Tile Circle 3 Act Math Task is a real world math problem that helps students to discover Pi through the relationship between the circumference of a circle and diameter. We also discover the area of a circle through inquiry and using concrete material and visuals.

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]]>In this 3 act math task, we will explore the relationship between circumference and diameter of a circle to “bump into” Pi, then show how Pi can be used to help us find the unknown **circumference of a circle** and **area of a circle** by teaching through task.

Thanks to Alice Aspinall and Chez Cetra from Walkerville Collegiate for inviting me to be a part of your Pi-Day that inspired the creation of this task. Looking forward to learning with you all again real soon!

In the first act of this **real world math problem**, we first ask students to create a chart with the words notice and wonder at the top.

Then, they write down all that comes to mind as they watch the following video:

Ask students to take a minute independently to finish writing down anything they notice and wonder.

Then, give them 2 minutes to discuss with their neighbours, before we share out as a group and I jot them down on the board.

Here’s some of what students noticed and wondered when I used this task for the first time at Walkerville:

- How big is the circle?
- Why is there a white spot on it?
- I noticed that there are square tiles around the outside.
- How many square tiles are there around the outer edge?

After having students share out their ideas and celebrating some of the creative thinking, we then narrow the question down to:

How many square tiles are wrapping this circle?

We will have students make a prediction based on what they see and share that out.

If students are reluctant to share initially, I might ask “who has a number higher than ____” or “lower than ____” and that almost always gets a few hands in the air. Students feel like you’re talking directly to them when their number is higher than the number I’ve tossed out there.

After sharing out predictions and hearing what students feel they need in terms of given information, you can show them this video or the animated gif below:

Alternatively, you can show this image:

Now that students have some information to work with, they can get to it!

Sharing and celebrating different strategies to solve the problem can really help elevate the lesson to a dynamic and innovative learning experience. I try to find a really “messy” solution to celebrate the need for students to brainstorm by writing **anything** and **everything** they are thinking rather than worrying about their work being a perfectly organized process.

Once students have shared out, it is time to experience the solution of this real world problem, share this video:

Or the animated gif:

If you would prefer to show a still image, here it is:

The intention of this first task is to really explicitly pull out the relationship between the **circumference and the diameter of a circle to reveal Pi**. For so many students, they are uncertain as to where Pi comes from, even if they engaged in a lesson previously intended to help them make sense of it. It is important for us to come back to these ideas and ensure that they truly understand this relationship so they can make better estimates when dealing with **circumference** or **area of a circle**.

While students might think the fun is over, that couldn’t be further from the truth.

Show them this image:

Then, ask them to make a prediction and share with their neighbours.

Be sure to have students share their strategies so we get an understanding of whether they are just making a gut shot guess, using a spatial approach or whether they have some prior knowledge in the area of a circle (i.e.: using procedural fluency with the formula) to come up with a number.

Show student this video:

Alternatively, you might consider using this animated gif:

Allow students time to discuss what is going on here. We want students to see that if we cut up the pieces of the circle smaller and smaller, we will eventually have a rectangle and finding the area of a rectangle is something we’re pretty good at.

Can students determine what the dimensions of this new rectangle is?

I’d hold off on showing them to see what they can come up with.

Using concrete manipulatives would be a great way to let them explore this.

Cut up a circle and using marker on the outer edges, where do those edges end up when you rearrange the pieces to make your rectangle?

After students have shared out their strategies, let them see the visual to ensure that they truly understand where the area of a circle formula comes from:

Alternatively, an animated gif can be used:

Want to use this task in your classroom?

Grab the Animated Keynote Slide Deck, Powerpoint Slide Deck, and all of the downloadable videos and images by clicking below.

Did you use this task in your classroom?

Please share your results in the comments section below!

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]]>For years, I was spinning my wheels trying to teach students how to make sense of mathematics through abstract representations, when the key was making math concrete and visual through the concreteness fading model or concrete, representational, abstract (CRA).

The post Make Math Matter With Concreteness Fading appeared first on Tap Into Teen Minds.

]]>During the first half of my teaching career, I would spend what seemed to be the first half of a math lesson teaching a new math concept by sharing definitions, formulas, steps and procedures. To make things more challenging for my students, I would simultaneously introduce the symbolic notation used to represent those ideas. Then, I would spend the remainder of the lesson attempting to help my students make sense of these very new and often abstract ideas. By the end of the lesson, I could help many students build an understanding, but there was always a group I felt who I would leave behind.

Like many other teachers, I was just teaching in a very similar way to that how I was taught.

I knew no different.

However, if we consider that new learning requires the linking of new information with information they already know and understand, we should be intentionally planning our lessons with this in mind. A great place to start new learning is through the use of a meaningful context and utilizing concrete manipulatives that students can touch and feel. When we teach in this way, we minimize the level of abstraction so students can focus their working memory on the new idea being introduced in a meaningful way.

When we intentionally start with concrete manipulatives to learn new math concepts, our goal is to help students better construct an understanding of the mathematics in their mind. The goal is not to burden students with a big bag of manipulatives that they must carry around with them anytime they are required to do any mathematical thinking, but rather to ensure that they can build their spatial reasoning skills physically – through the manipulation of concrete objects – so they can begin to visualize mathematics in their mind. When a student is able to “look up” as if they are peering into their mind to visualize their math thinking, we know students are thinking conceptually rather than simply following a memorized procedure.

While students are working with concrete manipulatives, it is helpful for the teacher to model visual representations of the student work for all to see. By introducing these visual drawings of the concrete representations students are creating, it will be easier for students to shift away from concrete manipulatives and towards visual (drawn) representations when they are ready.

When students have built an understanding both concretely and visually, we can then begin moving to the final stage called abstraction where we use symbolic notation. The goal here is that when students use the symbolic notation, they can visualize what the concrete representation of that mathematical statement represents.

Some know this idea as concreteness fading, while others have called this progression **concrete, representational, abstract (CRA)**. In either case, the big idea is the same. Start with concrete manipulatives, progress to drawing those representations and finally, represent the mathematical thinking abstractly through symbolic notation.

Let’s look at a couple different questions at different grade levels where using context and concrete manipulatives can lower the floor and help us progress towards more abstract representations.

If we were to ask students in a grade 2 class, I might have them look at the following image and ask them what they notice and what they wonder:

Then, I might have them predict how many doughnuts they believe fit in that box.

After students talk with a neighbour and share out their predictions, I would say:

This box of doughnuts has 3 rows of 4 doughnuts.

This question may seem quite simplistic after giving this new information, however for students who are just beginning to shape their understanding of number including place value and additive thinking, we are now throwing a very heavy and abstract idea at them.

In a perfect world, we could give them real doughnuts (or bagels, to be health conscious) so they could recreate the situation right in front of them:

Despite the fact that using square tiles or circular counters to represent doughnuts is more concrete than drawing doughnuts or using symbols (numbers and operations), we must understand that concrete manipulatives are still more abstract than using the actual items in the quantity being measured.

As students understanding of number increases, so too should their ability to begin using concrete manipulatives instead of real doughnuts to work through this situation.

As students use concrete manipulatives to build their conceptual understanding of a new idea, they will begin to feel burdened by the manipulatives and seek out less cumbersome tools and representations to show their thinking. If the teacher has been drawing visual representations of the concrete representations students share with the group along the way, many will eventually transition to drawing their representations rather than building them concretely. However, for other students who have seemingly mastered the concrete representations but are not shifting to visuals, we may need to help scaffold them along.

