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.
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]]>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:
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:
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:
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:
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:
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:
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:
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:
However, we also move towards finding more efficient ways to multiply factors greater than 9:
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”:
In this case, the standard algorithm breaks up 22 x 26 using the distributive property into the following smaller factors:
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.
What we see in the previous example is:
9 x 12 = 5 x 10 + 5 x 2 + 4 x 10 + 4 x 2
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. Many often use an acronym “FOIL” to help students remember this procedure where we extend distribution of one value into a brackets to distributing two values into a bracket:
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:
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:
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?
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]]>The underpinnings of proportional reasoning are born when students learn how to count and unitizing. Explore visual representations of this Big Idea.
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]]>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:
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:
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.
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:
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:
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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]]>The Solar Panels Problem is a 3 Act Math Activity asking students to predict how many there are, then determine how much greenhouse gasses they offset.
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]]>This is a 3 Act Math Task that focuses on “Green Energy” proportional reasoning questions related to solar panels in order to address proportions both visually using area models and proportions written as equivalent fractions. Special thanks goes to Dave Raney from CPE Inc. for allowing us to use his video and photos as well as Kathleen Quenneville, Energy & Environmental Officer at GECDSB for her feedback to create problems around solar energy.
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 we may explore some other wonderings, the first question I intend to address is:
How many solar panels are there?
I’ll flash this image on the screen and then ask students to make a prediction.
I’ll give students some time to think independently, then chat with a neighbour to discuss their predictions while encouraging them to share their thinking. Then, we share out to the whole group and jot down predictions.
When students have shared out their predictions, we will show students the blue prints for the solar panel system with the total number of modules (582).
Now that students have invested some thinking into this context, we are going to extend our thinking to this question which may or may not have come up in the notice/wonder rapid write:
How much greenhouse gas emissions can the solar panels offset?
Show students this image showing the carbon dioxide from 4,848 km of driving a passenger car that is offset for every 5 solar panels that are in service on the roof of the school.
Show students the act 3 animated solution video showing three different representations for solving this problem.
Consider asking students the following extension question:
How many homes can all 582 solar panel modules power?
Show students this image that states 4 solar panels provide 1,576 kWh of power per year and the average home consumes 9,000 kWh of power each year.
Click on the button below to grab all the media files for use in your own classroom:
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]]>Students are shopping for clothes with expensive retail price tags. Will the price be more reasonable after applying significant percentage discounts?
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]]>The following lesson resource material provides Real World Math Problems that were created with the Grade 6 Ontario Mathematics Curriculum in mind, but can be used in all intermediate grades to reenforce an understanding of percentages and discounts. Throughout the lesson, students are asked to estimate the cost of each item of clothing after a discount is applied while encouraging the use of friendly numbers and their understanding of percentage as a part (percentage) of a whole (out of 100). Students will then be able to take their estimate and complete the calculation with a calculator to compare their estimate with the actual result.
After our Real World Percentages Math Lesson, I will be able to:
Show students the act 1 video.
Then, ask the students:
What do you notice? What do you wonder?
Give students some time to jot down some of their noticings and wonderings on a piece of paper or on a whiteboard/their desk with non-permanent marker. I generally give a minute of time for students to do a “rapid write” of these ideas.
Then, I have students share out their noticings and wonderings while I typically list them in point form on the whiteboard or in a note on my computer on display for all to see. Attaching names to these ideas can be a nice way to build in some accountability and encourage sharing.
Since the price is clearly blocked out in the video, I hope someone is curious about the price and or discount price of the suit jacket with some ideas like these:
How much does it cost before the discount?
How much will the discount save you?
What will be the sale price you have to pay?
And many more…
While the questions I’m fishing for don’t always come out, that is O.K. The discussion is key in order to hook in my students and their curiosity can be moulded quite easily after they have shared out so many interesting ideas. We often spend some time trying to answer their other curiosities in order to ensure students know that their voice is valued.
Then, I ask students to make a prediction.
