This past week, Justin Reich referenced research and said "it is almost unethical to use lecturing in a control group when comparing with active learning."

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]]>This past week, I had an opportunity to hear Justin Reich (MIT Teaching Labs Researcher) speak for the second time and he referenced a white paper called: “*Active learning increases student performance in science, engineering, and mathematics*”. His interpretation of the paper was that the data indicated that *“it is almost unethical to use lecturing in a control group when comparing with active learning.”* It reminded me of a recent article I wrote inspired by the book, Teaching Minds by Robert Schank.

While you can access the entire 6-page paper here to come up with your own interpretations, I’ve quoted the abstract from the paper below, for your convenience:

## Significance

The President’s Council of Advisors on Science and Technology has called for a 33% increase in the number of science, technology, engineering, and mathematics (STEM) bachelor’s degrees completed per year and recommended adoption of empirically validated teaching practices as critical to achieving that goal. The studies analyzed here document that active learning leads to increases in examination performance that would raise average grades by a half a letter, and that failure rates under traditional lecturing increase by 55% over the rates observed under active learning. The analysis supports theory claiming that calls to increase the number of students receiving STEM degrees could be answered, at least in part, by abandoning traditional lecturing in favor of active learning.

## Abstract

To test the hypothesis that lecturing maximizes learning and course performance, we metaanalyzed 225 studies that reported data on examination scores or failure rates when comparing student performance in undergraduate science, technology, engineering, and mathematics (STEM) courses under traditional lecturing versus active learning. The effect sizes indicate that on average, student performance on examinations and concept inventories increased by 0.47 SDs under active learning (n = 158 studies), and that the odds ratio for failing was 1.95 under traditional lecturing (n = 67 studies). These results indicate that average examination scores improved by about 6% in active learning sections, and that students in classes with traditional lecturing were 1.5 times more likely to fail than were students in classes with active learning. Heterogeneity analyses indicated that both results hold across the STEM disciplines, that active learning increases scores on concept inventories more than on course examinations, and that active learning appears effective across all class sizes—although the greatest effects are in small (n ≤ 50) classes. Trim and fill analyses and fail-safe ncalculations suggest that the results are not due to publication bias. The results also appear robust to variation in the methodological rigor of the included studies, based on the quality of controls over student quality and instructor identity. This is the largest and most comprehensive metaanalysis of undergraduate STEM education published to date. The results raise questions about the continued use of traditional lecturing as a control in research studies, and support active learning as the preferred, empirically validated teaching practice in regular classrooms.

Active learning increases student performance in science, engineering, and mathematics. Scott Freeman, Sarah L. Eddy, Miles McDonough, Michelle K. Smith, Nnadozie Okoroafor, Hannah Jordt, and Mary Pat Wenderoth.

Two big questions that are still circling in my head include:

- What does “good” active learning look like in my subject area?
- How can I make progress to move away from a lecture-based approach without completely overwhelming myself?

Has this research sparked any questions in your head? I would love to hear about them if you’re willing to share!

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]]>Gimme a Break is a 3 Act Math Task that starts with a Kit Kat Candy Bar and goes on to explore adding, subtraction, multiplying and dividing simple fractions.

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]]>A colleague and good friend, Dave Burke, reached out to me recently for some ideas around **adding and subtracting fractions**. In particular, he was interested in trying to make the topic a little more enjoyable by adding some context. This was a challenge I was really eager to work on because working with fractions is such a sore spot for so many students and their teachers. In the past, I tried to help make dividing fractions less abstract by using the idea of dividing chocolate bars. With this in mind, I figured using chocolate bars for adding and subtracting would be a good starting point to move towards multiplication and division later.

Here are some expectations from the Grade 7 Ontario math curriculum that we’ll be aiming to touch on:

- Grade 7, NS2.01 – divide whole numbers by simple fractions and by decimal numbers to hundredths, using concrete materials (e.g., divide 3 by ½ using fraction strips; divide 4 by 0.8 using base ten materials and estimation);
- Grade 7, NS2.02 – use a variety of mental strategies to solve problems involving the addition and subtraction of fractions and decimals (e.g., use the commutative property: 3 x 2/5 x 1/3 = 3 x 1/3 x 2/5, which gives 1 x 2/5 = 2/5 ; use the distributive property: 16.8 ÷ 0.2 can be thought of as (16 + 0.8) ÷ 0.2 = 16 ÷ 0.2 + 0.8 ÷ 0.2, which gives 80 + 4 = 84);
- Grade 7, NS2.07 – add and subtract fractions with simple like and unlike denominators, using a variety of tools (e.g., fraction circles, Cuisenaire rods, drawings, calculators) and algorithms;
- Grade 7, NS2.08 – demonstrate, using concrete materials, the relationship between the repeated addition of fractions and the multiplication of that fraction by a whole number (e.g., 1/2 + 1/2 + 1/2 = 3 x 1/2);

I was surprised when I reviewed the grade 7 curriculum to find that the idea of dividing whole numbers by simple fractions was listed before adding and subtracting fractions. The more I thought about this, the more I like it. However, I stuck to adding/subtracting first.

So, let’s get started…

Show the students the video below:

Can’t see the video? Click here.

After watching the video, I have students take 30 seconds to discuss with their elbow partners what question(s) come to mind before sharing out with the group. I usually give them an open response question in Knowledgehook Gameshow to ensure all voices can be heard:

Can’t see the embedded Knowledgehook Gameshow question? Click here.

When the list of questions is exhausted, I then show them act 2…

Show the students the video below:

Can’t see the video? Click here.

Students quickly come to the conclusion that the question I am asking here is…

How many pieces of a whole KitKat bar are there?

Can’t see the embedded Gameshow question above? Click here.

I’ve been using Knowledgehook Gameshow as a way to keep my lessons interactive almost daily. The best feature is the upload solution tool (not available when embedded in websites) that allows me to share out student work instantly over the projector. I find using Gameshow makes it even easier for me to be the facilitator of the learning rather than the gatekeeper of knowledge. Typically, the solutions students share is enough to consolidate most tasks and I can minimize the amount of direct instruction I must deliver.

It is always nice to be able to show students the solution for a little celebration. This is where I see Gameshow could make an improvement. Have the option to add a “solution page” where you can insert media, rather than me having to open a video file or run a slide deck.

Can’t see the video? Click here.

Then, we continue on, slowly raising the bar on the complexity of the problems, but keeping the context the same.

Then, show students this video:

Can’t see the video? Click here.

Students can then evaluate the expression:

Can’t see the video? Click here.

Can’t see the video? Click here.

Ask students to talk to their neighbours about what question we might ask here. The goal is to have them realize that they are going to need to do some addition and subtraction of fractions.

Can’t see the video? Click here.

Students can then evaluate the expression in Knowledgehook Gameshow:

Can’t see the video? Click here.

