8.NS.A.2: Approximating Irrational Numbers to Compare, Locate and Estimate
In plain English: 8.NS.A.2 is the Common Core grade 8 math standard that asks students to use rational approximations of irrational numbers, such as 1.41 for √2, to compare irrational numbers, place them on a number line and estimate expressions such as π². Students trap a square root between closer and closer decimals. It follows 8.NS.A.1 in Grade 8 Math.
Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π²). For example, by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.
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.2 · Official standard
In 8.NS.A.1 students learned that irrational numbers such as √2 and π have decimals that never end or repeat. This lesson makes those numbers usable. Students trap an irrational number between two decimals, then keep making the gap smaller by truncating (cutting the decimal off without rounding) one digit later. With these rational approximations they compare irrational numbers, place them on a number line to scale, and estimate expressions such as π².
The official example is worked in full: truncating √2 shows it is between 1 and 2, then between 1.4 and 1.5, and each further digit gives a better approximation. Calculators are used to multiply, not to take square roots, so that students see where each digit comes from. Contexts stay concrete: rugs, patios, garden fences and circular tables.
Learning Objectives
By the end of this lesson, students will be able to:
Squeeze (trap) a square root between whole numbers, tenths and hundredths by comparing squares, and explain how to continue
Compare and order irrational and rational numbers using rational approximations
Locate irrational numbers approximately on a number line drawn to scale
Estimate the value of expressions with irrational numbers, such as π² and √2 + √10
Prior Knowledge Required
Students should already be comfortable with:
Knowing that irrational numbers have decimals that never end or repeat 8.NS.A.1
Squaring whole numbers and decimals as repeated multiplication 6.EE.A.1
Comparing decimals to the thousandths 5.NBT.A.3
Placing rational numbers on a number line 6.NS.C.6
Draw a square on the board and label its area 2 square meters, the size of a small rug. Read the prompt and give pairs three minutes with a calculator.
Warm-Up Prompt
"A square rug covers exactly 2 square meters. Is each side longer or shorter than 1.5 m? Than 1.4 m? Use multiplication, not the square root key, to decide. Can you find a decimal whose square is exactly 2?"
Pairs usually find that 1.5 × 1.5 = 2.25 is too big and 1.4 × 1.4 = 1.96 is too small, so the side is between 1.4 m and 1.5 m. No pair will find an exact decimal, because the side is √2, an irrational number (8.NS.A.1). Tell students that today they will pin such numbers down with decimals that are close enough to use.
Direct Instruction20 minutes
Define the key words and write each on an anchor chart:
Square root: √n is the positive number whose square is n, so √2 × √2 = 2. A perfect square is a whole number squared, such as 1, 4, 9, 16 and 25, and its square root is a whole number.
Rational approximation: a fraction or decimal that is close to an irrational number, such as 1.41 for √2 or 3.14 for π. It is close, never equal.
Truncate: cut a decimal off after a chosen place, without rounding. Truncating π = 3.14159... after the hundredths gives 3.14.
Squeeze (trapping a number): find a lower number and an upper number, called bounds, with the irrational number between them, then keep making the gap smaller. For a square root, compare squares: if 1.4² < 2 < 1.5², then 1.4 < √2 < 1.5.
Estimate an expression: replace each irrational number with its lower and upper approximation, compute both results, and report a value both results agree on.
Work through the five examples. Example 1 is the official example from the standard: each truncation step adds one digit, so the gap shrinks from 1 to 0.1 to 0.01. Use Diagram 1 to show the zoom. Students may use the calculator to multiply, but not the square root key, so they see where the digits come from.
Official example: truncating √2
Show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue.
Equation: 1² = 1 < 2 < 4 = 2². Then 1.4² = 1.96 < 2 < 2.25 = 1.5². Continue with hundredths: 1.41² = 1.9881 and 1.42² = 2.0164, so 1.41 < √2 < 1.42. Each new digit makes the gap 10 times smaller: √2 = 1.414...
Estimating π²
Estimate π² to one decimal place, using the digits of π = 3.14159...
Equation: 3.1 < π < 3.2 gives 9.61 < π² < 10.24, which is too wide. 3.14 < π < 3.15 gives 9.8596 < π² < 9.9225. Both bounds round to 9.9, so π² ≈ 9.9.
Comparing two irrational numbers
Which is greater, √10 or π?