I’ll have students share out these predictions, jot down their names and try to get a bit of friendly competition going on in the classroom to bring about student discourse in our non-threatening classroom environment.
We will then have students watch the act 2, scene 1 video to reveal the original retail price.
The teacher can then ask students to have a discussion in their table groups to determine ways that they can go about estimating the sale price after the discount. Some guiding questions:
After the discussion, students can share out via Apple TV or using chart paper in your classroom to model as many creative solutions as possible. Students can then check their estimates using a calculator and possibly encourage them to try and find a more efficient strategy as they move through the remainder of the tasks.
Students will watch the act 2, scene 2 video.
Students can then use estimation strategies to find the discount and the sale price of the item.
Students will watch the act 2, scene 3 video.
Students can then use estimation strategies to find the discount and the sale price of the item.
Students will watch the act 2, scene 4 video.
Students can then use estimation strategies to find the discount and the sale price of the item.
Now, students are asked to calculate a sub-total and determine the total bill we should expect when purchasing all four items.
Then, you can show what the bill would look like for students to confirm their thinking. Since I live in Ontario, I’ve shown a bill here using the Harmonized Sales Tax (HST) of 13%:
Something else I typically do with my students is have them figure out how much the bill would be in Michigan, since we are so close to the border and many families go across to Detroit and the surrounding area to shop:
Here’s an image with the final bill from Birch Run, Michigan:
Click on the button below to grab all the media files for use in your own classroom:
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]]>This year, I intend to speak less and listen more. I hope you'll join me here as I try to learn this new role - one day at a time.
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]]>Time flies, doesn’t it?
It is hard to believe that this is my first post of the 2016-17 School Year and it is the middle of October. After doing some travelling to present throughout the summer and spending the balance of my time with my family, the beginning of the school year came fast and furious and hasn’t slowed down yet.
There was definitely no “ease in” time for my new role as K-12 Math Consultant for my district. After our board assembled a Math Task Force to make recommendations to address falling EQAO standardized test scores, our Math Task Force Report was released and we are now entering the first year of our plan. The first two weeks of the school year involved providing mathematics professional development to build the capacity of our central office staff in mathematics content knowledge and pedagogy. Less than a week after, we dove into our first Administrator Capacity Training sessions in which all of our Elementary (K-8) principals and vice-principals engaged in similar learning. Shortly thereafter, we jumped into our Math Learning Team sessions where every school math team consisting of administrators, math liaison teachers, and learning support teachers would join us for mathematics learning. Tomorrow is our last session before we begin planning for our next round of professional learning.
While it might seem logical that my absence in the Math Twitter Blogosphere is purely due to a lack of available time, I think there is more to it. Since the early spring, I have made it a personal goal to speak less and listen more – both in face-to-face interactions and online.
For the past few years, I’ve thoroughly enjoyed sharing my thoughts as I work through ideas and beliefs on my blog and on Twitter. Many times, readers are positive and appreciative; other times, not so much. While I have learned so much from these experiences, my biggest take-away is the realization of how little I know.
We’ve all heard the quote from Socrates:
The more I learn, the less I know.
For me, it was always just a quote with no real meaning or connection to my own life. Then, when new learning prompts you to start rethinking what you thought to be true or believe, the quote suddenly makes perfect sense.
Redefining Mathematics Education and what learning math looks like is not easy. Sometimes in the past, when I would get an “ah-ha” moment or epiphany, I would jump to a conclusion about how to fix the problem or make it better. While deep inside I must have known that math is much more complex than that, my lens was probably too narrowly focused on these new ideas rather than widening my perspective to see the complex system as a whole. I think it is important to note that while mathematics education is complex, it is not complicated. One of My goals this year is to try identifying useful components of this complex system as well as the components that unnecessarily complicate the process.
While I do believe this latest revelation is a sign of personal and professional growth, the downside is that I now hesitate to write that next blog post when I stop to consider that there is always more to learn. I’m sure it will be difficult, but I will do my best to speak less and listen more by sharing my thoughts from a learning stance.
I hope you’ll join me here as I try to learn my new role – one day at a time.
