On Similar Triangles

By Ashli Black

The fact that a line has a well-defined slope—that the ratio between the rise and run for any two points on the line is always the same—depends on similar triangles.
(p.12, 6–8 Progression on Expressions and Equations)

As students are building their understanding of dilation at the beginning of grade 8 in Unit 2 of the LearnZillion Illustrative Mathematics 6–8 Math curriculum, an activity asks students to dilate different quadrilaterals using a given center and dilation factor on a square grid. Here are the results of two of the dilations in that activity involving triangles: Continue reading “On Similar Triangles”

Sometimes the Real World Is Overrated: The Joy of Silly Applications

By Charles Larrieu Casias

One of the cool things about math is that it can provide powerful new ways of seeing the world. Just for fun, I want you to open up this lesson from the grade 8 student text. Take a quick skim. What do you notice? What do you wonder?

When writing this lesson, I was guided by a few key questions:

  1. To paraphrase Dan Meyer: If I want arithmetic with scientific notation to be the aspirin, then how do I create the headache?
  2. What are some weird, silly comparisons involving really large numbers?
  3. Here, towards the end of 8th grade, what should students be doing to transition towards the high school mathematical modeling cycle?

Continue reading “Sometimes the Real World Is Overrated: The Joy of Silly Applications”

Warm-up Routines With a Purpose

By Kristin Gray

As a teacher, curiosity around students’ mathematical thinking was the driving force behind the teaching and learning in my classroom. To better understand what they were thinking, I needed to not only have great, accessible problems but also create opportunities for students to openly share their ideas with others. It only makes sense that when I learned about routines that encouraged students to share the many ways they were thinking about math such as Number Talks, Notice and Wonder, and Which One Doesn’t Belong?, I was quick to go back to the classroom and try them with my students. It didn’t matter which unit we were in or lesson I had planned for that day, I plopped them in whenever and wherever I could because I was so curious to hear what students would say. Continue reading “Warm-up Routines With a Purpose”

Why is the graph of a linear function a straight line?

By William McCallum

In my last post I wrote about the following standard, and mentioned that I could write a whole blog post about the first comma.

8.F.A.3. Interpret the equation $y = mx + b$ as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function $A = s^2$ giving the area of a square as a function of its side length is not linear because its graph contains the points $(1,1)$, $(2,4)$ and $(3,9)$, which are not on a straight line.

Continue reading “Why is the graph of a linear function a straight line?”

Why We Don’t Cross Multiply

By Kate Nowak
(co-authored with Kristin Gray)

“Ultimately, the goal of this unit is to prepare students to make sense of situations involving equivalent ratios and solve problems flexibly and strategically, rather than to rely on a procedure (such as “set up a proportion and cross multiply”) without an understanding of the underlying mathematics.”
Illustrative Mathematics 6–8 Math, grade 6, unit 2, lesson 12

Continue reading “Why We Don’t Cross Multiply”

Vocabulary Decisions

By Bowen Kerins

A wide-ranging team worked together to develop the Illustrative Mathematics Grades 6–8 Math curriculum. Many of the authors were and are experienced teachers of Grades 6–8, while others are experienced high school teachers.

My own experience is as a high school teacher, then a high school curriculum writer. One of the ways the IM team’s experiences led to a higher-quality product was the discussion around language and terms used throughout the three grades. Continue reading “Vocabulary Decisions”

Not all contexts have the same purpose

By Nik Doran

We sometimes use familiar contexts to understand new mathematical ideas, and sometimes we use familiar mathematical ideas to understand what is going on in a context. We do both of these things by looking for parallels between the familiar and unfamiliar structures. I want to highlight two places this happens in the Illustrative Mathematics 6–8 Math curriculum. (It’s easy and free to sign up to see the teacher materials.) Continue reading “Not all contexts have the same purpose”

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