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!


Math personified: a teaching story


My friend Mark R., his identical twin Expo, and I are sitting on the chalk rail. It’s not really a chalk rail but I think that’s what you call it even though we’re mounted to a white board. The three of us love exercise and we get a great workout in math support but sometimes we get abused. Sam will put me in choke hold and press my face against the white board. Ellie will forget to put the cap back on Mark R. so he’s all stuffed up. And poor Expo has lost his head completely. We now call him the Headless Horseman.

By this account you would think the students don’t like us. They actually do. Students can’t wait to get their hands on us because for the better part of the day they learn using an iPad.

Story I

This past week our caretaker, Mrs. Dooms, used us for a form of visual storytelling. Ellie, a seventh grader, struggles with mixed numbers to improper fractions. Her class is working on rational numbers, specifically adding and subtracting negative fractions. Mrs. D. picked me up to visually explain.

“Remember last week when I served you guys pizza and cut it up?”

“Yeah! That was good. Where was it from?” Ellie asked.

“It was a frozen pizza and I heated it up during my plan period. Let’s get back to fractions.”

Mrs. D. uncapped me and began to draw circles to represent 2 1/4.

Mrs. D. didn’t photograph our Expo marker work so we gave her permission to use the SAMR model of substitution using the iPad.

“This is where the short-cut comes from,” she said. “Two wholes cut into fourths plus the remaining one fourth equals 9/4. Ellie needs more examples. Mrs. D. writes two mixed numbers for Ellie to model. She hands me over to Ellie for more practice.

Story II

In sixth grade support, Sam picks me up with his usual choke hold. We’re multiplying decimals, 4.31 x. 2.2 and I brace for the decimal point. His math is fluent; the ink flows freely onto the whiteboard. Sam generally follows the algorithm but I hear Mrs. D. ask, “How do you know your answer is reasonable?”

“Because there are a total of 3 place values to the right of the decimal,” he said.

“But how do you know your answer is reasonable?” Mrs. D. asks.

“There are a total of 3 place values to the right of the decimal so I move the decimal three times.”

“If we were to estimate, to what whole numbers would we round 4.31 and 2.2?”

Sam’s eyebrows furrow, his hand grips me tightly. “We round 4.31 to 4.3 and we’d leave 2.2 alone?”

Mrs. D. picked up Expo from the chalk rail, saw his head was missing so she grabbed Mark R. instead. She lightly wrote next to Sam’s work ‘Estimate: 4 x 2’.

Sam reveals to Mrs. D. that he relies on memorizing a procedure and needs more support with the meaning of decimals and place value. My buddies and I are gearing up for next week’s lessons.


If Mrs. D. was writing this she would want you to know that most students enjoy math support. In fact some seventh graders drop in on their non-scheduled days.

As far as I’m concerned, students appreciate learning with us Expo Markers, even though I wish they would treat us with a bit more affection.

Favorite lessons create an intellectual need

As the math interventionist in our building I have the luxury of co-teaching with an incredible group of 6-8 math educators. When the 6th grade math teachers meet face-to-face much of our PLC time is spent examining student work and reflecting on what went well in today’s lesson and what can be improve upon.

Today we discussed with our counterparts the results of our attempt to give students a “headache” and create an intellectual need for vocabulary when learning about exponents. Our original plan was to go straight to the notes as a mini-lesson then introduce the vocab but, as my colleague and I were discussing over email, we decided to introduce exponents within the context of a real world situation.

Here’s what today’s lesson looks like from our lesson plans in Google Drive.


Here’s a summary of our discussion in our PLC tracking document.


Our improved instructional design of the lesson wasn’t perfect, but it was a much better learning experience than simply launching with notes followed by practice.

Here’s how we framed the learning. We began by posing to the class the problem in the image below and constructed meaning using a team teaching think-aloud. I read the problem, my co-teacher Cathy modeled confusion and read the problem again, and discovered a starting point. I interjected that there are many ways to represent problems and suggested we use a tree diagram to get us started.


As we created the tree diagram we eventually stopped to emphasize that the pattern continued. At that point we asked the class (and fell out of the think-aloud mode!) how to represent the continuing pattern for Thursday and Friday. Then we finished creating the tree diagram.

problem2 My co-teacher Kathy mentioned to the class that her eyes were beginning to look at the rows of “F’s and we counted the numbers in each row.


We then looked at the diagram, noted the values and introduced the vocabulary term “Standard Form” by labeling that column.


Next  we noticed two was a factor in 2, 4, 8, and 16. We introduced the vocabulary term “Expanded Form” and began with 2 x 1. Students quickly shared 2 x 2 = 4, but to write the expanded form of 8 most students wanted us to write 2 x 4 instead of 2 x 2 x 2. We had a discussion on only focusing on using 2.