With conceptual understanding continuing to deepen through the use of drawn visual representations, teachers can continue sharing student thinking through the use of visuals and begin introducing symbolic notation. Since students have had a significant amount of time to inquire, investigate and solve problems using both concrete and visual representations, they will develop the ability to visualize representations in their mind. At this stage, it would seem more efficient to use symbolic notation such as numbers and operations to represent mathematical thinking rather than building concretely or drawing visually.

It is important to note that while the concreteness fading model or concrete, representational, abstract (CRA) approach is a general progression that we want to keep in mind when teaching new concepts in math class, we don’t want to overthink it either.

For example, in the abstract / symbolic phase, you’ll notice the words:

“3 groups of 4 doughnuts is equal to 12 doughnuts”

By no means am I suggesting that we should wait until the concrete and visual phases are mastered before using those words. I would actually suggest that we are verbally saying those words during the concrete stage and even possibly writing down that sentence during the concrete stage since there are no new symbols or abstract ideas being introduced by doing so. With an idea like single digit multiplication, you might consider having students build the concrete representations and the teacher may draw the visual representation as well as the symbolic representations at the same time.

The key with concreteness fading is that we are aware of these three phases and we use our professional judgement to determine when to introduce each phase as to push student thinking forward without overwhelming them with too much abstraction too quickly.

We could introduce a similar question for say a grade 4 class by simply increasing the complexity of the question such as:

How many doughnuts are in 3 boxes?

If we are asking students to work with a problem that we could consider is a multi-step multiplication problem, the beginnings of volume or a double digit by single digit multiplication problem, my hope would be that students are now comfortable abstractly using concrete manipulatives (connecting cubes, square tiles, etc.) to represent how many doughnuts are in 3 boxes. If a student is struggling with the abstraction of using a concrete manipulative in place of the actual object – like doughnuts in this case – we might need to reassess the readiness of this particular student and do some more work with more accessible problems.

In this particular case, the progression of concreteness fading might look something like the following:

Or students might go about it using their knowledge of arrays and extend the idea to area models before finally developing a student generated algorithm:

Here’s a summary of the concreteness fading progression that may take place if students have been doing work with arrays and area models:

Assuming students have had a substantial amount of experience building concrete representations of multiplication, you may see students skipping right over the concrete phase to the visual stage creating drawn diagrams of this situation. This is absolutely fine as a student who is able to draw what the concrete representation should look like suggests that she could indeed build that representation if required. Furthermore, this also suggests that this student is now able to create a more abstract representation of that concrete model, which is what we are hoping to develop.

What I would not advocate is completely skipping over the first two phases and focusing only on the symbolic representation. Despite the fact that some students may have a visual of that concrete model clear in their minds, we don’t want to promote students relying solely on procedural fluency and risk forgetting all of that conceptual understanding we worked so hard to build. By giving students enough practice drawing visual as well as abstract or symbolic representations, they are utilizing their conceptual understanding and procedural fluency in tandem, where they can be used most effectively.

In a middle school classroom (end of junior/intermediate classroom in Ontario), the question might sound more like this:

There are 36 doughnuts in 3 boxes.

How many doughnuts are in 7 boxes?

While this may seem like a lot of doughnuts for students to represent concretely, having linking cubes, square tiles or other tools students can use to organize their thinking is important especially for those who have not yet built a conceptual understanding of what this task is asking of them. In this case, we are exploring a proportional relationship where the number of doughnuts is proportional to the count of how many boxes there are.

Despite the fact that proportional reasoning is introduced explicitly in the Ontario Grade 4 Math Curriculum, many of our grade 9 students continue to struggle with this type of reasoning. Many may not have fully conceptualized the prior knowledge necessary for them to be successful at that particular grade level. Before I understood the power of concrete and visual representations, I can recall trying to help students in my grade 9 (and sometimes grade 10) class by breaking down the symbolic representation with more symbols.

For example, with this particular problem, I might have attempted unpacking the problem with students by creating a proportion and solving for the unknown:

While working with and solving for unknowns in a proportional relationship was an expectation in my curriculum and in the curriculum prior to grade 9, I was stuck in my habit of trying to start with abstract symbols and unpacking them. However, the reality is that when we do this sort of work without building the necessary conceptual understanding at the concrete and visual phases of concreteness fading, students are forced to either memorize the steps and procedures or get left behind. For some students, they are able to make their own connections at the symbolic stage based on their prior knowledge from past experiences in school and at home, while other students are left scrambling to understand with stress and anxiety levels building with each passing class.

How might I have approached this same problem had I known and understood the **concreteness fading model**?

Well, I would definitely start with concrete manipulatives for all of my students. Just because a student is able to solve familiar problems using all the right steps and procedures does not necessarily mean that they have a conceptual understanding of the mathematics they are employing.

One possible idea could be giving students connecting cubes and having them model out the situation. They might start by grabbing 36 cubes and dividing them to the 3 boxes. Then, they could double the 3 boxes of doughnuts to get 6 boxes and add an additional box.

Some professional noticing you will want to engage in would be determining whether students are using additive thinking, multiplicative thinking or a combination of the two. To build an understanding of proportional reasoning, we must help students to think multiplicatively. So while thinking additively is not bad or wrong, we do want to try to prompt students to think multiplicatively.

For example, you might ask students:

How many times bigger is the quantity in 7 boxes than in 1 box (i.e.: 7 times bigger)?

How many times bigger is the quantity in 7 boxes than the quantity in 3 boxes? (i.e.: 2 and 1 one-third times bigger)

While I try to encourage all students to make a concrete model, some may be moving away from physical manipulatives and pushing towards a visual model which would suggest that they are ready to move one step deeper into abstraction.

Here’s an example of how a **double number line** could be used to help students visualize the situation and problem solve their way to a solution.

From both the concrete models and the visual models students use in the classroom, I can prompt students to attempt modelling their thinking using symbolic notation such as algebraic expressions and equations.

The more I can help my students link their concrete and visual models to more abstract representations, the stronger their conceptual understanding will be to help support any procedural approaches they wish to use to progress towards more efficient methods.

Had I known more about concreteness fading earlier in my career, the progression might have looked more like this:

While the above concreteness fading progression would have been a huge help to all students in my class to better understand proportional relationships, I would later learn from my colleagues in the AMP group that setting up a proportion of equivalent fractions is not a very powerful method mathematically, since it yields only the numerical answer to a single problem. A more powerful approach is to uncover the proportional relationship in the problem situation, since this allows us to immediately solve any problem based on that situation.

Let’s take a closer look.

I have been blessed to be a part of an amazing group of mathematicians funded through the Arizona Mathematics Project (AMP) to make sense of proportional relationships and this group of 18 mathematics education influencers have landed on some really useful definitions related to this very commonly encountered type of middle school math problem.

When we look at the animation of the concrete model using connecting cubes, you can see the two methods of attacking **proportional relationships** that the AMP group refers to as:

- scaling in tandem; and,
- using the constant of proportionality.

We can see the use of **scaling in tandem** when we see the doubling the number of boxes and number of doughnuts from 3 boxes, 36 doughnuts to 6 boxes, 72 doughnuts.

