Rigor in Proofs

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

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?”

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IM K-5 Math: Designing for Each Student

By Noelle Conforti Preszler and Kristin Gray

In the following activity, think about the students in your classroom. How might each respond?

What do you notice? What do you wonder?

This activity is the drafted warm-up of the first lesson in Grade 3, Unit 1: Introducing Multiplication. While we believe the structure of this activity — “What do you notice? What do you wonder?” — implicitly supports equity, it is the word each in the question at the top that has become central to our design of the IM K-5 Math curriculum.

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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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Realizing the promise of open resources

By William McCallum

All of our curriculum here at Illustrative Mathematics is released under a Creative Commons Attribution (CC-BY) license, which allows anyone to “copy and redistribute the material in any medium or format” and to “remix, transform, and build upon the material for any purpose, even commercially,” on the conditions that attribution is given and that others’ rights under the license are not restricted. This is both super scary and super exciting.   Continue reading “Realizing the promise of open resources”

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”

IM Preparing for the School Year

There are always so many things to do in preparation for a new school year.  At this point of the summer, to-do lists start getting made, materials get purchased, rooms are organized, and math class planning begins. Whether you are using the IM 6–8 Math curriculum for the first time or entering your second or third year with the program, there are always new things to learn. While the Illustrative Mathematics blog is packed with great information from curriculum authors, teachers, and coaches, it can often be a job in and of itself to narrow down what to read. Continue reading “IM Preparing for the School Year”

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