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# The Watermelon Problem

## Prologue

Curriculum and Connections Expectations: Fractions

• use fraction skills in solving problems involving measurement

## Part 1: Entry

To introduce the problem, I just brought in a watermelon and sat it on the table until students had enough questions. Once they were ready, I put "What do you want to know?" on the board, and as they asked questions, I gave them markers so they could put their questions on the board.

Journal entry #1: The Watermelon Problem.

Based on these questions, students decided to study the problem, "How can you cut up the watermelon so that each of the 27 students can have an equal slice?"

What's The Watermelon Problem? Here are the questions the students asked.

## Part 2: Attack

Here are some resources for the watermelon we chose. It's honestly better to get your own watermelon though.

Here are some examples of student work and ideas. Many of them came up with some form of idea to do a third of a third of a third.

## Part 3(a,b): Review (Check, Reflect)

After finishing this problem, it's time for students to check if the solution worked the way they expected, and to reflect on what went well and what can be improved on. You can see in the video that cutting the watermelon into thirds would require a very high level of precision to do well. There are a whole bunch of hindrances. For the sequel, encourage students to find a better way to divide the watermelon. Can they add the precision necessary to cut the slices precisely equally? Can they find a better way to distribute the watermelon?

## Part 3(c): Review (Extend)

Once students are ready, they can try to find a strategy that can work easily for any number of students.

## Elements of Solutions

Here is one possible such strategy. The result is pretty easy to generalize to any number of students. However, there are still a couple of precision-related problems with this strategy.

Here is a different strategy. Still pretty easy to generalize to any number of students, but with different pros and cons.

# Fractions, Decimals and Percents

We have officially started our first number talks as a class. I will need to update this post as the unit progresses, but here is what we have so far.

## Fractions, Decimals and Percents: Unit Expectations

Here are the expectations we will cover in this unit, which is based on the Curriculum and Connections resource. There are four pillars to what we going to be covering: (1) Number Talks, (2) Strategy Walls and Math Journals, (3) 3-Act Math Problems and (4) Exit Tickets.

## (1) Number Talks

We have done our first few number talks for this unit. The idea for Number Talks is simple: I present a math question, students take a moment to solve it individually, and then we discuss strategies. In my classroom, this sort of replaces "drill work" (usually taught through worksheets or a series of problems which become slowly more difficult). Instead of giving the students a bunch of questions to work on at once, we take the time to study one good question together.

## (2) Strategy Walls and Math Journals

"In order to learn from experience, in order to have fresh possibilities come to mind when appropriate, it is necessary to sensitize yourself to notice opportunities to respond to situations rather than to react, to choose to act rather than to be caught up in and driven by old habits. Thus, by offering tasks and follow-up prompts it is possible to promote awareness of the use of people's natural powers. As they become more sensitized to and aware of their own use of these powers, those powers develop in flexibility and usefulness." - Thinking Mathematically, Mason & Burton, p. xv

I believe that student learning is what happens in a student's head. The learning process is, in a sense, the action of a student organizing her or his thinking. With that objective in mind, the objective of strategy walls (as a group) and math journals (as individuals) is to give students an opportunity to consolidate their learning and communicate their thinking. While I enjoy working on problems with students, and may write comments and questions in their math journals, I will never give a math journal any type of grade.

## (3) 3-Act Math / Inquiry Problems

We just did "The Watermelon Problem" (click link for full article), which was long enough to merit its own post.

## (4) Exit Tickets

• The Security Camera Problem

Here is a copy of our first exit ticket. The objective here was to get students to start using their problem-solving methods to find the best place to place a camera. I am a big fan of problems that are easy to understand but difficult to solve. As students went through this problem, they began to understand its nuances, which allowed them to push their understanding further.

• The Square Mural Problem

This was our second exit ticket for this unit. I based it on my first attempt at a 3-act math problem last year, The Square Mural Problem. I showed students the video from that problem, but I didn't let them ask questions as a group, since I expected them to attack the problem independently.

The purpose of this problem is ultimately to bring students to an understanding of Least Common Multiples and Greatest Common Factors, even though neither of those concepts are immediately apparent in the problem. The extension for this problem was for students to find an efficient way to create the smallest square mural that could be made from any tile. For example, how would you find the smallest square mural made of tiles with a height of 180 mm and a width of 378 mm?

As students struggled with this problem, they started to realize that they didn't have enough information based on the video. As those students came up and asked about the extra dimensions, I gave them either (or both) of these additional resources.

# 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.

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.

## 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.

# 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.

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.

