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.

- 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.

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 this task 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?

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.

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:

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.

While many students will arrive at the answer of 600 donuts, I’ll then take a moment and 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.

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:

**Version #1: Friendly Number Animated Gif:**

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

**Version #2: Actual Dimensions Animated Gif:**

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.

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

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]]>Read about the author's work linking critical literacy to critical numeracy and how the work in literacy by Luke and Freebody can be applied to numeracy.

The post The Four Roles of the Numerate Learner appeared first on Tap Into Teen Minds.

]]>If one were to ask whether language literacy and mathematical literacy were alike, I would predict that the beliefs of the majority would be that they are very different. In The Four Roles of the Numerate Learner, authors Mary Fiore and Maria Luisa Lebar use the first lines of the Foreword to share their perspective of what teaching and learning means in any subject area:

“Teaching and learning in the twenty-first century – the now – are multi-faceted activities. We need students to become skilled critical thinkers, thoughtful problem solvers, and reflective communicators. To achieve this vision, educators strive to create a connected classroom culture that is built on trust and mutual respect, and where students are able to ask questions, pose problems, explore ideas, and make informed decisions. Building capacity for connectedness supports an environment that is empowering and engaging, where students are meaningfully involved, where relevance is key, and where their voice matters.”

This book is the result of the authors’ work linking critical literacy to critical numeracy and how the work in literacy by Luke and Freebody can be applied to numeracy.

The book begins with a chapter focused on developing an understanding of the Four Roles of the Numerate Learner including developing a new thinking framework in mathematics, why the framework matters, and the role of thinking in both literacy and mathematics.

Fiore and Lebar then dedicate a chapter to each of the four roles: Sense Maker, Skill User, Thought Communicator, and Critical Interpreter as well as a section in each of these chapters dedicated to explicitly connecting each of these roles to different grade levels in Ontario.

As I progressed through the book, I quickly made connections to much of the learning that we are engaging in at the GECDSB. For example, when reading about the role of Sense Maker, we learn that this involves conceptual and procedural understanding, using rich tasks to develop mathematical thinking and support through sense making, and what this might look like when we connect this work to learning goals and success criteria. A Skill User extends the procedural knowledge of operational skills, basic facts, rules and procedures, and the communication of that knowledge developed in the role of Sense Maker to procedural fluency where we include an understanding of why the procedures work. Other connections to our own district focus includes that of the Pedagogical System. Mathematical Discourse as well as Multiple Tools and Representations are referenced multiple times throughout the book to go along with Meaningful Rich Tasks as mentioned previously.

The Four Roles of the Numerate Learner is laced with many useful nuggets throughout. One such nugget was “The Elephant in the Room”, where the common incomplete debates about the role of basic math facts and the myths about the importance of having content knowledge specialists as math teachers. Both addressed very appropriately and meaningfully.

Overall, The Four Roles of the Numerate Learner is another excellent mathematics professional learning resource to add to your bookshelf as you continue to press for a deeper understanding of mathematics pedagogy.

Connect with Mary Fiore on Twitter.

Connect with Maria Luisa Lebar on Twitter.

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]]>Learn about why Number Talks Matter and access an 8 step routine that can be used to facilitate an effective number talk in your grade 4-10 math classroom.

The post Making Number Talks Matter appeared first on Tap Into Teen Minds.

]]>In my district, Number Talks: Helping Children Build Mental Math and Computation Strategies, Grades K-5 by Sherry Parrish has been a major resource for our primary and junior teachers to promote proficiency and fluency in number sense and numeration for the past five years. Now, Cathy Humphreys and Ruth Parker – who provided words of praise for Parrish in the front matter of her book – have released a book of their own with a focus on grades 4 to 10.

For those keeping track, Parker is one of the creators of Number Talks back in the early 1990s, and Humphries is known for her work extending them to intermediate and senior grades. Now, they have come together to continue the Number Talk love in the junior and intermediate grades on a quest to deepen student understanding and proficiency with number.

Jo Boaler writes in the foreword:

Teachers can watch a video of an expert teacher giving a Number Talk and believe that they are simple to teach, because they can appear deceptively easy. Yet when teachers begin to teach Number Talks themselves, many questions arise: What do they say when students share an incorrect solution, or when there is a mistake in their work? What do teachers do when students have no methods to share? How do teachers know the best problems to use for Number Talks? Where do they find examples of Number Talks for middle and high school students? These questions, and many more, are answered in this book.

In the first chapter, the authors explain what Number Talks are and why they are so important. They highlight some of the many student struggles in math and attribute them to how arithmetic is often taught as a set of rules and procedures to be remembered rather than concepts to be understood. With this, they are also quick to give respect to algorithms for their importance as reliable and efficient tools necessary to learn over time. However, they discourage a rush to the algorithm due to the “compactness” of algorithms that “hides the meaning and complexity of the steps involved” as quoted by the Hyman Bass work Computational Fluency, Algorithms, and Mathematical Proficiency, 2003. They give simple examples such as adding fractions, employing the subtraction algorithm, and even algebraic concepts such as expanding a perfect-square binomial.

The authors move on to presenting the reader with what is “as close to a recipe as you will find in this book” through an 8 step routine that can be used to facilitate an effective number talk in the classroom. With each step includes rationale and some key tips to remember, which are compact enough for teachers to keep in their daily lesson plan. Teachers are encouraged to begin using dot cards as a starting point as this is very non-threatening for teachers and students and can still provide an opportunity to share mathematical thinking through the use of numerical expressions incorporating different operators and their appropriate order. After providing teachers with an exemplar and a transcript to model the specific Number Talk, there are key ideas to promote successful use of Number Talks in the classroom.

The remainder of the book is split up into different strategies for different operators across grades 4 to 10 including addition, subtraction, multiplication, division, fractions, decimals and percent. Within each of these chapters, specific strategies are offered with exemplars and even suggestions for particular Number Talk questions that promote using one strategy over the other.

If I could provide any constructive criticism for this book, it would be around the use of visual (and concrete) manipulatives. While the use of arrays and area models is shown in some cases for multiplication and division for numerical and algebraic expressions and equations, I would argue that it would be very helpful for teachers to have more visual tools and representations at their disposal to assist students with visualizing (or spatializing) their understanding.

Overall, this is a great read, especially for those teachers who falsely believe that there is no place for Number Talks in their junior, intermediate or even senior level math classroom.

Connect with Ruth Parker on Twitter or at the Mathematics Education Collaborative (MEC).

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]]>Take a journey through a number of teacher stories about growing as a math teacher and learner, because Zager tells us: “good math teaching begins with us”.

