Category Archives: High School Math

Indirect Measurement #mtbos #made4math Monday!


In reading Building Thinking Classrooms and considering how to defront my classroom, I’ve found I need to purge lots of stuff! Getting ready for year 27, well, that means I’ve got a lot of stuff!

Several shoe boxes, most labeled correctly with what they contain, but a few are in disarray. I have run across many items from early years of having little to no money to spend, but I have always said, these are the times you are most resourceful and creative.

Homemade Hypsometer

I always said Geomtey was a favorite to teach because there are so many hands on tasks you can implement. I feel like this task likely came from a lesson in Discovering Geometry, Michael Serra.

I remember using one with a protractor attached in later lessons. And also having a mirror students placed on the ground to sight the top of their object to measure, and then shadows… we would use multiple approaches to measure then compare our results.

Even way back when it was obvious to me to allow students a time to reflect. Which tool was easiest, most challenging to use? Within their groups, they would discuss which meathod(s) seemed most accurate.

I always remember during these tasks students struggling with setting up their proportions. Reasong if their answer made sense within the context. It was a great opportunity to allow them to figure out corresponding parts as they would sketch the scenario on their recording sheets.

I loved these measurement tasks, then later using the protractor to make connections back to our previous indirect measurement tools. It gave them prior knowledge to have a foundation as we began a new way to measure with angles.

Cold Temps, Thermometers and a Vertical Number Line


Sara’s tweet reminded me of my hand made thermometers my first years of teaching. Oh goodness. It has been a bit since I thought about them.

I printed, literally used a ruler and drew a vertical number line, labeling with integer values, made copies, cut them into strips. Since laminating was not as readily available 25 years ago, I used wide packing tape to laminate the strips.

Lame example, but you get the idea.

Next, I hole punched at the top and bottom of the number line. I tied together a piece of red yarn and white yarn. Threaded lose ends through each hole and tied it as tightly as possible in the back.

Students could slide the knot up or down depending on the representation or operation. Even today, I still keep a physical vertical number line up in my classroom and kids will tell you it helps them with those signed numbers.

To the right of the door…

Shortly after returning to the classroom in 2001, I created this number line with calendar number cards, adding negatives to half of them. It has followed me from room to room to room ever since.

Kids will sketch them in the margins of their notes. I have a template we tape into our INBs for at home use. I’ve never really dug into the why or looked for research of vertical vs. horizontal. I just observe with my students they are more confident in their work. I’m not sure how it helps or alleviates space with one of their memories. I just see it works.

Thanks for the reminder, Sara. I thought about that old classroom and wondered about several of those kids, who are now in or nearing their 40s. 😃

Help! Creating a System for Individual Feedback #mtbos


If you don’t want to read – just watch the video of me asking you for ideas~~~ trying to give some individual feedback.

I muddled through the beginning of the semester -it ranged from writing on digital documents with my classroom smart board, to screencasting help,to opening a help MEET during classtime – and pulling students in one at a time.

I feel that I’ve gotten a start – a screen cast – very similar to feedback video on Flipgrid. I post a link to their feedback in Google Classroom as a question or assignment, only posting to that individual. I just find talking through their work as I am looking at it is fairly efficient. I have learned to include their name in the post to classroom.

When they respond, either by replying to the post, submitting the written task or emailing me – and I reply back, we set up time for more help / retakes. But once we are finished, how do I mark those assignments as “done” and move them out of the way?

I’m not sure that I want to delete them – I feel that I need to archive them somehow – but is that through having a new topic “Individual Feedback – Completed”?

This is strictly individual, formative work that leads to learning or developed skill – hopefully. It is ungraded, but can earn you the opportunity to retake an assessment after completing a reflection and/or more practice, showing growth.

My process – I open the quiz, their submitted work – Look for common errors/mistakes, misconceptions – just have a short convo with them in the screen cast explaining what I see/notice.

