Objectives

Students will explore notations through the use of functions and sets in this unit. Students will:  - examine patterns. - explore a variety of representations/notations of sets and functions, and establish connections between the two. - investigate the components and attributes of functions and sets. - learn notation and use and interpret them in context.

Core Questions

- What notations are widely accepted in mathematics? What is the function of notations in the context of mathematical comprehension?  - How do mathematical notations function within the problem-solving process?

Vocabulary

- Counting Numbers: Any of the positive whole numbers, 1, 2, 3,…. - Disjoint Set: Two sets are disjoint if there is no point which belongs to each of the sets, i.e., if the intersection of the sets is the null set. - Domain: For a function, the set of all values which the independent variable may take on. - Element: A single component found within a set. - Empty Set: A set without any elements, denoted with { } or Ø. - Finite Set: A set with a definitive number of elements listed, i.e., A = {1, 2, 3}. - Function: A relation in which each input element is mapped to a unique output element. A function describes a set and points to elements in the set. - Infinite Set: A set that is not finite, one whose members cannot be enumerated, i.e., F = {all real numbers}. - Integer: Any of the positive or negative whole numbers, including 0, i.e., 0, ±1, ±2, ±3,… - Intersection: The common ground for two sets; the elements common to sets A and B; the elements found in Set A AND Set B. - Interval Notation: A notation that uses the endpoints of the set to describe the elements, i.e., [4, ∞) represents the set of all real numbers greater than or equal to four. - Irrational: A real number not expressible as an integer or quotient of integers. - Linear Function: A function of degree one. - Mapping: Two sets related in such a way that to each element of set A there corresponds a unique element f(x) of a space B; then there is said to be a mapping or map f of the set A in the set B, and the point f(x) is said to be the image of the point x. - Natural Numbers: Any of the positive integers, 1, 2, 3,…. - Null Set: The set which is empty, has no members. - Proper Subset: A subset, in which the sets are not equal; for example, the factors of 6 {1, 2, 3, 6} are a proper subset of the factors of 18 {1, 2, 3, 6, 18}. - Range: For a function, the set of values the function may take on. - Rational: An algebraic expression which involves no variable in an irreducible radical or under a fractional exponent; a number that can be expressed as an integer or as a quotient of integers. - Relation: A subset of a set associated with another set. - Roster Notation: A notation where the elements of each set are simply listed, i.e., Set A = {1, 2, 3,…}. - Set: A group of elements. - Set Builder Notation: A notation where the set is described with symbols, formally, i.e., A = is the set of all x, such that x is an element of the natural numbers. - Subset: A set embedded in another set. - Union: The combining of two sets, such that all elements of both sets are included in the combined set; elements found in Set A OR Set B. - Universal Set: The set that contains all other sets, including the set itself. - Venn Diagram: A graphic organizer that shows the relationships between sets by encircling combinations of individual elements. - Whole Number: Any of the positive integers, 1, 2, 3,…

Materials

- copies of Set Relationships handout (M-A1-2-1_Set Relationships) - copies of Set Examples handout (M-A1-2-1_Set Examples) - copies of the Relation and Function sheet (M-A1-2-1_Relation and Function).

Assignment

- Assess the student's ability to differentiate between finite and infinite sets by listening to the class discussion; providing examples of null sets; and elucidating the relationship between sets, subsets, and proper subsets. - Assess the performance of students on the following:  - set illustrations. - Illustrations of Venn diagrams (by verifying the precision of the inclusion and exclusion of set elements). - creation of universal sets and subsets (by observing their appropriateness). - Function determination and rationale (by comparing relations that are and are not functions). - creation of particular relations. - Evaluation of the mathematics journal article's materials for clarity and precision.

