Spiralling mathematics to address multiple expectations / learning goals in one, authentic math task for grade 9 applied students. Linear Relationships.

The post Connecting Relationships to Multiple Strands appeared first on Tap Into Teen Minds.

]]>It’s been a couple weeks since posting, so I thought I need to get back in the swing of things. I’ve been enjoying a really busy start to semester 2. I’m loving the implementation of the gamified assessment approach I’ve modified off of Jon Orr and Alice Keeler’s work, the spiralled approach to teaching grade 9 applied math that Jon and I are trying after receiving guidance from Alex Overwijk and co., prepping for the third series of full-day PD sessions for the MYCI project, and practicing my first keynote speech coming up in just a couple short weeks. Unfortunately, my blog has been taking a backseat to all of the other “stuff” going on. So while this will be short, I wanted to give a quick update as to what we’re up to in room 1031 at Tecumseh Vista Academy (click here for a summary of my lessons to date).

As I mentioned above, I’ve been spiralling the content from my grade 9 applied course. In a nutshell, I’m breaking free from formal units and skimming the top of each strand, then spiralling back around to go deeper, then deeper, then even deeper. After only 12 classes, my students are doing things that I would traditionally wait weeks (or even months) before introducing.

Tomorrow, we will be exploring relationships and in particular, the relationship that exists between the diagonal length and side length of a square. Through this exploration, students will be demonstrating the following skills that we have already been working with this semester:

- calculate length of the hypotenuse of a right angle triangle using Pythagorean Theorem,
- identify independent and dependent variables,
- create a table of values,
- label and scale a graph,
- classify the relationship as rising/falling to the right, proportional/not, strong/weak/perfect, linear/non-linear, and positive/negative,
- make predictions using a graph and classify as interpolation/extrapolation, and
- if linear, create an equation to find the value of an unknown.

Since students have had some experience with all of these topics, students will be working on this task relatively independently. Here are some screenshots of the task template I’ve created:

We will consolidate the task by providing the visual below:

How can I make this better? Please let me know in the comments.

I hope to be sharing more new content soon, so keep an eye!

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]]>Applications are open for the Apple Distinguished Educator (ADE) Class of 2015! Apply to join our global community of learners before March 1st, 2015!

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]]>Every two years, the Apple Distinguished Educator (ADE) Program opens applications for educators using Apple technology in transformative ways to join the community. Since I became an ADE in the class of 2013, I have been blessed to participate in a large number of learning opportunities offered by Apple and outside of the community due to the global professional learning network I have established.

If you’re uncertain of the professional growth and value this program can offer you as an educator, here are just a few of the many learning opportunities I have experienced as a direct result of my being an Apple Distinguished Educator:

- Attended the Class of 2013 North American Institute – Austin, TX [read my reflection]
- Attended & presented at Apple Educational Leadership Institute (ELI) 2013/2014 – Toronto, ON.
- Joined the Apple Distinguished Educator Canadian Advisory Board in 2013.
- Became an Apple Education Trainer to deliver professional development.
- Attended the 2014 ADE Global Institute – San Diego, CA [see my ADE Showcase]
- Had my Learn Through Pythagorean Theorem iTunes U Course & iBook published on the ADE iTunes site.
- Currently participating in an Apple content writing project to create resources for struggling math students.

These are just a few of the many great learning opportunities I would never have had the chance to experience without the Apple Distinguished Educator program. Learning with ADE’s from North America and around the globe has been a life changing experience and I am so glad that I was pushed to apply back in 2013.

What is really great about the program is how they push you to always get better. Once you are an ADE alumni, you still must apply to attend future institutes. This means that everyone who attends any institute must demonstrate that they are still finding new ways to transform learning in their classroom. What an excellent idea.

Here’s my application video to attend the 2015 North American ADE Institute:

So be sure to apply before the March 1st, 2015 deadline and join in the learning!

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]]>Assessing students based on learning goals is great, but this new gamified approach I stumbled upon has potential to be a game-changer for my students.

The post Standards Based Grading GAMIFIED With Badges appeared first on Tap Into Teen Minds.

]]>Recently, Jon Orr shared his post called We Like Those Stinkin’ Badges that took Alice Keeler‘s idea of Levelling Up and Awarding Badges in her courses and gave it a twist to work with his math classes.

