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?

1.  What do you know about circles. (All circles have a center, a radius, a diameter, a circumference, and an area.)

2.  What are some things you know about squares. (All squares have 4 equal sides, 4 right angles, a perimeter, and an area.)

3.  How do you calculate the circumference of a circle? What information is needed? (How about the diameter. Pi x D = C.)

4.  How do you calculate the perimeter of a square? What information is needed? (You need the measurement of one side. 4 x a side = P.)

5. Draw a diameter from left to right through the middle of the circle, parallel with the bottom of the square. Is the measurement of the diameter the same as the width of the square?

6. Imagine that the diameter of the circle is 7 meters. What is the circumference? (Use 22/7 for Pi.)

7. If the width of the square is the same, what is the perimeter of the square? Which is longer, the perimeter of the square, or the circumference of the circle?

8. Imagine that the diameter of the circle is 10 meters. What is the circumference? (Use 3.14 for Pi.)

9. If the width of the square is the same, what is the perimeter of the square? Which is longer, the perimeter of the square, or the circumference of the circle?

10. Imagine that the diameter of the circle is “D” meters. What is the circumference? (Use 3.14 for Pi.) (D x 3.14)

11. If the width of the square is the same, what is the perimeter of the square? Which is longer, the perimeter of the square, or the circumference of the circle? (D x 4)

This week we have a different type of problem. In order for students to solve a problem in a given lesson, they often need an understanding of and fluency with prior concepts, procedures, definitions, or theorems.  If the needed concepts or procedures are missing, the path can be closed to them. Equally important, in order to support their students’ discovery of a successful path, teachers, parents, and tutors also need to know what prior knowledge and skills are required to solve a particular problem.

So for this week, what concepts and procedures does a student need in order to prove that the distance around the inner object is shorter than the distance around the outer object? We will post some suggestions next week. In the meantime, post some yourself by leaving a comment.

1. Put your left index finger on zero

2. With your right index finger, add 8 (Do not touch zero. Touch 1, 2, 3, 4, 5, 6, 7, 8.)

3. Say, “I just added 8.”

4. Slide your left finger from the zero to the 8 and say, “That’s one group of 8.”

5. With your right index finger, add 8 more (Touch 9, 10, 11, 12, 13, 14, 15, 16.)

6. Say, “I added 8 two times.”

7. Slide your left finger from the 8 to the 16 and say, “That’s two groups of 8.”

8. With your right index finger, add 8 more (Touch 17, 18, 19, 20, 21, 22, 23, 24.)

9. Say, “I added 8 three times.”

10. Slide your left finger from the 16 to the 24 and say, “That’s three groups of 8.”

11. How many times did you add 8? How many groups of 8 is that? How many is in each group? At what number did you end? So 3 x 8 = ?”

How many are in each group? How many groups are there?  How many are there altogether?

12. Try an experiment: Start at zero and add three, eight times. What do you notice?

Second-grade classes are usually taught to skip-count by twos, fives, and tens as preparation for learning to multiply and divide. That is, the teachers give instructions with that goal in mind, but typically not all of the students reach the goal. Some of them memorize the words, but fail to understand the idea behind them, which leads to failure with multiplication and division when they are presented. And some do not even memorize the words correctly. Why is that? I have observed several situations which shed light on this question, which I would like to share with you.

Consider the experience of Jose, a seventh-grader who wanted to show his study hall teacher how fast he could count by twos from zero to twenty-four. He enthusiastically began: “Two, four, six, eight, ten…, eleven?, no…, twelve?, no…, thirteen?, no…” He asked if he could begin again. He seemed to think that—like riding a bike up a steep hill—if he could get going really fast, the right words would flow and he would make it to the top (to twenty-four). He started over several times, but never got past ten, where his memory always failed him. He apparently had no idea how those counting words came to be in the first place; so with no concept of what he was doing, he could not extend the pattern. He could, however, quite successfully repeat from memory a limited number of words that had no meaning to him.

Consider the case of a high-school special ed student, who was attempting to count objects that were grouped in fives. She began, “Five, ten…,” and then went blank. The teacher’s aide mouthed the first sound of the next word in the pattern, to cue her next response: “Ffff…” “Ff…four,” said the girl. “No. Five, ten, ffff…,” intoned the adult. “Ff…fourteen?” replied the child. She could not successfully repeat from memory a limited number of words that had no meaning to her.

