Square numbers and square roots: a week of ready-to-teach maths

Five lessons for Year 7 Number. Print this pack and the week is prepared: each day has a one-page plan and a student worksheet with practice and a stretch section, plus square-builder pieces, root match and line-up cards, a mini-check and every answer with its working.

Start here: five minutes to Monday

1. Skim the week at a glance on the next page.
2. Print the five days. Each day is two A4 sheets: a plan and a worksheet.
3. Print the two cut-out sheets, one copy per pair, and cut them out before Day 2: the root match cards are for Day 2, the square-builder pieces for Day 3 and the line-up cards for Day 4.
4. Open the free interactive unit on the board. Every plan says which picture to show and which buttons to press.
5. Teach straight from the plan. Timings, talk prompts, misconceptions and worked answers are all on the one page.

The idea in plain words

A square number is the area of a square with a whole-number side: 7 rows of 7 tiles make 49, so 49 is 7 squared, written 7². The square root runs the same picture backwards: from the area 49 back to the side 7, written √49 = 7. Squaring and taking the square root undo each other, so every root can be checked by squaring it. Most numbers are not perfect squares, and their roots are not whole numbers, but the squares either side still trap them: 25 < 30 < 36, so √30 lies between 5 and 6. Squaring is not doubling: 7² is 49, while 2 × 7 is only 14.

One day, one lesson

The five lessons fill a week of maths, one lesson of about 60 minutes a day. Run them in order: each day stands on the one before. Every plan marks a split point for two shorter sessions, and every worksheet ends with a stretch section for students who finish early.

How we write squares and roots

Read 5² as “five squared”: the small raised 2 is the index and says how many fives are multiplied, 5 × 5. Read √25 as “the square root of 25”. Both 5 and −5 square to 25, but the root sign names the positive one, so √25 = 5; a side length is never negative anyway. Calculators stay in the drawer until Day 4, and even then students use them to square decimals, not to press the root key.

The week at a glance

One lesson a day for a week. Each day stands on the day before, so run them in order.

How the week builds

Day 1 makes a square number an area; Day 2 runs the picture backwards to find a side; Day 3 finds the odd numbers hiding between the squares; Day 4 traps the roots of numbers that are not perfect squares; and Day 5 puts squares and roots to work on problems. It builds on Year 6, where square numbers were one kind of number beside primes and triangular numbers, and it prepares the powers of primes later in Year 7 and, in Year 8, roots that are not whole numbers and Pythagoras’ theorem.

Materials for the week (one trip)

• From the classroom: scissors, pencils, coloured pencils, rulers, 1 cm grid paper, this pack printed.
• Cut-out sheet 1, one per pair: six L-shaped square-builder pieces (Day 3).
• Cut-out sheet 2, one per pair: 24 root match cards (Day 2) and 12 root line-up cards (Day 4).
• For Days 4 and 5: calculators, used for squaring decimals and checking, not for the root key. No other equipment to buy.

Words for the word wall

square number, perfect square, squared, index, index notation, square root, root sign, inverse operation, undo, odd number, consecutive, last digit, estimate, between, closer to, area, side length, perimeter

Dear families

This week in maths, Year 7 is learning about square numbers and square roots. A square number is the area of a square: 6 rows of 6 make 36, written 6². The square root goes backwards, from the area to the side: √36 = 6. We also look for patterns in the squares and estimate roots that are not whole numbers, such as √30, which lies between 5 and 6.

Try this at home

• Find something square with rows of squares: a chessboard (8 by 8), a tiled floor or a sheet of stamps. How many small squares, and what is that as a square number?
• Ask: is 6² the same as 6 × 2? (No: 36 against 12.)
• Ask: a square garden bed has an area of 25 m². How long is each side, and how much edging goes round it?
• Ask: between which two whole numbers is √50? (7 and 8, because 49 < 50 < 64.)

My squares this week

Fill one row a day. Find a square number in something real, or set someone at home a square-root question.

Printed from the free seegongsik Square numbers and square roots teaching pack · seegongsik.com/au/y7/number/AC9M7N01/pack

Square numbers are squares

A square number is the area of a square with a whole-number side. Students shade squares on grid paper, write them in index notation, table the squares to 15² and test numbers.

