Viewing a Square

By Jonathan Osters @callmejosters & Chris Robinson @isomorphic2crob, The Blake School 

This week, we were taking a look through How to Solve It, and came across this problem that intrigued us. The problem is as follows:

Point P is a point outside a square. Define the “viewing angle” as an angle that has P as a vertex and vertices of the square as points on its rays, as if P was on the side of a sculpture and viewing it by walking around it (Figure 1). What is the locus of points for which the viewing angle is 90˚? What is the locus for which the viewing angle is 45˚? (Paraphrased from p. 234 of How to Solve It)

viewing angle    Black Cube

This problem is really fun! Give it a try before you continue reading.

Just Do It

Solution for 90º viewing angle:

We began by creating convenient points in the locus. We reasoned that a point “half of a side length” from the midpoint of a side could create a 45-45-90 triangle.

90 degree viewing angle

Then we remembered Thales’ Theorem, that says that any point on a circle forms a right angle with endpoints of a diameter. Therefore, any point “half of a side length” from the midpoint of a side would create a right angle.


Thanks, Thales! Repeat that with all four sides, and you will get the following locus:

Screen Shot 2016-04-04 at 3.26.55 PM

Solution for 45º viewing angle:

We knew a few things more about this one, having solved the 90˚ version of the problem. We knew that in this scenario, we would have to take into account that you might be able to see one or two sides, rather than just one, like in the 90˚ version. Not being able to see the solution with our mind’s eye, we began by calculating a few special points in the locus.

Special point 1: On a side (extended)

On the extension of one side, you will only be able to see one side, but you will be on the cusp of seeing two sides. This gave us the impression that this was an important point to investigate.

Screen Shot 2016-04-04 at 10.05.40 PM

The point that forms a 45º viewing angle on the extended side forms an isosceles triangle which means that the point is a side length away from the vertex of the square. So the following 8 points are included in the locus.

Screen Shot 2016-04-04 at 10.12.18 PM

Special Point 2: On an diagonal (extended)

The points along the extended diagonal would include two sides in their viewing angle. By symmetry we know that the diagonal is the bisector of the 45º angle and that the diagonal also bisects the 270º outside of the square.

Screen Shot 2016-04-04 at 10.21.15 PM

Nicely, this leaves 22.5º, which means this a set of two isosceles triangles and that the point is a side length away from the vertex of the square. At that point we realized that to keep a 45˚ angle at the viewing angle, we could have the angle intercept a 90˚ arc on a circle. This circle:

Screen Shot 2016-04-05 at 8.19.47 AM

We can see that any point on this quarter circle creates a 45˚ viewing angle.IydgNE


There is such a quarter circle surrounding each vertex of the square.

Screen Shot 2016-04-05 at 8.21.44 AM

This leaves us with the task of discovering what the locus of points is when only one side is visible. After playing around with it some more, and trying more things, we saw that the endpoints of the arcs and the closest vertices of the square, form a square.

Screen Shot 2016-04-05 at 8.27.07 AM

If we create a circle at the center of that “side square,” then points on the outer edge will intercept a quarter of the circle at the viewing angle, thus making the viewing angle a 45˚ inscribed angle again!



There are these “bulbous bumpouts” on each of the sides, so the complete locus gives this “cloud” shape:

Screen Shot 2016-04-05 at 8.49.42 AM

After completing this problem the other day, Chris and I started discussing the “cosmic significance” of the problem. This problem could be a great problem for an honors geometry class working on loci or circle properties, but more importantly, this problem is a reminder of what Chris and I like so much about working at Blake. We will often find ourselves on flights of mathematical fancy where we toss out some random problem like this, and will work to solve it together. We are all eager and excited mathematicians, and are always looking for problems that energize us and help us “recharge our mathematical batteries.” This is one such problem.


Ready to recharge your mathematical batteries?

Next week, we will post the general solution to the problem. How do the loci change if the viewing angle is very small? What if it’s very large? See if you can sketch out a guess of what the loci look like, and snap a picture of it and send it to us on twitter! Our handles are @callmejosters and @Isomorphic2CRob. Have fun!

