Friday, December 14, 2018

Triangle in a circle

Show this first animation and ask, what do you notice?  What do you wonder?

Then ask the following:  If the diameter of the circle is 8 cm.  What (or when) is the largest area?  What are the dimensions of this triangle? 

After students solve that pose, When is the area exactly half the largest?  What are the dimensions of that triangle?  What is the angle?

After some work time and playing around with the above problem, show them this animated version

Wednesday, December 5, 2018

Trig Ratios animated

Two different triangles are animated.  Show your class this animation and simply ask:
  • What do you notice?
  • What do you wonder?

After some conversations show them this one, with the side lengths being shown and ask again:

  • What do you notice?
  • What do you wonder?

Lastly, show them this image and ask them "How could we analyze this?"

  Some possible routes:

  • Prove the sides hold true for Pythagorean theorem
  • What angle(s) would be in the triangle?
  • How are these 2 triangles similar?  How are they different?

Wednesday, November 28, 2018

Area and perimeter of a triangle

Another way to possibly introduce, or reinforce, the area of a triangle and perimeter could be to show this gif:

Ask, "What do you notice?  What do you wonder?".  After some discussion pose, if not already asked, the following questions:

  • When is the area the least/greatest?
  • When is the perimeter the least/greatest?

After some discussion show this gif.

Tuesday, September 4, 2018

Zombies and number talks....

Image result for the walking dead our world

Recently, during a walk I was playing The Walking Dead: The Zombies, and an awesome mathematical experience occurred.  

I noticed that my team had slaughtered 497 zombies out of the 800 required to complete a challenge.  Sadly, I commented to my friend, after killing 3 zombies, "Well at least we have 500....out of 800!", to which she asked "What percentage is that?"

I smiled, and replied "How would we do that?"

As she grabbed for her phone, I asked her to humor me and try it mentally....and as such began a wonderful minute of beautiful number play:

"Well I would make it 5 divided by 8.....then...."  

At this moment she begins to draw out 5 under a divisor sign and an imaginary 8 to the left of it, and continues with...

"Ok 8 goes into 5....oh...this doesn't work....Why can't I just use my phone?"

After smiling, I asked her, "And how else might you say '5 divided by 8'?"

Revisiting middle school in her head, she exclaimed, "5 eighths. but that does not help"

"Yeah eights are a pain, is there an easier grouping we could from eighths?"

Seconds pass, and she admits she does not understand my question.  I offer assistance with,

"Ok, eights is the denomination we are working with, but are tricky, so think about this....If I gave you unlimited eighths....what other fractions could you make?"

Eyebrow touch for a moment, and then she replies with "I could make quarters with 2 of them...."

At this point, I could not help from smiling as I knew to continue the journey of numerical delight all I had to do was ask, "So how many quarters is 5 eighths?" and so I did.

Tilting her head to the side, I could almost see the imaginary manipulatives going through her mind as she talked aloud, "Well 2 eighths is 1 quarter, and so 4 eighths is 2 quarters, and then 6...oh....2 and half eighths?"

I decided to ignore my Grade 6 teacher's rule of "NO DECIMALS IN FRACTIONS" (Which has never made sense!), and simply say "Hmmm...Interesting so what percentage would be 2 and half quarters?"

She had already started.."2 quarters is 50...half a quarter is... 12.5.....and so 50 plus 12.5 is 60....62.5 percent?"

I cheered, awarded her with a high 5, and asked "If it took us 2 days to do 5 eights and we have only 1 day left, are we going to make it?", to which he exclaimed "I don't love zombies enough..."

As I reflect on this experience, this is what I want my math classes to feel like for my own daughters.  Moments of exploration, followed with choices from different mathematical tools, and ended with delight as they solve problems that make sense.  

PS:  This would have lost all of its joy if I simply entertained the idea that dividing 5 by 8 mentally, is best done with the traditional algorithm

Monday, August 27, 2018

Happy and Sad Numbers

Image result for happy sad numbers
A happy number is defined by the following process:

Starting with any positive integer, replace the number by the sum of the squares of its digits in base-ten, and repeat the process until the number either equals 1 (where it will stay), or it loops endlessly in a cycle that does not include 1. Those numbers for which this process ends in 1 are happy numbers, while those that do not end in 1 are unhappy numbers (or sad numbers).[1]
For example: 19 is happy because:
4 lines . Line 1,  19 long right-arrow 1 squared plus 9 squared equals 82 . Line 2,  82 long right-arrow 8 squared plus 2 squared equals 68 . Line 3,  68 long right-arrow 6 squared plus 8 squared equals 100 . Line 4,  100 long right-arrow 1 squared plus 0 squared plus 0 squared equals 1

4 is sad because, following the process above, the pattern is:

4 long right-arrow 16 long right-arrow 37 long right-arrow 58 long right-arrow 89 long right-arrow 145 long right-arrow 42 long right-arrow 20 long right-arrow 4 long right-arrow period period period

After introducing this to the class some questions you may ask could be:
  • What other numbers are happy? Sad? 
  • Are there any certain numbers that are easy to determine if they are happy/sad? How do you know? 
  • Would there be an infinite or finite number of happy numbers? 
  • True or false: If a number is happy (sad), then all numbers of its sequence is happy (sad). 