Here are a few Knowledgehook Gameshow questions that might be useful for some practice prior to moving on:

Can’t see the video? Click here.

Ask students to talk to their neighbours about what question we might ask here. Some students might say we are adding, which is great. However, I wonder if any students will notice that we are not only adding, but we are doing repeated addition (multiplication).

Students can then give their take on what they think will happen:

In order to “show” students this, play this clip:

Can’t see the video? Click here.

Now, it’s time for students to take this new idea and run with it. Show them this image:

Let students chat it out and then, when they are ready, they can give an open response of their thinking:

After exploring some of the open responses submitted by students, we can ask them to determine the answer to the problem:

Show your students the video below:

Now, let’s give students an opportunity to try another problem involving multiplication. Show them this image:

Give them an opportunity to solve the problem by evaluating and interact with their classmates:

Show students the solution video:

Show students this video:

Students are then given some time to discuss what they think might be going on here and then they share out their thinking:

Now, you can show the students some more information to get their wheels turning:

Then, students can solve and share out their solutions:

Then, you can show students the video of the answer:

Show students this image:

Give them an opportunity to evaluate and share their solution:

Show students this video:

That’s it for now, but I do have a huge bag full of other chocolate bars that I intend to eventually use to make more problems involving fractions. Let me know how you like this one so I know if it is worth putting in the extra effort.

If you’re interested, you can try the full Knowledgehook Gameshow here or, you can copy the entire gameshow to your own Free Gameshow Account via the share link below:

Enjoy!

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

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]]>A summary of my NORCAN Experience from my week in Oslo, Norway from October 18th - October 24th, 2015 with colleagues from Alberta, Ontario & Norway.

The post My NORCAN Experience in Norway appeared first on Tap Into Teen Minds.

]]>I hope you enjoy a summary of my NORCAN Experience from my week in Oslo, Norway from October 18th – October 24th, 2015.

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]]>This week in MPM1D Grade 9 Academic, we explored Pythagorean Theorem with Dan Meyer's Taco Cart and Solving Linear Equations including distribution!

The post WIR #6 – Pythagorean Theorem, Equations, and Linear Relations appeared first on Tap Into Teen Minds.

]]>The semester is flying by and I’m really happy with the progress students are making in their mathematical thinking, communication, and confidence. I was especially pleased this week when a student who had very low levels of confidence and was not engaging in the work, began taking risks by participating and both his confidence and understanding are on the rise.

Here’s a quick glance at what we covered this week.

This week, Canada celebrates Thanksgiving and thus there is no school on Monday October 12th.

This week, I wanted to circle back to some measurement concepts. Since we had tackled some area and perimeter of composite figures, I thought it might be a good idea to tackle some concepts related to the slide lengths of right-angle triangles. In the Grade 9 Academic (MPM1D) course, there are two specific expectations related to Pythagorean Theorem:

- MG2.01 – relate the geometric representation of the Pythagorean theorem and the algebraic representation a2 + b2 = c2;
- MG2.02 – solve problems using the Pythagorean theorem, as required in applications (e.g., calculate the height of a cone, given the radius and the slant height, in order to determine the volume of the cone);

However, before I dove into the Pythagorean Theorem, I thought it was worthwhile to warm-up with a couple linear equations involving distribution for students to try.

When it comes to Pythagorean Theorem, I often look to Dan Meyer’s Taco Cart 3 Act Math Task as a great place to begin. Since students have experienced Pythagorean Theorem in elementary school, I typically dive straight into the task. For those who haven’t come across the Taco Cart Task, here’s the Act 1 video:

It’s great to let students have a quick discussion in their groups to try and predict who will get to the taco cart first; me or Ben. After sharing out predictions as a class, I want them to dive in so I give them some important details:

Something important to note is that I haven’t done any pre-teaching here. I haven’t *hinted* that **Pythagorean Theorem** can be used or even that they have tackled problems like this before in elementary. I just want to walk around the room and listen in on some discussion to see if any of that prior knowledge comes back. It didn’t take long before I saw a few students writing the formula for Pythagorean Theorem on their desk, while others were Googling terms like: “how do you find the side of a right angle triangle?”

I was surprised to see that some students remembered the geometric representation and began drawing squares to represent a-squared, b-squared and c-squared. All good stuff.

After some students were off and at it, while a few others were struggling to find a place to start, I decided to show them a couple animations and discuss some of what they may (or may not) remember from the past.

The first animation demonstrates the geometric relationship of the Pythagorean Theorem using a right angle triangle with leg lengths of 3 units and 4 units to find the length of the hypotenuse:

The second animation demonstrates a visual of a more generalized situation:

Finally, the third animation attempts to make a clear connection between the geometric and the algebraic representations:

Now, everyone was off and we had some fun using Knowledgehook Gameshow as a way for students to interact with the entire group throughout the problem.

Even though I have an iTunes U Course with a multi-touch book for iBooks that uncovers the Pythagorean Theorem through the Taco Cart 3 act math task, I chose to try a new way to introduce the problem via Gameshow. I’ve had much success with this task using the iTunes U Course and multi-touch book, but found that it was more difficult to tie it all together and make it a collaborative experience. Gameshow seems to fill that gap to allow students to truly collaborate with their peers not just in their groups, but with the entire class.

Here’s the Knowledgehook Custom Gameshow that I created for Taco Cart including some consolidation questions for students to practice their skills:

I know I’ve mentioned it before, but I think it is worth mentioning again that Gameshow allows students to upload photos of their work. So not only do we have an interactive way for students to engage in math class, but we can encourage students communicating their thinking so we can share it with the class and look at it later:

I should also note that you can grab the Taco Cart multi-touch book as well as a ton of other tasks by enrolling in the Curious Math iTunes U Course that I created in collaboration with Jon Orr:

I decided to move our assessment day to Wednesday because we had Monday off for Canadian Thanksgiving. This week had a heavy focus on solving equations, so we tackled the following learning goals:

- NA2.07 – I can solve first-degree equations, including equations with fractional coefficients, using a variety of tools and strategies.
- NA2.09 – I can solve problems that can be modelled with first-degree equations, and compare algebraic methods to other solution methods.

Even though we have been working with equations quite a bit over the past week, I was surprised at how many struggled with solving equations when variable terms were on both sides. Distribution also caused some issues that I clearly had not anticipated based on how students performed when we first worked with the problems. This would be a major concern back when I taught with a unit based approach; once you finish the unit, that’s it! But now that I’ve been spiralling the content and don’t feel so restricted by going really deep into a new concept over a short period of time, I can continue to bring back these concepts and not fear that I’m going to run out of time. For example, I used to teach solving equations as a complete unit. We would start by solving 1-step equations, then move on to two-step and multi-step equations. Then, we would solve equations that involve distribution. Finally, we would solve equations involving fractions. That can be intimidating and possibly too much to handle for some students. Now, we work on some one- and two-step equations, branch off to something else, then come back and move into some more difficult equations. Overall, I must say that I’m loving how my first time spiralling through grade 9 academic (MPM1D) is coming together.