Stars on the Flag – Solution

written by Alex Fisher, The Blake School

Hey folks!  If you’ve been eagerly awaiting a solution to the flag problem, perhaps while stitching your own replacements and making plans to kick Texas out of the Union, wait no more!

The easiest way to think about this grid is to actually separate it into two overlapping grids:

Screen Shot 2016-03-01 at 1.04.56 PM

From this picture, we can see two sub-grids: one in blue that has m rows and n columns (in this case, 5 and 6 respectively) and another in pink, that measures m-1 by n-1.  Add them together and I have T stars total.  It’s tempting to try to multiply (2m-1) by (2n-1), but that overcounts the total number of stars.  However, what if we split these two grids up?

Screen Shot 2016-03-01 at 1.05.41 PM

Now we have those same two rectangular grids, the blue m x n and the pink (m-1) x (n-1).  Together they almost make a larger grid, but not quite: there’s a column missing.  It’s tempting to just add those stars to the pink grid…  However, if I add a column, I’m changing the total number of stars by m or n, making it harder to determine whether a given number of stars T can be arranged in this pattern.  I want to find a way to make sure that the number of stars in my overall picture can be represented in terms of T and constants.

  Screen Shot 2016-03-01 at 1.08.33 PM Screen Shot 2016-03-01 at 1.08.48 PMScreen Shot 2016-03-01 at 1.09.11 PM

Now we’re getting somewhere!  By doubling the previous arrangement, we actually end up with that previously mentioned (2n-1) x (2m-1) grid!  Well, almost.  There’s one star of overlap.  Even so, this means that if I double T and take away that one overlapping star, I have a perfect grid.  Thus, this equation:  2T-1 = (2m-1)(2n-1).

This seems like an odd conclusion to draw, but it is actually the key to answering all of our questions.  Consider the situation where Puerto Rico, Guam, and Washington DC all are given statehood.  We now have 53 states, so T = 53, and 2T-1 = 105.  If I can factor 105, that means I can find numbers m and n to make that grid.  In this case, since 105 factors into 3*5*7, I can actually do this in three different non-trivial ways:  5*21, 3*35, and 7*15.  Solving for m and n in these scenarios gives us the following diagrams:

Screen Shot 2016-03-01 at 1.18.24 PM

And while the last one is a little silly, the 4 x 8 case is relatively nice looking.  Also, we can see that if only one state joined the Union, 2T-1 = 101, which isn’t factorable, so this arrangement wouldn’t be possible at all.  Likewise, if we lost a state, we’d be left at 97, also not a number with nice factors.  So, if you’re a big fan of this star arrangement, then bad news: you’re stuck with Texas.  At least, unless you convinced them to take Florida on their way out…

Also, note that while I did this work geometrically, there is also an algebraic approach: in the original set-up, T = the blue rectangle + the pink rectangle, or T = m*n + (m-1)(n-1) = 2mn-m-n+1.  I will leave the algebraic manipulation to get this to match our answer of 2T-1 = (2m-1)(2n-1) as an exercise for the reader.

Stars on the Flag

Written by Alex Fisher, The Blake School

One problem that caught my interest recently involves the US flag.  This is actually one I rediscovered from back in 2012 when Puerto Rico was considering becoming a state and there was much discussion online of how you add a star to the flag in such a scenario.  If you look at the stars on the current flag, you’ll see that there are 5 rows of 6 stars, with another 4 rows of 5 stars between them.  

Screen Shot 2016-01-29 at 1.35.13 PM

If you were to keep this pattern, where a larger grid contains a smaller grid nestled between its rows, what numbers can you represent?  Can you make one that has 51 stars?  How about 49?  And which numbers can be represented in more than one arrangement?

Answer to follow next week!

Noticing and Wondering– The Beginning of a Mathematical Journey

Written by Jonathan Osters @CallMeJosters, The Blake School

A few weeks ago, we began the journey of parallel lines in geometry. I have taught this course for several years now, and the first day of this unit is typically excruciating for me – a lot of bland definitions that don’t really have any connection to anything interesting. This year, I decided to change things up and have students create the definitions and conjectures on their own, but I would need a means to do that.

