Author: Samuel Paul Hickok
School/Organization:
James Rhoads School
Year: 2022
Seminar: What Makes Something a Number?
Keywords: Cantor, Cantor sets, cardinality, functions, infinite sets, infinity, irrational, natural numbers, rational
One of the most common mistakes eighth graders make when it comes to functions is not recognizing that a function only has to go one way. y=x^{2} is a function because for any given x value, there is only one possible y-value. It does not matter that when I invert it and say that x^{2} = y, for any given y there are two possible x values. Invertible functions are a special case, but, because we do not give language to them in eighth grade, students intrinsically recognize these invertible functions as special but assume that that means that for something to be a function it must be invertible, not that its invertibility is a special case within its function-ness. This unit puts a name to invertible functions by applying them to the case of the cardinality of infinite sets, while, at the same time, helping students identify more readily natural numbers, whole numbers, rational numbers, irrational numbers, real numbers, and to better differentiate between the sets, another much-confused aspect of the eighth-grade curriculum. The hope is that, by putting more formal terms to the concepts in this unit, some of the misconceptions at play are put to rest.
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When we say “ten,” what do we actually mean? And when we say that fourteen is one ten and four ones, or two groups of seven, what do those operations mean? If we say that a set has seven elements (and, by the definition of set, those are unique), what can we say about that set’s size? And how is that set different from a set with eight elements?
In the finite world, with tidy beginnings and endings, there is a concrete nature to our sense of numbers. A set of size (cardinality) seven has seven elements. If I combine it with a completely different set (no shared elements) of five elements, I now have a set with twelve elements. The set with seven elements is larger than the set with five elements, and the set of twelve elements that encapsulates them both is larger than any individual set. Similarly, when I say the number ten, it takes the metaphor of language and uses it to represent a concrete concept: ten individual items, popularly fingers (or toes) as early elementary students are building their understanding of numbers through counting.
As students move past counting, patterns emerge. In our base ten system, we can decompose numbers into multiples of powers of ten, and the addition and subtraction within each power of ten follows the same rules as they did when we first operated by counting our fingers and toes. This allows us to generalize addition past our own ability to quickly count, say thirty-nine billion, four hundred eighty-seven million, two hundred thirty-five thousand, eight hundred seventy-nine plus forty billion, three hundred twelve million, three hundred twelve thousand, one hundred ten. We no longer have to dole out some counter for each individual unit represented by the rather disgusting sums and can instead generalize to know that when we combine the nine ones in the first number and the zero ones in the second, we have nine ones. Similarly, when we combine the seven tens in the first number with the one ten in the second, we find ourselves now in possession of eight tens, just like if I held up seven fingers and then one more, I would be holding up eight fingers. I will not bore you with expanding this addition out to the tens of billions, only suffice it to say, that, in the finite world, we can generalize the rules for small numbers to numbers so large they are only concrete to the very richest among us (or our great-great grandchildren after generations of hyper-inflation).
The key, however, is that no matter how theoretical the number at hand, it has a specific value. Should I want to count it out, I could. For millennia, our main concern in math was its concreteness, its ability to describe and represent as much of the world as possible. Eventually, however, when working with large numbers, it becomes more useful to determine trends of all impossible large, rather than the results of one such number. In this realm of the impossibly large, one more or one less change very little in the macro-calculation. This very nature, though, begins to strain at our understandings of the basic operations in mathematics. Infinity plus infinity has no meaning outside of the context in which that infinite exists. We can all think of sets with infinite elements; we use them all the time. What does it mean to say that one infinite set has more elements than another? And how do I know that that is true?
One of the major standards in Eighth Grade, and, subsequently, Algebra, is the introduction and operation of functions. Where functional analysis can be confusing for eighth graders lies in the difference between invertible functions and non-invertible functions, as well as linear functions versus nonlinear functions. As we explored the infinite in our “What’s in a Number?” class, we were introduced to infinite sets and Georg Cantor’s ideas around the cardinality of those same sets. Cantor, the founder of Set Theory, defined sets of the same cardinality as sets where each element of one set has a one-to-one correspondence to an element in the other set.^{[1]} A function is defined as a relationship between two sets of numbers where each input corresponds to one, and only one, output. Under this construction, in both and , is a function of . However, in , is also a function of , but in , is not a function of . (When , is simultaneously and .) Therefore, is invertible, but is not.
In Eighth Grade, the focus is on linear functions, which are, by definition, invertible, even though non-invertible functions are introduced. In the Algebra sequence, however, nonlinear and non-invertible functions are used and analyzed more intensely. Additionally, in Eighth Grade, set notation is often used informally, but since there is no set theory standard, there is no formal definition. As a result, students are not always ready to differentiate between {3, 4}, a two-element set that contains the numbers 3 and 4, and (3, 4), a single point on the coordinate plane. Similarly, while {3, 4} and {4, 3} are interchangeable, (3, 4) and (4, 3) represent two different points and cannot be substituted one for the other.