We are learning to

• show a square number as the area of a square grid,
• write and read squares in index notation, such as 7² = 49,
• decide whether a number is a perfect square by finding the squares on either side.

Success criteria

• I can say why 36 is a perfect square and 24 is not, with a picture or a calculation.
• I can write the squares from 1² to 15² and explain why 6² is not 12.

You need

1 cm grid paper and coloured pencils, the worksheet, one per student. No calculators today.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part A; start the second with the board’s second picture.

Watch for these ideas

• Reading 6² as 6 × 2. The index counts how many sixes are multiplied: 6² = 6 × 6 = 36.
• Thinking any number of tiles makes a square: 24 sits between 4² = 16 and 5² = 25.
• Thinking squaring always makes a number bigger: 0.5² = 0.25.

Answers

• Part A: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225.
• Part B: 1: 1 and 2; 2: 4 and 4; 3: 9 and 6; 5: 25 and 10; 10: 100 and 20. Equal only at n = 2; past 2, n lots of n beats 2 lots of n.
• Part C: 81 = 9², yes; 36 = 6², yes; 48: 6² = 36, 7² = 49, no; 90: 9² = 81, 10² = 100, no; 121 = 11², yes; 144 = 12², yes; 150: 12² = 144, 13² = 169, no.
• Part D: 0.5² = 0.25, 1.5² = 2.25, 0.3² = 0.09, (½)² = ¼. Between 0 and 1 a square is smaller: half of a half is a quarter.

Square numbers

A. The squares from 1² to 15²

B. Squared or doubled?

For which n are they the same? ______ Which is bigger once n is more than 2, and why? ______________________________

C. Square or not?

Write the square it equals, or the two squares it lies between. Then tick.

D. Stretch: squares below 1

0.5² = ______    1.5² = ______    0.3² = ______    (½)² = ______

Squaring 0.5 made it smaller. Shade 0.5 by 0.5 inside a 1 by 1 square to show why.

The square root runs it backwards

The square root runs Day 1 backwards, from the area of a square to its side. Students find roots, check each by squaring, and give sides with the right units.

We are learning to

• find the square root of a perfect square as the side of a square with that area,
• check a square root by squaring it,
• say why √169 is 13 even though (−13)² is also 169.

Success criteria

• I can find √144 and show that 12² = 144.
• I can give the side of a square from its area, with the right units.

You need

Each student’s Day 1 table of squares, the root match cards from cut-out sheet 2 (one set per pair), the worksheet. No calculators.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after the root match; start the second with Part A.

Watch for these ideas

• Halving instead of rooting: √16 written as 8. Squaring catches it: 8² = 64, not 16.
• Losing the units. A square of area 36 m² has a side of 6 m, not 6 m².
• Thinking √169 is both 13 and −13. Both square to 169, but the root sign names the positive root, 13; a side length cannot be negative.

Answers

• Part A: √9 = 3, √25 = 5, √64 = 8, √81 = 9, √100 = 10, √121 = 11, √144 = 12, √196 = 14, √225 = 15, √400 = 20.
• Part B: 900 cm²: 30 cm; 49 m²: 7 m; 169 cm²: 13 cm; 2500 m²: 50 m.
• Part C: 11; 64; 13; 256; 169 and 13; 1 and 0.
• Part D: √144 = 12 = 3 × 4, but √25 = 5, not 3 + 4 = 7; √2304 = 48 = 6 × 8, but √100 = 10, not 6 + 8 = 14. A root splits over a product but not over a sum.

Square roots

A. Find the square root

Choose two and check them by squaring: ________________________________________________

B. From area to side

C. Undo and redo

Both 13 and −13 square to 169. Why does √169 mean 13? ________________________________

D. Stretch: can a root be split?

Which way of splitting a root worked both times, and which did not? ______________________________

Patterns in the squares

The gap from one square to the next is the next odd number, the first n odd numbers add up to n², and a perfect square can only end in 0, 1, 4, 5, 6 or 9.

We are learning to

• explain with L-shaped pieces why the gaps between squares are the odd numbers,
• add a run of odd numbers without adding every term,
• use last digits to rule a number out as a square, and check the numbers that survive.