Get the assessment below:

After some apparent struggles with distribution and most students using trial and error as the only approach for the last question on Assessment #5, I thought it was worth taking a closer look as a group. I’ve always despised taking up assessments because I always feel like I’m doing all of the work and the students I’m trying to take up the assessment for are typically disinterested. Regardless, I gave it a go. The results were similar to what I expected, which means I should probably consider taking up the assessment using another strategy. Maybe next time I could have students work on it in groups and present their solutions? However, the last question would probably need some help getting them started so that they would consider exploring other options besides trial and error. (I should note that trial and error isn’t bad, but I do want them to see that they could get there by solving the equation).

The remainder of class was used to get some more practice in on those pesky equations involving distribution and other multi-step equations involving variable terms on both sides. I have been really pushing students to draw out what the equation “looks like” so they can get a better intuitive sense as to what they need to do to isolate the variable.

The practice was in the form of another Knowledgehook Custom Gameshow and you can check out the questions here:

Here’s a few student exemplars.

The first student is trying to solve *k = 2(11 – k) + 14* and is creating a visual to help her organize her thinking:

This next student is using an algebraic representation to solve *2(x – 2) = 4x – 2*:

Today I was super stoked to get students working on a Desmos Activity I created using the new Activity Builder feature. Really, this activity is simply the activity builder version of my Representations of Linear Relations Math Learning Journey I had created by sort of hacking up a workflow using Desmos shared links. Something that originally took me a couple hours to do now took me about 40 minutes to put together. I should note that I typically overthink things when I create them. Who knows, maybe a more efficient lesson planner would take a fraction of that time. Either way, I was saddened when I clearly missed the memo that classes would be shortened for the school pep rally that was to take place later in the day. DOH!

My original plan was to start with a couple warm up equations to extend our work from the previous day, then shift gears into the Desmos Activity. With my 75 minute period shortened down to 50 minutes, I knew it wasn’t going to be possible to get through the whole activity. My instant audible was to do the warm-up questions and then give students a few more equation challenges to take them to the end of the class.

Here’s the equations warm-up I had created in Knowledgehook Custom Gameshow:

So that’s it for this week. Next week, we’ll start the week off with the Desmos Representations of Linear Relations Activity with a supply teacher taking the reign for the entire week as I head to Norway to meet with teachers from Ontario, Alberta (including John Scammel!) and Norway for the **NORCAN Project**!

Have a great weekend and an even better next week!

The post WIR #6 – Pythagorean Theorem, Equations, and Linear Relations appeared first on Tap Into Teen Minds.

]]>Watch the first act as you see the first three figures made with toothpicks. How many toothpicks are in the 6th figure? ...11th figure? ...the general rule?

The post Placing Toothpicks Part 4 appeared first on Tap Into Teen Minds.

]]>Yet another task in the Placing Toothpicks Series (Placing Toothpicks, Placing Toothpicks Sequel, Placing Toothpicks Part 3) I posted recently. The first task was proportional, followed by a quadratic in the second and a partial variation linear relationship in the third. This task is going to give my students another go at partial variation linear relations. As were the learning goals from the Part 3 Task, here’s the grade 9 academic expectations we can make connections to:

- LR2.02 – I can construct tables of values, scatter plots, and lines or curves of best fit as appropriate, using a variety of tools for linearly related and non-linearly related data collected from a variety of sources.
- LR2.03 – I can identify, through investigation, some properties of linear relations and apply these properties to determine whether a relation is linear or non-linear (by rate of change/initial value when described in words, by first differences in a table, straight/curved graph, degree of terms in equation).
- AG1.01 – I can determine, through investigation, the characteristics that distinguish the equation of a linear relation (straight line) from the equations of non-linear relations (curves).
- LR2.04 – I can compare the properties of direct variation and partial variation in applications, and identify the initial value when described in words, represented as a table, a graph, or an equation.

Show the students the video below:

Can’t see the video? Click here.

As we did in the previous toothpick tasks, the question(s) I want students to think about are:

How many toothpicks are in the 6th term? … the 11th term?

As usual, I like having students figure this out on their own, using any strategy and then consolidate the task. However, the following math task template might be a good option to consolidate student thinking:

You’ll notice that I tried to make the PDF file interactive as I was going to be absent the day my students were going to do this task. How I did it was by adding hyperlinks in the PDF that will allow students to jump straight to the Act 1 video:

After students have come to their conclusions, they can watch the Act 3 video by clicking the link embedded in the math task template:

The task template is structurally organized to ask students to do specific things like create a table, graph and answer some questions on the back. I’m not super thrilled with using this approach as of late, but these math task templates are really the only structure from my former, more traditional, teaching approach. I would be lying if I said I don’t worry about my moving to a less structured teaching style, but I feel deep inside that it is the right thing to do.

If you want to grab the template, click below:

Can’t see the video? Click here.

Have you tried this task? How can we make it better? Share your thoughts in the comments below!

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]]>This week in MPM1D Grade 9 Academic, we explored direct and partial variation through 3 act math tasks and solving equations visually with SolveMe Mobiles!

The post Week In Review #5 – Direct/Partial Variation & Solving Equations appeared first on Tap Into Teen Minds.

]]>After three days out of the classroom, it felt great to be back working on some math with my students. This week, I wanted to explicitly discuss the difference between direct and partial variation linear relations and then start moving towards solving multi-step linear relations that involve collecting like terms, distribution and (hopefully) equations involving fractions. Let’s see how well we did meeting those goals.

Although we have been looking at linear patterning on and off since the first day of school, most have been proportional where the initial value (or y-intercept) is 0. Since the first day, I have been asking students to identify the equation in order to find the value of the dependent when the value of the independent is large (like figure number 47) or visa versa. Occasionally, I have snuck in a linear pattern that is not proportional by changing the initial value to something other than 0 (like Placing Toothpicks Part 3). This occasional “sneakery” is because I wanted to make it easy and natural for students to tackle this learning goal:

- LR2.04 – I can compare the properties of direct variation and partial variation in applications, and identify the initial value when described in words, represented as a table, a graph, or an equation.

In Ontario, we refer to a direct variation as a linear relationship with an initial value of zero and a partial variation as a linear relationship with an initial value not equal to zero. Since we have tackled many different linear relations with descriptions, tables, and graphs, it makes sense that students are ready to tackle this not so complex learning goal. Something to note is that we have not explicitly started with the graph of a linear relation and found the other representations. We will be inching towards starting with a graphical representation in the coming weeks.

In order to get kids thinking right off the bat, my *Minds On* involves jumping straight into Jon Orr’s Flaps! 3 Act Math Task. I show them the act 1 video to get them thinking:

Can’t see the video? Click here.

