Author: Jon Orr
Chocolate Mania [3 Act Task]
This post and task was written and created by both Jon Orr and Kyle Pearce.
For about a year now Kyle Pearce and I have been travelling to schools and districts across North America sharing our techniques on how to Make Math Moments That Matter for our students.
In those live workshops we’ve been using a task without a name. On the first anniversary after creating that task we wanted to share it here with you and give it a name.
We’re all about creating tasks and then thinking about how they might be modified for use across a variety of grade levels. With a few modifications, you can successfully run this task in classrooms from K through 10. In particular, you could address the following expectations:
- building estimation skills;
- building multiplicative thinking and proportional reasoning using arrays;
- building multiplicative thinking and proportional reasoning using double number lines;
- making connections to the inverse relationship between multiplication and division;
- connecting double number lines and ratio tables to creating and solving proportions through algebraic reasoning;
- highlighting the value of the constant of proportionality (i.e.: unit rates) so students can “own” every problem possible in a proportional relationship;
- determining rates of change;
- representing linear relations in various ways;
- solving problems using the four representations of linear relations; and,
- many more.
Here is Chocolate Mania:
Act 1: Sparking Curiosity
Ask students to create a notice/wonder table or you can use one that we built for our online workshop Making Math Moments That Matter.
Ask your students to write down anything they notice and anything they wonder while viewing this video:
Note: There is no audio. Can’t see the video because you’re viewing this post in a rss reader? Click here to go to the post page.
Here are possible notice and wonders from our workshop participants and also some from our students:
- They’re both wearing plaid.
- The video is in reverse.
- How many chocolates will they eat?
- Did they get sick?
- How long did it take to eat all the chocolate?
- It looks like they’re spitting it out.
- Kyle is eating Kisses.
At this point the students’ responses are listed on the board during the class discussion.
After capturing all the notice and wonders on the board steer the class to working on the problem:
“How many chocolate did Kyle eat? How many did Jon eat?”
Have your students estimate how many each of us ate. What is too high? What is too low? Your students may be feeling uneasy about their estimates; that’s okay! The point here is we don’t have enough information. To help with estimates at this stage we disclose that all the wrappers of all the chocolates we ate are showing in the image above.
We encourage you to record many of the estimates in a chart as a class. This will put some pressure on making those estimates carefully.
Act 2: Revealing Information to Fuel Sense-Making
To avoid rushing to the algorithm we’ll push down the curiosity path some more. Instead of just handing over all the necessary information to solve a problem ask the students what they want to know more about. This process is key; student anticipation of what is needed is a gold mine for understanding where they are in their thinking. By having them ask for information they have to start problem solving!
Students may ask for the time it takes for the whole video and you as the teacher can then say, “And what would you do with that if I gave it to you?” Listen to how they answer this. You’ll gain valuable information about where that student is on this problem solving journey. You will know after that answer if the student is thinking proportionally or not.
Here is some information to share:
Ask students to share what this series of photos tells them. What do they notice? What do they wonder? Then share this photo. It reveals the total amount of ml each of us consumed.
At this point students will have enough information to determine how many pieces of chocolate each of us ate. Let them go at it!
Fuel Sense-Making to Consolidate Learning.
Note: You or your students may want to work with more familiar numbers compared to what you see above. For example, to get a close prediction to the actual number of chocolates each of us ate a student may round the 111.8 ml to 110 ml and similarly round the 17 ml for 3 chocolates to 20 ml.
Depending on the grade level or skill level of your students we can expect to see some of these strategies
- Counting with familiar numbers;
- Using arrays;
- Number line counting;
- Tables of value counting;
- Long division;
- Unit rates;
- Solving Proportions;
- Creating and solving equations.
Here are some of those strategies:
Counting Up Chocolates and ml.
Students may count up 17 ml every 3 pumpkins until they reach close to the total amount of ml. If they go over the total amount they may want to subtract a cup of chocolates so they can get more accurate.
Here’s that strategy in action
Working with Fractions:
To get more precise answers we can encourage students to work with parts of chocolates in decimals or fractions. Many teachers would be inclined to stay away from fractions because they feel it may “de-rail” the lesson. We say use this context to reinforce fraction work and understanding.
