Evaluating Statements about Rational and Irrational Numbers

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1 CONCEPT DEVELOPMENT Mathematics Assessment Project CLASSROOM CHALLENGES A Formative Assessment Lesson Evaluating Statements about Rational and Irrational Numbers Mathematics Assessment Resource Service University of Nottingham & UC Berkeley For more details, visit: MARS, Shell Center, University of Nottingham May be reproduced, unmodified, for non-commercial purposes under the Creative Commons license detailed at - all other rights reserved

4 Common issues: Does not properly distinguish between rational and irrational numbers (Q1, Q2) For example: The student does not write examples fitting one/both categories. Or: The student does not provide definitions of one/both categories. Or: The student gives the partial, incorrect definition that a rational number can be written as a fraction. Does not attempt the question (Q3) For example: The student does not identify relevant formulas to use. Does not provide examples (Q3) Does not identify appropriate examples For example: The student provides some examples but draws on a limited range of numbers. Empirical reasoning (Q3) For example: The student provides examples that show the statement is not true and concludes that there are no values of a, b that make the statement true. Suggested questions and prompts: A rational number can always be written as a fraction of integers. What kind of decimal representations can fractions have? An irrational number can never be written as a fraction of integers. It always has a nonrepeating non-terminating decimal. Which numbers do you know that have these properties? Which square roots are rational numbers? Which are irrational? How do you calculate the area of a rectangle? Now try some numbers in your formulas. What is the perimeter of your rectangle? Figure out some values for a and b to support your explanation. Suppose the rectangle were square. What area might the square have? What might the side lengths be? What would the perimeter be then? What irrational numbers do you know? Try some out with your formulas. What would you have to do to show that no values of a, b exist? Have you considered every possible type of example? Teacher guide Evaluating Statements about Rational and Irrational Numbers T-3

9 SOLUTIONS Assessment task: Rational or Irrational? 1. Check that the numbers written are rational. Students tend to define rational numbers as numbers that can be written as a fraction. The problem with this is that irrational numbers such as 3 2 are also fractions. A better definition is that rational numbers are numbers that can be written as a fraction of integers. 2. Following from Q1, irrational numbers might be defined as numbers that cannot be written as a fraction of integers. 3. It is possible to find values of a, b to make all these statements true. a. It is possible to choose a, b such that the perimeter and area are both rational numbers. For example, choose a = 4 and b = 3 so the perimeter is 2(4 + 3) = 14 units and the area is 4 3 = 12 square units. b. It is possible to choose a, b such that the perimeter is a rational number and the area irrational. The perimeter can be a sum of two irrational numbers in which the irrational parts have a zero sum. For example, if a = and b = 3 6. The perimeter is 2(a + b) = 2( ) = 2 7 = 14 which is rational. The area is ab = (4 + 6)(3 6) = = 6 6 which is irrational. c. It is possible to choose a, b such that both perimeter and area are irrational numbers, if you choose, for example, different irrational roots. For example, take a = 5 and b = 7. Then 2(a + b) = 2( 5 + 7)which is irrational, and ab = 5 x 7 = 35, which is also irrational. d. It is possible to choose a, b such that the perimeter is an irrational number and the area is a rational number. This occurs if a, b are multiples of the same irrational square root. For example, 3 5 and 5. You could also choose a = b = 5, say. Collaborative small-group work task The sum of a rational number and an irrational number is irrational This statement is always true. An irrational number can be represented as a non-terminating, non-repeating decimal. Any rational number can be written in non-terminating repeating form. Adding the repeating pattern to the nonrepeating decimal - this won t make it repeating. Alternatively, students might see this as a shift of the irrational number along the number line. Teacher guide Evaluating Statements about Rational and Irrational Numbers T-8

10 The circumference of a circle is irrational This statement is sometimes true. If the radius is rational, the circumference is irrational, because it is a rational multiple of π. If the radius is itself a multiple of 1, the circumference can be rational. π For example, if r = 4 π, then C = 2πr = 2π 4 π = 8 The diagonal of a square is irrational This statement is sometimes true. If the sides of the square are 2 units long, then the diagonal is irrational = = 8 which is If the sides of the square are 8 units, then the diagonal of the square is ( 8) 2 + ( 8) 2 = = 16 = 4. The sum of two rational numbers is rational This statement is always true. The rational numbers are closed under addition: you can t make an irrational number by adding two rational numbers. The product of a rational number and an irrational number is irrational This statement is sometimes true. Multiply any irrational number by the rational number zero. The product is rational. Choose any other rational with any irrational number and the product is irrational. The sum of two irrational numbers is irrational This statement is sometimes true. If the irrational parts of the numbers have zero sum, the sum is rational. If not, the sum is irrational. It is true, for example, for 2 7, 3. It is false, for example, for , 2 5. The product of two rational numbers is rational This is true for all rational numbers: the rationals are closed under multiplication. If a, b, c, d are integers with b, d non-zero, then a b c d = ac bd product is not defined, rather than irrational. is also rational. If b or d is zero, the The product of two irrational numbers is irrational This statement is sometimes true. For example, 2 8 is rational, but 3 8 is irrational. Teacher guide Evaluating Statements about Rational and Irrational Numbers T-9

