*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.

*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.

*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?”*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”*Kate Nowak, Director of 6-12 Curriculum*

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*?”

*By Greta Anderson & Patti Drawdy, IM Certified Facilitator*

I read the lesson three times through, but was still unsure why the number line below shows $3 – 7$. My aha moment arrived courtesy of the grade 1 standards.

Continue reading “Representing Subtraction of Signed Numbers: Can You Spot the Difference?”*Jennifer Wilson and Vanessa Cerrahoglu*

Having an extended period of time to teach a lesson can be an advantage in a problem-based classroom. Students and teachers can savor the questions that are asked. Activities can breathe in a way that they can’t in a shorter period of time. But questions about planning inevitably arise. We find ourselves asking questions like: Do I simply merge two lessons? What stays? What goes? How do we ensure that we engage our students in the right conversations that will prepare them for the next leg of the journey?

Q: What is the fastest way to get a heated debate going about some topic in the IM 6–-8 math curriculum?

A: Show people this graph from Lesson 4 in Unit 8.5:

**You will never have to subtract again**.

Students sometimes learn about addition and subtraction of integers using integer chips. These are circular chips, with a yellow chip representing +1 and a red chip representing -1. You start with the all-important rule that $1 + (\text-1) = 0$, so you can add or remove a red-yellow pair without changing the number. To calculate the right hand side of the equation in the title, $3 + (\text-5)$, you put 3 yellow chips together with 5 red chips, then remove 3 red-yellow pairs, leaving 2 red chips. So $3 + (\text-5) = -2$.

** “**At the end of the day, this wasn’t about focusing on the objective, it was about making the objective meaningful to him.”

Continue reading “Engaging All Students in Meaningful Mathematics”