Counting/Multiplying/Dividing Using Arrays:
Students may organize their counting strategy in a double array model. Simultaneously counting in groups of 3 pumpkins and 17 ml will allow them to see that they will need just over 6 cups of pumpkins, while showing the proportional relationship between the pumpkins and volume.
Double Number Line:
Students who solve the problem with a proportion will benefit from seeing it laid out on a double number line. By showing how to solve a proportion on a double number line we take a familiar concept (counting on the number line) and extend it to work multiplicatively. Students who solved the problem with an additive strategy will see the benefit of greater precision of using a scale factor.
Unit Rates:
Many students may use a unit rate to help solve this problem.
Note: This student will benefit from a conversation on notation, units and order of division.
Linear Relations:
You may choose to use this problem to either introduce or practice linear relations. I used this task to link the idea of finding the unit rate to determining the rate of change (slope) in a linear relation and then use it to build an equation to help solve the problem.
Reveal the Answer:
After consolidating the learning goals you wanted to bring out into the open for discussion with your class show them this reveal video of the actual number of chocolates each of us ate. Be sure to go back and validate those students who estimated the closest early in this task.
Is there a Volume relationship?
We want to leave you with some thinking here. We chose these chocolates for a very specific reason. In fact we hunted down the spherical chocolate that has the same height and diameter of that Hershey’s Kiss.
Your Task: What volume relationships can we pull from this image?
Did you notice the relationship between the amount of chocolate by volume Jon ate versus Kyle?
Look for an upcoming post on how we used this task to teach volume. But before we do that we want to know how you see a lesson on volume forming with this information. Use the comment section below to share your ideas, questions, comments, or even just snippets of what a lesson could look like.
How learning to ride a bike is like/not-like learning math, and why it should be!
What are the moments that truly matter?
For me they are moments where we learn or accomplish something we are exceptionally proud of. They are moments that make us stand back and say “Wow! I did it” They are moments that we say “wait, let me get a picture of this.” They are #Snapworthy moments.
For me they are, playing that first song on guitar, catching a fish for the first time, watching my daughters swim that full length of the pool, scoring a basket in basketball, scoring a goal, singing a song in front of an audience, or riding a bike for the first time.
Learning the skills needed to accomplish these feats takes a similar path. The process has a similar experience.
We learned these things through the process of productive struggle.
Take my daughter Lucie for example. She was the last to learn how to ride a bike (tough when you have a twin too). She finally learned in a similar way, most likely, to how you learned how to ride a bike; by getting on, trying to balance while coasting, then falling over! Then, trying again.
Every time she fell she learned something. She would adjust and try again. She was struggling productively.
The difference between just struggling and productive struggle is Feedback.
Going through the struggle, using feedback and then making small gains gives us a rewarding experience. It makes us want to keep going. We build perseverance. We want to do better. This is one of the key ingredients to make moments that matter.
You know, we learn to ride bikes this way but traditionally we don’t learn or teach math this way. Vice versa is also true. —> we don’t learn to ride a bike the same way we learn math.
For a moment Imagine that we did. Imagine we structured a course on riding bikes like we structure our traditional math classes.
Here’s what the syllabus of bike class might look like, especially if it was taught in our schools. (I’ve adapted an analogy here from Dr. William Rankin).
Day 1:
“Welcome to your first class on bicycle riding. It’s going to be a great semester! We’ll start off week 1 with learning all about the tires. Tires are super important they’re the life of the bike. Learning about tires is important because it will help us be ready when we ride a bike.
During week 2 and 3 we’ll go over how the pedals work. Pedals are vital, they help make the bike move. In those weeks we’ll learn how that happens so when we start riding bikes we’ll be ready.
In week 4 we’ll have a test on the tires and pedals and then we’ll move on to study the handle bars. We won’t revisit the tires and pedals again until the end of the year so make sure you study for this test!
Weeks 5-8 is for Brakes. Brakes are vital to controlling the bike. I know they are related to the handle bars but handle bars were last unit. We don’t want to mix the two.
Weeks 9-10 are for Gears! I know they’re part of the pedal, wheels, and handle bars, but we’ll just talk about gears those weeks. You’ll need to use them when you start riding your bike.
