8.NS.A.1: Rational and Irrational Numbers and Repeating Decimals
In plain English: 8.NS.A.1 is the Common Core grade 8 math standard that asks students to know that numbers that are not rational are called irrational, to see that every number has a decimal expansion, to show that the decimal of a rational number repeats eventually, and to turn a repeating decimal back into a fraction. It opens The Number System in Grade 8 Math.
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.
Common Core State Standards for Mathematics · Domain: The Number System (NS) · Cluster: Know that there are numbers that are not rational, and approximate them by rational numbers. Also written as 8.NS.1 · Official standard
Students have turned fractions into decimals with long division since grade 7. In this lesson they explain why that decimal must either end or repeat, and they run the process backward: they turn a decimal that repeats eventually into a fraction. Along the way they meet numbers that are not fractions at all. These irrational numbers, such as √2 and π, have decimal expansions that never end and never repeat.
The lesson keeps the ideas informal, as the standard asks. Students do not prove that √2 is irrational. They see that every number has a decimal expansion, that the remainders in long division must come back, and that the "multiply and subtract" method turns any repeating decimal into a quotient of integers. A companion standard, 8.NS.A.2, uses these ideas to estimate irrational numbers.
Learning Objectives
By the end of this lesson, students will be able to:
Define an irrational number as a number that is not rational, and classify numbers as rational or irrational
Explain informally that every number, rational or irrational, has a decimal expansion
Use the remainders in long division to show that the decimal of a fraction ends or repeats eventually
Convert a decimal that repeats, with or without digits before the repeating block, into a fraction
Prior Knowledge Required
Students should already be comfortable with:
Converting a fraction to a decimal with long division 7.NS.A.2d
Knowing that a rational number is a quotient of integers with a nonzero divisor 7.NS.A.2b
Solving one-step equations such as 99x = 48 6.EE.B.7
Multiplying a decimal by 10, 100 or 1,000 by moving the decimal point 5.NBT.A.2
Write 4 ÷ 13 on the board and have one student type it into a calculator. Read the prompt aloud and give pairs three minutes.
Warm-Up Prompt
"The calculator shows 4 ÷ 13 = 0.3076923077. Does the decimal really stop after ten digits? Do the long division by hand for twelve digits. What do you notice? Then think: could a decimal go on forever and never repeat?"
By hand, 4 ÷ 13 = 0.307692307692..., so the six digits 307692 repeat. The calculator ran out of room and rounded its last digit up, which hid the pattern. Collect ideas about the last question and leave it open: today students find out that such decimals exist, and that they are not fractions.
Direct Instruction20 minutes
Define the key words one at a time and write each on an anchor chart:
Rational number (recall from grade 7): a number that can be written as a quotient a/b of two integers, with b not 0. Examples: 3/4, -5 = -5/1, 0.2 = 1/5.
Decimal expansion: the digits of a number written in place-value form, as far as they go. Whole numbers and fractions have one, and so do numbers that are not fractions. For example 7 = 7.000..., and √2 (the positive number whose square is 2) = 1.41421356...
Terminating decimal: a decimal that ends, such as 0.5625. You can think of it as ending in 0s that repeat: 0.5625000...
Repeating decimal: a decimal where one group of digits, the repeating block, repeats forever, such as 0.185185185... (block 185). Books often draw a bar over the block. "Repeats eventually" means the block may start after a few digits that do not repeat, as in 0.3888...
Irrational number: a number that is not rational. It cannot be written as a quotient of two integers, and its decimal expansion never ends and never repeats. Examples: √2 and π.
Real numbers: all the numbers on the number line. Every real number is either rational or irrational.
Work through the five examples. For Examples 1 and 2, do the long division on the board and write each remainder in a column next to it, as in Diagram 1. A remainder is what is left after each division step. Ask: "Which remainders are possible when you divide by 27?" Only 0 to 26. So either a remainder of 0 appears and the decimal ends, or, after at most 26 steps, a remainder comes back and the same digits start again.
Fraction to decimal: it repeats
Write 5/27 as a decimal with long division.
Equation: 50 ÷ 27 = 1 R 23, 230 ÷ 27 = 8 R 14, 140 ÷ 27 = 5 R 5. The remainder 5 is the numerator we started with, so the steps repeat: 5/27 = 0.185185185..., block 185.
Fraction to decimal: it ends
Write 9/16 as a decimal with long division.