We Like Those Stinkin' Badges! Gamify your Assessment Recording http://t.co/3xBXEUrULu #mtbos #mathchat #edtech @MathletePearce @Regan_Bio

— Jon Orr (@MrOrr_geek) January 12, 2015

After spending most of the day with Jon in Toronto yesterday, I found some time to dig into his take on Alice Keeler’s work. I’ve shared some of the ways in which I’ve attempted to move towards assessing students based on learning goals (aka standards based grading) like some of the other folks in my district, but I’ve found that sharing one big public Google Sheet with students hasn’t inspired those who need it most to address the learning goals they are struggling with. For some time, I have thought about using cell referencing to create a personalized sheet for each student, but had no new ideas to really inspire making the change.

Jon’s spin-off of Alice’s gamification approach to assessment still focused on students being graded on individual learning goals, but the part that I really liked was the idea that students are to focus on consistently showing a deep understanding to earn 4-stars (level 4) in order to master the concept and earn a badge. Here’s what each of Jon’s students will have access to a skill evidence record google sheet that looks like this:

Jon created a Master Skill Record sheet that only Jon can view and edit that self-populates each of the student skill record sheets that looks like this:

Having found myself tossing and turning all night, I decided I should get up and give this a shot. The instructions by Jon and digging a little deeper on Alice’s site for new inspiration had me pretty much setup in 15 minutes. If you have used Google Sheets at all in the past, it won’t take you long. If you are new to Google Sheets, Google Drive, etc. then you’ll have a bit more learning, but nothing crazy. Here’s the steps I took:

- Read Jon Orr’s post.
- Followed up looking at Alice Keeler’s posts. This isn’t necessary if you’re just looking to replicate Jon’s idea.
- I started by copying Jon’s Master Spreadsheet (thanks, Jon).
- I added my learning goals for MFM1P Grade 9 Applied to my own copy of the GSheet.
- Added a few ideas I hope to continue building on throughout semester 2 (see below).

After making a copy of Jon’s spreadsheet that he created, I immediately poured in my learning goals from my MFM1P Grade 9 Applied course and came up with a few things that could enhance this great tool even more.

A few years back, I created a public spreadsheet with a bunch of related links to PDF files, tasks, etc. underneath each learning goal. For some reason, I never did this for my grade 9 applied class. I thought that with this new gamified way of assessing, it would be important to provide students with some next steps if I wanted them to actually take responsibility for their learning. So, I added 5 rows that I can add links to resources in. If you’re unsure how to create a clean text hyperlink in a GSheet, here’s the code:

=hyperlink(“http://www.google.com”, “Go To Google”)

The formula above will show the words “* Go To Google*” as a link sending you to http://www.google.com. Just as the rest of the features in the spreadsheet work, the resources will self-populate for each student sheet you create.

I loved the idea of badges and thought rather than just counting up how many badges the students had earned, what if we had hierarchical titles that can be earned as you get more badges? Like how some games start you off as a “townsperson” and move up to “King” or how in one of Alice’s classes you begin as a “Noob” and move up to “Super Genius”, I wanted to do something similar.

After a quick search on the Google machine, I stumbled upon a discussion on Quora where Arijit Lahiri suggested using polygons as the basis for the hierarchical structure:

- 0 Badges = 0-Sided Polygon
- 1 Badge = Line
- 2 Badges = Angle
- 3 Badges = Triangle
- and so on…

So, I created (well, am still creating) polygonal badges up to 20-sides with the last possible badge being an “Infinite-Sided Circle” for receiving all the badges in the course. They don’t look amazing, but I plan to spend more time making them look better once I get rolling with this.

You’ll notice that there is a “Badges & Titles” sheet where you can swap out badges for any images you’d like. I’m in the process of creating the polygonal badges and will add to the sheet as they are made.

**More Modifications Added Mon Jan 26th, 2015:**

Don’t have much time to add details right now, but I’ll list them so you can look for them…

For each learning goal that has been assessed, Jon had the sheet automatically add a badge. I slightly modified this and changed the Mastery Badge column title to * Next Step or Mastery*. Now, if the teacher has not awarded mastery for a learning goal by entering “M” in the column next to the mark on the master sheet, a link to a task related to that learning goal will pop up for the student to do. They can click on the task and print it, annotate with an iPad, or simply work it out on lined paper to be re-assessed. These tasks are added by the teacher on the Master spreadsheet in rows 16, 17 and 18 under each learning goal. Super easy to add as well as change up as needed. Don’t mind the tasks I have in there right now – they are just in there temporarily until I can add better ones.

