Patterns & Five Practices to Model

After having read "Orchestrating Discussion" by M.S. Smith, E.K. Hughes, R.A. Engle, and M.K. Stein, and really enjoying the idea of the five practices mentioned, I've decided to practice them for myself. Currently I'm not teaching in a classroom so all of the practices can not be applied, but I thought thinking about the practices and trying to apply them as far as I could in a given situation would be good practice and perhaps revealing. So I went and found a problem related to patterns on which to try some of them out(Sticky Triangles from NRICH: http://nrich.maths.org/88).

The five practices to model from the article are as follows:

1. Anticipate Student Responses:  As a teacher before introducing the activity in class, do it yourself, consider and push yourself to come up with multiple ways in which it could be solved and that your students may solve it. You may consider misconceptions students might have and how that could influence their work. This allows you to be better prepared to address answers students give.
2. Monitor Students Work & Engagement: Walk around the classroom, observe what students are saying and writing. Notice when students are nearly finished. Ask students about what they're doing and question them to think deeper or consider different aspects.
3. Select Particular Student Work to Share: Having been monitoring their work, you have a good idea of what each student is doing and pick certain students to share based on ideas you want the class to discuss and consider and highlight perhaps different techniques used in solving.
4. Specific Order to Student Sharing: Depending on what you want to highlight or address first, have the order students share in match this. Perhaps to help the flow of the discussion or to clear up misconceptions first.
5. Connect Student Responses to Another & to Mathematical Ideas: What's the take away of everything? What should students be learning from this? What can they learn from another?

So 2-4 I won't be able to do, but I'll be applying and practicing 1 and 5 in relation to the problem of the Sticky Triangles. The problem shows triangles composed out of match sticks like the pictures below:

The problem asks students to explore the pattern that is occurring by finding how many matchsticks are in each, how to make the next triangle, and how to know many matchsticks to create any size triangle. 

Anticipating Student Responses:

• All students will likely find the number of match sticks in the 3 triangles given: 3, 9, 18.
• Finding triangles of at the next sizes, students would likely draw to figure out or if they notice the pattern quickly would calculate.
• For triangles of any size, students then will have to find an equation or way to describe what is happening.
• Might see the center of the second triangle as being the original first triangle. So it's 3 + 2 (3) where the first 3 is the matchsticks in the first triangle, 2 is the number of matchsticks at each extended corner and 3 for the three corners. But then the student would hopefully realize that the small triangle is not in the center of the third. So this does not work
• Might come up with a recursive formula. You are adding a new row of triangles to the bottom of the previous. The triangles added to the bottom are the number of the size of the triangle and each triangles uses 3 matchsticks. So previous plus 3 times the size of the triangle.
• Might look at the numbers, not the pictures, to come up with a formula. From 3 to 9 is plus 6, from 9 to 18 is plus 9.
• Might look at the matchsticks around the edge and then the number in the center. So first is 3+0, second is 6+3, and third is 9+9. For the outside it is 3 times the size of the triangle. The inside is almost 3^s-1 where s is the size. Or again could use recursive.
• Might notice how the number of size 1 triangles in each size are changing.
• Likely some students will come up with formulas that apply to the first and second, but not beyond. And students who do not check to see if their formula can predict further answers.

Connecting Responses:

• From this activity, I would hope students would take away the multiple ways in which a pattern can be described within both words and equations to represent the situation. Students hopefully can learn from others and the different techniques their classmates used to solve the problem. Understand how we can represent a pattern with an equation, and how to use variables in this case (what do they represent, where can we see it). Highlight recursive formulas and how they are represented in an equation.
• The problem also at the end asks students to consider how their findings would change if we were composing squares in a similar manner. I might have students do this after the discussion or take home so that they can apply the ideas we talked about as a group and practice different ways they heard.

After looking at the problem in this way, I feel it really pushes me to consider and see what really can be taken away from a problem. It allowed to think about different ideas, some of which I might not have thought of, and maybe students wouldn't either. It also shows if a problem may or may not have multiple view points, if there would be anything to discuss about the problem. If a problem were fairly simple, using the five practices would reveal this. You likely would not be able to think up many different student responses. It gets you to consider your students and how they might think. I don't have students, but if I did I can see how you'd be able to think back to work they've done and how ideas they have may influence their techniques in the given situation. It also gets you to see what things should be discussed. If the problem is very computational, you can focus on different ways students think through the computations, what order do they do things in, and how can they check their work. You can get at why students are doing what they are doing, because you'll have an idea before hand and can prepare appropriate questions.

