March 2010

Today is 3.14 so we have a pi problem of the week. has been worshipped and maligned for at least 4000 years. And luminaries like Archimedes (3rd century BC – there on the left), Newton, Leibnitz, and Euler (18th century) all attempted their own precise approximations.  How about that for a number?!  is the (constant) ratio of the circumference “C” to the diameter “d” in any circle.  A ratio is a comparison of two amounts, which can be expressed as a quotient = C/d .

But this explanation can sound like Greek to many students! – who are also told and need to memorize that pi ≈ 3.14. Instead we offer this problem to guide students (with simple geometry and good questions) to reason for themselves why pi must be less than 4 and more than 3 – good mathematical thinking from the ground up.

Imagine that you were never told that ≈ 3.14. Use the figures below to prove that the value of must be less than 4 and greater than 3. We will post the solution next week.


1. Look at Figure #1.What is the length of the top? What is the length of the bottom? What is the width of the right side? What is the width of the left side?
2. Is Figure #1 a square?
3. Is Figure #2 exactly the same size and shape as Figure #1?
4. How many triangles are in Figure #1? Do they all have the same measurements? Are they all the same size? Are they all right triangles?
5. How many right triangles are in Figure #2? Are they all the same size as the triangles in Figure #1?
6. If the upper left triangle from each figure were removed, would the areas of the two figures still be equal?
7. If two triangles were removed from each figure, would the remaining areas still be equal?
8. If all of the triangles were removed from each figure, would the remaining areas still be equal?
9. How long is the shortest leg of each triangle? How long is the hypotenuse? How long is the other leg?
10. What are the dimensions of the smaller square in Figure #1? What is its area? Compare that square to one of the triangles; one side of the square is the same length as which leg of the triangle? Compare the length of that leg to the area of the small square.
11. What are the dimensions of the larger square in Figure #1? What is its area? Compare that square to one of the triangles; one side of the square is the same length as which leg of the triangle? Compare the length of that leg to the area of the larger square.
12. What are the dimensions of the large square in Figure #2? What is its area? Compare that square to one of the triangles; one side of the square is the same length as which leg of the triangle? Compare the length of that leg to the area of the large square.
13. If all of the triangles were removed from Figure #1, what would remain?
14. If all of the triangles were removed from Figure #2, what would remain? (8.) If all of the triangles were removed from each figure, would the remaining areas still be equal?
15. If you combined (added) the areas of the small square and the larger square in Figure #1, would that combined area (sum) be equal to the area of the large square in Figure #2?

And a tribute to someone with a brain, who could have used a few extra problems of the week on the Pythagorean Theorem. Ask your students if the diploma helped the scarecrow. Upon receiving his diploma he recited: “The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side. Oh, joy, oh, rapture. I’ve got a brain!”

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In an earlier post, we gave an outline for integrating assessment into a math lesson. And we said that in subsequent posts we would address several aspects in more detail. Here is a closer look at the first step – starting the lesson with a graphic representation of the concept and procedure handed out to each student.

In the post that followed, we put up a video (The Soccer Teams Graphic) that demonstrated exploring the graphic in numerical order and re-exploring it out of numerical order. Through tutoring and classroom application of the graphic pages, we have observed more effective and less effective ways of guiding a lesson using a graphic manipulative. While there are lots of manipulatives and number arrays out there, how they are implemented and integrated makes all the difference. As always, it comes down to effective teaching. Here’s an approach that works when guiding students with a graphic page and a blank piece of paper:

1. Use short phrasessay only what is needed for students to construct, count, and observe on the graphic page. AVOID explaining. “Now you see class, what you just did was…” Guide and allow for their discoveries. Short phrases move the lesson along, keep students actively engaged with the graphic, and help language learners and students who have a hard time focusing.

2. Start with an action verbcount (“Count how many…”), move (“Move your blank paper to show…”), cover (“Cover all but the first team”), uncover, and so on. This aids in fulfilling the first step. Students know right away what to do.

3. Model your directionswhen you ask them to uncover the next row of soccer players or members of a rowing team or weeks with the blank paper, do it with them at first – in a way that they can see your movements. A picture is worth a thousand words. This is helpful for everyone and especially for those learning the language – whether English or math! – and those who have a problem paying attention to aural directions. Your observations will tell you how long to continue modeling. It has proved more effective to model one step at a time rather than to give students an overview to recall and execute.

4. Observe (assess)look around to make sure everyone is engaged and following your direction on the graphic correctly. Since it involves physical activity, it is pretty easy to see who is where they need to be for the direction you gave and who is slow or looking around to see what to do. Assessing determines the speed and direction of the lesson – does a prior skill need to be addressed – who needs a little more support – who is ready for more of a challenge – and so on. Creatively using a graphic page enables you to derive multiple lessons and multiple challenge levels from the same page. Each student matters.

5. Correct student responses nowsince each student matters, something needs to be done when you see inaccurate or continually slow responses. But never give an answer. Guide individual students to correct themselves: “Show me on the chart how you got that answer.”  As they reconstruct and rethink their solution, they see for themselves what they did wrong, and therefore how to do it right. Some may require further guiding questions. It is not efficient to give them the answer and move on. You will have to retrace this same concept or procedure eventually. Do it now before it becomes an unlearned prior skill needed for another procedure the student is trying to learn.

