Tackling Tanton’s Pinwheel Area Problem

While our middle school math summer success program has ended I wanted to share how the students grappled with James Tanton’s Pinwheel Area problem. I was only working with seven students that day because the rest of the class was on an EL field trip.

I gave each student a copy of the problem and 10 minutes of quiet think time before solving as a group.

pinwheel area

After 7-8 minutes none of the students were able to get started so I asked, “What shapes do you see?” Their first response was the pinwheel and squares. Then I asked, “What other shapes to you see?” There as a very long pause. I sensed the students eyes were solely focused on the pinwheel and grid so I asked them to look at the entire image. Another long pause.

“Do you see any other shapes besides the pinwheel and squares?”

A few moments passed and a student responded, “I see triangles.”

I should have anticipated this task better. I was expecting to scaffold, but as we moved through the task, the students needed additional support.

“What triangles do you see?” My eyes naturally saw the larger triangles, but his eyes didn’t.  I asked the student to point to the triangles.


At this point I recognized the grid was a distractor–which is great in terms of adding rigor to the problem while providing another scaffolding opportunity.

Let’s draw this on the board.



Now we were able to discuss a strategy. It took some time, but we agreed that if we were able to subtract the non-shaded area from the 5×5 grid, we would find the area of the pinwheel.


Students enjoy working on the whiteboard and wanted to shade the triangles.


Students recalled how to find the area of a square or rectangle, but could not remember how to find the area of a triangle. I drew a rectangle, and labeled the length and width as 3×2. “What’s the area?”


I cut the rectangle in half horizontally and asked, “What’s the area of one section?”


I erased the horizontal line and replaced it by cutting the rectangle in half vertically. “What’s the area of this section?”


I erased the vertical line and cut the rectangle in half diagonally. “What’s the area of this section?”


So if we multiply the base and height of a triangle and cut it in half (divide by 2) we can find the area. We revisited the printout of the problem to find the measurements and began calculating.


Now we were able to put it all together.


To achieve productive struggle this group required more scaffolding than I originally anticipated. I need to reread Five Practices.


Students Evaluate Summer School Math

Yesterday I asked students to evaluate our four week Middle School Math Summer Success program which I previously wrote about here and here. One of the biggest challenges of teaching a multi-grade level class is, as you can imagine, meeting the needs of the students.


The makeup of this grade 6-8 class was quite diverse. It included 1) students recommended for summer school by their math teacher, 2) students enrolled in the summer EL program to receive math support, and 3) students who were enrolled by their parents to keep skills fresh.

When a teacher recommends a student for summer school, certain expectations are presumed. When parents enrolls their student, they have certain expectations as well. Given the make-up of the class, I needed to reach rising seventh, eighth, and ninth graders whose mathematical prowess ranged from developing to proficient.

I structured each 90 minute day to include number talks, individual and group tasks, number sense activities, and grade level review. I wanted student feedback on each of these areas plus one recommendation for improvement. Here are the results.

Students’ Evaluation

For the four focus areas students simply placed an “x” on a helpfulness continuum. I gathered the data and compiled it into four separate graphs. I also debriefed with each student individually using the feedback they provided.

number talk
Students who did not find number talks helpful already had strong number sense.
Individual group tasks
The one student who rated tasks less helpful was the most “skilled” student in the group. This student could also be considered the lead learner.
Here again, students who had strong number sense did not find these activities helpful.
grade level
One student, a rising 9th grader, felt the grade level practice was extremely helpful. A rising EL 7th grader gave the lowest rating.

It really didn’t surprise me how much students value the grade level practice. For several students in the class, this was exactly what they needed. Also, when working with intervention students, some do not recognize the value of foundational skills because, “I need to know how to do [insert grade level concept] that’s due tomorrow.”

Students’ Recommendations

I love the honest feedback the students provided.

In addition to the EL teacher and me, most days we had 1-2 high school student volunteers working amongst the students.
This feedback is from a rising 8th grader, the lead learner in the class, referring to the thumbs’ up sign when giving wait time during number talks.
Referring to number talks–from a rising 7th grader.
From a rising 9th grader.

