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.

The post Counting With Your Eyes: Subitizing 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**.

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.

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

]]>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.

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

]]>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!

The post Gummy Worms appeared first on Tap Into Teen Minds.

]]>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!

The post Gummy Worms appeared first on Tap Into Teen Minds.

]]>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

The post Year In Review: 2017 appeared first on Tap Into Teen Minds.

]]>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

The post Year In Review: 2017 appeared first on Tap Into Teen Minds.

]]>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.

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

]]>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!

The post Cover It Up! appeared first on Tap Into Teen Minds.

]]>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!

The post Cover It Up! 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 […]

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.

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

]]>Join our mission to engage 1,000,000 students around the globe in a joyous, uplifting mathematical experience with Exploding Dots!

The post Exploding Dots for Global Math Week appeared first on Tap Into Teen Minds.

]]>If you’ve ever checked out my 3 act math tasks, you will quickly notice that I am all about **sparking curiosity** in order to **fuel sense making** for our students. Well, Exploding Dots nails the curiosity AND sense making pieces of the puzzle and with Global Math Week coming up next week, what better time than now to jump right in!

It was only a few months ago when I really started exploring this idea called **Exploding Dots**, but I was immediately intrigued. Since then, I have fallen in love with this amazing story of mathematics that James Tanton and The Global Math Project are hoping to spread to over 1 Million students around the globe.

Why not join the rest of the world next week for a 15 minute exploration or days of math fun with **Exploding Dots**. Best of all, there are full technology, low technology and even no technology versions to enjoy! Registration is simple because it only involves a pledge of participation and you can do it in under a minute here.

Enough of me pitching this “joyous and uplifting mathematics experience for all.” Let’s get your feet wet!

DISCLAIMER: READING ON IS LIKE SPOILING THE END TO A MOVIE. I highly recommend just jumping into the first Exploding Dots activity (called an “Island”) instead of reading on. However, if you aren’t going to take the time to give it a shot, I’d rather you read in order to share this great experience with your students.

Watch this video.

In the “silent solution” style video, you are exposed to the “2-to-1” machine (written 1 <-- 2) and are left hanging at the end when they ask you to figure out how many dots would be required in the rightmost box to have a code of *10011*.

So. Give it some thought. Maybe draw it out?

It’s never too late to dive into Island 1 and see if playing with the “1 <– 2” machine can help construct some understanding AND I promise you’ll have fun doing it.

For those of you who still won’t go and just dive in to play with the Exploding Dots app, let’s have a closer look at the first island, Mechania.

I’d like to share a quick video at the beginning of the first Island, called “Mechania” to assist you with your exploration.

After watching the video, the **Exploding Dots** app has you jump into the action asking you questions that press you for understanding. It might seem tricky at first, but remember: this activity isn’t about finding the right answer; it is about sparking curiosity, using strategic competence and building a productive disposition towards mathematics.

Here’s a look at the first two questions:

Pretty simple, right?

The Exploding Dots app starts with a very low floor and has an extremely high ceiling.

Here are the next two questions to explore with the 1 <– 2 machine:

Before you know it, you’ll be tacking problems like this one:

Wait a second. Didn’t the question posed in the video at the beginning of this post look an awful lot like what we just did in the previous question?

Here it is again:

Do you think you can try it now? Go ahead. Take your time.

Heck, why not try the 1 <-- 2 machine in the Exploding Dots app to help you?

Here’s a strategy someone might use to answer the question:

Can you find a more efficient way?

Aside from the fact that curiosity is surging, strategic competence is spewing and productive disposition is oozing at all time levels, you *could* consider exploring how the 1 <– 2 machine helps computers work. While some may have picked up that the 1 <– 2 machine is the same language that a computer uses, many others may not. The base 2 number system – a series of “on” and “off” digits – is extremely important in computer science. It is the language computers speak!