I have seen a number of posts from the wonderfully knowledgeable teachers of #MTBoS (Math Twitter Blogosphere) on Twitter, and many of them had been beginning lessons asking students the questions, “What do you notice?” and “What do you wonder?” I have been skeptical about trying this in a high school level class, but I decided to give it a try anyway.

I used the following figure from Sketchpad and projected it on the board. Then I gave students a few minutes to notice and wonder (My only directions were that the “Notice” statements had to be statements, and the “Wonder” statements had to be questions).


The following were some of the “Notice” and “Wonder” statements I got throughout the day in my three sections of Geometry:

Notice Wonder

Then I asked if there was anything among the things we noticed that allowed us to answer any of the questions that we wonder. Some, like the distances, were directly measurable in Sketchpad, while “are the lines parallel?” required a different set of skills. These students have taken Algebra 1, and several pointed out that if these lines had been parallel, we would have been able to determine if they were parallel. I obliged them, despite the fact that discussions of slope weren’t an official part of the lesson.

parallel coordinate

So from here, we were able to re-familiarize ourselves with slope as a number that measures steepness, and that if two different lines have the same steepness, they will be parallel. At this point, many students were at least able to answer the question, as 6.63-1.131.87-(-14.08)=0.345, while 0.78-(-5.40)5.19-(-13.51)=0.330, so the lines are not parallel.

I asked if there was anything they could notice and wonder now that there is grid on the board. Several students wondered if the lines were making the same angle with the y-axis. I removed the grid and just placed a vertical line where the y-axis had been, and measured the angles.

parallel with vert
Many students then made the argument that had the two angles been equal in measure, the lines would have been parallel. They also argued that they could find the measures of the other six angles, and that if they could determine if “the angle at any corner” were congruent “to the angle at the same corner on the other line,” then the lines would be parallel. Some noticed that it would be enough to know that the “two angles on the inside but the other side of that axis” were congruent. At this point, it was they who decided we would benefit from some terms for such angles. And voila, the definitions for Alternate Interior Angles, Corresponding Angles, and the like. We were able to modify their original conjecture to: If Corresponding Angles on two lines are congruent, then those lines are parallel. This is exactly what I had hoped they would see, and while we ran out of time to prove the conjecture, we would begin there the next day, and the lesson was so much the better for it.

Noticing and wondering was a very powerful technique, especially when introducing something brand new. I will be utilizing this strategy frequently. Thanks to the teachers of #MTBoS for helping me learn something new!

A Comprehensive System for Student Assessment: Part 2

By Chris Robinson @Isomorphic2CRob & Jonathan Osters @callmejosters, The Blake School

What does a grade in your course represent? How does the way you calculate your students’ grades describe what you value?

Most of us think about two types of assessment on a daily basis. Formative Assessment should give students feedback on how close they are to meeting class expectations of knowledge and skill. Summative Assessment is a measure of what they know at the end of the learning process.

Another two pieces of the assessment picture that we often think about is the balance between a student knowing the skills of mathematics and students being able to solve novel problems. In the past, Mathematics education movements have swung far in each direction, at times over emphasizing skills and at times sacrificing skills practice.

As the Blake Mathematics Department renewed its discussion on teaching problem solving in 2009, we began thinking about how we could assess more effectively. We wanted to utilize frequent formative assessment but were unsure how to walk the line of making them worth enough that students took them seriously, but not worth an unfair amount as students are still in the learning process. We also wanted to assess authentic problem solving. The last post discussed our Skills Quiz System, while this post will discuss our Problem Solving Assessments.


Building a Culture of Problem Solving BEFORE Assessment

Being able to assess problem solving effectively requires that students are “practicing” solving novel problems on a daily basis. There is a great deal that can be said about buliding a culture in the classroom focused on problem solving and collaboration and every teacher’s experience will be a little different: see our blog 5 Things Teachers Can Do to Establish a Cooperative Classroom Environment and Carmel Schettino’s blog is a great resource as well. Carmel is a teacher at Deerfield and major proponent of problem based instruction. Suffice it to say that building a culture is more than selecting good problems but as we are trying to talk about assessment here, we shall move on.