The post Becoming the Math Teacher You Wish You Had appeared first on Tap Into Teen Minds.

]]>I just recently had the opportunity to read Tracy Johnston Zager’s new book, ** Becoming the Math Teacher You Wish You Had** and I’m so glad I did.

The Foreword begins with a reference to the author, Tracy Zager’s quote that “Good math teaching begins with us” and tells us that the book will take us on a journey through a number of different teacher stories about how they have grown as teachers and as math learners. There is a clear stance that mathematics is not black and white, nor based on speed, accuracy, or getting the answer correct, which builds nicely on the messaging Jo Boaler delivers in her book, Mathematical Mindsets.

The book also has a section dedicated to explicitly summarizing some of the many ways in which the book could be read, including:

- From start to finish
- In sections
- As a collection of standalone mini-books that can be read in any order
- On your own
- In a book study group or professional learning community
- With a colleague
- As part of a larger professional development course
- As a part of a math methods course

Similar to the explicit messaging about learning in math class: “there is no wrong way, as long as reading it is useful to you.” Although I read the book from start to finish, it was clear to me that I could have approached the book in any of the ways mentioned above, as the ideas are chunked nicely and do not depend on previous chapters.

The first chapter of the book, Breaking the Cycle, addresses the common beliefs and misconceptions many children and adults have developed based on their own negative experiences from math class. Zager goes on to help the reader better understand what mathematics really is. She includes transcripts of classroom discussions where her prompts help students describe what mathematicians actually do, then useful links to online resources are shared so that teachers can help expose the beauty in mathematics with their students.

Get ready for a rich learning experience as you read through subsequent chapters where Zager thoughtfully shares her thoughts and perspectives on risk taking; growth mindset; student voice; strategies and best practices; misguided approaches to precision; the Math-Twitter-Blog-o-Sphere (#MTBoS); low thresholds, high ceilings, and open middles; and many more great nuggets that you are sure to enjoy!

One of the most interesting pieces that I learned was where the Notice and Wonder approach commonly used in conjunction with a 3 act math task actually originated. My assumption was that it was born as a twin to 3 act math, but Zager cites a book by Max Ray-Riek from the Math Forum called Powerful Problem Solving: Activities for Sense Making with the Mathematical Proficiencies (2013, 42-55).

Who knew?

This is a definite on your “must read” math pedagogy book list, so pick it up and read it in any way you choose!

Buy your book here.

Connect with Tracy Johnston Zager through her website, on Twitter, or follow her book hashtag, #BecomingMath.

The post Becoming the Math Teacher You Wish You Had appeared first on Tap Into Teen Minds.

]]>Watch a video of a plane taking off and landing. What do you notice? What do you wonder? Take a trip from subitizing & unitizing to multiplication & algebra

The post Airplane Problem – Trip to Toronto appeared first on Tap Into Teen Minds.

]]>This **3 act math task** was designed specifically to have a very low floor in order to be useful from primary grades and a high ceiling with an opportunity for many extensions so the task can be used in junior and intermediate classrooms. While I believe this task can touch on many different specific expectations at many different grade levels, listed below are just a few possibilities from the Ontario curriculum.

Regardless of grade level, I recommend teachers show the task videos and and complete the initial “notice / wonder” to spark curiosity. After the initial task is completed, you may decide that jumping ahead to a grade appropriate extension is worthwhile.

- Grade 1 – NS1 – estimate the number of objects in a set, and check by counting;
- Grade 1 – NS1 – compose and decompose numbers up to 20 in a variety of ways, using concrete materials;
- Grade 2 – NS2 – compose and decompose two-digit numbers in a variety of ways, using concrete materials;
- Grade 2 – NS3 – solve problems involving the addition and subtraction of whole numbers to 18, using a variety of mental strategies;
- 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;
- Grade 3 – NS3 – multiply to 7 x 7 and divide to 49 ÷ 7, using a variety of mental strategies;

- Grade 4 – PA2 – identify, through investigation (e.g., by using sets of objects in arrays, by drawing area models), and use the distributive property of multiplication over addition to facilitate computation with whole numbers
- Grade 5 – PA2 – demonstrate, through investigation, an understanding of variables as changing quantities, given equations with letters or other symbols that describe relationships involving simple rates;
- Grade 6 – NS1 – estimate quantities using benchmarks of 10%, 25%, 50%, 75%, and 100%;

- Grade 7 – PA2 – model real-life relationships involving constant rates where the initial condition starts at 0 (e.g., speed, heart rate, billing rate), through investigation using tables of values and graphs;
- Grade 7 – PA2 – evaluate algebraic expressions by substituting natural numbers for the variables;
- Grade 8 – PA2 – evaluate algebraic expressions with up to three terms, by substituting fractions, decimals, and integers for the variables;
- Grade 9 Applied – NA2 – solve first-degree equations with non-fractional coefficients, using a variety of tools;
- Grade 9 Academic – NA2 – solve first-degree equations, including equations with fractional coefficients, using a variety of tools (e.g., computer algebra systems, paper and pencil) and strategies (e.g., the balance analogy, algebraic strategies);
- Grade 10 Applied – QR3 – solve problems involving a quadratic relation by interpreting a given graph or a graph generated with technology from its equation;
- Grade 10 Academic – QR4 – solve problems arising from a realistic situation represented by a graph or an equation of a quadratic relation, with and without the use of technology;

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.

Some noticings and wonderings that have come up when I’ve used this task include:

- Is that a prop plane or jet plane?
- I noticed the CN Tower in the background, so it is Toronto.
- Is that Billy Bishop Airport in Toronto?
- How fast does the plane have to go in order to get off the runway?
- How long is the runway?
- Where were you flying to?
- Is that the Porter flight to Windsor?

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

How many seats are on the plane?

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

Some students may ask clarifying questions like:

- Are we including the pilot(s) and the airplane steward(s)?
- Is the plane up on the screen the actual plane to scale?
- And others…

After students have shared out their predictions, I would show them this video or this image:

Just a word of caution, that I prefer to reveal the first row only and let them update their prediction. From here, some students might believe that there is a first class/business class section with a different seat size than the remainder of the plane.

Then, I’ll show them the first row and back row with one final update to their prediction before doing the big reveal.

Show students the act 3 video that reveals the total number of seats on the plane:

Alternatively, you can show this image.

While this is a fun and potentially challenging task for our primary friends, we can easily extend this task to be more mathematically purposeful for our junior and intermediate students. While I wouldn’t skip the portion of the activity we have completed thus far, I do think that we need to dive deeper into the mathematics to strengthen the intentionality of this task.