Well ……That’s all we have time for in this course….
If you take our next course we’ll learn all about balance, whoa, that’s a biggie when it comes to bike riding.
When do we actually ride bikes?
That’s when you graduate!“
Silly right!!
What did you do when you learned how to ride a bike? You just jumped on and rode! Just like Lucie did. You felt a purpose to what you were doing. You learned as you were riding. It was a memorable moment.
But that silly bike class is the way we traditional teach math class. We tell our students that a purpose of math learning is so they can solve problems in the real world! We hold it over their heads that real problem solving is only for when you’re in the real world — done all your schooling.
We’ve traditionally taught math concepts in siloed units as if one math strand isn’t connected at all to another.
We say now,
JUST RIDE BIKES
Teach through problem solving. Productive struggle teaches the resilience we are looking for in our students. Just get on the bike and ride it!
In many of my past “problem solving lessons” I wasn’t really teaching students how to become better problem solvers.
If we’re giving step by step guides to solving problems in our classes are we really teaching problem solving? How much genuine problem solving are we doing in our math classes?
Teach content through problem solving. It’s the productive struggle – feedback cycle that really teaches our students to build resilience and their problem solving skills. It’s the productive struggle – feedback cycle that will create moments that your students will feel pride in. Those are the moments that matter. Just get on the bike and ride it!
[UPDATE]
Carla, a participant from our Making Math Moments That Matter online workshop pointed me to a fascinating video –> “The Backwards Brain Bicycle” from SmarterEveryDay. The video illustrates the notion that we may have the knowledge of how something works but we don’t always have the understanding of making it work.
Watch:
How does the message of this video relate to math education? –> We may have the knowledge that we need to Just Ride Bikes so that our students can become better problem solvers while at the same time creating meaningful moments but we don’t understand exactly how to do that.
We ourselves need to Just Ride.
We have to unlearn what we understand about teaching math class so that we can build a new path towards Making Math Moments That Matter.
Resources to help “ride bikes”
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Creating Math Moments: How we can transform typical textbook problems into moments that matter.
In an ongoing effort to demonstrate that we can apply the 3 part framework of Spark Curiosity, Fuel Sense Making, Igniting Teacher Moves to any lesson in math class we’ll tackle the common problem of finding the equation of a line between two points. Like this basic problem:
Sometimes textbooks may even jazz it up a bit to give it some context like this one.
Let’s re make this lesson to fall under our 3-part framework.
Even though this is a grade 9 and 10 expectation here in Ontario you’ll find that this problem is quite accessible for many grades.
In particular you could use it to uncover:
- Find the slope (rate of change) of a line between two points;
- Find the equation (rate of change) of a line between two points;
- Model real-life relationships involving constant rates;
- Model linear relationships using tables of values, graphs, and equations.
Spark Curiosity
We’ve been arguing that instead of finding a context that will make students interested we should follow the curiosity path which we’ve described in the first two lessons in our 4 part video series instead.
Let’s consider the big idea here: We want students to build an algebraic representation of a linear relation using only two values.
To withhold information and build anticipation we will strip all the numbers and questions and ease into the lesson. To help create the classroom culture that values student voice, student thinking, and growth we’ll ask students to fill in the two blanks here:
Setting the floor low will help our students feel attached to the math problem that is coming. The more attached and invested they will feel the more internal motivation they will have to pursue the problem to the end.
In behavioural economics there is a theory known as The Sunk Cost Fallacy.
Or also known as Escalation of commitment.
From Wikipedia,
Escalation of commitment is a human behavior pattern in which an individual or group facing increasingly negative outcomes from some decision, action, or investment nevertheless continues the same behavior rather than alter course. The actor maintains behaviors that are irrational, but align with previous decisions and actions.
You see, we humans are inclined to avoid loss. We will continue with a project or line of thinking if we feel that if we abandoned it we would incur loss. Even if that abandonment was better for us.
