Pictured are two squares and a circle. The circle's perimeter is 10pi cm. What is the area of the small square?
(Here is some given info from the diagram: The large square is inscribed in the circle, and the small square is embedded in the larger square.)
We begin by noticing that since the circumference of the circle is 10pi, the diameter of the circle must be 10 -- and this equals the diagonal of the large square.
Now it's strongly implied -- but never directly stated, that the small square is in fact the midpoint quadrilateral of the larger square. Without this assumption, the question is ill-posed -- the area of the small square can be anywhere between that of the midpoint quadrilateral and that of the large square.
But once we assume that the small square is the midpoint quadrilateral, the question is easy. We know that the side of a midpoint quadrilateral equals half the length of the diagonal -- 10/2 = 5. And the area of a square with this side length is 5^2 = 25. Therefore the desired area is 25 cm^2 -- and of course, today's date is the 25th.
Chapter 7 of Sue Teele's Rainbows of Intelligence is called "Classroom Units That Incorporate Color and Multiple Intelligences." It begins as follows:
"As I was writing this book, I began to develop units on color that I thought might assist teachers as they integrate multiple intelligences into their classrooms. This chapter presents seven different units that can be used by teachers at any grade level."
In fact, I was originally going to post all seven of these units, one each day, so that our side-along reading would extend into next week. I've changed my mind -- I'm no longer going to write about Teele's units (although these might have been interesting). In other words, today will now be the last day of our side-along reading of Teele's book. Of these lessons, Teele writes:
"The standards referenced are from the California content standards in science, history-social science, mathematics, and language arts (1997-1998)."
Again, she wrote this book in 2000, before the advent of Common Core. Finally, she reminds us:
"I have discovered that when teachers develop lessons that represent all or most of the intelligences, they are able to become more creative and design lessons that provide opportunities for all their students to become actively engaged in the learning process."
Teele's book has forced me to revisit my own intelligences. Obviously, as an aspiring math teacher, I'm strong on logical-mathematical intelligence. But my problems with classroom management point to a deficiency in interpersonal intelligence -- after all, classroom management is all about how I interact with students. I could also stand to work more on intrapersonal intelligence as well -- I must look inward to figure out what I value in the classroom in order to become a more effective manager.
By the way, here are Teele's seven units:
- Primary, Secondary, and Complementary Colors
- The Many Dimensions of Color
- The Rainbow -- A Spectrum of Colors
- Exploring the Rainforest
- Pandas -- Endangered Species
- Butterflies and Moths -- Colorful Insects
- The Blending of Our Rainbow World
Notice that these units are all about Teele's favorite topics in science -- pandas and butterflies (and she draws pictures of these throughout her book), and of course color.
Today is a traditionalists post. But the main traditionalists haven't said much lately -- and besides, I've already alluded to traditionalists throughout our reading of Teele's book. Even though I'm adding the "traditionalists" label to today's post, the real discussion on Teele, multiple intelligences, and the traditionalists' debate has already occurred in the past six posts.
This is what I wrote two years ago about the answers to today's test:
1. C' is the same as C, but D' goes up diagonally to the left. This is tricky because the line of reflection is not perfectly vertical.
2. I' goes up diagonally to the right.
3. There are two symmetry lines -- the segment joining the two points and its perpendicular bisector.
4. The angle measures 62 degrees.
5. The angle measures 2x degrees.
6. The reflection image over line AD of ray AB is ray AC. This is tricky because it's been a while since we've seen the Side-Switching Theorem.
7. This is officially the Figure Reflection Theorem -- just make the right vertices correspond.
8. Reflections preserve distance.
9. The orientation is clockwise.
10. The orientation is counterclockwise, because reflections switch orientation.
11. There are three pairs: angles B and C, angles BAD and CAD, angles ADB and CDB.
12. There is one line of symmetry -- the line containing the angle bisector. This follows from the Angle Symmetry Theorem.
13. F' = E follows from the Flip-Flop Theorem. FG = EH is because reflections preserve distance.
14. Proof:
Statements Reasons
1. MO = MN 1. Given
2. M' = N 2. Given
3. MO = NO 3. Reflections preserve distance
4. MNO is equil. 4. Definition of equilateral
It's possible to add more details, such as O' = O, Transitive Property, etc.
15. The rectangle has two lines of symmetry, one horizontal, one vertical.
16. The isosceles triangle has one line of symmetry, and it's horizontal.
17. The images of the vertices are (1,3), (7,1), and (6,-2).
18. The image is (c, -d).
19. The angle measures 140 degrees.
20. The shortest distance is the perpendicular distance.
1. C' is the same as C, but D' goes up diagonally to the left. This is tricky because the line of reflection is not perfectly vertical.
2. I' goes up diagonally to the right.
3. There are two symmetry lines -- the segment joining the two points and its perpendicular bisector.
4. The angle measures 62 degrees.
5. The angle measures 2x degrees.
6. The reflection image over line AD of ray AB is ray AC. This is tricky because it's been a while since we've seen the Side-Switching Theorem.
7. This is officially the Figure Reflection Theorem -- just make the right vertices correspond.
8. Reflections preserve distance.
9. The orientation is clockwise.
10. The orientation is counterclockwise, because reflections switch orientation.
11. There are three pairs: angles B and C, angles BAD and CAD, angles ADB and CDB.
12. There is one line of symmetry -- the line containing the angle bisector. This follows from the Angle Symmetry Theorem.
13. F' = E follows from the Flip-Flop Theorem. FG = EH is because reflections preserve distance.
14. Proof:
Statements Reasons
1. MO = MN 1. Given
2. M' = N 2. Given
3. MO = NO 3. Reflections preserve distance
4. MNO is equil. 4. Definition of equilateral
It's possible to add more details, such as O' = O, Transitive Property, etc.
15. The rectangle has two lines of symmetry, one horizontal, one vertical.
16. The isosceles triangle has one line of symmetry, and it's horizontal.
17. The images of the vertices are (1,3), (7,1), and (6,-2).
18. The image is (c, -d).
19. The angle measures 140 degrees.
20. The shortest distance is the perpendicular distance.
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