Equation: 3.16² = 9.9856 and 3.17² = 10.0489, so 3.16 < √10 < 3.17. Since π = 3.14159... < 3.16, π < √10.
Locating on a number line
Place √55 on a number line marked in tenths, with a tick mark (a short line) at every 0.1.
Equation: 7² = 49 < 55 < 64 = 8². Then 7.4² = 54.76 and 7.5² = 56.25, so √55 is between 7.4 and 7.5, just past 7.4 (it is 7.416...).
Estimating a sum
Estimate √2 + √10 to one decimal place.
Equation: 1.41 < √2 < 1.42 (Example 1) and 3.16 < √10 < 3.17 (Example 3), so 4.57 < √2 + √10 < 4.59. Both bounds round to 4.6, so √2 + √10 ≈ 4.6.
After Example 2, point out that one decimal place of π was not enough: the bounds 9.61 and 10.24 did not agree. Estimating an expression means tightening the approximations until the answer you need no longer changes. Diagram 2 places the numbers from Examples 1 to 4 on one number line to scale.
Guided Practice15 minutes
Pairs work on four items, one at a time, and a pair explains each answer at the board.
Squeeze √40 between whole numbers, then tenths. (6² = 36 and 7² = 49; 6.3² = 39.69 and 6.4² = 40.96, so 6.3 < √40 < 6.4.)
Which is greater, √27 or 5.2? (5.2² = 27.04 is more than 27, so √27 < 5.2.)
Estimate 2π using 3.14 < π < 3.15. (6.28 < 2π < 6.30, so 2π ≈ 6.3.)
Place √90 on a number line from 9 to 10 marked in tenths. (9.4² = 88.36 and 9.5² = 90.25, so √90 is between 9.4 and 9.5, close to 9.5.)
Watch for students who square the wrong number, or who decide that √27 is about 13.5 by halving. Ask them to check every guess by squaring it.
Independent Practice15 minutes
Students work alone, then compare with a partner:
Squeeze √58 between two tenths. (7.6² = 57.76 and 7.7² = 59.29.)
Which is greater, √85 or 9.2? (9.2² = 84.64 < 85, so √85 > 9.2.)
Order from least to greatest: √24, 4.8, √23. (4.8² = 23.04, so √23 < 4.8 < √24.)
Estimate π/2 to one decimal place. (1.570 < π/2 < 1.575, so about 1.6.)
Choose a whole number that is not a perfect square, squeeze its square root to hundredths, and trade with a partner to check.
Closure5-10 minutes
Exit ticket: (1) Between which two tenths is √65? (8.0 and 8.1, since 8² = 64 and 8.1² = 65.61.) (2) Which is larger, √63 or 7.9? (√63, since 7.9² = 62.41.) (3) In one sentence, explain how to get a better approximation of √65 than 8.0. (Square 8.01, 8.02, ... and keep the last one whose square is below 65.)
Differentiation Strategies
For Struggling Students
Give a list of the perfect squares from 1 to 144 to keep on the desk for the first squeeze step
Use a three-row table (lower guess, upper guess, their squares) so each squeeze step has a fixed place to write
Start with number lines already marked in tenths, and ask only for whole-number and tenths bounds
For Advanced Students
Ask students to find √2 to four decimal places by truncation and count how many multiplications they needed
Ask: is (√2)² an irrational number? Is √2 × √8? Have them explain with rational approximations and then exactly
Challenge (beyond this standard): the diagonal of a 1 m by 1 m square is √2 m (8.G.B.7). Ask how many such tiles, laid corner to corner, make a line longer than 10 m
Assessment Guidance
What to Look For
Check that students justify every bound by squaring, for example "7.4² = 54.76 is less than 55." Watch for four errors: halving a number instead of taking its square root, rounding instead of truncating and then losing the lower bound, adding under the root (writing √6 + √8 as √14), and reporting an estimate that the two bounds do not agree on.
02
Classroom Activities
3 Activities
1
Squeeze Play
20 minPairs
Pairs squeeze 6 square roots between whole numbers, then tenths, then hundredths. The calculator may multiply but may not use the square root key. Partner A proposes a bound and Partner B checks it by squaring.