Since there is a column for each learning goal dedicated to awarding a badge for mastery with the letter “M”, I figured we could easily use those columns for descriptive feedback until the mastery badge is awarded. There is no obligation for the teacher to give feedback for each learning goal, but the option is there and the feedback will automatically populate in the appropriate student sheet next to the appropriate learning goal.

The teacher can add any resource links to the master sheet under each learning goal and they will self-populate to the far right on the student sheets. I’m actually considering using this as my course website to eliminate confusion from the organizational nightmare a blog or Google Site can become quickly. Rather than listing resources chronologically, all resources will be organized by learning goal and thus easier for students to find what they are looking for.

I’ve added a * Next Learning Goal* area in row 4 of each student sheet to clearly indicate a “next step” for each individual student to improve. What the sheet does is it searches for areas to improve and then returns that learning goal number, description, level, mark, link to a related task, and feedback right at the top of the sheet. Specifically, it will search for incomplete/did not submit learning goals, then those less than level 1, and so on and returns the first result. Goal here is for students to address the “oldest” learning goal with the greatest need for attention. It would be easy to modify it to the most recent and/or learning goals that are almost at a level 3, if you’d prefer.

I will make it clear to my students that they can select any learning goal requiring attention, but I think the Next Step information will make it easy for them to find something to work on very quickly.

You can quickly send a message to the entire class or a personal message to individual students that will appear at the top of each student sheet immediately under the *Next Step* row. To add a note to the entire class, simply write your note in cell E2 of the Master sheet and it will appear on all the student sheets. You can also write a personalized note in any cell in the D column next to a student name and only that student will receive the message. Note that a personal message takes priority over a class message.

I’m really loving the idea of gamifying this **standard based grading assessment** approach and want to find more ways to get students excited. I’ve been thinking about adding some more badges and/or a points system that rewards effort, clicks on resource links (won’t tell kids this), collaboration, participation, etc. in order to develop a leaderboard. I’ve had a lot of push-back from Tweeps in the Twittersphere about this and it is understandable. I won’t make knowledge the foundation of this additional community gamification setup as I know that you’ll always see the same folks at the top. I don’t know what this will look like yet, but when I do, I’ll be sure to share it.

So, to pay it forward in the same way that both Jon and Alice did with their great work on this spreadsheet, click the link below to make a copy of my modified version:

Click here to grab your own copy.

btw – when trying to find out how to force the user to make their own copy of a Google Spreadsheet rather than simply view mine, I found an older resource from Alice’s blog that suggested that you copy and paste the Google Spreadsheet share link and erase ** #gid=** and everything after it and replace with

Once you grab a copy, simply duplicate the “Student” sheet for each individual student and simply change their student number in the top-left corner of each sheet. After that, everything will be populated automatically when you make changes to the master sheet.

Good luck and let me know how it goes!

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]]>4-Part Math Lesson involves a contextual math task solved using inquiry to reveal a learning goal; connections are made to reveal algebraic representation.

The post The 4-Part Math Lesson appeared first on Tap Into Teen Minds.

]]>When it comes to teaching mathematics, we often hear about effective strategies that could be useful during a lesson, but there is little talk about what actually constitutes a complete and effective lesson from start to finish.

In the following video (or jump to the summary below), I break down the 3-part lesson into 4-parts:

(Can’t see the video? Click here)

In Ontario, John Van De Walle‘s 3-part math lesson has been promoted from the Ministry of Education to the districts, districts to school administrators, and from administrators to teachers. Despite efforts to provide resources for teachers, often times, guides provide only an outline of the lesson framework or a relatively basic example of what it may look like in the classroom. If a lesson example is provided, I’ve noticed the topics are geared towards primary or junior grade levels and the actual problem to be solved does not peak curiosity. Some examples of 3 part math lessons can be found here, here and here.

Just like a game of telephone, the message can get distorted as it moves down the line.

By the time the “3-part math lesson” message reaches the teacher, what is interpreted could easily look like the following:

- A 3-part lesson is an effective way to teach math.
- The 3-parts are named minds on, action, and consolidation or before, during, and after (or similar).
- Minds on is like a warm-up to the lesson.
- Action is where students learn the new concept.
- Consolidation is where we take our new learning and summarize it/practice.

