Sparking CuriosityStudents 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.
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.
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:
LEARN HOW TO FUEL SENSE MAKING
MAKE MATH MOMENTS ACADEMY
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:
This post and task was written and created by both Jon Orr and Kyle Pearce.
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 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.
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.
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.
DOWNLOAD THE TASK AND RESOURCES
New to Using 3 Act Math Tasks?
For the last few years I’ve shared out a Google Form for classes to record measurements around their pumpkins and make them explode! I shared that form on Twitter so that we could crowd source as many pumpkins as we could to make the sample size large enough. I was pretty shocked at how many schools from North America took on Pumpkin Time-bomb. By the time Halloween was over the spreadsheet had over 90 entries. That’s over 90 pumpkins exploded in the name of math and data collection.
[Update] – October 2018 – The form now has over 500 entries!!
This coming week let’s add to the data and use the it in our classroom to discuss: Scatterplots, Trends, Correlation strong, weak, no-correlation, lines of best fit, correlation coefficient, etc.
Here’s a sample lesson you could use on the day you make your pumpkin explode.
Play this video which shows Jimmy placing rubber bands around a pumpkin.
NOTICE & WONDER
Using a notice & wonder strategy, have your students record anything they notice and anything they wonder from the video.
Steer you class’ wonders toward the questions: How many rubber bands will make the pumpkin explode?
Have students write down a guess that is too low. Too high. Then estimate their best guess.
If you’re looking for your lesson goal to be around estimation then show the act 3 video next, but if you’re looking to go further and tackle a learning goal around Using scatterplots, lines of best fit, or linear regression jump down the post.
Show the Act 3 Video
Using Scatterplots & Trends to Improve Your Prediction.
WITHHOLDING INFORMATION to create ANTICIPATION:
Use the PAUSE tool on the activity to lock their screens while you show your students the video on your main screen. Encourage your kids to discuss what they notice and wonder from the video! In pairs, I have my students TALK first and then TYPE second when collaboratively working on a Desmos activity.
Consider pausing the screen again while you use the snapshot tool to grab student responses! This will lead into predicting how many bands will make Jimmy’s pumpkin explode. Have your students TALK first and TYPE second on screen 2 to make a prediction. Again, share students predictions using the conversations tools Desmos provides.
FUEL SENSE MAKING – IMPROVE YOUR PREDICTION:
Bring your students down the curiosity path a little more. Ask them about how we can improve our predictions? What other information would you like to know about the pumpkin or the bands?
Have a discussion on variables & relationships. Write all the variables on the board they come up with. Narrow down the list to items that are measurable with the pumpkin we have in the class. What affects the explosion the most? Height, diameter, circumference, thickness of the wall?
Using the PACING tool in Desmos move your students few the next few screens to make a scatterplot prediction of the relationship between the diameter of a pumpkin and how many bands will make it explode.
Screen 5 shows a scatterplot of pumpkins that have already been blown up and the relationship between diameter and bands (or non relationship). Have your students move the orange point to a place that helps them predict the number of bands. What placement would be wrong?
The next few screens ask your students to do that all over again while looking at the relationship between the height of the pumpkin and the number of bands.
Finally, reveal the answer after students have improved upon their predictions.
Now Bring out your pumpkin for the class to see! Have them predict how many rubber bands it will take before it will explode. Repeat the estimation process. Have them save their guess till the end of class. Where will YOUR pumpkin fit on the scatterplots shown in the Desmos activity?
If you are not planning on using the Desmos activity then you can use the original activity post from October 2015.
FUEL SENSE MAKING – Making A Model
Throw out the question: “What about the pumpkin do you think affects how many rubber bands are used to make it explode?” Let your students brainstorm a list of variables. Have a discussion on variables & relationships. Write all the variables on the board they come up with. Narrow down the list to items that are measurable with the pumpkin we have in the class. What affects the explosion the most? Height, diameter, circumference, thickness of the wall?
Have them choose a variable that they feel should have a relationship with the number of rubber bands. Fill out the prediction part of the handout.
As a class measure all variables needed. Write them on the board for all to see.
FUEL SENSE MAKING – Analyzing Data
Give students the link to the spreadsheet of all the pumpkins to date (You should copy and paste the data to your own sheet so you can filter/sort the results and share that sheet out to your students.)
Discuss with your students the lack of consistency in the selection of rubber bands from all over the country. How can we minimize this variable skewing our results? Filter the data with your students(or before hand) showing one type of rubber band (Most common is a rubber band of length 8.65 cm). This will only show all the pumpkins that have been destroyed using that type of band.
Get your students to grab the data that relates to their relationship.
If Kristen chose the relationship Circumference vs. Rubber bands she should copy and paste the circumference column and the rubber bands column into a new sheet side by side. Then copy and paste all that data into the pre-made Desmos File.
She can adjust the scale in Desmos as needed. Have her move the movable point and drop it where she thinks your class’ pumpkin will lie. Or you can have her find the line of best fit to help predict how many rubber bands it will take. Either way we want her to predict with more accuracy.
So Kristen would predict that if her circumference was 90.5 cm then it will take 272 rubber bands to blow up the pumpkin!
Now if Kristen chose a variable that it was clear there is no relationship then you get to have a discussion about correlation vs. no correlation. Have her choose new variables to predict on.
Once everyone in the class has a new prediction start wrapping bands around that pumpkin (You may want to start this as early as possible).
Watch your pumpkin explode and give congratulations to the student who predicted closest to the actual number of rubber bands.
Don’t forget to enter all your data to the sheet by filling out this form (you can also use the form to show the videos to the class).
[Updated] – You can use this Desmos Activity Builder Activity to facilitate the lesson. It includes only data for Diameter and Circumference.
From Oct 30. 2015
A few pumpkins from 2014 & 2015