Sticky Note Challenge 3 Act Math Task sparks student curiosity by asking how many sticky notes it will take to hold the weight of the man in the video.

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]]>This is a **3 Act Math Task** that focuses on ratio, rate and proportional reasoning related to a video shared with me recently on Twitter from Jerrold Wiebe:

Sticky Note Challenge @gfletchy & @MathletePearce, this would make a great 3act task! Lovin your work UMB EDUB 5220 https://t.co/lItnF2a49g

— Jerrold Wiebe (@Jerroldwiebe) May 18, 2017

In the video by Design Squad Global, a man sets up a Sticky Note Challenge involving the “shear” of the sticky notes to determine how many sticky note “slings” it would take for him to lift his own weight.

I immediately thought that could be a fun situation to mathematize in the classroom.

Show students the act 1 video.

Then ask students to do a rapid write of what they notice and what they wonder.

Students will then share out their noticings and wonderings while I jot their ideas down on the whiteboard.

While some great wonderings may arise, the first question we will address is:

How many sticky notes will it take to hold his weight?

Some of the best mathematical discourse can be had when asking students to make predictions without having enough information to know for certain. Extraneous factors such as how heavy you believe the person is, what brand of sticky notes we will use and how big each sticky note is can have students debating back and fourth over whose prediction seems most reasonable.

Once students have been given some time to think independently, discuss with neighbours and as a whole group, we’ll jot down some predictions.

Then, we’ll show students some more information in this video including how many sticky notes the man intends to try first as well as how much weight per sticky note sling that they will have to hold.

Now, you could have a class vote to see who believes that 10 slings consisting of 20 sticky notes (two sticky notes per sling) will hold him.

Show students the act 3 video to see whether 10 slings will hold his weight.

While I’m calling this an extension, I should probably call this part the true intention of this lesson. While we have already sparked curiosity with the video, now the real question I’d like students to ponder is:

How many sticky notes (or slings) will it take to hold your own weight?

This question might be too personal for students, especially for those sensitive to their weight. You might want to consider giving them set weights of items such as:

- Your own weight (if you’re comfortable with that)
- Weight of a dog
- etc.

If you (the teacher) is willing to participate in an experiment, you could have the class calculate how many sticky notes they believe you’ll need to hold your weight and then actually test it out!

As with all of the tasks I share out on my website, my intent is to help teach math concepts through rich tasks that allow students to engage in inquiry prior to making connections to prior knowledge and consolidating new learning goals.

These are just a few ideas of how you might use this task in your classroom. Please share your thoughts of how one might use this task in other creative ways by commenting below.

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

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]]>The EQAO Benchmark tool was added last week to Knowledgehook Gameshow and it is free for the rest of the 2016-17 school year! Find those gaps & address them

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]]>The Waterloo, Ontario based edtech startup “Knowledgehook” is at it again with a really cool new feature that is free for the remainder of this school year. With the EQAO Assessment of Mathematics Standardized Test just around the corner for grade 3, 6 and 9 students across the province of Ontario, I would have loved to know how my students stack up against students from those in prior years. Well, these guys have made that possible by taking the provincial data from previous EQAO tests and have created questions psychometrically valid for making comparisons.

Here’s what they had to say on their support page:

By using provincial data from prior EQAO exams, Knowledgehook’s EQAO Benchmark tool helps teachers easily identify the types of questions and areas that need special attention. By running our specially-made EQAO GameShows, teachers can measure how their own class is performing against provincial averages and be able to take action while there is still time.

Fast and informed EQAO preparation. Get started today!

I’m really curious to see what teachers think about this tool. I think there is a lot of potential here.

Let’s get started with some “how to” steps:

Create a free account and/or login at www.khmath.com.

Be sure you have already created a course for your students. If you have already done this, then move on to the next step.

Click on the **Gameshows** tab in the main navigation bar.

Click on the options button to the far right of the page. This should make the course selector dropdown menu appear.

Pull down the course menu to see all possible courses.

Select grade 3, 6 or 9 from the pull down menu in order to access EQAO Benchmark Questions.

Click on any EQAO pull down menu that appears in the list of mathematical strands for that course.

Click “View” in order to view/edit the questions in that gameshow. Click “Play” to play the questions as is. While you can edit the EQAO benchmark gameshows by adding/deleting questions as you see fit, note that if you actually edit the question itself, benchmark data will not appear for that particular question.

Select your options for the gameshow and then click “Invite Students”.

If you use gameshow often, your students will know what to do here.

If you plan to use Gameshow more than once, it is probably best to have them create an account and become students in your Gameshow “class”. However, if it is your first and only time, you might want to expedite the login process and have them login as a guest.

Click “Play” to begin.

All benchmark questions indicate which EQAO Year they came from and what strand they were related to.

This is one of the biggest features for me as a teacher. I love having students able to share their thinking with the rest of the group and we can really focus on multiple representations and ensuring students know that there are many ways to solve a problem with a high level of effectiveness.

