Why is 3 – 5 = 3 + (-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 $3 + (\text-5) = -2$. Continue reading “Why is 3 – 5 = 3 + (-5)?”

Planning Lessons for a Block Schedule

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?

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Professional Learning Through a Fraction Task Progression

DOWNLOAD THE IM FEATURED FRACTION TASKS

Teaching mathematics is a continuous cycle of identifying where each student is in their learning trajectory and determining meaningful ways in which to build on their current understandings. While we often have little control over students’ mathematical experiences before they walk into our classrooms, we do have complete control of our own learning.

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Parent Math Night Using Illustrative Mathematics

Open House night; cue anxiety and sweaty palms! Hope my students’ parents don’t mind.

Photo Courtesy of Robert Eiselstein

I just began my seventh year of teaching middle school mathematics. Middle school is a limbo land filled with prepubescent pre-teens, drama, and students trying to find their individual voice without drawing too much attention to themselves (sigh). There are sixth grade boys and girls in my class who are taller than me, 5’9”. Some of the boys have mustaches while others still look like they’re in third grade. It’s a difficult year for the students. This is their last year before moving onto the even weirder, and much more confusing junior high. Students are anxious about this being the last year of elementary school, and so are the parents; maybe even more anxious than their little boys and girls becoming young men and women. I think it is my job to help ease this transition, and to get them excited about what is to come.

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Planning for Meaningful Practice

There is no shortage of available math resources for teachers to use in their classrooms. The difficult and time-consuming job for teachers is weeding through all of the tools to decide which best supports students in learning mathematics. It is a difficult job because it first involves thinking about how students learn mathematics and then, after choosing a resource, ensure it is being used to best support students’ learning. Our team at Illustrative Mathematics has worked closely with partners such as Khan Academy to align their resources with the IM 6–8 Math curriculum so teachers can feel confident using them in their classrooms to support student learning. In aligning these resources, we keep the focus of how students learn mathematics at the forefront, while considering the type of support the additional resource is providing.    Continue reading “Planning for Meaningful Practice”

Say What You Mean and Mean What You Say

By William McCallum

Blog Post Bill 9.11.18
What patterns do you notice in the count? How would you explain why these patterns occur?

In one of our professional development workshops, there is an activity in which the facilitator asks teachers to skip count by $\frac34$. The facilitator records the count, $\frac34$, $\frac64$, $\frac94$, . . . and then asks for patterns they notice in the recording. In a recent workshop, a group of grade 5 teachers noticed that the numerator increased by 3 each time but that the denominator remained unchanged. When the facilitator asked why, they could easily explain why the number of pieces was increasing in the numerator, but couldn’t really give an explanation for the denominator other than “it is just always out of 4.” The funny thing is, they weren’t saying “3 out of 4, 6 out of 4, 9 out of 4” when they skip counted. They were saying “3 fourths, 6 fourths, 9 fourths” and writing it in fraction notation. The key to understanding why the denominator stayed the same was hidden in plain sight, in the very language they were using to name the fractions. Continue reading “Say What You Mean and Mean What You Say”

What is right about wrong answers?

When I first started teaching, at the end of each day, I would open my teacher’s guide, grab my pen, and thumb through the stack of completed worksheets. My eyes would dart quickly from the red answers in the teacher’s guide to the corresponding answers on each student’s page. I would dole out my x’s and checks with finality and authority. When I got to the end of a page, I would tally a percentage score and enter it into my electronic grade book. I approached every piece of student work as if it were a summative assessment.

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