Category Archives: Standards for Mathematical Practices

Better Questions Week 3 #MTBoS



I’ve pondered this challenge for a couple of days.  So many options!  But a tweet from @mathymeg07 led me to a post from @MrAKHaines blog Math Pun Pending.

The post was celebrating a variety of strategies his students had use to answer the question:



He wrote:  When I wrote the question, I had anticipated that students would use a couple of different strategies. What I didn’t know was that my 25 students would use a combined seven correct solution strategies to solve this problem.

Two parts to my post:  1. How can I make this an open question and 2. How can I use student samples to develop a better lesson in the future?

How can I make this an open question?

A.  Name a point that is NOT on this line.

B.  Name a point that this line passes through.

Thanks to @PIspeak‘s TMC14 session in Jenks, I urge students to “Support your claim with evidence/reasoning.  I want to see your thinking!”

How can I use student samples to develop a better lesson in the future for my classroom?

I appreciated the fact that he never explicitly taught “the teacher’s efficient strategy” but allowed group discussions and support to drive the lesson.  Students shared ideas.  The last paragraph  in his post says, “My students are acting like mathematicians, y’all. They’re using their toolkit of math ideas to solve problems flexibly. I couldn’t be happier.”

In the end, that’s what we all want – students thinking on their own, making sense and being confident enough to explore a problem with their own ideas.  So, how does this tie in with the Better Questions prompt?  My outline of the lesson feels a bit like those I’ve used from Formative Assessment Lessons, but I feel it lends itself to students doing the thinking, talking – I only provide the materials and support to make desired connections that will lead to the learning goal.

I’ve been following the #T3Learns chat from Wiliam’s book.  In chapter 3 of Embedding Formative Assessment, it suggests using student sample work. How might I structure a lesson, utilizing student samples of this question?  In Principles to Actions, MTP3 states Effective teaching engages students in making connections among mathematical representations to deepen understanding of mathematics concepts and procedures and as tools for problem solving.

  1. Begin with the same question. Allow students to write a response. (3-5 min)
  2. Place students into small groups and allow them to share their approaches. (5-7)
  3. Allow groups to explore student samples, making note of different ideas, what they like/ways to improve, questions they’d like to ask the student. *maybe as a gallery walk? (15-20)
  4. Discuss their findings as a whole group. (10 in)
  5. Connections to/between the different mathematical representations. (5-10 min)
  6. Written reflection:  (3-5 min. possibly use as a start up / bell ringer to begin class with following day – providing an opportunity for retrieval of previous days information?)
    • my strategy was most like:____
    • the strategy I liked most was ____ because ___
    • the strategy I found most difficult to understand was ___ because ___
    • Which approach was most efficient?  Why?
    • What do you think was the BIG IDEA your teacher intended for you to learn/understand?
  7. Transfer…provide a few, different contextual problems that allow students to connect the mathematics to something tangible, maybe in a problem posing situation (should this be small group?  individual? ) (5-15, would this be better as follow-up the following day?)

Timing is often an issue for me.  I want to provide students with enough time to make sense/discuss, but not so much time it feels long and drawn out.  Are the times I have listed appropriate?

Please offer suggestions.  How have you used a similar approach successfully in your own classroom?

Card Tossing & Spiraling Curriculum #tmc14


Awesome session Mary and Alex!  Thank you. Thank you. Thank you.

The session focused on their experiences with Grade 10 Applied students ( Canada).  The entire course is activity based which allows students to not miss out on big ideas as they would in a traditional unit by unit aligned course.
Students have repeated opportunities to experience big ideas. The tasks are rich  with multiple entry points and different approaches to solving.  It’s a collaborative environment with accountable talk.  There are fewer disciplinary issues with increased engagement.

Each 6 weeks a mini – exam over entire course up to that point takes place.  Questions are in context and tied to activities they have completed.

We began with beads and pennies on our desks and this task… Cole has 2 smarties and 3 juju bed for $.18 while Noah has 4 smarties and 2 juju be for $.20.  They shared that systems are presented this way – no algebraic forms- for the first several weeks of class.  I, personally, can see how effective this strategy could be.

The next activity shared was Sum of Squares (he doesn’t refer to it as Pythagoras Theorem, yet – or did he say ever?)

Students are asked to cut all squares from side length 1 to side length 26.  Each square is labeled with side length, perimeter, area.  Then they build with them.

Basically students explore and eventually they focus on triangles formed with question, are there 3 you cannot make a triangle with?   Which combinations form different types of triangles. Begin looking at 3-4-5 triangle families, similar triangles (Kate suggested dilations here), discuss opposite side and adjacent sides, then give them a TRIG table and allow them to figure it out.

Compare side lengths with perimeter, or side length with areas.  The possibilities of math concepts are endless.
We ended the day with Card Tossing by collecting data, then using rates to make some predictions.

Video of Alex & Nathan picture below is only a screenshot.


@AlexOverwijk downed by @nathankraft 75 to 72

Each person in the room completed several trials of tossing our cards for 20 seconds.  We found our average rate of success, then determined who we thought might beat King Card Tosser.

Alex asked us to predict how long they needed to toss if he gave Nathan a 35 (?) card advantage so it would be super close and exciting.  Our prediction 38 seconds about 75 cards. Many ways of making the predictions were possible. Not to shabby, huh?

This task was fun, exciting, engaging.  Definitely on the to-do list.

This approach is definitely something I would like to consider, if administration will allow it!

Powerful Problem Solving Winter/Spring Chat #ppschat


Here are links to the Storify for our chats:

Chapter 1 1/22/14 Introduction

Chapter 2 1/29/14  Communication & Community

Chapter 3 2/5/14  Learning Through Listening

Chapter 4 2/12/14 Noticing and Wondering

Chapter 5 2/19/14 Changing Representation:  Seeing the Big Picture

Chapter 6 2/26/14 Engaging Students’ Number Sense with Guessing

Chapter 7  3/5/14 Getting Organized

Chapter 8 3/12/14  Generalizing, Abstracting, and Modeling

Oops!  I somehow waited too long – having trouble rounding up these tweets for Chapter 8!

Chapter 9 3/19/14  Looking for Structure

Chapter 10 3/26/14  The Problem-Solving Process & Metacognition

Chapter 11 4/2/14  Reflecting, Revising, Justifying, and Extending

Chapter 12 4/9/14