This is great, especially around EQAO time for students to be able to explain their thinking and help others who may be a bit rusty on some ideas they haven’t played with for a while. Take the time and make it a learning experience!

Click Next to see the benchmark data.

Scroll down past the animated gif and other messages until you find the Benchmark data pull down menu.

How did your class do?

It is my understanding that this tool was created as a way for teachers and students to identify gaps in their understanding as they head for EQAO and the end of the school year. Hopefully, students will be able to see where they are weak comparatively to other students in the province in prior years and attempt to address those gaps with purpose and intentionality.

Get registered for a free teacher gameshow account!

Would love to hear how you found the tool. Was it helpful? Where might they improve the tool?

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]]>Set out on an adventure to discover early mathematics through context and interesting storylines to uncover the conceptual understandings of measurement.

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]]>Description

Zorbit’s Math Adventure apps actively engage students in math learning through entertaining storylines and a cast of charming space aliens. Based on rigorous educational research, the games integrate curriculum standards into gameplay and narrative to provide your students with a deep conceptual understanding of math.

The Zorbit games are created by an experienced team of teachers, educational consultants, academics, game developers, and experts in children’s entertainment.

KEY FEATURES:

+Real-time student assessment aligned to all major curricula.

+Memorable characters and reward systems motivate students to practice regularly and progress through their curriculum.

+Classroom activities that supplement the apps to provide a blended learning experience.

+Maximizes teacher instructional time, and makes lesson planning easier.

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]]>Set out on an adventure to discover early mathematics through context and interesting storylines to uncover the conceptual understandings of mathematics

The post Zorbit Kindergarten – Number Sense appeared first on Tap Into Teen Minds.

]]>Description

Zorbit’s Math Adventure apps actively engage students in math learning through entertaining storylines and a cast of charming space aliens. Based on rigorous educational research, the games integrate curriculum standards into gameplay and narrative to provide your students with a deep conceptual understanding of math.

The Zorbit games are created by an experienced team of teachers, educational consultants, academics, game developers, and experts in children’s entertainment.

KEY FEATURES:

+Real-time student assessment aligned to all major curricula.

+Memorable characters and reward systems motivate students to practice regularly and progress through their curriculum.

+Classroom activities that supplement the apps to provide a blended learning experience.

+Maximizes teacher instructional time, and makes lesson planning easier.

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]]>DragonBox BIG Numbers is the sequel to DragonBox Numbers where we now begin an actual gamified journey of discovering addition and subtraction without ignoring place value like so many do when using an algorithm. This app is great for kids aged 6 to 8, so let them loose for their next block of screen time fun!

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]]>Put your child in charge of building a wonderful world for the Nooms.

Learn how **big numbers** work through play and exploration.

Learn to perform long additions and subtractions.

Grow, gather and trade resources to unlock new worlds, and build houses, and shops.

***Perfect for kids between 6 and 9, or kids who have mastered DragonBox Numbers***

FEATURES

– An innovative interface that makes solving long additions and subtractions easy

– An infinite amount of additions and subtractions to solve.

– Over 10 hours of engaging gameplay

– No reading required

– 6 worlds to explore

– Learn to count in different languages

– 10 Different resources to collect and trade

– 4 Noom houses to furnish and decorate

– No third-party advertising

– No in-app purchases

DragonBox Big Numbers is based on the same pedagogical principles as the other games in the award-winning DragonBox series, and works by integrating the learning seamlessly into the gameplay, no quizzes or mindless repetitions. Every interaction in DragonBox Big Numbers is designed to heighten your child’s understanding of mathematics while motivating them to keep learning through play and exploration.

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]]>A great app from the creators of DragonBox Algebra, but this time for young children in Pre-K and Kindergarten to construct counting, quantity & magnitude.

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]]>Give your child the foundation they need for future math learning with DragonBox Numbers.

DragonBox Numbers will teach your child what numbers are, how they work, and what you can do with them. The game makes it easy and fun for your child to gain an intuitive understanding of numbers. It’s a great introduction to the wonderful world of math.

DragonBox Numbers brings math to life by turning numbers into colorful and relatable characters, called Nooms. The Nooms can be stacked, sliced, combined, sorted, compared and played with, any way your child pleases.

The app contains 4 different activites for your child to explore, each designed to challenge your child to use the Nooms and basic math in a different way.

The “Sandbox” section of the game is designed to let your child explore and experiment with the Nooms. It’s also the perfect tool for parents and teachers to explain basic math concepts to kids.

In the “Puzzle” section, your child will use basic math to create their own puzzle pieces, and place them in the right spot to reveal a hidden picture. Every move your child makes reinforces number sense. Your child will perform thousands of operations while solving the 250 puzzles.

In the “Ladder” section, your child will have to think strategically to build larger numbers. Your child will develop an intuitive understanding of how larger numbers relate to small numbers, and practice basic math strategies every step of the way.