My colleagues and I each take a different approach here but in all cases a majority of our lessons are centered around problems that explore new topics and are either scaffolded by printed leading questions or are scaffolded by interjected questions in small group discussions. Some problems are taken from the Phillips Exeter Academy Curriculum, some from Carmel Schettino, some from our current hodgepodge of curricular resources and many written by us. The goal in general is for students to encounter something approachable but new and for them to conjecture and test out approaches to solving a problem. Then students share their approaches in small group and then large group verbally or visually through doc cams or writing on the boards. There is a summary process of the approach and sometimes an urging of “if you want to remember the steps you just went through, you may want to take notes now.”


Problem Solving Under Pressure

The threat of assessment is real for all students, and although our skills quiz systems alleviates some pressure, solving novel or near-novel problems can be a hairy experience for most students and teachers. The constant practice of problem solving in the classroom should help prepare students. Our Problem Solving Assessments (PSA) come in a few varieties: Exploration Labs, Reflection Journals, In-Class PSAs and Take Home PSAs. Not all classes use all types but it is safe to say that all classes use at least two of the four.

Component 1: Exploration Labs

Exploration Labs come in many forms. A majority of them are guided questions utilizing a technology like Geometer’s Sketchpad, Fathom, or Desmos. In Geometry, students may be tasked with creating a diagram, manipulating the diagram and observing how things change. In algebra, students may use Desmos sliders to see how adjusting one part of an equation affects the graph of the equation. In statistics, we may create some statistic “from the ground up” that measures some particular facet of a distribution. Each of these activities puts students in new situations, where building something from scratch is required of them, but the stakes are low, so they can try different things until a “good,” “best,” or “most efficient” way to approach the problem comes up.

Pros: +Students have opportunities to conjecture in low risk situations (often using dynamic software).
        + Students often work together and develop math communication skills.

Con: – They can take a while to grade, depending on depth of expectation.


Component 2: Reflection Journals

Inspired by the work of Carmel Schettino and her frequent metacognitive journal assignments, we have experimented with a variety of writing assignments. My most recent Honors Algebra II written assignment was the following:

Which is your favorite representation for a line? (standard form, slope-intercept form, point-slope form) Be sure to compare and contrast your favorite to each other type regarding

  1. a) ease of graphing.
  2. b) ease of writing an equation given two points.
  3. c) ease in finding the intersection of two lines.

Students get a few nights to compose their first response. A grade and a written commentary is given to each student, and they can then revise and resubmit to regain half the points they lost.


Pro: + Reflection and metacognition are powerful learning tools.

Con:  – They can take a while to grade.


Component 3: In-Class Problem Solving Assessments

A traditional test has a mixture of skills and problem solving. Often coming in the form of 15 skill problems and 2-4 “word problems.” Those word problems were often the same as problems previously encountered with numbers changed; this is by necessity of students having so little time to approach them on a test filled with skills. And because there is so much to do, it was relatively common for students to skip or provide only a minimal attempt of the “word problems.”

It is important to have summative assessment on skills, but now that our skills quiz system is accomplishing both formative and summative assessment , we don’t need to do that on our tests. And thus was born the Problem Solving Assessment, or PSA. A PSA is essentially a traditional test with all the skill portions removed. It is a set of 2-6 problems that require synthesis of a number of skills.  

Our PSAs have novel problems in the sense that they may have the same theme as a previously seen problems, but there is a major twist. In an Algebra 2 class, for example, we may have a two-variable systems word problem, but instead of giving them the problem and they find the solution, we might give them a solution and a framework like a paint mixing problem, and ask them to write the question. In order to write the problem they will have to create a system of equations with the correct solution, and then create the sentences in the word problem. Another example might be that of a race between several racers, where speeds, head starts, and starting points all vary, leading to different equations of position for the different racers. We may ask them a straightforward problem like “who won the race?”, but we also could ask them more thought-provoking questions like “ which racers were in 2nd place at any point in the race? How long were they in 2nd place? Use the graph to justify.”