11 Extension Task An expression containing both This statement is sometimes true. For example, 6 + π is irrational but 6 and π is irrational 6π 6 = = 1 is rational. 24π Students might feel that including expressions not in their simplest form is cheating. It may help to emphasize that these are just different ways of writing the same number. Between two rational numbers there is an irrational number This statement is always true. Students might try to show this by providing a method for finding an irrational. For example, one could first find the difference d between the two rational numbers. This will be rational. Then one could add an irrational fraction that is smaller than this difference (e.g. The result will be an irrational that lies between the two. 0.5d ) to the smaller rational. Alternatively, represent the two rational numbers by two repeating non-terminating decimals, for example: and Find the first decimal place at which they differ. In this case it is the third digit after the decimal point. The digits are 1 and 3 respectively. Use all the digits so far to start the new number, in this case Add one to the third digit of the smaller number, in this case the third digit of the first number is changed from 1 to 2. Replace the digits after this 2 with a non-repeating, non-terminating decimal (e.g. all the digits of π): This method does not always work. Sometimes when 1 is added to the first differing digit, the resulting number is bigger than both numbers. For example: and In this case, add 1 to the next digit of the smaller number, 1 is added to the fourth digit. Then again the digits after this are replaced with a non-repeating, non-terminating decimal: Between two irrational numbers there is an irrational number This statement is always true. Students might try to show this by example. They may take the average of two irrational numbers. This will always be irrational, unless the irrational numbers sum to zero, e.g. π and π. In this case, they could divide the larger irrational number by a rational, say 2. This will result in an irrational number between the two rational ones. If the circumference of a circle is rational, then the area is rational This statement is never true. This is because the circumference and area involve different multiples of π. Teacher guide Evaluating Statements about Rational and Irrational Numbers T-10

12 π is a transcendental (non-algebraic) number, so it cannot be the solution of a polynomial equation. You cannot use multiples of different powers of π to give a rational solution. We are told that C is rational, so write C = k for some rational number k. We know C = k = 2πr. So r = k 2π which must be irrational. Then A = πr 2 = π ( k 2π )2 = π k 2 4π 2 = k 2 which is irrational. 4π If the area of a circle is rational, then the circumference is rational This statement is never true. This is because multiples of different powers of π cannot give a rational solution. A is rational, so write A = j for some rational number j. A = j = πr 2. So r 2 = j π and r = j π. Then C = 2πr = 2π j π = 2 j π. So C is a rational or an irrational multiple of π. Students could gain the sense of the last two examples without an algebraic proof or discussion of transcendental numbers if you ask them to pick a value for r and then try to adjust to make the statements true. Assessment task: Rational or Irrational? (revisited) 1. These numbers are rational: ( )( 27 3) 3 Students who claim that is rational may have incomplete definitions of rational and 27 irrational numbers, referring to fractions rather than fractions of integers, or ratios of numbers rather than ratios of integers. Students who do not pick out the products and quotients involving expressions with radicals may need help in manipulating radicals. 2. a. Fubara is incorrect. The student might show this by choosing two irrational numbers e.g. 5 and 3 5 whose irrational parts sum to zero and thus which have a rational sum. b. Nancy is incorrect. The student might show this by choosing two irrational numbers whose product is rational. For example, (3+ 5)(3 5) = 9 5 = If you divide one irrational number by another, the result is always irrational This is false. For example, 3 3 =1. Teacher guide Evaluating Statements about Rational and Irrational Numbers T-11

13 If you divide a rational number by an irrational number, the result is always irrational This is false. If the rational number is zero, the quotient is always rational. Otherwise the result is always irrational. If the radius of a circle is irrational, the area must be irrational This is false. For example, let r = 7 π. Then A = " πr2 = π 7 % \$ ' # π & 2 = 49π π = 49. Teacher guide Evaluating Statements about Rational and Irrational Numbers T-12

14 1. a. Write three rational numbers. Rational or Irrational? b. Explain in your own words what a rational number is. 2. a. Write three irrational numbers. b. Explain in your own words what an irrational number is. 3. This rectangle has sides lengths a and b. Decide if it is possible to find a and b to make the statements below true. If you think it is possible, give values for a and b. If you think it is impossible, explain why no values of a and b will work. a. The perimeter and area are both rational numbers. b. The perimeter is a rational number and the area is an irrational number. c. The perimeter and area are both irrational numbers. d. The perimeter is an irrational number and the area is a rational number. Student materials Evaluating Statements about Rational and Irrational Numbers S MARS, Shell Center, University of Nottingham

15 Always, Sometimes or Never True The sum of a rational number and an irrational number is irrational The circumference of a circle is irrational The diagonal of a square is irrational The sum of two rational numbers is rational The product of a rational number and an irrational number is irrational The sum of two irrational numbers is irrational The product of two rational numbers is irrational The product of two irrational numbers is irrational Student materials Evaluating Statements about Rational and Irrational Numbers S MARS, Shell Center, University of Nottingham