There are some problems, though.

Many teachers walk away without a clear understanding of what this should really look like in the classroom. Some believe they are “essentially” delivering a 3-part math lesson because they already do a warm-up problem, deliver a lesson, and students consolidate their learning (i.e.: do practice problems).

Other teachers are working hard to improve engagement in math class by using 3 Act Math Tasks in the style of Dan Meyer, but are using them only near the end of a unit. I think waiting until all of the learning goals for the unit have been delivered with traditional, context-less problems is a missed opportunity to engage students throughout the learning process.

Because of these misconceptions and others that may surround the use of the 3-part math lesson and missed opportunities when integrating 3 Act Math Tasks, I will deconstruct an effective 4-part lesson structure into finer parts and model what this might look like in a middle years/intermediate math classroom.

What I find is missing most often from most 3-part math lessons is the inquiry/discovery portion that should be embedded within the action/during phase. This is why I think it is really important that we break down the action/during portion into smaller parts:

Let’s jump into each part.

Like in a 3-part math lesson, we begin with a minds on. This may take 5 minutes some days or could take up to 25 minutes on others. While most are clear that the minds on/before phase is a warm-up, the biggest misconception I encounter is that it is any random problem to start the class off. John Van De Walle‘s 3-part math lesson suggested that the minds on promote activating prior knowledge in order to:

… get the students to be cognitively prepared for the lesson problem by having them think about ideas and strategies they have learned and used before. The teacher organizes a revisit to a concept, procedure or strategy related to the lessonâ€™s learning goal.

Sketch of a Three-Part Lesson. Ontario College of Teachers. March 2010.

Extending what John Van De Walle suggests, we should also attempt creating a minds on that extends the **context** used during the previous lesson to help scaffold us slowly towards the new learning goal for that day. If we can also maintain the context from the previous lesson, that is a huge benefit that will help students build their new knowledge on prior knowledge.

In a recent workshop, I used the Stacking Paper 3 Act Math Task and solving problems with proportions algebraically as an example of the learning goal:

For the minds on portion of the current lesson, we could then use something that will still keep the **content** and **context** the same:

After students have shared out, discussed and debated over solutions, the teacher then introduces the task for that day. Ideally, this task will maintains the same contextual scenario to allow students to easily leverage their prior knowledge while extending the previous learning goal to the new learning goal for that lesson. Making every effort to ensure the tasks offer real world context via a 3 act math task or similar will maximize the effectiveness of this part of the lesson.

The task used for inquiry/discovery is the **most important** part of a **4-part math lesson**. The overall effectiveness of the lesson is closely related to how well this part is planned. A well designed inquiry/discovery task carefully sends your students down a path where they will ultimately “bump into” the learning goal along the way. When possible, use the same contextual situation from one learning goal to the next to allow students to more easily build their new knowledge through connections made to their prior knowledge.

In the 4-Part Math Lesson video, I extend the Stacking Paper 3 Act Math Task to the Stacking Paper Sequel. In the first, students viewed a video of 5 reams of paper stacked on the floor and used information to determine how many reams it would take to reach the ceiling. The next day, during the inquiry/discovery part of the lesson, students see the same 5 reams of paper now stacked on a table with the total height given.

The question now becomes:

How tall is the table?

While students have never had to solve something like this before, they can use their prior knowledge and a bit of problem solving to come to a conclusion.

Here are some examples:

After students have shared out, discussed and debated over solutions, the teacher then prompts students to chat with their elbow partner(s) to decide which method they feel is most efficient. When discussing as a class, the teacher should prompt students by asking questions such as:

- when the context changes, will using arithmetic/logic always be as easy as it was here? What if we remove all context?
- how long does it take to create a table? …a graph?
- is there a way we can create some steps that will allow us to do problems like this more easily?

As a group with the teacher leading as facilitator, connections between student solutions and the algebraic method or procedure related to the intended learning goal can be established.

After the desired learning goal has been established as a group, we then formally introduce the intended learning goal for this lesson:

Learning Goal:I can determine the equation of a linear relation using algebra when given the slope/rate of change and a point.

