- Define the terms rational number and irrational number
- Determine whether a number is rational or irrational
- Approximate the value of irrational numbers and place these approximations on a number line
- Use irrational numbers to solve real-world problems
- It’s OK to Be Irrational!: Irrational Numbers printable
- Answer Key: Diving Into the Number System printable
- Standards Chart: Diving Into the Number System printable
- Whiteboard or chart paper and markers
- Make a class set of the It’s OK to Be Irrational!: Irrational Numbers printable.
- Print a copy of the Answer Key: Diving Into the Number System printable for your use.
Introduction to New Material
Step 1: Draw a circle on the board, labeling the radius as 3 meters. Ask students if they remember the formula for the circumference of a circle (C = 2πr). Then ask for a volunteer to calculate the circumference on the board (6π meters). Then ask if we can express the exact measurement of the circumference as a decimal instead of using π (we can’t, because π has a decimal that goes on forever without repeating).
Step 2: Indicate that the world of real numbers is made up of two components, rational and irrational numbers. A rational number is one that can be written as a fraction (where the denominator is not zero). A helpful mnemonic device for some students is that the word ratio is in the word rational (and ratios can be written in fraction notation).
Step 3: Indicate that rational numbers include all whole numbers (including 0 and negative numbers), fractions, and mixed numbers (a mixed number is a whole number accompanied by a fraction, and it is rational because it can be represented by an improper fraction).
Step 4: Demonstrate that a decimal that terminates is rational because it can be rewritten as a fraction. For example, 0.42 can be rewritten as 42/100.
Step 5: Demonstrate a repeating decimal is rational because it, too, can be rewritten as a fraction. For example, 0.33 is rational because it can be rewritten as 1/3.
Step 6: Indicate that irrational numbers are those that cannot be represented as a fraction. These include decimals that do not truncate or repeat (π is a good example). If necessary, indicate that the symbol to show a nonrepeating decimal is the ellipsis (…). For example, π can be written as 3.1415… Note that using the ellipsis is not the same as rounding. Using the ellipsis, π is represented as 3.1415…, but when rounding to four decimal places, we would represent it as 3.1416 because the digit in the hundred-thousands place is a 9.
Step 7: Indicate to students that many of the irrational numbers they will encounter will be found in geometry problems. In addition to π in formulas for radius and circumference of circles, volume of cylinders, etc., irrational numbers come into play in squares, other rectangles, and triangles. Draw a square, labeling the area as 36 units. Ask the class to find the length of a side (√ 36 = 6 units). Then change the area to 35 units and ask the class to find the side length. Indicate that the answer is √ 35, but that √ 35 is an irrational number, somewhere between 5 and 6 with a decimal that goes on forever without repeating. Indicate that all square roots are irrational numbers except for those of perfect squares.
Step 8: Demonstrate the Pythagorean Theorem by drawing a rectangle on the board with side lengths of 3 and 4 units. Then draw a diagonal and label the length of the diagonal as 5. Indicate that the diagonal creates two right triangles. Show that 32 + 42 = 52. Then show the formula a2 + b2 = c2, where c is the length of the hypotenuse and a and b are the lengths of the other two sides. Another way to show this is c = √a2+ b2. Show how unless the side lengths squared are perfect squares, the result will be an irrational number when the square roots are calculated.
Step 9: Show how to approximate an irrational number. In the case of π or another nonrepeating decimal, demonstrate how to round the decimal after a desired number of decimal places. For example, π = 3.14159… can be approximated as 3.1416 simply by rounding after four decimal places. In the case of square roots, write the example of √ 15. We know that the nearest square roots with perfect squares are √ 9 = 3 and √ 16 = 4, so √ 15 is somewhere between 3 and 4 and is closer to 4 because 15 is closer to 16 than it is to 9. Draw a number line and place √ 15 on the number line between 3 and 4, just short of 4. Show how a “guess and check” method can be used to calculate an approximate value for √ 15. Indicate that if we know √ 15 is less than 4, try finding its value by multiplying 3.9 x 3.9. Write on the board that 3.9 x 3.9 = 15.21. Then try 3.8 x 3.8 = 14.44. Since 15 is closer to 15.21 than 14.44, we will use 3.9 as our approximate value of √ 15. Indicate that the process could be continued to develop approximations that go beyond one decimal place if desired.
Step 10: Show how to make comparisons with irrational numbers. Indicate that 3.14 is a common approximation of π. Ask if the approximation is greater than or less than π. Show how to line up the decimal places with 3.14 and 3.1415… Since, in addition to the 3.14 amount that both numbers share, the number 3.1415… has a value in the thousandths and ten-thousandths place and 3.14 does not, π is greater than its approximation of 3.14. Similarly, if you rounded π to the ten-thousandths place, your approximation would be greater than π because 3.14159… rounds to 3.1416 when rounded to the ten-thousandths place.
Step 11: Group students in pairs and ask them to solve the following problems:
- Draw a right triangle with side lengths of 2, 3, and c for the hypotenuse. Ask students to find the length of the hypotenuse (expressed as a square root), approximate the value of the hypotenuse to the nearest tenth, and place the approximate value on a number line. Answer: The hypotenuse is √ 13 (the sum of √ 22+ 32), which is approximately 3.6 and should appear on the number line just past halfway between 3 and 4.
- Draw the formula for the volume of a cylinder (V = Πr2h). Indicate that the radius = 3 and the height = 4. Ask students to find the volume, using an approximation of π to two decimal places and place their approximate value of π on a number line. Answer: V = 113.04 units, because 3.14 x 9 x 4. π has an approximate value of 3.14, which should appear just to the right of 3 on the number line.
Step 12: Checking for Understanding: Review answers as a class and respond to any questions.
Step 13: Assign the It’s OK to Be Irrational!: Irrational Numbers printable for classwork or homework.
Step 14: Checking for Understanding: Review the answers to the It’s OK to Be Irrational!: Irrational Numbers printable, which are provided on page 1 of the Answer Key: Diving Into the Number System printable. Make sure students explain their mathematical thinking. Address any misconceptions that may arise.
- Grade 8: Irrational Numbers (CCSS: 8.NS.1, 8.NS.2, 8.G.7, 8.G.9)
- Grade 6–8: Making Sense of Problems, Modeling, Attending to Precision, Using Structure, and Regularity in Repeated Reasoning (CCSS Practice MP1, MP4, and MP6–8); NCTM Number and Operations and Geometry
For more information, download the comprehensive Standards Chart: Diving Into the Number System printable.