In the “Run” section, your child will have to direct the Noom down a path using quick mental calculations. Your child can use their fingers, Nooms or numerals to jump over obstacles. This activity reinforces your child’s number sense and trains their ability to quickly recognise and add numbers.

DragonBox Numbers is based on the same pedagogical principles as the other games in the award-winning DragonBox series, and works by integrating the learning seamlessly into the gameplay, no quizzes or mindless repetitions. Every interaction in DragonBox Numbers is designed to heighten your child’s understanding of numbers and strengthen his or her love of math, giving your child a great foundation for future math learning.

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]]>My daughter uses Endless Alphabet as well as Endless Numbers as a way to not only have some fun - and educational - screen time.

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]]>I love this app for young children. My daughter uses Endless Alphabet as well as Endless Numbers as a way to not only have some fun – and educational – screen time.

Watch as your kids grab numbers on the screen and hear them moan their name (i.e.: “one… one… one… one…”) until you place it in the correct spot. Soon enough, your child will be doing simple addition and subtraction, while also doing some subitizing to boot!

From the team at **ORIGINATOR** – the creators of the beloved Endless Alphabet, Endless Reader, and Sesame Street’s Monster at the End of This Book…. Originator is a team of passionate artists and engineers dedicated to the best education+entertainment apps for kids.*

As a follow-up to Endless Alphabet, set the stage for early numeracy learning with Endless Numbers! Kids will have a blast learning number recognition, sequences, quantity, numerical patterns, and simple addition with the adorable Endless monsters. Each number features interactive sequences and equation puzzles with numbers that come alive, and a short animation that provides context and meaning to each number.

Features:

- 5 numbers are free to try with numbers in Number Packs available for purchase.
- Delightful animations reinforce number recognition, quantity, and counting.
- Interactive number puzzles reinforce basic numeracy skills.
- Endless Numbers was designed with your children in mind. There are no high scores, failures, limits or stress. Your children can interact with the app at their own pace.

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]]>OAME 2017 Speech: The Beauty of Elementary Mathematics. In this talk, I reference the complexity hidden deep within the elementary math curriculum.

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]]>Earlier today I had an opportunity to speak for 5 minutes during the OAME 2017 Conference during a session where each presenter has exactly 20 slides that advance automatically every 15 seconds for a total of 5 minutes to discuss an educational topic of their choice. While most presenters would agree that being asked to do a talk like this is an honour, it can also feel like a curse. The constraints of slides and time really do create a pressure that is difficult to match in other presentation formats. But, just like last year, I felt a sense of accomplishment after doing the talk.

Feel free to watch my talk or check out the slides and transcript below.

Recently, Mishaal Surti tweeted out this Photo and it made me reflect on my learning from this year in my new role as K-12 Math Consultant. For years, I could only picture myself teaching high school math.

Spending my first few years teaching senior courses, I had the opportunity to explore some really interesting concepts like combinatorics, compound interest and probability distributions. I thought that elementary math lacked the complexity to be interesting enough to teach with passion each day for a 30 year career.

Boy was I wrong. Recently I’ve been spending the majority of my time in elementary schools and learning from my colleagues about kindergarten, primary and junior mathematics.

Everybody knows that elementary educators teach the “basics” of math, right? I now realize that anyone who calls the fundamentals of mathematics “basic” is ignorant to just how complex this stuff really is.

Take counting, for example. Until just recently, I thought “learning to count” was as simple as reciting a list of numbers; kind of like learning your A,B,C’s.

Well my kindergarten aged daughter, who can seamlessly list the letters of the alphabet AND recite numbers from 1 to 70 without skipping a beat, has helped me to realize that while she can LIST numbers to 70, she is yet to learn how to COUNT to 70.

While learning letters and their corresponding sounds is a big feat, the complexity packed into the early mathematics of counting and quantity is extremely complex for young children. For example, when students learn the alphabet, there is no known reason for why they are ordered the way they are. Yet, the list of numbers when counting are ordered by magnitude – something that does not apply to letters.

For example, the letter Z has the same magnitude (or no magnitude) as the letter A. Yet, children – over time – must learn that the number 26 is much larger than the number 1; be it larger by 25 units or 26 times as big as, kids are tasked with constructing some deep conceptual understanding in the first handful of years of school.

Don’t even get me started on how much there is to help kids unpack when we begin unitizing by counting groups of objects in order to begin exploring place value and making the leap towards multiplication and division. Oh and what do you think of when you hear multiplication? Math facts!

All of this thick content to conceptualize in the minds of our little ones and yet elementary teachers are constantly being distracted and possibly influenced negatively by groups in the media pushing “math facts” and “back to basics” as the holy grail of mathematical proficiency.