Let’s have a look.

Using an airplane like the one in this task or by going to company websites like Air Canada, you can grab seating plans of different aircrafts to promote counting principles like subitizing and unitizing:

As students begin to unitize, they can begin to start thinking multiplicatively by looking for multiple “groups” of seats to help them skip count and eventually, use multiplication.

As students look at an array of seats and head towards thinking multiplicatively rather than additively, we can encourage students to begin applying their ability to compose and decompose numbers with multiplicative thinking.

I love taking a PDF template of the array of seats into GoodNotes 4 on an iPad, SMART Notebook, or any other tech program you use for annotating and lead a number talk to encourage students to share different ways to compose and decompose the 18 seats.

You can grab a PDF template of different screenshot possibilities you might like by downloading here.

Here are some possibilities:

As I’ve mentioned in my Progression of Multiplication post, the distributive property appears for the first time in the grade 4 math curriculum and appears repeatedly even into grade 9 when we use distribution with variables.

This task can provide a great opportunity for educators to hit on this very important concept again by encouraging the use of distribution or “splitting the array” into more manageable chunks. Here’s a sample of how you might consider applying distribution and simplifying whole number expressions with your class.

Please note that I’ve taken a simple example for junior grades and extended to examples that are more complex for intermediate grades that include the use of brackets and can introduce conversations about order of operations.

Recently, I worked with the teaching staff at Roseland Public School and we did the Airplane Problem.

After the predictions for number of seats, I challenged the groups to:

Create any numerical expression that could represent the seat configuration (or array of seats) for the airplane and use your manipulatives to represent your expression concretely.

Then, in number talk fashion, we had each person share what numerical expression(s) they saw. Here’s a few screenshots of what came out:

We can extend this task even further to begin working in the Patterning and Algebra strand and even hit our Number Sense and Algebra strands in grade 9 applied and academic.

By simply creating a business class and economy class section on the airplane, we can have students predict how much revenue a sold out flight might generate:

By having students make predictions, the question is open and allow students to make predictions based on numbers that are friendly to them.

After sharing out, I might have students write a number sentence to represent the revenue for the flight.

I would encourage students trying to finding more ways to represent the same expression. I encourage students to look for ways to decompose their number sentence to allow for the use of mental math without a calculator.

Here is one such example:

Then, I might have students create expressions and/or equations to represent the Revenue for that flight. I might begin with a more open question where they set prices of seats, but then after sharing out, we might look at more specific questions like these:

Eventually, I might suggest to students that I want to keep the pricing for business and economy class at the prices indicated and ask:

What equation could represent revenue for this particular flight?

Then, we can work with substituting values into equations and solving for unknowns, which is a useful skill for all intermediate math classes.

I’ve put together a few consolidation silent solution videos, if you’re interested in helping you generate some ideas.

My last suggestion for a grade 10 applied or academic teacher would be to engage in some of the linear equation work sampled above and then head towards quadratics.

Here’s an example that could work nicely:

On a flight where all 18 seats have equal value on this particular plane and trip, research conducted by Air Canada has shown that a price of $200 per seat produces a very high probability of selling 10 seats. For every $10 increase in price, they sell 1 less seat. How much revenue can Air Canada expect on a flight where the seats are priced at $160 each?

Some follow-up work might include having students complete a table of revenue vs. seats sold to expose the second differences of this quadratic relation. Work can then be done to connect the table, to the graph of the parabola, to quadratic form of an equation and factored form of an equation.

I hope to find some time in the near future to create some visuals for this portion of the activity. If this is something you’re hoping for, please comment below so I can get motivated to get the work done!

One of the most common questions I receive when delivering workshops or via email is not around the actual use of a 3 act math task, but more about what to do after the task is complete. This is definitely a fuzzy area as I think this would most likely look different from classroom to classroom based on your own student learning needs. This is also tricky with this post in particular since I’ve tried to hit every grade level from grade 1 to grade 10. What I’ve tried to do was to give you an idea of what I might have my grade 9 applied and/or academic students work on after completing this task with the solving equations extensions. It might look something like this.

Access the goodies from this public Google Drive folder here or individually below:

Slide Deck [Keynote or Powerpoint]

Videos [Act 1 MP4 | Act 2 MP4 | Act 3 MP4]

Number Talk Template [PDF]

Purposeful Practice Template [PDF]

**I’m going to come clean here and tell you that my video and the plane we are exploring actually don’t match! A teacher noticed this because the plane we are finding seats for is an Air Canada flight and Porter is the only airline that actually flies from Windsor to Billy Bishop Airport. If any of your students pick up on this, they’re pretty slick!**

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

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]]>Counting and quantity might seem like a fairly basic concept, but you may be surprised at just how important and complex these principles are for students.

The post Counting Principles – Counting and Cardinality appeared first on Tap Into Teen Minds.

]]>As a former secondary math teacher and intermediate math coach, my new role as K-12 math consultant has led to a wealth of knowledge that I wish I had during my years spent in the classroom. My conversations about student learning needs with intermediate and senior math teachers always seems to come down to gaps in student understanding, however rarely were we able to dig back far enough in the math continuum of learning to determine exactly where those gaps began.

Recently, our Math Strategy Team focused on planning professional development for our math leads around composing and decomposing numbers all the way to addition and subtraction strategies. With this, Sharon Johnson shared her knowledge with the group around Basic Counting Principles that students must obtain for them to be successful composing and decomposing numbers. After taking some time to dig in and read more about counting principles and their importance in developing student sense of number and quantity, I realized that some of my intermediate students could still be struggling with some of these basic ideas.

Although researchers might differ in the number, naming or description of some of these counting principles, I find that this list of **9 Principles of Counting** seem reasonable and resonate with me.

The first principle of counting involves the student using a list of words to count in a repeatable order. This “stable list” must be at least as long as the number of items to be counted.

For example, if a student wants to count 20 items, their stable list of numbers must be to at least 20.

The order in which items are counted is irrelevant.

Students have an understanding of **order irrelevance** when they are able to count a group of items starting from different places. For example, counting from the left-most item to the right-most and visa versa.

Understanding that the count for a set group of objects stays the same no matter whether they are spread out or close together.

If a student counts a group of items that are close together and then needs to recount after you spread them out, they may not have developed an understanding of the **principle of conservation**.

**Abstraction** requires an understanding that we can count any collection of objects, whether tangible or not.

For example, the quantity of five large items is the same count as a quantity of five small items or a mixed group of five small and large things.

