### Archive of ‘instruction’ category

Today a friend emailed this math problem to me.

What *is* the “right” way for math anyway? Is technology inherently less “right” than algebra? Solving it graphically by Desmos (or by TI-84) still solves the problem, but maybe that doesn’t feel as elegant or satisfying. I did want to share the joy of Desmos (since my non-math teacher peers aged mid- to late-thirties did not grow up with it), so I sent the following screenshot (before I eventually solved it algebraically).

The response:

**“Ha ha, neat toy.**

**Since I forgot all the tricks to resolve mixes of square roots and variables, I looked at it as, it must be an integer since what awful brain teaser would have 4.87645372 as an answer, and it had to be a number with an integer square root, and that square root had to be less than half of 15, and the square of that number is 15 less than the square of (15 – that number). So I tried 7 first, 7 + 8 = 15, square of 7 is 49 which is 15 less than square of 8 (64) so that was it. 7 squared, 49. Had that not worked I would have tried 6, 5, 4 etc.**

**In the meantime, I’ve filled the backs of two envelopes with desperate attempts to solve it algebraically, going nowhere.”**

Poor friend! Sent this to him:

I then sent it to my math colleagues (and my boss, who is a former math teacher).

**Boss’ response, which I will ask to see in its original form, since email apparently translated it into gibberish.**

**Coworker:**

**Alternate Hint: Try putting the square roots on different sides of the equation and canceling 🙂**

**I took a pic of my work but don’t want to spoil it… email me if you want a look-y-loo. Thanks for the Thursday PM pick-me-up!**

I emailed her to trade answers and she solved it this way. It made us both happy to solve it algebraically, but differently!

So there you have it:

1) logical way

2) graphical way

3) algebraic way #1

4) algebraic way #2

…how many more ways?

As much as I dislike Facebook, I do find value in the “On this Day” feature. From April 21, 2011, I wrote this note. Six years later, 3) through 6) are still so important for teaching (and now parenting). Six years later, we have Google Classroom, SeeSaw, Workplace, ClassDojo, Khan Academy and countless other technologies that increase our ability to access content or transmit content to our colleagues or students. Six years later, we have Amazon PrimeNow to get whatever baby product we need within two hours. We have similar access to “wisdom” via countless online mommy/baby forums, sleep consultants, and ScaryMommy/Pregnant Chicken-esque blogs. We can also transmit our content in those forums, in mommy Facebook groups, or on group iMessage threads during late-night feeding sessions.

Increasing access to or ability to transmit content doesn’t make teaching or parenting any easier if the emotional aspects aren’t addressed.

“From my roommate, who attended a talk about happiness by an HBS negotiation professor tonight. The HBS students voted for the three professors who they wanted to hear from, and these are the main points of the first talk.”

1) **Quit early and often**. Save up enough to make it monetarily possible for you to do so, and make your own choices.

2) **Create value and worry about monetizing it later**. The opposite is thievery (going where the money is and trying to create value).

3) **Cultivate empathy**. See the world through gentler eyes, because you will be better for it.

4) **Learn humility**. Either humility or arrogance is not enough; you must have a combination to succeed.

5) **Learn from unlikely sources and don’t judge your teachers.** Everyone around you can teach you something. Even if it’s harsh, if you are willing to take it, you can’t go wrong.

6) **Make time for reflection.** Ask questions about who you’ve become and who you were. Build in this time to keep reevaluating. You don’t need to aim for being stagnant and stable. Inevitably, things will change about you.

I’ve been thinking a lot about how to better balance work and life so that teaching is sustainable for the long haul. Balancing the demands of parenting a toddler with the demands of teaching and IB coordinating has made year 9 of teaching much more challenging than the first few years. I have often told my senior students to make the most of college because “you will have the most free time you have ever experienced in your life, and that will be a blessing and a curse.” I wish I’d learned that lesson about teaching, because until my “free” time got cut by about 80%, I didn’t realize how important it is to find efficiencies versus just throwing more time at lesson planning. I take a lot less work home now, but I’d love to reclaim more night and weekend time (especially during end-of-term grading or around major IB events).

Here goes today:

**Before School**

4:40 a.m. Alarm goes off. Snooze when contemplating starting Day one of the Blogilates PIIT28 challenge.

4:49 a.m. Alarm goes off again. Get dressed, put contacts in, and wander out to the living room. Look up “how to work out in the morning” and find a Parenting magazine 10 minute workout.

5:00 a.m. Warm up with some dynamic stretches. Start kettle for French press. Do the workout.