Supports

Scaffolding, Active Engagement, Modeling, and Explicit Instruction  W: The course relates sets and functions, analyzing their components through various notations and representations. Students are allowed to investigate notations in a highly open-ended manner through class discussions and involvement in discovery-oriented activities, both independently and as part of a group.  H: Connecting sets to the real number set and subsets helps students understand the importance of sets in mathematics. The presentation of a range of representations, forms, and notations will highlight the numerous ways in which sets can be represented.  E: The lesson is separated into two parts. Part 1 covers the concept of a set, the notations and representations used when discussing sets, the many types of sets, and set operations. Part 2 delves into the notions of relations and functions, connecting them to the concept of a set. Several notations and representations are studied.  R: Open-ended activities provide many opportunities for students to reflect, review, revise, and reconsider ideas. Classroom discussions provide students more opportunities to express and dispute their ideas.  E: The lesson format allows students to self-evaluate their learning through scaffolding and modeling. Participating in both individual and group activities enables self-evaluation.  T: The presentation and study of various notations is designed to accommodate varied learning styles.  O: The lesson progresses from abstract to concrete concepts and generalizations.

Procedures

Part 1 Provide students with the following information: "We will be discussing sets, elements and function today. A set is a collection of elements. An element is a component that is contained within a set. A function is used to describe a set and point to the elements in the set. In a bit, we will discuss functions in greater detail."  Display the box below using a projector overhead or the board:<figure class="image"><img style="aspect-ratio:598/450;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_6.png" width="598" height="450"></figure>"Now, let's think about our system of real numbers. Does anyone know the two disjointed number sets in our actual number system?" (Allow all responses, directing the response to both rational and irrational numbers. When summarizing responses, it is important to observe that rationals can be expressed as the numerator divided by the denominator, whereas irrationals cannot.) "We have the set of rational numbers and the set of irrational numbers. Particular embedded number sets exist within the rational number set. A subset is a set that is embedded within another set. In this case, since the sets are not equal, we have what is called a proper subset." "What numbers are there in the set of rational numbers?" (integers, whole numbers, counting numbers) "Which sets of numbers are subsets of the rational numbers?" (integers, whole numbers, natural numbers) “The rational numbers include all of the typical numbers that we consider, such as integers, whole numbers, and natural numbers (counting numbers).”<figure class="image"><img style="aspect-ratio:522/513;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_7.png" width="522" height="513"></figure>"Observe that the rational numbers also include other numbers, such as those that terminate or repeat. All rational numbers can be expressed in the following form: \(a \over b\), b ≠ 0.” "We can also represent each of these numbers as sets. Shown below is the roster method, whereby we simply list elements of each set." Roster Method<figure class="image"><img style="aspect-ratio:595/195;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_8.png" width="595" height="195"></figure>Ask, "Is there any another way to illustrate the sets?" (Yes; we frequently use a special type of notation to convey certain mathematical concepts.) “Certainly, let's first create a Venn Diagram to illustrate these three subsets of the real number system.”<figure class="image"><img style="aspect-ratio:219/203;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_9.png" width="219" height="203"></figure>"Next, we will record the number sets using a method known as set builder notation. The set is described with symbols in set builder notation, using a formal notation. For example, to denote the sets of natural numbers, whole numbers, and integers, respectively, using set builder notation, we would write: A = {x| x ∈ N} is the collection of all x, where x is a component of the Natural Numbers. B = {x| x ∈ W} is the set of all x, where x is a Whole Numbers element. C = {x| x ∈ Z} is the set of all x that are elements of the Integers. It is important to note that Z is used instead of I, which distinguishes it from the I notation for “irrational numbers.” "Observe that we assign a letter to each set. The letters A, B, and C serve to name the sets. This allows us to refer to them as Set A, Set B, and Set C. Up until now, we have encountered both sets with and without names. {3, 4, 5,…} is an example of a set that lacks a name. We know it's a set. It is simply does not a name. In general, we do wish