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Fold and Cut

Fold and Cut

Prologue

  • Ontario Grade 8
    • Curriculum and Connections: Patterning and Algebra
      • "Sort and classify quadrilaterals by geometric properties, including those based on diagonals, through investigation and using a variety of tools"
    • Ontario Curriculum
      • 8m40 (Overall Expectations) demonstrate an understanding of the geometric properties of quadrilaterals and circles and the applications of geometric properties in the real world
      • 8m41 (Overall Expectations) develop geometric relationships involving lines, triangles, and polyhedra, and solve problems involving lines and triangles
      • 8m43 (Geometric Properties) sort and classify quadrilaterals by geometric properties, including those based on diagonals
      • 8m45 (Geometric Properties) investigate and describe applications of geometric properties in the real world

Act I

  • Is there a way to fold a triangle so that it can be cut out by using only one cut?

Act II

  • Suggest attempting a simpler problem. Ironically, both of the following are easier.
  • Or maybe watch one failed attempt at cutting a triangle out in one cut

Act III

  • One possible solution for cutting out the triangle in one cut (view as image)

Sequel

  • Try cutting out a parallelogram in one cut. Try other polygons.
  • Can you cut out the letters of the alphabet using this theorem?
  • Check out Erik Demaine's awesome site
  • Numberphile explains the "Fold and Cut Theorem"

https://www.youtube.com/watch?v=ZREp1mAPKTM

Ontario Math Continuum and Connections

Introduction

I've been trying to find a more flexible way to approach teaching mathematics to students, one that shows students that everything they learn over the course of a year is interconnected. The Continuum and Connections resource does a really good job of breaking the grade 7, grade 8, grade 9 and grade 10 math curriculums into units containing all five strands. However, the documents are a bit too large to work with on a daily basis. This is my attempt at making this resource into something that is immediately useful in the classroom.

It is still important to refer to the curriculum to make sure that all expectations are covered, but on a day-to-day basis, I really find this document easier to work with.

Overview

There are six units for each of the grades. In each of those units, I have placed the plans for grades 7, 8, 9 and 10. This makes it a bit easier to compare across grades, but unfortunately takes a couple extra moments if you are looking for all the plans for just one grade.

  1. Fractions;
  2. Integers;
  3. Patterning and Algebraic Modelling;
  4. Perimeter, Area and Volume;
  5. Proportional Reasoning; and
  6. Solving Equations.

Here are the download links for the full document:

Curriculum and Connections

The "Continuum and Connections" (link to page) documents are definitely worth consulting in their original context. In case that ever goes offline, I have also mirrored the documents here:

 

Fractions

 

Integers

 

Patterning and Algebraic Modelling

 

Perimeter, Area and Volume

 

Proportional Reasoning

 

Solving Equations

 

From Patterning to Algebra

From Patterning to Algebra

Learning Goal and Success Criteria

Over the course of this chapter, we want students to understand that pattern rules, tables of values and graphs are simply different ways of visualizing patterns. There are a number of specific expectations that we want to cover, listed in the following photos.

Vocabulary

Before starting, we added a number of new words to our strategy walls. Students don't need to memorize the words, but they do need to understand them and be able to use them when they are working.

Graduation

The first project we worked on for this section was, "Graduation". Here, we wanted to start really using tables of values, pattern rules and graphs to solve problems. Here is a variety of answers, with comments on each. I kept one hypothesis, and then a bunch of graphs and tables of values.

New Success Criteria

After completing the "Graduation" problem, we constructed new success criteria based on what we learned. Here they are.

New Success Criteria

Exit Ticket

This exit ticket tested each student's ability to use what they learned in small groups, by applying it to a specific problem. Students were allowed to use everything from their strategy walls, including the success criteria, to do this problem. However, the problem was done individually, in class.

Word: 2016-03-04 From Patterning to Algebra - Pattern Rule PDF: 2016-03-04 From Patterning to Algebra - Pattern Rule

2016-03-04 From Patterning to Algebra - Pattern Rule

Understanding the Relationship Between Percents, Fractions and Decimal Numbers

Understanding the Relationship Between Percents, Fractions and Decimal Numbers

In this chapter, students are supposed to understand the relationship between percents, fractions and decimal numbers, and that they are three ways to represent the same thing.

Dueling Discounts

The first problem we did together was called "Dueling Discounts" by Dan Meyer. We studied the problem on the attached page. Students learned how to deal with percentages and decimal numbers.

Nana's Lemonade

The next problem we studied was "Nana's Lemonade", also by Dan Meyer. Here, students were studying how ratios, equivalent ratios and proportions work in real life situations.

