Zoom – Promoting Clarity in Communication

Roughly fifty percent of my time as an engineer was spent on communication in one form or another.   Writing documents, engaging in meetings, and giving presentations are an inherent part of the job.   In order to succeed, one has to be able to communicate well.   The same is true in my classroom.


Since experience is a better teacher than telling, I let me students experience the value of communication during the first week of school.    I use the wordless picture book Zoom.   This book shows a series of pictures, each of which zooms out a little bit from the previous page:   you start with the comb of a rooster,  then you see the rooster, then you see some children looking out a window at the rooster, and so on.


I have removed the binding from the book so that the pages are separate.    The pages are such that the front side has a picture and the back side is black.    I laminated the pages to protect them a little bit.

The Activity

Each student is given  a page from the book and told not to show the picture to anyone.   Students are then told that the pages form a book and they must put the book back together in the correct order.   However, they may not show their picture to anyone.   They must figure out the order simply by talking about what they see on their page and by listening to what others say.   When they think they are done, they lay the pages (picture side down) on the floor side by side in order.    When all the pages are on the floor, they reveal their picture and step back to see how they did.

The Debrief

The final stage of the activity is to discuss the activity.    Why did some of the pictures end up in the correct order while others did not?    What worked?    What didn’t?    What was essential for success?   How does that relate to communication in general?


Students learn the importance of detail and clarity in communication.

I learn a lot about my students.   I get to see who the leaders in the group are and to see their leadership style.    I get to see who has great ideas but needs to find a voice.   I get to see who the followers are.   I get to see who the questioners are.    I also get to hear how students think and how they communicate.


Accommodating Imperfection – Proportional Relationships Cards with Multiple Variations For Play

Successful design is not the achievement of perfection but the minimization and accommodation of imperfection. – Henry Petroski

I know that I will not ever design the perfect lesson any more than I will ever create the perfect design as an engineer.  I’m not sure that perfect exists in this world.   While Dr Petroski (a civil engineering and history professor at Duke University) focused his work on failure analysis in an engineering context, the underlying principle he espouses applies to my work as an educator as well.    On any given day, I know that some students will walk away  not having fully mastered the concepts addressed in class.   So, I plan ways to revisit concepts in small chunks of time until everyone does “get it.”  When I plan ways to revisit concepts, I try to create activities that I can use a lot of different ways because I want to be able to use them more than once rather than having to create an endless array of materials.

Proportional Relationships – What I Want Them To Know

I want students to know that proportional relationships are linear and go through the origin.

Proportional Relationships – What I Want Them To Be Able To Do

I want to ensure that students see proportional relationships in tables, graphs, equations, and word problems.   In each representation, I want them to see the constant rate of change and that there is no “start-up” value (the y-intercept is 0).     I want to incorporate Trail’s work (Twice-Exceptional Gifted Children) to support conceptual learners, so I envision a whole-part-whole instructional sequence:    what is a proportional relationship; how do you see it in a table, how do you see it in a graph, how do you see it in an equation, how do you see it in a word problem;   how is the constant rate of change shown in each representation, how is the “no-start up” shown in each representation (compare and contrast these in the different representations).

Proportional Relationship Cards

I created a set of 48 cards.   There are twelve cards for each of the four representations.   The cards can be used separately or together.   That is, I can use just the cards relating to a single representation if I want to focus on that representation.   Alternatively, I can use cards from all four representations if I want students to make connections across representations.   You can download the cards and game instructions by clicking on the link below the photos.

Proportional Relationship Card Sort and Game

Activity 1 – Quiz-Quiz-Trade

Quiz-Quiz-Trade is a Kagan Cooperative Learning Structure.   In this structure, students partner and quiz each other.   Then, they find a new partner and repeat the process.   Marzano’s research (Classroom Instruction That Works) shows that using cooperative learning structures produces gains of 27%    It also shows that incorporating movement increases levels of engagement (The Highly Engaged Classroom).