We can see the use of **the constant of proportionality** when we look at the number of doughnuts (12) in a single box – often referred to as the **unit rate**.

First, we will head a bit further down the concreteness fading continuum by taking our horizontal double number line and represent it as a vertical number line. This is a nice way to progress towards a table of values without losing the relative magnitude between each quantity on the number line.

For years, I would teach my students with an end goal of setting up and solving a proportion rather than focusing on helping them “own the problem” as Dick Stanley put it at our recent AMP meeting.

What we are referring to here is the limited usefulness of setting up a single proportion for a “rule of 3” problem.

When we set up a proportion of equivalent fractions, we have set out to solve a single problem and often times, we unintentionally rush to a procedure by setting up and solving for a single unknown. While this might be efficient for finding a single answer to a closed problem, it does not help us efficiently solve multiple problems nor does it promote a deep conceptual understanding of the proportional relationship that underpins this situation.

While I do not want to advocate that we avoid proportions altogether, I would much prefer giving students the opportunity to explore these problems more deeply and allow for **the students** to stumble upon some of the procedures we see taught explicitly in middle school classrooms.

Let’s look at where we might start.

We can see from the animation below that **scaling in tandem** is responsible for allowing us to solve a proportion using any of the procedures we see taught in many middle school math classrooms:

By utilizing **ratio reasoning** by **scaling in tandem**, students are explicitly introduced to the power of the proportion, but in a much more powerful way.

If students are encouraged to utilize scaling in tandem with double number lines, tables and equivalent fractions, over time we can help students see that some of this scaling can be done more efficiently:

Over time, students may progress from a vertical number line to a table of values. When students are ready, they may begin disregarding the magnitude of number allowing them to “skip over” some of the values on the number line.

We can use this **scaling in tandem** strategy to find any unknown in this proportional relationship, but it will take a bit of work.

For example, we can find the number of doughnuts in 9 boxes of doughnuts by scaling in tandem by multiplying both 36 doughnuts and 3 boxes by 9/3 or 3:

As students become more fluent using scaling in tandem as a strategy for proportional relationships, we can then begin making generalizations:

What we see through this generalization is the conceptual understanding for **why** the common procedures we see in math classrooms actually work.

One of the most over-used, but misunderstood tricks from the middle school math classroom is **cross multiplication**. When we look at the generalization of scaling in tandem, we can see where ideas like cross multiplication comes from:

q/p = d/c

qc = pd

c = pd/q

or

d = qc/p

While I taught tricks like cross-multiplication, “the magic circle” and “y-thingy-thingy” before I constructed a firm conceptual understanding of proportional relationships, the reality is that they provide a dead end pathway to a single answer rather than an understanding that allows you to own the problem.

So rather than simply teaching a trick using steps and procedures, let’s give students an opportunity to build a conceptual understanding of scaling in tandem and challenge them to come up with their own procedures and algorithms.

While we can use **ratio reasoning** and **scaling in tandem** to up the conceptual understanding over solving a “rule of 3” problem using a proportion, we still don’t “own” the problem yet.

Under the hood of every **proportional relationship** lies a constant that we can use to solve **ANY** problem related to the proportional situation. This **constant of proportionality** can be found by taking the quotient of any two covarying values in the relationship.

Many know this constant of proportionality as the unit rate.

Unlike the **ratio reasoning** strategy of **scaling in tandem** where one must determine a new scale factor to find each unknown quantity in a proportional relationship, the **rate reasoning** strategy of finding the **constant of proportionality** allows one to use the constant to find any unknown from the relationship.

In the case demonstrated here where the number of doughnuts is proportional to the number of boxes, we can determine the number of doughnuts in any number of boxes by multiplying the number of boxes by 12 doughnuts per box, while we can determine any number of boxes by multiplying the number of doughnuts by 1/12 boxes per doughnut (or dividing by 12 doughnuts per box).

So again, while I see huge value in students understanding how they can use ratio reasoning to scale in tandem to solve problems involving proportional relationships, only when we unlock the conceptual understanding behind rate reasoning and the constant of proportionality do we own every problem related to that proportional relationship.

So rather than suggesting that the concreteness fading progression should end at the creation of a proportion of equivalent fractions and solving for an unknown, I would much rather see students exploring both ratio reasoning by scaling in tandem and rate reasoning through the constant of proportionality. Therefore, a suitable progression might look something like this:

For years, I was spinning my wheels trying to teach students how to make sense of mathematics through abstract representations. However, even musicians are aware of the importance of marketing through a concreteness fading model.

Take the musician Prince for example.

Every successful musician knows that the best way to build a true fan base is to begin at the concrete phase. If you go to see Prince live in concert for example, you will quickly understand why he and his music are so great.

After seeing the show, you might rush to grab the next best thing to seeing him live in order to bring back the energy and positive feelings you had while watching live. Buying albums, videos, posters and magazines are great examples of how we can listen and see Prince in our minds as if he were there in the flesh.

For those who are fans of Prince, it is highly likely that you know that after years of being known as Prince, he would legally change his name to a symbol:

This would be a career ending move if he didn’t already have millions of fans who had watched him live in concert (concrete) and enjoyed his music and memorabilia (visual). But yet, in math class, we so often begin with symbols and try to make meaning of them after.

My call to action here is to be more Prince and think about concreteness fading during the planning process of each and every lesson. If we do this, we stand a much better chance of Making Math Moments That Matter for students.

How are you using **concreteness fading** in your lessons? I’d love to hear from you in the comments.

The post Make Math Matter With Concreteness Fading appeared first on Tap Into Teen Minds.

]]>Have you ever looked at a group of items and just knew how many there were without actually counting? This ability to "see" how many items are in a group without counting is called subitizing. Read to learn more.

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]]>Have you ever looked at a group of items and just knew how many there were without actually counting? This ability to “see” how many items are in a group without counting is called **subitizing**.

The ability to subitize is an important part of developing a strong mathematical foundation and understanding of number (Baroody 1987, 115).

Playing with dice, dominoes, and asking children to find a specific number of items will help them develop subitizing skills and a sense of quantity. Asking to guess how many items you are holding will help develop estimation skills, which is another very important skill that will help children develop their mathematical skills.

An interesting activity to do with children and adults is to have them look at the image of the dots below for just a few seconds and then look away (or remove the dots from their view).

Ask them to make a picture in their mind of what they saw.

Then, describe what you saw in your mind to someone else.

It’s highly likely that they will “see” it differently than the person next to them.

Even though we are looking at the same dots, it is quite possible that the way you visualized these dots in your mind was different than the next person. This is because the number of dots you are visualizing is too difficult to subitize in a single group.

Here is a video of just a few of the many ways people describe how they visualized the dots:

When the number of items we are counting is small, we perceptually subitize to “see” the count suddenly.

Most can develop the skill to perceptually subitize quantities of 5 items or less.

When the number of items we are counting is too large to “see”, we conceptually subitize to “know” the count suddenly.

When quantities are larger (say, 5 or more), our brains decompose the group into smaller “chunks” and then add them together.

You can help develop your student’s and/or child’s foundational mathematics skills in school and at home by making use of the following games and tools for subitizing:

Use fingers, dice, playing cards with the corners cut off, dominos or dot plates to “make 5” or “make 10”.

Using dice, playing cards, or dot plates, two players roll a die, flip a card or dot plate and each player says their number. Player with the higher number wins the round.