This unit will introduce students to a more formal definition of sets and set operations, hopefully clearing up the confusion between a two-element set and an ordered pair, and then look at finite sets as functions of other finite sets, determining whether these sets are invertible or non-invertible. As students develop comfort with finite sets, they are introduced to their first infinite set, the one against which all other infinite sets are measured: the set of natural numbers. Once they have been introduced to the set of natural numbers, they apply their knowledge of functions (and invertible functions) to other sets to determine whether other infinite sets (including the set of integers and the set of rational numbers) are countable, by Cantor’s definition. Finally, we will look at the fact that the set of all rational numbers is countable, but the set of all irrational numbers is not countable, meaning that the set of all real numbers is not countable.
^{[1]} Emanuel Lazar, Ideas in Mathematics, Lesson 3: Cardinality, p. 19, https://www2.math.upenn.edu/~mlazar/math170/notes03.pdf, February 2016.
The lessons in this unit are built on the current framework for School District of Philadelphia Mathematics lessons, which are based on a constructivist model for Math education. Each lesson starts with an opening routine, an open-ended question designed to encourage student thinking without requiring intense mathematical thought. Often, it begins to provide a potential insight into the formative task. The formative task is a single question, designed to be accessible to a wide variety of students while building to the day’s learning target. The guided instruction portion is split into two: connect and explain, a few questions designed to help students make the connection between the formative task and the learning target, and apply and practice, another problem designed to help students test their independence on the learning target. Following that is the inclusive student activities, which are independent practice problems, and the reflective closure is akin to an exit ticket. Attached to each part of the lesson is a suggested timing as well as, in italics, teacher notes to help with implementation and answers to the questions.
This format is designed to be student-centric. Student voice is paramount in this model; part of the implementation notes includes potential student responses and notes on ways to order and display student work. This work is difficult for eighth grade, and so more teacher voice than is ideal may be necessary to help students access the objectives. However, when possible, elevate student voice wherever your students have provided it.
The goal of this lesson is for students to understand what a set is, set notation, and how we define set cardinality.
8.F.A.1: Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.
Select any (or all) of the following that share something in common and explain what that thing in common is.
Apples
Lemons
Oranges
Strawberries
Raspberries
Tomatoes
Carrots
Celery
Broccoli
Spinach
Eggs
Kiwi
Cherries
Milk
Pears
Plums
Asparagus
Chocolate
Chicken
Chipotle
Teacher’s Note: This opening routine is designed to get students thinking about sets as groups with something in common, groups that follow a rule. This list of 20 items is designed to have multiple possibilities for grouping, making the students’ rationales vital. Once you collect their sets (and pay attention to their notation for these), an extension question might be to ask students if they can think of anything else not from the original 20 that would belong in their list.
Launch (3-5 minutes):
Build off of student responses from the opening routine. If students used any of the criteria listed below (fruits, used in a pie, vegetables), acknowledge that. If not, ask students to come up with a couple examples together for each group and to think about whether there are any elements that will work for two or more of the criteria. Before sending students to work on these sets on their own, make sure they pay attention to how they are displaying this information.
Task (10 minutes):
From the list in the opening routine, Angel created a set, A, of all the fruits. Barry created a set, B, of all the items used in making pie, and Comfort created a set, C, of all the vegetables.
Debrief (5-7 minutes): This formative task is designed for students to begin to transition to a more formal idea around sets, including unions, intersections, elements, differences, deltas, and complements. As students present their answers, begin translating their answers to set notation, including (2a), (2b), (2c), and or (2d). Use cardinality notation to show the number of elements in each set: .
Today, we will understand what a set is, set notation, and how we define set cardinality.
Teacher’s Note: While the objective is at the beginning of this lesson plan because all elements need to point back to the objective, this is the point of the lesson where students should be formally introduced to what they are learning today. As you review the learning target with them, check on their comfort with the vocabulary (this is the first time they are formally introduced to the concept of sets, set notation, and cardinality. While they may be able to use context clues to develop an informal definition for some or all of these concepts, this is where we will formalize their understanding.
Set – a finite or infinite set of objects where order has no significance.^{[1]}
Set notation – the symbols we use to represent sets and the operations of sets.^{[2]}
Set: Let {a, b, c} where a, b, and c are all elements of the same set.
Union: , the set of unique elements of two or more sets.
Example: Let {a, b, c} and {a, c, e}. {a, b, c, e}.
Intersection: , the set of elements that exist in all of the referenced sets.
Example: Let {a, b, c} and {a, c, e}. {a, c}
Difference: , the set of elements in set that are not in set .
Example: Let {a, b, c} and {a, c, e}. {c}, while ={e}.
Cardinality – the number of elements of a set, notated with straight lines (like absolute value).
Example: Let {a, b, c}. .
Teacher’s Note: Begin this section (also Guided Instruction – Connect and Explain) by asking students to informally define these terms, as they appeared in the learning target. Once students have informally defined the terms, provide formal definitions with examples. If necessary, work other examples before the Apply and Practice section, though there will be opportunities to continue to clarify notation and vocabulary during that section.
Let our world be the set of integers greater than 0 and less than 10.
Let the set of prime numbers in this world.
Let the set of even numbers in this world.
{2}
{2, 3, 4, 5, 6, 7, 8}
{3, 5, 7}
{4, 6, 8}
4
If students are struggling with set notation:
{2, 3, 5, 7, 11, 13, 17, 19, 23}
9
{3, 6, 9, 12, 15, 18, 21, 24}
{2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24}
The set of multiples of 6 between 0 and 25.