Another example may include a student being able to count linking cubes that represent some other set of objects like cars, dogs, or bikes.

Understanding that each object in a group can be counted once and only once. It is useful in the early stages for children to actually tag or touch each item being counted and to move it out of the way as it is counted.

Understanding that the last number used to count a group of objects represents how many are in the group.

A student who must recount when asked how many candies are in the set that they just counted, may not understand the **cardinality principle**.

The ability to “see” or visualize a small amount of objects and know how many there are without counting.

Since it becomes increasingly difficult to **subitize** as the number of items increases, you’ll notice that five- and ten-frames are common in early years mathematics education.

Understanding that as you move up the counting sequence (or forwards), the quantity increases by one and as you move down (or backwards), the quantity decreases by one or whatever quantity you are going up/down by.

Unitizing involves taking a set of items and counting by equal groups (i.e.: skip counting).

For example, if there is a large group of candies on a table, one might choose to create groups of five (often doing this by subitizing these groups) and skip counting up by five.

Unitizing is also important for students to understand that objects are grouped into tens in our base-ten number system. For example, once a count exceeds 9, this is indicated by a 1 in the tens place of a number.

As we move through the counting principles and get to unitizing, I quickly see how important this understanding is for students when we explore place value, fractions, unit rates, and other big ideas connected to proportional reasoning.

Have I missed any? Have some great insight to add to these descriptions? Please share in the comments!

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]]>Let's take a journey exploring fractions from a simple definition all the way through four key fraction constructs we use in the K-8 Mathematics Curriculum.

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

]]>Fractions are a beast of a concept that causes struggles for many adults and students alike. While we all come to school with some intuition to help us with thinking fractionally and proportionally, the complexity quickly begins to increase as we move from concrete, to visual, to symbolic and from identifying, to comparing, to manipulating. Fractions are formally introduced in the Ontario Math Curriculum when students begin dividing whole objects into pieces and identify these pieces using fractional names (e.g.: halves; fourths or quarters) and continues to promote the development fractional fluency concretely through each grade. Interestingly enough, it is often said that students struggle much more with numbers represented in fraction notation than those represented in decimal notation, yet the word “fraction” appears in the curriculum document 99 times beginning in grade 1, while the word “decimal” appears only 69 times beginning three years later in the 4th grade. While it might be true that fractions tend to intimidate, I wonder if our dependence on the calculator has tricked us into believing we are more fluent with quantities represented in decimal form than is reality.

As I did with the Progression of Proportional Reasoning, I’d like to reference the Paying Attention To Mathematics series released by the Ontario Ministry of Education Literacy and Numeracy Secretariat (LNS) called Paying Attention to Fractions. These guides are a great start to help you wrap your head around big ideas in mathematics and thus, this post will attempt to expand on the ideas shared in this particular document.

As if the struggles our students experience when working with fractions aren’t enough justification, I like this quote shared in the guide:

“No area of elementary school mathematics is as mathematically rich, cognitively complicated, and difficult to teach as fractions, ratios, and proportionality. These ideas all express mathematical relationships: fractions and ratios are ‘relational’ numbers. They are the first place in which students encounter numerals like ‘ 3/4 ’ that represent relationships between two discrete or continuous quantities, rather than a single discrete (‘three apples’) or continuous quantity (‘4 inches of rope’).”

(Litwiller & Bright, 2002, p. 3)

A fraction is a number.

While fractional notation is typically used to represent quantities that are **not whole**, it is possible for all quantities to be represented as a fraction.

While these descriptions are simple on the surface, they do not appropriately communicate the complex constructs that lie within this big idea.

If you recall from my last post, **proportional reasoning** is takes place when one compares two numbers in relative terms rather than absolute terms. Thus, it can be said that a fraction provides a means for representing that relationship between two numbers.

It is reasonable to believe that young learners begin working towards proportional reasoning even before they enter school when they make comparisons between two quantities such as:

- this bag is heavier than that one;
- I’m “this much” shorter than you are; and,
- there are 5 more red candies than green.

While these early comparisons are likely **absolute** (i.e.: **additive**) in nature, we are one step closer to comparing these quantities **relatively** (i.e.: **multiplicatively**).

Let’s take some time to explore four common **fraction constructs**.

Have a look at the image below:

I might ask students to determine how many eggs there are. This question is quite clear that I would like to know the quantity of eggs, however they get to decide how they would like to represent that quantity. While many may simply choose to represent the number of eggs as 18, others may get creative and **unitize** (i.e.: 3 groups of 6; 9 groups of 2; etc.) or even reference the number of eggs relative to egg cartons.

After students share out with a partner, I might ask students:

how many cartons of eggs are there?

After students have some time to think independently, I would have them share out with their table groups. Some interesting responses are sure to come up such as:

- 18/2
- 18/1.5
- 3/2
- 1.5
- 2

You may notice that by asking students to state how many cartons of eggs there are, there is room for interpretation. Some students might simply state that there are two egg cartons without considering the number of eggs present, some might give a quantity relative to two egg cartons, while others might state the number of eggs relative to a single carton.

After students have been given an opportunity to defend their thinking, I might ask students to state **how many cartons are full of eggs**?

This fraction construct would be an example of a **part-whole relationship**. Students might consider how many whole and partial cartons there are in the form of a mixed fraction (1 and 1/2), improper fraction (3/2) or in decimal form (1.5). Some common representations for these different fractional forms might be using a **set model**, **area model** or **number line**. The following animated gif shows each of these representations, sometimes more than one simultaneously:

You might have noticed the double-number line included in the animated gif. This idea can be very useful when making the jump from thinking fractionally to proportional reasoning.

In the Doritos Roulette 3 Act Math Task, students are asked questions that stem from the relationship of “hot” chips to “not hot” chips based on the image on the front of the bag:

I might ask students to consider the image above and ask them this question:

What fraction of hot chips to not hot chips are there in a bag of Doritos Roulette?

Here are some common responses I see when I ask students (and teachers) this question:

- 1/6
- 1/7
- 1:6
- 1:7

Despite the explicit request for a part-part relationship between “hot” chips (1) and “not hot” chips (6), we often see the relationship between “hot” chips (1) and the whole (7).

Something I might ask the group to discuss with their neighbours is whether we are “allowed” to represent a part-part relationship as a fraction. Great (and sometimes heated) discussions can arise from this question, which makes this exciting to facilitate.