Like any exam, these in-class PSAs have a time element to them. Students must complete the problems during the period. At times we don’t predict properly how long students will take to “solve a problem,” and so we at times let them take them home to revise and at times allow them to revise in the class after teacher comments. This has worked well for us so far, since the novelty of the problems make them such that students can’t find a similar problem in their book or online.

Pros: + Students have the extra time to problem solve, compared to a traditional test.

          + They are faster to grade than a journal, students gain comforter in a timed situation.

Con:  – It’s a timed situation, and problem solving is tough with limited time.


Component 4: Take Home Problem Solving Assessments

Take-home PSAs are essentially the same as in-class PSAs, the only difference being the location. The advantage is students have more time to work through the problem if needed. The disadvantage is an increased risk of academic dishonesty. We might give students only one in depth problem rather than a couple of shorter problems as in the in-class PSAs.

Pro: + Students have more time to complete it.

Con: – Managing academic honesty becomes much trickier.

A teacher can use all or some of these components for a successful assessment of problem-solving. But what happens if a student has trouble even getting started? Or what if their work is haphazard and difficult to follow? Are these PSA’s graded differently than traditional exams? We will discuss those questions in next week’s post!


A Comprehensive System for Student Assessment (Part 1)

by Jonathan Osters @callmejosters & Chris Robinson @Isomorphic2CRob, The Blake School

What does a grade in your course represent? How does the way you calculate your students’ grades describe what you value?

Most of us think about two types of assessment on a daily basis. Formative Assessment should give students feedback on how close they are to meeting class expectations of knowledge and skill. Summative Assessment is a measure of what they know at the end of the learning process.

Another two pieces of the assessment picture that we often think about is the balance between a student knowing the skills of mathematics and students being able to solve novel problems. In the past, Mathematics education movements have swung far in each direction, at times over emphasizing skills and at times sacrificing skills practice.

As the Blake Mathematics Department renewed its discussion on teaching problem solving in 2009, we began thinking about how we could assess more effectively. We wanted to utilize frequent formative assessment but were unsure how to walk the line of making them worth enough that students took them seriously, but not worth an unfair amount as students are still in the learning process. We also wanted to assess authentic problem solving. The remainder of this post will discuss our Skills Quiz System, while the following post will discuss our Problem Solving Assessments.



If you are reading this blog then it is fairly safe to assume that you have heard of Dan Meyer, and his blog dy/dan. He has made a name for himself by deconstructing problems into 3-acts – peaking students interest in solving real math problems. Dan Meyer is also a regular presenter at math teacher conferences like NCTM (as well as this fantastic TED talk) and is now the Chief Academic Officer at Desmos. In 2008, he wrote a blog about how he helps students develop and perfect their skills using a retakeable quiz system ( Our department began discussing this idea in 2010 and invited Anna Maria Gaylord (A.G.A.P.E. High School) to a department meeting to share her skill system. Finding someone who had experimented with developing a skills quiz system was instrumental in our process of developing our system.

Both Dan’s and Anna Maria’s system began with a list of skills students need to master for success in their course. The teacher then develops a set of questions that assess that particular skill, with as little utilization of other skills to reach a solution. Students are then assessed a number of times on each skill, and allowed to retake each individually to show improvement and hopefully show mastery.  Dan and Anna Maria would then modify student grades based on retake scores. After we reviewed and discussed these two approaches a number of teams of teachers at Blake decided to implement a similar system in 2010-2011 school year.


Our Skills Quiz System

Our system takes and refines some of the details of Meyer’s and Gaylord’s systems. Under our system, there are several components. There is an in-class quizzing component, a remediation component, and a redemption component.

Component 1: Assessment

Students take a quiz each week. That quiz will contain some newer skills they have learned that week, but also older skills they have learned in previous weeks. That skill will also appear on the following week’s quiz, and the quiz the week after that, for a total of three times the student will have seen a question on that skill. Each time the student sees the question, it will be assessed on a scale from 0-4, inspired by the same scale used in the AP Statistics reading. And, like the AP Statistics reading, the scores themselves have descriptors that we can use to decide on a student’s score (4 = Complete, 3 = Substantial, 2 = Developing, 1 = Minimal). And rather than scoring the quiz as a whole, we follow the scores earned by the students on each skill.


