Objectives

This unit requires students to study and compare real numbers. Students are going to:  - explore and compare real numbers using a number line and coordinate grid. - analyze and compare subsets of real numbers. - connect real number subsets to real-world occurrences.

Core Questions

- How can graphing help students get a conceptual knowledge of algebraic and/or number theory topics?

Vocabulary

- Dependent Variable: The range of a function; the set of values associated with each value of the independent variable. For example, in the function y = 2x + 5, y is the dependent variable. - Independent Variable: The domain of a function; the set of all possible values the function may take on. For example, in the function y = 2x + 5, x is the independent variable. - Integer: Includes the positive counting numbers and additive inverse of each (negative counting numbers), as well as zero. - Irrational Number: A number that is not rational, or cannot be written as the ratio of a/b. An irrational number is nonterminating and nonrepeating. - Natural Number: A number that shows one-to-one correspondence between the number and the amount shown. - Opposite: The additive inverse; for example, the opposite of 5 is −5 and the opposite of −x is x. - Ratio: An expression of the relative sizes of two quantities by division; the quotient of two numbers. - Rational Number: A number that can be written as the ratio of a/b, where b ≠ 0. A rational number is terminating or repeating. - Real Number: The number system consisting of rational numbers and irrational numbers. - Square Root: The number that when multiplied by itself yields the original number (the number under the square root sign); one factor of a given number, which when squared gives the original number. - Universal Set: In set theory, a set which contains all objects, including itself. - Venn Diagram: A graphical representation of all hypothetically possible logical relations between finite collections of sets. Developed around 1880 by John Venn, it is used to teach elementary set theory, as well as illustrate set relationships in multiple disciplines. - Whole Number: Includes zero and all natural numbers, or counting numbers.

Materials

- Lesson 1 Exit Ticket (M-A1-3-1_Lesson 1 Exit Ticket)

Assignment

- Assess whether students—individually and in groups—demonstrate that number lines are infinitely long in both positive and negative directions and that, regardless of how small the interval is, there are infinitely many spaces between each one by keeping an eye on their participation in class discussions and math-related activities.  - When evaluating student replies on the Lesson 1 Exit Ticket, look for comparable classifications as genuine and rational on \(2 {11 \over 13} \) and \(3 {5 \over 8} \). Recognizing that both terminating and repeating decimals are rational is an indication of how well students understand the difference between numbers that can be expressed as numerator and denominator and those that cannot.

Supports

Scaffolding and Active Engagement W: The lesson guides students through an exploration of real numbers, their equivalencies, real numbers as they appear in the real world, and exercises that require them to put their knowledge of graphing real numbers into practice.  H: Students will become engrossed in the inquiry-based first talk about real numbers and the study of converting rational numbers—repeating decimals—into fractions. Students will probably be engaged in additional open-ended activities as well as the tactile-kinesthetic ones.  E: The lesson combines concrete and abstract elements; it begins more concretely and concludes in a highly abstract way.  R: During each exercise and class discussion, students are given the chance to go over, reconsider, think again, and revise. In particular, students need to reconsider their comprehension of the human number line actions.  E: As students participate in deeper investigations of real numbers throughout the course, they must evaluate and reflect on themselves.  T: The session incorporates many techniques, including tactile-kinesthetic techniques.  O: The lesson fosters abstract thinking about real numbers through active participation, scaffolding, and encouragement. For example, students ought to stop thinking, "Where can I find integers in the world today?"