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]]>Fast Clapper is a Three Act Math Task by Nathan Kraft involving proportional reasoning and rates of change. Will he beat the record for fastest clapper?
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]]>The following 3 act math task shared by Nathan Kraft (@nathankraft1) involves proportional reasoning with opportunities to address learning goals around rates, proportions, rates of change and creating equations.
Jon Orr and I have used this task in presentations a number of times over the past few months and many have asked us to share our slide deck and other resources with a summary. So, here we go!
Show the following video:
*Note that this is an edited version of Nathan’s video where I have blacked out the number of claps. You can always add. You can’t subtract.
I typically give students a prompt to jot down what they notice or wonder. Giving a minute of rapid writing in point form or otherwise can be a great opportunity for students to get sucked into the problem.
Students might notice:
Students might wonder:
While we will engage in a healthy amount of discourse around these noticings and wonderings, my main focus is to pull out the question:
Will he beat the current record?
I’m then going to ask students to make a prediction. Will he beat the record? How many claps per minute do you think he’s going to get based on what you saw? I might even show the Act 1 Video a few more times to give students an opportunity to use a mathematical strategy to help with their prediction.
We will record student predictions up on the board next to their names for an opportunity to celebrate later.
Show Nathan’s video:
I am very explicit with my words after showing the video. I say something like:
Do you think you can improve your prediction?
The reason I do this is so that every student has an entry point to the problem. I don’t necessarily care about the exact answer the “math” says you should get. I’m happy with a student using friendly numbers to help them get closer. I’m happy with students using some sort of strategy we might have seen in the past (proportions or maybe trying to create an equation from previous years). The best part by doing this is everyone can get closer to the expected actual result.
What do you think? Will he beat the record?
Here’s one question from a Knowledgehook Gameshow that could be used to support this task:
Allow students to check their solutions:
Additional resources can be downloaded from the download link and the Knowledgehook Gameshow can be grabbed here to modify and edit.
How are you using the problem? Please share in the comments and be sure to thank Nathan for sharing on his blog or on Twitter.
Click on the button below to grab all the media files for use in your own classroom:
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]]>Are students suffering in math because we no longer memorize multiplication tables?Understanding multiplication is much more valuable for students to apply...
The post [Updated Post] Does Memorizing Multiplication Tables Hurt More Than Help? appeared first on Tap Into Teen Minds.
]]>Post update: Saturday May 21st, 2016
This post was originally written two years ago and admittedly took an angle that downplayed the importance of automaticity of multiplication skills which was not the original intention. Please read on as I attempt to clarify the original intent of the article.
As you may have heard in the Globe and Mail or in my recent post, Ontario Education Minister, Liz Sandals recently tossed a comment into the media about the need for students to know their math facts:
That’s actually a great homework assignment: Learn your multiplication tables.
Liz Sandals – Ontario Minister of Education
It seems that whenever things aren’t going well in the world of math education – or more poorly than usual – people are quick to claim that it is because students don’t know their basic math facts; namely, memorizing multiplication tables. There is an obvious benefit to knowing your multiplication tables when solving problems, but is it only the memorization of basic math that students lack?
If we recall what memorizing multiplication tables looked like when we were in school, you might picture flash cards, repeating products aloud or writing out your 7’s times tables repeatedly until it was engrained in your mind. There is no doubt in my mind that having multiplication tables memorized allows for making calculations without taxing working memory.
Although the debate between traditional and reformed mathematics has been going on for decades, I think both groups ultimately want the same thing: students to be proficient in mathematics. However when people make statements about memorizing math facts or “going back to the basics”, people likely have very different interpretations as to how we should get there.
There are no silver bullets in math education and the memorization of multiplication tables alone will not solve all of the problems our students face. That said, many suggest that Ontario needs to “go back to the basics”, but did they ever leave? While the memorization of multiplication tables is not explicitly stated in the Ontario math curriculum, there is a huge push to develop a deep understanding of what it really means to multiply and in time, this should promote automaticity. As I mentioned in this post, shortcuts in math are only effective when you know how to take the long way; learning multiplication is no exception.