to name the set, particularly when comparing multiple sets. For instance, the following are three sets: {3, 4, 5,…} {1, 2,… } {♥, ♦, ♠, ♣}” "We must determine which set we are talking about. It is crucial to identify the set later on when we examine relations, and more specifically, functions of sets." "Additionally, observe how each notation for the number sets is expressed in English. Words could be used to write each set. A = {the set of natural numbers greater than two} B = {the set of whole numbers} C = {the suits of a deck of cards}” "We can go even further. We could provide the following example to demonstrate that a specific number is a member of the set of integers: 4 ∈ Z.” "Suppose we want to identify only a part of the set of real numbers? Please remember that the real numbers include both rational and irrational numbers." "Suppose our focus is only on real numbers that are greater than or equal to 4? This set can be represented in a variety of ways, including the roster method, set-builder notation, and interval notation." Summary of Notations for Real Numbers Greater Than or Equal to 4<figure class="image"><img style="aspect-ratio:600/785;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_10.png" width="600" height="785"></figure> Interval Notation Tell students: "Interval notation is a new one concept that we haven't explored yet. To describe the elements, interval notation makes use of the set's endpoints. Again, we can discuss specific elements by stating that 7 ∈ A. We are claiming that 7 'is an element' of set A.” Activity 1 “It is time to practice using multiple representations of various sets. Consider these sets: 1)      The four most recent presidents of the United States 2)      Integers less than −2 and greater than −9 3)      The four seasons of the year 4)      All polygons with four or fewer sides 5)      Rational numbers greater than \(1 \over 2\)” (Answers: 1. Barack Obama, George W. Bush, Bill Clinton, George H. W. Bush; 2. –8, –7, –6, –5, –4, –3; 3. Winter, Spring, Summer, Autumn; 4. Quadrilateral, triangle; 5. {n|n > \(1 \over 2\)}) "The information you have recently learned is beneficial for the general purpose of illustrating sets for a variety of applications. You may have observed key differences in some of the sets. There are two types of sets: finite sets and infinite sets. A finite set has a definitive number of elements listed. For instance, B = {1, 2, 3} represents a finite set. An infinite set is a set that is not finite. The set can be either a countable infinite set, as the set of all natural numbers demonstrates, or an uncountable infinite set, as the set of all real numbers demonstrates. Examples of infinite sets include D = {1, 2, 3,…} and F = {all real numbers}." “The focus of this lesson is on different types of number notations. Thus, it is important that you recognize and have the facility to work with all different representations of sets. We will now look at some of the common conventions of sets, using an example set. Suppose we are interested in the set of all whole numbers less than 8. We can represent this set in the following ways.” Notations for the Set of All Whole Numbers Less Than 8<figure class="image"><img style="aspect-ratio:598/738;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_31.png" width="598" height="738"></figure><figure class="image"><img style="aspect-ratio:597/216;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_32.png" width="597" height="216"></figure>Activity 2 Explain to the class, "Select one numeric and one non-numeric set to represent. For each set, incorporate all of the representative forms that we have previously discussed, as well as any additional forms that you may think of." Allow students to work on each set in pairs or as a small group. Encourage as many different ideas from students during the discussion and instruct them to evaluate each other's recommendations. Subsequently, ask: "So, which set notation is the most generally accepted and frequently used?" "Regarding the discussion of a set, what is the most prevalent convention?" "We have already examined a variety of versions; however, is there one or more that are simply superior?" "What is the standard notation?" "Are the others just loose forms of the formal convention? Are they accepted?” "What are a few agreed-upon sets of identification conventions that we could include in a file for future students to review?" Inform students that there are four commonly used notations for sets: Words containing the set name and brackets (A = {The set of all whole numbers less than 8}) Roster notation with name (A = {0, 1, 2, 3, 4, 5, 6, 7}) Set-builder notation with name ( A = {x ∈ W | x < 8}) Interval notation ([0,8)) "Notice that the most commonly used notations are formal and include the name of the set. Now, we will examine the relationships between sets and specific categories of sets. We have not explored two sorts of sets directly. These are the empty and universal sets. The empty set, which is represented by empty brackets, { }, or Ø, is a set that has no elements. This may appear to be a contradiction. An example should clarify." Instruct students to provide examples of empty sets. Guide the conversation to assess the examples they have provided. "Now, what if we engage in a conversation about a universal set? We have already done so. The real number system is an example of a universal set. A universal set is a collection of sets that includes all other sets, including itself. The real number system is indeed the universal set, as it encompasses all other real numbers as well as the sets of rational and irrational numbers." "However, a universal set can be related to any encompassing set. I suggest that we examine the Venn Diagram provided below." Note: A Venn diagram is a graphic organizer that illustrates the relationship between the sets.