I mean, don’t get me wrong – I think anyone with an interest in math education would love students to have automaticity of math facts – but I believe the real debate is over how to best help students arrive at this automaticity. If it is my belief that mathematics proficiency consists of memorizing math facts, rules and procedures without first building a conceptual understanding to underpin these ideas, then it is plausible that I may also believe that counting is as simple as singing the Alphabet song.

However, great elementary teachers know how their little students’ big brains work.

They know that they thrive on context and thirst for using real objects they can see and hold in their hands. In time, they might use manipulatives to represent objects from the context, like square-tiles to represent donuts, carrots, or goldfish.

Their experience and familiarity with the manipulatives grows so strong that students can then begin to make drawings on paper and create visualizations in their minds once they begin to feel that the manipulatives are too cumbersome and time-consuming. At this point, their teacher helps them formalize these ideas using symbols that represent these concrete and visual representations.

At this point, their teacher helps them formalize these ideas using symbols that represent these concrete and visual representations.

Some know this idea I speak of as “The concreteness fading model” – but this model is something that’s completely new to me and once again, highlights how wrong I was about the simplicity of elementary mathematics.

I mean, even Prince understands the concreteness fading model. Fans start by hearing some great music by Prince on record or live. Then, they go out and they buy posters, magazines and other visual representations of the music legend to bring back memories of the great music they’ve experienced.

Then and only then could that (point at screen) “symbol” actually mean anything to you. If you’re a prince fan, you’re probably feeling some tingles, goosebumps or thinking of your favourite song when you look at that symbol.

If you’re not a prince fan, then these 15 seconds are probably a complete waste of time – just like many math classes for so many students who stare blankly at a meaningless symbol.

So let’s steal these great ideas from our elementary colleagues and “BE MORE PRINCE” to give our secondary students an entry point into math…

by using relatable context so they can make the math concrete and visual before we introduce symbols and structure.

Although I still look forward to my opportunities to teach in secondary math classrooms, my math heart has forever grown to love and respect the great work that elementary teachers do to share the beauty and richness that is locked inside so many concepts in the K-8 Ontario Math Curriculum.

**End of transcript**.

Special thanks to Jon Orr for providing much needed feedback throughout the planning process of this talk as well as filming it for me.

Also be sure to check out Jon’s awesome talk as well as a great poem by Jimmy Pai.

Once others are available, I’ll add them to this post.

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]]>Have you ever seen someone share the Japanese multiplication method on social media? Often times, it looks like nothing more than a math trick. It's not!

The post Why Japanese Multiplication Works appeared first on Tap Into Teen Minds.

]]>Have you ever wondered why **Japanese multiplication** works?

I’ve heard some call it Chinese multiplication, multiplication from India, Vedic multiplication, stick multiplication, line multiplication and many more.

While many might argue as to the origin of this **multiplication trick**, I’m going to argue that it very well could have originated right here in Ontario, Canada, considering how our Ontario grade 1 to 8 math curriculum suggests we might go about teaching multiplication.

But just for the record, I really do have no idea where it came from and nor do I care. I do, however, REALLY care about WHY this method works. If you think it’s simply a trick, that’s because you’re likely considering this method from a procedural perspective alone.

Check out a full explanation in the video or jump to a written/visual summary underneath.

In order to understand how **Japanese multiplication** works, we must start back at the good old, reliable method of organizing equal groups in rows and columns. You’re right, I’m talking about an **array**:

When we say “3 times 2”, that is the same as saying “3 groups of 2” and we can show these three groups as 3 rows and 2 columns or 3 columns and two rows.

As numbers get bigger, like 6 groups of 7, it can often be helpful for students to show the number of groups and number of items in each group (also known as factors).

Note that the arrangement might look familiar as this is often how we traditionally organize our multiplication tables or multiplication charts.

We can also use arrays to practice skip counting, visualizing “doubles”, “triples” and all kinds of other useful skills that many “back to basics” advocates would love to see improved in our students.

While arrays are super cool, we’re not here to just discuss the benefits of using arrays when learning how to multiply. They can also help us understand why **Japanese multiplication** actually works.

We’ll get closer to the reason when we start looking at larger factors like 13 groups of 14. But man, it would really suck if we had to build an array of 13 rows and 14 columns with individual tiles!

Luckily, somebody out there thought of base 10 blocks to make building arrays with large factors easier!

If you really want to go deep with base ten blocks, consider reading this post.

To build an array of 13 groups of 14, we can use base ten blocks to represent 13 as a “10-rod” plus 3 “unit tiles”. This reduces the number of manipulative pieces from 13 pieces to represent the number 13 to only 4 pieces and from 14 pieces to represent the number 14 to only 5 pieces.

Now, we can multiply in parts, focusing first on our 10 rods.

Just like you would with a multiplication table, we can multiply 10 times 10 and see that the space the product occupies is 100. With base ten blocks, we can use a “100 flat” instead of 100 individual units, or 10 ten-rods.