For example, my first car, a 1993 Ford Escort – you know, this is the car that had the automatic seatbelts. When you sit down and turn the car on the seat belt came up and automatically moved over your shoulder. One winter the heater in the car stopped working and I paid over $1000 to have it fixed. Then not long after something else broke on the car and instead of saying enough with this car I said, “Well, I just paid $1000 to fix it if I don’t fix it now then it’s like my $1000 was wasted.” This bias I just exhibited is an example of the sunk cost fallacy. I wanted to throw bad money after good. The $1000 I previously spent was a sunk cost and there’s no way I could get that back so the $1000 shouldn’t play a roll in my new decision to fix the car. I should decide to fix the car or not fix the car without letting that $1000 affect this decision.
The sunk cost fallacy makes us feel that if we invest time, money, resources into a project or decision that we should keep going with that project or decision so we avoid loss. The escalation of our commitment keeps us in the game.
In math education we can use our students own tendencies of avoiding loss for their own good. By setting a low floor in activities, we are easing our students into those activities and lessons so that it will be harder for them to just quit and give up once they are deeply invested in the activity . They won’t want to feel that what they’ve done so far in the activity was a waste of time and resources. They’ve sunk a cost into the activity and will continue with it to avoid loss. You can read another application in education of the Sunk Cost Fallacy from Robert Kaplinsky.
Your students will fill in various items and values for this problem. In my class this was a fun moment as we shared out what they wanted to buy and for how much.
Fuel Sense Making by Revealing Information
So now we’ll move down the Curiosity Path and narrow the focus to give a little more information.
How much would 12 shirts cost?
Students can make quick predictions before revealing the information slowly…
We don’t want to waste all the work we’ve done on escalating our students commitment so we’ll move down the curiosity path a little bit more and avoid the rushing to the algorithm. Students will use the given information and their prior knowledge to build a strategy to solving this problem.
Fuelling Sense Making by Anticipating
We are strong believers and practitioners in the PDF or the book 5 Practices For Orchestrating Productive Mathematics Discussions. So in preparation for this lesson we used our Anticipation, Selection, and Sequencing template to brainstorm possible solutions and strategies our students will try.
You can grab a blank copy of this template here.
To maximize your mathematical discussions you may want to sequence the strategies from most common to least common.
For example;
You can expect many students to try to find a unit rate to solve this problem. This is quite natural! It makes sense to find the price per shirt. However, not all situations are directly proportional. We can ask our students: How do we know this is a direct proportional relationship?
When students find the unit rate for 12 shirts at $122 and then again for 24 shirts at $209 they will see that it doesn’t cost the same per shirt! WHAT!?
Something else is going on here. You may want to give a small hint here asking, “hmmm, If 12 shirts cost $122 does 24 shirts – which is double the amount of shirts cost $122 x 2? How much more does 24 shirts cost? What would 36 shirts cost?
Students who noticed this right away may draw a double number line to show the changing prices and eventually determine the cost per shirt.
Students who have found the cost per shirt will still notice that simply multiplying the cost per shirt by the number of shirts STILL doesn’t get the cost — there is some other value that consistently needs to be added – The initial value or fixed cost.
Have a discussion at what this fixed cost could be — shipping charges? Overhead costs? Printing rental fee? ect. With this new calculation rule students can move on to verify that it does indeed work with 200 shirts, and then finally find the cost of 1100 shirts. You may even want to steer your discussion towards finding an algebraic representation of this relation.
Some students may represent this pattern as a table instead of a double number line. Depending on your grade level you may also want to use the word slope to represent the cost per shirt. If you see this solution from your student you’ll want to push for an algebraic representation
You may see some students turning toward Desmos and graphing the points to find an algebraic representation. We definitely anticipated this having taught this lesson in a grade 10 applied class.
The order you present these strategies/solutions will depend on your lesson goal. If you are trying to achieve the goal from the top of this post (Finding an equation of line between two points) then you most likely will want to end with finding the algebraic representation and then showing how you can use Desmos to verify that representation.
Finally we can show students that if the relation is linear, we really only need two points.
We feel that if we can take this particular learning goal and modify the delivery and teacher moves to create a math moment that matters we can do this with any textbook problem. What lesson should we make over next?
If you haven’t checked out our 4-part video series yet, get over there now:
We also have a course inside our Math Educator PD Academy all about how to transform textbook problems into curiosity machines. Best of all, you can join the Academy free for 30 days and cancel anytime.
That should be enough time for you to dive in and learn a ton from that course.