We now solve the Stacking Paper Sequel Task using the algebraic method:

The consolidation phase is where we begin to remove the context from the problem and move towards the typical types of problems you see in traditional textbooks and on standardized tests. It might seem as though you are moving backwards from great real world problems that peak curiosity and perplex students, but unfortunately, the types of problems students are currently expected to solve in traditional textbooks and on standardized tests are text-heavy and lack any meaningful context or purpose.

Now that students have an understanding of the learning goal and we have made connections from an authentic real world problem to the algebraic representation, students can use this knowledge to solve problems with little or no context.

Here is an example of a problem we could now use to consolidate the learning:

I think it should be noted that the problem above used to be the FIRST problem I would introduce in the “Action” phase of the 3-part math lesson. It was as simple as I could make it; yet, students still struggled with the concept. Now that they have been given a contextual problem and were enabled to “bump into” the learning goal through the inquiry/discovery portion of the lesson, this is not a troublesome topic.

Upon completing these 4-parts of your math lesson, students should now have the understanding and confidence to do the necessary practice to master the intended learning goal.

I mentioned earlier that building on the same context from each learning goal to the next can provide a huge benefit. In order to continue the deconstruction of this particular lesson, the task I would use for the Inquiry portion of the next lesson would be the Thick Stacks 3 Act Math Task. In this task, students are provided with an image of two stacks of paper on a shorter table and are given the total height of each stack on the table. Students are then to determine:

What is the height of this table?

Through the inquiry process, students can determine the height of the table and we follow through by making connections when it is realized that we have been given two points in disguise. The learning goal for the next day is:

Learning Goal:I can determine the equation of a linear relation using algebra when given two points on the line.

In my experience, this can be one of the most challenging learning goals for grade 9 academic students until connections are made. This task allows students to make these connections quickly and in a seemingly natural way.

Here are some possible solutions to this task as well was the consolidation (i.e.: problem with no context) I would use for that lesson:

While I will continue experimenting to make learning math more engaging for my students, the 4-part math lesson seems to provide a good balance and offers opportunities for accessing prior knowledge, learning through inquiry/discovery, building new knowledge by making connections to what we already know, and opportunities to practice traditional math problems. One of the other added benefits is the opportunity to benefit from Dan Meyer 3 Act Math-style problems on a regular basis, without having to wait until the end of a unit to engage learners. Engagement should not be on a once a unit or even a once a week schedule; we need to engage our students on a daily basis by offering contextual problems that peak curiosity and excite learners.

Would love to hear your experiences, both positive and negative, in the comments section!

The post The 4-Part Math Lesson appeared first on Tap Into Teen Minds.

]]>Will the gumballs from the short and wide jar fit into the tall and thin jar? Use your knowledge of volume of a cylinder and sphere to find out!

The post Guessing Gumballs Sequel appeared first on Tap Into Teen Minds.

]]>In this 3 act math task, students will extend the **proportional reasoning** and **3D-measurement** skills used in the previous task, Guessing Gumballs, to determine whether the gumballs from the short and wide jar will fit into the tall and thin jar. The learning goals for this task include:

- calculating the volume of a cylinder and applying their knowledge;
- calculating the volume of a sphere and applying their knowledge;
- applying their knowledge of proportional reasoning to solve problems.

Show students the following video or this photo:

Ask students to talk to a neighbour and come up with some possible questions.

The question I’m looking for here is:

Will the gumballs fit into the tall and thin jar?

Prompt students to make a prediction and be prepared to backup their prediction by discussing with a partner. Will all the gumballs fit? Will it overflow? Will there be a lot of extra space? Will it be a perfect fit?

After students make a prediction, have them discuss with their partner what information they need to make a more accurate prediction.

Then, show this video clip or show these photos:

- Download Photo 1: Jar Dimensions

- Download Photo 2: Gumball Diameter

At this point, students should be able to improve their prediction after using the dimensions to calculate the volume of both jars and determine how much volume the gumballs could “theoretically” occupy.

Once students have shared out their work, updated their predictions based on their calculations, show them the solution:

Real World Applications of 3D Measurement Proportional Reasoning With Volume of a Cylinder and Sphere In this 3 act math task, students sharpen their proportional reasoning and 3D-measurement skills as they try to determine how many packages of gumballs as well as how many gumballs (individually) will it take to fill the cylindrical jar. The learning goals for this task include: cal...