I ask students to talk to their groups about what questions they have about the slow-motion video. Ultimately, we are hoping to get somewhere close to this question:

How fast are the hummingbird’s wings flapping?

Usually, I have students individually make their predictions and share out. Today, I had students make their own prediction, discuss with their group, and compromise to make a “group prediction” they could share out as a team. It was interesting because now students felt like they had to better justify their own prediction in order for the group to agree on a number closer to their individual prediction. I’ll definitely try this again in the future.

After sharing out the group predictions, I showed students Act 2 of the Flaps! task and then started a Knowledgehook Custom Gameshow that I made to use for students to interact throughout the tasks for that day. Here’s a couple of the questions from the gameshow:

Keeping with the theme of spiralling the course, I always want to make sure we don’t take for granted that things like identifying independent and dependent variables are common knowledge. So, let’s keep bringing them back throughout the year:

If you’ve been reading my other Week In Review posts, you’ve probably picked up on the fact that I haven’t been delivering a traditional lesson with a lecture and note. I have been trying to introduce new concepts as tasks are being delivered rather than formally defining these new concepts, giving examples, and then trying to have students use the knowledge throughout a task. By flipping this idea on its head, I get into the task, sort of have the students prove to themselves that they can solve any problem that is put in front of them using intuition and prior knowledge, then help them define, identify and consolidate what skills they were actually using to do the work. During the Custom Gameshow, I formally introduce the concept of direct and partial variation:

Students have been encountering direct and partial variation since the beginning of the year, yet we never had a label on it. Now, I’m hoping to start using this terminology moving forward. Some other terms we are starting to use more frequently, even though I haven’t formally introduced them in the traditional sense is ** initial value** and

The task (and custom gameshow) go on to extend the concepts here by having them identify an equation that models this relation, determining the number of flaps in 5 minutes, and finding how long it would take the hummingbird to flap its wings 1 million times.

Then, I toss them another contextual task from Jon Orr called Crazy Taxi with the intention of giving students an example of partial variation. Here’s the first act:

Can’t see the video? Click here.

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.

Even though I created math task template files in PDF form in the past for Flaps! and Crazy Taxi, I let the Knowledgehook Custom Gameshow do all of that work. I still have the urge to give a template for everything we do, but often times, that sort of “tells” the students that they must do a certain series of things in order to solve the problem. I want them to truly make the call on how they go about things. I have enough structure on my Tuesday Assessments, I figure.

With both of these tasks and the custom gameshow, this brought us to the end of class. Tomorrow, it is Assessment Day!

Here’s the learning goal lineup for Assessment #4:

- LR2.02 – I can construct tables of values, scatter plots, and lines or curves of best fit as appropriate, using a variety of tools for linearly related and non-linearly related data collected from a variety of sources.
- LR1.04 – I can describe trends and relationships observed in data, make inferences from data, compare the inferences with hypotheses about the data, and explain any differences between the inferences and the hypotheses.
- LR2.04 – I can compare the properties of direct variation and partial variation in applications, and identify the initial value when described in words, represented as a table, a graph, or an equation.
- NA2.07 – I can solve first-degree equations, including equations with fractional coefficients, using a variety of tools and strategies.
- LR1.01 – I can interpret the meanings of points on scatter plots or graphs that represent linear relations, including scatter plots or graphs in more than one quadrant.
- LR3.01 – I can determine values of a linear relation by using a table of values, by using the equation of the relation, and by interpolating or extrapolating from the graph of the relation.
- MG3.01 – I can determine, through investigation using a variety of tools and describe the properties and relationships of the interior and exterior angles of triangles, quadrilaterals, and other polygons, and apply the results to problems involving the angles of polygons.

As usual, we aren’t trying to completely assess each learning goal in its entirety. Because we are spiralling the course, we are slowly digging deeper and deeper through each expectation with the hopes that student depth of knowledge will increase each time.

Overall, the overwhelming majority of the class is right on where I would hope they would be. We do have 3 students who did struggle in a couple small areas (mixing up independent/dependent variables, uncertainty with creating an equation, etc.) but those issues can now be tackled on an individual basis prior to the start of class with a little one-on-one guidance.

Feel free to snipe the assessment below:

The grade 9 academic course has three expectations related specifically to solving linear equations:

- NA2.07 – solve first-degree equations, including equations with fractional coefficients, using a variety of tools (e.g., computer algebra systems, paper and pencil) and strategies (e.g., the balance analogy, algebraic strategies);
- NA2.08 – rearrange formulas involving variables in the first degree, with and without substitution (e.g., in analytic geometry, in measurement) (Sample problem: A circular garden has a circumference of 30 m. What is the length of a straight path that goes through the centre of this garden?);
- NA2.09 – solve problems that can be modelled with first-degree equations, and compare algebraic methods to other solution methods (Sample problem: Solve the following problem in more than one way: Jonah is involved in a walkathon. His goal is to walk 25 km. He begins at 9:00 a.m. and walks at a steady rate of 4 km/h. How many kilometres does he still have left to walk at 1:15 p.m. if he is to achieve his goal?).

Thus far in the course, I have had students creating direct variation (or one-step equations) quite routinely and I’ve slowly been adding in partial variation (two-step equations). Almost all students can intuitively solve these two linear equations and many in that group can also do this using opposite operations, which I have not explicitly taught. Great start, for sure. You’ll note that the first expectation involves solving first-degree equations including fractional coefficients, which can be interpreted in many different ways. I’m going to stay away from the fractional coefficients equations until we get some good confidence under our belts. I plan to tackle the second expectation a little later in the course, while continuing to hit the third expectation as we do our daily contextual tasks.

In the past, I would jump straight into rules of solving equations including the concept of opposite operations. Actually, my first “day” of solving equations had a note that looked like this:

Now, along with attacking simple equations intuitively using trial and error when they come up, I try to give students different equation puzzles to work with that make them think. The puzzles I used today were SolveMe Mobiles. Students are given a puzzle consisting of different shapes that hang in balance like the items hanging from a mobile over a baby’s bed. Here’s the first one this free website serves up:

A pretty easy place to start, in my opinion. I start the kids off by having them complete the first puzzle without any guidance. As is usual when I have the students start something without any sort of detailed instructions, it seems that the strongest students are most vocal about their initial confusion with the task. I hear a lot of “what the heck are we supposed to do?” or “what is this?” and “I have no idea what I’m doing.” This is perfect, because I want my strong students to get used to the feeling of not always knowing the answer immediately. My hypothesis is that my strong students have been grouped this way because they are really good at following steps and procedures. When I don’t give them explicit steps and procedures, they start to get a bit squirmy and what I perceive to be a fixed mindset emerges. In this case, it doesn’t take long for the entire class to “get the point” of the SolveMe Mobiles. I gave them about 25 minutes to work through problems after they created a free account to track their progress.

It didn’t take long for much of the class to be up to Puzzle #30 with some students making it even farther than that.