While you might think the fun is over, it’s just begun as you continue through this activity to explore the 1 <-- 3 machine as well as the 1 <-- 10 machine. Something you may notice rather quickly with the 1 <– 10 machine is that it is the “base 10” number system; the standard number system we use in our everyday lives.

Once you arrive at the Exploding Dots landing page, you’ll notice that there are quite a few activities (called “Islands”) for you to explore. I believe the best experience would involve students diving in from the beginning, however it might be useful for you to see where students will eventually land as they traverse through each of the Islands.

Since we explored the first island, Mechania in detail above, let’s take a quick look at each of the remaining 5 islands that are active on the Exploding Dots website to expose you to the other mathematical connections that can be made from Kindergarten through Grade 12.

After students explore and discover in Mechania, students will be brought to Insighto to begin unpacking the conceptual pieces that make the different dot machines work.

After building some of the conceptual understanding from the Insighto Island, students will be brought to Arithmos to begin applying our understanding of the 1 <-- 10 machine to addition and later, multiplication. The logic students can build using the dot machines really helps them grasp an understanding of place value and why standard algorithms work.

Have a look at a sample of addition:

And later on, multiplication:

As students work through the Island of Arithmos, they will eventually arrive at commonly used or “standard” algorithms for addition and multiplication wrapped up with a nice bow of conceptual understanding.

What? Integers before subtraction?

YES! When students arrive on the Island of Antidotia, they will immediately be introduced to “dots” and “antidots” which intuitively build an understanding of integers prior to looking at subtraction. This is because the concrete and visual representations for both will look identical.

Students will eventually be led to standard algorithms we commonly see in elementary school for subtraction while also building an understanding of integers and the zero principle.

Get ready to look at division in a brand new light. Imagine possibly getting to a point where you could pretty accurately divide large numbers by focusing just on digits in each place value column.

While I love the entire exploding dots experience, I think the conceptual understanding this particular activity builds in students around division alone is worth the time and effort!

Ok, secondary math teachers. This is the island you’ve been waiting for. This is the reason why you put in the time and effort with your senior secondary math students with the Exploding Dots experience.

Have you ever imagined multiplying or dividing in any base in a way that was not only procedurally possible, but also conceptually understandable for students?

Well, here’s your opportunity. Students can play with big, long, BEAUTIFUL polynomials and they can quickly discover how to divide, handle remainders including The Remainder Theorem and multiplication of polynomials.

Ready to explore?

If this blog post hasn’t inspired you to join us during Global Math Week with Exploding Dots, then clearly I have done you wrong.

Take the time to register your class and spread the word with your colleagues.

Together, we can reach over 1,000,000 students!

Or, 0111011100110101100101000000000 students if we use the 1 <– 2 machine.

Or, 2120200200021010001 students if we use the 1 <– 3 machine.

Or, 323212230220000 students if we use the 1 <– 4 machine.

Or…

Ok, I think you get the point.

The post Exploding Dots for Global Math Week appeared first on Tap Into Teen Minds.

]]>Learn the Progression of Division where we will explore fair sharing, arrays, area models, flexible division, the long division algorithm and algebra.

The post The Progression of Division appeared first on Tap Into Teen Minds.

]]>Over the past school year, I have had an opportunity to work with a great number of K to 8 teachers in my district with a focus on number sense and numeration. As a secondary math teacher turned K-12 math consultant, I’ve had to spend a significant amount of time tearing apart key number sense topics including the operations. While I often hear teachers concerned about multiplication skills of their students, an operation that doesn’t come up too often in discussion is **division**. However, what I found interesting this year was how much of a struggle it was for teachers to attempt representing division from a conceptual standpoint instead of simply relying on a procedure. After spending quite some time diving into division independently as well as collaboratively with educators through workshops, I will attempt to share what I believe to be some pretty important pieces along the **progression of division**.