<figure class="image"><img style="aspect-ratio:239/234;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_12.png" width="239" height="234"></figure>Explain how a Venn diagram is a very general representation of a universal set and its subsets. Activity 3 Inform students, "Let's create a Venn diagram to represent the universal set and any subsets using any sets you desire." "Frequently, we want to illustrate the connections between sets and/or to compare them. It is now necessary to discuss methods for representing relationships between sets. Initially, we should consider the potential meanings of these representations. How could you connect two sets?" "We could examine a combination of the components of both sets. Common elements may be examined. We could examine the elements found in one but not in the other. In what way could we illustrate these connections?" "We have the option of using any of the methods that we have previously examined." Display the following statement on the board or an overhead projector:          Let U be the set of integers from −4 to 8.          Let A = {−1, 0, 3, 4} and B = {2, 3, 6, −3} "We should use Venn diagrams to illustrate the relationships we are examining, discover the meaning of the symbolism, and represent the solutions in a variety of ways. Consider the table below." Distribute the Set Relationships handout (M-A1-2-1_Set Relationships) to students for their reference.<figure class="image"><img style="aspect-ratio:637/738;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_13.png" width="637" height="738"></figure><figure class="image"><img style="aspect-ratio:636/723;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_14.png" width="636" height="723"></figure> Note: The concept of complement is identical to that of "negation." For instance, the symbols p and ~p denote "p" and "not p," respectively. Activity 4 Assign students to groups of three or four. Instruct the groups: "Create a universal set and at least two subsets. Select the most effective representation of each relationship that has been illustrated above. Give a rationale for the use of that representation in this instance." "Develop a brief PowerPoint presentation that outlines the circumstances in which different representations of union, intersection, complement, and difference are useful. What appears to be the most widely recognized convention for indicating set relationships?" Part 2 "Now that you have an in-depth knowledge of the concept of sets, we will use a function to generate a set." "Before we go into functions, let's talk about relations. All functions are relations, but not all relations are functions. It is crucial to recognize this fact." "Does anyone understand what a relation is? Aside from mathematics, what is a relation?" (Students may provide examples of different types of relationships, such as husband and wife, child and mother, friends, and grandparents.) "Now, let's consider some relations. Particularly, mapping. A mapping relationship is a relationship in which each element in the first set corresponds to one or more elements in the second set." The following examples are included in the Set Examples pamphlet (M-A1-2-1_Set Examples), which should be distributed:<figure class="image"><img style="aspect-ratio:597/334;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_15.png" width="597" height="334"></figure><figure class="image"><img style="aspect-ratio:147/122;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_23.png" width="147" height="122"></figure><figure class="image"><img style="aspect-ratio:208/364;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_17.png" width="208" height="364"></figure> R = {(2,9), (3,4), (3,0), (8,2)} “Based on the provided examples, what definition of a relation would you provide? A subset of a set that is associated with another set is referred to as a relation. A capital R is used to represent a relation." "A relation does not limit the capacity of an input value to be mapped to more than one output value. In other terms, a relation would enable an element in Set A to be mapped to more than one element in Set B." "A function operates in a different way. In reality, functions are employed to generate sets by applying a rule that is specified by the function. A function is a relation that maps each element in the domain to exactly one element in the range. In other words, each input value (or value from Set