumference of the circle. Figure ADE is an isosceles right triangle. One side measures 1 meter.

What is the area of the square? What is the area of the circle?  (Use 3.14 for ∏.)

What prior knowledge/skills do students need to bring to this problem? 1) List all the features of a circle: (All circles have a center, a radius, a diameter, a circumference, and an area; the radius is half of the diameter; A = πr2; C = πD)

2) List all the features of a rectangle: (All rectangles have 4 sides and 4 right angles, a perimeter, and an area; the top and bottom sides are parallel and equal in length; the right side and left side are parallel and of equal length; L•W = A; P = 2L + 2W)

3) How can you figure out the length of Line DC?

4) How can you create a right angle in this drawing?

5) If you draw a line connecting points A and C, how would you figure out its length? (Hint: Pythagoras…)

6) Would Line AC pass through the center of the circle? How do you know?

7) Is Line AC the same as the diameter of the circle?

8 ) What information do you need to figure out the circumference of the circle? Do you have that information? Can you figure it out?

9) When calculating the circumference, use 22/7 for π, and use cross-canceling so that you can work with smaller numbers,

10) What is the length of the circumference of the circle?

Building on the last two problems, what do students need to notice in the diagram, what shapes, properties, and relationships do they need to consider to solve this problem? Then solve it.

The area of rectangle ABCD is 588 cm². The left side is 21 cm. What is the circumference of the circle? Use Π = 22/7. Warm-up Question #1:
1)  Look at the first circle (upper left), which has a rectangle above the diameter. Are all 4 corners touching the circle? Which corners are touching the circle?

If the rectangle is made longer by stretching it to the right and to the left, so that the bottom corners touch the circle, will the top corners still touch the circle?

If the bottom right corner of the rectangle is stretched to the right so that it touches the circle, and the bottom left corner is stretched to the left so that it touches the circle, and the top corners stay where they are, will the figure still have four 90° angles? Will it still be a rectangle? What kind of figure would it become?

Apply these same questions to the second circle, which has a rectangle below the diameter.

Warm-up Question #2:
2)  Look at the third circle, which has a rectangle that is mostly below the diameter. Are all 4 corners touching the circle?

Which corners are touching the circle?

If the rectangle is made longer by stretching it to the right and to the left, so that the top corners touch the circle, will the bottom corners still touch the circle?

If the top right corner of the rectangle is stretched to the right so that it touches the circle, and the top left corner is stretched to the left so that it touches the circle, and the bottom corners stay where they are, will the figure still have four 90° angles? Will it still be a rectangle? What kind of figure would it become?

If the top of the rectangle is stretched upward, until the top corners touch the circle, will the figure still have four 90° angles? Will it still be a rectangle? Will most of the rectangle still be below the diameter? Would the diameter appear to cut the rectangle into 2 equal parts? Would the diameter cut through the middle of the rectangle?

The main question:
3)  In Figure 1, which has a rectangle inscribed in a circle, is Line AB the same length as Line CD? How do you know? Is Line BC the same length as Line DA?

Is Arc AB the same length as Arc CD? How do you know? Is Arc BC the same length as Arc DA?

If you add Line AB to Line BC, would the combined length be equal to the combined length of Lines CD and DA?

If you add Arc AB to Arc BC, would the combined length be equal to the combined length of Arcs CD and DA?

Is Arc AC (ABC) the same length as Arc CA (CDA)? Does Line AC cut the circle into 2 equal parts? Is Line AC a diameter of the circle? Does Line AC pass through the center of the circle?

Is Arc BD (BCD) the same length as Arc DB (DAB)? Does Line BD cut the circle into 2 equal parts? Is Line BD a diameter of the circle? Does Line BD pass through the center of the circle?

Is Point E located on Line AC? Is it also located on Line BD? Are both lines diameters? Do both lines pass through the center of the circle? Is Point E the exact center of the circle? How do you know?

4)  Look at Figure 2 and ask the same questions as in (3) above. Do the same for all the other figures.

Are all the rectangles the same size? Is Line AB the same length in Figure 9 as in Figure 1? Line BC the same length in Figure 9 as in Figure 1?

The circle in Figure 9 is the same size as the circle in Figure 1. Are the diagonals in Figure 9 the same size as the diagonals in Figure 1? Are all the diagonals in all the figures the same length?

Do all the diagonals pass through the center of the circles?
Are all the diagonals the same length as the diameter?