4
{2, 3, 5, 7}
{11, 13, 17, 19}
{23, 29}
{31, 37}
{5, 7}
{11, 15}
{15}
They have the same cardinality (9).
For students exhibiting a greater comprehension of sets and set notation:
Yes. As part of our definition of a set, order has no significance. Therefore, where the
values are the same, there is equivalence regardless of order.
The cardinality of is greater than the cardinality of . In fact, it is one less
than twice as much as the cardinality of B.
Only one element in B can represent the absolute value of any element in .
With the exception of 0, each element in is represented by two values in .
8: {1, 2, 4, 5, 6, 7, 8, 10}
3: {2, 4, 8}
2: {6, 10}
3: {1, 5, 7}
5: {1, 5, 6, 7, 10}
1;
have a total of 8 elements.
There are an infinite number of solutions to this. The key elements: and
have 9 unique elements between them, and they share 5 of those. has 8
elements altogether, and has 6. has 3 elements that are not in and has
1 element that is not in .
A and B are the same set, if in a different order. Therefore, A-B is an empty set, either {}
or .
What is something you understand about sets after this lesson?
Let the set of odd numbers between 0 and 10. Let the set of multiples of 3 between 0 and 16.
{3, 9}
They have the same cardinality (5).
The goal of this lesson is that students apply their knowledge of functions to determine whether functions between finite sets are invertible and see that finite sets that are invertible functions have the same cardinality.
8.F.A.1: Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.
How are these the same? How are they different?
The goal of this same/different is that students begin to notice that the magnitudes are the same in both, the number of elements is different, and that there is some relationship between the two sets.
Launch (3-5 minutes): Let’s think about the opening routine: is there a rule that will take set a to set b? What about set b to set a? We can see a relationship; the question is can we define that relationship? For the formative task, you will be given two different versions of two related sets. Your task will be to determine the relationship between the two sets and, if you reversed the “input” and “output,” you could still define the relationship between the two sets.
Task (10 minutes):
Below are two sets.
B: {0, 1, 4, 9, 16, 25} D: {-4, -2, 0, 2, 4, 6, 8, 10, 12, 14, 16}
Debrief (5-7 minutes): If students are struggling with the format in which these sets are presented, encourage them to create an arrow diagram. Students should see that squaring each element in set A will give you a corresponding element in set B, but, when you take the square root of B, you don’t get all of A. Similarly, the rule that gets you from C to D is 2x+6, while the rule that takes you from D to C is ½x-3. C and D have the same cardinality. During this time, as you are showing student answers, encourage students to think about why C and D have the same cardinality while A and B do not, and how it relates to them as functions.
Today, we will apply our knowledge of functions to determine whether functions between finite sets are invertible and see that finite sets that are invertible functions have the same cardinality.
What do you think the difference between a function and an invertible function is?
A function only needs each input to have a unique output. Multiple inputs can map to the same output. For an invertible function, if the inputs and outputs are reversed, it preserves its status as a function. In an invertible function, each input maps to a unique output.
Why do you think that a function has to be invertible for the sets to have the same cardinality?
A function that is not invertible will have more elements in the input than the output, because at least two of those inputs will map to one output, meaning that whatever the cardinality of the output, the input will be at least one greater. An invertible function requires that each set has the same number of elements so that each input can map onto one output and each output can map onto one input.
Is it necessary for there to be a function rule for two sets to be related as a function?
No, but, in large sets, being able to define the relationship between the sets by a rule allows us to show a one-to-one relationship without enumerating every element of each set.
Feel free to supplement the set notation with other functional notation shown in eighth grade, most helpfully graphs and arrow diagrams. Additionally, encourage students to think about why they can find an answer for number 3, but couldn’t for the similar version in the formative task. What changed? In all these examples, students may find other rules to match two sets. If the rules are true, then they are also valid. The rules listed are the simplest rules.
Let A represent all-natural numbers x where 0<x<10. Let B represent all y where y = x^{2}.
A: {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} B: {1, 4, 9, 16, 25, 36, 49, 64, 81, 100}
Yes. Each element of A matches to an element of B. Rule: y = x^{2}.
Yes. Each element of B matches to an element of A. Rule: y=√x.
Yes. In the domain and range defined, each input maps to a unique output, and each
output maps to a unique input. Additionally, both sets have 10 elements.
The most likely response here is an arrow diagram. The main requirement is that the student shows a one-to-one relationship between the elements of A and the elements of B.
Again, the most likely response here is a similar arrow diagram. The main requirement is still that the students show a one-to-one relationship between the elements of B and the elements of A.
While both sets are definable (A is the set of prime numbers between 1 and 11, inclusive, and B is the set of even numbers between 1 and 11, inclusive), they are not definable in terms of each other.
Yes. There is a function that takes A to B. There is also a function that takes B to A, so they have the same cardinality. Additionally, they have the same number of elements.
Yes, y=|x|.
No. I can say y=+/-x, but that is two rules, not one.
This is an open-ended question. Make sure that there is a function that can take A to C and C to A.
Yes, they are related to each other by an invertible function, so they have the same cardinality.
Yes, an example of a function from A to B is y=x^{2} and the reverse function from B to A is y=√x.
The easiest change would be to expand the domain from 1-100 to -100 to 100 and the range from 1-10,000 to 0-10,000. However, if the students find a different way, they can.