I appreciate the students’ suggestions and will take them to heart. It would be lovely to have even more high school volunteer in the classroom facilitate the learning. I had one volunteer, “Annie”, who was exceptional. She knew how to give hints without “giving it away” and gave students a growth mindset pep talk–“I was where you were in middle school and through hard work I’m now in Calc III.”

Overall I’m pleased the students found the class valuable. When I debriefed, I asked the lead learner, “Bradley”, why he enrolled in the class. To paraphrase his response, “I was recommended for reading and that was the first time slot. My mom didn’t want to pick me up right away so she signed me up for the this.”


Reflections on a newly created 6-8 math summer school curriculum

No matter the content area, one of the greatest challenges in implementing curriculum with fidelity is securing the qualified personnel to teach it. This is even more evident when it comes to summer school programs. Since teaching summer school is optional, perhaps some middle school math summer school programs are limited in scope in order to attract more, but not necessarily math, teachers to fill the position.

As a middle school math interventionist, I offered to teach our district’s Grades 6-8 Math Summer Success program. I also wanted the students to experience a more robust, tailored curriculum so I created one. I just wrapped up teaching week three of the four week program and I continue to ask myself, “Did the curriculum I designed best meet the needs of students? Is this curriculum viable and sustainable?”

The question: did what I designed best meet the needs of students—is directly related to the purpose and goals of summer school math. I deviated from the Origo Grade 5 workbook supplied by the district and instead created a curriculum based on the input from our grades 6-8 teachers. They are the ones who recommend students to the summer success program, they know the students’ strengths and growth opportunities. And for those students whose parents enrolled them into the program as a refresher, I contacted their teachers to confirm this class was still a good fit even though they were not formally recommended for the program.

The curriculum combines differentiated grade level review based on the key concepts recommended by the grade level teachers, along with mixed, multi-grade group tasks and daily number talks. This is a combined EL Math and Summer Success class of rising seventh to ninth graders; daily attendance ranges between 22-25 students.

Am I meeting the needs of the students?

Week 1

Week 1.JPG

  • Daily number talks are extremely valuable in improving students’ number sense.
  • Every student benefited from reviewing GCF/LCM using the MARS individual and group tasks.
  • I overplanned and did not get to grade level practice two out of the four days that week.
  • Grade level practice using dice to review fractions and decimals were difficult to quickly monitor when checking student work.
  • Routine grade level practice problems were easy to monitor. Having worked solutions for the EL co-teacher and high school helpers were beneficial.

Week 2

Week 2

  • Number talks using percent number strings are fabulous.
  • MARS Fractions, Decimals, and Percents group task generated a lot of small group discussion. I inadvertently handed out the area model set before the fraction set which generated even more discussion!


  • Students weren’t invested in The US Congress: Is It Representative? task as I would have liked. The students didn’t connect with the task. Both the EL teacher and I are social studies buffs, and perhaps more time should have been spent activating prior knowledge and discussing current events.
  • Grade level review was worthwhile this week.

Week 3

Week 3

  • Students are really getting the hang of the percent number strings. They are moving much more quickly through the strings.
  • Tanton’s Counting Jelly Beans task was challenging. Students were not able to execute the strategy when shown the video. It required a whole class discussion of the solution.
  • The MARS A Sense of Scale task was fruitful. A few students were able to complete the initial task independently using a variety of methods. Students were then grouped and shared strategies on giant whiteboards and later successfully completed the follow up task individually.


  • Again, the grade level review is proving worthwhile.

Anticipating Week 4

Week 4

  • I’m anticipating the two Tanton tasks, Pinwheel and Mixed Colors, will continue to challenge students.
  • Finishing out the program with two math art activities will be a nice wrap up.
  • Continued grade level review will continue to benefit students.

Is this curriculum viable and sustainable?