The discourse this question can promote is a great way to address the fact that part-part relationships are commonly represented as **a ratio**, but that both part-part relationships and ratios can also be represented as a fraction. In this case, the numerator represents the number of “hot” chips; the denominator represents the number of “not hot” chips; and the sum of each part, the numerator and denominator, is the whole:

After exploring part-whole and part-part fraction constructs, you may begin to notice how important clarity around what a given fraction represents prior to attempting any comparison or manipulation of that fraction.

While the use of a **set model** and **area model** remain fairly similar regardless of whether you are dealing with a part-whole or part-part relationship, the use of a **number line** changes significantly.

Recall how the number line used in the part-whole counting cartons example made identifying the part and the whole fairly intuitive:

When we attempt to use a number line with a part-part relationship, it becomes more difficult to determine the size of the second part without modifying how we use this representation:

Note that in the representation above, two parts are clearly shown and the sum of those two parts represents the whole.

Although the number line representation might not be as intuitive in the part-part case, it does help you visualize both parts proportionally:

Extending from a single number line to a double-number line can help shine a light on the meaning behind the **part-part relationship**. In this case, there are one-sixth as many “hot” chips as there are “not hot” chips. Alternatively, there are 6 times as many “not hot” chips as there are “hot” chips.

When considering **fractions as quotient**, we are referring to the times when we are dividing two numbers. The Paying Attention to Fractions document uses an example of kids fair sharing some brownies. Let’s keep the context the same, but I’ll add some visuals to enhance the experience.

Suppose there are 6 brownies that are to be shared amongst 4 friends fairly. How much should each person receive?

Like many situations in life, there are multiple ways to share these brownies fairly. Let’s represent a few of these methods.

Since there are 4 people to share the 6 brownies, we can divide the 6 brownies into fourths and then pass them out, one-by-one:

After splitting the brownies into quarters, you’ll notice that each of the four people should receive 6 quarters. Hence why we can represent this fraction as quotient as the improper fraction, 6/4.

Some people might find it useful to share whole brownies equally and then when there are not enough whole brownies to share fairly, consider partitioning up the remaining brownies appropriately:

In the case with 6 brownies, we can share one brownie each and then partition the remaining brownies into four halves to share. This method would be equivalent to the mixed fraction of 1 and 1/2 brownies per person.

This method involves considering all 6 brownies and where cuts must be made to create four equal groupings:

This method would be another representation of the mixed fraction, 1 and 1/2 brownies for each person.

For each of the methods above, we could also consider the use of a number line to support the partitioning of the brownies to distribute a fair amount to each person. Here are a few examples:

Another common construct is the fraction as operator.

Let’s have a look at another problem. We mind as well keep the context as food, because that is just yummy.

There are 7 pieces in every full roll of Rolo chocolate.

Two partially eaten rolls are found in a drawer; one with 5 pieces and the other with 4 pieces.

How many full rolls of Rolo are there?

If we use concrete manipulatives or visuals, like we will in this instance, it is much easier to see how to add and subtract fractions. In this case, we’ll show a few different representations including how using a number line can be helpful:

So in this particular example, you can see that 5 pieces plus 4 pieces will yield 9 pieces, which is more than a full 7 piece roll; 9/7 or 1 and 2/7.

If you’re a 3 act math fan, you might be familiar with my Gimme a Break 3 act math task involving the use of fractions as operators. I should note that while there are some useful pieces in that task, it has really evolved over time and I just haven’t had the time to update the post yet. Here’s some of the pieces I’ve added along the way.

The task begins with an opportunity for students to notice/wonder after watching a video of me opening a KitKat bar.

Later in the task, the following video is shown asking students to consider what operation could be taking place:

Since the video is open ended, I’m always hoping that all students will be able to share something mathematical about the situation. For example, we might see some responses like these:

However, the new learning I’m hoping to spend some time on is the multiplication of a fraction by a whole number. In this case, a student might think of this situation as 4 groups of one quarter of a whole bar:

Alternatively, we could think of this situation as one quarter groups of four whole KitKat bars:

In either case, the use of arrays and/or area models can be very useful when trying to help build an understanding of multiplication with fractions. In the following example, we extend this idea to the multiplication of two fractions:

While I find that multiplication of fractions seems to be the one operator I hear the least noise about, my gut tells me that it is more about being able to remember the algorithm and not due to any conceptual understanding. In my experience, many students can manage to remember to “multiply the tops, multiply the bottoms” without necessarily having any real conceptual understanding to fall back on in times of doubt.

On the other hand, division of fractions seems to be a sore spots for many students (and adults) because the algorithm is a bit more complicated and less intuitive. Pair that with a very low level of conceptual understanding as to why the algorithm actually works and you’ll end up with confusion and frustration for many. The gap between the pure memorization of steps and the conceptual understanding of what dividing fractions really means is often so large that many educators simply follow suit and focus purely on the algorithm. While I believe it is so important to build the conceptual understanding behind a complex idea like dividing fractions, I know that the task can be quite difficult and many teachers aren’t so sure how to approach it. When we rush to an algorithm before constructing conceptual understanding, students (and adults) are often very difficult to engage in the heavy thinking required to connect the dots. This leaves many teachers believing that successful use of the algorithm is enough (or all they have time for) and they just move on to the next topic.

Let’s look at the Gimme a Break context in terms of division of fractions. With manipulatives on the table such as pattern blocks or relational rods, I might ask students:

What would 1 ÷ 1/4 “look like” in terms of the KitKat task?

Can you show a neighbour what that looks like to you?

Hopefully students come up with some representation to model what a whole KitKat bar would look like an a quarter of a KitKat bar:

Then, I would ask them to determine how many quarter pieces there are in a whole KitKat bar. With this in mind, students should be able to “see” and manipulate their representation on the table to come up with an answer of 4.

Then, I might consider asking students to represent a couple more similar situations using whole numbers divided by a certain number of quarter pieces to build some confidence.

For example, 2 divided by 1/4:

3 divided by 3/4:

When ready, students might be ready to start stepping outside of whole numbers to situations like 3/4 divided by 2/4:

Then, once students are really comfortable dividing fractions with common denominators, you might start moving to situations where the fractions have uncommon denominators.

Note that context is always very important for students to build a conceptual understanding. If we are trying to explain abstraction through abstraction (i.e.: explaining a rule algebraically based on another algebraic rule), then it is unlikely to stick. However, if we, the educators, spend the time necessary to understand mathematics conceptually, it will be a huge pay off for the understanding of our students. Only once students are able to visualize what mathematics “looks like” does it make sense for us to move towards more efficient methods like algorithms or more abstract problems where context is stripped away.