No. In the obvious function rule, y=x^{2}, for all values in A that are greater than 10, there are no elements in B that would match y. Note: in set theory, this does not matter, because order does not matter. The main concern is that one set has more elements than the other. However, because we are relating it to functions, and measuring cardinality from those functions, the idea is that I can map every element from one set to the other, but not the other way around.
No. Again, in the obvious function rule, y=x^{2}, for all values in A that are greater than 100, there are no values in B that would match y. Again, in sets we do not have corresponding values, simply elements, but in functions, we do. The important concept here is that there are more elements in A than in B.
Let A be the set of all integers between -10 and 10. Let B be the set of all perfect cubes between -1000 and 1000. Do A and B have the same cardinality? Prove it.
There is a function from A to B in the form y=x^{3}. There is a function from B to A in the form y=^{3}√x. Because cubic functions are invertible (the negative values are preserved), there is a unique output in B for every element in A, forcing A and B to have the same cardinality.
The goal for today’s lesson is for students to transfer their knowledge of finite sets to the concept of infinite sets and understand that the set of Natural numbers is an infinite set.
8.F.A.1: Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.
If I were to hand you a handful of sand, how would you count the grains?
Teacher’s note: This opening routine is designed to get students thinking about infinity, and how infinity relates to counting numbers. Students may have a variety of solutions, including that it can’t be done, count them by dropping each grain, one by one, into a container and counting, weigh the handful and then weigh one grain, or many others. The push is for students to begin thinking about large numbers, and infinity as both a concept and, in some ways, a number.
Launch (3-5 minutes): The opening routine leads directly into the formative task. Make sure to document students’ hypothetical strategies, and then connect them to the task: they are to count an “uncountable” (actually 10,000) number of dots. Ask if they think the image represents all the grains of sand at Rehoboth Beach (or some other local beach; feel free to replace with one more local that your students would recognize).
Task (10 minutes):
Kathleen has decided that she will do the unthinkable and count, not just the number of grains of sand in a handful, but also the number of grains of sand at Rehoboth Beach. We are going to help her out. With whatever methodology you can come up with, count the grains in the sample shown below.
^{[1]} Adapted from the Wolfram-Alpha Definition, https://www.wolframalpha.com/input?i=set&assumption=%7B%22C%22%2C+%22set%22%7D+-%3E+%7B%22MathWorld%22%7D.
^{[2]} Cue Math has a good and quick explanation of the operations and notations listed here, https://www.cuemath.com/numbers/set-notation/
Debrief (5-7 minutes): Look for student strategies; some potential strategies: find a smaller subsection and find the number of circles within that subsection and multiply that out; count each individual one, give up and call it infinity. If students call it infinity, challenge them with the idea that there is an area to the “grains” and an area to the region, so there has to be a finite number of grains in a finite space. The goal is for students to think about large numbers and how they notate them.
Today, we will explore the concept of infinite sets and understand that the set of Natural numbers is an infinite set.
When you read “infinite set,” what came to mind?
Some students might reference the set of Natural numbers, or integers, or, perhaps some set with an infinite number of items or a set with infinity as an item. The actual answer is a set with an infinite number of elements.
As a class, list as many Natural numbers as we can in one minute. Have we listed all the Natural numbers?
Record students’ answers as a set; they should be looking at positive whole numbers
Can you count the number of counting numbers?
Students may come up with a variety of different answers to this, some may think that because counting numbers are, most literally, the numbers we count with, then we should definitely be able to count them. Others may say that there are an infinite number of counting numbers (we can always go one higher), so that we cannot count the counting numbers. Allow students to live in this paradox. If only one side is presented, present the other side. This paradox is the key to understanding and the countability of infinite sets.
In the 19th Century, German mathematician Georg Cantor said that, while we can never “count” the elements in the set of Natural numbers, notated from now on as , we can consider that to be a countable infinite set. He also expanded our definition of cardinality. In our last lesson, we defined cardinality as the number of elements in a set. Cantor expands this definition to say that two sets have the same cardinality if the elements of one set can be mapped in a one-to-one relationship.^{[1]} In essence, if there is a function that takes A to B and there is a function that takes B to A, then and have the same cardinality.
Let the set of counting numbers. Let represent an element of () and the set of values that are equivalent to . Do and have the same number of elements?
Is an infinite set? Explain.
Is a countable set? Explain.
Let students discuss this. is a function of , and is a function of , so, while they are both infinite, they have the same number of elements, and so are a countable set. This is also a good time to introduce the concept of an invertible function: if y is a function of x, f(x) and x is also a function of y, g(y) then f(x) is invertible.
is an infinite set. There is no such thing as a “greatest” odd number, so the
set of odd numbers has no upper bound, making it infinite.
is an infinite set. There is no such thing as a “greatest” even number, so the set of even numbers has no upper bound, making it infinite.
can be written as a function when , namely .
Conversely, can be written as a function when , namely
.
{1, 3, 5, 7, 9}. Because has an upper and lower bound, as well as a
predetermined distance between elements, is not an infinite set.
{1, 2, 3, 4, 5, 6, 7, 8, 9}. Because has an upper and lower bound, as well
as a predetermined distance between elements, is not an infinite set.