5:20 a.m. Pour coffee. Shower and get dressed. Cook breakfast (egg scramble for me and husband, milk, sliced banana, and peanut butter toast for toddler).

5:49 a.m. Toddler wakes up. Sing good morning song and change his diaper. Consider logging the wrestling holds used during the diaper change as part of my morning workout. Eat breakfast with toddler and husband. Clean up and pack daycare bag.

6:37 a.m. Leave for the T (husband is taking car into shop to repair a broken headlight). Sometimes the T is more enjoyable than driving–it’s fun to read and to walk vs. sitting in the car listening to Spotify or a podcast.

6:46 a.m. Barely miss a Braintree train, so spend some time writing down notes for this post.

6:56 a.m. Catch Ashmont train. Run into teacher friend on his way to his school and chat until Central Square. It’s good to chat about peer feedback and commiserate over the challenges of teaching before break.

7:11 a.m. Walk from Charles MGH to school via the Public Garden. Fun sights on this walk: gingerbread houses in the Hungry I restaurant and knitted hats and scarves on the Make Way for Ducklings statue.

**School**

7:31 a.m. Arrive at school. Make copies of reassessments, peer grading rubrics, and progress reports. Reserve Chromebooks. Start looking for teacher with backup key because the teacher who normally has the cart is out. Put up plans on the board.

7:50 – 8:37 a.m. 9th grade system of equations coffee project work day. Hand out progress reports and feedback to students. Think about how to revise this project for next year with Hacking Assessment and Strength in Numbers.

Revision ideas, from Hacking Assessment: *Meet with groups to hear ideas. Help them ensure they stay on target and complete the systems of equations tasks on Google Slides and Desmos. Use feedback to troubleshoot, not provide answers. Help them discover knowledge on their own. When they work, discuss progress and observe group dynamics to see that all students are contributing. Try individual feedback (check out Grading and Group Work). When students are done, do a gallery walk on Chromebooks. As groups, to take notes, make questions and comments (on paper? on the docs themselves?) Provide feedback based on the IB Rubrics using a Google form. Have all students submit a reflection and self-evaluation about what he or she learned as compared against the standards so we can discuss growth. Students should reflect on what they had hoped to get out of the project and share what they learned.*

8:40 a.m. – 9:00 a.m. Answer emails and make slight revisions to lessons.

9:00 a.m. – 10:17 a.m. Meet with other IB coordinator and leadership consultant to plan out our school leadership team meetings for January as well as some longer-term Diploma Programme and Middle Years Programme initiatives for Terms 3 and 4. It feels good to “backwards plan” our leadership work like we do to unit plans.

10:20 a.m. – 11:07 a.m. 12th Math Studies class. Students are to finish trig fairy tale work, but it turns out that four of the eight students are out for dance rehearsal. Turns out fine because of having to attend a parent meeting from 10:26 – 10:40 (with sub coverage from our office assistant).

11:10 a.m. – 11:57 a.m. 9th Math Enrichment. Groups finish up gingerbread house building and photography. They start reflections. Half the class came in late after dance even though rehearsal was only supposed to be from 8:40 – 10:17.

12:00 p.m. – 12:20 p.m. One group stays through lunch to finish their house.

12:23 p.m. – 1:10 p.m. Work on more school leadership team planning. Write emails. Make project adjustments for both 9th & 12th grade.Set up sub for two PD days in February. Add Sierpinski Christmas tree and Kirigami snowflakes to Math Enrichment plans for the week.

1:13 p.m. – 2:00 p.m. Repeat the 9th A-block plan from 7:50 a.m., but in a more energetic atmosphere. Afternoon and morning classes are so different in terms of motivation challenges.

2:03 p.m. – 2:50 p.m. Tidy up Chromebook cart and charge it. Check with headmasters if it is ok to build a Sierpinski tree. Change it to non-denominational tree. Obtain pink card stock from secretary and scrounge up some ecru card stock in my cabinet (at least it won’t look like a typical Christmas tree). Research Mandelbulb ornaments and log this idea for a potential 3-D printing project next year. Update coffee project progress in Jupiter Ed. Make copies of the snowflake patterns and Sierpinski nets.

3:19 p.m. Sign out. Walk to Charles MGH.

**After School**

3:39 p.m. Catch train, respond to emails, and read Glamour.

4:06 p.m. Arrive at home. Check mail, put away laundry and dishes, and prep French press while listening to Dinner Party Download podcast.