After expanding each of the numbers, we worked from the bottom to complete the Exponential Form column noting 2 x 2 x 2 x 2 is equivalent to 2^4. Students quickly caught on as we worked our way up the column. Comprehending that 2^0 is equivalent to 1 was a bit troublesome for a few students. I don’t have the image but we then attached meaning to the vocabulary words base, power and exponent.

The goal for the original lesson was to acquire new vocabulary. We could have accomplished it using only notes followed by independent practice, but providing a context for learning and creating an intellectual need for the vocabulary made the lesson more meaningful.

I wonder, however,  if too much vocabulary was shared in the lesson. I hope not. Students were only asked to identify the base, power, and exponent in their practice. The next day they examined more closely the terms standard form, expanded form and exponential form.


Classroom Management or Classroom Environment?

As the school year begins I’ve been thinking a lot about classroom environment. Our district uses the Danielson Framework as part of our teacher evaluation and one of the four domains identified is Classroom Environment. As an aside, when The Framework was first published it was designed to be a reflective tool not an evaluative one.  If you are not familiar with The Framework for Teaching, Charlotte Danielson identifies four domains of teacher responsibility. Domain 2, Classroom Environment, includes five components that make up this domain.


For some reason whenever I hear “classroom management” I immediately think of procedures, student behavior, and physical space. Do the students know where the supplies are? Are they efficiently transitioning between learning activities? Do students know the procedure for when they return from an absence?  That’s just the business of learning.

Two components I want to focus on are the first two from Danielson’s list: 1) respect and rapport and 2) a culture for learning. I LOVE how the wording below captures the essence of both while merges the two ideas into one. It even addresses the expectations of both teacher and student as it relates to learning:

respect learning

I came across this language three or four years ago at a summer technology workshop hosted by our district. The words never escaped me. Kate Kieres traveled from Pennsylvania to lead the PD and referred to it several times.

When sharing this with students we obviously discuss examples of respect, but instead of posting a list of do’s or don’ts,  I prefer to capture it in question form as the basis for our classroom norms.

Creating a culture for learning requires students to be in an engaging, collaborative environment that empowers all learners. That’s my responsibility. When students ask…

What am I learning from others?

When students consciously ask themselves, What am I learning from others? attention is immediately drawn to the learning experience. It demands critical thinking and processing. It invites inquiry, debate, and feedback. An example could be a simple turn and talk where students share a problem solving strategy, but the caveat is the intentional self reflection while the conversation is taking place, “What am I learning from you?”

How am I contributing to the learning of others?

Similarly, students should be cognizant of how they contribute to the learning of others. Either in pairs, small groups, or whole class discussion it is my responsibility to create a learning environment where every student voice is valued.  That means every student needs to be into a position where they can participate and not feel they have nothing to offer. A group-worthy task with roles and responsibilities or an open middle problem are ways to give students the opportunity to contribute to others’ learning.

As classroom norms

My support classes are only 30 minutes. Time is precious. By referring to the norms What are you learning from others? and How are you contributing to the learning of others? I can quickly redirect behavior. It makes the statement that this is a learning environment and these are the expectations, on both the students’ part and mine.



Intervention Goals and Goal Setting

I’m so grateful to @jreulbach and her blog  for resurrecting #SundayFunday.   My position as a 6-8 math interventionist (a relatively new position) has prevented me from blogging regularly but I hope the weekly prompts will get me to write more often. My responsibilities continue to evolve, but my main projects are providing grades 6-8 homeroom math support, co-teaching four classes, and overseeing our Title I after school math program.

Our homeroom intervention program is a work in progress. It is NOT a formal class, but the identified students (below 30% percentile MAP and teacher recommendation) are scheduled to attend math support twice weekly. The two previous years, support time was dedicated to re-teaching and supporting the daily classroom instruction. This year I will also be targeting specific skills for intervention.

One goal I have for myself is to spend time identifying quality intervention programs that best suit our students. One resource I’ll be looking at more closely is this document from Hanover Research. I’m also quite fond of Marilyn Burns Math Reasoning Inventory .  I’m not sure if my district will be purchasing a program or if we will be creating our own. Either way the work is cut out for me. If you have a favorite or have struggled with a particular set of resources please add your thoughts in the comments.

Along those lines, one goal I have for my homeroom support students is for them to set their own goals and monitor their progress. I created this google spreadsheet for them to complete.

Below is a screenshot. The student would fill in the blue and green and the graph would self generate.

Goal setting

Students can use the continuous learner characteristics as the basis for the goal setting plan.

If the electronic version becomes too cumbersome I guess I’ll go back to paper.

I’m interested in learning how goal setting has worked in your classroom as well so please chime in!