And there is the case of Robin, a sixth-grader who recognized that an array of objects was arranged in groups of five, and counted them like this: “Five, ten, fifteen, twenty, twenty-two, twenty-four, twenty-six, twenty-eight!” She understood that she should be skip-counting by fives, but did not realize that she had suddenly switched from counting by fives to counting by twos. A similar thing happened with a middle-school boy, who counted, “Five, ten, fifteen, twenty, thirty, forty, etc.” He unknowingly switched from fives to tens.

I had two experiences myself with memorizing as a high-school student. Our cheerleaders came back from a special summer camp all fired up to teach us some new cheers. The first one they taught us was composed entirely of nonsense syllables. It was crazy, funny, and we could say it! What did it mean? Nothing. Not less crazy was the day that my Algebra teacher demanded that our class memorize a formula, which she wrote on the board: “y = mx + b.” I raised my hand and asked what it meant. “Don’t worry about that,” she said. “Just memorize it now, and we’ll learn what it means later.” Like the nonsense cheer that we learned during the pep rally, we could say the formula. And what did it mean? Nothing. It was crazy, but it wasn’t fun.

Memorizing musical words can be quite charming and engaging—although equally nonsensical. Many Beatles fans can still sing from memory the French words to “Michelle,” but they don’t know what they mean. Thousands of elementary school children enthusiastically sing Frère Jacques every year without understanding the lyrics they repeat. They have fluent recall of the words, but cannot use them to develop their understanding of the language. In fact, millions of American children annually memorize the words to the Pledge to the Flag—which is in English—but they don’t know what many of those words mean, either. And when they proudly recite, “…and to the republic for Richard stands,” they don’t think to ask who Richard is.

I once witnessed an entire class of kindergartners perform a feat of instantaneous linguistic mimicry. Their music teacher told them to listen quietly while he sang them a new song. To his surprise, they started singing the new song right along with him—even though they had never heard it before. As the words of the song rolled out of his mouth, they also came out of the children’s mouths! Somehow those little five-year-olds were able to instantaneously imitate both the words and the tune of the new song! Did they also understand the meaning of the words? Not necessarily. Just as reading comprehension does not automatically follow accurate decoding (reading a sentence aloud correctly, without understanding its meaning), the universal talent for linguistic mimicry and memorization does not always connect with the mental realm of curiosity, meaning, and understanding.

What do the stories of Jose, Robin, and the other students have in common? They were all skip-chanting, not skip-counting. Their words were mere regurgitations of memorized linguistic patterns—like Frère Jacques—not indicators of perceived amounts. They recalled them the same way that students recall the words to the Pledge to the Flag, through sheer rote memorization—just as I had memorized the nonsense syllables at my high school pep rally. But without an understanding of the concepts behind them, fluently recalled words do not provide a sure basis for developing mathematical understanding. Unfortunately, classrooms throughout the country regularly engage in whole-class unison skip-chanting. When all the students are saying the right words together, the teachers are too often satisfied with the deceptive evidence that learning has taken place—not realizing that if only one student in the class can count by fours, the rest of the class can simultaneously imitate the words, just as the kindergartners instantaneously mimicked the words of their new song.

However, rote memorization is only one of many kinds of memory. Just as there are many different learning styles or modalities, there are also many ways of developing memory. I have found that the model most applicable to the development of conceptually connected mathematical (not merely linguistic) memory is the way in which we learn our way around a new city. We do not use flash cards, timed tests, rhymes, tricks, and unison chanting to accomplish that task. Instead, we look at maps and travel repeatedly around the city, gradually building a mental map of landmarks, streets, and locations. And as we continue traveling to meaningful destinations, our need for maps and directions gradually diminishes until we know our way around with such familiarity that we can go where we want without even thinking about it.

This everyday approach to memory development is normally referred to as remembering—as opposed to memorizing. Even though they both have to do with memory, they are clearly not the same thing. Memorization is purely verbal, sometimes has no connection with experience or conceptual understanding, and is much more stressful. Remembering, on the other hand, is generally derived from multi-sensory kinesthetic experiences within a spatial/conceptual context, with a connected verbal component, and generates relatively little stress.

