# Minions in Math – My Favorite Thing

Miniature robots zoomed across the floor, lights flashing. Students bent over computer screens, writing code. Parents sent emails telling me that their kids couldn’t stop talking about the Ozobots. My students were reviewing linear relationships, being introduced to systems of equations, and learning some coding skills. It was the best way I could have hoped to end the semester with my 7th graders.

My students came into this project with a lot of background knowledge. This is their second year with me. Last year, I introduced them to some basic coding by having them create video games using SCRATCH. The Ozobots use Ozoblockly, which is very similar in structure to SCRATCH. (Both languages are object-oriented visual programming language. I think this is absolutely the best way to introduce coding. They allow the user to focus on the big ideas of coding without getting bogged down in the syntax of the language.) The added challenge this year is having the code control a miniature robot. This required students to work through a few challenges that had nothing to do with coding, but everything to do with problem solving and making things work. I documented the journey on my twitter account . There are videos there showing each step of the projects (posts dated Dec 8 – 15). Unfortunately, you will have to go to the twitter account to see the videos if are interested, because my blog doesn’t support videos.

**Building Background Knowledge **

InitiallIy, I focused on building knowledge. Students had to learn how to calibrate the bots, how to write code, and how to load the code onto the bots.

Next, they worked on getting the bots to do what they wanted. They explored the different capabilities: getting the bot to move in specific ways, playing with the light features, etc. During this early phase, I had students work in pairs. Because there is a certain amount of frustration inherent in the process of figuring out how to make things work, it helps to have someone to share the journey.

**The Project**

Once I was confident that everyone had a little experience, I assigned a two-part project that would require them to review the work we had done earlier in the year with linear equations and also act as a launch into the work solving systems of linear equations that we will do at the start of the spring semester.

**Linear Relationships in Tables and Graphs**

For the first part of the project, I assigned each pair of students a different linear relationship. Each pair was given two ordered pairs, an ordered pair and the y-intercept, or an ordered pair and the slope. They had to find the linear equation representing their relationship and create a table for the relationship. Then, they had to create a large scale coordinate plane and write code so that their Ozobot would graph the linear relationship. The Ozobot had to flash lights on three ordered pairs and perform a different action at the y-intercept (flash different lights or spin on the spot).

**Solving a System of Equations Graphically**

Next, each pair had to use a second Ozobot to graph a second line intersecting the first line on their coordinate axis. They had to calculate the point of intersection and time the two Ozobots so that they would not collide at the point of intersection.

While there was plenty of math in the project, the most important piece of math was not what one usually thinks of as math. It was learning how to make something real actually work. Students had to learn how to frame a task and then to persevere in the face of challenge (one of the math practice standards).

**My Favorite Thing**

My biggest measure of success is that the kids dinner time conversations were all about math for the entirety of the project (a number of them ended up getting their own bots for Christmas as a result).

# Constructing an Understanding of Integer Addition

Making sense of integer addition and subtraction is hard. The algorithms are complicated and hard to remember. On their face the results sometimes seem counter intuitive. Sometimes the answer is positive, sometimes it is negative. Sometimes the answer gets “bigger” when you subtract and sometimes it gets “smaller” when you add. Because of all this messiness, students have a really hard time knowing whether an answer makes sense if they haven’t had the chance to build some conceptual understanding before jumping to an algorithm.

I start helping my students make sense of the process using a chip model. They use black chips to represent positive numbers and red chips to represent negative numbers. They combine the chips, making zero pairs (one red and one black chip) to find the sum. I start with the chip model because most students think of addition as combining sets of objects.

After students have mastered the chip model, I move on to a number line model. The curriculum that I use (Connected Math) does a really nice job introducing the number line model. It talks about addition as combining sets by providing a context in which two kids each have a number of video games and then talking about the combined number of video games. It illustrates the problem using a chip model. Then, it goes on to talk about addition also being representative of a context in which one “adds on”. It provides a context in which there is a temperature at sunrise and that is “added on” to as the temperature rises over the course of a day. They model this problem using a number line.

After introducing students to the number line model, I take them out into the hall to walk the number line. Prior to this, I have had my student aides create a number line for each table group. Each number line is created using painters tape (for ease of removal when the time comes). The numbers on each number line range from -10 to 10. I begin by modeling a couple of integer addition problems on the line. I walk forward for positive numbers and backward for negative numbers.

After modeling several problems, I have each group complete a set of problems by walking on the number line. Each group gets a laminated index card with the problems to be completed and a dry erase marker to record their answer. The first group member walks the number line to solve the first problem. The other members of the group check his or her work. The next person in the group walks the number line for the second problem and the others check the work. Each problem is completed by a new student until everyone in the group has walked the number line. At that point, the group rotates through again until all the problems have been completed.