Visually Understanding Area of a Circle and Volume of a Cylinder Over the past year, I have been on a mission to try and make some of the formulas we use in the intermediate math courses in Ontario (Middle School for our friends in the U.S.). I think it can be difficult for math teachers to explain where formulas come from because we often think of deriving formulas algebraically. Unfortu...

How Many Pyramids Does It Take To Fill a Prism? In this multi-step 3 act math task, the teacher will show three sets of 3 Act Math Style tasks involving comparisons between rectangular prisms and pyramids, triangular base prisms and pyramids, and cylinders and cones. While the intention has been to leave Act 1 of each set very vague to allow for students to take the problem in other direct...

How Many Cones Does It Take To Fill a Sphere? In this 3 act math task, the teacher will show short video clips to help students understand where the Volume of a Sphere formula comes from. Similar to the last Volume 3 Act Math Task: Prisms and Pyramids, the intention has been to leave Act 1 of each set very vague to allow for students to take the problem in more than one direction. The tea...

Measurement: Volume of Cylinders and Cones This is an attempt to better develop the question by splitting the problem into more than 3 acts, since I found it difficult to make the intended learning goal obvious enough through visuals. Act 1 - Introduce The Problem Act 1 is split into two very short videos. The first, simply shows an empty spray bottle: http://youtu.be/K8d0Ynu1h_Y A...

If you use this task in your classroom, please share your experiences in the comments section! Always appreciative of any improvements that can be made including resources you might want to share for inclusion.

Click on the button below to grab all the media files for use in your own classroom:

The post Guessing Gumballs Sequel appeared first on Tap Into Teen Minds.

]]>There are bags of gumballs and a cylindrical jar. How many packages and how many individual gumballs will it take to fill the jar?

The post Guessing Gumballs appeared first on Tap Into Teen Minds.

]]>In this 3 act math task, students sharpen their **proportional reasoning** and **3D-measurement** skills as they try to determine how many packages of gumballs as well as how many gumballs (individually) will it take to fill the cylindrical jar. The learning goals for this task include:

- calculating the volume of a cylinder and applying their knowledge;
- calculating the volume of a sphere and applying their knowledge;
- applying their knowledge of proportional reasoning to solve problems.

Show students the following video or this photo:

Ask students to talk to a neighbour and come up with some possible questions.

This task will focus on two questions:

Q1 – How many packages of gumballs will it take to fill the jar?

Q2 – How many gum balls (individually) will it take to fill the jar?

You might want to show the students the following video before or maybe even after they discuss with a partner and make a prediction:

After students make a prediction, have them discuss with their partner what information they need to make a more accurate prediction.

Then, show this video clip or show these photos:

- Download Photo 1: Jar Height

- Download Photo 2: Jar Diameter

- Download Photo 3: Gumball Diameter

At this point, students should be able to improve their prediction of how many individual gumballs it would take to fill the jar by calculating the volume of the jar and volume of a gumball.

You can also challenge them by telling them how many gumballs on average are in each package:

Once students have shared out their work, updated their predictions based on their calculations and some good ‘ol debating happens in your classroom, show them these two clips:

Visually Understanding Area of a Circle and Volume of a Cylinder Over the past year, I have been on a mission to try and make some of the formulas we use in the intermediate math courses in Ontario (Middle School for our friends in the U.S.). I think it can be difficult for math teachers to explain where formulas come from because we often think of deriving formulas algebraically. Unfortu...

How Many Pyramids Does It Take To Fill a Prism? In this multi-step 3 act math task, the teacher will show three sets of 3 Act Math Style tasks involving comparisons between rectangular prisms and pyramids, triangular base prisms and pyramids, and cylinders and cones. While the intention has been to leave Act 1 of each set very vague to allow for students to take the problem in other direct...

How Many Cones Does It Take To Fill a Sphere? In this 3 act math task, the teacher will show short video clips to help students understand where the Volume of a Sphere formula comes from. Similar to the last Volume 3 Act Math Task: Prisms and Pyramids, the intention has been to leave Act 1 of each set very vague to allow for students to take the problem in more than one direction. The tea...

Measurement: Volume of Cylinders and Cones This is an attempt to better develop the question by splitting the problem into more than 3 acts, since I found it difficult to make the intended learning goal obvious enough through visuals. Act 1 - Introduce The Problem Act 1 is split into two very short videos. The first, simply shows an empty spray bottle: http://youtu.be/K8d0Ynu1h_Y A...