Disclaimer:

This is by no means a complete progression and would welcome other pieces in the comments that I could add in to build on this post over time. I have found thinking about these pieces as pivotal in my own understanding of how division is constructed over time, but will likely continue changing as my own understanding deepens.

I would recommend first exploring the progression of multiplication prior to jumping into this post focusing on the opposite operation, division.

Before we begin diving into division, I feel it is important for students to be very efficient with unitizing which I discuss in a separate post with counting principles.

To summarize, **unitizing** is:

Understanding that every quantity we measure is relative to another pre-measured group we call a unit. For example, our base ten place value system.

Before students can successfully unitize, they must be able to count via one-to-one correspondence. For example, a student successfully counting a group of items, one at a time.

After learning **one-to-one correspondence** and working on other principles of counting and quantity, teachers can begin encouraging students to skip count by 2s, 3s, and so on. This might be considered the beginnings of having students unitize implicitly. For example, by counting by 2s: 2, 4, 6, 8, and so on, students are counting up by a group larger than 1. Over time, students can begin counting the groups of 2 (or whatever unit they are skip counting by) with their fingers to really bring out unitizing explicitly.

Or, maybe in groups of 3 (i.e.: 3-to-1 correspondance):

This ability to create equal groups and keep track of the count is important for students to really begin their journey towards **division**.

Once students are able to count groups of 10, they are not only well on their way down the progression of division, but also on their journey to conceptually understanding our base ten place value system.

Something that comes quite natural to young children is the ability to **fair share** a group of items. For example, sharing a handful of candies between siblings isn’t something that is typically taught explicitly, but rather students develop this sense of fairness through play.

Prior to attempting to formalize division as an operator, students should have extensive experience fair sharing items amongst friends, both concretely (by sharing to real people like their classroom peers) and when ready, visually/pictorially (by sharing to groups organized on their desk, on paper or on a whiteboard).

An example of such fair sharing is given in the Ontario Grade 2 Mathematics Curriculum in the Number Sense and Numeration strand:

represent and explain, through investigation using concrete materials and drawings, division as the sharing of a quantity equally (e.g., “I can share 12 carrot sticks equally among 4 friends by giving each person 3 carrot sticks.”);

While most students will likely fair share the carrots through **one-to-one correspondence** (i.e.: grabbing one carrot at a time and giving it to one of the friends), something to note here is that they are engaging in an early form of **repeated subtraction**. In other words, the student is repeatedly subtracting one carrot from the total and then will count how many each friend has once they run out of carrots to share.

As students become more comfortable with the idea of fair sharing, they may begin unitizing the amount of carrots that they share fairly. For example, in this same situation, an experienced student may notice the large group of carrots and begin to share two carrots at a time. You can understand why skip counting backwards can be helpful in completing this task.

Once students are comfortable with fair sharing using repeated subtraction in units of 1 or more, we can begin formalizing this idea as an operation we call **division**.

In the Ontario curriculum, we begin formalizing division in Grade 3 using tools and strategies up to 49 ÷ 7:

And extend our work with division in Grade 4 to using tools and mental math strategies:

Something that is not stated explicitly in the Ontario mathematics curriculum and can easily be overlooked by elementary math teachers is that there are two types of division: **partitive** and **quotative**.

The first type of division we will explore is called **partitive division**. This type of division is possibly the first type of division students intuitively experience when they are young by sharing a group of items with their friends as we did earlier by sharing 12 carrots amongst 4 friends. In other words, partitive division occurs when a scenario requires a student to divide a set of items into a given number of groups, where the number of items in each group is unknown.

To give another example, we could look to the following question from Alex Lawson’s book, What to Look For, where a student is asked to answer the following question:

You buy 15 goldfish. You are going to put them in 5 jars evenly. How many goldfish will you put in each jar?

Since the student knows how many jars he must divide the 15 fish into, students might fair share by units of 1 or more at a time until all the fish are shared.

After viewing the visualization above of the student distributing the goldfish by assigning one fish to each bowl at a time, it becomes more obvious as to why we also call partitive division “**fair sharing**”.