Once students have mastered integer addition on the number line, they use models to construct an algorithm for integer addition. They begin with a group of four problems. They solve these problems using a model. Then, they identify what is the same about the problems and create two more problems to fit the group. (All of the problems in the group have addends in which the signs match.) Finally, they come up with an algorithm for the adding problems within the group. They repeat this process for a second group of problems in which the signs of the addend pairs do not match. If students have trouble figuring out the algorithm, I remind them that each number has a magnitude (size) and a direction (sign) and suggest that they consider the two parts separately. This usually helps them get to the algorithm.

After creating the algorithm, students test it on rational numbers (mixed numbers) and verify its efficacy using a number line model. Finally, they test rational number addition for commutativity.

After students have created an algorithm, I use the foldable shown below to summarize the lesson.

I follow up with this Always, Sometimes, Never exit card.

integer-addition-always-sometimes-never-v2

# Plotting Points on a Coordinate Plane

I’ve always thought that plotting points on a graph is one of the most boring things in life. It just seems so tedious, necessary, but tedious. I am not a fan of tedious and neither are the 11 and 12 year old students in my class. Our collective view is that math should be interesting and engaging (my view because I want them to learn to love math, their view because all things should be interesting and engaging when you are 11).

As I planned this lesson this year, I decided to try out a life-size coordinate plane. I had my student aides tape it onto the tile floor out in the hallway (since I have carpet in my room) a couple of weeks prior to the lesson. I wanted it to just sit there so that kids would see it and see the quadrant numbering every time they walked past the door for a few weeks. (My Algebra teacher did this with the quadratic formula. He wrote it in a corner of the board and just left it there. He didn’t say anything about it but we sat there looking at every day for a solid two weeks before he introduced it. Without even realizing it, most of us knew it before he even started the lesson.)

On the day of the lesson, students completed a foldable summarizing how to plot points on the coordinate plane.

Afterwards, they were each given an index card with an ordered pair. Each student walked over to their x coordinate and then up or down for their y coordinate.

When everyone could correctly plot the points, they returned to the classroom and played battleship on coordinate planes. Each player was given two coordinate planes. They were required to plot their ships on one of the coordinate plane ranging from -5 to 5 in both directions. Each player had a five point ship, a four point ship, a three point ship, and a two point ship. Ships could be aligned horizontally, vertically, or diagonally. They used the second coordinate plane to record their “shots” at their opponent.

# Every Graph Tells a Story – A Lesson Interpreting Graphs

Every graph tells a story, but can you “read” that story? Do you see the beginning, middle and end (if there is an end)? Do you see how the story builds and changes? These questions are at the heart of a unit I teach my 6th grade students. (The unit is a Connected Math unit called Variables and Patterns). They were also at the heart of a lesson I taught recently in which student were asked to interpret a graph, to read its story.

To begin, I wanted students to zero in on the key elements in a graph that reveal the story but I didn’t want to tell them too much. I decided to try using a chalk talk, which is a technique in a book I have been reading with my PLC, Making Thinking Visible . In the chalk talk, students respond to a prompt on a large sheet of paper. Students “talk” about the prompt by writing on the paper. This allows them to respond to the prompt and also to the responses that other students have written, hence making thinking visible. I decided that my prompt would be each of the six graphs that my students would need to interpret.

I drew each of the six graphs and posted them around the room, each hanging above a blank piece of poster board. As students entered, I assigned them to a specific graph station. I asked them to look at the graph and write what they noticed about the graph. After everyone had completed the task, I had them rotate to another graph station. This time, they could either write about the graph or respond to what someone else had written.

The six graphs were large scale versions of the graphs shown below.

In this phase of the lesson, I was hoping that students would look for markers that tell the story of the graph: what is the initial value, how does the graph change (e.g., is it a constant rate of change or a variable rate of change), is the graph increasing or decreasing in value, does the pattern in the graph repeat, is there an “end”. I wanted to draw out student thinking on these ideas as a launch into the lesson. Since this was the first time that I was using a chalk talk, I didn’t really know what I would get, though. It was a new process for my students, so they wouldn’t have the benefit of experience with the process to guide them and I wasn’t sure if I was presenting the task well since it was my first attempt with it. As I looked at their work, I could see the evidence of our inexperience. The responses varied widely Some students gave responses along the lines that I expected. Some students made up a story for the graph. A few students responded to other students’ thinking, but most used the thinking of other students to create their own elaborate stories. Since I got results I hadn’t anticipated, the debrief of the task was a little different than I expected and didn’t really focus on some of the things I had hoped it would. I didn’t try to force it though because I wanted to give students the chance to make meaning rather than me giving them meaning. I liked the technique as a launch, I just need to pose the prompt a little better next time.