I give them an option to do more practice / reflect. When they have completed that, they notify me (submit evidence or work, reply to question or email).

After we have completed that cycle – whatever it may look like, I want to shuffle completed tasks out of the Individual Feedback topic/tab so that only what is current appears there. Does that make sense? Suggestions, please?

Growing Toys #mtbosblaugust Post 12


Many years ago, we used some smaller versions of these toys. Somewhere in the cave of files, I’ve got a lesson someone shared with me. They used alligators and the plastic box was referred to as the Gator Garage. Anyone have recollection of that?

Virtual Class Pets?

So I wonder if I just share a picture with students, get some input from them…quantities we could measure, relationships we might expect -will it be linear or otherwise, make predictions.

Will cola make it grow differently than water? Will it really reach 600%? How big would it have to g we t for that to be true?

We could collect data daily. Create a visual model, description the relationships and attempt an algebraic model.

Just jotting some thoughts down so I have them stored outside my brain. Any input or suggestions would be awesome and appreciated!

Interesting Ways for Students to Be Right & Wrong #MTBoSBlaugust Post 11


I have looked literally ALL summer for notes I took from Eli Luberoff’s NCSM Session on March 30. I have thought about it numerous times and because I took notes on loose leaf as opposed to in a notebook, the loose leaf ended up in a stack of unrelated papers for some reason. I usually try to start a notebook for my summer learning and into the next school year. I had not started one yet – I mean, we would be back in school right after spring break, right?

I wanted to find those notes though – because all that he said really resonated with me. As we did the task that day and he shared – I truly realized the benefit of how they plan the Desmos tasks. I knew this was a structure I wanted to follow to benefit my students. The task was Turtle Races (title is actually Eli NCSM20, Code: F2TX97 – if its still open).

Warm-up: Tell a Story.

Benefit #1

After watching the Turtle Race clip, we were asked to tell a story. By lowering the entry point, it will let more students ENTER the door. Start with tell a story – this allows underrepresented students to join-in.

Next we were asked to create our own Turtle Race by sketching graphs for up to four turtles, using a different color for each turtle on the graph.

Create your own Turtle Race.

Then we could click play and the clip modeled our graphs on the race track.

Benefit #2

Have students to create their own by changing the shapes/slopes, shifting shapes with transformation, etc. So often math is about getting information created by someone else – but we all know when the actual work is student created in classtime, it becomes more meaningful students become the thinkers and creators, not just consumers of mathematics. Yes!!!

We spent some time dragging points and creating shapes, then observed and described the shape shifters and how they affected our shapes. All of the students had their own shapes, but the transformations were the same.

He shared a cool math picture and then a Paiget quote – but I did not get either saved/written down. But went on to say Mathematics is diverse. Mathematicians are diverse. Math is still alive.

Our final task began with students dragging points to create a parabola that would pass through the red gate. Our first challenge was to create an equation that make a parabola that would pass through a different set of gates. Then press TRY IT. What I have loved about watching students in Desmos through the years – they are okay with getting a wrong answer – when they know they can try again. And most of the time, they do not give up. Or when they get stuck, they have a conversation with someone… perseverance and communication.

Benefit #3

These types of problems, like Parabola Slalom have built in differentiation.

Students get to “try it” without penalty. You can change the challenges to have them use 1) more points, 2) try equations, 3) try with more gates. Looking at several student examples:

  • y = -x2+3
  • y = -5x2+3
  • y = -.25x2+3
  • y = -.2x2+3
  • y = -0.02x2+1
  • y = -(x+4)(x-4) – 10.5
  • y=-(x2-16)-10.5
  • y = -.4(x-1)2+5

What do we notice? What do we wonder?

The final tasks was asking people to build their own challenges. Again, the built in differentiation continues. John Merrow- typically, students work on different tasks, but at what cost? This types of task allows students to join at their comfort level – “the same task, but at different depths.” I love this idea.