# 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!)

## 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

## 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.

# Grade 8 Data Management: Designing and Analysing a Survey

## Data Management Strategy Wall

This is what our data mangement strategy wall looks like now, at the end of our unit. A lot of the resources we co-created were done in Excel using Microsoft Office Online. I would have printed more off and put them on the wall, but most of our wall space right now is being used for French or another math strand.

## Curriculum Expectations and Student Reformulation

At the beginning of the chapter, we looked at the curriculum expectations, and the students reformulated the expectations in their own words.

## Introduction to Data Management: Barr Homeroom

Here is the first example of a quick, in-class survey that we did in the Barr homeroom. In the first survey, we realized that if we offer people too many choices in a survey, it can be difficult to get useful results.

After completing this activity together, we divided the class up into four groups, and each group proposed a hypothesis they would like to study, and some questions they could use to test their hypothesis.

## Introduction to Data Management: Kelley Homeroom

This is a similar example to the last one, but this time with the Kelley homeroom. We divided into four groups again, and each group proposed a hypothesis and some questions that would help test the hypothesis.

## Draft Survey Questions

Once we had all the questions finished, we compiled them into one big survey. We did this survey ourselves, for troubleshooting purposes. We realized that some questions were biased, others were vague and yet others didn't account for all the possible answers students could give.

We weren't always able to find a solution to every problem. For example, should we ask if students needhave or wear glasses? Each of those questions is slightly different, and can influence results. We said that we need to make questions as clear as possible, and remember to add our thoughts to the final analysis.

## Final Survey Results

Here are the final survey questions and results. We were able to do almost the entire school (455 students out of about 480). We used these results during the final evaluation.

## Evaluation

Here is the final evaluation. We didn't put in every success criteria, but I reserved the right to write questions on the page in response to student work, and allow students the chance to continue to work after. Below, there are two examples of survey analyses we did in class (though they are missing the conversation we had while doing the analyses).

# 1.8 Order of Operations

Order of Operations is the eighth and final section of "Chapter 1: Number Relationships". Introduction to the Order of Operations

When I took this picture, we hadn't yet established the learning goal or success criteria. We usually do those after completing a couple exercises, since if we try to do it right at the beginning, all the students can really guess is that "we need to understand the order of operations".

Ultimately, the learning goal is to understand that multiplication is repeated addition, division is a grouping of a number into groups, and both of those take priority over addition and subtraction. After that, we will try to understand why brackets and exponents have an even higher priority.

Questions 2 and 3 (modelled with algebra tiles) encourage students to visualize what is happening with the positive and negative numbers. In question 2, they should come to the conclusion that the \$latex -(4+5)\$ means that the four and five need to be added together, but that the resulting number (nine) will be negative.

In question 4, students should realize that multiplication is repeated addition, and as such, should be either expanded first or done first. The yellow algebra tiles are the visual representation we used for question 4(a). The other two parts of the question are done in a similar fashion.

## Support Questions

Do any five of the following: p. 36, #4, 6, 10, 11, 12, 15.

# 1.6 Square Roots

Square roots is the sixth section of Chapter 1: Number Relationships. The learning goal for this section is to be able to determine the square roots of numbers using visual representations, or by estimating or using a calculator. Students know that they are successful if they can visualize square roots in their heads and understand the differences between the square and square root of a number.

## Introduction to Square Roots

We examined squares and square roots using linking cubes in this section, to help students visualize the relationship between the two. Since linking cubes are three-dimensional, we ended up talking about squares, cubes, square roots and cubic roots, although only squares and square roots are in the curriculum.

## Support Questions

The support questions for this section are on p. 30, #3, 4, 5, 7, 8, 9 (choose any five).

# 1.5 Expanded Form and Scientific Notation (and Mid-Chapter Review)

This is the sixth section of Chapter 1: Number Relationships.

## Introduction to Expanded Form and Scientific Notation

The Learning Goal for this section is to: Develop a strong understanding of expanded form and scientific notation, and to be able to compare large numbers in different formats.

There are two ways that students can tell that they are being successful:

1. They can convert confidently between expanded form, standard form and scientific notation; and
2. They can compare numbers using multiples of 10

In the introduction, we defined Scientific Notation and Expanded Form, and gave examples of each. There were a number of additional examples on the board, but they didn't make it into the notes.

## Activity: A Lifetime of Heartbeats

In this activity, students needed to compare the total number of heartbeats in the lifetime of an old man and a parrot. The objective was to show students how exponential notation can come in useful.

## Section 1.6 Support Questions

The support questions for this section were on p. 22-23, #4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (choose any 6).

## Mid-Chapter Review Support Questions

The support questions for the Mid-Chapter Review were on p. 24-25, #4, 5, 6, 7, 8, 9, 10, 11 (choose any 5).