After the debrief of the chalk talk, I had students work in table groups to match the same set of six graphs to a set of seven different stories. I gave them graphs and stories on a set of cards, so it was essentially a card sort. As students worked in their groups, I told them that they must work silently. One person would match a story to a graph. The next person could either match another story to a graph or could change one of the matches on the floor. I did this to ensure equity of voice in the group, the silence and turn taking ensured that no one could dominate the group and that each must contribute. I also did this to encourage students to analyze the choices that other students were making. The card sort used the graphs and stories shown below.

After each group had “completed” the matches, I had the groups do a gallery walk. During the gallery walk, they went around to each of the other groups and considered the choices that the other groups had made as they matched the stories to the graphs. During the gallery walk, they could discuss what they saw with their group members but could not make changes to the work that they saw. At the conclusion of the gallery walk, each group was given time to discuss the choices they had made with their own card sort and make any changes they wished to their group’s card sort.

Finally, we debriefed the card sort. We returned to the large scale graphs on the walls that we had used for the chalk talk. I read one of the stories and asked one of the groups to share which graph they believed it matched. They had to identify the independent and dependent variables and explain how the changes in the dependent variable reflected the story. They also had to give the graph a title. Other groups then had the opportunity to comment or question the group’s response.

One of the things that I really like about this lesson is that students grapple with hard questions. There is the potential for them to wrestle with misconceptions. As they discuss the graphs and stories, their misconceptions are exposed. For example, someone always suggests that one of the parabolas is the amount of daylight over the course of time. Someone always then raises the question of what it means for the amount of daylight to be zero (either initially or in the middle of the graph). At that point, there is always this lovely “oh!” moment in which students realign their thinking.

Next year, I do want to make a couple of changes to the way I did this. First, I will re-work my directions for the chalk talk. In addition to the graph, I will write a prompt asking them to respond to specific components of the graph (initial values, how the dependent variable changes as the independent value increases, etc). Second, I will change the way that I do the card sort a little bit. I think I will give each group a set of dry erase markers and require them to add labels and a title to each of the graph card as they match it to the story. I am curious whether seeing those things written on the graph will help them see their misconceptions before the whole class debrief.

# Identical Triangles Card Sort

Identifying identical triangles was a little bit of a challenge for a few of my 7th grade students. They could correctly identify a property (Angle, Angle, Side), but then would completely disregard the fact that the sides in the two triangles were not corresponding. I want to revisit the concept with these students as a quick review activity even though our lessons have moved on to other topics. To do so, I created a card sort.

In the card sort, each card has a pair of triangles. Students will sort the cards into categories: “Identical”, “Not Identical”, or “Not Enough Information To Tell”. If the triangles are identical, they will be required to identify the property shown. In order to address the fact that students are not looking to see if the sides are corresponding, I will probably require them to use colored dry erase markers to indicate corresponding sides.

I will pair students who did not master the concept with students who did. I think I will implement a strategy that I read about on another blog (I don’t remember which one, only that it was a great idea), in which students within the pairing are required to take turns. The first partner sorts a card into a category. The second partner then either agrees or disagrees with the sorting. If he or she disagrees, he or she can move it but must explain why it is being moved. The second partner then sorts another card and the first partner can agree or disagree.

# Making Graphs Anecdotal Records Form

Knowing what my students know is important to me, but remembering who is struggling with what aspect of a given concept from day to day is impossible. There are just too many students and too many variations. I absolutely have to write down what they know if I want to act upon it the next day. Unfortunately, there isn’t a lot of time in a given class period to take those notes. As I result, I have started creating anecdotal record forms that are specific to a big idea that I am addressing in class. On these anecdotal records, I have a single box for each kid. In each box, I have specific items of which I want to ensure mastery. I use one sheet for each period. I put a different student’s initials in each box. Then, I make copies of the sheets so that I have enough to use for more than one day. Alternatively, I use a different color pencil each day. I keep them on a clipboard for ease of use. As I circulate around the room, I circle an item in a student’s box if it is an area that I need to address with him or her. The next day, I simply look at the sheet from the previous day and I know exactly what I need to address with specific students.

I have just started a unit in which students will be representing and analyzing data in tables and graphs. During this unit, I will be using the anecdotal records shown below.

I use the” Ind/Dep” category to indicate whether students are correctly selecting the correct axis for the variables (independent variable on x, dependent variable on y). The “Intervals” category indicates whether the student is maintaining uniform intervals on the axis (a very common error in the early days of making graphs). The “Continuous/Discrete” category indicates whether a student is correctly determining whether or not to connect the points on the graph. The “Plot points” category indicates whether a student is correctly plotting points (x,y vs y,x). The remaining categories are more minor errors, but errors that I want students to clean up.

You can download the form by clicking on the link below the photo.