Equation: The remainders are 10, 4, 8 and then 0, so the division stops: 9/16 = 0.5625, which is also 0.5625000... with 0 repeating.
Repeating decimal to fraction
Write x = 0.484848... (block 48, two digits) as a fraction.
Equation: 100x = 48.484848..., and subtracting x = 0.484848... gives 99x = 48. So x = 48/99 = 16/33.
Repeats eventually
Write x = 0.3888... (only the 8 repeats) as a fraction.
Equation: 10x = 3.888... and 100x = 38.888... Subtract: 90x = 35, so x = 35/90 = 7/18. Check: 7 ÷ 18 = 0.3888...
Rational or irrational?
Classify √64, √20, 3.14, π and 0.7070070007... (one more 0 before each 7).
Equation: Rational: √64 = 8 and 3.14 = 314/100. Irrational: √20 (20 is not a perfect square, a number such as 16 or 25 that is a whole number squared), π, and 0.7070070007..., whose digits never settle into a repeating block.
Explain why the method in Examples 3 and 4 works. Multiplying by 10 moves the decimal point one place, and by 100 two places. Choose the powers of 10 (10, 100, 1,000 and so on) so that the two numbers you subtract have exactly the same digits after the point. Then the endless tails cancel and a simple equation is left. Use Diagram 2 to place every number from the examples. Stress that 3.14 is a rational number that is close to π, not equal to it.
Guided Practice15 minutes
Pairs do four items, one at a time. After each one, a pair shows its work at the board.
Write 7/16 as a decimal and list the remainders. (0.4375; remainders 6, 12, 8, 0, so it ends.)
Write 3/7 as a decimal and list the remainders. (0.428571428571...; remainders 2, 6, 4, 5, 1, 3, then 2 again. The block has six digits, the most possible for a denominator of 7.)
Write 0.151515... as a fraction. (100x - x = 15, so x = 15/99 = 5/33.)
Write 0.6222... as a fraction. (100x - 10x = 62.222... - 6.222... = 56, so x = 56/90 = 28/45.)
Watch for students who multiply 0.6222... by 100 only and subtract x, which leaves 99x = 61.6. Ask them to line up the digits after the point before subtracting.
Independent Practice15 minutes
Students work alone on five items, then compare with a partner:
Write 9/11 as a decimal and name the repeating block. (0.8181..., block 81; remainders 2 and 9.)
Write 0.575757... as a fraction in lowest terms. (57/99 = 19/33.)
Write 1.2555... as a fraction. (100x - 10x = 125.555... - 12.555... = 113, so x = 113/90.)
Classify each number as rational or irrational: √60, √16, 0.060060006..., -7/4. (Irrational, rational (4), irrational, rational.)
Make up your own decimal that repeats eventually, trade with your partner, and convert each other's decimal to a fraction.
Closure5-10 minutes
Exit ticket: (1) Write 0.393939... as a fraction in lowest terms. (39/99 = 13/33.) (2) Name one irrational number and explain in one sentence why it is irrational. (3) Without dividing, explain why the decimal for 4/19 must end or repeat. (Each remainder is one of 0 to 18, so a remainder of 0 appears or one comes back within 18 steps.)
Differentiation Strategies
For Struggling Students
Give a long-division template with a remainder column, and have students circle a remainder as soon as it appears a second time
For conversions, give a fill-in frame: x = ..., 10x or 100x = ..., subtract, solve. Start with one-digit blocks such as 0.111... before two-digit blocks
Keep a card on the desk: "Ends or repeats: rational. Never ends, never repeats: irrational."
For Advanced Students
Ask students to predict, before dividing, which of 1/40, 1/42, 1/45 and 1/48 will end, then test the predictions and describe the pattern in the denominators
Have students convert 0.999... to a fraction with the same method and explain what the result means
Challenge (beyond this standard): the sum of a fraction and an irrational number is irrational, a fact students prove in high school (HSN.RN.B.3). Ask them to test it with 1/2 + √2 on a calculator and say what they notice
Assessment Guidance
What to Look For
Listen for reasons, not only answers. A strong answer about a fraction names the remainders and says that one of them came back. A strong answer about an irrational number says the decimal never ends and never repeats, not just that it is long. Watch for four errors: thinking every square root is irrational, thinking a calculator display shows the whole decimal, dividing by 99 when the decimal only repeats after one digit, and treating 3.14 as equal to π.