One player shows how many counters they have in total. Then, hide some of the counters under the cup while the opponent closes their eyes. How many are under the cup?

Using dice, take turns rolling 1 or 2 dice. Say the number rolled and record using a tally chart. First player to 20 wins.

Take turns rolling a die. Find the same number of dots and cover it with your colour counter. Get 3 of your counters in a line and you win!

Download the game board here.

Want to grab a subitizing cheat sheet that you can print to keep handy in your classroom or as a resource to provide parents to raise awareness of the importance of subitizing at home?

The post Counting With Your Eyes: Subitizing appeared first on Tap Into Teen Minds.

]]>Struggling to find a way to make math more accessible for all students in your classroom? In this post, we'll give examples why using concrete manipulatives and visual representations is a great place to start!

The post Lower the Floor in Math Class appeared first on Tap Into Teen Minds.

]]>What comes to mind when you think back to learning math in school? It would seem that most people I ask typically respond with a negative or neutral response and very few with something positive. Since many of us were taught primarily using procedures and steps, it is unlikely that too many of us could see math as anything more than rules, steps and symbols despite the fact that mathematics was created to help us better understand the world around us.

If this is so, then why aren’t we learning math first with concrete objects that we can touch and feel in order to allow students to co-construct and develop the rules, steps and symbols that represent those real world situations. By doing so, we are helping students develop the ability to **visualize the mathematics** they are engaging in and they will have an opportunity to see mathematics very differently to that of our generation.

Have a look at the visual below.

I bet you see 18, right?

Here’s the fun part.

How many different ways can you write a numerical expression to represent those 18 seats. I’m going to guess that you all can come up with at least these two:

9 + 9

and

2 x 9

An assumption I’ll go with in this post is that “2 x 9” is read “2 groups of 9”. However, there are other interpretations that would match a different visual.

How many others can you come up with?

While this is a fun activity to give students practice writing expressions, the most important element here is the concrete representation (if you were using square tiles) or the visual representation (say images of the seats as we are doing here).

There are just a few of the many representations you could come up with:

By using concrete manipulatives like square tiles for this activity and allowing students to progress towards drawing visual representations when they are comfortable and able, we can give students the opportunity to build a conceptual understanding of how mathematical expressions are created and make conjectures as to what generalizations can be made about simplifying them.

While my representations are based on the assumption that a single seat represents the whole, you could also explore other scenarios such as having the entire plane represent the whole for exploring expressions with fractions, decimals and percentages.

If you’ve been trying to find a way to make math more accessible for all students in your classroom, using concrete manipulatives and visual representations is definitely a great starting point.

I plan to come back to this idea on a regular basis, so be sure to stick around for that. In the meantime, you might consider exploring some of my previous posts related to visualizing mathematics.

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]]>Are you a school administrator or math leader looking for tools to support planning your professional learning in mathematics with your colleagues? Check out Knowledgehook's new PLC Planning Tool!

The post Knowledgehook Math PLC Planning Tool appeared first on Tap Into Teen Minds.

]]>Have you been using Knowledgehook’s Free Gameshow Tool or the Premium Mission Feature? I was pleased to find out today that Travis, Lambo, James, Arthur and the rest of the team at Knowledgehook have released version 1 of their **Professional Learning Communities (PLC) Tool** to assist district math leaders, administrators and school math leads to easily and more effectively plan their professional learning sessions.

Here’s a quick overview with some screenshots to give you a heads up on what you can expect from this really valuable resource.

Upon logging into Knowledgehook with your school and/or district login, you’ll arrive at the Administrator Dashboard where you will see a graph showing usage for Gameshows (teacher paced / whole group activities) and Missions (student paced / independent activities).

Underneath, we see the number of student gaps triggered, custom questions created by teachers and Mathalon medals earned thus far as well as the option to view Trending Gaps across the district:

My suggestion to the Knowledgehook team is to also show how many students have remediated the gaps in their learning after reattempting problems in Missions. I have also suggested that the dashboard “auto-magically” suggests which gaps should be on our “short list” to focus on for informing our professional development planning with some sort of confidence scale indicating how confident the algorithm is in their recommendation.

From the dashboard, we can also see the usage from each school as well as gaps triggered in those specific schools.

When we explore Top Student Gaps in the district, we can explore – grade by grade – the top 3 gaps:

Upon clicking on “View Teacher Support”, we can access instructional guidance including “Math Background” and “Remediation” documents which are downloadable PDF files:

When I clicked on “Math Background”, I got the following document which has a great background around Representing Whole Numbers to 10 000:

The 6-page math background document outlined:

- Why some schools struggle with representing whole numbers to 10 000
- A background of the base ten place value system
- Useful models that can help students develop number sense
- Strategies for comparing whole numbers (and decimals)
- Glossary

In the remediation document, you are presented with a 23-page document that highlights:

- Common misconceptions and what you can do about it
- Remediation questions and solutions
- A Teacher Guide
- Black Line Masters including blank and scaled number lines, place value charts and more.

Another new option includes the Administrator PLC Tool:

It is listed currently as “PLC Polls” where administrators can choose PLC content with teacher input (hence the poll) and they can optionally use their PLC Guide to organize how the learning will take place.

By Creating a Poll, we are actually creating a PLC Plan.

Here, we select all the different grades we are working with in the PLC. Let’s say it is a grade 4, grade 5 and grade 6 PLC. After checking those grades and clicking “NEXT”, the administrator can choose as many topics as he/she would like to offer for possible learning during this PLC cycle.

This school has been focusing primarily on fractions thus far, so the administrator decides to keep only topics related to fractions as options.

Then, we fill out some details about the PLC including a potential format in order to receive a structured guide for how we might organize the learning, select the date of the first PLC and add a note for teachers.

You’ll then get to preview the PLC Poll and CREATE IT!

Then, you’re provided with a link you can share with your staff members so they can share their thoughts on topics they would be interested in learning more about.

Alternatively, you can opt not to send out the link and just access the PLC content if you are comfortable selecting the topic or if the group has had a verbal conversation and you’ve all agreed on a specific topic.

Then, you can click on “VIEW” to see the poll results as well as access the content for your PLC:

You’ll be able to see who has voted and finalize which topics you’d like to explore. Note that you don’t need any votes in order to select the topics. Just hit “choose your final topics” to close the poll and select.

If I’d like to focus on comparing fractions visually and numerically, then I would select those topics and hit “Submit your Final Topics (2)”.

You are then provided with a sample email with links to content that teachers can access to bring to the PLC or even to read prior to the PLC.

On the PLC Polls page, I can click the 3 dots to the far right of the desired Poll and select “PLC Resources” to access the resources for that PLC and the facilitation guide.

Overall, I was really impressed with version 1 of the PLC Tool. It sounds like there will be many more features added as they continue developing it out further.

If you’re interested in trying out the FREE Knowledgehook Gameshow tool, click here.

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]]>This "sweet" 3 act math task asks you to first guess how many gummy worms are in a jar Estimation 180-style, then use a part-part-whole model to solve!

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]]>In this task, we will start with something for our **kindergarten to grade 3** friends focusing on early addition and subtraction with a **part-part-whole model**.

In the first task, we will take it Estimation 180-style to spark curiosity and build number sense through estimation. However, in the sequel, we will fuel sense making by diving into subtraction promoting visual models such as the part-part-whole model.