It depends. This curriculum requires a certain skill set. The EL teacher with whom I share responsibilities for this class readily admits she is learning alongside the students. So the ultimate question is what is the purpose and goal of middle school summer school math? Perhaps my expectations are unrealistic given the way summer school currently operates.

I spoke to one of our building’s music teachers and he mentioned that our district middle school music teachers in both buildings voluntarily rotate summer school responsibilities in order to maintain a quality summer music program.

Perhaps that’s a commitment our team of middle school math teachers would consider to make this curriculum viable and sustainable.

Multi-age Middle School Summer Math

On Monday I will be welcoming a diverse group of learners to our  Middle School Math Summer Success Program. Thirteen EL students will join my class of 12 for two hours of collaborative learning Monday-Thursday for the next four weeks. Here’s the breakdown by grade level:

  • Nine rising seventh graders
  • Nine rising eighth graders
  • Seven rising ninth graders

The middle school math summer program previously had no curriculum, so for the past month I’ve been planning daily learning experiences. Students in the middle school EL summer program historically did not receive math instruction so combining the groups makes sense. It will be a large class but thankfully my EL colleague will help facilitate.

While the curriculum is not perfect, my goal is to provide a collaborative multi-age learning environment and meet each students’ grade level academic needs.

To give you an idea, here’s the weekly structure:


As you can see every day begins with a number talk. The number talk resource I’m using is from Math Solutions. To help monitor the learning, my EL colleague and I will be using a classroom observation checklist from Formative 5. Many of the group lessons/tasks are from either the Mathematics Assessment Project, Stenhouse’s Well Played middle school series, or James Tanton’s Curriculum Inspirations. Grade level review is primarily skills practice pulled from similar homework problems students encountered during the year.

When putting this together my original thought was to load all electronic copies on Canvas and have students work from their iPads. Then I said to myself, “Nah, this is going to be technology free.” As a result I spent an hour and a half at the copier preparing two weeks worth of copies.

I’m looking forward to teaching this multi-age summer school program. The students enrolled  perhaps never had a chance to be a “lead learner.” Now they’ll have that opportunity.



Playing in the IM sandbox: G6 Expressions and Equations

Our sixth graders recently experienced a series of lessons from Illustrative Math.  We played in the “sandbox” by delving into Unit 6 Expressions and Equations, Lessons 6.1- 6-11. One huge eye-opener for me was the emphasis on equivalence. Since the expressions/equations unit we’ve taught in the past is very procedural based one-step equations, the IM lessons made me think much more deeply about the math. It made the students think more deeply too.

Adapting IM to fit our needs

Since our current curriculum emphasizes solving one-step equations, we embedded more rote practice than what IM provides in the unit. But we so fell in love with IM’s emphasis on equivalence we modified their end of unit assessment to include more problems of this nature. For lessons 6.1-6.11 we created a Socrative review activity based on problems similar to the upcoming assessment. Feel free to modify it to fit your needs. As an aside, there is a growing repository of teacher created resources that can be found here.

To give you an idea of what I was most impressed with, I hastily tweeted out this question (don’t laugh at the misuse of the word illicit. I meant elicit! I’m human!)


I appreciate @mathminds’ feedback as well as @geonz’s who provided additional insights.


For our students, one lesson that might require future adaptation is Lesson 6.9 Distributive Property 1. Students struggled completing the table. Either they needed more time (our periods are 41 minutes long) and we rushed it, or more scaffolding is needed. Perhaps we’ll get some advice from the IM Professional Development we’ll receive this summer.

task statement

One reason we are dabbling with IM’s curriculum is to identify resources that can supplement our teacher created curriculum. Frankly, after experiencing IM’s lessons, I would be in favor of letting go of what we’ve worked so hard to create.



Rates, Unit Rates and the Cape Town Water Crisis

A couple of weeks ago the Cape Town water crisis caught my eye. This is a slightly different article than what Dan Meyer referenced in his post  in that the NBC News article includes a graphic and a link to a water usage calculator.


This was a great opportunity for my sixth grade math students to: 1) explore estimation and rates while examining a real world issue, and 2) ponder what they can do to conserve water. As an aside, the 6th grade team is looking to expand this two day lesson into a larger conservation effort.