A) is assigned a single output value (or value from Set B). It is crucial to acknowledge that input values are equivalent to domain values." "Could anyone provide an example of a function?" Give students ample time to offer a variety of examples. "Suppose that a cow is mapped to a giraffe, a giraffe is mapped to an elephant, and a cow is mapped to an elephant? Is that relation a function? No, because the cow corresponds to multiple animals. Let's examine the relations between the animals listed above." (M-A1-2-1_Set Relationships). Set Examples<figure class="image"><img style="aspect-ratio:591/228;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_33.png" width="591" height="228"></figure><figure class="image"><img style="aspect-ratio:590/741;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_36.png" width="590" height="741"></figure> Activity 5 Spread copies of Set Examples (M-A1-2-1_Set Examples). Students are required to complete the columns labeled "Function?" and "Supporting Reasons" in the table. The activity's objective is to enable students to understand the concept of functions. Say, "Now, we will use a function to generate a set. Suppose, for instance, that we have the function. Suppose that our set is the set of output values. We will call this set Set G. Read this as g of x, not g times x." "What elements would be included in Set G?" Allow students to express their ideas. "Set G would consist of −5, −3, −1, 1, and 3. Are there any additional elements? How many? Could you please list them all? Count them. This set is an uncountable infinite set, as any input value for x could result in a different output value. An input value (or domain) may be any real number, as there is no restriction in this context." "Have you observed that no input value is converted to more than one output value? If not, we will examine a few representations: a) a table, and b) a graph." a)<figure class="image image_resized" style="width:19.11%;"><img style="aspect-ratio:240/383;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_20.png" width="240" height="383"></figure>b)<figure class="image image_resized" style="width:21.9%;"><img style="aspect-ratio:280/325;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_21.png" width="280" height="325"></figure>Instruct students to explain why the graph represents a function. Ensure that the explanation includes a description of the graph's appearance. "Has anyone observed the function that was demonstrated above? Are there various categories of functions? How are they different? How are you able to determine this?" Allow for a period of discussion. "What is special about a linear function? Is it possible to identify this component through an equation, table, graph, or words?" Initiate a discussion regarding the multiple representations of linearity and rate of change. Activity 6 "We should investigate the representation of functions by examining certain relations from Set A to Set B. Suppose we have Sets A and B. Set A is defined as {2, 5, 6}, while Set B is defined as {3, −1, 9, 0}." Distribute copies of the Relation and Function page (M-A1-2-1_Relation and Function). Instruct students to complete the "Reasons" and "Function?" columns in the table below.<figure class="image image_resized" style="width:51.84%;"><img style="aspect-ratio:597/442;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_22.png" width="597" height="442"></figure>Activity 7 Assign students to groups of three or four. "Create four relations using any manipulatives that you prefer. Two of the relations should be solely relations, while the other two should be both relations and functions. Be prepared to present the relationships and discuss the rationale behind your claims." Explain to students that: "Your task is to write an article for a lesson mathematics journal to review the lesson. Briefly describe the key representations that are associated with functions and sets. Discuss any similarities or differences and provide a consensus regarding the "best" notation and varying representations. Provide a minimum of three illustrations to support your ideas." Extension: Ask students to design certain types of functions, both linear and nonlinear, using a variety of representations. Using manipulatives to make nonlinear functions would be very helpful. Students could illustrate the numbers and objects that belong to each set in accordance with the function. You have the option of advising them on the appearance of certain nonlinear functions. For example, y = \(x^2\), y = \(x^3\).

Layla Nguyen

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Sets and Functions (M-A1-2-1)

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Layla Nguyen

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Description

Students will explore notations through the use of functions and sets in this unit. Students will:
- examine patterns.
- explore a variety of representations/notations of sets and functions, and establish connections between the two.
- investigate the components and attributes of functions and sets.
- learn notation and use and interpret them in context.

Grade

Grade 9 Grade 10