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

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?

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.

Zombies and number talks....

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....no 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

Happy and 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 is sad because, following the process above, the pattern is:

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?

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
• BEYOND!

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.