I blogged a few years ago about starting geometry with developing definitions. I’ve made some changes since then, and I have additional ideas for next year that I want to remember.
I’ve found it more useful to start the school year introducing geometry as art, and developing a need for definitions as students struggle to describe the process to create their designs.
Once there is a need for developing agreed upon definitions for terms, I want to use this video to motivate how and why definitions develop:
To make a definition is to highlight and call attention to a feature or structural property. Historically this comes out of working on a problem, not as a prelude to it. The point is you don’t start with definitions, you start with problems. Nobody ever had an idea of a number being “irrational” until Pythagoras attempted to measure the diagonal of a square and discovered that it could not be represented as a fraction. Definitions make sense when a point is reached in your argument which makes the distinction necessary. To make definitions without motivation is more likely to cause confusion.
Then, students will develop their own terms and definitions (We ended up with of holes, tubes, and bubbles – you can see the fun thread here.) and we can see how complete it is by trying to classify different objects: a sock? a slice of Swiss cheese? a block of Swiss cheese? etc…
After this introduction of what definitions are and how they work, I will use examples and counterexamples for students to work in small groups and develop definitions of other geometry terms, as described here.
I have been trying to use CPM’s new Precalculus textbook as a guide for my pre-calculus class. During my planing period, I opened to the next lesson to try to put together a plan for my upcoming class.
I only had about one hour to prepare for this class period.
This was the opening prompt:
Gerrit wants to simplify the complex fraction
but is overwhelmed by the fractions within a fraction. Work with your team to help Gerrit write an equivalent expression that is a rational expression instead of a complex fraction. Be ready to share your strategies with the class.
I stopped to think about how this would work with my class:
I thought back to when I addressed this topic last year – and how poorly it went. Students were frantically writing steps to memorize procedures. Although I am experienced enough to avoid saying “copy-dot-flip” or “invert & multiply“, I know this is how my students learned to divide fractions and it is how they approached these problems.
I decided I did not want to teach this the same way this time.
I considered just doing the lesson as described by CPM because I really didn’t have much time to plan and it was good enough. They are HS seniors. I can’t change their perspective on math and learning at this point – right? I tell myself this sometimes. To just keep it simple, but I never listen.
I read Dan Meyer’s blog post “If Simplifying Rational Expressions Is Aspirin Then How Do You Create The Headache?” I pictured myself asking students to evaluate a complex expression for specific values and whether or not they would be surprised (or even care) if I could evaluate the expression quickly. They have been beaten down by math for most of their lives. I thought this would just be another instance where a math teacher made them feel less competent, and that did not seem like a productive way to pique their interest.
“I don’t see how it’s doing society any good to have its members walking around with vague memories of algebraic formulas and geometric diagrams, and clear memories of hating them … Mathematics should be taught as art for art’s sake. These mundane “useful” aspects would follow naturally as a trivial by-product.”
I thought about Tracy Zagar’s session that I recently attended at ATMNE’s Conference where she talked about how connections through multiple representations made explicit can help students to develop understanding. Tracy also asked participants to think about related topics, often taught separately resulting in learners thinking of math topics as separate concepts. We identified related math topics that are taught separately such as:
graphing lines: f(x)=mx+b and transformations of functions f(x)=a f(x-h)+k
“There are two kinds of simplicity: that of naivety; and that which, by penetrating beyond superficial differences, brings simplicity by unifying.”
At this point I was losing valuable planning time, but I decided it was more important for me to make sense of dividing fractions for myself, and help students to experience the joy in understanding, than it was to have a flawless detailed lesson plan, so I found and worked through Graham Fletcher’s Making Sense of Invert and Multiply. During my lunch, I created and thought through as many cases of dividing fractions as I could come up with. Then I selected a few to use with my students.
I decided against starting with the opening question provided in CPM’s textbook above, I thought I would close with that question.
I decided to open the lesson asking students to think about and share how they would represent the number of groups of 1/2 that are in 3/4. Instead of beginning my class with these learners feeling intimidated and overwhelmed, they were curious.
Students discussed and compared representations, made connections and got genuinely excited at the silliness of being in a college credit pre-calculus class and that we were making sense of fourth and fifth grade mathematics. Mid-discussion, one high school senior yelled “I am in 12th grade and I just now understand how dividing fractions works!” Shaking her head with a mix of frustration towards how math is taught and satisfaction that she understood division of fractions.
Once they were ready, students worked in pairs on whiteboards to think through CPM’s opening question. They took time, consulted and corrected each other and all ended with the same simplification in different ways. Then the craziest thing happened:
They asked me for more complex fractions to simplify!