The 6 Cards (with answers)
Card 1: √28: between 5 and 6, then 5.2 and 5.3, then 5.29 and 5.30
Card 2: √42: between 6 and 7, then 6.4 and 6.5, then 6.48 and 6.49
Card 3: √75: between 8 and 9, then 8.6 and 8.7, then 8.66 and 8.67
Card 4: √110: between 10 and 11, then 10.4 and 10.5, then 10.48 and 10.49
Card 5: √200: between 14 and 15, then 14.1 and 14.2, then 14.14 and 14.15
Card 6: √0.5: between 0 and 1, then 0.7 and 0.8, then 0.70 and 0.71
Procedure
Write each bound with its square, for example 5.29² = 27.9841 < 28 < 28.09 = 5.30²
Switch roles after every card
When all 6 cards are done, check one answer with the square root key and see that its first digits match your bounds
Discussion Questions
Card 5 gives √200 = 14.14..., and √2 = 1.414... Why do the digits look the same? (Hint: 200 = 100 × 2.)
Card 6 is less than 1. Why is √0.5 larger than 0.5?
How many more multiplications would you need to get each root to the thousandths?
Modification for Distance Learning
Use a shared spreadsheet with columns for the guess and its square. Partners type guesses and watch the squares move toward the target number.
2
Human Number Line
15 minWhole class (9 volunteers at a time)
Tape a number line from 0 to 10 on the floor, about 3 m long, with a sticky note at every whole number. Nine students each get a card, estimate its value, and stand where it belongs. The class checks the order and the spacing.
The 9 Cards (with answers)
√2 (about 1.41)
π (about 3.14)
√32 (about 5.66: 5.6² = 31.36 and 5.7² = 32.49)
2π (about 6.28)
√40 (about 6.32)
√57 (about 7.55: 7.5² = 56.25 and 7.6² = 57.76)
√80 (about 8.94: 8.9² = 79.21 and 9² = 81)
√95 (about 9.75: 9.7² = 94.09 and 9.8² = 96.04)
π² (about 9.87)
Procedure
Each student writes the whole-number and tenths bounds for the card on the back before walking to the line
Students stand in order, then adjust their spacing: 0.1 on the line is about 3 cm
The class may challenge any position, but only with a squared number as evidence
Discussion Questions
2π and √40 end up closer together than any other pair of cards. Were tenths enough to decide which one comes first?
√95 and π² are the next closest pair. Which one is larger, and how do you know?
Which card was easiest to place, and why?
3
Estimation Stations
20 minGroups of 4
Groups rotate through 4 stations, about 5 minutes each. At each one they estimate a real quantity with rational approximations, state their bounds and decide what the answer means in context.
The 4 Stations (with answers)
Station 1: a square poster has an area of 3,000 cm². About how long is each side, to the nearest cm? (54² = 2,916 and 55² = 3,025, and 3,000 is much closer to 3,025, so about 55 cm; √3000 = 54.77...)
Station 2: a circular flower bed has a radius of 4 m, so the fence around it is 8π m long. Fencing is sold by the whole meter. How much should you buy? (25.12 < 8π < 25.20, so buy 26 m.)
Station 3: estimate 10π - 30 to the nearest tenth. (3.141 < π < 3.142 gives 1.41 < 10π - 30 < 1.42, so about 1.4.)
Station 4: is √70 + √2 more or less than 9.8? (8.36 < √70 < 8.37 and 1.41 < √2 < 1.42, so 9.77 < √70 + √2 < 9.79: less than 9.8.)
Procedure
Write the lower and upper approximation for every irrational number before computing
Compute the expression with both approximations and write the result as an inequality
If the two results do not give the same answer to the question, tighten the approximations and try again
Discussion Questions
At Station 2, why do you buy 26 m and not 25 m, even though 8π rounds to 25?
At Station 4, the tenths bounds 8.3 and 8.4 for √70 were not enough. Why?
Which station needed the most digits, and what made it hard?
Challenge Variation
Add a fifth station: a square window has a diagonal of 100 cm, and each side is √5000 cm. Estimate the side to the nearest cm and measure a real window to compare.
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: Squeezing √2 by Truncating Its Decimal
The official example, drawn to scale. Each number line is a 10-times zoom of the shaded piece of the one above: √2 is between 1 and 2, then between 1.4 and 1.5, then between 1.41 and 1.42. Keeping one more digit each time gives better and better approximations: √2 = 1.414...