Throughout this post we have tackled the idea of fraction constructs by using a similar approach to how I introduce most big ideas in math class: using tasks that are contextual, visual and concrete in order to build a conceptual understanding.

This post is by no means the “fractions rulebook”, but rather a journal outlining my own journey to understanding fractions conceptually. As a secondary mathematics teacher, much of what our curriculum is built on is the assumed understanding of topics like fractions. However, the more I reflect on my own mathematical understanding, the more I realize that I never truly had a conceptual understanding of most concepts that I was teaching.

Let’s avoid the rush to the algorithm and slow down to let kids truly experience and understand mathematics. I’d appreciate any pieces I should add to this post in the comments section.

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]]>Let's look at how important arrays and area models are in building our understanding of the standard algorithm and to abstract concepts in secondary math.

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

]]>Did you know that the words “**array**” and “**area model**” appear in the Grade 1-8 Math Curriculum a combined **22 times**?

Not only do arrays and area models help to support the development of proportional reasoning when we formally introduce **multiplication** in primary, but they also help us understand how to develop strategies that lead to building number flexibility and the automaticity of math facts.

Arrays and area models should be used as a tool and representation for many big ideas in mathematics including, but not limited to:

- Multiplication
- Distributive Property with Whole Numbers
- Finding Area with Whole Number Dimensions
- Perfect Squares & Square Roots
- Multiplying a Binomial by Monomial
- Multiplying a Binomial by Binomial (aka FOIL)
- Factoring (Common, Simple/Complex Trinomials)
- Completing the Square

For many, the term “array” is not a familiar one. Luckily, the definition is fairly straightforward:

In mathematics, an array is a group of objects ordered in rows and columns.

Seems pretty simple, but they are extremely powerful in building a deep conceptual understanding as students learn multiplication and begin applying that knowledge to more abstract ideas requiring fluency with procedures.

In grade 3, students are asked to:

- 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);
- multiply to 7 x 7 and divide to 49 ÷ 7, using a variety of mental strategies (e.g., doubles, doubles plus another set, skip counting);
- identify, through investigation, the properties of zero and one in multiplication (i.e., any number multiplied by zero equals zero; any number multiplied by 1 equals the original number) (Sample problem: Use tiles to create arrays that represent 3 x 3, 3 x 2, 3 x 1, and 3 x 0. Explain what you think will happen when you multiply any number by 1, and when you multiply any number by 0.);

Consider 3 x 2, or “3 groups of 2”:

This may seem like a simple and maybe even unnecessary representation, but having a visual that multiplying two numbers will always yield a rectangular array is an important concept that not only shows the interconnectedness of the Number Sense and Numeration strand to Measurement, but also implicitly provides students with insight into why more abstract mathematics works later on in grade 9 and 10.

In grade 3, it is reasonable to believe that students are still working on **counting and quantity** including unitizing in order to skip count more fluently. By working with arrays, we can allow students to continue developing their ability to unitize and work with composing and decomposing numbers.

Not only is it beneficial for students to understand that multiplying two quantities will yield an array covering a rectangular area, but it is also useful for students to discover without explicitly stating that when we make an array where the number of groups and number of items in each group are equal, the array is now a special rectangle; a square!

While we don’t specifically discuss **perfect squares** until intermediate when students represent perfect squares and square roots using a variety of tools in grade 7 and estimate/verify the positive square roots of whole numbers in grade 8, I think it would be much easier for students to identify a perfect square and estimate the square root of a number if they have four years of concrete and visual work with perfect squares.

As we have witnessed in the previous examples, with every array a student builds, we are implicitly providing them with a window into the measurement strand with opportunities to think about perimeter and area. Not only can we easily make connections between an array and the area of a rectangle, but we can also better serve specific expectations involving estimation such as this grade 3 Measurement expectation:

- estimate, measure (i.e., using centimetre grid paper, arrays), and record area.

With the use of arrays in the Number Sense and Numeration strand, I can better serve my Measurement strand with problems that force students to estimate using visuals and then improve their predictions using concrete manipulatives.

Something worth noting is the wording of the specific expectation in the grade 3 curriculum related to multiplication and division which states students are to *multiply to 7 x 7 and divide to 49 ÷ 7, using a variety of mental strategies* without any reference to memorization or automaticity. While I agree that knowing multiplication tables for intermediate and senior level math courses is a huge asset, I think working with multiplication early and often with concrete manipulatives is a great way to get there over other more traditional and/or rote strategies.

When students enter grade 4, we extend our multiplication and division through the use of a variety of mental strategies to *multiplying to 9 x 9 and dividing to 81 ÷ 9*. As the factors and divisors get larger, we begin creating a need for some new strategies that will help us as we approach two-digit multiplication. In the grade 4 Patterning and Algebra strand, students are expected to:

- identify, through investigation (e.g., by using sets of objects in arrays, by drawing area models), and use the distributive property of multiplication over addition to facilitate computation with whole numbers (e.g.,“I know that 9 x 52 equals 9 x 50 + 9 x 2. This is easier to calculate in my head because I get 450 + 18 = 468.”).

Sounds kind of complicated. Ultimately, what the distributive property is trying to offer students is a way to “chunk” their factors into friendlier numbers to make multiplication easier.

Let’s have a look at a basic example:

If a student does not know how many objects there are in six groups of seven, they can use distributive property to “chunk” this multiplication problem into two or more smaller multiplication problems and add the products. In the example above, the student may have a comfort with multiples of 5. In the example below, the student might like working with 5 groups of 5 because the product, 25, is a friendly number to add up afterwards.

Although not explicitly stated in the grade 4 curriculum, I think it is worth noting that the introduction of the distributive property is the first situation where a bracket could be used symbolically in a mathematical expression:

6 x 7

= 6(5 + 2)or

6 x 7

= 5(5 + 2) + 1(5 + 2)

While a bracket **could** be used symbolically, I don’t think it would be developmentally appropriate or useful. However, when brackets do show up on the scene later on, how awesome would it be to connect them to our prior knowledge of arrays and distributive property?

In grade 5, we continue to promote the use of mental strategies for addition, subtraction, and multiplication:

- solve problems involving the addition, subtraction, and multiplication of whole numbers, using a variety of mental strategies (e.g., use the commutative property: 5 x 18 x 2 = 5 x 2 x 18, which gives 10 x 18 = 180);

However, we also move towards finding more efficient ways to multiply factors greater than 9:

- multiply two-digit whole numbers by two-digit whole numbers, using estimation, student-generated algorithms, and standard algorithms;

Unfortunately, I think it is more common for educators to miss the first expectation about mental strategies as well as a huge portion of the second expectation, and rush straight for the standard algorithm.