Great classroom discussion was taking place as some students got stuck at different parts of their learning journey. I encouraged them to talk it out and try to find other ways to represent the situation (hoping for some algebra to pop up).

At this point, I stopped the class and we took one of the SolveMe Mobile puzzles and explicitly represented the puzzles as equations. First, using the shapes in the puzzle and then eventually, using variables.

So for example, this SolveMe Mobile:

…would be represented as:

hexagon + hexagon + hexagon = triangle + triangle + triangle + triangle + triangle + triangle

When a student says “do I really have to write it out like that?” I immediately ask them for an easier way. Doesn’t take long before we see stuff like this:

3 hexagons = 6 triangles

or

3h = 6t

And since we know that a triangle = 5, then:

3h = 6(5)

3h = 30and thus, h = 10.

At this point, we went to some “puzzles” that were given to them algebraically instead of visually, like the SolveMe Mobiles. We did this through gameshow and I asked students to upload their solutions so we could discuss potential strategies that might be useful for others to see. Here’s the gameshow I used:

Students made the transition fairly easily from the SolveMe Mobiles to straight algebra. I suggested that they draw it out as a SolveMe Mobile if it would help them organize their thoughts.

I had a stack of my old “Pearce Pence” currency I created on my desk and some students had asked for some. Here’s what I got in some of the Gameshow Solutions:

Pretty slick, those kiddies!

I was pleasantly surprised with how well my students took to solving equations algebraically last day. I often hear from teachers that there just “isn’t enough time” to include manipulatives and investigations that allow students to discover new math concepts and construct their own understanding. This thinking is 100% accurate if there are no plans to adjust the remainder of a lesson. For example, if I was unwilling to ditch my traditional note where students would copy the steps and then engage them in some examples – I mean, students would sit passively while I did the examples – then I would definitely run out of time. However, by approaching my lesson as an interactive experience that is student-centred rather than teacher-centred, my lesson is no longer something that my students have “done to them”. To follow up the activity from yesterday, I wanted to make sure my students didn’t believe that the SolveMe Mobile thing was just some sort of scam to sucker them into thinking equations were fun. So, today, I wanted to give them an interactive way to continue the idea of equations requiring balance.

Here’s what I showed them:

Can’t see the video? Click here.

Next, I had students download the first of three Explain Everything project files for iOS I had created to complement the sour patch kids equation analogy in the first video clip above. Here’s what the first slide in the file looks like:

The goal of the Explain Everything file is to allow students to intuitively solve the equation. What I typically expect is to see something like this take place:

Can’t see the video? Click here.

Students then move to the next three tasks in the Explain Everything file that look like the following:

**Task #2:**

Check out a sample solution:

Can’t see the video? Click here.

**Task #3:**

Check out a sample solution:

Can’t see the video? Click here.

**Task #4:**

Check out a sample solution:

Can’t see the video? Click here.

You can download the first Explain Everything Project File below:

Although the expectations of this course do not involve solving systems of linear equations algebraically, I always try to stretch the limits when possible. With the Sour Patch Kids tasks leveraging student intuition, it seems logical that students might be able to extend their thinking to a system of equations. In a second Explain Everything file, I give students the following two tasks:

I hope students can use their problem solving skills to come up with a solution. Here’s a possible approach:

Can’t see the video? Click here.

Here’s the second task:

And finally, if there is time, students can download the third Explain Everything file to create their own tasks using sour patch kids as well as other items they find on the internet:

I was pretty psyched to keep on the solving equations train and begin sneaking in distribution into the process. Not long after arriving at school in the morning I learned that half of my class would be gone on a field trip for the day. DOH! Oh well, I still went on with the lesson for the most part, as we will be spiralling back to this content later in the semester anyway.

The Minds On today was SolveMe Mobile Puzzle #71. My challenge to my students was to solve for the values of each shape in any way they wanted, but then to attempt modelling the puzzle algebraically. Students struggled to do this as I think they thought it was more complex than it really was. They could all solve it, but most were intimidated when it came to representing with algebra. Here’s a silent solution I created to share with them after students shared their solutions:

Can’t see the video? Click here.

After that, I asked students to solve the following equation:

4x + 10 = 90

After sharing out their strategies, I offered another possible way to look at solving an equation that students found helpful:

Can’t see the video? Click here.

Then, it was on to challenging the class with some tasks via the Knowledgehook Custom Gameshow below:

Can’t see the gameshow? Check it out here.

When we got to the third question, some students were stumped trying to figure out what they needed to do:

Find the root of the equation:

2(x – 2) = 4x – 2

Note: Think of “root” like getting to the “root” of the problem or the “SOLUTION”.

Interesting enough, most students were struggling with “BEDMAS” and the idea that they had to do something inside the brackets first. We had a great chat about why just remembering rules in math can be cause for more confusion than help.

Here’s the crash course on the distributive property that we looked at on the fly:

I gave students a bit more time to work out a solution and then suggested that maybe they try to draw their own SolveMe Mobile to help them organize their thoughts:

Finally, we consolidated the problem using algebra:

Overall, it was a fun class with some great learning moments. We definitely have more work to do on distribution and how it applies to solving equations, but I know it will come in time.

That’s it for this week! Looking forward to heading into a four-day week with the Canadian Thanksgiving Holiday on Monday. Enjoy the weekend!

The post Week In Review #5 – Direct/Partial Variation & Solving Equations appeared first on Tap Into Teen Minds.

]]>This week in MPM1D Grade 9 Academic, we explored some more geometry of angles as well as identifying linear and non-linear relations.

The post Week In Review #4 – Geometry & Linear/Non-Linear Relations appeared first on Tap Into Teen Minds.

]]>I can’t believe the fourth week of school has already come and gone. This week was a bit hectic as I was away on Tuesday and Wednesday engaging in professional development as a part of my role as Middle Years Collaborative Inquiry (MYCI) Lead for my district and again on Thursday delivering a keynote for the Carleton Ottawa Mathematics Association (COMA) in Ottawa, Ontario. With me out of the classroom for three days in a row, that leaves the majority of this week a “self-directed” experience for my students. Here’s the recap:

The day before I know I will be out of the classroom is always a struggle for me. I am always a bit too excited to do too much on a regular day, so knowing I’ll be gone leaves me wanting to do way too much to compensate for the lost time. Either way, I wanted to use this class to give students a bit more exposure to some of the Geometry of Angles topics we had introduced in the previous class.

The warm-up was a Knowledgehook Custom Gameshow to get students thinking as well as some spaced practice. Here’s the three questions they did:

I know, I know… the warm-up questions aren’t breathtaking. However, I do feel that my students need some practice working with the new skills we cover, so I try to make it as enjoyable and collaborative as possible via Knowledgehook Gameshow.