Diagram 2: Irrational Numbers on a Number Line
√2, π, √55 and π² placed on a number line from 0 to 10 using the approximations 1.41, 3.14, 7.42 and 9.87 (each rounded to hundredths). The positions are computed to scale, so π² sits just below 10 and √55 just past 7.4.
Directions: Show every bound with its square, for example 6.1² = 37.21. You may use a calculator to multiply, but not the square root key. Draw number lines with a ruler.
Part 1: Squeezing and Locating (Problems 1-3)
Between which two whole numbers is each square root? Show the squares that prove it. (a) √38 (b) √89 (c) √120
Use truncation to squeeze √19: first between two whole numbers, then between two tenths, then between two hundredths. Explain how you would find the next digit.
Draw a number line from 4 to 7 marked in tenths. Place √21, √33, √45 and π + 3 on it. For each number, write the tenths it lies between.
Part 2: Comparing and Estimating (Problems 4-6)
Write < or > and justify each answer with a square or a decimal: (a) √60 and 7.7 (b) √35 and 5.9 (c) π and √9.5
Estimate each expression to one decimal place. Show the lower and upper bounds you used. (a) π + √2 (b) 2√17
A square vegetable garden has an area of 72 m². (a) Squeeze the side length √72 between two hundredths. (b) The fence goes all the way around, so it is 4√72 m long. Fencing is sold by the whole meter. How many meters should the family buy? Explain.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Squeezing
Every bound is correct and proved with a square
One or two bounds wrong or unproved
Bounds missing or guessed
Number Line
Drawn to scale with each point in the correct tenth
Order right but one or two points in the wrong tenth
Order wrong or no number line
Comparing
Every comparison is correct with a reason
One comparison wrong or reasons vague
Most comparisons wrong
Estimating in Context
Bounds agree, and the answer makes sense in context
Bounds right but the context answer is missing or wrong
No bounds or wrong method
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. Use scrap paper, and multiply instead of using a square root key.
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
Between which two whole numbers, next to each other, is √150?
Answer: B
12² = 144 and 13² = 169, and 150 is between them, so 12 < √150 < 13. Choice C halves 150 instead of finding a number whose square is 150. Choice A stops at 11² = 121, and choice D starts at 13² = 169, which is already more than 150.
Question 2 of 20 · Multiple Choice
Kiara knows that 1.7 < √3 < 1.8. Which pair of hundredths squeezes √3 next?
Answer: C
1.73² = 2.9929 is less than 3 and 1.74² = 3.0276 is more, so 1.73 < √3 < 1.74. Choice A is wrong because 1.73² is still below 3, so √3 is above 1.73. Choice B is wrong because 1.74² is already above 3. Choice D guesses the halfway point instead of testing squares.
Question 3 of 20 · Multiple Choice
Which statement about √15 and 3.9 is true?
Answer: B
Squaring 3.9 gives 15.21, which is more than 15, so √15 is less than 3.9. Choice A proves only that √15 is more than 3.8, which says nothing about 3.9. Choice C compares 15 itself with 3.9 instead of its square root. Choice D treats two nearby numbers as equal.
Question 4 of 20 · Multiple Choice
Which list is in order from least to greatest?
Answer: D
√8 is less than 3 because 8 < 9. π = 3.14... Since 3.3² = 10.89 < 11, √11 is more than 3.3, so it comes last. Choice A puts √8 after 3, as if √8 were more than 3. Choice B puts π before 3, and choice C starts with π, the second-largest number.
Question 5 of 20 · Multiple Choice
Using 3.1 < π < 3.2, between which two numbers is 5π?
Answer: A
Multiply both bounds by 5: 5 × 3.1 = 15.5 and 5 × 3.2 = 16, so 15.5 < 5π < 16. Choice B adds 5 instead of multiplying. Choice C divides by 5. Choice D moves the decimal point one place too far.
Question 6 of 20 · Multiple Choice
A number line runs from 0 to 6 with tick marks every 0.5. Where does √12 belong?
Answer: C
3² = 9 and 3.5² = 12.25, so √12 is between 3 and 3.5. Since 3.4² = 11.56, it is between 3.4 and 3.5, close to 3.5. Choice B is wrong because 3.5² is already more than 12. Choice D halves 12 instead of taking its square root.
Question 7 of 20 · Multiple Choice
Which number is between 7 and 8?