Procedures

Part 1: Real Numbers: Overview Ask students to describe the number system that we use most of the time. Students should provide the answer: the real number system. Next, pose the following questions to the class to start group discussions (with sufficient time between questions for discussion): "What is a real number? Which kinds of numbers are part of the real number system? Do these figures represent subsets of the actual numbers? How do we identify particular categories of real numbers? What tools of comparison do we have available? How would you respond if asked to give an example of a real number? Where may one find actual numbers in the real world?" A real number is either rational or irrational. Ask the students, "What is a rational number? What is an irrational number?" Allocate time for students to offer definitions and examples before moving further. After giving students time to define rational and irrational numbers, give them the following explanations: "A rational number can be expressed as the ratio a/b, when b ≠ 0. A rational number may also be said to be repeating or terminating. Any number that cannot be expressed as the ratio of a to b or is not rational is considered irrational. Nonterminating and nonrepeating numbers are examples of irrational numbers." Ask the class: "What does it truly mean when a rational number can be expressed as a/b, with b not equal to 0? Why don't all numbers correspond with that meaning?" Give the class some time to talk about this, and ask them to consider specific numbers. 2 and \(\sqrt{2}\) are good options for students to consider. 2 is a rational number, despite its appearance. Before moving forward, have students explain why the number makes sense. (2 is rational because it can be expressed as \(2 \over 1\), but \(2 \over 1\) simplifies to 2.)  "When we question if a number is rational or not, what we truly mean is this: If we have a decimal number, can we represent that decimal number as a fraction or ratio? If so, constraints are placed upon the decimal? The decimal number must either repeat or terminate in order to be a rational number. Let's examine a few examples to illustrate this." "Suppose we had a repeating decimal of 3.\(\overline{92}\). Could this be expressed as a fraction? Yes, that is correct, but what is the fraction? To answer this, we first give the decimal's repeating portion—that is, just that portion—a variable. It doesn't matter what we call this variable, so call it a. Assume that a =.9292929…, which is the repeating part of that decimal." (Explain that the only requirement is that you have a complete repetition; therefore, a =.92... or a =.9292..., which is two repetitions, would both be acceptable.) "We now need to do a trick with a. We must find a way to eliminate the recurring portion because a by itself will not reveal the fraction. Here's how to go about it: Find \(10^n\) · a, where n is the total number of digits that are below the bar. Here, n is 2, therefore, we compute 100a. Since a =.929292, we may get 100a = 92.9292.... You may then compute 99a, which provides us with:<figure class="image"><img style="aspect-ratio:158/70;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_63.png" width="158" height="70"></figure>which no longer has a repeating component because the subtraction cancels out all of the repeated numbers." We can solve for a to obtain the following and turn this into a fraction: a = \(92 \over 99\) We can write 3.\(\overline{92}\) as \(3 {92 \over 99} \). As a result, the number has been expressed as a ratio where b does not equal 0. Thus, the number is a rational number." Ask the class now, "Suppose we have the terminating decimal, 12.68. Could this be expressed as a fraction?" (Yes, either \(12 {68 \over 100} \) or \(12 {17 \over 25} \).) "Here's another illustration of a rational number. These examples demonstrate how we might take a decimal and determine whether or not it is rational." "Now let's examine an example of an irrational number. Assume we have the decimal number 2.9345612.... Nothing repeats and it doesn't terminate." Ask students, "How can we determine if we can write the decimal as a fraction and whether this number is rational?" Continue by stating that "We can apply the guess and check method. The previous repeating decimal conversion method works only when a piece of the decimal repeats, which is not the case in this example. Furthermore, since the number doesn't terminate, we are aware that we cannot just express the decimal part as a numerator over a particular denominator or power of 10. Let's attempt to determine a fraction that represents this number now." "We understand that 0.9 = \(9 \over 10\). We also know that 0.94 = \(94 \over 100\). Our portion would therefore fall between \(2 {9 \over 10} \) and \(2 {94 \over 100} \)." “Trial and error gives:<figure class="image"><img style="aspect-ratio:128/166;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_64.png" width="128" height="166"></figure>Keep investigating and try to find a fraction that is equivalent to 2.9345612.