The five practices really gets you to focus in on what you are having students do, what are they doing, and why does this matter? You can better plan and be better prepared.

Repeating Decimals

Awhile back as a class we investigated repeating decimals. After some work and discussion, we came to see that when doing the long division of the fraction, if you arrive at the same remainder twice then you have a repeating decimal. For example consider 1/6:

Each reminder will keep being a multiple of 4, which 6 will continue to go into by the same amount each time. Pretty cool.

Then we all went home with the task of investigating repeating decimals further on our own.

What immediately came to my mind to look into was if there were a way to look at the fraction and tell if it would be repeating decimal without doing the long division or typing it into your calculator. My first thought was just that maybe it dealt with the denominator being prime, since I had been focusing on a 1/3 for some time. But this was quickly, like the second after thought, proven wrong. There's a 1/5 and a 1/2, and 1/6 repeats but 6 is not prime. Then I realized I should probably come up with a couple more examples of repeating decimals, so know what the numbers look like. Here's the list I got: 1/3, 1/6, 1/7, 1/9, 1/11, 1/12, 1/13, 1/14, 1/15, 1/17, 1/18.

I also knew that certain multiples of these would become non-repeating (so I would have to look at fractions in simplified form). I noticed from my list that 1/6, 1/9, 1/12, 1/15, and 1/18 related to a 1/3 since multiples of them could be reduced to a third. They are also was of breaking down a 1/3 into smaller pieces, so it makes sense that they would be repeating decimals as well. I then looked at 1/7 and thought to myself that anything with a denominator a multiple of 7 will be a repeating decimal. I then checked: 1/14, 1/21, 1/28. What do you know, it worked. So it would likewise follow for 11, 13, 17, and 19 (the denominators of other repeating decimals I recorded).

I then noticed something all these numbers had in common, their denominators were composed of more than just 5 and 2 (the prime factors of 10). Those composed of only 5 and 2 or that could be simplified to such were not repeating: 1/5, 1/2, 1/4, 1/8, 1/10, 1/20.

I was curious as to why and might explore that later. I had a thought that since the numbers the denominator is compose of go into 10, when doing the long division you would eventually have a point when that value goes into the reminder (a multiple of 10) completely. But I was content with my findings and had fun processing through it: asking different questions, gathering more examples, analyzing what I had, and then being able to make a conclusion and reach an understanding on what I had set out on,

Understanding Division by Fractions

After reading an article by Richard R. Skemp discussing relational vs. instrumental understanding, in class we discussed how we had learned fraction operations (addition, subtraction, multiplication, division). For me and the people in my group, most of us had learned through instrumental understanding. We were taught a rule to memorize and apply in the given situation in order to solve the problem. A "rules without reasons" way of learning.

As we went over the different operations, for all of the first 3, addition, subtraction, and multiplication, we felt we had a good understanding of them. The answers we would get from problems made sense to us and we could explain to someone verbally why. However, when it came to division with fractions, we were a little stumped. Most of us were taught the rule of flipping the denominator and numerator of the fraction you are dividing by and then multiplying that with the other fraction. We knew the rule well and could explain to someone how to solve it. But why was the answer the way it was?

 I thought this was sort of interesting. Why was it so difficult?

I think part of the difficulty comes from the fact that we often don't talk about dividing by fractions. Another part might be that we did not have a good idea of how to visualize what was happening or to describe the process. After looking into division with fractions, these two things coming together really helped.

1). Verbally:
Knowing how to read a division by fractions problem is really important. This helps you to understand what you are solving for. For example, given: 1\2 ÷ 1/4, what is being asked is how many fourths are in a half. This relates right back to division with integers, like 6 ÷ 3 is asking how many 3's are in 6. Once we see that this is what is being asked, we can better represent it with a picture and also understand our answer.

2) Pictorially:
Keeping with the same problem (1\2 ÷ 1/4), it's helpful to begin with drawing what a whole is. After this, we can mark in what a half is and what a fourth of our whole will look like.Then it simply becomes finding out how many fourths will fit into the half.

We see that two will fit, so our answer is two.

3) Making sense of it all:
We can reason to see that the answer we got is correct. Remember, we're finding how many groups of a 1/4 go into a 1/2. We already know that a 1/4 is less than a 1/2, so we should expect our answer to be at least greater than one. Thinking in the same way, we can reason through other division by fraction problems. Like 4 ÷ 5/6 for example. We know that 5/6 is less than one whole, so we should expect our answer to be at least greater than one. We also know that we have four whole pieces, so we should expect for each whole that at least more than one 5/6 will fit into it, so our answer should be at least more than 4.
What if the dividend is smaller than the divisor? Let's look at the example of 1\3 ÷ 1/2. Since 1/2 is larger than 1/3 we should expect our answer to be less than 1, since not even one 1/2 will fit into the 1/3.