No, the cardinality of is 5, while the cardinality of is 9.
is an infinite set because there is no maximum?
is an infinite set because there is no minimum?
can be expressed as the function where and can be
expressed as the function where where . Therefore, they
have the same cardinality.
Let . where and the sign of is the same sign of .
Because , there are an infinite number of possible values of ,
making an infinite set.
Let . where and . Because there are
an infinite number of possibilities for b, is an infinite set.
The function where , can take A to B, and the function
where , can take B to A, and since only contains
rational squares and only contains positive numbers, the function is invertible.
Let and where . Let where and are perfect
squares and . but does not have to be an element of .
Therefore, the cardinality of is greater than 0.
Let and where . Let where and are perfect
squares and . . Therefore, the cardinality of is equal to
0.
Let . Let . , and, since is an infinite set, is an infinite
set.
Let . Let . , and, since is an infinite set, is an infinite
set.
Let and . can be represented by the function while
can be represented by the function . Because we have invertible
functions, and have the same cardinality.
In essence, we determine the countability of an infinite set by determining whether set
can be expressed as some function and whether can be expressed as some
function .
Yes, there is no maximum for .
Yes. Let . Let . because is a positive integer.
Therefore, and .
The goal of today’s lesson is that students will transfer their knowledge from and develop the set of integers, determining whether the set of integers is countable.
8.F.A.1: Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.
Which of these does not belong? Why?
Potential responses: a is the only response where 0 is not included in the set; b is the only response where 0 is included but it has no starting or ending point; c is the only infinite set that starts at 0 and then “telescopes” out to infinity and negative infinity, and d is the only finite set. Students are going to begin expanding their “standard” infinite sets in this lesson, though not counting them yet, and these are also designed to get students thinking about the definition of an integer.
Launch (3-5 minutes): Ask students what they notice about the infinite sets. Ask if they can come up with a “set” rule for each set in the opening routine. Write these rules down. Then ask them the difference between whole numbers, natural numbers, and integers. This may be a place of confusion for them; it was introduced in prior grades (about 4th), but keeping each definition straight is often a struggle for students.
Task (10 minutes): Write the definition of integer, as precisely as you can, using words. Then, define the set of all integers as precisely as you can using set notation.
Debrief (5-7 minutes): Monitor student definitions for precision: many dictionaries, including the Oxford and Cambridge dictionaries, define an integer as “a whole number, not a fraction.”^{[2]} Since whole numbers make up a subset of integers, rather than the complete set, this is imprecise. Wolfram-Alpha defines an integer, colloquially, as “one of the numbers …, -2, -1, 0, 1, 2, ….”^{[3]} We are gaining precision, but are looking for something more, potentially something to the effect of a number that is either a whole number or the additive inverse of a whole number. In set notation, it might be notated . Some students might also write the set the way Wolfram-Alpha did. Build from the Wolfram-Alpha notation to the set unions. We will be using both tomorrow.
Today, we will develop definitions and notations for: the set of all integers, the set of all rational numbers, the set of all real numbers, and the set of all irrational numbers. We will also show that the set of all integers is countable.
Do you think that the set of all integers, , has the same cardinality as
This is a point for students to begin to grapple with the idea that, objectively, there should be more integers than counting numbers, by the order of 2n+1. After all, if I had the set of counting numbers with an absolute value less than or equal to 4, there would be 4 counting numbers. If I had integers with an absolute value less than or equal to 4, there are 9 elements in that set.
Can you create function to define as a function of . Show at least two attempts to create it.
While is a function of , they are not related by an arithmetic function rule. Look for solid student attempts.
If you can’t create a function rule to define the relationship between each element of and each element of , does that mean that is not a function of ?
This is the place to clear up that potential misconception: we do not need to have a function rule to have a function; it simply helps us generalize more readily.
Using what we know about sets, show whether and have the same cardinality. Start with {…, -2, -1, 0, 1, 2, …} and {1, 2, 3, 4, 5,…}.
Show students that if we look at as starting at negative infinity and positive infinity, the function that would describe to is not invertible. However, if we reorder to start with 0 and telescope out ({0, -1, 1, -2, 2, -3, 3…}), we can predict the nth value of the set, and we can assign a counting number to each element.
We can show that is represented by {1, 2, 3, 4, …} and the set of all whole numbers is represented by {0, 1, 2, 3, …}. Thus, the function y=x+1 can take each element of to an element in the set of all whole numbers, while the function y=x-1 can take each element from the set of all whole numbers to a corresponding element of .
This is an open-ended answer. Look for students to understand that they will have to show that each set is a function of the other, either through a formula or a predictable arrow diagram.
The set of all prime numbers begins {2, 3, 5, 7, 11, 13, 17, 19, …}. I can assign in the form {1, 2, 3, 4, 5, 6, 7, 8, …} to count the set of all prime numbers, and every time I add a prime number, there is a new counting number to count that prime.
Now that we have shown to be countable, the function y=x^{3} will map each element from the set of perfect cubes to an element of , while the equation y=^{3}√x will map each element of to an element in the set of all perfect cubes. Because we have an invertible function, there is a one-to-one relationship, and so they have the same cardinality. If the cardinality of the set of perfect cubes is equal to the cardinality of the set of integers, which is in turn equal to the cardinality of the set of natural numbers, then the cardinality of the set of perfect cubes is equal to the cardinality of the set of natural numbers.
