Not going to jinx it but…

CafKo6RUkAAGETRThe fall term finished amidst a lot of anxiety – students anxious about their grades, me anxious about their reactions to their grades. Regents week was surprisingly restorative, and I was able to give a lot of thought and attention to what I wanted to change/try/implement in the spring term.  I am teaching, in addition to Algebra 2/Trig, two sections of the ‘non-Regents’ Geometry course.  This was a course I developed last year, but the pacing and units have been changed to be better aligned with the Common Core standards.  Thankfully, the composition of the class has changed as well.  Last year, the roster included a combination of sophomores who were perceived to be unlikely to succeed in the Regents class, off-track juniors and seniors who needed a math credit, as well as an over-compliance-limit number of students with IEPs and English Language Learners.    This mixture made for some very interesting times in class, but you can look in my blog archive if you want to read more about the past.  This year, the students programmed for the non-Regents Geometry class have come through a four term Algebra 1 class (don’t ask), and, for the most part, are on track for graduation.  Because of my Problem-Solving elective, I didn’t teach the first term of Geometry, but the reports on the fall classes from the two teachers who did were good.  I want to keep this momentum going, so I spent a lot of time visualizing lessons that were accessible and rigorous, with students doing the heavy lifting in class rather than me.  The first unit includes right triangles and trigonometry, and begins with a rather very dry topic – radical expressions.

Not only can radicals be a difficult topic to motivate, but the range of student experience and understanding can be very wide; this is a topic this is included in the Algebra I curriculum.  My lessons needed to be intriguing enough to keep those students who remember the content engaged, while opening the door for those who do not remember, or perhaps never understood it in the first place.  Thanks to the master of creating interesting activities out of even the driest of topics, Don Steward, my first two lessons were well-structured, well-scaffolded, and engaging.  And thanks to starting the term with the 100 number activity shared by Sara Vanderwerf and Megan Schmidt, and classroom seating using  Visibly Random Groupings every day, my students are already talking to each other mathematically, as well as beginning to work cooperatively and supportively.

UntitledBy opening with a warm-up which related square roots to area, I was able to give the topic a visual context, as well as introduce the idea of a cube root.  We reviewed perfect squares through 144, and I was impressed to see how seriously the students were taking this content that I am certain they have seen before – without being prodded or asking “do I need to copy this?”, they all created tables of the perfect squares in their notebooks.  [Side Note: I am going to do a modified INB in this class (I’ve been given notebooks and supplies thanks to  a Donorschoose grant which was funded Untitled2in record time!), easing up on page numbers and exactly copied content.] We then looked at determining between what integers some irrational square roots fell, which helped the students understand that not all numbers have perfect squares, but still have square roots.  The final activity on the Untitled3first day of the lesson was an exploration of the representations of square roots.   It was great to hear the students working together on each of these small activities.  I had some challenge worksheets in hand for a few students who knew the content fairly well, but needed both review and to be more deeply engaged so as not to hijack all class participation.  This intervention, too, was successful. (Dare I say ‘score!’?  I don’t want to tempt fate.)

Picture8Today we continued with simplifying non-perfect square roots.  Again, student understanding at the outset of the lesson ran from a truthful “I know how to do this already” to evident misconceptions.  Despite the fact that a lot of the content on this warm-up previews next week’s work, it got most of the students working – even if they didn’t remember exactly how to simplify radicals, they began messing around with the first example.  The increasing difficulty of the problems kept the students who were ready interested.  I wrote the perfect squares through 144 on the board for reference in looking for perfect square factors, although [and I’ve seen this before] most students resist using that support and instead use guess and check with a calculator.  When they worked on p204some practice examples, I gave them the worksheet in a page protector with a dry erase marker, and distributed only enough so that they needed to share.  Friday afternoon before a long weekend, after a very snowy morning – they worked all period.  I’ll take it.

An interesting note regarding the puzzle worksheet (I’m sure you’ve seen these or some like them; a successful solution to all problems rewards the student with a silly pun.  However, for my English Language Learners (13 of which were put in the same class without any kind of heads up to me), the puns are meaningless.  I think there’s a literacy lesson in here for me to develop.

A huge amount of thought went into these lessons on my part; I’ve taught radical expressions almost every year of my teaching career (now numbering 10!) but these lessons were more engaging, I think, for the students AND for me.  I like that I was able to improve on something I’ve done so many times before, even with borrowing all of those wonderful materials, and I’m pleased that I saw an appropriate result.  I’m looking forward to revamping more lessons for this group of students – their involvement and effort is a wonderful reward.



  1. Marisa Aoki (@MarisaAoki)

    It’s funny – my post this week is about revamping how I teach simplifying radical expressions as well! I thought it was serendipitous that you happen to be just a few comments above me so that I stumbled across your thoughts on this topic!

    I agree that this topic tends to be tedious and dry, so finding ways to make it more applicable, relatable, and interesting can be a challenge! My students are learning it for the first time, and my approach was a little different, but I like your ideas and will probably steal some for next year! 🙂


  2. Pingback: Day in the Life: 2 First Days | Her Mathness

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