Cinnamon Toast Crunch

Cinnamon Toast Crunch

cinnamon-toast-crunch-3.jpg

Prologue

Play this video for students. You may have to play it more than once before they are ready to ask questions.

[wpvideo EdgORK0d]

Act I

  • Write, "What do you want to know?" on the board
  • Allow students time to ask questions. All questions are valid. Don't answer any of them just yet.
    • Optional: After each question ask, "Who wants to know the answer to this question?". Write that number beside the question.
  • Answer all questions on the board.
    • If a student asks for information about the price or the number of grams, give them access to the photos in Act II.
    • If a student asks a good, math-related question, circle it and tell students you would like to know the answer to that. A good starter question is, "Which box of cereal is the best deal?", but there are many other problems that students can explore based on this video.
    • Hopefully someone asked something like, "What does this have to do with math?" Give moving speech about how math is life. You get bonus points for moving kids to tears during this speech.

Act II

Cinnamon Toast Crunch 1

Cinnamon Toast Crunch 2

Cinnamon Toast Crunch 3

Act III and Sequel

[Placeholder text. Will be updated once students are finished the problem.]

Grade 8 Mathematics: Multi-Strand Long-Range Plans

Grade 8 Mathematics: Multi-Strand Long-Range Plans

The Problem

I was trained to do math in a very classical fashion. Start by what you are certain that you know, and build from there. Pedagogically, this makes a certain amount of sense: once you can figure out where students are, you can build on it.

Using traditional textbooks to teach a math course, implies that you should start with a chapter that students understand well, and from there, go to a chapter that would (a) use what they've already learned, to (b) extend their understanding. In theory, this sounds like a good idea. In practice, there are nuances. With each new group of students (and with each new chapter), you need to figure out which students know what, and how to best get them where they need to be. And using a traditional textbook, it can be hard to figure this out.

I like planning long-term. I've used Nelson Math to create grade 8 long-range plans, and to create grade 7 long-range plans. I've used Jump Math for grade 8 long-range plans as well. And I've compared Nelson Math and Jump Math for grade 8 and for grade 7. There is something comforting about having a plan and knowing where you want to go next.

But every year, as the realities of teaching the students in my class have set it, I've either heavily modified or completely abandoned those plans. (Helmuth von Moltke might have something to say about that!)

Download for Word: 2016-01-25 Grade 8 Math Multi-Strand Long-Range Plans

Download as PDF: 2016-01-25 Grade 8 Math Multi-Strand Long-Range Plans

A Possible Solution

So, here's my next idea. I am certain it is at best an imperfect solution, but it's better than the other solutions I've tried so far. These plans are based on the EduGains' "Continuum & Connections" resources, which date from 2010. I've mirrored them here, in case that website goes down at any point.

The original documents are:

While I have only been working with this system for a few months, I have found a number of benefits:

  • It implicitly covers all of the grade 8 Ontario math curriculum (though I find I still need to refer to the curriculum regularly to make sure).
  • It allows for flexibility in what I teach next. For example, while focusing on fractions, I could cover algebraic expressions and experimental and theoretical probability as two different facets of the same idea, rather than as two different "chapters".
  • It allows me to evaluate all of my strands simultaneously. Hopefully, I'll never be stuck again trying to do just one more chapter so that I can have a mark for report cards.
  • It allows for more fluidity in my strategy walls. I don't have to throw out everything from the last chapter in order to make room for the new chapter.

The downside I have seen with this method is that some students perceive this approach as less structured. They miss the reassurance of knowing they are going to do chapter 10, starting with section 10.1 and going through section 10.8, and knowing that there will be a quiz at the halfway mark and a test at the end. And I have to say that I do miss that structure as well to a certain extent. But so far, I feel like the tradeoff is worth it.

 

Dot Talks: Starting Number Talks

Dot Talks: Starting Number Talks

Making Number Talks Matter

We just started number talks in our grade 8 math classroom back in January. I started preparing the idea after a workshop where I was given the book, "Making Number Talks Matter" (not an affiliate link). I tried it because it's always fun to try teaching in new ways, but I have found an interesting shift in student work since we have started doing math this way.

Subtraction Number Talks

It took us about 3 weeks of daily, 20-minute number talks to develop the strategies in this gallery. Once students had developed their subtraction abilities sufficiently, I gave them names for the different strategies, and then the students chose which problem best represented each strategy.