Further Extensions:

  • A happy prime is a number that is both happy and prime. Determine any happy prime numbers. What would be common among all happy primes?
  • Change the base (from 10 to 2, or any number) are the numbers still happy/sad, or does it change how the "feeling" of the number? 
  • Try cubing the digits instead of squaring. Are the numbers still happy/sad, or does the number change "feelings"? 
  • Computer science challenge: Could you write a code or algorithm that determines if the number is happy or sad?

Monday, June 18, 2018

Dice Chats

Another way to bring conversations alive in your classroom is with "Dice Chats"!!  These are similar to Dan Finkel's Unit Chats.

Simply show these images and ask your students "How many do you see?"

The students could:
  • Count the total dots
  • Number of dice
  • Number of colors
  • Number of dots on a certain dice
  • Number of dots on a certain part of the pattern

I would provide students white boards (or have them at non permanent table tops) and have them write down how they know.

After 2 or 3 minutes, simply start the conversations and reply to every answer "How do you know?".

Remember, during this time try to ask more questions than give answers.

Here is a slideshow showing different possible pictures.

Wednesday, May 16, 2018

Differentiating Assessment

“The primary purpose of assessment is to improve student learning” -Anne Davies, PHD.
The above quote sums up why I challenged my assessment practices. In 2013, as a High School Math teacher, here is the model I used:
  • Teach concept 
  • Various quizzes during the learning journey.
  • Summative unit exam.
  • Repeat 
After 8 years of teaching in this model, I realized there was an issue...students who entered my class with passion towards mathematics, were leaving the class beaten down, sometimes dropping out, and ultimately not having the desire to learn more math.

Originally, I thought this was normal!! When I went to high school, math classes always ended with less students than it started with. As a student, I remember daily expectations of having to do the odd (or even) numbered questions on page X, multiple worksheets, and having to prepare for weekly quizzes or tests. This was my normality. This was the machine I wanted to perpetuate when I entered teaching.


Because this worked for me. I am intrinsically motivated by mathematics, and I find prime, fibonacci, and complex numbers inherently interesting...because they are!! However, too many people have not had the chance to struggle, discover, and play with these awesome, and other, mathematical ideas.

In 2013, as an educator, I saw the true assessment style was more about ranking, sorting and grading, not at all about learning. Even furthermore, I was more focused on preparing students for AP, or diploma exams, instead of creating an environment which allowed students to bring their passions and interests in, next to their pencils and paper.

Similar to the comic, my grades were focused on what was easy to test, grade, and report on, instead of what was important.

This had to change. If I was differentiating my instruction, why was I focusing on standardizing my assessment?

In 2013, I made a stand:
I will only assess in a way that increases learning; if my assessment isn’t increasing learning then the assessment needs to change.
In this year, my late friend Joe Bower reminded me “the word ‘assessment’ comes from the latin word ‘Assidere’: to sit beside”; an action that was rarely taken when I was assessing my students.

Here is my journey, and the steps I took, to explore what it mean to provide differentiated assessment.

1. Manageable outcomes:
In consultation with University professors, colleagues, and teachers across the province, I looked at every course outcome through the “Rock, Sand, Water” analogy: If you plan for the rocks first, then sand and then water, it will all fit, however if you simply plan a course to cover all outcomes equally, all the outcomes will rarely fit.

During this process I combined parts of one outcome with another, broke up some outcomes into smaller chunks, and then I created a list of “Rock, Sand and Water” objectives:

  • Rock outcomes (outcomes that pass the endurance, leverage and readiness test) - Expect ALL students to master. 
  • Sand outcomes - Expect MOST students to master. 
  • Water outcomes - Expect SOME of my students to master. 
I then ensured that these decisions reflected in my long range outcomes, course outlines, and daily plans. I planned my courses in a way to ensure that the essential learning outcomes were weaved throughout the entire year, while less essential outcomes were covered through the lens of a higher leverage outcome. Of course, I still taught all the outcomes, but I decided to only report on the essential ones, regardless of how difficult it might be to do so.

2. Change the tests
After having a smaller list of outcomes to report on, I decided to ensure my summative assessments matched this philosophy. Instead of giving tests grouped by question type, I grouped questions based on outcome. Any assessment that covered more than 1 outcome would be given back to students with more than 1 grade. Each grade represented the learning of the student on a specific outcome; no longer did I average 2 or 3 outcomes into one mark and call it “Unit X Test”. I then changed the categories on the online reporting program to “outcomes” instead of “Quizzes, tests, homework, etc”. Every mark, on a specific outcome, was reported in the corresponding “outcome” category.