02
Classroom Activities
3 Activities
1
Remainder Detectives
20 minPairs
Pairs use long division on 8 fraction cards. For each card they record every remainder until a remainder of 0 appears or a remainder comes back. Then they write the decimal and the length of its repeating block.
Partner A divides and Partner B writes each remainder in a column; switch roles for each card
Stop as soon as a remainder of 0 appears or a remainder repeats, and mark the digits that repeat
Record the denominator, the number of different remainders you saw, and the length of the block
Check two cards with a calculator and explain why the calculator display alone would not prove the pattern
Discussion Questions
Card 6 has denominator 7 and a block of 6 digits. Why can a block for a denominator of 7 never be longer than 6 digits?
Card 8 has denominator 41, but its block has only 5 digits. Does the remainder argument say the block must be 40 digits long, or at most 40?
Cards 3 and 7 start with digits that do not repeat. Are they still rational numbers?
Modification for Distance Learning
Share the cards on a slide. Partners take turns typing one division step at a time into a shared table with columns for the step, the digit and the remainder.
2
Multiply and Subtract Relay
20 minGroups of 3
Each group converts 6 repeating decimals into fractions. Student 1 writes x = the decimal and chooses the power of 10, Student 2 writes the two lined-up numbers and subtracts, and Student 3 solves, writes the fraction in lowest terms and checks it by long division. Roles rotate for each card.
The 6 Cards (with answers)
Card A: 0.636363... = 63/99 = 7/11
Card B: 0.216216... = 216/999 = 8/37
Card C: 0.8666... = 78/90 = 13/15
Card D: 3.090909... = 306/99 = 34/11
Card E: 0.0777... = 7/90
Card F: 0.142857142857... = 142857/999999 = 1/7
Sample Record
For Card C, one group wrote x = 0.8666..., 100x = 86.666... and 10x = 8.666... Subtracting gave 90x = 78, so x = 78/90 = 13/15. The check 13 ÷ 15 gave remainders 10, 10, 10, so the digits are 0.8666...
Procedure
Before multiplying, underline the repeating block and count its digits
Line up the two decimals on the paper so the digits after the point match exactly
The group earns a card only after the long-division check matches the original decimal
Discussion Questions
For Card B you multiply by 1,000, and for Card A by 100. How does the length of the block tell you which power of 10 to use?
Cards C and E need two multiplications. What do they have in common?
Card F turned out to be 1/7. What does that tell you about every repeating decimal?
3
Rational or Irrational? Sort and Defend
15 minPairs
Pairs sort 10 number cards into "Rational" and "Irrational" and write a reason on each card. A reason for "Rational" shows the number as a quotient of integers. A reason for "Irrational" explains why the decimal never ends and never repeats.
The 10 Cards (with answers)
√49 (rational: it equals 7)
√40 (irrational: 40 is between the perfect squares 36 and 49)
0.25 (rational: 1/4)
0.252252225... with one more 2 each time (irrational: no block repeats)
π (irrational)
3.1416 (rational: 31416/10000, a decimal close to π)
-11/3 (rational: a quotient of integers)
0.717171... (rational: 71/99)
√(9/4) (rational: it equals 3/2)
2.030030003... with one more 0 each time (irrational)
The Calculator Trap
A calculator shows √40 = 6.32455532. Ask: "Does this show that √40 is a terminating decimal?" Have students square 6.32455532 on the calculator. A calculator may even show 40, but the exact product is 39.9999999957..., just under 40. A quick way to see that it cannot be 40: the last digit 2 times 2 makes the product end in the digit 4, not in 0s. So 6.32455532 is only a rounded approximation of √40.
Discussion Questions
Six cards are rational and four are irrational. Which card was hardest to decide?
√49 and √40 both have a square root symbol. Why is only one of them irrational?
The cards 0.25 and 0.252252225... begin with the same digits. Why can the first few digits never decide the question?
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: Why the Decimal of 5/27 Repeats
Long division of 5 by 27. Each step multiplies the remainder by 10 and divides by 27. The remainders go 23, 14, 5, and 5 is where we started, so the digits 1, 8, 5 repeat forever. Any remainder must be less than 27, so a remainder has to come back (or be 0) within 27 steps.
Diagram 2: Rational and Irrational Numbers
Every real number is either rational (its decimal ends or repeats) or irrational (its decimal never ends and never repeats). Whole numbers sit inside the integers, and the integers sit inside the rational numbers. Several examples come from this lesson's worked examples.