Then, we can look at the empty space in the top right of our array and note that we now have to multiply 10 (from the factor of 13) by the remaining 4 units (from the factor of 14) to get 4 ten-rods, or 40.

Repeating the same logic for the remaining 3 units form the factor of 13, we then multiply 3 by the ten-rod to get 3 ten-rods or 30.

Finally, we multiply 3 units by 4 units to get 12 for a final product of 182.

So, let’s do one more, then make the connection to **Japanese multiplication**.

This time, we’ll look at 12 x 15. Notice that the same logic applies:

Fun stuff, right?

Now, let’s make the connection to the **Japanese multiplication method**.

I’m going to hide the values of the base 10 blocks in order to clean up the screen and get rid of the clutter. Now, I’m going to highlight the “gaps” between each base ten block piece with lines (see where this is going?):

Moving forward, I will separate our factors from the array a little bit more so we don’t get confused. As you’ll see below, the **Japanese multiplication** is simply skipping the step of drawing out the base 10 blocks by having you focus on the intersection of the base 10 blocks (or the sticks / lines). As you can see in the animated gif below, each base ten block is replaced by the intersection of the lines situated between each base ten block:

Each step can be broken down as follows:

- In the top left corner, we have a ten-rod multiplied by a ten-rod to give 100. Notice it is the intersection point of the two ten rods that represents the 100 flat.
- In the top right corner, we have 5 units multiplied with a ten-rod to give 5 ten-rods or 5 intersection points to represent 50.
- In the bottom left corner, we have a ten-rod multiplied by 2 units to give 2 ten-rods or 2 intersection points to represent 20.
- Finally, in the bottom right corner, we have 5 units multiplied by 2 units to give 10 units or 10 intersection points.

Looking at both the array with base ten blocks or Japanese multiplication, both methods are automatically chunking our factors of 12 and 15 to make use of the distributive property; 12 = 10 + 2 and 15 = 10 + 5.

Now that you’ve had a chance to experience using base ten blocks through this post or more in-depth here, you can probably visualize the base ten blocks sitting between the lines that are used in the **Japanese multiplication method**.

Pretty cool, eh?

I’ve seen a bunch of posts floating around social media suggesting that **Japanese multiplication** is a multiplication trick or some sort of “magic” or “voodo trick“. This statement is only true if you never seek out to understand why it works. While I’ve taught many math tricks such as cross multiplying for solving proportions and sum and product for factoring in the past, these past few years I have completely abandoned this approach from my teaching. I have to be careful here because I’m not suggesting that cross multiplication or sum and product are bad methods to use in math; it is more about when and how they come about in math class.

It is my belief that there is no such thing as a “trick” in math class when a deep conceptual understanding is constructed prior to introducing procedural fluency. In the case of solving proportions, students should be able to solve a proportion using opposite operations and their understanding that the equivalent relational quantities are multiples of each other. Understanding “how many times bigger” one “piece” of a fraction is than another is very important prior to simply giving students a tool like cross multiplication to simply “get to an answer” as fast as possible. I’d like to think that if students have built a deep conceptual understanding prior to moving towards procedures and algorithms, it is likely that they will better understand how to use the procedure efficiently and will also be able to get themselves out of a jam if troubles ever arise.

In the case of **Japanese multiplication**, I would argue that it is only a **multiplication trick** if you are teaching this method without students having had the opportunity to work with the conceptual underpinnings that make it work flawlessly. In particular, students should have the opportunity to spend a significant amount of time working with concrete materials like square tiles and base ten blocks to build arrays in order to build strong multiplication fluency prior to pushing students to an iconic or visual representation like drawing the base ten blocks or using a more abstract representation like drawing intersecting lines.

You might be asking yourself:

Why do I always see the lines in the Japanese multiplication method on a diagonal?

Well, that’s likely because the majority who are using and sharing the **Japanese multiplication method** may have no idea why it actually works. If I’m not too certain why it works and I’m trying to teach somebody else how to do it in a procedural fashion, I may need some assistance to organize the solution for both myself and the student.

By showing the lines diagonally, the base ten block array now organizes the intersection points in order of place value. Have a look below:

As you can see above, an opportunity to circle back to place value and the importance of understanding that in base ten, we cannot have any number greater than 9 in any place value column. You’ll notice that the 10 one’s must be swapped out for a ten rod.

So while many might consider this to be a pretty cool “trick”, it is much more powerful if students can articulate where procedures like these come from and why they work.

Better yet, after students have a thorough understanding of arrays with base ten blocks, I’d much rather challenge them to see if *they* could come up with an easier way to visually represent their two-digit multiplication on paper without having to draw a bunch of rectangles and squares. Some might use sticks for base ten blocks and maybe, just maybe, someone in your class might come up with something similar to this stick method. How cool would that be?

Oh, and before you go, you should know that using base ten blocks or the Japanese multiplication method is a great way to explain why partial products and the standard algorithm for multiplication works.