3. Ensure learning is the focus on every assessment
During this time, my summative assessments where 1 part multiple choice, 1 part numerical response and 1 part written. Simply sorted by outcomes not by question type.

I quickly realized that when my students answered a multiple choice question wrong (or even when they guessed right) I was clueless as to how to support them from their current understanding to mastery. If I wanted learning to be the primary focus, I could not administer multiple choice exams.

In 2014-2015 I moved to an entire written response assessment strategy grouped by outcomes. Instead of 1 part multiple choice, 1 part numerical response, and 1 part written, I only assessed with questions that forced students to make their understanding visible. It was in this year, I truly started to sit next to my students and provide them written and verbal feedback that pushed their learning forward instead of simply saying “here are the X questions you answered incorrectly and here are the correct answers”. My feedback was focused on learning, not on which questions did they answer wrong.

I was writing grades and comments on everything my students handed in. This was the inherent problem; I was giving both grades and comments.

Every time I handed back an assessment with a mark, I quickly noticed that students focused on their individual grade, their friend’s grade; and how they ranked within their peer group; and most completely ignored the comments. Students were not asking “How do I understand this better?” but instead “How do I get an A (or 90% or excellent)?”

I had invested a lot of my time into giving useful and effective feedback however these comments were being overshadowed by mark. Grades were the commodity of my classroom not learning. This had to change.

4. Challenge the grading system

Simply put, I stopped writing grades, learning levels, or any other ranking system on student work. Instead I only provided feedback and asked questions that pushed learning forward. Even if a student demonstrated “mastery” of an outcome, I would still provide feedback or leave them with a question that pushed them beyond the scope of the outcome.

This was the most profound transformation I have ever experienced in my entire career.

Students truly became engaged in their learning, not their grade (or ranking). As well, I was able to truly push my students forward when they made mistakes. When I looked at the work of my students, I simply focused on 3 essential questions:

  • What does success on the essential learning outcome (rock outcome) look like? 
  • Where is the student now? 
  • How do we close gap? 

This ensured that the feedback I was providing to students was truly “learning focused”. Every written comment was also input into our online reporting system.

This meant that when a parent, or a student, logged into our online reporting tool, they didn’t see grades, but instead comments for any outcome. Instead of seeing “80%” on an outcome, parents (and students) would see what they need to work on to close the gap...without the support of a grade.

Even the conversations I was having with parents were learning focused and not grade focused. Incredible shift! At the beginning of the year, parents were apprehensive of not receiving any marks as feedback, however when parents saw the products their children were bringing home from my courses, parents were quick to become allies of this new model of assessment.

5. Differentiate the assessments
I finally started a “Differentiated Assessment” model. After teaching outcome X, I would have an assessment on the outcome, however, I started to use this time to also assess each students’ understanding on a previous outcome. For example, students who were also being retaught ideas on outcome 2 would have questions around outcome 2 on their sheet, the ones who were working on outcome 3 would see questions around this outcome...truly every single assessment was tiered to the individual student and what he/she has been working on in the previous weeks.

The assessment Jimmy and Jane received on this day, would only match if they were working on identical material with identical errors and misconceptions; which was rarely the case. Even when designing the questions from previous outcomes, the questions were focused around the feedback the learner had received on their last assessment. For example:
  • Jane might be tasked to demonstrate understanding of a specific part of a certain essential outcome, because I saw only a minor learning gap when I previously assessed her on this outcome. 
  • Jimmy, however, might have more questions around the same essential outcome, because when I assessed him previously I saw major learning gaps. 
This is when learning became the focus of every single assessment I gave. I can honestly say that every assessment had learning as the only priority!

Looking back, I have always believed that every child can learn math to the highest levels, but only in the past 3 years did I take a differentiated approach to what happens when they don’t.

Monday, January 29, 2018

Folding Paper Activity

How high will a piece of paper get if you fold it 3 times in half? 7? 10? 20? How long until you get to space??

Check it out digitally with "Folding Paper digitally"  Check out possible uses below

Early grades: Ask the question I did, wonder, try it, wonder more, and then to the site...wonder before you click each time.

Middle grades:  Ask the question I did, wonder, try it, wonder more, see if you can create an actual equation for the height vs times folded..and then to the site...wonder before you click each time.  Ask then, how big would the original sheet have to be to be able to stand on the stack after X folds?

High Grades:  Ask the question I did, wonder, try, wonder more, ask why type of relation would be best to model this type of growth?  How would you change the question so it grew as linear, quadratic (would be fun to explore this), sinusoidal, etc, functions? and then to the site...wonder before you click each time...Ask then, how big would the original sheet have to be to be able to stand on the stack after X folds?