Enjoy!

Let’s break this task down into the 4-part math lesson model. In order to get started, we are going to introduce a task that is **contextual, visual and concrete**.

Show this video.

Then, ask students:

What do you notice?

What do you wonder?

Give students some time (maybe 60 seconds?) to do a rapid write on a piece of paper.

Then, ask students to share with their neighbours.

Then, allow students to share with the entire group.

A question that I’m sure you’ll hear from the notice and wonder portion of act 1 is the following:

How many gummy worms are there?

Let’s give them an opportunity to **make a prediction**!

Consider using Dan Meyer’s “too low” then “too high” strategy to help them come up with a more reasonable estimate. Let them chat with their neighbours and challenge them to a prediction duel.

After allowing students to share and writing them down on the whiteboard, let’s show them the act 3 video!

Celebrate the closest prediction in the way that you typically do in class. Also make a special note to congratulate some of the students who weren’t so close and ensure that they know that we are building our estimation skills through this process.

While it is great to do this Estimation 180-style task to spark student curiosity while also building their estimation skills, I’m always seeking out ways to extend tasks in order to **fuel sense making**.

Since we’ve already taken some time to set the context for this problem and student curiosity is already sparked, we have them in a perfect spot to help push their thinking further.

I also find that once students are already “into” the task, we don’t necessarily have to spend a ton of time building up the curiosity and anticipation that we did in the initial task.

Let’s give them an opportunity to inquire.

I might show them this video next where I put all of the gummy worms back into the jar and then I remove some.

The question we’ll try to figure out is:

How many gummy worms are left in the jar?

Alternatively, we could also ask:

How many gummy worms were taken from the jar?

Having students predict is always fun, but it might not be necessary to have them all share out as we did in the previous portion of the task. Play it by ear to see how into sharing they are at this stage.

To me, this portion of the task is the most important. I invest a lot of time sparking curiosity and making predictions in each of my lessons for the payoff of knowing I can dive into the sense making portion using the part-part-whole model for early addition and subtraction. The best part is, you could be using this model for the first time and introducing it in the consolidation of the task or revisiting if your students have already been actively using this model. The benefit of using a part-part-whole model for addition and subtraction is that they can quickly see that addition and subtraction are intrinsically related. They can also see that addition and subtraction word problems can be attacked by determining whether they have been given two parts or a part and the whole right from the start.

Assuming you’ve given students actual gummy worms or alternatively, concrete manipulatives (too much sugar!), you can walk around the room as they work to sequence how you’d like to make some connections using student work.

Here are a few animated gifs that might help when making these connections.

In the first, students might line up 25 gummy worms (or square tiles) and physically remove 8 before recounting.

Alternatively, if students have been exposed to a part-part-whole model, they might choose to go that route. If they haven’t, then you definitely want to include this model in the consolidation of the task as we press for understanding from conceptual to more procedural in nature.

After sharing out student solutions, you will want to consolidate the task as well as the key learning for the lesson. While there are many different consolidation possibilities depending on your grade level, student readiness and the time in the school year, I’d like to think that an anchor chart outlining the importance of a part-part-whole model would be a good possibility in this case.

If this is not the first time students have seen a part-part-whole model, you might consider using this task as a way to fuel sense making around the 4 types of addition and subtraction problems and maybe ask students to create their own problems to match each type:

- Join –
*I had 15 gummy worms in the jar and I added 10 more. How many gummy worms do I have altogether?* - Separate –
*I had 25 gummy worms in the jar. After removing some gummy worms, there were 10 left in the jar. How many did I remove?* - Part-Part-Whole –
*There are 25 gummy worms in the jar total. 8 are green the rest are red. How many are red?* - Compare –
*One jar has 25 gummy worms while another has 18 gummy worms. How many more gummy worms does the first jar have?*

If you haven’t read any of my previous posts that mention concreteness fading, be sure to give them a read.

You’ll notice that in the visuals I’ve posted above, I’ve shown visual representations of subtraction. It is actually really important that students have an opportunity to manipulate, experience and feel gummy worms or manipulatives that they can imagine are the gummy worms to build their fluency with numbers and operations.

In this case, students should have that concrete manipulative experience prior to having them draw the gummy worms visually on paper.

Then and only then should we move on to symbolic representations when they can build a visual in their mind of the math they are engaging in.

Hope you enjoyed the task! Let me know in the comments how it went!

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]]>Year in Review 2017: Looking Back and Planning Forward. This past year has been exciting with over 675,000 pageviews! Let's look at the top content accessed

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]]>What an amazing year 2017 has been!

For those of you who have been with me since the beginning, you likely remember this blog as a place to share my learning from a Teacher Learning and Leadership Program (TLLP) project where I went paperless in my math classroom.

Boy, have I learned a ton since 2012.

Over time, this blog has undergone a complete transformation focused on technology based content to specializing in supporting teachers interested in building their K-12 math content proficiency and pedagogical knowledge.

When I look back to the numbers from 2012, I remember being elated to see that I had just over 38,000 pageviews in that first year of blogging. At the end of 2017, I’m shocked to see that the website has had over 675,000 pageviews from educators like yourself all around the world!

Let’s take a quick look back to some of the most popular **3 act math tasks**, **blog posts** and **math visual videos** from this past year and I’ll leave you with my reflections from 2017 as well as **what you can expect for 2018**.

Donut Delight was originally crafted to provide an opportunity to spark curiosity around multiplication and fuel sense making around different strategies involving concrete manipulatives, visual representations and finally, standard algorithms.

This task has since expanded in both directions; stretching backwards to helping primary students who are working on early multiplication as well as pushing forwards to providing opportunities for students working on building a conceptual understanding of division and later, proportional reasoning.

This task was created to provide an opportunity to implicitly (then explicitly) introduce arrays when working with early multiplication, however many teachers are using it as a way to access students spatial reasoning skills by conceptually subitizing, using strategies such as skip counting and introducing the distributive property by “splitting the array”.

Teachers in later grades can use this task as they introduce order of operations as well as algebraic expressions and equations.

Gimme a Break is a task that was intended to give students a more contextual situation and visual experience to tackle the idea of operating on fractions. This problem starts with a really low floor by using unit fractions like 1 one fourth and builds to doing some multiplication and even division of fractions.

Be sure to give this one a look and modify to suit the needs of your students.

Earlier this year, I had been seeing a Facebook post going around showing how people in Japan supposedly multiply using sticks. The video made it appear as though this was some sort of magic trick. Anyone who reads this blog knows that I don’t like leaving people to believe that math is just a bunch of tricks, so I tried to figure out why it works.

Turns out, the reason it works is fairly obvious when we explore multiplication concretely using base 10 blocks as a starting point.

Spending so much time working with Kindergarten and primary math teachers has opened my eyes to how important early development of counting and quantity for our young children really is. Having come from the secondary world, I had no idea how complex the ideas behind counting and quantity really are.

This post summarizes 10 principles with visual animations that are really important for students to build their counting skills and their understanding of quantity in general. Give it a read.

The third most popular post on the blog is a continuation from where we leave off with unitizing in the counting and quantity post and building into early multiplication, through the standard algorithm as well as how we can leverage these important skills in grade 8 and 9.

Check it out!

In this video, we show the summary of a 3 act task called Cones and Spheres where we learn that a sphere can hold twice the volume of a cone with the same radius and height.

