I’d like to begin by thanking those involved with Emergent Math for organizing and compiling (and creating?) great tasks. This task originates from John Berray. He originally designed this activity for an “Intermediate Algebra class, which is a lighter form of Algebra 2. Really, this activity is a first year algebra activity that can extend deeper depending on the course.” I have modified his activity based on his suggestions from his blog, and made it fit 7th grade standards for Ratios and Proportions.
Day 1: Notice and Wonder
This was a minimum day strictly dedicated to noticing and wondering about the scenario cards. What do YOU notice? Math concepts they notice and wonder about involve slope, unit rate and y-intercept and we haven’t even started the activity yet!
Day 2: Data Collection
If you are doing with Skittles, buy 3 – 2 pound bags for two classes of 38. It was a bit crazy at first. The students realized how hard it was to actually consume the amount of Skittles they were supposed to in their given scenario. I may try M&M’s next year. However, this led to different strategies for collecting data. Some shoved their faces full of Skittles to the point of being sick, while others strategically set aside the amount of Skittles they were supposed to eat to be able to count the total a little easier.
Day 3: Unit Rates
They were asked to figure out who ate the most skittles per second. I was a little disappointed with the results. Of course I had 5 or so students in each class who knew exactly what to do and immediately had the right answers with realizing that something was wrong with scenarios A and B. Then the typical mistake was to mix up the order of the quantities in their unit rate. But everyone else was lost. Blank papers, weird strategies I hadn’t even anticipated and the questions I was asking them to move them along weren’t getting them anywhere (which means I was asking the wrong questions or I needed to come up with something else). I had anticipated incorrectly. We didn’t get to finish this as planned so I needed to re-evaluate questioning/other strategies.
Day 4: Unit Rates cont…
Revisiting the idea of Unit Rate, I knew I had to get them to the meaning of their answers. We spent the entire period talking about these 4 strategies. Some of the questions I asked during the sequence referred to the last two strategies at the bottom:
- What do you notice?
- How are the similar?
- How are they different?
- Why divide?
- What does this mean in terms of seconds and Skittles?
- How do we know which one is right?
- Is there another way we could write this?
- How can we use a Giant One?
The students struggled with defining what their answers meant. We focused just on person A when we were talking about the 0.6. Is this Skittles or seconds? The class was split 50/50 and it led to a great debate about why it represented Skittles per second. I wanted them to write ratios do help develop the idea of rate vs. unit rate and they just weren’t going there with their strategies. Finally after many (and I mean many) table talks, they could remember what ratios were (i.e. 3:4, 3 to 4 and 3/4). And then we decided which form of the ratio would be most efficient to use (they chose fraction form). They did come up with the idea that if I’m eating 3 Skittles every 5 seconds, then that is less than one Skittle per second, so the answer has to be less than 1 (which cleared up the error in the representation of the division). To link back to their prior knowledge, I asked them about the “Giant One” a concept from the CPM text.
This led to a discussion about whether or not we should multiply or divide for equivalency and what that meant in terms of our answer as it applied to the context of the problem. This tied back to their understanding of equivalency with rational numbers and they talked about how the to two answers could be the same, which we defined through table talks. Now they could apply this to the rest of the scenarios. Finally, Success.
A student in my 6th period noticed that person A and B both start with a number, and that when we try to find the unit rate, we get a different answer each time. This wasn’t the case with scenarios C and D where no matter what we divided we got the same answer. So then another student asked, “What would happen if we re-did the chart without the Skittles that they started with?” I was SO HAPPY that this question came from a student and not me. So she made a new chart without the starting amount of Skittles and started to look for patterns in numbers. Now they noticed that they were getting the same answers as when they were dividing with the rate. This took 2 full days plus a minimum day to get here. But I have to remind myself…
“Teach at the speed of learning.” Phil Daro
And sometimes learning is slow.
Day 6: The Blank Graph
Unfortunately I had a sub when the kids analyzed the blank graph, but she is a seasoned sub and could trust her not to just give the kids the answers. First the kids did Notice & Wonder. I reviewed the discussion questions with them again the next day to determine their level of understanding. Discussion questions were:
- How do you know you labeled the graphs correctly?
- Who takes the longest to eat? How do you know?
- How is that related to what we know about the rates and unit rates?
Students were able to connect that if you eat slower, your unit rate would have a smaller value. We already realized that if we eliminated the Skittles that person A and B started with, that we get the same unit rate, so 0.6 for Person A had the smallest value, and therefore would not be a steep as the other lines. If this were an Algebra 1class, I would have loved to go more in depth about what it means when Line 2 intersects Line 3. The kids thought this was interesting and noticed that Line 3 was eating more than Line 2 at first, but then Line 2 ends up eating more. They related this to the fact that Line 3 which was person A, started with Skittles which accounted for why they were eating more at first.