Having a better understanding of what fraction division problems are saying allowed me to think through the problem and reason where the answer should be, as well as to better understand how to visualize and model what is happening. Building a relational understanding of fraction division really helps to better understand it and now for me to help explain to others how to do it and incorporate the use of models effectively.

Confusing Contexts

A few weeks back during one of our previous classes, we had a brief discussion on what contexts we would choose to use in teaching fractions. Many of us mentioned food, baking, measuring time and distances, money. These we believed were things students knew about and had back ground knowledge on, and potentially things our students would find interesting or care about - who doesn't like food?

This past week during an observation, I realized more so just how important the context is that we choose to use. I was in a seventh grade classroom. The students were reviewing for their upcoming test the next day over percentages (comparing percents, fractions, and decimals, finding percentage change, what's the percentage of a whole?). I was walking around the classroom, helping students who had questions. Most of the students in the class seemed unafraid to ask questions if they were uncertain about something - which seemed to be a result of the sort of classroom the teacher had cultivated. Anyway, as I was walking around I had a girl ask me over to see if she was solving a problem correctly. The problem gave her the price of a good, the percent of a mark up on the good, and asked for the new price of the good after the mark up. When I asked her what she was thinking, she explained to me how she had solved the previous problem (given the price of a good, and the percentage of discount, find the new price). She asked if she would then solve this one in the same way - after all she was given a price and a percentage and asked to find the new price. So to her the two problems seemed very similar, if not the same. I asked her if she knew what "mark up" meant and she shook her head no. I helped to explain to her the concept and then she was able to solve the problem on her own.

Originally, viewing the problem, it sounded like the previous one. While she may not have known or simply had forgotten, not knowing what a mark up was, she made an assumption that felt logical to her so she could keep going and solve the problem. Luckily she had asked a question, but not all students will. On a test, she might have gotten points off, even though once she understood the situation she knew how to correctly solve it.

Context is important. We want to see what students have learned and understand, instead of whether they know what the term "mark up" means. We may think students know what a mark up is (it uses the word up, so wouldn't students think increasing?), but they may not. Lots of students haven't had jobs yet and are more familiar with ideas of sales and discounts, than the marking up of prices by suppliers. When using contexts, we need to be careful and ensure they are relate-able to students and that students understand them and any vocabulary or ideas that accompany it.

Proportion Sense

Recently in class we began discussing fractions. We began by looking and discussing proportions without the use of numbers. It seemed like it would be simple, but without the use of fractions at our disposal it became difficult.

We were given a picture of two boxes of chocolates(like below) and asked which one was more nutty?

It was difficult to stop yourself from using and comparing fractions - since we already have that knowledge. Our future students, however, will be just developing how to use fractions and relate them. Instead, we had to rely on the picture and ways in which we could manipulate it. What we ended up finding most helpful was the idea of buying enough of each box so you have the same number of chocolates and then comparing how many nuts each one has. An idea  that relates back to finding a common denominator. Once you had the same number of each, making a comparison between the two was logical.

However, after accomplishing this, when faced with another proportional problem we fell into mistakes. We were given a problem comparing mixtures of blue dye with water. Given certain a number of beakers of dye and water, we had to decide which would produce a stronger blue color. It was interesting to see how different we reacted to this problem. Many of us ignored the idea of getting to the same number of beakers and began to cancel out blue dye and water combos. The result was a false belief that the two mixtures shown below would be the same color.

We failed to take into account the ratio of water to blue dye. It was startling to find ourselves making this mistake.

These two activities really helped me to come to understand the importance of proportional sense. My future students will likely be making similar mistakes, but having a discussion on these ideas and working through proportions without numbers will be helpful. Introducing students first to proportions and making sense of the relationships between them will get students asking important questions:

• What quantities are being compared?
• What other factors do I have to consider? 
• What roles do these variables play/what's their relationship?
• How is it affecting the goal(if it's nuttier or more blue)?

Through this, students can begin to develop a sense of how to compare proportions and what to take into account. Then, when dealing with the numbers and fractions they should to able to reason if the answer they arrive at is a logical one or not. They can rely on asking questions to led them to the correct calculations and hopefully dealing with fractions will not seem as daunting.