If we have a look at the array and the standard algorithm, side by side we can clearly see each step of the algorithm. Check it out:

If you’re interested in more about how arrays, area models and the standard algorithm connect, see this post.

Concreteness fading is a theory suggesting that mathematical concepts are best learned in three stages; the enactive stage, where students use concrete manipulatives that represent the mathematical concept they are working on.

Over time, after students have had enough experience physically working with the concrete manipulatives, they move to the iconic stage, where they begin to (often naturally) draw a visual representation of the concrete manipulative instead of having to physically hold and manipulate the object in their hands.

As students become increasingly comfortable with the iconic or visual representations, does it make sense for them to begin using symbols that represent the meaning behind the previous visual and concrete representations. This stage is thought to be the most abstract of the three stages because now numbers and symbols are used as a more efficient way to represent the work and experiences that have been developed in the previous stages.

So what does the multiplication we just explored today look like relative to the three stages of concreteness fading?

When it comes to single digit by single digit multiplication using individual unit tiles as we did at the beginning of this post, the stages might look like this:

- Enactive/Concrete: Physically arranging square tiles into an array.
- Iconic/Visual: Drawing squares or dots in an array on paper or using spatial reasoning to visualize the array in your “mind’s eye”.
- Symbolic/Abstract: Using numbers and symbols to represent your thinking, with the hope that you can visualize what those symbols mean in your mind.

As we move to two digit by one digit or two digit by two digit multiplication, the stages of concreteness fading *might* look like this:

- Enactive/Concrete: Using physical base ten blocks to create arrays and over time, possibly moving towards free virtual manipulatives like Number Pieces by the math learning centre, the Ontario Ministry of Education’s Mathies Colour Tiles app or the interactive manipulatives offered through the free Knowledgehook Gameshow tool.
- Iconic/Visual: Drawing the array using a base 10 configuration on paper and/or visualizing in their mind.
- Symbolic/Abstract: Connecting the concrete and visual to symbolic notation such as this “conceptual” multiplication algorithm (or “partial products”).

Another possibility might include different visual and symbolic representations such as this:

- Enactive/Concrete: Using physical base ten blocks to create arrays.
- Iconic/Visual: Drawing an area model on paper to show partial products and/or visualizing in their mind.
- Symbolic/Abstract: Connecting the concrete and visual to symbolic notation such as the standard algorithm for multiplication.

Finally, another possibility might be:

- Enactive/Concrete: Using physical base ten blocks to create arrays.
- Iconic/Visual: Drawing a modification of a base ten block array using the Japanese multiplication method on paper and/or visualizing in their mind.
- Symbolic/Abstract: Connecting the concrete and visual to symbolic notation by using mental math strategies like decomposing and re-composing numbers. In this case, mentally using the distributive property to multiply 10 by 15 and then 2 by 15.

While my initial intent with this video and post was a quick animation to show how the Japanese multiplication method really isn’t a trick, but rather a simplification of what we are asked to do in the Ontario mathematics curriculum, it blew up into a monster. I hope the time and effort spent at least has you thinking about how we might work to deepen our student understanding of multiplication in conjunction with Concreteness Fading.

I strongly believe that as we are exposed to more ways to represent concepts in mathematics, our understanding of those concepts will continue to deepen and produce more and more connections over time. I am living proof that this is true, because I am shocked routinely at the new connections that seem to present themselves to me with less and less effort with each passing day. Let’s all keep an open stance to learning and continue to build more and more connections in mathematics that we can leverage as tools in our classrooms to address student learning needs.

Do you know any other interesting ways to multiply? Please share some more (links welcome, also) in the comments for others to enjoy!

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]]>Check out this video of Krispy Kreme Donuts and that huge box! What do you notice? What do you wonder? Explore multiplication and division strategies!

The post Donut Delight appeared first on Tap Into Teen Minds.

]]>When one of our district math leads, Brennan Jones asked me to brainstorm some ways we could help his staff engage in some professional development around division and incorporate a 3 act math task into the learning, I immediately thought of some contexts where arrays, base ten blocks and area models could be used to help attack this concept. I was thinking about a box of Coca-Cola, jars of peppers and many other ideas prior to settling on the idea of donuts in a box. It was then that I remembered that Graham Fletcher, Mike Wiernicki, YummyMath and others had explored the Krispy Kreme Double Hundred Dozen box of doughnuts in the past (read about it here). However, these innovative mathletes had approached the problem from the angle of multiplication and possibly some extension questions related to proportional reasoning. Brennan and I thought that we might be able to take that idea and make some connections to division.

This **3 act math task** was designed with the idea of accessing student prior knowledge of multiplication and then connecting that knowledge to division including the use of open area models, repeated subtraction and then connecting these to a flexible division algorithm that is considered to be a more accessible algorithm for use by students with varying abilities.