Figure 1: This is a set of quizzes a student might take. On Skill #2, this student received a 3 the first attempt, a 2 the second attempt, and a 3 the third attempt. This student would receive an 8 on Skill #2.

The students receive scores for each attempt they make at a skill, and at the end of the three times they have attempted that skill, they receive a score, which is the sum of the three attempts (for example, if a student scores a 3 the first time a skill is on a quiz, a 2 the second time, and a 3 the third time, then the student receives a score of 3 + 2 + 3 = 8 for that skill). The highest score a student can receive in this system is 12. These scores are recorded in a google sheet to which each student has viewing access to only their own scores.


Figure 2: This student’s scores for Skill #2 have been entered into a google sheet.


Component 2: Remediation

If a student is not satisfied with their mastery (score) of a particular skill, they have the opportunity to take what we call “Redemption Quizzes” in order to show an increased level of mastery. But first, they need to to remediate their skill in order to show improvement on another assessment. Students are encouraged to review their past attempts on that skill with their teacher. In addition, on our course websites, we post “Redemption Assignments,” extra work that a student must complete in order to take a Redemption Quiz. They then check their answers against keys we post along with the assignment. We make it plain to them that if they don’t put in extra work to master a skill they have yet to master, then it is quite likely they will continue to earn the same scores they previously earned.

Component 3: Redemption

Once a student completes the Redemption Assignment, they can take a one-question Redemption Quiz, which covers the same skill as they have just remediated. This question is scored on the same 0-4 scale as the original quiz questions, and if it is an improvement over the previous scores, then the Redemption Quiz score replaces the lowest quiz score for that skill (for example, if the student who scored a 3 + 2 + 3 = 8 takes a redemption quiz and score a 4 on it, their score will improve to 3 + 3 + 4 = 10). These Redemption Quizzes are generally taken outside of class; our school has a few tutorial periods built into the schedule, but students can also take Redemption Quizzes during Study Hall, free periods in their schedule, or before or after school, contingent on there being someone available those times to proctor the student. We also allow for periodic “Days of Redemption” about once every six weeks, where students can use the class period to take Redemption Quizzes. We allow up to three Redemption Quizzes per skill, allowing for a student who scored poorly on all three in-class attempts to completely redeem themselves. We do have a rule that a student can take only one Redemption Quiz per skill, per day. We want to see sustained excellence rather than a one-time performance. This improvement gets recorded in the google sheet, and often we make a big fuss over entering a good score, as the student can actually see their score improving in real time. This incentive to improve and celebration of improvement is what made us want to create this system in the first place.


Figure 3: This student has taken a redemption quiz on Skill #1. His score has improved from 9 to 12.


To make sure students stay current with their remediation and redemption, Redemption Quizzes for a particular skill are only available for about 6 weeks after the last in-class attempt. This is about two months after a student first learns the skill.


Reflection: Assessing the Assessment System

Benefits of using this system over a traditional quiz system
  • Students have a clear message about what they know and don’t know, and a clear path to improvement.
  • Students take ownership of their learning more willingly when they know what they need to study.
  • The system produces increased feedback among students, parents, and teachers (and makes for rather easy conferences, since the path to improvement is so clear!).
  • Not only can we pinpoint the skills with which a particular student struggles, but we also gain a clear understanding for which skills the class as a whole is struggling.


Challenges/Costs of this system and how we address them
  • Enumerating the Skills – this takes some time upfront, but once it is done, then the only remaining big job is to write the questions.
  • Writing good questions – This can be tricky, too, since we want to only test one skill, not multiple skills at once. It’s also important that each question for each skill be of similar difficulty without being nearly identical.
  • Security of the Redemption Quizzes – we have tried a few different strategies for this over the years. The current way we try to keep the Redemption Quizzes secure is to have students turn in the Redemption Quiz, then place the graded Redemption Quiz in a file folder for each student, so they can request their folder from their teacher in order review any of their previously taken redemption quizzes. This has worked better than any other system we have used in the past.
  • To be honest, it’s kind of a lot of grading and bookkeeping- but the results are impressive. Our students do significantly better on retakes, and it’s way more fun to grade if they can show mastery as a result of their hard work. One can reduce the grading by reducing the number of skills assessed or in finding another way to summarize mastery beyond the sum of the three best attempts. This is simply a system that has worked well for us.

