engagement

No Bikes Allowed

by Jon Orr

Sparking Curiosity

Students will extend their use of patterning to connect different representations of linear relations. More specifically, students will be exposed to rates of change, calculating the slope of a linear relation, building linear equations, and solving linear equations.

The purpose of this Task is:

  • to help students develop an understanding of how various representations of linear relations are connected;
  • create a linear equation given two points; and,
  • solve linear equations.

Intentionality

As is true for any task, the intentionality or learning objective can vary depending on what mathematical thinking you are hoping to elicit.

The mathematical ideas we are trying to elicit when we use this task are connect various representations of linear relations, build linear equations from two or more points, and solve linear equations.

This will be achieved by first having students watch how the cost of renting a scooter is related to the time rented. You can learn more about the actual scooters and the pricing structure at lyft.com.

Spark Curiosity

WARNING!!! Be sure to share with your students the importance of wearing Bike Helmets while riding; we certainly should have. We realize that you may not want to do this activity with your students as a result of our poor judgement.

Show students this video:

Ask students to engage in a notice and wonder protocol. ANYTHING and EVERYTHING that comes to mind is fair game.

Here’s some of the “everything and anything” students noticed and wondered on chart paper:

  • I noticed that they were riding scooters;
  • I noticed that they weren’t wearing helmets;
  • I noticed the map;
  • I noticed that the cost changes;
  • I wonder where that was?
  • I wonder how much the scooters cost?
  • I wonder what the range means?

Now you can focus in on the big question of the task.

How much does it cost to ride the scooter the entire length of the outlined route?

We can now ask students to make a prediction using their estimation skills. Ensure you use the Too high and too low strategy. Ask them what is a wrong answer? How much would be too high? How much would be too low?

Students will also be uncomfortable here because the length of time of the trip or how the scooter charges customers has not been revealed yet. We encourage you to hold off on revealing these answers because it will build anticipation. Anticipation is what students need so they can start formulating a plan.

At this point, we want to give students the opportunity to improve their predictions by engaging in developing a problem solving strategy.

Ask them: If we are going to improve our predictions what information will we need? Have students share with an elbow partner before sharing this information with the entire class.

As students voice the information they wish to see.

Ask them: “And what would you do with that information if I gave it to you?”

Listen in very closely here. Their responses will give you allow you to assess their prior knowledge and also their thinking into solving this problem.

Slowly Reveal More Information

Once students have asked for information reveal the information that you do have.

Reveal a key fact:

The cost of renting a scooter depends on the time the scooter has been rented for (You may want to reveal depending on your students’ prior knowledge that there is a flat fee to get on the scooter).

Reveal three snapshots showing the cost at different points in the trip. We include three here so that we can verify that the relationship between the cost and the time is linear.

Reveal the total cost of the entire trip.

With this information students can start to develop strategies to determine how much it will cost for a 15 minute and 9 second trip.

Fuel Sense Making:

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After consolidating the learning using student generated solution strategies and by extending their thinking intentionally, we can share what the actual cost of the trip was:

Chocolate Mania [3 Act Task]

by Jon Orr

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.

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Creating Math Moments: How we can transform typical textbook problems into moments that matter.

by Jon Orr

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:

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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.

Hour Glass Multiples

by Jon Orr

Sparking Curiosity & Fuelling Sense-Making with the Least Common Multiple.

In this 3-Act Task students will be presented with a puzzling video of 3 “hour glass” sand timers. They’ll solve a brain-teaser like problem while ultimately learning about common multiples and the least common multiple (LCM).

In this particular, this task can be used for

  • estimation;
  • spacial sense;
  • volume;
  • counting in multiples;
  • least common multiple;

Act 1: Sparking Curiosity

Ask students to create a notice/wonder table or you can use one that Kyle Pearce and I 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:


Then have them share with elbow partners and then finally with the entire class.

Some possible notices and wonders:

  • I see three different colour timers.
  • Is that sand?
  • Whose house is that?
  • Are they timing the same amount?
  • What times will they time?
  • Will all three timers ever end at the same time? If so, when?
  • Is the timer in minutes?
  • I think the yellow timer times for 3 minutes.

After capturing all the notice and wonders on the front board steer the class to working on the problem

“Will all three timers ever run out of sand at the same time? If so, when? If never, why not?”

Assume that we will keep turning over a timer after the sand runs out.

Take a few minutes to have your students estimate when the timers will all run out at the same time –> “Predict with reasoning”.

Act 2: Reveal Information to Fuel Sense-Making.

To avoid rushing to the algorithm 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. For example student 1 might say “I’d like to know the times of all the timers”. As a teacher your next question should be: “I see, and if I gave you that information what would you do with it?” We can learn what our students understand and are thinking with their response to one prompt. By asking them to anticipate what they need forces them to develop a problem solving strategy.

After hearing a few students out, give them this information:  But make them guess first. What time does each timer time?

Reveal the timers:

After this reveal send students to their vertical spaces to explore the strategies they began in the anticipation stage to determine when the timers will run out of sand at the exact same time.

Strategies you may see:

  • Drawings that show how much time is left every time one timer runs out.
  • lists of the multiples of 2, 3, and 5.
  • tables that track minute by minute.

Fuel Sense-Making to Consolidate Learning.

Depending on your grade range and student ability you’ll want to frame your consolidation so showcase your target learning goal.

I’m sure most learning goals will include a triple number line showing how multiples of 2,3, and 5 overlap.

Clearly show using the lines how the 2 and 3 minute timer will be turned over at the same time at the 6 minute mark. Then show them all the common multiples between 2 and 3.

Finally bring in the multiples of 5 to the mix.


As part of your consolidation show this video which overlays the common multiples as they occur in the reveal video. Students can clearly see that when the timers are turned over at the same time we have a common multiple.

Here is a reveal video without the number line overlay.

Try this lesson out in your class and report back here in the comments to tell us how it went.

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VIDEOS & IMAGES

Download the lesson files so you can run bring out great moment around least common multiples. 

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Are you new to 3-Act Math problems? Grab our guide to running these problems in your classroom. Learn tips, suggestions, and avoid common mistakes of using these types of tasks.

New to Using 3 Act Math Tasks?

Download the 2-page printable 3 Act Math Tip Sheet to ensure that you have the best start to your journey using 3 Act math Tasks to spark curiosity and fuel sense making in your math classroom!

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Acknowledgements.

I want to thank Michael Jacobs for turning my thinking towards thinking about the least common multiple. The creation story of the above task comes from this hour glass timer I bought from David’s Tea

Mike said,

Bryan also was thinking it was screaming LCM.

Which made me start thinking about how that couldn’t work with all three timers attached. So I set off to buy some new timers. I found the ones you see in the problem above.