Answer: D
7² = 49 and 8² = 64, so a square root between 7 and 8 must be the root of a number between 49 and 64: √52. Choice A is about 6.8 because 46 < 49, and choice B is about 8.1 because 66 > 64. Choice C, √7.5, is less than 3; a student who reads the 7 in 7.5 as the answer picks it.
Question 8 of 20 · Multiple Choice
Which is the best estimate of √6 + √8?
Answer: B
2.44 < √6 < 2.45 and 2.82 < √8 < 2.83, so 5.26 < √6 + √8 < 5.28: about 5.3. Choice A adds under the root: √14 is about 3.7, but √6 + √8 is not √14. Choice C adds 6 + 8 and forgets the roots. Choice D halves 6 and 8 and adds 3 + 4.
Question 9 of 20 · Multiple Choice
Which number is the greatest?
Answer: D
Compare squares: 5.1² = 26.01 is more than 26, so 5.1 > √26. Next, 5.09² = 25.9081 is less than 26, so √26 > 5.09, and √25.5 is less than √26. The order is √25.5 < 5.09 < √26 < 5.1. Choice A is a trap for students who think a square root of 26 must be larger than any decimal close to 5.
Question 10 of 20 · Multiple Choice
Jon has squeezed √2 between 1.41 and 1.42. What should he do next to get a better approximation?
Answer: A
Keep one more digit: test the thousandths between 1.41 and 1.42. 1.414² = 1.999396 is less than 2 and 1.415² = 2.002225 is more, so 1.414 < √2 < 1.415, a gap 10 times smaller. Choice B goes back to tenths that are already too big (1.5² = 2.25). Choice C tests one number: it shows √2 < 1.415 but gives no new lower bound. Choice D squares numbers near √200, not √2.
Question 11 of 20 · Multiple Choice
A square patio has an area of 30 m². About how long is each side?
Answer: C
The side is √30. 5.4² = 29.16 and 5.5² = 30.25, so the side is between 5.4 m and 5.5 m, about 5.5 m. Choice A divides the area by 4, confusing area with perimeter. Choice B halves the area. Choice D stops at 5² = 25 and ignores the rest of the area.
Question 12 of 20 · Multiple Choice
Point P on a number line is at about 4.7. Which number could P be?
Answer: A
4.6² = 21.16 and 4.7² = 22.09, so √22 is just below 4.7: about 4.69. Choice B is about 4.90 and choice C is about 5.20, both too far to the right. Choice D takes the square root of 4.7 itself, which is only about 2.17.
Question 13 of 20 · Multiple Choice
Which number is closest to π = 3.14159...?
Answer: D
22/7 = 3.142857..., only about 0.001 more than π. The others are much farther away. 31/10 = 3.1 is about 0.04 less than π. Since 3.1² = 9.61 is more than 9.6, √9.6 is less than 3.1, even farther below π. Since 3.16² = 9.9856 is less than 10, √10 is more than 3.16, about 0.02 more than π. Choice A is a trap for students who remember that √10 is close to π but do not check how close.
Question 14 of 20 · Multiple Choice
Using 2.23 < √5 < 2.24, between which two numbers is 4√5?
Answer: B
4√5 means 4 times √5. Multiply both bounds by 4: 4 × 2.23 = 8.92 and 4 × 2.24 = 8.96. Choice A adds 4. Choice C multiplies by 2 instead of 4. Choice D puts the decimal point in the wrong place.
Question 15 of 20 · Short Answer
Use truncation to squeeze √7: first between two whole numbers, then between two tenths, then between two hundredths. Show the squares.
Draw a number line from 6 to 7 marked in tenths. Mark √39 and √47 approximately, and explain how you chose each position.
6.2² = 38.44 and 6.3² = 39.69, so √39 is between 6.2 and 6.3. Since 6.25² = 39.0625 is more than 39, it sits in the first half, closer to 6.2 (it is about 6.24). 6.8² = 46.24 and 6.9² = 47.61, so √47 is between 6.8 and 6.9. Since 6.85² = 46.9225 is less than 47, it sits closer to 6.9 (about 6.86).
Question 17 of 20 · Short Answer
Order from least to greatest: √50, 7.2, 50/7, √48. Show how you compared them.