…" (Students should understand that the nonrepeating, nonterminating decimal cannot be precisely replicated; there must always be a second number between two near fractions.) "Since we are unable to discover a fraction that accurately describes this number, it is irrational. It is not repeatable, nonterminating, and cannot be expressed as the ratio of \(a \over b\)." Activity 1: Rational/Irrational Number Examples Divide the students into groups, and ask each group to name five rational and five irrational numbers. The rational numbers must have at least one repeating decimal. Incorporate the fractional and decimal versions. Help students find the fractional forms of the decimals that repeat as needed. When they're ready, ask some students to present their examples and discuss whether the numbers they chose are rational or irrational. After finishing the exercise, tell the students, "We have been approaching the real number system by providing an overview of the two primary subcategories of numbers: rational and irrational. There are other numerical categories inside the rational subset." Ask the class, "What are these additional numerical categories? What are a few instances?" (natural numbers, whole numbers, and integers) Ask students to present sets, and then ask them if they can give an example of an integer. After a few examples, teach the class the official definition of an integer. An integer comprises zero and the additive inverse of each (negative counting numbers) as well as the positive counting numbers. Thus, the set of integers appears as follows: …., -4, -3, -2, -1, 0, 1, 2, 3, 4,… The numbers continue indefinitely in both directions. Ask the students, "What is a whole number? Could someone please provide me with an example of a whole number?" After responses have been given, define a whole number. A whole number contains 0 and all-counting numbers, often known as natural numbers. This is how the set of whole numbers looks like: 0, 1, 2, 3, 4,… "We used the word 'counting number' in the previous two definitions. A counting number is just a number that indicates a one-to-one relationship between the shown quantity and the number. As a result, a negative counting number is not possible. Since we begin counting with the number 1, we are likewise unable to have 0. The number zero (0) stands for "nothing," or "zero pieces." But 0 is not used in counting. This is the set of counting, or natural numbers: 1, 2, 3, 4,…" Activity 2: Relationships Between Numbers Ask students to use the real number system's definitions of subsets to develop a diagram that shows how each subset relates to the universal set of the real system. "Overall set" or "umbrella set" is what is meant to be understood when one refers to a "universal set." Provide at least three numerical representations for each set. (Students are to create a Venn diagram illustration.) Verify that the diagram contains integers, rationals, irrationals, whole numbers, and natural numbers. With this diagram, you might need to help certain students who are having trouble. Tell the students that not all numbers, whether rational or irrational, are represented by fractions or decimals after the activity is finished. "In some cases, you may be shown a square root. You must decide whether the square root is rational or irrational, as well as, where applicable, what type of rational number it is. Let's take an example where we write the following square roots at random: \(\sqrt{22}\) \(\sqrt{18}\) \(\sqrt{565}\) \(\sqrt{1021}\) Are these numbers rational or irrational? How are we able to know?" Let the class have some time to talk before moving on. Inform the students that because these are nonterminating and nonrepeating, they are all irrational. Ask the students now, "When would we get a rational square root, and can you give us an example?" Give the class some time to respond, and then state, "When we have a perfect square under the square root sign, we will have a rational square root. As a result, we'd require either a perfect square natural number or a perfect square real number in decimal. For example,  \(\sqrt{121}\) = 11 and \(\sqrt{68.89}\) = 8.3 are both instances of rational square roots.” Activity 3: Exit Ticket Request that students fill out the Lesson 1 Exit Ticket (M-A1-3-1_Lesson 1 Exit Ticket). After completing a chart, students must indicate which categories—real, rational, irrational, integer, whole, and natural—apply to a particular number. Square roots are included. Part 2: Graphing and Comparing Real Numbers The lesson focuses on two things in this section: 1. Plot actual values on a coordinate grid and a number line. 2. Arrange and compare the actual numbers. "We will also go over one of our initial questions, which was about the prevalence of real numbers in the environment. More specifically, where are integers to be found? Where are irrational numbers found? Where are rational numbers to be found?" and so on. Ask the students to arrange the following numbers on the number line, using estimation where necessary, to begin this portion of the lesson. (Irrational number placement will require estimation.)<figure class="image"><img style="aspect-ratio:110/196;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_65.png" width="110" height="196"></figure>Draw this number line on the board without specifying the place of each number. Have students mark each location in turn. <figure class="image"><img style="aspect-ratio:464/152;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_66.png" width="464" height="152"></figure>Activity 4: Human Number Line The first step in this exercise is to turn a region into a number line. Set up desks or seats in a straight line with numbers taped to each one using masking tape to create straight lines. After assembling the grid, assign the class to the following human number line exercise: Give each student a number and direct them to the appropriate area on the number line. (Note: If you use desks, you may encounter spacing issues with fractions; nonetheless, you might keep them in the activity and