Changing Methodology

Up to now, I have always spent a lot of time and energy agonizing over precisely which strategies students are ready for, and which ones I should teach, in specifically which order. Since starting number talks, I've been forced to change. I still have an idea of which strategies I want students to develop, but with Number Talks, it is no longer possible to directly teach specific strategies. Rather, I have to develop a series of good questions that will point the class in the direction of developing efficient strategies.

This is simultaneously more and less challenging that planning which strategies to teach.

  • Number Talks are less challenging, because I don't have to worry that a lesson will fall completely flat. If I make a problem too easy, I can always work another problem into the same number talk. If it's too challenging, we can still have a good discussion, and we can hit the same concept using a different question the next day. At the same time;
  • Number Talks are more challenging, because I have to force myself not to give away the answer. The pleasure of doing mathematics is in the discovery, and as a teacher, it is very difficult not to chime in with my own ideas. It can also be challenging to try to invent questions that will encourage students to develop strategies that are more efficient than what they currently have.

Follow-Up: A Month of Number Talks

Here is a gallery of some of the other Number Talks we have done since. Some of the questions are here twice I teach two classes of grade 8 math. While I was doing these number talks, we started off with a unit on "Fractions" (where the goal was to understand that percentages, fractions and decimal numbers are three ways of representing the same quantity) and moved into a unit on "Patterning to Algebra" (where the goal was to understand that tables of values, graphs and equations are three ways of representing the same pattern). That shift is reflected somewhat in the series of Number Talks.

Stick or Switch? The Missing Ace Problem

Stick or Switch? The Missing Ace Problem

Prologue

[wpvideo Ytdf27NB]

Act I

Is it better to stick or switch?

Act II

  1. What's your hypothesis: stick or switch?
  2. What's the theoretical probability of your hypothesis?
  3. How would you set up an experiment to test the experimental probability?
  4. Will the experimental probability exactly match the theoretical probability?

Act III and Sequel

Here is the problem in its original context. The history of this problem is fascinating. The Time Everyone "Corrected" the World's Smartest Woman

Grade 8 French: Writing a Fairy Tale / Écrire un conte

Grade 8 French: Writing a Fairy Tale / Écrire un conte

This is an overview of our unit on writing a fairy tale (Écrire un conte). Before starting this unit, we first worked on analyzing fairy tales. The big idea was to write a fairy tale (1) using the four steps of writing, (2) using our understanding of the French language and (3) using the format of a fairy tale.

Strategy Wall

As the unit progressed, we started adding some of our collective work to our strategy wall. The first thing to go on the wall were the success criteria we came up with together. Those success criteria eventually became the rubric that we used to evaluate student work.

We also had an example from our unit, "Lire et analyser des contes", co-constructed examples (which you can find below) and occasional examples of student work.

2016-01-26 Écrire un conte

Evaluation: Rubric with Learning Goal and Success Criteria

This is the rubric we used to evaluate student work. If I had to do it again, I would specify which curriculum expectations were being addressed. Also, because this is such a big project and takes so long, I would have set specific dates for student conferences, and I would have given a minimum grade during that conference. For example, if a student completed their "remue-méninge" and did a really good job, I might be able to give them a minimum of a level 1 or 2 (equivalent to the letter grade D or C). The advantage to this is that if student work goes missing along the way, at least there is a note that we met, and I can more accurately gauge the student's progress.

Download for Word: 2015-12-03 Module 3 Des textes qui font réagir Écrire un conte

Download as PDF: 2015-12-03 Module 3 Des textes qui font réagir Écrire un conte

2015-12-03 Module 3 Des textes qui font réagir Écrire un conte

Example: Barr Homeroom

We also modelled how to outline a fairy tale and started the writing process together. Due to time constraints, we didn't finish the fairy tale, but we completed enough that the students understood what was expected.

Download for Word: 2015-11-26 Remue-méninges pour mon conte (Barr)

Download as PDF: 2015-11-26 Remue-méninges pour mon conte (Barr)

2015-11-26 Remue-méninges pour mon conte (Barr) 1 2015-11-26 Remue-méninges pour mon conte (Barr) 2 2015-11-26 Remue-méninges pour mon conte (Barr) 3 2015-11-26 Remue-méninges pour mon conte (Barr) 4

Example: Kelley Homeroom

This is the same activity (modelling how to outline a fairy tale), but with the other group of grade 8 students.