Again, Eli sharing that day was a big a-ha for me. Things I’ve done on occasion, not necessarily together with intention – that can truly benefit all learners.

Always, Sometimes, Never: Getting to Know Your Students #mtbosblaugust


I think my first true memory of Always, Sometimes, Never was a second semester Geometry course during undergraduate. My professor’s exams ALWAYS had true false and you were asked if it were always, sometimes or never true. If sometimes, you explain when and if it were never, you rewrote the statement to make it always true.

As a teacher, I remember experiencing these from the Discovering Geometry book. But was fully aware of using them… Was it Making Thinking Visible? More Good Questions? 75 Formative Assessment Classroom Techniques? or the FALs from Mathshell? Uhm. Likely all of the above.

Always, Sometimes, Never #75facts

As an anticipation guide prior to lesson or unit

As a carousel activity to share thinking.

But these tweet from @MrsSheenanMath and the thread to follow it…

… made me want to jot down shared ideas – file them in a doc or keep so I can quickly add to as I see or think of more. I love the idea to use non-math ASN to get to know students, what they are thinking, their experiences. I created a doc where we can share ideas, build a list! Please add to it!

Always Sometimes Never, NON Math

Overlooking the Needs of Students Who Are Significantly Behind: (Un)Productive Practice #MTBoSBlaugust


While watching the webinar last spring, this one the one that caught my attention most. You can watch here – Juli K. Dixon addresses this one around the 45:00 min. mark. I cannot find any blog posts on the site, but I am sure this is one she will be sharing more about.

I appreciate her acknowledging that teachers are well-intentioned feeling that if they work hard enough, they can catch students up. However, teachers will burnout and students will fall further behind.

What are we doing to support these students? With RTI and Multi-Tier Support Systems, we often focus on basic facts – this is not the best used of our time, not the most important. Juli shares, students with extended time will get those facts. We we want to reteach everything – we simply cannot do it. She suggests focusing on strategies that will extend beyond the basic facts, strategies that can be used to figure things out as opposed to “just knowing.”

Instead, she encourages us to look at our current year – and focus on prerequisites and teach them for understanding. This is where our energy and time can make a difference. For example – figure out the 3 most important topics, consider what is prerequisite here asking what do we need to be able to do to get there? Here is a slide with examples she shared.

As I rewatched this segment, I was reminded of ideas shared in our Routines for Reasoning chat last spring. We challenged each other to revisit the last chapter in August to choose 1 routine we would consider using in our classes as we started back. I committed to starting with Capturing Quanitites during the first few weeks of our classes. I feel that this would definitely be a supporting strategy for learners who are significantly behind.

Capturing Quantities supports Mathematical Practice 2 by focusing students io consider the important quantities and relationships in problem situations, we are helping them develop their ability to reason quantitatively and abstractly.

The info-graphic from Fostering Math Practices site captures the flow of the routine.

We begin by giving students a scenario and asking

  • What can I count or measure in this situation?
  • An important quantity is…
  • A relationship we know is…
  • Create a diagram – how can I represent the quantity and relationship?
  • Share with someone – ask, how did you represent? Then I did this… and explain your diagram.
  • Together, come up diagram that best represents the scenario. Share diagrams.
  • A gallery walk in a sense is where I visualize this – students are asked – Do you see the quantities and relationships in this? Where do you see the quantities and relationships in this?
  • Then students are asked to meta-reflect…
From Capturing Quantities, Fostering Math Practices

This. This is what I struggled with as a student. How the heck do I make sense of planes, trains and automobiles leaving stations at different times going it different or same directions. I saw immediate value in this routine. Giving students the question stem – but now question and letting them make sense of it.

Juli shared the 3-Reads strategy also and it will be a part of our learning environment this year as well. However, for 9th grade students – CQ is a step in that direction – it is actually embedded in 3-Reads. Give students the scenario, removing the question stem.