04
Homework Assignment
~30 min
8.NS.A.1 Homework: Decimal Expansions and Repeating Decimals
Directions: Show all long division with the remainders written next to each step. When you convert a decimal, write the two lined-up equations you subtract. Give every fraction in lowest terms.
Part 1: Decimals of Fractions (Problems 1-3)
Use long division to write each fraction as a decimal. List the remainders, say whether the decimal ends or repeats, and name the repeating block if there is one. (a) 7/15 (b) 13/32 (c) 4/37
A calculator shows 9 ÷ 17 = 0.5294117647. (a) Maya says 9/17 is exactly 0.5294117647. Explain why she is wrong. (b) Without finishing the division, explain why the decimal of 9/17 must repeat, and what the longest possible repeating block is.
Decide whether each number is rational or irrational, and give a reason: (a) 0.8 (b) 0.888... (c) 0.808808880888... with one more 8 each time (d) √8 (e) 8.08
Part 2: Repeating Decimals to Fractions (Problems 4-6)
Write each repeating decimal as a fraction in lowest terms: (a) 0.555... (b) 0.171717... (c) 0.306306...
Each decimal repeats only after its first digits. Write each one as a fraction in lowest terms: (a) 0.2333... (b) 2.1444...
Kai wrote: "x = 0.3666..., so 100x - x = 36 and x = 36/99." (a) Divide 36 by 99 to show that Kai's answer is wrong. (b) Find Kai's mistake and write the correct fraction. (c) Check your fraction with long division.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Long Division
Every decimal and remainder is correct and the block is named
One or two arithmetic errors
Division missing or mostly wrong
Reasoning About Remainders
Explains that remainders are less than the divisor, so one must come back or be 0
Reason given but incomplete
No reason
Conversions
Correct power of 10, lined-up subtraction and fraction in lowest terms
Method right but one error or not in lowest terms
Method missing or wrong
Rational or Irrational
Every classification is correct with a reason
One or two wrong or reasons vague
Most wrong or no reasons
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Work through the questions in order. Your score updates as you answer, and Reset quiz clears everything so you or your students can try again. Keep scrap paper handy for long division.
Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.
0 of 20 answered · 0 correct
Question 1 of 20 · Multiple Choice
Which number is irrational?
Answer: C
30 is between the perfect squares 25 and 36, so √30 is not a whole number, and its decimal 5.4772255... never ends or repeats. Choice A is a trap for students who think every square root is irrational: √81 = 9. Choice B is a quotient of integers, and choice D is 1/8.
Question 2 of 20 · Multiple Choice
Which sentence gives the meaning of an irrational number?
Answer: B
An irrational number is a number that is not rational, so it cannot be written as a/b with integers a and b and b not 0. Choice A is not enough: 1/3 = 0.333... never ends but is rational. Choice C fails for √81 = 9, and choice D fails for -4 and 0.5, which are rational.
Question 3 of 20 · Multiple Choice
Nora divides 3 by 13 with long division. At most how many different nonzero remainders can appear before one of them repeats?
Answer: D
Every remainder is less than 13, and a remainder of 0 would end the division, so only 1 to 12 are possible: at most 12 different ones. In fact 3/13 = 0.230769230769... uses just six (4, 1, 10, 9, 12, 3). Choice C counts 0 as well, choice B thinks of the ten digits, and choice A uses the numerator.
Question 4 of 20 · Multiple Choice
What is 13/18 written as a decimal?
Answer: A
130 ÷ 18 = 7 R 4, then 40 ÷ 18 = 2 R 4, and the remainder 4 keeps coming back, so only the 2 repeats: 0.7222... Choice B repeats the block 72, which is 8/11. Choice C stops after two digits. Choice D divides 18 by 13.
Question 5 of 20 · Multiple Choice
Which statement about decimal expansions is true?
Answer: C
Every number has a decimal expansion. A whole number such as 6 is 6.000..., and an irrational number such as √2 has one that goes on forever without repeating. Choice B confuses "never ends" with "does not exist". Choice D forgets that 1/3 = 0.333... is a decimal expansion too.
Question 6 of 20 · Multiple Choice
Which fraction equals 0.242424... (the block 24 repeats)?
Answer: B
Let x = 0.242424... Then 100x = 24.2424..., so 99x = 24 and x = 24/99 = 8/33. Choice A is 0.24 as a terminating decimal. Choice C divides 24 by 90 instead of 99, a mix-up with decimals such as 0.3888... that repeat after one digit. Choice D divides 24 by 999, the divisor for a three-digit block: 24/999 = 8/333 = 0.024024...