The Human Number Line

One of my current classes is centered upon teaching middle school math - which looking back I now realize began a decade ago for me. Most of what and how I was taught, I honestly don't remember in much detail. However, I can recall that for the most part I was in my seat and taking notes - likely doodling as well. Maybe once in a while I remember forming groups, but really infrequently. 

During one of our past meetings as a class, we were introduced to the idea of using a human number line - a number line down the center/front of the classroom on which students could walk/stand. Our line was centered at zero and went from -15 to 15. As a class we illustrated by walking various equations and story problems. We played games that involved adding and subtracting of negative integers - trying to beat your opponent by making it to your end first (whether -15 or 15). Even as a college student it was enjoyable to work out how to play the games and interact with another in a different way. 

Instead of having students just sitting and taking notes, students can make connections between action/movement, visuals, and the math algebraically. For example, when given the problem 8-(-5), the student would begin on the number 8 facing the class, then turn toward the negative end of the number line (representing the subtraction) and then walk backwards (representing the negative number) 5 paces. The student would land on 13, the solution to the problem. In the process, students will begin to notice that subtracting a negative (though different) results in the same solution if you were to have added a positive 5. Other students in the classroom not walking out the problem can follow by using a printed out number line paired with an object to move (like a chip or plastic cricket). 

In addition, number lines are useful and helpful later on in students mathematics careers. Allowing them to become familiar with them and their usefulness, just gives students another tool with which to work. Such as later encounters with inequalities and graphing the solutions. 

Doing the same old, same old, is boring. The use of the human number line seems like a good way to break this. Students are up and moving, and shown through multiple modes of representation - connecting ideas and figuring out what works best for them.

All of the Lights

One of my family's favorite Christmas movie is National Lampoon's Christmas Vacation. So since it's getting close and it's that time of the year, I thought I would look into something Christmas related. This weekend, since the weather was nice (no more snow and not too cold!), my family dug out the Christmas lights to put them up. And it got me thinking of that movie and all those lights. How many and how much would it cost to decorate a house as done in the movie - the entire roof and all around the entire sides.

First, I went to find out what sort of lights were used. Based on a picture in a movie, they're larger than the sort I'm used to. So then I searched Christmas lights. I know that there are all sorts of places to buy lights but I choose Home-Depot. The lights cost $8.48/each for a strand of 25 lights at a length of 25 feet. You can however only connect at most 2 sets. But I'm guessing by the looks of it with any lights you wouldn't be able to create that long of a strand. So we're just going to assume that we can attach them all together no matter.

So now how to figure out how many strands are needed. After a little searching, hoping to find the house dimensions (and then go round trying to figure out how many strands), I was able to find that house was decorated with 25,000 lights! That's a ton and no wonder. So divide that by the nice easy 25 lights per strand and you would need 1,000 strands of lights. And based on the website, you'd have to travel to several stores to pick them up - the one near me only has 20 on stock. These lights being bigger than usual, already cost a lot. Multiplying the cost of a strand at $8.48 by the number of strands, it ends up costing $8480.00 dollars. No thank you. I could pay to study abroad instead!

And the length of all those strands, at 25 feet per strand ends up being the same as the number of lights - 25,000 feet.

So then I thought: how tall of a tree would you need to use that many lights? I'm going to use a cone to represent a tree to find this out. Since lights go around the tree, I'm going to look at the surface area of a cone minus the base: pi*r*l. The variable "l" of the cone represents the slant height, which in terms of height equals sqrt(h² + r² ). So our formula becomes: pi*r*sqrt(h² + r² ). Just looking at my own tree, the height is about 3 times the radius of the tree. Using this I can narrow the formula down to having only one variable, height: pi*h/3*sqrt(h² + (h/3)² ). Now to simply the equation a bit:

= pi*h/3*sqrt((4h/3)² )

= pi*h/3*4h/3

= pi*4h²/9

To make things simple, just like the house we'll cover the entire tree with lights! Which I think would hurt to look at up close haha. The width of the light strands is about 2 inches or 1/6 of a foot, so the area of all the light strands is  feet squared. Setting this equal to the surface area and solving:

4166.67 = pi*4h²/9

37500 = pi*4h²

2984.15 = h²

54.63 = h

So the height of the tree is about 55 feet. A little too small to be the tree at Rockefeller Center (69 to 100ft). If the spacing between the lights was increased from nothing to something, the height of the tree would just continue to grow. So again, like I thought, I'm going to pass. I'd rather keep with tradition and continue to put up the fake tree and string a few lights.