see how creative your students get.) Start with a simple one. Give one student the number \(5 \over 2\) and the other student \(\sqrt{6}\). Request that the two students line up between the two seats that indicate their placements in the correct order. Talk about the similarities and differences between the two numbers with the class as the two students are deciding where to stand. Similarity: Their values are very close. Contrary to the square root of six, which is irrational, the five halves are rational. This is sufficient information to demonstrate that the first student should be standing closer to 3, and the second student should be standing closer to 2. By this point, one or more students will have estimated the value of \(\sqrt{6}\) as 2.449. Once the project is finished, ask the students, "When might we need to graph ordered pairs of real numbers?" After giving students a chance to answer, say, "In any situation where we relate the value of one variable to the value of another variable, we need to graph ordered pairs of real numbers. For example, if we look at the cost of a particular vehicle over x number of years, we may determine how much the vehicle costs for each year. Real numbers, rational numbers, and natural numbers are all involved in this example." Present the following queries to the class now: "We might also chart the distance that has been traveled over time. What numbers are involved in this case?" (rational numbers) "What if we wanted to check the weight of babies at various months? Which numbers are involved?” (positive rational numbers) "What if we wanted to look at the quantity of money in a checking account throughout a given week? What numbers are most likely involved? What numbers could be involved?" (rational numbers.) Once the students have responded to these questions, ask them to graph the following pairs in order on a coordinate grid. <figure class="image"><img style="aspect-ratio:147/155;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_67.png" width="147" height="155"></figure>Assist students in graphing these points as necessary. Activity 5: Human Point Plotting Show another exercise that resembles the number line and human exercise. Before the exercise begins, arrange a tiny grid with one origin point designated. You have two options for this grid: make a grid out of desks and chairs or use a lot of duct tape. With the requirement that points be plotted as closely together as possible on a big grid, students will be assigned an x-value or y-value. It's an open-ended question that calls for more consideration. In this exercise, students will compare and organize real numbers, providing them with an indirect experience with the distance formula. Explain this problem to the students after the exercise is finished: "Let's say we wanted to graph the weights of infants at different months of age. We may end up with irrational or rational weights, depending on the scale. It is conceivable, for instance, to weigh 18.473521… units. Yet, weight can be expressed fractionally (either as a precise weight or rounded) or rounded to the closest tenth of a decimal place. Zero or negative weights are not allowed. They can be both rational and irrational. They can also be natural numbers, such as a baby weighing exactly 18 pounds, or they can be rounded." Activity 6: Ordered Pairs Ask students to compile a data set of 20 ordered pairs that represent a single baby's birth weight during 12 months . For example, month 1 and a weight of 7 pounds and 3 ounces might be plotted as (1, 7.2). Let's say the baby's weight was recorded using a different technique by the medical staff. Thus, for rounding purposes, include rational and irrational numbers as natural numbers, rational numbers as fractions and decimals, and irrational numbers. Write a summary that contrasts the baby's growth from month to month. Between which two months did the infant experience the greatest weight gain? Gain the least ? Sample data is available at <a href="http://www.infantchart.com">http://www.infantchart.com</a>. Activity 7: Real-World Data Points Choose a practical instance where integers—positive and negative—can be gathered. Obtain a sample of twenty data points with two different variables. As an illustration, during 20 weeks, note 20 average weekly temperatures. The week number will be the independent variable, and the temperature will be the dependent variable. Discuss any questions, difficulties, or revelations that came to light throughout the lesson in class as a way to review the material. Higher-level tasks that call for analysis and critical thinking are the final activities. Consequently, learners can have multiple questions regarding the subjects covered here. Extension: Ask students to draw a chart that lists the various real number subsets that can be encountered in everyday life. Students should provide at least five real-world examples for each number type. For instance, a bank encounters integers, real numbers, whole numbers, rational numbers, real numbers, and natural numbers.

Layla Nguyen

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Real Number Comparisons (M-A1-3-1)

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Layla Nguyen

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Description

This unit requires students to study and compare real numbers. Students are going to:
- explore and compare real numbers using a number line and coordinate grid.
- analyze and compare subsets of real numbers.
- connect real number subsets to real-world occurrences.

Grade

Grade 9 Grade 10