16 Rational and Irrational Numbers p p p p Student materials Evaluating Statements about Rational and Irrational Numbers S MARS, Shell Center, University of Nottingham

17 Poster Headings Always True Sometimes True Never True Student materials Evaluating Statements about Rational and Irrational Numbers S MARS, Shell Center, University of Nottingham

18 Extension Task An expression containing both 6 and p is irrational Between two rational numbers there is an irrational number Between two irrational numbers there is an irrational number If the area of a circle is rational, then the circumference is rational If the circumference of a circle is rational, then the area is rational Make up your own statement Student materials Evaluating Statements about Rational and Irrational Numbers S MARS, Shell Center, University of Nottingham

19 Rational or Irrational? (revisited) 1. Circle any of these numbers that are rational: ( )( 27-3) 2. a. Fubara says, The sum of two irrational numbers is always irrational Show that Fubara is incorrect. b. Nancy says, The product of two irrational numbers is always irrational Show that Nancy is incorrect. 3. Complete the table. Make sure to explain your answer. Statement True or false Explanation If you divide one irrational number by another, the result is always irrational If you divide a rational number by an irrational number, the result is always irrational If the radius of a circle is irrational, the area must be irrational Student materials Evaluating Statements about Rational and Irrational Numbers S MARS, Shell Center, University of Nottingham

20 Poster with Headings!"#\$%&"'("&)(*++"#\$%&"'(,-./0+1!"#\$%&'()*+ 0-/+(1/+&'()*+ 2+3+)'()*+ 45+'6)-7*8(' -9'(#-' )\$(1-.\$"'.*/:+)&'1&' 1))\$(1-.\$";!"#\$#%&'()*"' "+,-)%#%&')%' \$.+'/)-\$+"0,-*)'.\$/+& Projector Resources Evaluating Statements about Rational and Irrational Numbers P-1

21 Instructions for Always, Sometimes or Never True 1. Choose a statement. Try out different numbers. Write your examples on the statement card. 2. Conjecture: decide whether you think each statement is always, sometimes or never true. Always true: explain why on the poster. Sometimes true: write an example for which it is true and an example for which it is false. Never true: explain why on the poster. Projector Resources Evaluating Statements about Rational and Irrational Numbers P-2

22 Always, Sometimes or Never True? The sum of a rational number and an irrational number is irrational True for: False for: Projector Resources Evaluating Statements about Rational and Irrational Numbers P-3

23 Always, Sometimes or Never True? The circumference of a circle is irrational True for: False for: Projector Resources Evaluating Statements about Rational and Irrational Numbers P-4

24 Always, Sometimes or Never True? True for: The diagonal of a square is irrational False for: Projector Resources Evaluating Statements about Rational and Irrational Numbers P-5

25 Always, Sometimes or Never True? The sum of two rational numbers is rational True for: False for: Projector Resources Evaluating Statements about Rational and Irrational Numbers P-6

26 True for: Always, Sometimes or Never True? The product of a rational number and an irrational number is irrational False for: Projector Resources Evaluating Statements about Rational and Irrational Numbers P-7

27 Always, Sometimes or Never True? The sum of two irrational numbers is irrational True for: False for: Projector Resources Evaluating Statements about Rational and Irrational Numbers P-8

28 Always, Sometimes or Never True? The product of two rational numbers is irrational True for: False for: Projector Resources Evaluating Statements about Rational and Irrational Numbers P-9

29 Always, Sometimes or Never True? True for: The product of two irrational numbers is irrational False for: Projector Resources Evaluating Statements about Rational and Irrational Numbers P-10

30 Mathematics Assessment Project Classroom Challenges These materials were designed and developed by the Shell Center Team at the Center for Research in Mathematical Education University of Nottingham, England: Malcolm Swan, Nichola Clarke, Clare Dawson, Sheila Evans, Colin Foster, and Marie Joubert with Hugh Burkhardt, Rita Crust, Andy Noyes, and Daniel Pead We are grateful to the many teachers and students, in the UK and the US, who took part in the classroom trials that played a critical role in developing these materials The classroom observation teams in the US were led by David Foster, Mary Bouck, and Diane Schaefer This project was conceived and directed for The Mathematics Assessment Resource Service (MARS) by Alan Schoenfeld at the University of California, Berkeley, and Hugh Burkhardt, Daniel Pead, and Malcolm Swan at the University of Nottingham Thanks also to Mat Crosier, Anne Floyde, Michael Galan, Judith Mills, Nick Orchard, and Alvaro Villanueva who contributed to the design and production of these materials This development would not have been possible without the support of Bill & Melinda Gates Foundation We are particularly grateful to Carina Wong, Melissa Chabran, and Jamie McKee The full collection of Mathematics Assessment Project materials is available from MARS, Shell Center, University of Nottingham This material may be reproduced and distributed, without modification, for non-commercial purposes, under the Creative Commons License detailed at All other rights reserved. Please contact if this license does not meet your needs.

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