Next, I wanted to get my students to begin extending their thinking about patterning from direct variation linear relations to partial variation, where the y-intercept is not equal to zero. We did this by introducing the Placing Toothpicks Part 3 task.

Here’s the act 1 video:

Can’t see the video? Click here.

Students would go on to determine the number of toothpicks in the 6th figure and 11th figure. Then, I had them think of a general rule that would help them find the number of toothpicks for any figure number. I also asked them to determine the figure number that would require 162 toothpicks to see how they would go about that. Good discussion and thinking here.

Here’s some student work:

After sharing out some interesting approaches and discussing a general rule in words and using algebra, we moved on to the final challenge of the day called Railing Reproduction (will share task online soon). Students watch a short video clip of some railings on a stairway and are asked “What’s the question?”

Can’t see the video? Click here.

In this case, I make up a short story about how Paisley, the Bull-Puggle gets pretty worked up and put her tubby body through the railing. I only had one angle to work with and had to find the other missing angles prior to making my cuts with the saw.

Can’t see the video? Click here.

Students worked diligently on this problem and it was clear that they had been able to recall some of what they learned back in grade 7 math. From what I’m seeing with this group, I should be able to jump into interior and exterior angles of n-sided polygons from an inquiry standpoint without much trouble.

On the assessment for this week, students will be tackling portions of the following learning goals:

- LR1.02 – I can pose problems, identify variables, and formulate hypotheses associated with relationships between two variables.
- LR3.01 – I can determine values of a linear relation by using a table of values, by using the equation of the relation, and by interpolating or extrapolating from the graph of the relation.
- MG2.03 – I can solve problems involving the areas and perimeters of composite two-dimensional shapes.
- NA2.02 – I can solve problems requiring the manipulation of expressions arising from applications of percent, ratio, rate, and proportion.
- MG3.01 – I can determine, through investigation using a variety of tools and describe the properties and relationships of the interior and exterior angles of triangles, quadrilaterals, and other polygons, and apply the results to problems involving the angles of polygons.

You’ll notice I said we will tacking “portions” of those learning goals. This is because I decided to use specific curriculum expectations for my standards based grading approach due to the huge number of learning goals that ballooned last year when I broke them all down. I think somewhere in the middle between how I’m doing it this year and how I did it last year is probably the best bet. Maybe identifying the specific expectation with learning goals as sub-categories of that. More thinking to be done with that one.

Here’s the assessment, if you want to snag it:

As mentioned earlier, I was away today and always struggle to create a meaningful learning environment that will be easy enough for a substitute teacher to lead and beneficial to the students. In a spiralled classroom, you’ll see us jumping around from strand to strand quite a bit and constantly introducing new concepts as we go. Days when the teacher is not present might not be the best way to introduce new concepts because it is hard to determine how many students are putting in their usual effort.

When I am away delivering or receiving professional development, the lesson will typically involve a few tasks students can handle independently or collaboratively with their table groups with a realistic timeline of finishing prior to the end of the period. My instructions to the supply teacher will be explaining the tasks and the protocol in which students can tackle the tasks, followed by the opportunity for students to re-address learning goals from their gamified learning log like this sample learning log and/or they can complete some practice problems from Knowledgehook Homework that I am experimenting with this year.

The first task is called Snowy Winters, where the following situation is laid out for students:

The last snow storm lasted 6 hours. Over that time, the amount of snowfall was measured hourly, except for at the 5th hour when Mr. Pearce watched the 3rd period of the Toronto Maple Leaf game.

Student are then asked to create a scatter plot, line of best fit, make some predictions using interpolation and extrapolation based on the data, and classify the relationship. I know, I know… not the most interactive or creative task you’ve ever seen. It’ll have to do while I’m away.

Then, I had attempted to make a semi-interactive task that continues the Placing Toothpicks tasks (original and part 3) that students had worked on over the past week and a half called: Placing Toothpicks Part 4 (original, eh?). I had taken the sequel task and actually saved to use it for tomorrow as a way to introduce identifying linear and non-linear relations. Maybe I should just re-name all the tasks? Ah well.

The Part 4 Task is another partial variation linear relation pattern that is formed using a design with toothpicks.

Here’s Act 1:

Can’t see the video? Click here.

Students are going to be looking at the 6th and 11th figures as they did in previous tasks, however they will be going through this task in a self-led fashion by downloading a math task template I created for them:

How I made the task template interactive was by adding hyperlinks in the PDF that will allow students to jump straight to the Act 1 video:

After students have come to their conclusions, they can watch the Act 3 video by clicking the link embedded in the math task template:

The task template is structurally organized to ask students to do specific things like create a table, graph and answer some questions on the back. I’m not super thrilled with using this approach as of late, but these math task templates are really the only structure from my former, more traditional, teaching approach. I would be lying if I said I don’t worry about my moving to a less structured teaching style, but I feel deep inside that it is the right thing to do.

If you want to grab the template, click below:

This would be the third day in a row that I’m out of the classroom. Luckily, I’ve had the same supply teacher who is actually the long-term occasional teacher that teaches my afternoon schedule while I do Middle Years Collaborative Inquiry (MYCI) work for the district. While there is some consistency while I’m away, this is the first time he has covered the group. I asked him to keep me posted on how things were going while I was away and it sounded like most students are on pace, but there was a school-wide interruption planned to take up a portion of the period today.

With that in mind, I left a new topic (identifying linear and non-linear relations) for students to explore with the thinking that we would be spiralling back to it a number of times throughout the semester. This would give students the opportunity to construct their own understanding of what makes a relation linear or not through an investigation. This is where the Placing Toothpicks Sequel comes in as it is quadratic. I was interested to see how students reacted to it, since they have only seen linear patterns for the most part thus far.

Here’s Act 1 of the Placing Toothpicks Sequel:

Can’t see the video? Click here.

Again, I thought I would create an interactive math task template that would allow students to link directly to the videos independently:

Students were not in school today as teachers had a scheduled professional development day.

That’s it for this week! Looking forward to being back in action next week as we spiral into our next few topics. Toss me some feedback on what you’re seeing. Any ideas how I can release more of the responsibility to my students? On my mind for this coming week is how I can better “let go” of the formal structures (i.e.: procedures) that most of our learning goals tend to lead to. Until then, have an awesome week of learning.

The post Week In Review #4 – Geometry & Linear/Non-Linear Relations appeared first on Tap Into Teen Minds.

]]>Watch the first act as you see the first three figures made with toothpicks. How many toothpicks are in the 6th figure? ...11th figure? ...the general rule?

The post Placing Toothpicks Part 3 appeared first on Tap Into Teen Minds.

