Continue reading “Co-Creating Classroom Norms with Students”Establishing norms is critical to creating an environment where all students see themselves as knowers and doers of mathematics. Reflecting on the Illustrative Mathematics mission statement,

Creating a world where learners know, use, and enjoy mathematics,how can we contribute to this mission as first year teachers implementing the IM curriculum in our classrooms?

# Introducing IM Certified 9–12 Math

IM Algebra 1, Geometry, and Algebra 2 courses are now available to all.

**Alright, folks, this is not a drill: IM 9–12 Math is now available to all.**

*By Ashli Black*

So now what? To help folks dive into the curriculum, we’ve put together some links to the curriculum and some relevant blog posts here. No matter what your experience with IM curricula, this post will give you a place to start.

Continue reading “Introducing IM Certified 9–12 Math”# Making Peace with the Basics of Trigonometry

Six months ago, I hated trigonometry.

*By Becca Phillips*

In fact, when my daughter missed a week of school, she announced on her first day back, “Someone has to teach me trig because I missed the whole thing.” Her father jumped in, “That’ll be me. Your mother hates trig.”

At least that used to be true. I have since made peace with my least favorite topic, in large part because of my experiences with the Illustrative Mathematics Geometry course. Let me tell you ways that the IM Geometry course has helped.

Continue reading “Making Peace with the Basics of Trigonometry”# Making Sense of Distance in the Coordinate Plane

*By Linda Richard, Curriculum Writer*

I used to teach my students a catchy song to memorize the distance formula. We all had fun goofily singing this song. My students hummed it to themselves during tests and successfully calculated distances. I was pleased with this outcome—but what did my students actually *understand *about distance in the coordinate plane? In retrospect, very little.

# Developing Conceptual Understanding and Procedural Fluency

*By Melissa Schumacher, Curriculum Writer*

Which is more important for students to have: conceptual understanding or procedural fluency? Does one have to be taught before the other can emerge?

# NCSM NCTM Recap

*Illustrative Mathematics*

It was great to see so many of you at NCSM and NCTM in San Diego. If we missed you, or you weren’t able to attend, read our NCSM and NCTM round-up below.

# What is a Measurable Attribute?

*By Kristin Umland,VP Content Development*

A great conversation I had with the IM elementary school curriculum writing team got me thinking: What *is* a measurable attribute? That is, when given an object, what can we measure about it? Before you jump in with your own answer, consider these questions:

Is “redness” a measurable attribute? Why or why not? Does this picture help you decide?

Continue reading “What is a Measurable Attribute?”# 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.

Continue reading “Rigor in Proofs”# 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*?”

# 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?