Activity Two – Give One, Get One

Like Quiz-Quiz-Trade, students work with a partner and quiz each other.   I will use this activity when I want students to make connections across representations.   It incorporates movement and a cooperative learning structure.   It is outlined in Marzano’s The Highly Engaged Classroom.   I have students form two lines facing each other with about three feet in between the lines (the structure does not specify this, but I find it works well this way).   Each student is given a card.   I will give one line a single representation and the other line a different representation.   The cards for partners will be different representations of the same problem.   Each partner will have to find the rate of change and the y-intercept using their card.   When a partner has found them, he or she steps forward.   When both partners are in the middle, they quiz each other on what the rate of change is/how it is shown on their card and on the y-intercept/how it is shown in their card.  When they are done, they step back into the line.   When all the pairs are done,  have one line pass their card to the next person and the other line shift (line one passes the card down one, line two shifts up one).

This is an activity that I do for 5 minutes at the end or start of class.   I end it based on time.   I don’t try to have every student do every problem.

Activity Three – Rummy

Students play in groups of 2 to 4 players.   They use the entire set of 48 cards to match the table, graph, equation, and word problem.


This is an activity I will use so that students compare and contrast the different representations.   Marzano’s research (Classroom Instruction That Works) shows that finding similarities and differences can produce gains of 45%.   His research (The Highly Engaged Classroom) also shows that using a game increases levels of engagement.

I may have students play the game in heterogenous groups as a general review activity.

I may use this as a differentiated instruction activity.  I will have students play the game in groups according to their level of mastery.   Students who have not attained mastery play the game.   Students who have attained mastery play a different game reflective of their own skill gaps.

I will have students play the game for 5-10 minutes at the end of class.   If the game is not over, the player with the most sets wins the game (and a piece of candy)

Activity Four – Card Sort with Three Variations

Students work individually, in pairs, or in table groups to sort the cards.   In the first variation, they work with a single representation and sort them into proportional/not proportional categories.   In the second variation, they work with a mixture of representations to sort them into proportional/not proportional categories.   In the third variation, they work with a mixed set of representations and find the matching cards (same situation represented in a table, graph, equation, word problem).

I may have students work in table groups as a general review activity.

I may have students work with intentional pairing.   In this scenario, I pair a student who is struggling with the concept with a student who has mastered the concept.    As they sort the cards, the discussion is scaffolded for the student who has not yet attained mastery.

I may have students work individually and use this as a formative assessment.





Necessity Is The Mother Of Invention – #ILookLikeAnEngineer

Necessity is the mother of invention.   Unfortunately, the “necessity” can be all too easily forgotten as an essential component in education.    I teach what I teach, in part, out of necessity but it is my necessity not that of my students.   I need to teach the curriculum that I teach because it aligns with the standards set forth by the state but it is not a burning necessity for my students no matter how many times I tell them the essential questions and how they will use it in the future.   Knowing something only becomes a burning necessity in the mind of an eleven year old when they see a need to know it so they can do something they want right now.

So how do we create that need to know?   I think we give kids real problems that they really want to solve.   It’s not something that I can do every day, but I try really hard to find time and space to do it every year.   To do this, I  compact lessons and I accelerate where I can.   This year, I managed to squeeze out almost a month at the end of the year to do an engineering project with my students.

Request For Proposal

Students were presented with a Request For Proposal (RFP) from a fake toy company.    The proposal indicated that this fake toy company was seeking to expand market share to include more girls in their customer base for motorized toys.    The toy company wanted those bidding on the contract to conduct market research and build a toy to meet that need.    The toy company indicated that the toy must meet one of three different criteria:  travel 3 m in 3 s, climb 1 m at a 15 degree slope in 2 s, or climb 1 m at a 30 degree slope.

Creating a Team and Conducting Market Research

Students were assigned teams and formed mini-companies that would bid on the RFP.   They created a team name, logo, and slogan.   Then, they conducted customer surveys with both adults and children in the target age range.    They analyzed the data and determined the type of toy the customer was seeking.

