Square numbers and square roots
Squaring a whole number, finding the square root of a perfect square, and estimating roots that aren't whole numbers
About two to three lessons of 45 to 60 minutes
How do you find the exact side length of a square, knowing only its area?
A square rug has an area of 121 square feet. What is its side length? You could guess and check, but there is a faster way: since a square's area is side x side, the side length is whatever number, multiplied by itself, gives 121. That number is the square root of 121.
Squaring and finding a square root are inverse operations, exactly like adding and subtracting undo each other. Squaring a whole number always gives a perfect square. Going the other way, from a perfect square back to its whole-number side length, is finding a square root. Not every number is a perfect square though, so this unit also covers estimating a square root that falls between two whole numbers.
- A square rug's area is 121 sq ftthe side length is the square root of the area
- A square photo frame's areasquaring the side length gives the area; a square root undoes it
- A square tile's area from its side lengthsquaring is just multiplying a number by itself
- Estimating sqrt(50)50 is not a perfect square, but it falls between 7² = 49 and 8² = 64
What students will be able to do
Students will find the square of a whole number, find the square root of a perfect square, and estimate the square root of a non-perfect square by identifying the two consecutive whole numbers it falls between.
- I can find the square of a whole number by multiplying it by itself.
- I can find the square root of a perfect square number.
- I can explain why squaring and finding a square root are inverse operations.
- I can estimate the square root of a non-perfect square by finding the two consecutive whole numbers it lies between.
Standards this unit teaches
- AC9M7N01Australian Curriculum v9 (ACARA)Squares and square roots
Describe the relationship between perfect square numbers and square roots, and use squares of numbers and square roots of perfect square numbers to solve problems.
- 8.EE.A.2Common Core (US)Square roots and cube roots
Use square root and cube root symbols to represent solutions and evaluate roots of small perfect squares and cubes.
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
Words to teach and display
- Square number
- the result of multiplying a whole number by itself, such as 6² = 36
- Square root
- the value that, multiplied by itself, gives a number; the inverse of squaring
- Exponent
- the small raised number showing how many times a number is multiplied by itself, as in 3²
- Perfect square
- a whole number that is the square of another whole number, such as 1, 4, 9, 16, 25
- Radical sign
- the symbol √ used to show a square root, as in √36 = 6
Teach it: concrete, pictorial, abstract
The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.
1. Squaring a number: finding area from a side length
ConcreteSquaring a number means multiplying it by itself. Written with an exponent, n² means n x n. It is called 'squaring' because that many unit squares, arranged n rows of n, tile a perfect square shape exactly.
A 4-by-4 array of unit squares has 4 rows of 4, or 4 x 4 = 16 squares in total, so 4² = 16. The same array also shows why squaring only ever needs ONE number: both the number of rows and the number of columns are the same value.
A square tile has a side length of 9 cm. Find its area.
- The area of a square is side x side, written side².
- Substitute the side length: 9² = 9 x 9.
- 9 x 9 = 81.
Answer: 81 cm².
- Why does squaring a number connect to the area of a square shape?
- What is the difference between 5² and 5 x 2?
2. Square roots: undoing a square to find the side length
PictorialA square root undoes a square. If side² = area, then the square root of the area gives back the side length. The radical sign √ asks the question directly: what number, multiplied by itself, gives this result?
√64 asks 'what number times itself gives 64?'. Since 8 x 8 = 64, √64 = 8. A perfect square, like 64, always has a whole-number square root; the first several are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144.
A square photo has an area of 121 cm². Find the side length of the photo.
- Finding the side length means finding the square root of the area, since side x side = area.
- √121 asks: what number, multiplied by itself, gives 121?
- 11 x 11 = 121, so √121 = 11.
Answer: 11 cm.
- Why is finding a square root the same as 'undoing' a square?
- How could you check that √121 = 11 is correct?
3. Estimating square roots that aren't whole numbers
AbstractNot every number is a perfect square, so not every square root is a whole number. When a number falls between two consecutive perfect squares, its square root falls between the matching two consecutive whole numbers, and you can judge which one it is closer to.
√70 is not a whole number, since 70 is not a perfect square. The nearest perfect squares are 8² = 64 and 9² = 81. Because 64 < 70 < 81, √70 must be between 8 and 9. Comparing distances, 70 is only 6 more than 64 but 11 less than 81, so √70 is closer to 8 (√70 ≈ 8.37).