I recently attended a lecture by Kelly Wickham Hurst about systemic racism in America’s School system. The following is a combination of my notes from her lecture as well as a few resources that really helped paint a comprehensive picture for me of the current state of schooling in America.
When the Supreme Court decided on the case of Brown vs Board of Education mandating that schools be desegregated, wonderful black schools were closed and over 30,000 competent, qualified black teachers and administrators were fired. Partially as a direct consequence of this action, here is what students and teachers in American public schools currently look like, according to APM reports:
Malcom Gladwell has a great podcast called Revisionist History, and last season it included an incredible episode about the negative impacts of Brown vs. Board of Education:
The Brown v Board of Education might be the most well-known Supreme Court decision, a major victory in the fight for civil rights. But in Topeka, the city where the case began, the ruling has left a bittersweet legacy. RH hears from the Browns, the family behind the story.
Adam Ruins Everything made this 6 minute video that explains and summarizes the impact of red-lining in creating segregated suburban communities:
Nikole Hannah-Jones is a force. She was recently named a MacArthur Genius Fellow for 2017, partially as a result of this series of podcasts:
Right now, all sorts of people are trying to rethink and reinvent education, to get poor minority kids performing as well as white kids. But there’s one thing nobody tries anymore, despite lots of evidence that it works: desegregation. Nikole Hannah-Jones looks at a district that, not long ago, accidentally launched a desegregation program.
Be aware of your own biases. We all have biases, but increasing our awareness of them can help up to make conscious adjustments to ensure that our prejudices have minimal impact on our decisions. Try taking Harvard’s implicit-association tests.
Where do we go from here? Below are the ideas presented by Kelly Wickham Hurst at her talk as well as a link to a article she wrote in 2016 on the topic.
My biggest takeaway from Hurst’s talk was to stop getting stuck analyzing data and start acting and having hard conversations. This requires you to critically consider who is served when implementing school and local polices with which you may have been participating and complying in for your whole life.
When you see something in schools that marginalizes a population, or a well intentioned policy that you recognize as being harmful to students, do something. Don’t just passively comply. Speak up. Change the rules. Run up the flagpole and take that shit down.
I am starting the school year in precalculus having students develop both their collaboration skills and their ability to model real world scenarios with mathematics. It is the beginning of the school year and many learners are still a little uncomfortable with each other and with their confidence in math class. I wanted to have a productive class discussion, and to make students feel safe to engage in dialogue around the content. I decided to create a discussion that appeared natural as a gateway to get some of the less confident learners engaged in the content.
At the start of class I presented a scenario to optimize by modeling with mathematics. Learners worked in randomly assigned small groups to develop an optimal solution and then, in the last 20 minutes of class, each group was to present their solution and reasoning. I wanted the learners in the audience to ask challenging questions to learn more about the groups thinking after each presentation.
Here is what I did:
While listening to learners work on a task in small groups, I circulated, listened, and wrote questions for each group on separate index cards. I was also thinking about how to sequence their presentations based on their approaches as described in the book, 5 Practices for Orchestrating Productive Math Discussions.
When each group was setting up to present, I would discretely hand the questions written on index cards to a less confident learners. After the group presented their thinking, I would ask, “Does anyone have any questions for this group?”
The learners would look around nervously in an awkward silence while I glared uncomfortably at a learner with an index card. Eventually, uncomfortable with the silence, the learner would ask the question on their card. Then, the presenting group would respond, which led to genuine student questions, thinking and further discussion.
By the last groups presentation, many of the learners seemed comfortable to ask questions and engage in thoughtful whole class discussion!
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This is my first year teaching AP Computer Science Principles. I am super excited to bring the first AP class to my tiny alternative high school. I attended AP training through code.org’s TeacherCon in Phoenix, AZ in July 2017. It was very helpful and so far, I am enjoying using their curriculum.
I was contacted by Chalkbeat to answer a few questions for their “How I Teach” series. They asked some difficult questions and it took me a lot of thought to answer them. I appreciated having to stop and think about what I do.
Chalkbeat is a nonprofit news organization committed to covering one of America’s most important stories: the effort to improve schools for all children, especially those who have historically lacked access to a quality education.
I attended Twitter Math Camp for only 2 of the 3.5 days this year. I learned so much. Each of these items below could be a separate blog post, but for now I am just listing them in order to put them somewhere and remind myself of what stood out to me this year.