Building Technical Knowledge

ChJmkKSUkAEA9Q-During the same time-frame, students built knowledge of how gear trains work.   They began by building gears on a frame and exploring relationships between the rotations of the gears and the number of teeth on the gears (gear ratios, teeth ratios).   Next, they added a motor and wheels so that they could calculate the rate on a 3 m course and measure the rim force on the wheel.   They repeated this process with gear ratios ranging from 1:3 up to 225:1.   As they did this, they were building important skill in construction as well as an understanding of the different kinds of performance they might expect from different kinds of gear ratios.    From there, they measured rim force on the tooth of a gear connected to the motor.    They did so for different sized gears and then learned how to calculate torque.    With this knowledge, they could explain why certain gear ratios would not move and why certain gear ratios would be well-suited to climbing.   At this point, they had built sufficient knowledge to answer the first stages of that burning question of how to build a toy that would meet each of the criteria.

Making a Prototype

Each team began construction of a basic prototype to meet their desired criteria.   This amounted to attaching the motor and the desired gear train along with the wheels on the frame structure.   Students then tested their motorized frame to see if it met the criteria.   Once they had a basic working prototype, they started constructing a body to give the toy the desired aesthetics.    As they constructed the body, they continued to test the toy to make sure the additional weight did not place them out of compliance with the criteria in the RFP.    They repeated tests multiple times and used median values in order to eliminate outlier trials resulting from poor testing technique.

Sealing the Deal – Writing a Written Proposal and Giving an Oral Presentation


When the toy was completed, each team wrote a written report in response to the RFP and prepared an oral presentation.   The final stage of the project required each team to present their toy to a panel of judges representing the fake toy company.   I recruited 3 engineers and a soon-to-be lawyer to represent both the technical and business interests of the company for the panel of judges.   (I am lucky enough to have Sandia National Laboratories nearby and willing to provide this kind of support to encourage excellence in math and science.)   The judges selected a winning team based on the presentation and a demonstration of the toy.   (The winning team members each got a gift card to Cold Stone Creamery).

While this last stage is not “math”, it is very much a part of what engineers do and I wanted my students to appreciate the importance of being able to communicate effectively as an engineer.  Reading, writing, and speaking are just as much essential skills for an engineer as are math and science mastery

Why It Mattered

  • Students got to experience the engineering process, which is so much more powerful than hearing about it.
  • Girls had to learn how to make something and how to make it work.   It’s not that they are any less adept, but many of them are much less experienced.   This results in a certain amount of hesitancy, initially,   Having to make it work pushes them past this hesitancy and they discover just how good they are at it.    Giving girls this experience and confidence is important in leveling the playing field when it comes to engineering.
  • Students used the math that they have learned this year to do something real that mattered to them (finding unit rates, conducting surveys, making data representations, analyzing data to make decisions, finding medians, using equations to calculate torque, measuring radii).
  • Students had to find ways to work together – teams could not shift part way through the month long project.
  • Students who lacked confidence as speakers learned that public speaking is a learned skill and that you get better at it with practice.   (I made each team do a dry run of their presentation in front of their classmates and get feedback the day before the final presentations.  They took the feedback and were so much better the second day.)

Gallery of Toys


It’s All Greek To Me – Managing Cooperative Learning

Group project.   Words that would make my sixteen year old self silently scream.  Yet again, I was going to have to do 95% of the work and three other people were going to just go along for the ride.   That was my best case scenario.   Worst case scenario, I was going to have to undo/redo their work so that I would get the A I wanted.   I was definitely not a fan.

Fast forward past college, graduate school, and years of working as an engineer (sometimes still not a fan of the whole group work thing but recognizing it was a reality with which I had to live) to my graduate licensure classes.  Naturally, the topic of cooperative learning was addressed.   The voice in my head was grumbling “Great.   New name, same old story.   No way am I doing this to my students.”    I firmly pushed the whole idea aside and focused on the important thing: math.