*Kindergarten, grade 1 and grade 2 are not explicitly covered in this task, however it would be reasonable to modify this task to make useful for these levels as well.*

- Grade 3 – NS1 – divide whole objects and sets of objects into equal parts, and identify the parts using fractional names (e.g., one half; three thirds; two fourths or two quarters), without using numbers in standard fractional notation;
- Grade 3 – NS3 – relate multiplication of one-digit numbers and division by one-digit divisors to real-life situations, using a variety of tools and strategies (e.g., place objects in equal groups, use arrays, write repeated addition or subtraction sentences) (Sample problem: Give a real-life example of when you might need to know that 3 groups of 2 is 3 x 2.);

- Grade 4 – NS3 – multiply two-digit whole numbers by one-digit whole numbers, using a variety of tools (e.g., base ten materials or drawings of them, arrays), student-generated algorithms, and standard algorithms;
- Grade 4 – NS3 – divide two-digit whole numbers by one-digit whole numbers, using a variety of tools (e.g., concrete materials, drawings) and student-generated algorithms;
- Grade 5 – NS3 – multiply two-digit whole numbers by two-digit whole numbers, using estimation, student-generated algorithms, and standard algorithms;
- Grade 6 – NS1 – solve problems involving the multiplication and division of whole numbers (four-digit by two-digit), using a variety of tools (e.g., concrete materials, drawings, calculators) and strategies (e.g., estimation, algorithms);

- Grade 7 – NS3 – demonstrate an understanding of rate as a comparison, or ratio, of two measurements with different units (e.g., speed is a rate that compares distance to time and that can be expressed as kilometres per hour);
- Grade 8 – NS3 – identify and describe real-life situations involving two quantities that are directly proportional (e.g., the number of servings and the quantities in a recipe, mass and volume of a substance, circumference and diameter of a circle);
- Grade 9 Applied – NA1 – solve for the unknown value in a proportion, using a variety of methods (e.g., concrete materials, algebraic reasoning, equivalent ratios, constant of proportionality);
- Grade 9 Academic – NA2 – solve problems requiring the manipulation of expressions arising from applications of percent, ratio, rate, and proportion;;

Consider showing students the act 1 video from an episode of Fast Food Mania. The original, longer version is here.

Once the video is complete, show students this image if they are in an Ontario grade 5 classroom and ready for multiplying two-digit by two-digit numbers using a variety of strategies:

If students are in an Ontario grade 4 classroom or lower, you might consider using this image:

Then ask students to do a rapid write of what they notice and what they wonder.

Students will then share out their noticings and wonderings while I jot their ideas down on the whiteboard.

Some noticings and wonderings that have come up when I’ve used the task with the huge box of donuts include:

- How many donuts are in that box?
- How heavy is the box?
- Those people look really small.
- Is this a real picture?
- How many calories are in that box?

For those who wish to go there, you can check out this article that covers the event at which this donut box was prepared for.

While we may explore some other wonderings, the first question I intend to address is:

How many donuts are in that box?

With manipulatives and/or paper/whiteboards already out on their tables, I would then give students some time to make a prediction and discuss with their neighbours and/or group.

If you’re using the small box of donuts with your class, you might show this image for predictions instead.

After students have shared out their predictions, I would show them some information, depending on the group I’m working with.

If I want to use friendly numbers for students who are just beginning to multiply two-digit by two-digit numbers, I might use this image:

If students are ready for more of a challenge (and the actual dimensions of this box of Krispy Kreme doughnuts), I might use this image:

If students are in an Ontario grade 4 classroom or lower, you might give them a look at these images:

With these dimensions, you will offer students an opportunity to utilize multiplication strategies that might differ from student to student. Depending on where students are relative to concreteness fading, they may choose to concrete materials like base ten blocks; a visual representation such as drawing base ten blocks, an area model, or Japanese multiplication; or a symbolic representation such as partial products or the standard algorithm.

After students share out their strategies (see some exemplar strategies in “version 2” below), it is likely they will have arrived at a total of 600 donuts:

Students may look confused wondering how the answer could not be 600. As they discuss with their groups, I’ll proceed to show them this image which shows a zoomed in photo of the box that states: “DOUBLE HUNDRED DOZEN”.

Then, I let kids discuss and decide if they want to take some time to update their answers.

Hopefully, after consulting with their peers, they will arrive at an answer of 2,400 donuts.

If you used the actual dimensions, students should arrive at a final answer of 800 donuts.

Here are some possible tools and representations I’ve observed when using this task with students and teachers. Hopefully, you see some of them in your classroom. If not, what a great opportunity to expose students to new strategies and representations to broaden their mathematical perspective.

In this **multiplication strategy**, some students make the connection that there are 4 groups of 25 in 100. Therefore, with the base ten blocks on the table, they can quickly use skip counting or other multiplicative thinking strategies to determine how many groups of 100 are in 32 groups of 25.