From there, we take the formula for volume of a cone and explore what happens when we double it and simplify. Definitely a fun way to go about introducing the Volume of a Sphere instead of just writing down the formula like I did (regretfully) for the majority of my career.

This video had over 74,000 views this year alone and has over 160,000 views total.

I really had a blast focusing most of my attention on K-8 mathematics over the past 18 months. From all of that learning, I tried to summarize it all in a 5-minute Ignite Speech.

Finally, another visualization where I try to help conceptually develop the formula for volume of a triangular prism. Nothing super fancy here, but it did receive over 11,000 views in 2017.

Not only did I prepare this Year in Review post to give you a quick summary of some of the big ideas shared over the past 365 days, but also to give me a reason to go back and analyze what content provided the most value for the most people. For those who know me personally, you’d probably agree that I have many ideas and I can sometimes struggle to decide what is most important to focus on.

While I’ve learned more than I can share in a single blog post about myself and my own mathematical journey, I’d like to mention a couple goals for this website for the year ahead.

With two growing children and a very demanding position as K-12 Math Consultant with my district, I have less time to commit to sharing online. Rather than posting less, I want to make it a priority to try avoiding some of the perfectionist qualities that create more stress and anxiety than productive content.

As I mentioned in a recent post, I have committed much of the past 5 years blogging focusing on sparking curiosity alone. I now know that a ton of effort can go wasted if we just get students interested in a problem without having a solid plan to help them build their conceptual understanding around the learning goal for that day.

Based on the top tasks, posts and videos from this year, it is clear that there is a thirst for more of this focus.

While my blog has over 250 posts, 52 of my own 3 act math tasks and a ton of Ontario specific course resources, I can see how it would be difficult for anyone to know where to start. I’d like to commit to creating a course that helps summarize all of what I’ve shared thus far and add the piles of content that sits dormant on my hard drive for more to benefit from. I’d love to hear your feedback on how I might be able to accomplish this moving forward to provide the most value possible for my colleagues in the math education community.

With these 3 goals in mind, I am excited to continue learning as we head into 2018 together!

Here’s wishing that you have a Happy New Year with your Family and Friends.

We’ll be in touch soon!

Kyle

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]]>Here's how I went from teaching my students lessons so they could do tasks to using tasks that teach students lessons about mathematics.

The post Using Tasks to Teach Lessons appeared first on Tap Into Teen Minds.

]]>Over the past 5 years, I’ve been exploring the use of Dan Meyer’s **3 Act Math Task** approach in my math classroom and share many of my own tasks when facilitating workshops. After participants experience these tasks in the role of the student, they quickly understand ** why** 3 act math tasks are useful. After their own curiosity is sparked, it would seem reasonable that this type of task protocol would also likely spark curiosity in their students as well. However, what is less obvious to teachers is

For some, they want to know ** when in the unit** they should use 3 act math tasks?

- Do I use a 3 act math task once a unit?
- Once a week?
- Everyday?

Then, after deciding when in the unit to use a 3 act math task, the question now becomes ** when in the lesson** should I use the task?

- Do we start with the task at the start of class?
- In the middle after I teach the lesson?
- At the end after I’ve given enough examples?

Because we are all human and humans prefer when decisions in life are black and white, you might be sad to learn that the answer to the above questions is almost always: “it depends”. Since all teachers are unique individuals – just like our students – that means we could (and probably should) have our own thoughts and beliefs around how our own perfect lesson might be delivered.

That said, my intention here is to share some of the ideas that have been developed collaboratively with Jon Orr over the past couple of years around how the structure of our math lessons have changed over time, often involving the use of a 3 act math-style task. Luckily for Jon and I, we were fortunate to have crossed paths at a time when we were both just starting to shift our practice from a largely ** teacher directed lesson** where we would teach at our students to what we might consider more of a

For the first 7 years or so of teaching, my lessons looked a whole lot like the way I remember math class from my K-12 experience:

**Take Up The Homework**: to ensure everybody “got it”**“Teach” the Lesson**: to give definitions, rules, formulae, procedures and algorithms**Give Examples**: in order to show the tips, tricks and common misconceptions**Assign Practice Problems**: to ensure they would be ready for the next day

I would spend hours each evening planning these lessons in order to feel like I was going to give my students the best chance at them succeeding with that concept the next day.

Know what happened?

Some kids still didn’t “get it”.

Don’t get me wrong, I always had a group of students who were well on their way, but I think they would have been fine regardless of how I delivered the lesson.

It was the group of students who “didn’t get it” that I was concerned about.

How could I reach them?

It wasn’t until I came across Dan Meyer and 3 act math tasks that I began shifting my thinking about how I delivered my math lesson. You’ll notice that in the previous sentence I intentionally avoided using the word “taught”, because I now know that I can’t “teach” my students math, but rather create the conditions where students can construct their understanding of the learning goal I have set out for them each day. This is where I see rich tasks like those using a 3 act math structure can be extremely helpful.

When I first began using 3 act math tasks, I thought that these tasks could only be used after I “taught” students everything they needed to solve the problem. In the first couple of years, this would have been at the end of a unit – maybe on review day – and I thought that I only had “time” to use 1 or 2 per unit of study. Despite the large amount of time and effort I put into seeking out these tasks, planning how I would “fit them in” and figuring out how to best deliver them in class, the response from students wasn’t much better than that of any old task I would typically use from the textbook.

They would be intrigued initially by the problem, but when it came to “doing the math”, the classroom vibe reverted back to the lethargic state we would sadly deem as normal. After the curiosity that was sparked during act 1 had fizzled, we were back to students who believed that they couldn’t get started without a significant amount of my guidance and scaffolding.

I know some of the reasons the tasks flopped had to do with poor delivery; I wasn’t very smooth due to my lack of practice and I was also missing some key elements like giving students time to notice and wonder. However, I now realize that the biggest problem I created in my math class was my pre-teaching of all the math throughout the unit and waiting to ask students to do any of thinking until review day. By then, students were lost in a sea of disjointed mathematical ideas, rules, formulae, steps and procedures that they hadn’t yet conceptualized because I hadn’t provided them with the opportunity to construct that understanding.

Now, I’ve come to realize that I can use rich tasks delivered in 3 acts to spark curiosity in order to fuel sense making around a new mathematical idea. Rather than pre-teaching all of the math, let’s use tasks to create a need for the math.

While I know this might seem really scary to some teachers (especially those who tend to teach in a similar fashion to how most of us learned), but I’m going to argue that we can teach math concepts through the use of a really interesting task that sparks curiosity and opens the door to fuel sense making as we attempt to connect prior knowledge to new learning.

The best part is that most 3 act math tasks can be used to fuel the sense making of many different mathematical ideas.

Consider the act 1 video from the Airplane Problem, for example:

Not only does the act 1 video of this task spark curiosity and generate some great discussion, but it also opens the door for getting at mathematical ideas including (but not limited to):

- subitizing,
- early multiplication using arrays,
- distributive property,
- order of operations,
- algebraic expressions,
- and many more!

By avoiding the urge to pre-teach all of the most efficient strategies we math teachers believe students should know how to do, we allow for students to use their prior knowledge as a way to help us assess where they are and where our teaching of the new learning goal should begin that day while consolidating the task. After the consolidation is when I can then shift into teacher directed mode if necessary to address any misconceptions, specifically target gaps in prior knowledge, and build on student solutions to press them for deeper understanding and more efficient and/or effective strategies.

