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., π). 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.
Official wording from the Common Core State Standards for Mathematics (© 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers). View on thecorestandards.org
Because an irrational number has no exact decimal, the practical skill is pinning it down closely enough to use. A student who knows 4² = 16 and 5² = 25 can say at once that √20 lies between 4 and 5. Squaring a few decimals narrows it further: 4.4² = 19.36 and 4.5² = 20.25, so √20 sits between 4.4 and 4.5, and a little closer to 4.5. Repeating the squeeze one decimal place at a time gives an estimate as precise as anyone needs.
With these rational stand-ins students can compare irrational values, order mixed lists (such as √50, 7.2 and 2π), and mark them on a number line in roughly the right place. They also estimate expressions, for example π² is a bit less than 10 because 3.14² is about 9.86. The point is reasoning from perfect squares and known values, not reading a calculator, so students can justify every placement they make.
Some students estimate √20 as 10 because they divide by 2. Ask them to square their answer: 10 × 10 = 100, which is nowhere near 20.
√20 is not exactly 4.5 just because 20 is between 16 and 25. Squaring 4.5 gives 20.25, so the true value is slightly below 4.5.
Students sometimes plot √30 near 30. Remind them the root is the side length of a square with area 30, so it lands between 5 and 6.
Comparing √50 with 7.2 needs a common form. Square both: 50 versus 51.84, so 7.2 is larger, even though 50 looks bigger than 7.2.
Find the two whole numbers √30 lies between, then estimate √30 to one decimal place.
Answer: √30 is between 5 and 6, and is about 5.5 to the nearest tenth.
A list of perfect squares from 1 to 225 on the wall is the single most useful support here. Build a number line together and add √2, √3, √5 and √8 as a class, testing each guess by squaring. This turns the topic into a game of hot and cold and makes the squeeze method feel natural before students meet it in notation.
Test items commonly show a number line with labelled points A to D and ask which point best represents √45, or ask students to order three or four mixed values. Teach students to convert everything to decimals to one place, or to compare squares, and to write a one-sentence justification.
Original questions written for this standard. Choose an option or type your answer, then press Check. Every question has a worked explanation.
Answer: C) 7 and 8
7² = 49 and 8² = 64. Since 55 is between 49 and 64, √55 is between 7 and 8.
Answer: 9
9² = 81 and 10² = 100. 90 is closer to 81 than to 100 (9 away versus 10 away), and 9.5² = 90.25, so √90 is just under 9.5 and rounds to 9.
Answer: A) π, √10, 3.5
π ≈ 3.14, √10 ≈ 3.16 (since 3.16² ≈ 9.99) and 3.5 is largest. So the order is π, √10, 3.5.
Answer: 1.96
1.4 × 1.4 = 1.96, which is just under 2. That is why √2 is a little more than 1.4.
Answer: C) 4.5
√5 ≈ 2.24 because 2.2² = 4.84 and 2.3² = 5.29. Doubling gives about 4.47, so 4.5 is the best estimate.
Answer: 4.1
π ≈ 3.14, so π + 1 ≈ 4.14, which is 4.1 to the nearest tenth.
A full lesson with slides, activities and an exit ticket on approximating irrational numbers, pitched to grade 8 and editable in PowerPoint or Google Slides.
Make a lesson →A printable, differentiated worksheet on 8.NS.A.2 with an answer key, ready in about a minute.
Make a worksheet →Turn approximating irrational numbers into a quiz students answer online that marks itself, with a class summary for you.
Build a test →The standard is about reasoning with rational approximations, so the core skill is estimating from perfect squares. Calculators are useful afterwards to check how close an estimate was.
Usually to the nearest whole number or tenth, plus the ability to explain how to get closer by testing the next decimal place.