As part of the definition of sets, order does not matter. So, when I change the order but leave a pattern where the next number is clearly discernible, the sets remain the same.
What is the difference between and , and why do they have the same cardinality?
includes positive numbers, negative numbers, and 0, while only includes positive numbers.
The goal of this lesson is that students will be able to define the set of rational numbers, , , and the set of all irrational numbers and determine whether each set is countable based on Cantor’s rules of cardinality.
8.F.A.1: Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.
What do you notice about this graph?
What do you wonder about this graph?
^{[1]} Emanuel Lazar, Ideas in Mathematics, Lesson 3: Cardinality, p. 19, https://www2.math.upenn.edu/~mlazar/math170/notes03.pdf, February 2016.
^{[2]} “Cambridge University Press, “integer,” https://dictionary.cambridge.org/us/dictionary/english/integer, accessed 12/13/22.
^{[3]} Wolfram-Alpha, “integer,” https://www.wolframalpha.com/input?i=integer, accessed 12/13/22.
Teacher’s note: We will be using this graph to make visual sense of the fact that we can count the set of all rational numbers. In this graph, each line represents an element of , while each of the green dots represents a unique rational number. This opening routine is designed to help students begin making sense of this graph.
Launch (3-5 minutes):
Point out that the green dots represent all the rational numbers between 0 and 1, while the lines represent our natural numbers. Ask where rational numbers between 1 and 2 live. Ask students how we know that this is true.
Below is a graph of “all” rational numbers as graphed by the vertical access representing the denominator and the horizontal axis representing the numerator. (Note: we are only seeing a window of the actual graph).
Students may notice that the green dots form triangles, that there are different numbers on each row, that there seems to maybe be a pattern about the number of dots in a given row.
Our last row was a prime denominator, meaning that every intersection had a dot. Our
next row, with a denominator of 12, has only 4 new unique values (1/12, 5/12, 7/12, and
11/12). 13, however, will have 12 new values, and 14 will have 6 new values. In general,
the number of new values is increasing, but oscillating on a given row.
Subdivisions Green Dots Number in Row
1 0 0
2 1 1
3 3 2
4 5 2
5 9 4
6 11 2
7 17 6
8 21 4
9 27 6
10 31 4
11 41 10
There are a number of ways to do this. There does seem to be a visual pattern of threes developing. We should always be able to count the dots.
In general, each new subdivision or row adds a finite number of new rational numbers to the set. If I am adding a finite number of new rational numbers with each subdivision, then I am adding a finite number to a finite number which is, by definition, countable. I can create (or imagine) a one to one relationship between the number of rational numbers between 0 and 1 at a subdivision so large I must represent it with infinity and the set of all counting numbers.
If I increase the domain of my numerator at any given range, it increases the number of points, but not the inherent countability of those points. By extension, with any arbitrary increase in domain and/or range, I can continue to find numbers from to map to the new point(s). Therefore, as the domain and range approaches infinity, has the same cardinality as , and so the points remain countable.
Today, we will define the set of rational numbers, , , and the set of all irrational numbers and determine whether each set is countable based on Cantor’s rules of cardinality.
How can we define, in words, a rational number?
Rational numbers should have been defined in 7th grade. At its most basic, a rational number is any number that can be expressed as the ratio (or quotient) of an integer (dividend/numerator) and natural number (divisor/denominator). This definition automatically eliminates division by 0 and, since one of the two numbers can be positive or negative, the other one does not. As students give answers approaching the level of precision for which you are looking, write them all down, and write down counterexamples that might fit their definition but are not rational numbers.
How can we define, in words, a real number?
Eighth graders have not yet had a formal introduction to real vs. complex numbers, but the definition is necessary to understand irrational numbers. WolframAlpha defines a real number as “a number corresponding to a point on the real number line.”^{[1]} If students are confused by this definition, point out that, while both the x- and the y- axes exist in two-dimensional space, they can exist independently. However, in the complex world, i, which takes the place of the y-axis, cannot exist independently of the real axis (x). Therefore, if I can place a point on either the y-axis or the x-axis, it is a real number.
How can we define, in words, an irrational number?
The reason we are not going to define a real number as a number that is either rational or irrational is that we need that difference here: an irrational number is any real number that is not a rational number. Some students may say, colloquially, that an irrational number is any number where the decimal form neither terminates nor repeats (which is true for real numbers, as the decimal form of any rational number will eventually terminate or repeat.)
How can we use set notation to define the set of all rational numbers?
Introduce students to . Let .
How can we use set notation to define the set of all real numbers?
In this case, because there is neither an algebraic form nor a symbol for the set of irrational numbers (which is why it comes last), we cannot use a set union to define the real numbers (just as we did not in our verbal definition). That means that we will just say that represents the set of all real numbers.
How can we use set notation to define the set of all irrational numbers?
We have difference notation, represents the set of all irrational numbers. Otherwise, we would have to use complement notation, in the world of .
Looking at the graph from the formative, why is it impossible to count the irrational numbers?