So while I think the answer to “when should I use a 3 act math task” is still “it depends”, I truly believe we should at least consider using a 3 act math task approach to introducing new mathematical ideas as often as possible. For some, that might mean once a week, while for others that could be once a day; it really does depend. I might also recommend changing your definition of what a 3 act math task is and what makes them so great. For me, the key is finding ways to spark curiosity with tasks as a means to fuel sense making around a new mathematical idea. These really interesting tasks can definitely be your typical 3 act task with a great act 1 video and act 3 “solution” video, but I’m learning more and more each day that the video itself is not what makes them awesome.

It’s so much more than that.

Check back to the blog for more on this topic as we dive deeper in future posts.

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]]>Dive into a fractions task where we spark student curiosity asking them to predict how many pieces of different shapes it will take to Cover It Up!

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]]>Last year, our district focused our system wide math content learning on number sense and numeration including counting and quantity principles, composing and decomposing numbers, addition and subtraction as well as multiplication and division while exploring these concepts through a spatial perspective. This school year, we continue to our work in number sense and numeration by deepening our understanding of whole number as well as introducing fractions.

Thus far, much of our work has been grounded in the ideas in the Paying Attention to Fractions document, the recently released Fraction Learning Pathway and some well known books including Uncomplicating Fractions.

Recently, Heidi Horn-Olivito and I were at a Math Knowledge Network meeting in Toronto where we were lucky enough to bump into Beverly Caswell, co-author of Taking Shape. Heidi and I were planning a task for an Uncomplicating Fractions book talk and Beverly was kind enough to help us out during the brainstorming process. As a group, we came up with some great ideas and here’s how it might look in different classrooms.

Before this lesson, you will want to print and cut out the following materials:

You can download the printable PDF template here.

Hold up the small square and the big square and ask them:

What do you notice? What do you wonder?

After allowing students some time to generate some things they notice and some wonderings, allow them to share with their neighbours. Finally, select some students to share with the entire group.

As usual, we will acknowledge all that the students share.

Our first wonder we will go with is:

How many small squares do you think it will take to cover the big square?

Since my 5-year-old daughter Taliah found this activity in my bag, I figured I’d try it out on her and her 3-year-old brother, Landon.

Here’s a video of what they did.

I was actually really impressed with how Taliah used her spatial reasoning skills to determine how many of the small squares she needed to cover the big square!

At this point, it might be useful to consolidate with students that the 4 small squares required to cover the big square can be considered 4 parts of the whole. These 4 parts can be called “fourths”. While we might use the word “quarters” when referring to fourths quite often, I would suggest holding off on that word at least for now.

If your students have had exposure to whole numbers on a number line, then this could be a great spot to relate the area model we are working with to a linear model using a number line.

You will notice that I have intentionally avoided the use of standard notation (i.e.: 1/4, 2/4) of a fraction to highlight the unit fraction of “1 fourth”. You can read more about the importance of the unit fraction through the work of Cathy Bruce and the Fraction Learning Pathway. Helping students understand that when we are working with fractions, we are counting a certain “unit” or equal partition of the whole. In this case, we are counting fourths as “0 one fourths, 1 one fourths, 2 one fourths, 3 one fourths, …” and so on. If students can recognize early on that they are counting fractional pieces in the same manner they would count candies, cheerios, or books, it would seem logical that they could better connect the idea of fractions as simply a unit of measure.

Over time, you might gradually move from a unit of “one fourth” to “fourths”:

And when students are ready, we can make a smooth transition from writing the unit in words to standard fraction notation.

In Ontario, this transition happens in Grade 4.

Next, I asked the same question, but this time with triangular pieces that are half the size of the small squares:

How many triangles would it take to cover the big square?

Here’s a video of what Landon did initially.

So, not quite what I was hoping for, but he did manage to cover the entire square using a combination of 4 triangles and 1 square. I thought that was pretty telling as to the spatial reasoning he was using in order to cover up the whole.

And, as children often do when another is receiving attention, Taliah copied his design initially.

However, as you’ll see in this video, Taliah determined the number of triangles it would take to cover the square by only placing 1 on the big square. She then visualized how many other pieces she would need coming up with an answer of 8.

I attempted to help her make the connection that 4 pieces of a whole are called “fourths” and 8 pieces of a whole are called “eighths”, but when she bumped her triangles off the square, her attention was lost. Will have to try to come back to this in one of the later activities.

Although I’m not set on a specific order of the next few tasks, I thought that sticking with a smaller square would be best for them despite there being more partitions (16 total).

So, the question now is:

How many of the really small orange squares will it take to cover the big white square?

Here’s a video of what Taliah and Landon did.

As you saw in the video, Taliah again used only one of the parts to determine how many it would take to cover the whole.

We even managed to make the connection between a whole partitioned into 4 parts being called “fourths” and when we partition the whole into 16 parts, we call them “sixteenths”. You may have also caught her say a “oneth” – referencing what I assume was the big white square. Makes me wonder whether we want to maybe use a bit of slang calling wholes “oneths” and halves as “twoths” (prounounced “tooths”). Worth thinking about, I think.

I then asked her to prove it by handing her a big pile of small squares (sixteenths).

Something worth noting is that in the Ontario curriculum is the fractional pieces students are expected to work with in each grade level.

For example, the specific partitioning mentioned in the following grades are as follows:

- Grade 1 – halves; fourths or quarters
- Grade 2 – halves; fourths or quarters; eighths
- Grade 3 – halves; thirds; fourths or quarters;

*specifying the use of more than one fractional piece (e.g.: one half; three thirds; two fourths or two quarters)**without**using numbers in standard fractional notation - Grade 4 – halves; thirds; fourths or quarters; fifths; tenths;

*specifying the use of standard notation and the use of the word denominator to represent the number of partitions of the whole for the first time in the curriculum - Grade 5 – denominators of 2, 4, 5, 10, 20, 25, 50, and 100
- Grade 6 – denominators of 2, 4, 5, 10, 20, 25, 50, and 100

In Grade 7 and 8 there are no explicit references to which denominators should be used, which suggest that students should be working to become flexible with any denominator.

With this said, by no means does the curriculum suggest that we hold students back from pressing their thinking. For example, while my daughter is not expected to work with sixteenths for quite some time, it is to my advantage as a teacher to press her for understanding when she is ready. I can certainly assess her on her progress based on her own learning needs, but should not formally evaluate her via a mark, grade or report card on this specific learning.

Interestingly enough, when I gave Taliah the whole partitioned into rectangular pieces, her visualization / spatial reasoning skills let her down.

She initially believes that it will take 16 to cover the white square. So, I toss a pile of blue rectangles at her and let her get to work. Here’s a video showing what she came up with.

I thought this task was a great example of how important concrete materials come into play when we are discovering new ideas in mathematics. Taliah was using her visualization skills to help guide her thinking earlier. Now that the number of partitions are increasing and the shape of the partitions are becoming more difficult to visualize for her, she is forced to go back to the concrete manipulative (cutouts of the shapes) in order to guide her thinking.

It would be really easy for me to walk away from this task thinking my kids have done some great work with fractions (and I think I’d be right). However, we are constantly leaving money on the table in our classrooms and I almost left some here in this case.