6. Begin the transition to print say “show this many teams” as you write a “5” on the board. Don’t say the word “5.” Or say, “show this on your page” as you write on the board, “7 x 11.” By doing this, students see numbers and symbols as action prompts and begin making the connection between manipulation and computation. It prepares them for the problem pages that follow.

7. Ask questions transition to mini-word problems. “3 teams are coming to the practice tomorrow, how many water bottles will we need?” After the students have some experience with a concept and procedure, then begin to make the process conscious by asking them questions about what they did.

Our experience has been that new vocabulary and notation should be introduced gradually in conjunction with the work on the graphic pages and reiterated during practice with the partner pages that follow. It is better to solidify (talk about) concepts with the correct nomenclature after students have actually experienced using them with the graphic pages. They own something concrete then to hang those words on. We will look more closely at working with the partner pages in a later post.

Summary: Guide work on a graphic manipulative using short phrases. Model how to do it. Make sure they’re doing it correctly; help them if they aren’t. Guide the work using numbers and math notation on the board. Give further written and aural challenges at appropriate levels. Ask them questions about what they did.

Some may wonder why there are all the steps and detail in the Guided Solutions? It looks like a lot of work for students who are not doing well in math and may already have some resistance. Wouldn’t it be easier for them and more time efficient if we just explained the answers clearly?

Maybe your students are different, but the students we’ve worked with who are not doing well tend to miss important details and fail to derive some of the needed implications from the given information. They also fail to make logical connections on their own, even when the solutions are explained to them.

In order to get more out of the given in a geometry problem, it’s helpful to understand the problem (what are students trying to solve) and think about some of the possible routes to a solution (definitions, postulates, theorems, procedures). It’s also helpful to be able to ask questions of the given (what information do they have or can they reasonably derive). Asking these questions can open up the given and remind students of solution paths. It is this latter experience that is behind the guided solution approach.

Why guided solutions?
1. It is how we teach and tutor anyway.  Rather than telling what we know, we guide student discoveries as they build their mathematical thinking, their skill and knowledge.
2. It models the kinds of questions a student can ask of a diagram.
3. It models looking for connections between different parts of the given.
4. It models a path from the given to the solution.
5. It models an approach for stimulating students’ emerging sense of mathematical curiosity and logic.

Depending on the need, you can break the lesson down into logical and accomplishable sub-lessons, so the students feel a sense of success each step of the way. Or when doing the whole lesson, don’t show them all of the questions at once. Succeeding with each individual question and consequently seeing the diagram open up before them, is a motivator to keep going. While for other students, seeing all of your guiding questions up front, finding patterns in the questions as you ask them, and realizing at the end that they did get through all of them, can make the process less daunting in the future.

Students who know that the answers will eventually be given to them have less of a need and so are less motivated to develop their own analytical skills. To the extent that analytical reasoning is developed through guiding and not telling, future problem-solving becomes more efficient. The long-cut in the beginning is the short-cut in the long-run.

To wrap up this series of Pythagorean Problems of the Week, here is one that often presents difficulties for students at first. Where are the numbers?! What is the length of a side that is only assigned a letter!!?

There has been a purposeful flow to our weekly Pythagorean problems. We began with whole number measurements and solutions, the second one using the information to solve another problem, and moved to whole number measurements with an irrational number solution. We now end with a problem that preserves complete generality.

Can you use the “measurements” in these two pictures to prove that a2 + b2 = c2 is true for all right triangles? The length of “a” is the same in both figures, as is the length of “b” and “c.” For this problem, all corners that look square are square. The solution will be posted next week with a special tribute to the scarecrow from the Wizard of Oz.

1. How many degrees is angle ADC?
2. What kind of triangle is triangle ADC?
3. How many degrees is angle DAC?  …angle DCA?
4. How many degrees is angle ABC?
5. How many degrees is angle CAB?  …angle ACB?
6. How many degrees is angle  BAD?  …angle BCD?
7. Is figure ABCD a square?
8. What is the area of figure ABCD?
9. What is the area of triangle ABC?
10. Are triangles ABC, CBF, FBE, and EBA all the same size?
11. What is the area of figure EFCA?
12. Is figure EFCA a square?
13. What is the area of figure EFCA?
14. What is the measurement of each side of figure EFCA?
15. What is the length of  line segment AC?

Following on the previous two posts, here is the Partner or Folding Pages podcast. With the partner pages, students rediscover, reinforce, and extend the work they did on the graphic pages. They connect manipulation with computation and continue building memory by taking more spins around the city.

Four additional features of the partner pages that support students who are not doing well in math are:
1. They involve cooperative learning with individual accountability
2. Students receive immediate and ongoing feedback
3. They engage teacher, peers, and students in student self-correction
4. They use no-stress timing focused on improving rather than merely measuring performance (See the earlier postings on fluency for a discussion of this.)

Again, because of the lesson design, we can observe and assess how students are doing each step of the way – and make the adjustments our observations indicate.

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