I usually use the last six week of the school year in geometry to teach circles: Arc lengths, chords, secants, tangents, etc. Which I have outlined in the Geometry Planning Guide
The following school year students begin Algebra 2 and promptly forget these properties of circles. They also start the new school year frustrated and overwhelmed because they forgot all about the quadratics they learned in Algebra 1 during their year in geometry.
This school year I decided to make some changes to the last unit in order to set students up to be more successful in the transition from Geometry to Algebra 2:
I distributed an index card to each student and gave them students 5 minutes to make a beautiful work of art on the card. When time was up, students turned in their art, I shuffled the cards and redistributed them to the class.
I cut orange paper to the same size as 3×5 inch index cards, and told the students it was “Real Gold” it was very expensive and we could not waste any of it (they referred to orange paper as “Gold” for the rest of the school year). They had to try to make a frame for the art and tape it down to the assignment sheet that I distributed to them with an explanation of their thinking and strategy.
It worked. They begged me to help them find a better way to complete this task.
2) Complete the Square with algebra tiles.
I used this activity from Salt Lake City Schools to guide student thinking. Students worked in small groups to create squares using algebra tiles and relating the squared and factored forms of perfect square quadratic equations. We referred back to this task often throughout the unit.
3) Solving equations by Completing the square notes and practice
I used Sarah Hagan’s foldable for completing the square. The following day I planned a basic practice sheet for students to just build fluency with solving equations by completing the square, but it was so nice out, we decided to do this on the sidewalk instead. (Note: Always keep sidewalk chalk in your classroom for beautiful day emergencies)
4) Applications of completing the square
I returned students framed art task from day 1 above, and they wrote and equation and found the appropriate frame size using completing the square, then measured their estimated solution and reflected on their work.
I printed and laminated sets of the domino cards from the Shell Center task: Representing Quadratic Functions Graphically and students completed the loop and filled in the blanks. They then were able to summarize the relationship between standard, vertex and factored form of a quadratic equation and understand what each of the forms of the equation illustrated about it’s graph.
The next day, students worked on the Desmos Activity: Match My Parabola. I was able to pause and pace this activity as needed and monitor students understanding in order to support students understanding of the various forms of a quadratic equation.
6) Converting a quadratic equation between vertex, standard and factored forms
Now that students understood the usefulness of the various forms of quadratic equations, they wanted to be able to convert a quadratic equation between the various forms:
6) Develop the equation of a circle
I started with this question:
There were a few students who shouted out “Pythagorean Theorem!” and students realized they could construct a right triangle and find the length of the hypotenuse/radius.
Next, I gave students individual whiteboards and asked them to draw a circle centered at (0,0) with a radius of 5. Then I asked them to name coordinates of points that they know for sure were on the circle and I would have them explain how they knew. Eventually we identified 12 points on the circle.
After this, I asked students what the relationship was between the x and y coordinates on the circle. Most groups were able to explain that they were all related by the Pythagorean theorem because x² + y² = 5².
At this point, I had students complete the first side of the “Going Round in Circles” sheet from this Shell Center lesson in order to see how well students understood the discussion and to see how they could apply their learning.
The next class period began by asking students to find the radius of a circle given the coordinates of the center and a point:
We used this to determine the radius of this circle as (7 – 2)² + (15 – 3)² = 13²
After further discussion, students were able to generalize the equation of a circle centered at (h, k) to be: (x – h)² + (y – k)² = r²
Then students worked in small groups on the vertical white boards to complete the task included in the Shell center lesson: Sorting Equations of Circles 1
I printed and laminated the cards to make them easier to use on vertical dry erase boards and for facilitate discussion.
Finally, students completed the “Going Round in Circles, Revisited” sheet included in the lesson linked above.
7) Find the center and radius of a circle by completing the square
I created a foldable for students to summarize the equation of a circle that included examples of using completing the square to put an equation of a circle into a standard form. We followed this with additional practice.
<foldable below should be printed on legal sized paper>
Every time we referred back to completing the square I made visual diagrams connecting it back to their initial development of their understanding of how to complete a square. I never stated any shortcut like, “just divide by 2 and square it.” Students developed a genuine understanding of the process, which will hopefully lead them to increased success as they begin algebra 2 next school year.
I want good feedback from my students, but I also want them to take it seriously & I think this survey in its current state is way too long for students to endure the entire thing. I’m considering it my question bank at this point & I plan to delete questions depending on the class. I searched for something like this to start with because I did not want to re invent the wheel, but I could not find one, so I am sharing mine so that hopefully other teachers can use it as a starting point for their own survey. (I cannot figure out how to make Google forms visible for you to create a copy without changing the original, so request access if you would like it to use as a staring point for…