√48 is less than 7 because 48 < 49. 7.07² = 49.9849 and 7.08² = 50.1264, so √50 is about 7.07. 50/7 = 7.142857... So the order is √48, √50, 50/7, 7.2.
Question 18 of 20 · Short Answer
A circular rug has a radius of 1.5 m, so its area is 2.25π m². Using 3.14 < π < 3.15, find bounds for the area and estimate it to the nearest tenth of a square meter.
2.25 × 3.14 = 7.065 and 2.25 × 3.15 = 7.0875, so 7.065 < area < 7.0875. Both bounds round to 7.1 m².
Question 19 of 20 · Short Answer
Is √20 + 1 more or less than 5.5? Use rational approximations to decide.
4.47² = 19.9809 and 4.48² = 20.0704, so 4.47 < √20 < 4.48 and 5.47 < √20 + 1 < 5.48. So √20 + 1 is less than 5.5.
Question 20 of 20 · Short Answer
A square tile has an area of 18 cm². Maya says each side is exactly 4.24 cm. Is she right? Find the side length to the nearest hundredth and explain.
No. 4.24² = 17.9776, not 18, so 4.24 is only an approximation. Since 4.25² = 18.0625, the side √18 is between 4.24 and 4.25. To round, test the halfway point: 4.245² = 18.020025 is more than 18, so √18 is below 4.245 and rounds to 4.24 cm. No decimal is exactly right, because √18 is irrational.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does 8.NS.A.2 mean?
8.NS.A.2 means students use rational approximations, fractions or decimals close to an irrational number, to work with irrational numbers. They compare them, place them on a number line and estimate expressions such as π². The official example squeezes √2 between 1 and 2, then between 1.4 and 1.5, and keeps going.
Is 8.NS.A.2 taught in grade 8 or in Algebra I?
It is a grade 8 standard, usually taught right after 8.NS.A.1 and near 8.EE.A.2 on square roots. Some schools teach Algebra I in grade 8 and cover it there. High school courses assume students can already estimate square roots this way.
What is a rational approximation?
It is a rational number, such as a decimal that ends, that is close to an irrational number. 1.41 and 1.414 are rational approximations of √2, and 3.14 is one for π. An approximation is never equal to the irrational number, but you can make it as close as you need.
What is the difference between truncating and rounding?
Truncating cuts the decimal off; rounding may change the last digit you keep. For 2/3 = 0.666..., truncating to hundredths gives 0.66 and rounding gives 0.67. Truncation is useful for squeezing because the truncated value is always a lower bound, and adding 1 to the last digit gives an upper bound.
Can students use a calculator for 8.NS.A.2?
Yes, for multiplying. The goal is to understand where the digits of √2 come from, so many teachers ask students not to use the square root key while they learn to squeeze. Later, the square root key is a fine way to check their bounds.
How do you estimate π² without a calculator?
Square the approximations of π. Since 3.14 < π < 3.15, π² is between 3.14² = 9.8596 and 3.15² = 9.9225, so π² is about 9.9. The standard lists π² as its example of an expression to estimate.
How do I place a square root on a number line accurately?
Find the whole numbers first, then the tenths. For √83: 9² = 81 and 10² = 100, then 9.1² = 82.81 and 9.2² = 84.64, so √83 is between 9.1 and 9.2, a little past 9.1. Draw the line to scale with equal spaces for equal steps, and place the point in the right tenth.
Is √a + √b the same as √(a + b)?
No, and it is a common mistake. √9 + √16 = 3 + 4 = 7, but √(9 + 16) = √25 = 5. To estimate a sum of square roots, approximate each root separately and then add the approximations.
Why does the standard use √2 as its example?
√2 is the simplest irrational square root, and it appears in real life as the diagonal of a square with sides of 1. Squeezing it shows the whole method in a few steps: whole numbers, then tenths, then hundredths, with each step 10 times more precise.
What comes after 8.NS.A.2?
In grade 8, the Pythagorean Theorem (8.G.B.7) often gives side lengths such as √34 cm that students must estimate. In high school, HSN.RN.B.3 explains why sums and products of rational and irrational numbers are rational or irrational.
07
Related Standards
6 standards
These standards connect to 8.NS.A.2: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
5.NBT.A.3Prerequisite
Read, write and compare decimals to thousandths
Lesson coming soon
6.NS.C.7Prerequisite
Understand ordering and absolute value of rational numbers