Day 7: Labeled Graph
One of the major goals for Day 7 is for the kids to use the graph to determine the Unit rate, and find the relationship between the Unit Rate and the starting amount of Skittles. Again, we “noticed and wondered” about the graphs and then they had to label which person belonged to which line.
It was nice to see that some students used the amount they started before the time, the amount of Skittles eaten total, and a connection to the Unit Rates to justify their thinking. The student below found a coordinate from the graph and matched it to data in the table to justify his answer.
Overall, I have been pretty pleased with my students’ ability to read the graphs and make connections between the graph, data, rates and unit rates. Monday we will formalize the unit rate from a graph as the point (1, y). Depending on how long that takes, it might be fun to push them into using similar triangles to see the rates! A girl can dream right? Ha!
Day 8 or 9…? (I’ve lost count)
Finally we come to the equation part of the activity. Something I didn’t anticipate was students writing equations in both minutes and seconds. First the tried to figure out how many Skittles person C would eat in 5 minutes. I used the strategies from 5 Practices on sequencing work.
Then I asked them to come up with a more general equation that would work with any amount of time…
It didn’t take them very long to come up with the equations, and as you can see they did it in seconds and minutes. Then they applied what they did for person C to person D. We had to review a little bit the different forms of the Unit Rates and what they might look like. This seems to keep throwing them off which means they haven’t really grasped what the unit rate is, as they keep confusing it with the rate. After realizing this, I know my next task has to deal with Unit Rates. It’s like I keep asking them, “And where did that number come from again?”
What is interesting is what came next. I asked them whether or not these types of equations would work with Persons A and B.
They knew what the format of the equation would look like and they clearly defined what each part of their equations meant. I couldn’t take pictures of their work because they were honestly arguing about what to write down, so I just recorded the class discussion. They knew that the equation in seconds had to contain some type of constant. They remembered that there wasn’t a constant when we divided the values in the chart until we took out the amount that they started with. This led them to the 0.6. I could see now why they were confusing the eating 3 skittles every 5 seconds as a unit rate. Originally this was the rate at which person A ate the skittles. However when dividing the values in the chart, they didn’t get .6 until removing the amount that they started with. So Person A ate at a constant rate of 0.6 Skittles per second AFTER they had already consumed 6 skittles. They even noticed this on the graph by drawing slope triangles (We didn’t call them that but they noticed it). We tested our equations to see if they worked for figuring out Skittles consumed in 5 minutes, and noticed that they answers didn’t match. Therefore they started tweaking the equation in minutes. They realized that the 42 already included the 6 Skittles already consumed which led them to change the equation from y=42x + 6 to y=(42 – 6)x. When we tested it again, we still got a different answer from what he had with the equation that was in seconds. Then someone realized we needed to add the 6 back in (which is the blue 6 I added when this came up). So now our equation went from y=(42 – 6)x to y = (42 – 6)x + 6. So I asked them, “Why subtract the 6 and add it again later?” They knew that in the equation y=(42 – 6)x they were accurately figuring out how many skittles were eaten at one minute, however they still needed to add the 6 to find total amount of Skittles consumed for any given amount of time. We checked it again with 10 minutes and it worked. When I asked them to apply this to writing an equation for person B, it was a piece of cake.
Without giving them a formal definition of what it means to be proportional, for their HW they had to list characteristics of what it means for quantities to be proportional to each other with regards to the Scenarios they were given in the activity, knowing that if I just give them the definition this whole activity would be a waste. They did this on the very bottom of the handout in the chart.
So when I checked the HW that they were supposed to complete (filling in the chart), only 5 out of 37 students completed the assignment in one of my classes. I was so frustrated and linked this to lack of perseverance. They want me to just give them the definition which isn’t meaningful to them at all. Then when collaborating with a colleague, I realized that maybe my mistake was assigning that part for HW. My 6th period did a much better job on pulling their thoughts together (or maybe they heard how upset I was and made sure they did it before class). They do a better job at persevering through mathematics and making connections in general.
The 5 students that did the assignment came up with the ideas on the left. For the most part they had the right ideas but it wasn’t as specific as I had anticipated.
Below are the responses from my other class, which are a bit more thorough. I felt like I didn’t even need to do direct instruction at the end of this class because everything I would have said what was included in these student responses.
Because of their lack of understanding Unit Rate, I’m going to do a task from illustrative mathematics called Track Practice. Then use the MARS activity Proportion or Not Proportion to solidify their knowledge of proportional relationships.
In a perfect world, there would be no weekends and no absences so we could have just continued the lesson straight through. Me being out one of the days and two (or 3?) long weekends made it a bit more difficult at the end when I was trying to help the kids pull all of the pieces together. The struggle is real right? The lesson is posted here and I am always open to suggestions on how to make it better. I made some notes in the PD column about possible changes to the activity I may make next time. So what do you think?