]]>This task is a follow up to the Placing Toothpicks and Placing Toothpicks Sequel tasks I posted recently. The first task was proportional, followed by a quadratic in the second. This task is going to extend the original task from a proportional (direct variation) linear relation to a partial variation linear relation. That means this task could be used for patterning in elementary or for linear relations in grade 9 academic and applied. Here’s the grade 9 academic expectations we can make connections to:

- LR2.02 – I can construct tables of values, scatter plots, and lines or curves of best fit as appropriate, using a variety of tools for linearly related and non-linearly related data collected from a variety of sources.
- LR2.03 – I can identify, through investigation, some properties of linear relations and apply these properties to determine whether a relation is linear or non-linear (by rate of change/initial value when described in words, by first differences in a table, straight/curved graph, degree of terms in equation).
- AG1.01 – I can determine, through investigation, the characteristics that distinguish the equation of a linear relation (straight line) from the equations of non-linear relations (curves).

Show the students the video below:

Can’t see the video? Click here.

If you have already used the Placing Toothpicks and Placing Toothpicks Sequel, asking them what questions they have might be a bit redundant. Sure, I could change what figure number I want them to find, but this process might be more forced than it is worth.

As we did in the previous toothpick tasks, the question(s) I want students to think about are:

How many toothpicks are in the 6th term? … the 11th term?

I would likely have students figure this out on their own, using any strategy and then consolidate the task. The following math task template might be a good option to consolidate student thinking:

You can now let students see their solution in action!

Can’t see the video? Click here.

Here’s some work I captured when I tried this task for the first time the other day.

Have you tried this task? How can we make it better? Share your thoughts in the comments below!

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

The post Placing Toothpicks Part 3 appeared first on Tap Into Teen Minds.

]]>When we talk about people who are "Good at math" are we really referring to people who are good at memorization? Let's dig into this a bit deeper...

The post Tips Moving From Math Procedures to Understanding appeared first on Tap Into Teen Minds.

]]>I opened my inbox to the following email this morning:

Hi Kyle,

Scott Miller here. We met at the DVC Math Conference in February.

In reading/watching your presentation “Two Groups of Math Students I Created in My Classroom” have you found students that are

Not Good at Memorization, but do not struggle with unfamiliar problems?Have you gotten flack from parents that their child does not learn “that way” because you are “not teaching” them? I have used the terms

Good at ProceduresandNot Good at Procedures. I have found that by creating more challenges for a student to think through has benefited more of theNot Good at Proceduresbecause they try different things. TheGood at Proceduresstudent shuts down and is unwilling to try because he or she does not recognize a memorized pattern.

What Scott is describing is very common when you start making a shift to teaching for understanding instead of teaching for procedures.

I would agree that challenging my students to think is initially more more beneficial for my *not good at memorization* or *unwilling to memorize* students. I see this group of students giving up when they are presented with an algorithm that they must follow. However, when I challenge them to solve a problem without attempting to give them a procedure, they are more willing to get creative in their thinking. The opposite appears to be true for our *good at memorization* friends likely because they are aware of their skill to replicate a given set of steps. When we don’t offer up those steps, they are concerned that they might not get the right answer on the first try and thus, they wouldn’t be considered “good” at math.

I believe this issue has a lot to do with mindset. I often find that the typical *good at procedures* group has a fixed mindset (I’m good or not good at something and I don’t have much control over it) and they have consciously or unconsciously figured out the “game” of getting good grades in math class. It is much easier for someone who can memorize easily to follow a procedure without having to understand what they are doing. Even in a classroom where a teacher is introducing topics through investigation, if the assessment routinely requires only the use of an algorithm, students naturally pick up on this. Why bother with understanding the “why” if I can just remember this handy set of steps?

If my assessments focus primarily on steps and procedures, then we are subconsciously telling our students to treat math concepts like a job on an assembly line:

- Grab two bolts from the bin.
- Fasten one (1) bolt to the threaded hole at the top left of the vehicle.
- Fasten one (1) bolt to the threaded hole at the bottom left of the vehicle.
- Apply grease with the grease applicator to the hinges on the front passenger door.
- Repeat for each vehicle on the line.

The worker may or may not understand why she is doing those steps in that particular order to complete her job successfully. Observing the worker might make one believe that she has a deep understanding due to her competence in completing the steps. However, only when you move her to another random job on the assembly line without any training will you truly observe how much she knows about building a vehicle.

This assembly line approach is what I see in my math classroom on a regular basis when we allow students to rely too heavily on an algorithm. Unless a problem looks familiar to what they have tackled previously, students lacking a deep conceptual understanding will experience difficulty. While it is difficult to pinpoint the specific cause, my hypothesis is that students in the *good at procedures* group have not been asked to think critically often enough in the math classroom.

I think there is a connection between what we are describing here and all of the posts on social media that describe “new math” as a simple question with a solution that appears more complex than necessary vs. “old math” where the same problem is tackled with a seemingly simple algorithm. Parents often get more frustrated than the student because they believe they understand math, but can only tackle addition/subtraction and multiplication/division using the standard algorithms.

When a *good at procedures* student is frustrated because I do not give them an algorithm to memorize, I make sure that I discuss why I am teaching from a task based, inquiry approach over simply giving them the steps. The frustration will not go away immediately and we shouldn’t expect it to as we have literally ripped the mathematical carpet from under them. You can imagine how a student must feel when they come into my grade 9 math class and quickly realize that they’ve been misled into believing they are “good at math”. The first instinct of the student and his/her parents is to blame the teacher because they have made it through eight years of elementary school math without a hitch. However, I find that you can get both the student and the parents on your side by being clear about why the student has been successful in the past. I try to help the student and the parents understand that knowing procedures alone is like being a “mathemagician” with a limited number of math tricks that are only useful in very specific situations. Helping students develop a deep understanding by thinking critically on a daily basis is the only way students can truly understand math and apply their knowledge to unique situations.

We can also help ease any anxiety that this change to learning math might cause by using assessment practices that promote student learning and growth. Since learning math is a process and every student is at a different place on their learning journey, offering multiple opportunities to demonstrate learning will help the student understand that they will not suffer if they struggle with a concept initially.

Have you had a similar experience in your math classroom? Have you found a way to help students understand why you don’t want them to simply memorize? Help us out by leaving a comment below!

Here is a timely YouTube video that I came across just hours after I published this post. Fits in nicely.

http://t.co/6ZMXcmHTUq

Can’t see the video? Click here.

The post Tips Moving From Math Procedures to Understanding appeared first on Tap Into Teen Minds.

]]>Get a look at what we're up to in Grade 9 Academic as I attempt spiralling my MPM1D curriculum for the first time using a task-based approach.

The post Week In Review #3 – Measurement, Linear Relations and Geometry appeared first on Tap Into Teen Minds.

]]>Today, I had intended to use a Knowledgehook Gameshow as a warm-up related to area of composite figures and then move into the Gas Guzzler 3 Act Math Task. Unfortunately, my timing was completely off as I had 7 questions and some students who were pretty eager to solve them with their complete solutions uploaded to the system (bonus!).

Here’s the gameshow that we used:

Can’t see the Gameshow on the page? Click here.