4:45 p.m. Walk to daycare. Toddler’s face lights up when he sees me at the window. He runs over to his cubby to get his jacket. Talk to the teachers about his day while trying to get him into his vest and jacket. Get toddler into stroller and walk home.

5:19 p.m. Get home. Feed toddler mac and cheese, one small avocado, an applesauce pouch, and water. Play with toys, read books, and chase him around the house.

6:20 p.m. Start bedtime routine.

6:42 p.m. Toddler is asleep. Make pizza. Watch three episodes of Happy Endings. Do dishes. Prep toddler lunch and put quinoa out to cook in the morning for part of our lunches.

8:17 p.m. Write this post.

9:00 p.m. Read a little and go to sleep.

*Disclaimer: I was sent a complimentary sample of Atlas Coffee in exchange for my thoughts on the product. I was not paid to write this post. All opinions are my own. There are no affiliate links in this post.*

I’ve written about coffee math before and recently got inspired again after learning about Atlas Coffee Club. After my beginning of semester poll on “what would help you do better in math?” some of the students in my math enrichment class said “real-world projects!” I love doing those anyway, so I got to thinking about topics that are relevant to them. My first period 9th grade class doesn’t seem to have the same coffee addiction as my past 11th and 12th grade classes, but that may change for them in the future. In past years, my early morning 11th or 12th grade classes often looked like a Starbucks (minus the hipster clothing and MacBooks) because of the sheer number of Frappuccinos, iced passion teas, and coffees. The teachers, however, still come in with various types of coffee (homemade in a travel mug, Dunkin iced, Starbucks, or local cafe) and often go out for a mid-day coffee break or use the Keurig machine in the teachers’ lounge.

Thus is born my new project idea: **Have students do “market research” and create a “make coffee at home” plan for teachers who spend way too much on buying coffee out.** Atlas Coffee Club offers a wide variety of coffees from around the world along with subscription options that suit different frequencies of coffee drinking.

1) Interview a teacher about their favorite type of coffee and their coffee habits. For example, if they like Ethiopian reserve coffees at Starbucks, they might like the Ethiopian Sidamo. For coffee habits, find how often they drink it, how much it costs per day, and their habits (e.g., are they always rushing in the morning? do they have to make coffee for a husband/wife/signficant other? do they have patience for using a Chemex? do they even know what a Chemex is?).

2) Find a coffee that matches their flavor preferences and coffee-making style (clearly explained by the Atlas coffee brewing guide)

3) Price out how much related equipment will cost if the teacher does not already own it (travel mug for taking to school, French press, Chemex).

4) Price out their subscription (including if they use the 5% lifetime discount).

5) Figure out when the teacher will break even given the cost of the equipment and subscription versus the daily purchase.

6) IB bonus: research the coffee industry in the country of origin of the chosen coffee.

7) Extension: find a k-cup brand and price out how much it would cost for the teacher to bring those to school to brew in the teachers’ lounge machine. Assess the environmental impact and time savings of this option.

I’m interested to see how this plays out with the kiddos. I like that this problem is open-ended and that they’ll have to apply the skill of “reaching out to an adult and actually speaking to them” as well as using linear equations and systems of equations in a context that’s not just a math problem that comes out to nice integer answers.

It feels good to be back at school! Here’s my 9th grade math lesson from day 2 (the first day of Coordinate Geometry), for the two 47-minute sections.

**Day 1: 1-a Distance Formula**

1) Name Graphs: Check off completion. [This assignment was to have students write their names on graph paper using only straight lines that begin and end at coordinate points on a grid.] Have 3 students draw a letter of their names on the board with points labeled.

2) Practice Quiet Coyote [my routine for getting attention–his ears are open but his mouth is shut]

3) Mini-Lesson: Distance Formula [taught using the 3 student examples]

– Look at how to solve for vertical and horizontal line distances.

– Look at diagonal distances to introduce the distance formula. Elicit the right triangle shape and Pythagorean Theorem.

4) Practice using Kuta worksheet (start with lines drawn on a grid, then ordered pairs). Introduce distance formula.

5) Formative Assessment: Plickers Exit Ticket (1 multiple problem with two ordered pairs) via Kuta multiple choice. Students turn their sheets and cards to the inbox.

6) HW: Haese & Harris Exercise 5A1 (answer key included)

7) Support for Math Enrichment: Extra WS practice. Give feedback on ET.