Download for Word: 2015-11-26 Remue-méninges pour mon conte (Kelley)

Download as PDF: 2015-11-26 Remue-méninges pour mon conte (Kelley)

2015-11-26 Remue-méninges pour mon conte (Kelley) 1 2015-11-26 Remue-méninges pour mon conte (Kelley) 2 2015-11-26 Remue-méninges pour mon conte (Kelley) 3 2015-11-26 Remue-méninges pour mon conte (Kelley) 4 2015-11-26 Remue-méninges pour mon conte (Kelley) 5

Example of Student Progression

This is an example from a student who finished somewhere in a level 4. There were a number of students who wrote excellent fairy tales during this unit, and I really wish that I had taken more photos.

Rubric

I wish I had pictures of the evolution of this rubric over time. The checkmarks that are here were added over the course of several weeks, as the student completed more and more sections of the project. If I had to do this again, I would have left a little bit of space in between each section, so that I could make comments on each step of the writing process.

2016-01 Écrire un conte Rubric

 

Outline

Students were allowed to work in pairs (if they wanted to) on the outline. This allowed students to generate ideas together for their story more quickly and get feedback on what might work and might not work more quickly. It also allowed me to meet with groups in a more efficient manner and help students get started on the right track. This group actually did two outlines, hence the "(2)" in the title.

2016-01 Écrire un conte Remue-méninge 1 2016-01 Écrire un conte Remue-méninge 2

First Draft

Students worked individually from the draft onward, even though they may have generated a rough copy together. This allowed students to have a common jumping-off point, but be creative in how they solved the problems in their own story. We spent a lot of time going over grammar and spelling. I tried to group students together when I noticed that they were working on similar skills (for example, all the students who were trying to add good metaphors), so that I could teach those students what I wanted and follow up more effectively. This student actually did several drafts before handing in the final copy, but I haven't included them.

2016-01 Écrire un conte Brouillon 1 2016-01 Écrire un conte Brouillon 2 2016-01 Écrire un conte Brouillon 3

 

 

Final Copy

And here's the final story.

2016-01 Écrire un conte Copie finale 1 2016-01 Écrire un conte Copie finale 2 2016-01 Écrire un conte Copie finale 3 2016-01 Écrire un conte Copie finale 4

Word Study

This is slightly separate from the fairy tale, but still related. I asked students to show me what they learned while writing. Some students just defined the new words they used (as this student did). Others also added in new grammar rules, homonyms that they needed to look out for, or verb tenses that they worked on. In future projects, I really want to develop the success criteria for the word study, since it really shows how much actual French a student learned during a project.

2016-01 Écrire un conte Copie finale 5

Ontario Grade 9 Academic Level Mathematics (MPM1D): Principles of Mathematics

Ontario Grade 9 Academic Level Mathematics (MPM1D): Principles of Mathematics

This is an overview of what the Ontario grade 9 academic-level mathematics course (MPM1D) contains. I debated whether or not posting this would be useful, because the course can be taught in so many different ways. However, it might be useful as a guide to what kind of content students might see over the course of the year. Please take it with a grain of salt, (1) this resource is over 10 years old as of the date of this post, and (2) it only lists content, and can't say anything for how it would be taught, presented or evaluated in a classroom. Here is the grade 9 and 10 Ontario math Curriculum (PDF): OntarioGrade9-10MathCurriculum2005

Here are the PDF's for each of the four units:

Grade 9 Academic Mathematics MPM1D Unit 1 1 Grade 9 Academic Mathematics MPM1D Unit 1 2 Grade 9 Academic Mathematics MPM1D Unit 1 3 Grade 9 Academic Mathematics MPM1D Unit 1 4 Grade 9 Academic Mathematics MPM1D Unit 1 5 Grade 9 Academic Mathematics MPM1D Unit 1 6 Grade 9 Academic Mathematics MPM1D Unit 1 7 Grade 9 Academic Mathematics MPM1D Unit 1 8 Grade 9 Academic Mathematics MPM1D Unit 1 9 Grade 9 Academic Mathematics MPM1D Unit 1 10 Grade 9 Academic Mathematics MPM1D Unit 1 11 Grade 9 Academic Mathematics MPM1D Unit 1 12 Grade 9 Academic Mathematics MPM1D Unit 1 13 Grade 9 Academic Mathematics MPM1D Unit 1 14 Grade 9 Academic Mathematics MPM1D Unit 1 15 Grade 9 Academic Mathematics MPM1D Unit 1 16 Grade 9 Academic Mathematics MPM1D Unit 1 17 Grade 9 Academic Mathematics MPM1D Unit 1 18 Grade 9 Academic Mathematics MPM1D Unit 1 19 Grade 9 Academic Mathematics MPM1D Unit 1 20 Grade 9 Academic Mathematics MPM1D Unit 1 21 Grade 9 Academic Mathematics MPM1D Unit 1 22 Grade 9 Academic 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