Question 7 of 20 · Multiple Choice
Which fraction equals 0.1333... (only the 3 repeats)?
Answer: D
Let x = 0.1333... Then 100x = 13.333... and 10x = 1.333..., so 90x = 12 and x = 12/90 = 2/15. Choice A treats 13 as the repeating block: 13/99 = 0.1313... Choice B subtracts correctly but divides by 99: 12/99 = 0.1212... Choice C stops the decimal at 0.13.
Question 8 of 20 · Multiple Choice
A calculator shows √11 = 3.31662479. Ana says √11 is rational because the decimal stops. Which response is correct?
Answer: A
11 is not a perfect square, so √11 is irrational. The calculator shows only as many digits as fit and rounds the last one. Choice C describes a rational number close to √11: 3.31662479 squared is 10.9999999976..., not 11. Choice D gives a false reason: √9 = 3, and 9 is odd.
Question 9 of 20 · Multiple Choice
The digits of 0.202002000200002... follow a pattern: one more 0 each time before the next 2. Which statement is true?
Answer: B
The runs of 0s keep getting longer, so no fixed block ever repeats, and the decimal never ends. That makes the number irrational. Choice A confuses a pattern with a repeating block. Choice D forgets that every number is either rational or irrational.
Question 10 of 20 · Multiple Choice
Which fraction has a decimal expansion that ends?
Answer: C
70 ÷ 40 = 1 R 30, 300 ÷ 40 = 7 R 20, 200 ÷ 40 = 5 R 0. The remainder 0 ends the division: 7/40 = 0.175. Choice A is a trap for students who think a small denominator gives a short decimal: 2 ÷ 3 leaves the remainder 2 every time, so 2/3 = 0.666... Choice B is 0.2727... and choice D is 0.1666...
Question 11 of 20 · Multiple Choice
A terminating decimal can also be written as a repeating decimal. Which repeating decimal equals 0.375?
Answer: A
After the digits 375 the decimal has only 0s, so the block that repeats is 0: 0.375 = 0.375000... Choice B repeats the whole 375, which is a larger number, 375/999. Choices C and D repeat the last digits as if they were a block.
Question 12 of 20 · Multiple Choice
Which fraction equals 2.777... (only the 7 repeats)?
Answer: D
Let x = 2.777... Then 10x = 27.777..., so 9x = 25 and x = 25/9. Check: 25 ÷ 9 = 2 R 7, then 70 ÷ 9 = 7 R 7 again and again. Choice B forgets the whole number 2: 7/9 = 0.777... Choices A and C are the terminating decimals 2.7 and 2.77.
Question 13 of 20 · Multiple Choice
Which list contains only rational numbers?
Answer: C
√25 = 5, 0.444... = 4/9 and -3/8 are all quotients of integers. Each other list has one irrational number: √12 in A, π in B and √7 in D. A student who treats every square root as irrational rejects choice C because of √25.
Question 14 of 20 · Multiple Choice
Why must the decimal of every fraction of whole numbers either end or repeat?
Answer: B
In long division by the denominator, only a limited set of remainders is possible. If a remainder of 0 appears, the decimal ends; if not, a remainder must come back, and from then on the same digits repeat. Choice A confuses the display with the number. Choice D is false: 7/4 has a numerator greater than its denominator and still ends as 1.75.
Question 15 of 20 · Short Answer
Write 0.405405405... (the block 405 repeats) as a fraction in lowest terms. Show the two equations you subtract.
Let x = 0.405405... Then 1000x = 405.405405... Subtract: 999x = 405, so x = 405/999. Divide the top and bottom by 27: x = 15/37. Check: 15 ÷ 37 = 0.405405...
Question 16 of 20 · Short Answer
Use long division to write 17/22 as a decimal. List the remainders and name the digits that repeat.
170 ÷ 22 = 7 R 16, 160 ÷ 22 = 7 R 6, 60 ÷ 22 = 2 R 16, and then 16 comes back. 17/22 = 0.77272...: after the first 7, the block 72 repeats. The remainders go 16, 6, 16, 6, ...
Question 17 of 20 · Short Answer
Classify each number as rational or irrational, and give a reason: (a) √121 (b) 5.020020002... with one more 0 each time (c) √45 (d) -12/5
(a) Rational: √121 = 11. (b) Irrational: the runs of 0s grow, so no block repeats. (c) Irrational: 45 is between the perfect squares 36 and 49, so √45 is not a whole number or a fraction. (d) Rational: it is a quotient of integers, equal to -2.4.