In summary, skills quizzes are both formative and summative assessments. They provide effective feedback to the students, while encouraging remediation and redemption. A student’s skills quiz average is 40% of their grade, 5-10% is homework, while a variety of problem solving assessments complete the other half of their grade. Please read our next post to read about how we measure synthesis of the individual skills into a deeper understanding of mathematics.
Thanks for reading until the end!

Chessboard (General Solution)

By Chris Robinson, The Blake School, @Isomorphic2CRob

Let’s begin with our assumptions:

  • A square board with even side length of two squares or greater.
  • Chessboard has the typical alternating color pattern.


If we remove a square of each color, there is only one rectangle, which contains those two squares as its corners.

Chess2_1The key concept is that this red rectangle is always even by odd or odd by even.

Imagine for a moment that the rectangle was of even width, then the top left and top right-most squares would be different colors, so the height would have to be 1, or 3, or 2n+1 to maintain opposite colored corners.
Imagine for a moment that the rectangle was of odd width, then the top corners would be the same color, so the height of the rectangle would need to be 2, 4, or 2n to ensure a different color in opposite corners.

So what’s so important about this rectangle?

We shall approach the tiling of the exterior and interior of the rectangle separately.

Once we remove the red rectangle we can divide the frame into up to 4 rectangles. Without loss of generality we shall assume that the odd-length of the red rectangle is the height. We take full vertical strips of the board on either side of the inner rectangle; these vertical strips have even height. They also leave even-width rectangles above and below the red rectangle as shown.

The fact that each of the blue border rectangles has at least one even side allows each rectangle to be easily tiled in an array.

Thus the exterior can always be tiled.

Keeping the orientation the same, we can cut vertical strips of even length and width 1 above or below the two removed squares. This leaves a rectangle with an even width as we removed one square from each side of the original square. This remaining rectangle can be tiled with an array again.


Thus the interior can always be tiled. 

So with both the interior and the exterior of our red rectangle tiled, we have shown it to always be possible to tile the chessboard when one tile of each color is removed.

A sample application of this technique is shown below.





Indirect Reasoning and the Chessboard

By Jonathan Osters, The Blake School

I have always been a fan of the movie The Prestige ( The opening scene of the movie shows how to do a good magic trick. I think that sometimes, good teaching is similar to the outline described in the movie. Dan Meyer uses three acts frequently, and this problem uses three variations of the same problem.

In class this week, we have been studying indirect proof, which is a very challenging topic for beginning Geometry students. We show that a situation must be true by eliminating the chance that it is false. To illustrate this idea, we explored a classic problem in mathematics involving a checkerboard. The framework of the problem is very simple: you have a standard 8×8 checkerboard, and dominoes that cover two squares each.

The Pledge

The first question I ask them is, can you tile the checkerboard with dominoes? It does not take long for the first students to show a solution, and I ask if there are other solutions. Below are several such solutions.

Tiling 1      Tiling 2     Tiling 3

You can see that there are a variety of different solutions to this problem. They all come to the conclusion that there are 32 dominoes to cover 64 squares, and that for the dominoes to cover the board, there must be an even number of squares.

The Turn

The second question I ask is: suppose the opposite corners of the checkerboard were cut away. Would it still be possible? Many emphatically express that it is, since there are still an even number of squares (62). I turn them loose to explore.

Chess 1

Some students insist, after failing the first time, that it can’t be done. I encourage them to continue trying; maybe they just haven’t found the solution yet! After attempting and failing numerous times, I call the class back together and ask if anyone came up with a solution. I contrast their early, frequent, and easy success with the current problem, and give them two choices: (1) A room full of intelligent, capable students failed to solve a solvable problem when they were able to solve a similar problem minutes ago, or (2) the problem is not actually solvable. Students become inclined to believe the latter, and I insist that if this is the case, then we should be able to explain why the problem is not solvable.