A funny thing happened, though.

As I focused on math, I discovered the importance of mathematical discourse.   If students were going to have discourse, they had to sit in groups, so I arranged my room accordingly.   It kind of worked, but it still seemed like the higher functioning students were doing a disproportionate amount of the thinking and talking and the lower functioning students were sort of “along for the ride.”     Not fair.    Not equal.   Not good.   I stuck with it, but was not completely happy.

About that time, a friend introduced me to Kagan Cooperative Learning Structures.   She had PD on them in another state and shared some of the structures.   I decided to try a couple of them out.   I started with Numbered Heads Together.   In it, each student works independently on a problem.   When he or she has solved it, he or she stands up.   When the whole group is standing, they discuss their thinking.    When everyone is in agreement, they sit down.   A student is called upon at random to speak for the group.   This resonated with me on a lot of levels.   Each student had to work through the problem.   The structure provided think time, no one rushing any one.  Discourse was embedded in the structure.   There was mutual accountability, no one knew who would speak for the group so everyone made sure everyone understood the problem.   This could work.

I am still not a fan of group work, but I use Cooperative Learning all the time.     It has been great.

These days, I still have the desks arranged in groups of four.


I use colored index cards to put labels on each desk in the group.   Since I teach math, I use Greek letters that students see in mathematics (Delta, Sigma, Epsilon, and Pi) for the different tags.  (I make all of the Deltas one color, all of the Sigmas another color and so on).    When I call upon someone to speak for the group, I randomly select a seat position to speak  (think pulling a stick with Delta, Sigma, Epsilon, or Pi).


When I arrange the desks, I put all of the Deltas in the same seat position within each group, all the Sigmas in the same seat position, and so on.   When I make my seating chart, I am intentional.   I place stronger students in the diagonals at the table group.   Then, I fill in the students who need more support between them.   That way, they have a strong partner to scaffold the discussion if needed.

I also use a task chart to assign responsibilities to each member of the group.








It’s Really Not All About the Rules – Creating a Collaborative Community

Growing up, my parents weren’t really grade-obsessed.   It was all about effort.  They always said a C would be fine if I had done my best but a B wouldn’t be if they thought I hadn’t.   I never really tested them on it, but I am fairly certain that they would have held true to that.   Behavior, on the other hand, was an entirely different matter.   I knew that if I got into trouble at school, I could count on getting twice as much trouble once I got home.  Maybe this is why I’ve always been a rule follower.   I guess it is an inherent part of my being.   I don’t know whether there is something genetic about it or if it is the way that I was raised.

Whatever the case may be, I started teaching with a mindset that there really was no excuse for bad behavior.   I created a set of rules that I was pretty sure covered everything I wanted without being too overwhelming for students to remember.   I posted them on the wall.   I intentionally taught them at the start of the year.    I enforced the rules when necessary.


A funny thing happened, though.   As I worked to build a collaborative classroom environment where students engage in meaningful discourse about mathematics, I started thinking more about norms.    I spent a lot of time thinking about how I wanted kids to act and how I wanted them to interact.   After a lot of thought, I came up with a set of norms.

  • We are a community.   Everyone has to contribute.
  • Everyone has good ideas to share.
  • Treat others the way you want to be treated.
  •  Everyone makes mistakes.   That is how we learn.
  • Anything worthwhile is hard work.   It takes lots and lots of practice.

I had one of my student aides create a poster with the norms.   I posted the poster at the front of the room.   I intentionally taught the norms.

It’s been about five years since I started using these norms.   I still have the class rules posted, but the only time I ever talk about them or do anything with them is the first day of school.   Instead of enforcing rules, I find myself focusing a lot more on norms.  On those rare occasions when things go awry, I find myself reminding students of norms.   The funny thing is, this works so much better than rules ever did.   So, for this rule- follower, it’s all about the norms.