Question 18 of 20 · Short Answer
Write 0.4111... (only the 1 repeats) as a fraction in lowest terms.
Let x = 0.4111... Then 100x = 41.111... and 10x = 4.111... Subtract: 90x = 37, so x = 37/90. 37 is prime and does not divide 90, so the fraction is in lowest terms.
Question 19 of 20 · Short Answer
A calculator shows 6 ÷ 13 = 0.4615384615. Jaylen says the decimal of 6/13 ends after ten digits. Explain what the whole decimal expansion looks like and why.
Long division gives the remainders 8, 2, 7, 5, 11 and then 6, the starting numerator, so the digits repeat: 6/13 = 0.461538461538..., with the block 461538. The calculator shows only ten digits and rounds the last one, so the display does not show the whole decimal. Because 6/13 is rational, its decimal must end or repeat, and here it repeats.
Question 20 of 20 · Short Answer
Write 1.363636... (the block 36 repeats) as a fraction, and explain why your answer shows that the number is rational.
Let x = 1.3636... Then 100x = 136.3636..., so 99x = 135 and x = 135/99 = 15/11. The number is rational because it equals a quotient of two integers, 15 and 11. Check: 15 ÷ 11 = 1.3636...
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does 8.NS.A.1 mean?
8.NS.A.1 means students know that a number that is not rational is called irrational, and that every number has a decimal expansion. They show that the decimal of a fraction ends or repeats, and they turn a repeating decimal back into a fraction. The standard asks for an informal understanding, not formal proofs.
Is 8.NS.A.1 a grade 7 or a grade 8 standard?
It is a grade 8 standard. In grade 7 (7.NS.A.2d) students convert fractions to decimals with long division and learn that the result ends or repeats. In grade 8 they explain why, convert in the other direction, and meet irrational numbers.
What is the difference between a rational and an irrational number?
A rational number can be written as a quotient of two integers; an irrational number cannot. In decimal form, a rational number ends or repeats, as 0.75 and 0.1666... do. An irrational number, such as √3 or π, has a decimal that never ends and never repeats.
Is a terminating decimal like 0.75 also a repeating decimal?
Yes, you can think of it that way. 0.75 is the same as 0.75000..., with the block 0 repeating. That is why the standard says the decimal of every rational number "repeats eventually": decimals that end are the case where the repeating digit is 0.
Does 0.999... really equal 1?
Yes. Let x = 0.999... Then 10x = 9.999..., and subtracting gives 9x = 9, so x = 1. Another way to see it: 1/3 = 0.333..., and three times that is 0.999..., while three times 1/3 is 1. Many students find this surprising, and it makes a good discussion question.
How do I know whether to multiply by 10, 100 or 1,000?
Count the digits in the repeating block. A block of one digit needs 10, two digits need 100, and three digits need 1,000. If some digits come before the block, as in 0.58333..., first multiply to move them to the left of the point (100x = 58.333...), then use one more power of 10 for the block (1000x = 583.333...). The goal is two numbers with the same digits after the point.
Are all square roots irrational?
No. The square root of a perfect square is a whole number: √36 = 6 and √100 = 10. The square root of a whole number that is not a perfect square, such as √5 or √50, is irrational. Students learn to find square roots in 8.EE.A.2, which also asks them to know that √2 is irrational.
Is π equal to 22/7 or 3.14?
No. Both are rational numbers close to π. 22/7 = 3.142857142857... repeats, and 3.14 ends, while π = 3.14159265... never ends and never repeats. They are useful approximations, and 8.NS.A.2 shows how to choose and use approximations like these.
What mistakes do students make with repeating decimals?
Common ones are dividing by 99 when only one digit repeats after a non-repeating digit, forgetting a whole-number part such as the 3 in 3.1666..., and not writing the fraction in lowest terms. Another is trusting a calculator display: calculators round the last digit, so a repeating decimal can look as if it ends.
What comes after 8.NS.A.1?
The next standard, 8.NS.A.2, uses rational approximations to compare irrational numbers, place them on a number line and estimate expressions such as π². In high school, HSN.RN.B.3 explains why sums and products of rational and irrational numbers are rational or irrational.
07
Related Standards
5 standards
These standards connect to 8.NS.A.1: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
7.NS.A.2Prerequisite
Multiply and divide rational numbers, and convert fractions to decimals by long division