The second type of division we will explore is commonly known as **quotative division**. This type of division occurs when a scenario requires a student to divide a set of items into groups with a given amount in each group, where the number of groups is unknown.

To rearrange the previous question as an example of quotative, we might ask the student this question:

You buy 15 goldfish. You are going to put them in jars, with 3 in each jar. How many jars will you need?

Since a **quotative division** problem tells the student how many items should be in each group, it would seem reasonable to assume that the student would unitize the 15 goldfish into units of 3 until all of the goldfish have been used and then count the number of groups created. When a student completes this problem, possibly using cubes or square tiles to represent each of the goldfish, they are implicitly using repeated subtraction to take away 3 goldfish at a time from the total of 15.

The visualization here shows the student **measuring** groups of 3 fish and then **repeatedly subtracting** those groups from the set of 15 fish until there are none left. Hence the names “**measured**” and “**subtracting**” division.

In summary:

- Number of groups is known; and,
- Number in each group is unknown.

- Number in each group is known; and,
- Number of groups is unknown.

While I personally am less concerned about teachers being able to name these two types as quotative and partitive, it is important that teachers are aware that two types exist to ensure that they are exposing their students to contexts that address both of these types.

When using division without context like fair sharing carrots or placing goldfish in jars evenly, the student can decide whether to use a partitive or quotative strategy based on what they are most comfortable with. While convenient, a potential pitfall may arise if students divide procedurally using only one type of division while ignorant to the fact that another type of division exists.

For example, if a student were to model *56 ÷ 8* using square tiles, a student could approach this using **partitive division** by fair sharing the 56 tiles evenly into 8 groups like this:

Or, the student could choose to model this problem by using **quotation division** by repeatedly measuring out groups of 8 and subtracting them from the set of 56:

If students are given large quantities of straight calculation problems and only few contextual problems like the goldfish problem shared above, they may not build the necessary fluency to approach contextual problems successfully when they do arise.

You’ll probably notice that regardless of which type of division students are using, they often make circular piles of the item they are working with. I believe this to be an important stage in the progression as it seems fairly intuitive to simply create your groups without ordering or organizing the items in each group.

As the dividends and divisors that students work with get larger, it can be helpful to think about organizing the items to help promote unitizing as well as building conceptual understanding and procedural fluency of division using **arrays**.

For example, if we look at an extreme case of dividing *120 ÷ 8* using individual square tiles, we can still identify partitive and quotative division:

While I don’t want students spending too much time trying to divide large dividends using square tiles, it **could** assist in showing students why base ten blocks are helpful.

Just like with multiplication, I think students should have a significant amount of experience dividing with arrays up to 81 ÷ 9 if you hope to help them conceptually understand division with dividends greater than 100.

Not sure what is going on in the above image? Be sure to check out the progression of multiplication before going on where we explicitly address arrays and area models which we will be diving into pretty quickly here.

It doesn’t take long for students to become annoyed by trying to use square tiles to represent division with large dividends. Thankfully, as we saw in our Progression of Multiplication post, we can turn to base ten blocks to cut the hassle.

For example, if a student wants to model 120 ÷ 12, they could use 1 hundred flat and 2 ten rods to represent 120 and 1 ten rod and 2 units to represent 12. The representation would be similar to multiplication, except in this case, you have one factor (12) and the result (120) and you must find the unknown factor:

It is easy to see the benefit of using base ten blocks when dividends get large. Instead of using 120 individual square tiles, we can use 3 base ten blocks to represent the same dividend. I intentionally selected a fairly easy example to begin with in order to allow students the opportunity to become familiar with dividing with concrete materials. While it might not seem obvious at first, dividing becomes increasingly difficult when more challenging examples are attempted.

Let’s try another example like 112 ÷ 8.

Your dividend will consist of 1 hundred flat, 1 ten rod, and 2 units, while your divisor consists of 8 units.