Let the plane defined by the grid represent the set of all real numbers. If the intersections of the grid lines represent some form of a rational number, then the space between the grid lines on a given row represents the set of uncounted rational numbers and the set of irrational numbers. Unlike we did with the rational numbers, not every can be named using the set of natural numbers. Intuitively (and this is an intuition and not a proof), I cannot show that there is a one-to-one correspondence between rational and irrational numbers because I cannot derive the set of irrational numbers from the set of natural numbers. Cantor’s diagonalization method proves the uncountability of irrational numbers.^{[2]}
If we cannot count the irrational numbers, what does that say about the set of real numbers, ?
, as the union of a countable infinite set and an uncountable infinite set, is an uncountable infinite set.
Assume that we are only looking at the infinite sets .
Students should write a set of natural numbers.
Students should write a set of negative integers.
Students should write a set of fractions and/or decimals, as long as they cannot reduce to integers.
Students should write irrational numbers, including e, π, square roots of non-perfect squares, etc.
Neither, they have the same cardinality.
Neither, they have the same cardinality.
Neither, they have the same cardinality.
; it is uncountable where is countable.
How does our knowledge of functions help us to make sense of the world of infinite sets?
Functions are, in essence, how we determine the cardinality of an infinite set. As a result, they help us see relationships between infinite sets that we might not catch were we simply relying upon finite logic.
Any set with 7 elements.
{-10, -9, -6, -1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 26,
39, 54, 71, 90, 111, 134, 159, 186, 215, 246, 179, 314, 351, 390}
{6, 15}
{1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20}.
Any two of , because we can map the elements from one to another using a one-to-one relationship.
The two sets that we have defined as uncountable in this unit are the set of all real numbers, , and the set of all irrational numbers.
The easy way to do this would be to use an arrow diagram going both ways. Some students may pick up on the fact that y=x^{2}-2, so x=√(y+2).
^{[1]}WolframAlpha, “real numbers,” https://www.wolframalpha.com/input?i=real+numbers&assumption=%7B%22C%22%2C+%22real+numbers%22%7D+-%3E+%7B%22MathWorld%22%7D, accessed 12/13/2022.
^{[2]} Jeremy L. Martin, Cantor’s Diagonal Argument, https://jlmartin.ku.edu/courses/math410-S09/cantor.pdf, accessed 12/23/22
Cambridge University Press. “integer.” Cambridge English Dictionary.
https://dictionary.cambridge.org/us/dictionary/english/integer. accessed 12/13/22.
While WolframAlpha is one of the main mathematical references used for vocabulary in
this unit, its definition of integer was incomplete. The Cambridge dictionary is another
authority, but their definition for integer was also slightly incomplete.
Cue Math. “Set Notation – What is Set Notation?” Cue Math.
https://www.cuemath.com/numbers/set-notation/.
For people who have not used sets recently and need a refresher on set notation, this is a
good resource to review. I used it as a reference to make sure my notations were correct.
Lazar, Emanuel. Ideas in Mathematics, Lesson 3: Cardinality, p. 19.
https://www2.math.upenn.edu/~mlazar/math170/notes03.pdf, February 2016.
This was the article that we used in our session as the introduction to set cardinality and
Cantor’s Theorem. It is a good overview at the topic at hand, while still being detailed.
Martin, Jeremy L. Cantor’s Diagonal Argument.
https://jlmartin.ku.edu/courses/math410-S09/cantor.pdf. accessed 12/23/22.
WolframAlpha. www.wolframalpha.com.
This site is the main math dictionary used in this unit. It is one of the preeminent math
resources on the internet.
All activities in the lesson plans are original to this unit, as are all images, created by the author using the Desmos Graphing Calculator.
The main standard this unit addresses is 8.F.A.1: Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. This unit connects the one-to-one correspondence that determines whether two sets have the same cardinality to functions, particularly invertible functions. This builds on the eighth grade understanding of functions to determine cardinality. Not only does each element of the input have to map to only one output, but each element of the output has to map to only one input.
While this is the main standard addressed in this unit, 8.NS.A.1: Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number. The main infinite sets explored in this unit are and the set of all irrational numbers. As part of this, students will need to be able to differentiate between the different categories of numbers. We do not look at decimal expansion, but we do look at the difference between rational and irrational numbers.
On next page.
Standard:
8.F.A.1: Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.
Opening Routine:
Select any (or all) of the following that share something in common and explain what that thing in common is.
Apples
Lemons
Oranges
Strawberries
Raspberries
Tomatoes
Carrots
Celery
Broccoli
Spinach
Eggs
Kiwi
Cherries
Milk
Pears
Plums
Asparagus
Chocolate
Chicken
Chipotle
Formative Task:
From the list in the opening routine, Angel created a set, A, of all the fruits. Barry created a set, B, of all the items used in making pie, and Comfort created a set, C, of all the vegetables.
Learning Target:
Today, we will understand what a set is, set notation, and how we define set cardinality.
Vocabulary:
Set – a finite or infinite set of objects where order has no significance.
Set notation – the symbols we use to represent sets and the operations of sets.
Set: Let {a, b, c} where a, b, and c are all elements of the same set.
Union: , the set of unique elements of two or more sets.
Example: Let {a, b, c} and {a, c, e}. {a, b, c, e}.
Intersection: , the set of elements that exist in all of the referenced sets.
Example: Let {a, b, c} and {a, c, e}. {a, c}
Difference: , the set of elements in set that are not in set .
Example: Let {a, b, c} and {a, c, e}. {c}, while ={e}.