Here is an example of how to teach skip-counting by fives in a way which incorporates a mathematical way of remembering, rather than a linguistic way of memorizing. Present the students with their own copy of a vertical array of pea pods, with each pod containing five numbered peas.

[   (1)     (2)     (3)     (4)     (5)  ]

[   (6)     (7)     (8)     (9)    (10) ]

[  (11)   (12)   (13)   (14)   (15) ]

etc.

Giving very brief directives, accompanied by clarifying gestures…

• Have the pupils touch each pea and count them aloud in a quiet voice—except for the last pea on the right in each group, which should be counted in a loud voice:

“One, two, three, four, FIVE;

six, seven, eight, nine, TEN; etc.”

Watch the students carefully, and do not allow the class to go faster than the slowest student can manage.

If the speed accelerates to a pace comfortable only for the fastest students, the slower ones are likely to stop touching-and-counting, and will cope with the situation by reverting to rote verbal imitation of their peers.

When the special ed student mentioned above was given these directions, she followed them easily, then smiled and said “I get it!” It is important that the objects (the peas, in this case) in such an array be numbered; without that visual imprint, some students do not make the connection between the numbers that are saying aloud and the written numbers that teachers hope they are imagining (because they are not imaging them!).

• Next, have the students whisper the counting words as they touch the peas, saying each fifth number in a regular voice:

“One, two, three, four, five;

six, seven, eight, nine, ten; etc.”

When the girl mentioned above followed these directions, she laughed and said “This is easy! I can do it!”

• Then have the pupils touch-and-count the peas again, thinking the first four counting words as they touch the peas, and saying each fifth number in a regular voice:

“(One, two, three, four,) five;

(six, seven, eight, nine,) ten; etc.”

The girl’s reaction conveyed the pleasure engendered by understanding: “This is fun! I like learning this way!”

• Then have the pupils slide a finger across each group of peas, smoothly gliding from left to right as they again think the first four counting words, and say each fifth number in a regular voice:

“(One, two, three, four,) five;

(six, seven, eight, nine,) ten; etc.”

• Now present the students with another vertical array of pea pods, with each pod containing five unnumbered peas (see graphic above).

[   (  )     (  )     (  )     (  )     (  )  ]

[   (  )     (  )     (  )     (  )     (  )  ]

[   (  )     (  )     (  )     (  )     (  )  ]

etc.

Then have the pupils again slide a finger across each group of peas, as they think the first four counting words, and say each fifth number in a regular voice:

“(One, two, three, four,) five;

(six, seven, eight, nine,) ten; etc.”

The students are now skip-counting by fives. The same approach can be used to teach skip-counting by other numbers—only different graphics are required. Each child must be given the opportunity to show that they can do the last step (skip-counting groups of unnumberd objects) by themselves, without the possibility of verbally imitating other students. The first three steps should be reviewed (with teacher support) by students who display uncertainty, hesitation, or inaccuracy. (“You sound like you’re not certain about that. Count them for me again so we can see if you’re right.”)

What distinguishes skip-counting from skip-chanting? Skip-counting requires physical interaction with numbered objects; skip-chanting requires no actual counting at all. Skip-counting requires mathematical thinking; skip-chanting can be accomplished with no thinking whatsoever. Many teachers rely on slow-paced counting activities with physically constructed grouped amounts to develop the concept of skip-counting, but then revert to skip-chanting to develop fluent recall of the number-facts. They assume that since their students understood the conceptual part of the lesson, they will automatically invest the following rote verbal exercise with understanding. Experience shows that some students do make that connection. Other students, however, are not rapidly recalling the previously counted groups during the group chanting exercise; for them, skip-chanting is a near trance-like state with only one goal: successful verbal duplication. Verbal imitation does not require any conceptual connection in order to be successfully performed. For that reason, some students make the connection while others do not; during the same group activity, some students can be truly skip-counting, while others are merely skip-chanting. The pea-pod lesson modeled above provides all students with the means to remain conceptually connected during the memory-building phase of the skip-counting lesson.