Check out the wording of an overall expectation in the Grade 2 Number Sense and Numeration Strand of the curriculum:
solve problems involving the addition and subtraction of one- and two-digit whole numbers using a variety of strategies, and investigate multiplication and division.
We could easily change that overall expectation to say:
know multiplication tables up to 5 times 5 without a calculator.
Which one would be more valuable to the student?
The grade 1-8 Ontario math curriculum contains the word multiplication a total of 47 times, which hardly suggests that multiplication is not a focus throughout elementary.
The curriculum doesn’t suggest that students should not know their multiplication tables, but instead has a greater focus on developing multiplicative skills and understanding. Digging deeper into the document, the words “variety of” occurs in the curriculum a total of 179 times and suggests that much more attention should be spent on exposing students to a variety of strategies, using a variety of tools, to show understanding a variety of ways. If done well, I would have to believe students would have a very deep understanding of math concepts including multiplication.
Over time, I’d like to think that an effective implementation of the Ontario Grade 1-8 Math Curriculum would allow for committing multiplication tables to memory over time.
The curriculum document doesn’t explicitly state the memorization of multiplication tables as an expectation, but it does require that concepts be delivered using a variety of strategies and tools/manipulatives. Where the problem may lie is how the curriculum expectations around multiplication are interpreted by the teacher.
What if the teacher isn’t comfortable teaching math? What if they only really understand one way to multiply? Does “a variety” mean three strategies? Four strategies? … Ten?
These are just a few of the questions that pop into my head when I read the curriculum and ponder some of the challenges we continue to experience in mathematics education.
Tom is a teacher, who admits not having a real passion for teaching math. When he teaches multiplication, it might look like this:
Great for patterns and provides a strategy for students to “get there” if they are stuck, but might not be enough for students to build a deep conceptual understanding.
We all know the algorithm and it is likely that teachers will use it moving forward as students begin working with larger numbers. Works like a charm if you do each step correctly. Something very interesting about the algorithm is that it can often be taught as a procedure and nothing more. Knowing how to use the algorithm without a conceptual understanding of how it works can be useful, but possibly only slightly more useful than typing the expression into a calculator. Miss a step or press the wrong button without a deep understanding of how the algorithm works and you might be out of luck.
Rather than debating over whether memorizing multiplication tables is necessary, maybe a better question might be:
What is the most effective way to have our students build automaticity with multiplication and other math facts?
Can we promote the automaticity of multiplication facts by building a deep conceptual understanding and spacing the practice meaningfully throughout our math curriculum? Would helping students visualize how multiplication works allow for this memorization to build over time?
Here are just a few visual strategies that might help students build a deeper conceptual understanding than simply learning their times tables through mass practice.
Chunking through the use of an area model is a skill that can be used not only to lower the bar for all learners at every level of readiness to begin multiplying with confidence.
If we promote the use of multiple strategies to complete mental math rather than turning to the calculator, then I believe students will value the skill of multiplication automaticity and feel the need to use it as an efficient strategy. I must be clear in saying that this will not happen unless we promote the use of mental math and multiplication strategies throughout the entirety of our math courses.
If multiplication strategies are used consistently and with purpose, students can build their multiplication automaticity in order to create “friendly chunks”. The model above would suggest that this student may be comfortable multiplying by groups of 10 and thus can save a ton of work by chunking in such a manner until they feel comfortable with multiplying by groups of 12.
While I think there are huge advantages to having multiplication tables memorized, I think we need to be cautious about how we plan to get to that end goal. If we do not take time to clarify the how, we risk promoting the use of repetition through mass practice without promoting a deep conceptual understanding of what multiplication really represents. Even worse, promoting memorization without meaning could push students to dislike math and deter them from building a productive disposition towards the subject area we enjoy so much.