By the time we tackled each problem while sharing student solutions over the projector using the “Share Solution” feature as we went, we had 15 minutes remaining in class. Definitely not enough time to tackle what I had planned for Gas Guzzler.

Instead, I assigned an activity related to linear correlation via Knowledgehook Homework mode and students were off to the races.

Tomorrow, we will have Assessment #2 to see how we are making out thus far with our learning goals that we have covered.

Tuesday is Assessment day where I take a few questions from **any** learning goal from the year to see what point students are at in their learning. Students are not told what they will see prior to the assessment, but also know that the purpose is to promote growth, not cement marks into a grade book. If students struggle on a concept, now they know what to look at for the following week and can submit work from given “Growth Opportunities” from their online Learning Logs. Students are also permitted and *encouraged* to create their own “Growth Opportunities” to improve.

Assessment #2 will cover the following learning goals:

- I can solve problems involving the areas and perimeters of composite two-dimensional shapes.
- I can construct tables of values, scatter plots, and lines or curves of best fit as appropriate, using a variety of tools for linearly related and non-linearly related data collected from a variety of sources.
- I can describe trends and relationships observed in data, make inferences from data, compare the inferences with hypotheses about the data, and explain any differences between the inferences and the hypotheses.

We will also introduce, for the first time, a new learning goal:

- I can determine values of a linear relation by using a table of values, by using the equation of the relation, and by interpolating or extrapolating from the graph of the relation.

Students get a little squirmy when I introduce a new concept on an assessment, however I do give enough information for them to apply the new concept. I consider this like a diagnostic to determine where students are in relation to the new concept. I simply provide students feedback on how they handled the new concept and ensure they know that the concept will begin coming up regularly in class and on future assessments.

You can check out the assessment below:

Today, I used the Gas Guzzler 3 Act Math Task as a way to help students build their problem solving skills while connecting **proportional reasoning** to **direct variation linear relations**.

If you’ve never come across Gas Guzzler, here’s the Act 1 Video:

Can’t see the video? Click here.

Grab the whole task here.

After watching the video, we have students share out their questions. The question I want to focus on first is:

How much does it cost per litre of gas?

We then get kids making their predictions and then I give them some more information about the task in Act 2.

Students then work in their table groups to come up with a solution prior to seeing the Act 3 video to check their answer.

For the first time this year, I decided that we could really extend this task to other learning goals we’ve covered this semester and even introduce new concepts along the way by creating a Custom Gameshow. So I started with questions pretty closely tied to proportional reasoning:

Tomorrow, I’ll do a warm-up gameshow with more questions that extend to direct variation linear relations.

Here’s some student exemplars from today:

As mentioned above, I had intentions to extend the Gas Guzzler task to direct variation linear relations by introducing initial value and rate of change. Here’s the warm-up Gameshow that I created to support the extensions:

This was the first time I’ve attempted to “embed” a new concept into practice. It seemed to work well. I attempted to define the new concepts in a question to get students to think critically about what we have introduced. Here’s one of the questions, in particular:

After we finished up the warm-up, I had the Placing Toothpicks 3 Act Math Task lined up and ready to go. Students watched this video:

After hearing the questions students had about the video, I showed them this:

Click here if you can’t see the video.

The questions I want students to think about are:

How many toothpicks are in the 6th term? … the 11th term?

I would likely have students figure this out on their own, using any strategy and then consolidate the task. The following math task template might be a good option to consolidate student thinking:

You can now let students see their solution in action!

Can’t see the video? Click here.

Can’t see the video? Click here.

This week, the focus has been on area of composite figures, proportional reasoning and linear relations. Some areas I haven’t spiralled into yet are angle geometry, exponent laws, and algebra. When I used traditional units to teach this course, I typically left angle geometry until very late in the course. This year, I wanted to slowly introduce them early on in order to give students multiple opportunities to build a deep conceptual understanding.

To this point I have struggled to find a compelling reason for why we learn the properties of interior and exterior angles of a triangle. I know that we can apply triangles to many real world situations, but how can we hook students in to want to engage in a task requiring their use?

This time around, I decided to use Dan Meyer‘s Best Triangle 3 act math task as a way to get kids thinking about triangles and allow me to branch off into properties of interior and exterior angles.

Here’s how it went down:

I showed students this video:

Can’t see the video? Click here.

The question asked in the video is:

Draw three points in the shape of an equilateral triangle.

I asked students do this on their iPads in a whiteboard app like GoodNotes 4 or Explain Everything. You’ll see why this is important later.

I then asked students to rank the triangles created by Andrew, Nathan, Chris and Timon. **Who, do they believe, created the best equilateral triangle?** Do they think that their triangle is better than all four?

We then had a discussion about how they could prove or disprove a claim that one triangle was “better” than another. Students were quick to recall from elementary school that equilateral triangles have equal sides, but it took a little while longer to get someone to state that the angles are also equal.

At this point, I had intended to use a pre-made interactive Geogebra worksheet that has a screenshot of the four triangles so students could measure the angles. My thinking was that this would be a pretty pain-free way to introduce students to this free online resource and also serve a purpose to help us measure angles. Check out the sheet below:

Can’t see the Geogebra Worksheet? ” target=”_blank”>Click here.

Unfortunately, about half of the students had some pretty significant struggles using this web based tool on their iPad. I had no such struggles on my iPad Air 2, however my students are using 5 year old iPad 2 devices. Maybe these relatively old iPads didn’t have the processing power necessary? After an abundance of troubleshooting around the room, I wiped the sweat from my brow and finally pulled the plug on this portion of the activity. I went straight for this image to reveal the angle measures of each triangle:

At this point, I asked students to work in their table groups to come up with an agreed upon order from “Best Triangle” to “Worst Triangle” and we would share out in 3 minutes.

As a whole, we managed to come up with the following order:

- Timon
- Nate
- Chris
- Andrew

The discussion was rich as students tried to justify why they ordered Nate, Chris and Andrew the way they did. Students agreed that Andrew had the “greatest” angle and thus they felt he should be last. While they noticed that Chris had the “least” angle, they failed to make a comparison to determine what else might affect their decision.

Students were really surprised to see that Chris was last, but we had a good discussion as to why that made sense. While we do not explore some of the analytic geometry required to explore this task further in grade 9, I did mention to students that we could revisit this problem in grade 10 using more mathematical tools to help us come to this same conclusion.

Once we completed this task, I had students jump into the Explain Everything Angle Journey I had created for my students last year. The intention with this task was to allow students to revisit some of the concepts they have experienced in elementary school. The screencasting features of the app also allowed them to consolidate some of their learning with the interior and exterior angles of triangles while helping me determine what we need to touch upon before moving on.

If you haven’t checked out the full blog post, you can click here to jump to the post.

The post Week In Review #3 – Measurement, Linear Relations and Geometry appeared first on Tap Into Teen Minds.

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