I have students generate a list of given information. There are only two main givens: first, is that each domino will cover two squares, and the other is that there are 62 squares. I remind them, only at this point, that the checkerboard has different colored squares, and ask if they can refine the givens with this new information. The given information is finally refined to: (1) each domino covers a white square and a black square, and (2) there are 32 white squares and 30 black squares on this board.

I then have them prove this checkerboard cannot be covered with dominoes.


Assume that the board can be covered with dominoes. Since dominoes cover one black and one white square, n dominoes will cover n black and n white squares. This statement is a contradiction to the given information that there are more white squares than black. Therefore, this board cannot be covered with dominoes. QED

The Prestige

The key to the follow-up is to get the students to drive it with lots of “what if” questions. In class this week, I got the following “what if” questions as follow-up.

“What if you tried this on a 7×7 board, where the opposite corners are different colors?”

“What if, instead of removing a square from two corners, you added a square to two corners?”

“What if you removed a square from each corner instead of just two?”
“What if the squares you removed were different colors but not corner pieces?”

“Will you be able to cover the board if you remove ANY two squares, so long as they are different colors?”

These are absolutely wonderful questions, and get at the heart of what a true mathematician does. They are able to see that the first question, about the 7×7 board, does not work based on the odd number of squares. They come to realize that the adding of two squares creates a problem that reduces to the one we just solved, which is a mathematical technique that requires a very sharp mind.

The last question is the trickiest. It is the most general, to be sure. For instance, the scenario below is one such instance of removing two squares and being able to tile with dominoes.

Tiling 4

As to whether it is always possible to do? Well, I will tell you that my friend and I sat down and we have a proof as to whether or not it is always possible… but a magician never reveals all his secrets. [We will post the proof next week 🙂 ]

A Profound Respect for 1 Million

By Chris Robinson (@isomorphic2crob), The Blake School

How long will it take for your heart to beat ONE MILLION times?

Before you read on, imagine that the ball just dropped, January 1st, 2015, when will your one millionth heart beat occur? Will it be later that day? Some point in January? In July? In 2016? 2017? How long does it really take to reach ONE MILLION heartbeats? Before doing any figuring, take a guess, then after you have committed to that conjecture, go ahead and calculate it.

This question was posed by Marjory Baruch (Syracuse University) to a group of high school math teachers attending PROMYS for Teachers at Boston University this summer. The initial conjectures for this group of math teachers ranged from a couple months to a couple years. All of us were astounded when we calculated the result to be slightly less than two week’s time.

million heartbeat calc

The questions stuck with me as I began to prepare for my year at Blake, and so I decided to ask my class of Algebra 1B ninth graders, my Honors Geometry ninth graders, and my Honors Algebra II tenth graders. Their answers were a much wider range, including a student or two in each class that guessed less than a month. But not too many predicted it to take more than a couple of years. There was very little difference in the guessing from one class to the next leading me to believe from a very small sample size that students ability to estimate the size of large numbers was not necessarily connected with their current success in mathematics. There was however a difference between the teachers and the students.

Why do math teachers have such a profound respect for the size of 1 million? Maybe its related to our salaries…. But I digress.

Here’s Hank Green discussing a very similar idea (How long was it a million seconds ago?) on his YouTube channel.

I also had my students place the number 1 million on a number line from 0 to 1 billion, like the one shown below.

billion number line

Students struggled to see how much larger 1 billion was compared to 1 million, with one student commenting that they always felt like millionaires and billionaires were the essentially the same.

OK, so what’s the big deal (get it? Big.)? I am not sure I have a punch line here, so I will leave you with some observations, and the hope that you find this intriguing as well.

  • For most students, and perhaps most teachers, our initial impression of numbers beyond 10,000 take on the same amount of “bigness” until they are compared to other large numbers.
  • Asking students to reflect on why they guessed what they did is a useful exercise, and having them write about it for a few minutes can help them develop habits of explaining their mathematical reasoning.