In this **multiplication strategy**, students create a multiplication array consisting of rows and columns with the factors 25 and 32. This is a great strategy that progresses nicely to the standard algorithm for multiplication.

In this **multiplication strategy**, a connection can be made from a multiplication array using concrete manipulatives like base ten blocks to a visual representation, such as Japanese multiplication.

Follow this link to watch a full video explanation of Japanese multiplication.

In this **multiplication strategy**, we look at the visual Japanese multiplication representation and recognize that this method does not take care of place value for us. We have 10 units in the ones column and 19 groups of 10 in the tens column. Let’s take care of that. The animation below shows this.

In this **multiplication strategy**, we dive into the most common representation of Japanese multiplication where the multiplication is typically completed using sticks or lines without any connection to base ten blocks (where this method is derived from).

In this **multiplication strategy**, we move away from the concrete manipulative completely and work with a visual representation where students represent multiplication as a series of partial products. Students can decompose the factors into friendly “chunks” and then multiply to find the area of each section of the donut box.

In this **multiplication strategy**, we make a jump from a visual representation using an area model to a symbolic representation, the Standard Algorithm as we know it here in Ontario. You’ll quickly notice that base ten blocks, Japanese multiplication and area models all help support the conceptual understanding of the standard algorithm.

Check out my post on the Progression of Multiplication for more tools and representations that can be useful when working with multiplication in your classroom.

If you used the primary version of this problem, you’ll want to reveal the following so your students can get their moment of glory:

If you’re a primary teacher, jump to the primary sequel here.

The best part is that the fun has just begun.

If you chose **friendly numbers** for this task, I’ll show students this image:

If you chose the **actual dimensions** (less friendly numbers), I’d show students this image:

Next, I challenge them to determine how many layers of donuts their must be based on what they have done so far. Students already know that one layer is 600 (if friendly numbers were used) or 800 (if the actual dimensions were used), so now they must determine how many layers there are.

So while we know that the double hundred dozen box has 2,400 donuts in it, the number of layers will be different in the friendly number case and in the actual dimensions case.

If students are quite fluent with division and/or the long division algorithm, then the solutions will likely be less than fun to explore. However, if you hit students with this task before introducing long division, it could be a great way to build conceptual understanding using repeated subtraction and open area models:

In the following **division strategy**, I’ve attempted to show what students (and teachers) typically resort to when we think about physically dividing objects regardless of whether they are real objects (like donuts) or manipulatives being used to abstractly represent the real object. In the Ontario Grade 2 Math Curriculum, we explicitly use fair sharing as an introduction to division where students are asked to take groups of objects and split them into equal groups. Here’s what that might look like in a classroom where students use the same strategy, but use large units (or groups) for each round of sharing:

In both versions – using friendly numbers or the actual dimensions – we utilize division as repeated subtraction to help make a connection to a flexible division algorithm which can also be found in our Ontario Guide to Effective Instruction Number Sense and Numeration, Grades 4 to 6: Division resource on page 21. I’ve attempted to make a clearer connection between repeated subtraction using a visual representation and show how it can be connected directly to a symbolic representation that looks mighty close to the long division algorithm.

Here’s the version if you used friendly numbers:

Or, consider checking out the friendly number animated gif as a silent solution video here.

Here’s the version if you used the actual dimensions:

Next, we ask students:

If your school bought this box of doughnuts to split between 8 classes, how many would each class get?

If you used friendly numbers for this task, here are some possible strategies that students might consider including using an open area model and flexible division:

If you chose the actual dimensions in order to raise the floor on this task with less friendly numbers, the extension question will result in the same number of donuts for each of the 8 classes. However, some of the representations may look the same, while others may not.

For example, if a student chooses to use repeated subtraction, flexible division or the long division algorithm, both the process and result may look the same.

However, if the student approached the problem by visually dividing each layer into parts, the process may look significantly different than if a student were to use a similar approach in the friendly numbers case.

In the animated gif below, you’ll see an example where a student might attempt dividing each layer into smaller and smaller pieces until there are enough pieces to fair share with all 8 classes.

Show your students this image:

Allow them to make some predictions. Possibly show them this image as well:

After allowing them to use a strategy of their choice, you can reveal the following image:

Thanks to Chase Orton for sharing out the following extension image:

@MathletePearce Did you see what @tawny_malone @Sjmarshall415 and I have been geeking out on? How many donuts? They say 1836. Agree? Disagree? pic.twitter.com/dPigziaRmM

— chase orton (@mathgeek76) April 11, 2017

I’m immediately wondering some of these:

- How many donuts of each colour?
- What fraction of the flag is red? blue? white?
- What might the ratio / fraction be if you did this with your province / state flag?

The full article, Krispy Kreme Celebrates Texas Independence Day with 1836 Donuts Arranged into the Texas Flag can be found here.

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

The post Donut Delight appeared first on Tap Into Teen Minds.

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