Should We Stop Making Kids Memorize Times Tables? – Jo Boaler – US News
Automaticity and why it’s important to learn your ‘times tables’ – Dr Audrey Tan
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]]>Angle geometry including the Opposite Angle Theorem and patterns involving parallel lines cut by a transversal can be boring. Spice it up with some context!
The post Railing Reconstruction appeared first on Tap Into Teen Minds.
]]>From Grade 7 to 9 in the Ontario Math Curriculum, student understanding of angle geometry is extended to include (but not limited to) the following specific expectations:
Grade 7
Grade 8
Grade 9 Applied
The expectations extend nicely through these three courses, but the topics alone can often be a bit of a drag. While I’m still going to approach finding missing angles like a “puzzle” for my students as they seem to enjoy it, I have felt pretty uninspired when trying to introduce the idea of finding what would seem to be completely random missing angles.
When I was asked by Keri K. from Kingsville PS to join her class as they prepared to introduce some angle geometry, I figured I had better put my thinking cap on to attempt making the intro to this unit of study more meaningful for students than I might have in the past. So, I thought we would try to get kids thinking about what they notice and wonder via the 3 act math task approach.
Here we go.
Before we started, I ask students to think about what they notice and what they wonder when they watch the following video.
I played the video a second time. I then asked students to take 60 seconds to create a list of what they noticed and wondered on a piece of paper.
Depending on the task, sometimes students nail the question you’re looking to focus on that day, but other times they won’t. Either way, we can always try to answer some of the questions they have shared and nudge them towards the question we are looking to tackle for the day.
After some good discussion, I move on to the act 2 video.
Or, here’s a screenshot:
Students are now fully aware of the question to ponder:
How big is the angle?
You can frame this as a challenge or possibly play up a story involving the need to reconstruct the railing in your home. I think I prefer posing it as more of a challenge since students have not yet been exposed to any angle theorems involving parallel lines with an intersecting transversal line.
Because this was not my own classroom, I thought I should start with a low floor and ensure students were comfortable with benchmark angles prior to moving on. So, we did a few warm-up questions in Knowledgehook Gameshow prior to attempting to tackle our main question from the video.
Here’s the gameshow I used: [play as student | view/clone as a teacher]
Note that I didn’t do the KH Gameshow prior to introducing the task because that would have got them thinking immediately about angles and the whole notice/wonder piece would probably be a dud. If your students have some knowledge of benchmark angles, then you might consider skipping the warm-up.
Already on the desks of the students were bins including paper, scissors, markers, etc.
I said the following:
Friends:
Take a sheet of paper and fold it twice to create an “X” with the folds.[I held up a piece of paper and folded it over twice to create an “X”]
It doesn’t matter how wide or thin your “X” is and it will probably look different than your neighbour.
Now, take a marker, start at any angle you’d like and number the angles in order, clockwise.
[I modelled this.]
Cut out your four angles and piece them together on the desk.
Take one minute on your own to ‘play’ with the pieces. Jot down anything you notice. Do you notice any relationships?
Now, have a conversation with your neighbour. Ensure both of you have a turn to speak.
After sharing out with the group, students noticed that the sum of the angles is 360 degrees and that the opposite angles were equal. And so, the Opposite Angle Theorem (OAT) was born.
Then, I asked students to do the following:
Friends:
Take another sheet of paper and fold it twice to create two parallel lines with the folds.[I held up a piece of paper and folded it over twice.]
It doesn’t matter how far apart your parallel lines are and they may look a bit different than your neighbour.
Now, fold the paper once so a single fold cuts through both of the parallel lines any way you’d like.
Now, take a marker, start at any angle you’d like and number the angles in order, clockwise.
[I modelled this.]
Cut out your four angles and piece them together on the desk.
Take one minute on your own to ‘play’ with the pieces. Jot down anything you notice. Do you notice any relationships?
Now, have a conversation with your neighbour. Ensure both of you have a turn to speak.
Similar discoveries to the previous paper fold activity were discussed. We consolidated as a group and then determined that if we have parallel lines cut by a transversal, we basically have two “groups” of opposite angles.
Then, I showed the act 2 video again. Students were now ready to head out on their way to solve for the missing angle.