As students are given opportunities to explore and discover, they will make observations and I would encourage them to discuss these observations with their peers to come up with rules. For example, some students come to realize that you must convert hundred flats to ten rods when working with a divisor with a value less than 10. Other students may come to realize that since multiplying two numbers yields a rectangle, that we must continue “trading down” the base ten materials until you can make a rectangle with the dividend with one factor being the divisor.

This experience might look something like this:

While I will attempt to be as clear as possible in this post, it is important to note that students will need a significant amount of experience at each stage of the progression. Do not attempt to rush or you will find both you and your students frustrated.

Due to a shortage of concrete base ten materials or to make life a bit easier, students may begin gravitating to digital base ten block manipulatives. While it might seem easier to just jump straight to digital manipulatives, I don’t recommend rushing to this stage as students should really have the opportunity to physically manipulate the base ten materials prior to moving to a digital alternative.

When students have had significant experience manipulating physical base ten materials, you might consider using the Number Pieces iOS app or the web-based version from Math Learning Center.

While using a digital manipulative can be more efficient for a student who is experienced using physical base ten blocks to model division with 3 or more digit dividends, it can be a huge hinderance for students who have not been given an opportunity to build their conceptual understanding in this area.

When I deliver workshops specific to the progression of division, I find teachers quickly jump to the conclusion that dividing with base ten blocks is simply too difficult and unnecessary.

However, I believe quite the opposite.

I agree that using base ten blocks is definitely not the most efficient method a student should use when trying to divide two numbers, but they are very useful for building a conceptual understanding of division as well as a unique opportunity to build strategic competence by problem solving their way to a solution. Less obvious is the experience students are gaining around conversions when they trade in a hundred flat for 10 ten rods, a ten rod for 10 units as well as implicitly building the foundation for factoring quadratics – a grade 10 concept here in Ontario – all the way down in grade 5.

Let’s check out another example: 189 ÷ 9

Check out the conversions from hundreds, to tens, to units.

We could go ahead and rush to the **long division algorithm**, but why would we want to rob students of the opportunity to look at mathematics as a puzzle waiting to be solved?

Still want more practice?

Try 221 ÷ 13 using base ten blocks or Number Pieces app.

Once you’re done, see if your result looks something like this.

Like anything we do in mathematics to build conceptual understanding, we don’t want the learning to stop there. Ultimately, we hope that the rich learning experiences we offer our students will begin to solidify by creating procedural fluency and automaticity. The next step on our way to the long division algorithm is the **area model**. This model is very useful when dividing and helps set us up for a clear connection to the long division algorithm, which is our end goal for this progression.

If we think about the distributive property from our multiplication post, you’ll remember that we could use this property by “splitting the array”. We can do much the same with division and it is convenient to do so by using open area models. In other words, rather than using square tiles or base ten material to represent an array for multiplication or division, we will use a not-to-scale rectangle in its place. A little less precise, but still gives us the same visual that an array can offer leaving our minds to visualize the rest.

Let’s have a look at *195 ÷ 15* using an **area model** to represent division.

When drawing an area model for division, students are able to essentially unitize their own “chunks” to partitively (divide x items into 15 groups) or quotatively (divide into x groups of 15) approach this division problem.

In this case, I’ll give an example of a student who decides to approach this quotatively by repeatedly subtracting groups of 15 from 195 until running out of items:

Wow, that isn’t a super efficient way to go about things. However, over time, your students may begin to notice more efficient approaches like the example below where a student notices that when there is 150 remaining, that is 10 times larger than 15.

If we’ve taken away 3 groups of 15, then another 10 groups of 15, we know that we’ve taken away 13 groups of 15 total.

Let’s try another one.

Have a look at a possible approach to solving 888 ÷ 24.