Estimate √70. Between which two consecutive whole numbers does it lie, and which is it closer to?
- Find the perfect squares closest to 70: 8² = 64 and 9² = 81.
- Since 64 < 70 < 81, √70 is between 8 and 9.
- 70 is 6 more than 64 but 11 less than 81, so 70 is closer to 64, meaning √70 is closer to 8 than to 9.
Answer: √70 is between 8 and 9, and closer to 8 (√70 ≈ 8.37).
- Why must √70 be between 8 and 9, rather than exactly one of them?
- How do you decide whether an estimate is closer to the lower or upper whole number?
Common misconceptions and how to address them
Misconception5² means 5 x 2, not 5 x 5.
Why it happens: Students confuse the small raised exponent with an ordinary multiplication by that number.
How to address it: An exponent tells you how many times to multiply the BASE by itself, not what to multiply it by. 5² = 5 x 5 = 25, never 5 x 2 = 10. Say '5 squared, 5 times itself' every time.
MisconceptionEvery whole number has a whole-number square root.
Why it happens: Students have only met perfect squares so far and assume the pattern always continues neatly.
How to address it: Only perfect squares (1, 4, 9, 16, 25...) have whole-number square roots. Most whole numbers, like 70, fall between two perfect squares and need an estimate, as in section 3.
MisconceptionEstimating a square root just means picking the nearer-sounding whole number, without checking.
Why it happens: Students guess from familiarity with the target number rather than comparing it to the two surrounding perfect squares.
How to address it: Always find the two consecutive perfect squares the number sits between FIRST, then compare how far the target is from each one, as in the worked example, rather than guessing directly.
Guided practice (with answers)
1. Find 7².
Answer: 49, because 7 x 7 = 49.
2. Find 12².
Answer: 144, because 12 x 12 = 144.
3. Find √81.
Answer: 9, because 9 x 9 = 81.
4. Find √144.
Answer: 12, because 12 x 12 = 144.
5. Estimate: between which two consecutive whole numbers is √55?
Answer: Between 7 and 8, because 7² = 49 and 8² = 64, and 49 < 55 < 64.
Independent practice worksheets
Practise squaring, finding square roots and estimating with computed, never-wrong answer keys.
Differentiation
- Provide a printed list of the first 12 perfect squares (1 to 144) as a reference chart while estimating roots.
- Use physical square tiles or grid paper to build n x n arrays before introducing the n² notation.
- Start estimating only with square roots very close to a perfect square (e.g. √50 is very close to 49) before less obvious ones.
- Practise squaring and square-rooting as a matched pair each time, so the inverse relationship stays visible.
- Introduce square roots of larger numbers (3-digit perfect squares) requiring a longer search for the right factor.
- Ask students to estimate a square root to one decimal place using linear interpolation between the two nearest perfect squares.
- Explore why the square root of a number between 0 and 1 is actually LARGER than the number itself (e.g. √0.25 = 0.5).
- Investigate the pattern in the DIFFERENCES between consecutive square numbers (1, 3, 5, 7, 9...) and why that odd-number pattern exists.
Assessment: exit ticket
A three-question exit ticket sampling squaring, exact square roots, and estimating.
1. Find 11².
Answer: 121, because 11 x 11 = 121.
2. Find √100.
Answer: 10, because 10 x 10 = 100.
3. Estimate: between which two consecutive whole numbers is √40, and which is it closer to?
Answer: Between 6 and 7 (6² = 36, 7² = 49, and 36 < 40 < 49), closer to 6, since 40 is 4 more than 36 but 9 less than 49.
Teacher notes and timings
- Rough timing across two to three lessons: Lesson 1 squaring (section 1), Lesson 2 exact square roots (section 2), Lesson 3 estimating non-perfect-square roots (section 3) plus the exit ticket.
- This unit assumes comfort with multiplication facts up to 12 x 12, since both squaring and finding square roots depend on them.
- Language to repeat: 'squaring multiplies a number by itself'; 'a square root undoes a square'; when estimating, 'find the two nearest perfect squares first, then compare distances'.
- Curriculum note: AC9M7N01 (Australian Curriculum v9) covers this whole unit at Year 7. The Common Core crossover, 8.EE.A.2, covers the same square-root skill one year later in the US sequence, alongside cube roots (not covered here, since AC9M7N01 is squares and square roots only).
- Present and print both work: use Present to build the 4x4 array live and count the squares with the class, then print the worksheet for independent practice.