  • Read 1: What is the situation?
  • Read 2: What do the quantities describe?
  • What mathematical questions could you ask?
  • Then add the question stem back and allow students to compare their questions with the actual.

Now to consider ways I can implement Capturing Quantities remotely. Sharing ideas, diagrams. It is doable. I just need to ponder it a little longer.

Bitmoji Fun #MTBoSBlagust


Everyone can have their opinion. I need a little break from the logistics of formulating my plans for this semester and this is a fun little thing for me to play with.

Room 148

Today I decided to put on a plain t-shirt and put on a little school spirit! So I screen shot my Bitmoji, zoomed in and added my own lettering and artwork. I copied the image to KEEP, opened in the browser, copied the image, pasted it onto and voila! Then pasted my mini-me into my Bitmoji Classroom.

Small Group Instruction – Another Possible (Un)Productive Practice #MTBoSBlaugust


2018 Ignite at 21:40 (5 minutes)

Global Math Department (1 hour)

DNA Math (Blog Posts) Discussion of the TQE structure.

I feel that Juli’s description of small groups looks like small groups, grouped by ability, often/everyday, teacher calling up each group to work with them on “their level” – maybe moving the questioning to a lower level, leading a lot of the discussion.

Some things I jotted from her webinar –

  • at times whole group may be better
  • have concurrent small groups where they actually work on the same task, teacher visiting the groups (this looks like my classroom often) – as opposed to pulling groups up for help
  • ask students “as a team, justify/explain…”
  • having time as a teacher to observe evidence of learning as well as gaps in understanding.
  • moderate heterogeneous grouping – no outliers – in ability
  • worthwhile tasks / questions to engage student reasoning

I am not sure what small groups looked like in my classroom pre-2002, if I even had any. When I did my NBCT, it was the first time I really remember planning small groups with intentional discussion. Even then, the activities were still very leading. I have no idea how I actually grouped students. Most of my small groups were about collecting data and modeling with it.

I remember using an Amusement Park task from Key Curriculum’s Algebra I. I modified it to use geoboard to map out the park. Students found was to find the distance between rides/attractions. Later, each student was given a set of points to plot, found lengths of sides and slopes of the rectangles. Then they shared their information with the group and they looked for/discussed found patterns. The goals were really about parallel and perpendicular slopes. That was likely the first time the group had to actually “find” something on their own. I feel like I’ve grown since then.

Around 2010, I was part of the KLN and began reading A LOT. I began paying attention to Wait Time and Wait Time 2. I started actually planning the use of THINK-INK-PAIR-SHARE. I discovered the Math Shell Site and FALs. The Making Thinking Visible and Five Practices really began reshaping my practices and planning.

The past few years, I’ve been using Visible Random Grouping, some classes daily. We were not necessarily working in groups for every task, but it gave them a varied group to discuss with when I asked them to turn and talk. I found that VRG removed some of the class status. It sometimes “forced” students to work with and talk with new people. I rarely had issues with anyone having to work with someone they did not get along with – they just did it, without complaint – it was the norm and it really wasn’t questioned.

As I mentioned FALs – one thing I love about those lessons is the idea to group kids homogeneously by their approach. Not ability. Not correct solutions. But by thinking. I am often amazed at how the groups play out when I consider their approaches. It really does provide a good opportunity for richer discussions.

As I began planning for the possibility of virtual classroom, using Kirch’s WSQ structure – I see the idea of choice in student practice – maybe Here is what I want you to be able to do, here are 3 options –, a small group working on a similar task (maybe a virtual card stack), or meeting with me to ask questions, get some one-on-one help. I am still formulating how that might look within MEET. Again, I am mentioned Flipgrid as a possible back and forth discussion of how-to, answering student questions.

