Rigor in Proofs

By Tina Cardone, Geometry Lead, & Gabriel Rosenberg, Curriculum Writer

There is no doubt that proof plays a central role in the human endeavor of mathematics, but there remains much debate on what role it should play in high school mathematics. At least two standards for mathematical practice in the common core focus on this concept. Certainly MP3, “Construct viable arguments and critique the reasoning of others”, is about the need for students to be able to write their own proofs and to analyze the proofs of others. MP6, “attend to precision” goes deeper, though, by noting the need for precision, including the use of clear definitions, when communicating their reasoning. This is what we mean by rigor in mathematical proof.

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Presenting IM Algebra 1, Geometry, Algebra 2

By Kate Nowak

When I was teaching high school mathematics, my local colleagues and I spent a whole lot of time creating problem-based lessons. We were convinced that this style of instruction was a good way to learn, but the textbooks in use at our school simply contained definitions and theorems, worked examples, and practice problems. One day I was talking to my dad about how much time I had been spending lesson planning. His response was, “People have been teaching geometry for, what, 3,000 years? Shouldn’t the lessons be, like, already planned?”

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How do you start the year?

By Ashli Black, Algebra 2 Lead

Students need a chance at the beginning of the year to shake off the summer dust. Learn how IM’s curricular design builds in opportunities for review while starting the year with inviting, grade-level mathematics.

One of the things I love about the IM 6–8 Math curriculum and the forthcoming HS curriculum is that none of these courses start with review. While starting with a short unit of review was something I tried when I first started teaching, I later decided on a different route because the review time wasn’t helping my students in any way I could measure. If a student already had graphing and fractions and factoring down, how did the review help? If a student didn’t have those sorts of skills down, how could a single week or two be enough? And how long would the classroom status issues revealed during the review time last?

I eventually changed from review to giving a short quiz during the first week of class. This quiz always had two purposes: to introduce students to what my quizzes looked like and communicate to my classes how I interpret the results of a quiz. Question 1 was always to plot a few points on the coordinate plane. I never had a class get over 80% on the first go. Not even the precalculus classes. Does a result like that mean pausing to spend a day on graphing points?

The next day, after handing back the quizzes—ungraded, comments only—I would display the class average for that question. There was always at least one student who would shout out that they forgot what negatives were, or mixed up $x$ and $y$. Others would then murmur agreement or argue that their brains were still in summer. They would do better next time.

And they did. Always. One thing I know about myself is that it takes time to get back into a groove. The same was true for my students and their mathematical selves. I ask a straightforward question like “plot these points” and then have a discussion about the results to communicate to my students that I knew their brains were still a bit in summer mode and that I wasn’t going to expect they come into class right where they left off at the end of last year. The quiz gave me an opportunity to clearly make the point that review of a variety of skills would be built into our daily work. I also needed that quiz to remind me that the faces I was looking at were not the same as the students I’d last seen sitting in those chairs after a full year of instruction.

The results of that question on the quiz didn’t mean to stop everything and work on that skill, they meant that I needed to look for an opportunity within the grade-level work to highlight things to remember about graphing points. Each course in the IM curricula starts with a unit that has built in opportunities for students to shake off the summer dust and show who they are as mathematicians while keeping the focus on grade-level mathematics. In IM 6–8 Math, that meant a focus on geometry at the start of the year. In Algebra 1 it’s one-variable statistics. In Geometry it’s Constructions and Rigid Transformations. In Algebra 2 it’s Sequences and Functions.

In all of these courses, the first unit gives time to introduce students to the instructional routines they will use throughout the school year, in particular those they are likely less familiar with such as Info Gap Cards. Students have fewer preconceptions about their abilities (and those of their peers) when we focus on new ideas at the start of a year, so these units give an opportunity to set classroom norms for communication, collaboration, and making connections.

In the first unit of Algebra 2, attention was given to the fact that a student who has followed the “traditional” sequence of Algebra 1, Geometry, Algebra 2, may be out of practice with some key algebraic ideas. This unit takes a look at the familiar concepts of linear and exponential functions through the new lens of geometric and arithmetic sequences. The activities ask students to represent relationships using words, tables, graphs, and equations. They learn to define sequences recursively using function notation in a new way, giving contrast to how they have written equations for linear and exponential functions in the past.

As they build new skills while honing older ones, students also get to experience the beginnings of an important part of Algebra 2: modeling. In viewing the IM high school pacing guide, you may notice that Algebra 2 looks a bit shorter than the courses preceding it. This was purposeful to give space for teachers to make use of the modeling prompts that will be part of the course. Here in Unit 1, students address an aspect of mathematical modeling when they make the decision for what input value goes with the first term of the sequence. Often this boils down to choosing 0 or 1, which is highlighted in the task A Sierpinski Triangle. While this choice can appear trivial, making a decision and recognizing the consequences that decision has on different parts of the modeling process is necessary training for students if we want them to be powerful mathematical modelers. And while this is going on, students are plotting points on graphs and thinking critically about equations for linear and exponential functions, blowing summer dust away.

Next Steps

  • Check out the Unit 1s for the high school courses (Algebra 1, Geometry, and Algebra 2). Where do you see embedded review in these lessons? Where could you build in review to your Unit 1 for the skill practice some students need more time on?
  • Give your students an ungraded quiz on something you would expect them to get 100% on. What do you notice about the results?
  • Check out the demo of IM 9-12 Math Algebra 1, Geometry, Algebra 2 which is now available on our website. To access the full curriculum, please see our IM Certified™ partner page
  • See unit 1s for the high school courses. Where do you see embedded review in these lessons? Where could you build in review to your Unit 1 for the skill practice some students need more time on?
  • Give your students an ungraded quiz on something you would expect them to get 100% on. What do you notice about the results?

What is problem-based instruction?

By William McCallum

When I was a child, I used to get puzzle books out of the library. One of the puzzles was the twelve-coin problem, the most difficult of all coin weighing problems. My mother and I worked on it separately at the same time, and she solved it first. Some time later that evening she came into my room to find me in tears of frustration. Instead of helping me, she asked: “Do you want me to tell you the solution?” I said no and she left. I will never forget the joy when I finally figured it out.

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Extra Supports for Algebra 1: The Gateway Resources

By Sadie Estrella

Illustrative Mathematics’ high school curriculum is scheduled to be released this summer. This is an exciting time for Algebra 1, Geometry, and Algebra 2 teachers. I honestly am ready to take a job at a school just to have the opportunity to teach with this material (and everyone knows I am always dreaming of being back in the classroom). However, I want to bring light to a hidden gem I think not too many people are aware of that is also part of our high school materials.

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Truth and Consequences Revisited

By William McCallum

What are extraneous solutions?

A while ago I wrote a blog post about solving equations where I talked about seeing the steps in solving equations as logical deductions. Thus the steps
\begin{align*}3x + 2 &= 5\\3x &= 3\\x &= 1\\ \end{align*}

are best thought of as a sequence of if-then statements: If $x$ is a number such that $3x + 2 = 5$, then $3x = 3$; if $3x = 3$, then $x = 1$. Continue reading “Truth and Consequences Revisited”

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