By continuing to use our knowledge of the distributive property with two-digit by two-digit multiplication, we can begin making things super friendly by splitting our arrays into chunks of 10. However, it can take a whole lot of square tiles and a whole lot of time:

That’s where base ten blocks come in!

What a genius idea to respect the theory of concreteness fading and allow students to begin this new layer of abstraction with the aide of concrete manipulatives to build a deep conceptual understanding.

Now, we can do the same problem with much less concrete “pieces”, but still gain all the benefits of manipulatives. Also, how cool is it that base ten blocks are what I call “forced distribution” into chunks of friendly 10’s.

As we introduce base ten blocks, we also begin to transition more explicitly from an array to an area model. Since we are no longer using single tiles to represent each single object or unit of an array, the “ten rods” and “hundred flats” are covering areas of 10 units-squared and 100 units-squared, respectively.

The fun doesn’t stop here. Multiplying two-digit by two-digit numbers is extremely helpful using an area model with base ten blocks:

Speaking from experience as a math teacher in my own classroom in Ontario and having an opportunity to travel to observe in classrooms both near and far, I see a common trait where we **rush to the algorithm**. This race to the procedure might be motivated by teacher beliefs, the anxiety we feel due to limited time and a thick curriculum, or simply a lack of awareness to other approaches and strategies. In any case, I think using arrays and area models can serve as a great precursor to promoting students to begin creating their own algorithms and eventually, connect to the standard algorithm.

Let’s make some connections between the standard algorithm and using arrays and area models.

Consider 22 x 26 using the **standard algorithm for multiplication**:

While there are many educators who do a great job of breaking the standard algorithm down into its working parts including doing their best to explain the impact place value has on each step to build a conceptual understanding, I think students who can use arrays and area models to multiply have an advantage to building a deeper conceptual understanding.

If using arrays, area models and base ten blocks for multiplication is a new idea for you, then it might be worth making some connections between these representations and the standard algorithm.

Have a look at the area model representation and the standard algorithm method for 22 x 26.

Can you make any connections between the two representations/methods?

What did you come up with?

It might not be super obvious, but if we consider the two products created using the algorithm (132 and 440), we can see these products by looking vertically down the area model:

Some might assume that because we end up with two products that the standard algorithm uses distribution to chunk the multiplication of 22 x 26 into two sets of factors. However, if we dig deeper, we will notice that both products are the result of two smaller chunks.

When multiplying two-digit by two-digit factors with the standard algorithm, our base ten place value system allows for chunking into smaller “chunks”:

*x*units of 1 times*x*units of 1*x*units of 1 times*x*units of 10*x*units of 10 times*x*units of 1*x*units of 10 times*x*units of 10

In this case, the standard algorithm breaks up 22 x 26 using the distributive property into the following smaller factors:

*6*units of 1 times*2*units of 1; (6 x 2)*6*units of 1 times*2*units of 10; (6 x 20)*2*units of 10 times*2*units of 1; (20 x 2)*2*units of 10 times*2*units of 10; (20 x 20)

If we consider all of the deep thinking required to understand how the standard algorithm works conceptually, it might make sense to make the connections more explicit. Let’s come back to the grade 5 expectation *multiply two-digit whole numbers by two-digit whole numbers, using estimation, student-generated algorithms, and standard algorithms*. Note that student-generated algorithms sits in there and often times, we don’t provide enough opportunities for students to truly create their own strategies and procedures.

Rather than using arrays and area models and then suddenly flying into the standard algorithm, what if we tried to guide students to develop their own “conceptual” algorithm by having them organize the smaller products they are creating when using base ten blocks?

Consider this “conceptual” algorithm that might assist students in making the leap to the standard algorithm when developmentally appropriate:

The connections don’t stop here, either. By explicitly addressing the conceptual understandings behind why the standard algorithm works, students can then apply that same thinking to create their own friendly chunks outside of those limited to 10s created when using base ten blocks.

You may recall the acronym “FOIL” which is commonly used for students to remember how to multiply two binomials. While I am guilty for teaching this memorization tool in my math class until only a few years ago, I now understand that using tricks like “FOIL” to teach important math concepts is not helpful (and maybe even harmful).

What if instead of simply teaching students “FOIL” or “double-distribution”, which is a skill limited to the very specific case of multiplying two polynomials with two terms, we actually helped students to visualize what multiplying binomials really looks like?

What we see in the previous example is:

9 x 12

= (5 + 4)(10 + 2)

= 5 x 10 + 5 x 2 + 4 x 10 + 4 x 2

= 50 + 10 + 40 + 8

= 108

It might not be obvious to those who have never worked to make a connection, but what the standard algorithm we teach in grade 5 is actually the same procedure we teach students when multiplying binomials in grade 10.

As we head into grade 9 and 10, the thinking becomes more abstract due to the use of variables.

Some Expectations from Grade 9 Academic:

- multiply a polynomial by a monomial involving the same variable [e.g., 2x(x + 4), 2x^2(3x^2 – 2x + 1)], using a variety of tools (e.g., algebra tiles, diagrams, computer algebra systems, paper and pencil);
- expand and simplify polynomial expressions involving one variable [e.g., 2x(4x + 1) – 3x(x + 2)], using a variety of tools (e.g., algebra tiles, computer algebra systems, paper and pencil);

And here’s a couple examples of what these might look like if we use arrays and area models from grade 3 onwards:

An Expectation from Grade 10 Academic:

- expand and simplify second-degree polynomial expressions [e.g., (2x + 5)^2,

(2x – y)(x + 3y)], using a variety of tools (e.g., algebra tiles, diagrams, computer algebra systems, paper and pencil) and strategies (e.g., patterning);

Here’s an example of what this might look like:

If we are helping students understand what math looks like whenever and wherever possible as I have tried to do in this post for the **progression of multiplication**, then it would seem logical that some of these quite challenging expectations would be much less complex if we use arrays and area models prior to rushing to the algorithm.

How are you learning in order to better understand what math looks like concretely and visually?

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

]]>The underpinnings of proportional reasoning are born when students learn how to count and unitizing. Explore visual representations of this Big Idea.

The post The Progression of Proportional Reasoning From K-9 appeared first on Tap Into Teen Minds.

]]>Proportional Reasoning is a big idea that is connected in some way to all mathematical strands and stretches across many grades in the Ontario mathematics curriculum. While many focus on the importance of proportional reasoning for students in junior and intermediate grades, I believe that we can see the early development of proportional reasoning in kindergarten and primary grades when students begin unitizing, grouping, and fair sharing. In this post, I’d like to briefly explore some of the **progression of proportional reasoning** across the Ontario mathematics curriculum.