Next, I wanted to see if Taliah was thinking in absolute or relative terms. If she was thinking in absolute terms, she might believe that the green squares were **ALWAYS** fourths and the small orange squares were **ALWAYS** sixteenths. If she is thinking in relative terms, then she would know that the name of each piece would change based on what we were comparing it to. This is super heavy, so I wasn’t really expecting much.

Boy, was I wrong. Check out the video here.

As you see in the video, she seems to be able to think in relative terms. Something that is so important for students as they develop their fractional, multiplicative and proportional reasoning skills over the next handful of years.

Check out all of the great samples of student thinking shared from around the web! Keep them coming!

First, Lisa Burke submitted a video of her son, Henry engaging in some pretty awesome fractional thinking:

Then, we had Ms. Cruickshank from John Campbell Public School submit a great video series of student thinking. Definitely worth the 9 minute watch:

Thanks to everyone for submitting student work samples. Would love to see more!

I find that teachers in junior and intermediate grades tend to want something a bit more “robust” and they might consider jumping straight to this image grabbed from Marian Small’s Uncomplicating Fractions book:

Many teachers I’ve worked with have found that students actually struggle quite a bit with this problem. Often times, what students miss in this particular visual is to define the whole. Some students will see the entire “big” square as the whole, while others might look at each fourth of the big square as a whole.

The good part is that as long as the student clearly articulates the whole they are referring to, the name of their fractional pieces can be different than another student in the class.

For example, if one student says the rectangular pieces are “thirds”, then they should also be able to articulate that we can visually see 4 wholes since you could fit 12 thirds over the entire image.

As you can imagine, playing with the idea of comparing fractions and equivalent fractions could easily be tied in as well as extending to operating on fractions.

Some other interesting visuals to use and address such as:

Even in older grades, it would be worth giving students the cut out of this visual so students can predict and then prove by folding and/or cutting. They’ll find out in different ways that each partition is a fourth.

Here is another to consider using:

Thinking about using this task in your classroom? Download the printable PDF below:

Would love to hear how you use these tasks in your classroom. Let us know in the comments!

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]]>Recently, Jon Orr and I received some descriptive feedback from James Francis from Knowledgehook after watching us co-present a workshop titled “Making Math Moments That Matter” at the GECDSB Math Symposium. After sharing some of the pieces he really enjoyed, he also shared some constructive criticism: What I personally didn’t enjoy was the really general […]

The post Why I Ask Students to Notice and Wonder appeared first on Tap Into Teen Minds.

]]>Recently, Jon Orr and I received some descriptive feedback from James Francis from Knowledgehook after watching us co-present a workshop titled “Making Math Moments That Matter” at the GECDSB Math Symposium. After sharing some of the pieces he really enjoyed, he also shared some constructive criticism:

What I personally didn’t enjoy was the really general “what did you wonder” questioning that I have experienced in other workshops as well. I feel like if a teacher asked me to notice and wonder, I would be annoyed knowing that it is very likely this task will have nothing to do with what I come up with, so why waste my energy? When people ask for your opinion and they don’t do anything with it, they might become resentful that you would even ask in the first place.

If the idea of “Notice and Wonder” is new to you, check out Annie Fetter from the Math Forum who has done a great job developing this idea and sharing it with the math world.

This isn’t the first time I’ve had workshop participants question the utility of asking students to notice and wonder. Sometimes, I can see a few eyes roll and every now and again I come across some who are reluctant to participate in this portion of the task. However, I feel that this portion of the lesson can often make or break a task. Let’s explore why.

If you’ve ever been to workshops led by Jon and I, we make a significant effort to get participants talking as much as possible in non-threatening situations just as we would when working with students. For example, in this past workshop, we asked the group to think about memorable moments in their lives and the math moments they remember from their educational experience as a student in order to share with the group. Taking the time early on in a math lesson for students to talk and share their thoughts where the stakes are low can be helpful to build trust and confidence, while also showing them that we value their voice regardless of their ranking in the invisible – yet very apparent – math class hierarchy. A well led notice and wonder discussion can really go a long way to creating a classroom of discourse that will hopefully over time, develop into mathematical discourse.

Not only does asking students to notice and wonder give them an opportunity to have a non-threatening discussion with their peers, but it also helps to feed their natural curious mind. I will never forget the first couple of years attempting to use Dan Meyer-style 3 act math tasks in my classroom and how often I felt like the lessons were a flop. What I eventually realized was that I didn’t take enough time to spark the curiosity in my students by developing the storyline of the problem. After taking much time to reflect on what my lessons were missing, I realized that I wasn’t giving my students a reason to get excited about the task or give an opportunity to engage in any thinking until they were ready to actually solve the problem. They knew that I was going to show some sort of video or photo and I would then tell them what to do next. When we ask students to notice and wonder, we are asking them to think, discuss and share their thinking which builds more interest and anticipation for more. And while the teacher should always have a specific direction in mind for where the learning will lead, we can still make each student feel like a contributor to the class discussion and the direction of their learning by writing down their noticings and wonderings for possible extensions and for future lessons.

That said, asking students to notice and wonder isn’t something all students will enjoy at first. Some have said they “feel silly” or that “this is stupid” likely because they aren’t accustomed to being involved in the development of a problem and thus, they aren’t quite sure what they are supposed to do. However, I think that this temporary struggle can be a good thing. One of the reasons I want students to notice and wonder when they think about mathematical situations is so they aren’t so dependent on me telling them everything they are supposed to see or do in math class. Over time, many students learn to enjoy the process however like in other areas of life, some may not. An observation I have made over time is that I often find that my “go-getter” students are the largest group of students who hold out the longest on the notice and wonder – much like workshop participants who dislike the process – because they just want to get to the point. However, if you were to watch a movie or read a book that jumps straight to the conclusion, you’d be pretty let down. We have come to expect that sort of uninspiring and emotionless experience in math class and it shouldn’t surprise me when students push back when I try to push them to get involved in the development of the problem.

Something Jon and I have discussed in the past is about how most teachers would likely fit in the “go getter” category since we were likely the students who understood the game of school and specifically, how to succeed in math class. It can be easy for us to believe that all students think and feel the same way we did in the math classroom. However, the reality is that many students do not feel as comfortable or confident as many of their teachers may have when they were in math class. When our experiences learning math are very different than that of many of the students in our classroom, it is easy for us to develop an unconscious bias. This might influence our thinking around whether or not there is a need to create non-threatening opportunities for students to talk and discuss in math class.

Interestingly enough, it is not uncommon for those who oppose the notice and wonder portion of a lesson to also become uncomfortable making predictions when required information is withheld. For example, if I ask a group to make a prediction about how many passenger seats there are in the plane below, some get anxious and a bit scared to throw out a prediction that may be way off despite the fact that they don’t have enough information.

While I don’t have a definitive answer as to why the high achievers in my class most commonly tried to side-step the notice, wonder and predicting portions of the lesson, my hypothesis is that this process may be perceived as a threat – either consciously or unconsciously – to their position in the math class hierarchy. By no means is it my intent to make any group of students feel uncomfortable, but I do believe that this protocol assists in the levelling of the playing field. By providing more opportunities for all students to participate and feel as though they have something valuable to offer the group, we are taking steps to remove the math class hierarchy and build a learning environment that is equitable for all students.

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