While I’m a huge advocate for using context in math class, I’ve kept things fairly contextless for the majority of this post. However, I want to explicitly show that whether there is context or not, these strategies to promote students conceptual understanding and procedural fluency with division are very helpful.

Let’s have a look at a problem with some context involving a pool.

A pool has a width of 14 m and an area of 700 metres-squared.

What is the length?

When using open area models, you’ll find students will quickly jump on the use of friendly numbers like multiples of 10, for example.

You may notice that when I’m using area models, I’m using a symbolic approach to keep track of the repeated addition. That method is used in a number of different places with different names. In Ontario, the Guides to Effective Instruction call that strategy “**Flexible Division**”, since it is very similar to the **long division algorithm**, but puts the power in the hands of the student to select how many groups of the divisor to subtract with each iteration.

Eventually, students can opt to skip drawing the **open area model** and using what looks to be the long division algorithm or a variation like flexible division in order to solve division problems without a calculator.

As students enter grade 9 and 10, they will be offered an opportunity to put the conceptual understanding and strategic competence they have been developing through division with arrays and area models to use.

When common factoring, students will encounter problems like this one:

A pool has a width of 3 metres and an area of 12 times a number.

What is the length?

It could be helpful for students to create themselves an open area model as we did in the previous example:

When moving from working with multiplication and division to algebra, we rename our concrete materials from **base ten blocks** to **algebra tiles**. Instead of units, ten rods and hundred flats, we use very similar tools, but call them units, x-rods and x-squared flats. It should be noted that a hundred flat could also be called a “10-squared” flat.

So, instead of using ten rods like we use with base ten blocks, a student may opt to draw in “x-rods” representing the missing number, x. While there are many ways to approach this problem conceptually, I often see students approaching this additively by adding groups of 3 x-rods until reaching an area of 12 x-rods total. Students might also keep track of their work by using repeated subtraction (or flexible division) to the right of their area model:

Recall our work using base ten blocks and arrays for division earlier in the progression. In our next example, we will look at a similar context using the dimensions and area of a pool to show how all that conceptual work back from grade 5 and 6 can be utilized in Grade 10 to factor both simple and complex trinomial quadratics.

In this example, we’ll explore the following:

A pool has an area of 3x^2 + 11x + 6.

What are the dimensions?

While many students in grade 10 struggle with the idea of factoring quadratics, they may not experience the same level of struggle if they have any experience multiplying and dividing with base ten blocks.

When factoring quadratics, students can simply grab the number of tiles that represent the quadratic they are factoring and then attempt to create a rectangle. Once they find a complete rectangle, they can quickly identify the factors that yield that area. If you can’t make a rectangle, then you know your quadratic cannot be factored with integer coefficients.

Let’s have a look:

Factoring quadratics with algebra tiles is actually much easier than dividing with base ten blocks due to the fact that you are not required to convert from hundred flats to ten rods and ten rods to unit tiles! Who would have thought that grade 5 was tougher than grade 10?

As I’ve mentioned in previous posts, there is definitely an argument for starting new ideas in mathematics with concrete manipulatives, slowly moving towards visual (or drawn) representations and finally to abstract representations that use symbols to represent the concrete. Although the name suggests that the concrete stage fades away over time, it is important to note that we should be returning to concrete manipulatives with every new layer of abstraction.

For example, when we first introduce division, we might be working with two digit dividends and single digit divisors. Concreteness fading for this idea might look something like this:

When progressing to three digit by one digit division, the stages of concreteness fading may look something like this:

Years later, when factoring complex trinomials in grade 10 academic math courses, the stages of concreteness fading might look something like this:

So while this progression of division may not be “the” progression, I certainly hope it shines some light on how important understanding division conceptually through the use of concreteness fading is for promoting the development of a complete understanding for our students.

Interested in checking out some 3 act math tasks that can be used in conjunction with the progression of multiplication and division? Be sure to check out the tasks below:

The post The Progression of Division appeared first on Tap Into Teen Minds.

]]>