PLC tracking document

I began the lesson by asking students how much water they consume in a day. “Six hundred gallons,” one student suggested. “Ten gallons,” said another.

I jotted down their estimates, then asked, “Where are those numbers coming from? Are they wild guesses? How could we possibly make our guesses a bit more informed?”

One student asked, “How do we use water?”

“Great question! In what ways do we use water?”

Students started calling out several uses that were captured on the water usage calculator–bathing, flushing the toilet, hygiene, etc. I then asked, “How much water do you think you use when you shower or flush the toilet? Can you think in terms of liters?” I needed the students to think in terms of liters because the calculator is metric based. Fortunately a student had a one liter Nalgene water bottle on her desk so I held it up for reference.

“Now, open up this document in Canvas and complete the first column, (there were 10 categories). When you’re finished, go to the website and calculate your daily usage.

water estimate

actual amount

At the end of day one students reflected on their estimate vs. actual water usage.


On day two we read the NBC News article aloud in class, stopping periodically to discuss the reading. At the end students again reflected–by describing at least two ways residents in Cape Town are conserving water, along with a closing statement about what you can do to conserve water.


This lesson addressed Dan’s question, “How can we put students in a position to appreciate, replicate, and even adapt those calculations for their own contexts?”

With more time, I hope a future rendition of this lesson will allow students to delve deeper by creating a cross-curricular study focusing on conservation, advocacy, and service.


Taking care of me

For the most part I’ve been blessed with good mental and physical health. In 2010 I was diagnosed with an early stage, indolent type of Non-Hodgkin’s Lymphoma and took a leave of absence from school for treatment. That period of my life put things very much in perspective for me. One way I take care of myself is to be more mindful by actively noticing new things. I think less about what might happen in the future and focus more on what is happening now.

Events don’t cause stress. What causes stress are the views you take of events–Epictetus

Recently the On Being podcast with Krista Tippett featured Social Psychologist Ellen Langer and her studies on mindfulness. If you are interested in learning more head over to the podcast.

I’ve always been quick to laugh and maintain a positive outlook on life. Besides my family, here’s what brings me joy.

Outdoors and Exercise

This summer I spent a lot of time with a dear friend walking an 8 mile loop in the local forest preserve. We hiked 4-5 times a week. Between that and eliminating gluten from my diet I lost 25 pounds this summer.

Since our children are grown, my walking buddy and I also found time to take day and overnight trips hiking a variety of state parks in Illinois and Wisconsin.

At left, Mississippi Palisades State Park, IL; right Governor Dodge State Park, WI.

I’ve rediscovered tennis. I played the sport competitively in college but dropped it entirely when my priorities became parenting. I am slowly getting my timing back and adjusting to the modern game by drilling weekly, hitting against the backboard, and playing pickup games. I even analyze YouTube tennis videos to help me modernize my serve and one-handed backhand!

Love the backboard!

With the weather turning colder, my walking has moved to the treadmill. Time passes much more quickly listening to Live from the Poundstone Institute or All Songs Considered.

Film and Television

My husband and I love going to the movies. If I had the money I’d open an independent movie theater, but I do enjoy many mainstream films too. We just saw Blade Runner 2049, which had a complex plot I really liked, and American Made which was less compelling, but somewhat entertaining.

I also escape by tuning into certain programs on Netflix. Since we don’t get cable, I satisfy myself with series such as Luther and Black Mirror. I particularly like foreign series such as 3%, Fauda, and Nobel.



I used to read a lot of fiction, but non-fiction is where I spend my time these days. Besides Weapons of Math Destruction, I’ve read Dream Hoarders, The Color of Law, No is Not EnoughDemocracy in Chains and Stamped from the Beginning. As a middle schooler, my daughter loved Philip Pullman’s Dark Materials trilogy and she recently recommended his new novel The Book of Dust.

Everyone takes care of themselves differently. This works for me. Do what’s best for you!