Sunday, June 24, 2018

Tribute to a very special woman...

The following was written by one of my daughters. It expresses what each of us feels every day after six years...

A little boy walks by alone on the sidewalk. A woman stops gardening looks up and sees this little boy. She asks the little boy to help her plant seeds in the ground. The little boy puts down his bag and sits next to the woman to help her dig a hole in the ground. He is intent in digging the perfect hole in the ground to fit the seeds. The woman tells him to open his hand and she gives him one seed. He looks at her and asks if he should have more, so it could grow big and strong. The woman looks into the little boy’s dark brown eyes and explains that all you need is one seed for it to grow into the most beautiful, strong, and tall tree that will live a very long life. The boy didn’t believe her. She took the little boy for a walk around the yard and showed him all of the trees that were planted. He asked the woman who planted all these trees. The woman told him that they were each planted by someone special. The little boy looked at all the trees and couldn’t believe how many people she thought was special. This little boy asked if he could come back and visit this tree. The woman said this tree is yours and will always be here for you. The woman showed the little boy how to pat down the dirt to make sure the seed was completely covered and would be protected. Then the woman gave the little boy a watering can to begin the process of bringing the seed to life. She explained that taking care of the seed in the beginning of its life is the most crucial time, so that it grows strong roots. The roots will hold the tree nice and tall and make sure it doesn’t ever fall. She explained that when the sun shines down upon the tree, the tree will feel its warmth and as the branches grow so will the leaves. All of a sudden a little bird flew by and landed right next to where they were planting. The woman told the little boy when the tree grows, she will hang a bird feeder off of it, so the bird can have a place to eat. The little boy questioned what if many birds want to eat. The woman said we can put as many bird feeders on it as we need. Everyday, the woman worked in her garden and everyday the boy would pass and stop to check on his tree. As the tree grew, so did the little boy. Years later, a man was taking a walk and stopped and saw the biggest, strongest, most beautiful tree he has ever seen. The tree was covered with bird feeders hanging all over with bluejays, robins, and squirrels climbing on the branches. The tree had a blanket of sunshine upon it. He looked closer and the tree had a word carved in it. He looked even closer and it said… “Home”. He looked around for the woman but she was nowhere to be found. He put his hand on his tree and felt her spirit, which lifted his heart. As he walked through the garden, he now realized all the trees were different sizes and types, but they all said.. “Home”. It was at this moment, the man realized this was where he belonged all along. He was home. I am proud to say that this woman was my mother and the little boy represents all the children my mom loved and took care of her whole life.. The tree represents stability, strength, and life. My mom would always say it doesn’t take much to love someone if you just open your heart and let the sunshine in. I miss her so much. I try to carry on her spirit, but it is almost an impossible feat since she was superwoman, but all I can do is always be kind and open my heart. I know that would make her proud. I also know that home is where your heart lives. Love you mom today, tomorrow, and always. Use contact form at top of right sidebar to contact me directly. Note: If you'd like detailed solutions and Common Core strategies for my Twitter problems emailed daily, see info at top of sidebar.

Saturday, April 16, 2016

Wednesday, April 13, 2016

Tuesday, April 12, 2016


Also posted on
Let me know how you use/modify this activity for your Algebra and Geometry students! Feel free to comment or send email via Blogger Contact Form in sidebar.

Thursday, July 30, 2015

Balanced Learning is not Blended or Flipped

Haven't been up to one my passionate rants in a long time so buckle up...
Technology has enabled educators to reimagine the  traditional classroom, how students learn and how we facilitate this process, aka teach. Incredible new opportunities to empower students to take more control over their own learning in a "space-time continuum" sort of way. Not to mention providing powerful tools to analyze data to individualize and maximize learning. Are there any buzzwords I omitted!
We have strayed from NCTM'S central message from over 25 years ago:
Technology changes the landscape in a fundamental way but the best source code cannot quite replace the critical dialog and face-to-face interaction that is needed to accomplish the above goal. The spontaneous give-and-take of questions and ideas. Interaction vs interactive...

Oh, Dave, you're so 90s, 80s,70s,... You just don't get it, Dave....
The hexagon/triangle problem in the diagram above can be approached using dynamic software like Geogebra & Desmos. You could develop an extraordinary exploration with carefully crafted questions enabling the student to discover relationships in the figures. I love doing that. I used to do this in the classroom long before it was fashionable. Anyone who follows me knows I'm a techie geek at heart.
BUT I came to realize that there was something missing. If you believe I'm not knowledgeable enough of how these new tools can accomplish the BALANCING I speak of, then challenge my premise! Let the New Math Wars begin...
And I haven't even addressed the myriad of approaches to "solving" this multifaceted geometry problem. Most students/groups will find their own solution paths but it is human nature to CHOOSE THE METHOD THAT FITS YOUR OWN WAY OF THINKING.
To develop the deeper ideas of geometry - symmetry, transformations, dissecting, combining and rearranging pieces of a puzzle, students need to be TAKEN OUT OF THEIR COMFORT ZONE and experience others' ideas and we need to fill in the gaps. That is part of teaching, yes?
And, oh yes, there certainly are algebraic/geometric approaches here with lots of nice formulas like (x²√3)/4...

Tuesday, July 28, 2015

Modeling - What Algebra Looks Like on the New SAT/PSAT and the Common Core

A bear population, P(t), after t yrs,  is modeled by
P(t)=M-k(t-20)², 0≤t≤20.
Initial population:356
Max pop'n:500
Estimated population after 10 yrs?
Answer: 464
Is this the "new" algebra? Students given a function with PARAMETERS which "models" real world data? Questions like this have appeared on SATs for a few years now and, based on the sample new SAT/PSATs released by the College Board, they will become even more common. Students will be asked to analyze the function and use it in application.
The Common Core also emphasizes algebra models - "using" algebra to solve applied problems.
Middle and secondary math educators are not surprised by any of this as these changes have been around for a while but textbooks may need to include even more examples and homework problems of this type.
The real challenge, IMO, is to find that proverbial BALANCE between traditional algebra skills and  applications.
How much knowledge of quadratic functions is needed for this question? Will most students relate the form of the model to f(x) = a(x-h)²+k? Will they immediately recognize that M must be 500 since (20,500) is the vertex or maximum point? Try it and let me know!
Students should be allowed to explore this function using powerful software like Desmos and Geogebra. Sliders in Desmos allow for considerable analysis when parameters like M and k are given. 
BUT they also need to develop a fundamental knowledge of quadratic functions.
A key question for me is:
Should some background be developed BEFORE exploring with technology or AFTER or something in between?
I included a screenshot from Desmos for the bear population problem but this does NOT show how to IMPLEMENT this powerful tool in the classroom. I'll leave that for the real experts like John Golden! (@mathhombre).