After students solved and shared out their work, we could consolidate the learning from this task via the Act 3 Video.
Or the animated gif:
If you try this out, let me know what you did differently in the comments!
Click on the button below to grab all the media files for use in your own classroom:
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]]>When delivering teacher workshops, I almost always include a 3 Act Math Task as a way to model the 4-part math lesson framework and for teachers to experience the power of introducing new concepts by leveraging curiosity. Although my workshops spend a significant amount of time exploring interesting lessons, teachers often want a formal description […]
The post What Makes A 3 Act Math Task So Effective? appeared first on Tap Into Teen Minds.
]]>When delivering teacher workshops, I almost always include a 3 Act Math Task as a way to model the 4-part math lesson framework and for teachers to experience the power of introducing new concepts by leveraging curiosity. Although my workshops spend a significant amount of time exploring interesting lessons, teachers often want a formal description of what defines a 3 Act Math Task from any other effective task. I thought it was about time that I pull apart what I believe makes an effective 3 act math task and why I have found them so useful in my classroom.
As many are aware, the Three Acts of a Mathematical Story approach shared by Dan Meyer when he began creating media rich math tasks that were structured using the effective storytelling technique of “acts” where:
The hook that introduces the storyline, often leaving you curious with questions you are interested in answering and rising tension.
Tension continues to rise to its highest point as more clues are revealed to help lead you to the climax of the story.
Curiosity is satisfied with answers to your questions and tension is restored to its original state prior to the first act.
Click here for an example of how we can apply the three acts storytelling technique to a math task.
It’s no secret that math isn’t high up on most people’s list of things they enjoy to do in their spare time. Worse yet is trying to teach new concepts to students when more often than not, there are no immediately apparent reasons why we need them. As students move from junior grades into intermediate (or from elementary to middle school) where concepts become more abstract, the purpose of mathematics is less obvious to the learner and interest in the subject often decreases. As student engagement in mathematics falls, teachers are left wondering how they can recapture that abundance of natural curiosity younger children seem to possess.
When discussing the topic of student engagement, I often hear math teachers firmly state: “teachers are paid to teach, not to entertain” – something I’d firmly agree with. However, I do believe that it is our duty as teachers to seek out ways to capture the attention of our students by making math class a compelling environment. Just as storytellers and filmmakers are forced to work hard to hook in their audience, it would make sense that math teachers, especially in the middle to high school grades, would have to work even harder to hook in their respective audience. It is easy to forget that many students never willingly signed up to sit in our classroom – they had to. I think educators need to work just as hard to engage our audience as those in other industries do – some for good, others not so much – if we want to have any sort of influence on our learners.
While there are many beliefs as to why the 3 act math task approach is beneficial, here are just a few of many reasons I’ve compiled over the past couple years as a frequent user of this approach.
While I believe the 3 act approach could be used to introduce any question in any subject area, they are best known as a way to deliver media-rich questions in the math classroom. By showing an image or short video, you can quickly spark curiosity in all learners regardless of their mathematical ability.
Dan Meyer and others using this framework typically encourage students to share their thinking throughout the process. Sharing what they notice or wonder, what questions come to mind, predictions and extraneous variables that could affect outcomes are just some of the common discussion starters that can be used.
Because most 3 act math tasks model math in the world around us, often times applying a standard formula or algorithm may not accurately calculate what will actually happen in the given scenario. Encouraging students to think outside of the box and consider what would happen in the real world vs. in the “fake world” math questions often have us living in.
Those who are lucky enough to have the sense of sight are constantly processing images of the world around them. It seems logical that providing students with a visual to better understand mathematics, especially when the content becomes increasingly abstract over each grade level.
Most tasks provide an opportunity for all students to participate and can be extended to more open ended (or open middle) questioning via “sequels” to the original question.
While there are many different ways to engage students in math class, it is clear to me that using the art of storytelling such as Dan Meyer‘s 3 act math approach is a solid way to raise student interest, curiosity and engagement in my mathematics classroom.
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