If you use this task in your classroom, please share your experiences in the comments section! Always appreciative of any improvements that can be made including resources you might want to share for inclusion.

Click on the button below to grab all the media files for use in your own classroom:

The post Guessing Gumballs appeared first on Tap Into Teen Minds.

]]>Students engage in a real world problem involving volume of cylinders and cones to determine where we should stop filling the spray bottle with vinegar.

The post Mix, Then Spray appeared first on Tap Into Teen Minds.

]]>This is an attempt to better develop the question by splitting the problem into more than 3 acts, since I found it difficult to make the intended learning goal obvious enough through visuals.

Act 1 is split into two very short videos. The first, simply shows an empty spray bottle:

At this point, some discussion regarding where the problem may go could be initiated. In order to lead students down the right path, show the next short clip:

While the question might seem obvious to students at this point (i.e.: how much vinegar/water do you need), we are actually going down a slightly different path.

Show this video:

Students now know that they must predict:

Where on the bottle should I stop filling with vinegar?

They can make this prediction by drawing a line on an image of the bottle.

Looking closer at the spray bottle, you’ll notice that it is a 3D composite figure consisting of a cylinder and a cone. You might want to consider asking students:

Will the amount of vinegar stop before reaching the top of the cylinder, after, or will it stop right on the dividing line between the cylinder and the cone?

After asking for students to determine what information they need to make their best prediction, you can show them this video:

Alternatively, you could show the following images:

Height of the spray bottle:

Height of the cylindrical portion of the spray bottle:

Diameter of the spray bottle:

Now, your students can see how close they came based on their prediction and their mathematically calculated prediction:

I haven’t tried this task out yet, so if you do, please share how it went in the comments. Any ways to make it better? Let me know!

Click on the button below to grab all the media files for use in your own classroom:

The post Mix, Then Spray appeared first on Tap Into Teen Minds.

]]>I lowered the bar and wore out my students before reaching the intended learning goal prompting a student to ask "Why are we doing this?" Here's what I said

The post How Lesson Failures Can Still Provide Value appeared first on Tap Into Teen Minds.

]]>Today, we extended our learning goal from the previous day from:

I can find the point of intersection of two linear relations on a graph and interpret the meaning of the intersection point related to a real world situation.

to the following learning goal:

I can solve multi-step linear equations using a variety of tools and strategies. (i.e.: 2x + 7 = 6x – 1)

For years, I would simply extend our work solving linear equations in Slope/Y-Intercept (*y = mx + b*) Form by adding additional terms on both sides of the equation. However, I found that we were quickly moving from a deep conceptual understanding of rate of change and initial value to no meaning at all. Although solving systems of equations algebraically is not introduced in the Ontario applied math curriculum until grade 10, I decided it was worth the extension:

I can find the point of intersection of two linear relations using the algebraic method of substitution.

Today, we extended the concept of Jon Orr’s 3 Act Math Task Crazy Taxi by adding a second option, Insane Cab:

Crazy Taxi: C = 0.50d + 5

Insane Cab: C = 1d + 2

We started with the bar really low, offering an opportunity for students to use their prior knowledge of linear relations:

Students then used that information to create a table of values and graph in order to identify the point of intersection:

Nothing groundbreaking by any means.

However, at this point the students are feeling good and all can satisfy the original learning goal related to solving for the point of intersection graphically. They have identified the point of intersection as the point (6, 8) and we had a discussion about what each value, 6 and 8, represent in this scenario.

My plan now, was to have the class solve using the value of the dependent variable (C = 8) from the point of intersection in order to prove that the distance would be 6 km using both original equations:

I think having the students create the table, graph and identify the point of intersection first gassed them. Engagement was definitely lost as we finally approached solving using the original equations for both taxi cabs. I suppose I should have predicted that students wouldn’t be entertained to find an answer they already had (i.e.: distance of 6 km and cost of $8).

Regardless, I moved on as planned by using an animation created in Keynote to introduce the idea of substitution:

Students recognized that they were dealing with two equations that were equivalent at the point of intersection and we then managed to address the intended learning goal involving solving multi-step linear equations:

While this was an extension to the expectations outlined in the Ontario Curriculum for this course, I felt it was necessary to give students a reason to solve linear equations involving more than two-steps.

That being said, this was also the first time in a while that a student said “why are we doing this?”