The Ontario Ministry of Education has done a great job releasing documents in the Paying Attention to Mathematics series and the Paying Attention to Proportional Reasoning document is no exception.

This document does a great job of providing a great overview of proportional reasoning from a research based perspective without getting too heavy. I will share a few take-aways below and build on some of the ideas from the document with my own spin.

When students begin approaching problems multiplicatively instead of additively, such as thinking that 10 is two groups of five or five groups of two rather than one more than nine, students are said to be reasoning proportionally. This multiplicative thinking allows for students to begin comparing two quantities in relative terms rather than absolute terms.

The essence of proportional reasoning is the consideration of number in relative terms, rather than absolute terms.

Paying Attention to Proportional Reasoning Document

Ontario Ministry of Education

Scenario #1: You invest $100 and it grows to $400.

Scenario #2: You invest $1,000 and it grows to $1,500.

Your opinion will change based on whether you make your comparison of the quantities in each scenario in relative or absolute terms.

If you view this comparison in **absolute terms**, you might believe that Scenario #2 is the best investment since you have earned more money.

However, if you view this comparison in **relative terms**, you might believe that Scenario #1 is the best investment since you have earned more money relative to the initial investment amount.

As are most big ideas in mathematics, proportional reasoning is an idea that connects to many key ideas including:

- Partitioning
- Understanding rational numbers
- Multiplicative reasoning
- Scaling up and down
- Relative thinking
- Understanding quantities and change
- Spatial reasoning
- Measuring, linear models, area, volume
- Unitizing
- Comparing quantities and change

Although the ideas of multiplicative thinking begin developing in grade 1 and 2 when students are encouraged to group quantities into units and then formally as multiplication in grade 3, let’s take a look at a couple expectations in the Number Sense and Numeration strand in Grade 4 of the Ontario Grade 1 to 8 Mathematics Curriculum and some sample problems for each:

- multiply to 9 x 9 and divide 81 ÷ 9, using a variety of mental strategies (e.g., doubles, doubles plus another set, skip counting);

Question:

There are 4 fish bowls with 5 fish in each. How many fish in total are there?

Each bowl can be considered as both 1 bowl (unit) or 5 fish, simultaneously.

- divide two-digit whole numbers by one-digit whole numbers, using a variety of tools (e.g., concrete materials, drawings) and student-generated algorithms;

Question:

You buy 15 goldfish. You are going to put 3 fish in each bowl. How many bowls will you need?

Each bowl can be considered as both 1 bowl (unit) or 3 fish, simultaneously.

Near the end of the Grade 4 Number Sense and Numeration strand, we run into the first reference to **proportional reasoning** in the Ontario elementary math curriculum. Under this overall expectation, we see the following:

- describe relationships that involve simple whole-number multiplication (e.g.,“If you have 2 marbles and I have 6 marbles, I can say that I have three times the number of marbles you have.”);
- determine and explain, through investigation, the relationship between fractions (i.e., halves, fifths, tenths) and decimals to tenths, using a variety of tools (e.g., concrete materials, drawings, calculators) and strategies (e.g., decompose 2/5 into 4/10 by dividing each fifth into two equal parts to show that 2/5 can be represented as 0.4);
- demonstrate an understanding of simple multiplicative relationships involving unit rates, through investigation using concrete materials and drawings (e.g., scale drawings in which 1 cm represents 2 m) (Sample problem: If 1 book costs $4, how do you determine the cost of 2 books? … 3 books? … 4 books?).

Notice the multiplicative thinking and comparison of quantity in relative terms. We also encounter the formalization of unitizing through whole number unit rates. It is at this stage where we see the ground work being laid for ratios and rates. While I don’t believe it would be developmentally appropriate to make the leap to ratios or rates at this junction, I do think it is important for the teacher to be aware that the tasks students are asked to solve here are ultimately very simple ratio and rate problems in disguise.

For example, each of our previous unitizing problems for multiplication and division involving fish and fish bowls can be represented as a rate of 5 fish to 1 bowl and 3 fish to 1 bowl, respectively.

In Grade 6, we formally introduce the comparison of two quantities with the same unit as a **ratio**:

- represent ratios found in real-life contexts, using concrete materials, drawings, and standard fractional notation (Sample problem: In a classroom of 28 students, 12 are female. What is the ratio of male students to female students?);

It is fairly common to see a rush to the algorithm via an equation of equivalent fractions in order to meet this expectation while minimizing or avoiding the use of concrete materials, drawings and other representations.

Let’s take a look at a sample problem with some possible concrete and/or visual representations in order to promote the idea of concreteness fading as we build our conceptual understanding:

For every 2 red candies in a package, there are 3 green candies. How many red candies would there be if you have 12 green candies?

While it might be tempting to jump straight to representing this problem in standard fractional notation (i.e.: 2/3 = x/12), let’s consider some concrete and visual approaches to promote a deep conceptual understanding prior to jumping to the symbolic.

The “**double array model**” is a representation that I feel strongly about as it builds on the idea that proportional reasoning is multiplicative and a relative comparison of two quantities. I am uncertain if I came across this representation somewhere along my learning journey or if it came about organically. In either case, I have yet to find this representation online or in common Ontario resources such as The Guide to Effective Instruction in Mathematics.

Here’s what a double-array model might look like in this case:

While I especially like the explicit connection that we can make from multiplication and division as inverse operations to proportional reasoning that a **double array** can provide, double number lines (or “clotheslines“) and ratio tables are also common representations that can make proportional relationships easier to conceptualize.

If you’ve read some of my other posts or attended a workshop, you’d probably know that my vision for teaching math involves making tasks **contextual, visual and concrete**. So, let’s flip back to a 3 act math task I created a while back called Doritos Roulette.

Check out the act 1 video if you aren’t familiar with the task.

After much mathematical discourse amongst students, I always aim to settle on:

How many “hot” chips should we expect in a bag?

Assuming students are developmentally ready to take the leap to a symbolic strategy using a proportion via equivalent fractions, it can be useful to connect our understanding of double arrays, ratio tables and double number lines like you see in this video:

While I do believe that building a conceptual understanding through multiple representations of proportional reasoning is very important, I think it should be explicitly stated that taking this approach will not necessarily speed up the learning process. Realistically, it could take longer due to the depth of knowledge we are striving for. I’m a firm believer that anything worthwhile takes time and effort. Our mathematical understandings are no exception.

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