My small group activities are what I worry about most with virtual instruction. Is it possible to communicate electronically? Yes. Students do it all the time. As teachers, we collaborative electronically as well. But how effective will it be academically? I just don’t want to waste time when I have students synchronously. I want to have that part ironed out and working prior to doing it with the students. Maybe I need to let them share some ideas on how to accomplish this task as well. They are quite tech savvy.

A few things Juli has me thinking about as we set up Norms for Discourse Virtually:

  • Ask students to provide explanation / justification with their solutions.
  • Make sense of others solutions (encourages discourse)
  • Communicate when you don’t understand or don’t agree

I’ve been skimming the Distance Learning Playground and it really stresses the importance of setting norms online – just as within the classroom. It is challenging to think about what I want it to look like, my end goal and what norms/procedures will help us get there. Planning is more crucial than ever before.

Possible (Un)Productive Practices #MTBoSBlaugust


2018 Ignite at 21:40 (5 minutes)

Global Math Department (1 hour)

DNA Math (Blog Posts)

Rigor, as it relates to the shifts associated with recent state standards, is often defined as the need to include conceptual understanding, procedural skill, and application in mathematics teaching and learning.

Considering this definition, Juli shares the idea that Conceptual is the why and Procedural is the what and that conceptual should be taught ahead of the procedures. One example she shares that I found helpful was division of fractions. Most of us have been taught flip the second and multiply. But as a kid, I always wondered – Why does this work? And who knew to do this?

Juli uses the context of baking cookies and butter, great illustration. I’ll use the idea of money and coins. 3 divided by 1/4. Okay. Lets use the context, $3, how many quarters (1/4) are in $3? 12 quarters. Let’s try how many quarters are in $4? 16 quarters. And again, how many quarters (1/4) in $5? 20 quarters. Then pause and allow students to notice, share what they see, what’s happening mathematically. THEN we can model the procedure… 3 / (1/4) -> 3 * (4/1) = 12, etc.

The big idea is to allow students to play with some math ideas in a context they can relate to, then bring in the procedure and using their thinking and ideas, help them see the connections.

There are times in Algebra the context is a little hazy for me. But even just looking at big ideas and allowing students the space to notice patterns, describe those patterns and then generalize them with the math is a win. (Sara!) What Juli shares as being possibly Un-Productive is that fact that we soft often neglect the opportunities to help students make these connections. And that is so easy to let go. We can sometimes plan an awesome discovery lesson, but without the end discussion to wrap things together, students will walk away, “HUH?!?” and frustrated.

For me, I see this as intentional planning to include time to help them make those connections. I mentioned in an earlier post – closure to class or a learning task is vital. I set a silent alarm on my fitbit that allows me to wrap up class rather than yelling things to kids as they walk out the door. I do a quick review of their responses, then I begin the next day by addressing the previous days take-a-ways. Its where we take many big ideas, reflect on them and decide what we can take away from it all. I feel like the FALs (MathShell) AND Five Practices (Smith & Stein) were great resources to help me formulate how I do this in the classroom.

For this year – I have a few ideas of how to still accomplish this task.

Simply within the MEET chat window, or within a shared jamboard, allowing students to post sticky notes and type the reflection in there. I can easily schedule a question in gClassroom that will appear toward the end of instructional time. I see flipgrid eventually becoming a great tool as well.

My big “new” thing this year will be the “homework” of watching the instructional video prior to classtime. This prepwork will be followed with WSQ gForm. I will encourage students to have these completed by midnight of the day, which will allow me to grab a quick look at them prior to our classtime online together. I can take their questions and ideas and incorporate them into our WSQ discussion we will have at the beginning of class to ensure everyone is moving in the right direction.

If you are unfamiliar with WSQ – Watch, Summarize, Question – check out Flipping with Kirch blog. Ultimately, I want students to do more with the Questioning, but in the first few weeks, I plan to encourage them to use it as a way to communicate their trouble, need for help or clarification.

Again, Juli’s posts have given me a chance to reflect on my practices and process how exactly/possibly I can modify and continue productive practices in a remote environment.