When a student comes out with the “why are we doing this” question, I know that I have work to do in order to add context to the problem and maintain the understanding of that context throughout.

In this instance, I went back and counted how many words/numbers/terms/points/etc. were required to find the solution via substitution:

I counted 28, which included some items that were not required, but added for clarity.

I then did the same for the table and graph:

I counted over 60 items.

We discussed how algebra is intended to save us time and effort by using the language of math, but we first required an understanding of how it works. A few students argued that although the table and graph required more writing, they could probably do it in less time than using the algebraic approach.

I then asked:

Which is better: crawling or walking?

All students agreed that walking is better.

I followed up with:

Which is better: walking or biking?

They all agreed with the latter.

I then asked students to raise their hands if they remember getting hurt when they were learning how to ride a bike and how long it took them to become proficient. Many student hands were in the air.

We discussed how babies and young children don’t have a fear of failure, but somehow we develop this fear as we get older.

Imagine you gave up on biking because you couldn’t do it right away? It’d be a shame if we gave up on math simply because we had to put some hard work in and fail a few times to understand it.

Working to ignore this fear of hard work and the possible failures that result is a major focus I try to embed in my lessons.

While this “work-in-progress” lesson didn’t produce the “ah-ha” moments I was hoping for, it did give me an opportunity to help my students grow by moving a little bit further away from the fixed mindset that many bring with them into my classroom.

As for this lesson, I’ll go back to the drawing board and attempt to maintain the context offered early in the problem, but lost once algebra took over. Suggestions appreciated in the comments…

The post How Lesson Failures Can Still Provide Value appeared first on Tap Into Teen Minds.

]]>Hop into a taxi cab with a fixed rate of $5 and a rate of change of $0.50 per kilometre travelled to practice solving linear equations and partial variation

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]]>If you’re an educator on Twitter or other social media, you probably hear a lot about gamification. Well, when you don’t have a reasonable option to “gamify” your math class, you can always turn to finding the **perplexing math** in a game. This is where Crazy Taxi by Jon Orr comes in.

This 3 Act Math Task begins with a scene from a “Grand Theft Auto-esque” video game where a man jumps into a taxi and begins what looks to be a joy-ride. The cost of the taxi ride and the distance travelled are displayed; yes, the foreshadowing is probably killing you.

After travelling a few kilometres, the game fast forwards and asks the viewer to determine:

How much would it cost to travel 30 km?

Some key information include the **initial value of $5** on the meter before the taxi begins moving and a **rate of change of $0.50 per kilometre** travelled.

Check out the portable document format (PDF) file below to grab a quick resource that asks the same question as well as an extension requiring the student to determine:

How far you could travel if you had a $50 bill in your pocket?

Click on the button below to grab the full task for use in your own classroom:

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]]>Address direct variation and multiple representations of linear relations through this 3 Act Math Task relating a hummingbirds number of wing flaps vs. time

The post Flaps! appeared first on Tap Into Teen Minds.

]]>This is another great task by Jon Orr.

Students watch a video of a hummingbird lowering itself to a feeder in what most students will realize is slow-motion.

Teacher will prompt the students to guess what the question is… The question we are trying to get them narrowed in on is:

How fast are the wings flapping?

I prompt students to chat about independent/dependent variables, rate of change and initial value. Once students are comfortable with the independent variable (time) and dependent variable (number of flaps), we then have students ask for more information.

Show the video again, this time with the time and number of flaps counting on the screen.

The video ends at **27 flaps** after **0.50 seconds**.

In this case, I usually have students show the multiple representations of this linear relationship in a table, graph and equation in order to solve some problems.

The discussion allows for us to substitute different values for time and number of flaps in order to organically introduce solving equations for a purpose.

I usually use this task for both Grade 9 Academic (MPM1D) and Grade 9 Applied (MFM1P) to address learning goals like:

- I can determine values of a linear relation using a table of values.
- I can determine values of a linear relation using the equation.
- I can determine values of a linear relation using the graph by interpolation/extrapolation.
- I can describe the effects on the table, graph and equation of a linear relation when the initial value and rate of change are varied.

I created this math task template to assist in addressing the learning goals listed above. Note that this task could easily be altered to address such learning goals as proportionality and solving ratios/rates.

Click on the button below to grab the full task for use in your own classroom:

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