**Reflection**

Section 1: Students were slow to take out paper for notes and turn in HW. We need to make this process more automatic. I noticed that many needed prompting to label the notes with the skill and the date, or even to use their binders (which seem to have grown into a pile in the back of my room over the semester I was out on maternity leave). Some students do have some excellent, organized notes from the past semester, so I might start having students volunteer to share their notes for my “absent work” binder to help absent kids catch up. I forgot to practice Quiet Coyote explicitly but the quieting was better today. For the practice, students really had enough time for the first four problems (mostly because they got stuck with simplifying radicals–the answer key had listed the square root of 8 as 2 square root of 2). I had to write more directions for the close of class on the board to clarify what I wanted for the exit ticket (for example, students didn’t realize they had to turn in their work as well as the Plickers cards). I need to label the Plickers cards with the student names for both sections and make sure they go into a separate bin.

Section 2: This lesson went smoother overall because of anticipating cueing of the HW checking process (in which I handed Plickers cards out to save time later), prepping them to segway from a picture to two ordered pairs, and completing the exit ticket. More students completed the ordered pairs questions, but we will still review. A principal intern came in to observe my class, and it was good to hear this feedback: “Thank you for allowing me to come observe your class today and welcome back!! I liked that you put the answers on the back of the sheet so that they could self check, it fostered more discussion than I would have thought. The students were so engaged. It was interesting to listen to a couple of conversations where the student answer didn’t match the back and so the next step was to ask a partner about what they had come to and why. Plus those Plicker cards were awesome – such a great way to use technology without the lag time. I had never seen that before but I will be sure to remember it for a high tech/low complication formative assessment measure.”

You’re planning a lesson and you try to come up with super good question to ask to get kids to think about something. What is that question? Why did you phrase it the way you did? Why do you think it will prompt discussion/thinking?

I’m returning to work at my International Baccalaureate school on Monday after seven months of maternity leave, and teaching Middle Years Programme (MYP) Math 9 for the first time. I’d written a few MYP-style units before in spite of teaching only Diploma Programme (DP) classes. I’m trying to be more thoughtful and intentional about unit planning now to create more effective assessments that tie into the MYP global contexts, key concepts, and approaches to learning. I felt that my previous MYP units tried to fit those IB components versus having the assessments arise organically from them. I also didn’t want to plan in the disorganized way that I’d done before–thinking by day or by week rather than backwards from assessments. That leads to more daily work in spite of being less work up front. I also didn’t want to develop a creative project as a summative assessment rather than weaving it into formative assessments. I’ve often done end-of-term projects in the past, but this stressed students and me out because the projects were stand-alone and not designed before the unit started.

I listed out the skills for the first unit I will teach (Coordinate Geometry) and brainstormed project ideas that would assess those skills.

1-a Distance Between Two Points

1-b Midpoints

1-c Gradient

1-d Using Coordinate Geometry

1-e Equations of Straight Lines

1-f Distance from a Point to a Line (though after the fact I realized that systems of equations are an embedded skill; not sure how my students are with this)

I thought about trying a water park project like this, but realized that the types of skills I wanted to assess didn’t fit quite right with the way my students would likely design their water parks, so I’d end up making up some artificial constraints that limited their creativity. I was intrigued by the idea of placing benches at the midpoints of walking paths or by designing slides with constraints on the slope, so I tried to explore urban planning projects. Urban planning seemed so perfect for the MYP global context of globalization and sustainability, in which students “explore the interconnectedness of human-made systems and communities; the relationship between local and global processes; how local experiences mediate the global; the opportunities and tensions provided by world-interconnectedness; the impact of decision-making on humankind and the environment.” However, I couldn’t come up with an urban planning project whose scope would be appropriate for this particular 9th grade unit.

I kept coming back to the idea of having students draw pictures with lines of given slopes. I first experienced this in 7th grade (the 1992-1993 school year, eek) when my Algebra 1 teacher gave us an art project in which we had to create a picture using a given number of linear equations (our choice on the equations). I still remember my friend Gwen’s for its clever name. She made a kaleidoscope pattern named “Kaleidoslope.” Mine was “Starry Night on the Slopes”–my take on Vincent Van Gogh’s Starry Night. There are lots of slope picture projects and cool ideas out there (like this from Math Equals Love).

After I impulse-bought this magazine at Whole Foods, I began to explore the idea of artistic expression more.

I started to focus on the global context of personal and cultural expression, in which students “explore the ways in which we discover and express ideas, feelings, nature, culture, beliefs and values; the ways in which we reflect on, extend and enjoy our creativity; our appreciation of the aesthetic” the key concept of form (the “shape and underlying structure of an entity or piece of work, including its organization, essential nature and external appearance. Form in MYP mathematics refers to the understanding that the underlying structure and shape of an entity is distinguished by its properties. Form provides opportunities for students to appreciate the aesthetic nature of the constructs used in a discipline.”), and the approach to learning of communication.