Cardinality – the number of elements of a set, notated with straight lines (like absolute value).
Example: Let {a, b, c}. .
Guided Instruction – Apply and Practice:
Let our world be the set of integers greater than 0 and less than 10.
Let the set of prime numbers in this world.
Let the set of even numbers in this world.
Inclusive Student Activities:
If students are struggling with set notation:
Let the set of prime numbers between 0 and 15, and let be an element of . Let be the set of numbers that can be represented by .
For students exhibiting a greater comprehension of sets and set notation:
Reflective Closure
What is something you understand about sets after this lesson?
Let the set of odd numbers between 0 and 10. Let the set of multiples of 3 between 0 and 16.
Standard:
8.F.A.1: Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.
Opening Routine:
How are these the same? How are they different?
Formative Task:
Below are two sets.
B: {0, 1, 4, 9, 16, 25} D: {-4, -2, 0, 2, 4, 6, 8, 10, 12, 14, 16}
Learning Target:
Today, we will apply our knowledge of functions to determine whether functions between finite sets are invertible and see that finite sets that are invertible functions have the same cardinality.
Guided Instruction – Connect and Explain:
What do you think the difference between a function and an invertible function is?
Why do you think that a function has to be invertible for the sets to have the same cardinality?
Is it necessary for there to be a function rule for two sets to be related as a function?
Guided Instruction – Apply and Practice:
Let A represent all-natural numbers x where 0<x<10. Let B represent all y where y = x^{2}.
Inclusive Student Activities:
Reflective Closure:
Let A represent the set of all integers between -10 and 10. Let B represent the set of all perfect cubes between -1000 and 1000. Do A and B have the same cardinality? Prove it.
Standard:
8.F.A.1: Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.
Opening Routine:
If I were to hand you a handful of sand, how would you count the grains?
Formative Task:
Kathleen has decided that she will do the unthinkable and count, not just the number of grains of sand in a handful, but also the number of grains of sand at Rehoboth Beach. We are going to help her out. With whatever methodology you can come up with, count the grains in the sample shown below.
Learning Target:
Today, we will explore the concept of infinite sets and understand that the set of Natural numbers is an infinite set.
Guided Instruction – Connect and Explain:
When you read “infinite set,” what came to mind?
As a class, list as many Natural numbers as we can in one minute. Have we listed all the Natural numbers?
Can you count the number of counting numbers?
Guided Instruction – Apply and Practice:
In the 19th Century, German mathematician Georg Cantor said that, while we can never “count” the elements in the set of Natural numbers, notated from now on as , we can consider that to be a countable infinite set. He also expanded our definition of cardinality. In our last lesson, we defined cardinality as the number of elements in a set. Cantor expands this definition to say that two sets have the same cardinality if the elements of one set can be mapped in a one-to-one relationship. In essence, if is a function of and is a function of , then and have the same cardinality.
Let the set of counting numbers. Let represent an element of () and the set of values that are equivalent to . Do and have the same number of elements?
Is an infinite set? Explain.
Is a countable set? Explain.
Inclusive Student Activities:
Reflective Closure:
Standard:
8.F.A.1: Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.
Opening Routine:
Which of these does not belong? Why?
Formative Task:
Write the definition of integer, as precisely as you can, using words. Then, define the set of all integers as precisely as you can using set notation.
Learning Target:
Today, we will develop definitions and notations for: the set of all integers, the set of all rational numbers, the set of all real numbers, and the set of all irrational numbers. We will also show that the set of all integers is countable.
Guided Instruction – Connect and Explain:
Do you think that the set of all integers, , has the same cardinality as
If you can’t create a function rule to define as a function of , does that mean that is not a function of ?
Guided Instruction – Apply and Practice:
Using what we know about sets, show whether and have the same cardinality. Start with {…, -2, -1, 0, 1, 2, …} and {1, 2, 3, 4, 5,…}.
Inclusive Student Activities:
Reflective Closure:
What is the difference between and , and why do they have the same cardinality?
Standard:
8.F.A.1: Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.
Opening Routine:
What do you notice about this graph?
What do you wonder about this graph?
Formative Task (20 minutes):
Below is a graph of “all” rational numbers as graphed by the vertical access representing the denominator and the horizontal axis representing the numerator. (Note: we are only seeing a window of the actual graph).
Subdivisions Green Dots Number in Row
1 0 0
2 1 1
3 3 2
4 5 2
5 9 4
6 11 2
7 17 6
8 21 4
9 27 6
10 31 4
11 41 10
Learning Target:
Today, we will define the set of rational numbers, , , and the set of all irrational numbers and determine whether each set is countable based on Cantor’s rules of cardinality.
Guided Instruction – Connect and Explain (5 minutes):
How can we define, in words, a rational number?
How can we define, in words, a real number?
How can we define, in words, an irrational number?
Guided Instruction – Apply and Practice:
How can we use set notation to define the set of all rational numbers?
How can we use set notation to define the set of all real numbers?
How can we use set notation to define the set of all irrational numbers?
Looking at the graph from the formative, why is it impossible to count the irrational numbers?
If we cannot count the irrational numbers, what does that say about the set of real numbers, ?
Inclusive Student Activities:
Assume that we are only looking at the infinite sets .
Reflective Closure:
How does our knowledge of functions help us to make sense of the world of infinite sets?