My unit inquiry question will be: **How does math influence personal artistic expression?** I’ll weave in drawing activities throughout the unit and tie in the current trend of adult coloring books (like this, this, and this). For the final assessment, I’ll have them design their own quilt panel to represent themselves for a hypothetical time capsule that would get buried to commemorate their class (perhaps in a new building) and explore the mathematical properties of the quilt design. Maybe I’ll even get to tie in the history of quilts in the Underground Railroad. I’m psyched about trying these lesson and assessment ideas with my students and to see what they will come up with!

One of the best things about my husband is that he loves creating curriculum ideas with me. He helped me develop SAM animation math stories years ago, and recently had his high school interns make some of their own.

Complex Commutes

I originally chose to teach math because I thought that it’d be so cut-and-dry, with one right answer for everything. Eight years later, I find myself looking for ways to assess that have multiple right answers. A recent conversation with two of my best college friends reminded me of that, and led me to pick trig tale for my favorite. One of those friends is a German teacher, and she likes to give open-ended quiz questions that are unlike the typical matching ones. In a recent class, some students got quite agitated at the question “How was your last report card of freshman year?” One student said “but what do you want me to say? that the report card was good?” because she hadn’t studied for that (and probably couldn’t ask fellow students “so what’d you get for number 5?” after class). My friend could tell if they knew the vocabulary if the response made sense, and would have been fine with multiple answers. Another best (but non-teacher friend) piped up and said that in one of her favorite high school English classes, you could get 100% on any paper as long as you could justify your reasoning.

That’s why I like to do these projects in spite of the bigger grading effort that they require. Sometimes students protest the open-ended nature of this and claim they want to just do math problems instead, but they end up getting really into the story lines and figuring how to use the trig in ways that advance the story well. My mentor teacher used this during my student teaching year, and I find that creativity and fairy tales stand the test of time even as technology permeates ever more of the classroom. Though I do appreciate the many ways this could be done on the computer, sometimes it’s just fun to kick it old-school with construction paper and crayons.

**The Description**

The goal of this project is to have you develop a creative fairy tale that shows your understanding of the trigonometry skills from this unit. You may work with 1 or 2 classmates to outline the story, draft it, and illustrate it in class.

1) Unit Skills (32 points)

Your trig tale must demonstrate the following skills (4 points each, similar to quiz rubric).

Right Triangle Word Problems:

1. Find Missing Angle in Right Triangle

2. Find Missing Side in Right Triangle

3. Find Missing Angle in 3-D Figure

4. Find Missing Side in 3-D Figure

Oblique Triangle Word Problems

5. Apply Sine Rule to Find Missing Angle

6. Apply Sine Rule to Find Missing Side

7. Apply Cosine Rule to Find Missing Angle

8. Apply Cosine Rule to Find Missing Side

2) Writing, Illustrations, and Organization (22 points)

Elements of Fairy Tales (from http://www.readwritethink.org)

– Set in the past—usually significantly long ago. May be presented as historical fact from the past.

– Include fantasy, supernatural or make-believe aspects.

– Typically incorporate clearly defined good characters and evil characters.

– May include objects, people, or events in threes.

– Focus the plot on a problem or conflict that needs to be solved.

– Often have happy endings, based on the resolution of the conflict or problem.

– Usually teach a lesson or demonstrate values important to one’s culture.

**The Rubric: **TrigTaleRubric

*“Want to live next door to the famous “round house?” ;-)” *

A lot of condo listings and sublets often pop up on the Somerville Moms email list that’s so popular around these parts. This one intrigued me, so I hit the Google. Turns out the round house has some interesting history.

There is also a hexagonal house in Jamaica Plain and Octagon House in D.C.

I’ve had students design their dream houses before–might add on a historical research or current real estate / house design aspect!

**Day 2**

*Write about one piece of technology that you would like to try this year, and why. You might also write about what you’re hoping to see out of this edtech integration.*

I first heard about Plickers at EdCamp Boston and put them on my “to-try” list for 2015-2016. This year, two awesome math bloggers posted about them on one good thing (misscalcul8 and Sam Shah). I’ve found it hard to incorporate edtech that needed constant access to the computer lab, Chromebook cart, and school Internet. I’m hoping to cut down on the transition time usually inherent with such integration. I’m also hoping to improve my formative assessment.