Success criteria

• I can show with the pieces why 6² = 5² + 11.
• I can explain why 7742 cannot be a square, and why a last digit of 6 does not prove that 4096 is one.

You need

The square-builder pieces from cut-out sheet 1 (one set per pair), coloured pencils, the worksheet. No calculators.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part B; start the second with the last-digit table.

Watch for these ideas

• Thinking the gap between squares stays the same: 4 − 1 = 3 but 9 − 4 = 5.
• Treating the last-digit test as proof. A last digit of 2, 3, 7 or 8 rules a number out; 0, 1, 4, 5, 6 or 9 only means it might be a square, as 1000 shows.
• Adding twenty odd numbers one at a time. The pattern gives 20² = 400 straight away.

Answers

• Part A: gaps 3, 5, 7, 9, 11, 13, 15, 17, 19, the odd numbers from 3. 11² = 100 + 21 = 121; 15² = 196 + 29 = 225.
• Part B: 25, 64, 100, 400 (5², 8², 10², 20²). 225 = 15², so the run ends at the 15th odd number, 29.
• Part C: squares of 0 to 9 end in 0, 1, 4, 9, 6, 5, 6, 9, 4, 1, so a square ends in 0, 1, 4, 5, 6 and 9. 2209 = 47²; 3248 ruled out (ends in 8); 5327 ruled out (ends in 7); 4096 = 64²; 1000 not a square (31² = 961, 32² = 1024); 7742 ruled out (ends in 2).
• Part D: 36 has 9 factors, 24 has 8 factors, as 6 × 6 pairs with itself. To 30: 1, 4, 9, 16, 25.

Patterns in the squares

A. From one square to the next

10² = 100, so 11² = 100 + ______ = ______

14² = 196, so 15² = 196 + ______ = ______

B. Adding odd numbers

C. Last digits

D. Stretch: counting factors

List the factors of 36 and of 24. Which has an odd number of factors, and why? ______________________________

Which numbers from 1 to 30 have an odd number of factors? ______________________________

Trapping a root

Most numbers are not perfect squares. Students trap such a root between two whole numbers, decide which it is closer to, and narrow it to tenths by squaring decimals.

We are learning to

• find the two whole numbers a square root lies between,
• decide which of the two it is closer to, and say why,
• narrow a root to tenths by squaring decimals.

Success criteria

• I can say that √30 lies between 5 and 6, closer to 5, because 5.5² = 30.25 is more than 30.
• I can put roots and decimals in order by squaring them.

You need

The Day 1 table of squares, the root line-up cards from cut-out sheet 2 (one set per group of three or four), calculators for Part C only, the worksheet.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after the line-up; start the second with Part A.

Watch for these ideas

• Answering with the squares: “√30 is between 25 and 36”. The squares trap it; the answer is the sides, 5 and 6.
• Halving: √30 is not 15. Squaring checks it: 15² = 225.
• Taking the root key’s 5.477225575 for √30 as exact: 5.477² = 29.997529.

Answers

• Parts A and B: √8: 2–3, 2.5² = 6.25, so 3; √20: 4–5, 4.5² = 20.25, so 4; √45: 6–7, 6.5² = 42.25, so 7; √60: 7–8, 7.5² = 56.25, so 8; √90: 9–10, 9.5² = 90.25, so 9; √130: 11–12, 11.5² = 132.25, so 11; √220: 14–15, 14.5² = 210.25, so 15.
• Line-up: 2.5, √8, √12, 4, √20, √30, 6, √40, √50, 7.5, √70, 9.
• Part C: √50: 7.0² = 49, 7.1² = 50.41; √10: 3.1² = 9.61, 3.2² = 10.24; √70: 8.3² = 68.89, 8.4² = 70.56.
• Part D: 6.5² = 42.25 > 40, so less than 6.5 cm; 7, 7.2, √52 (7.2² = 51.84); √1000: 31 and 32 (961 and 1024).

Trapping a root

A and B. Trap it, then say which side it is closer to

Mark each root on the line, as close as you can.

C. Narrow it to tenths

Use the calculator to square decimals only, not the root key.

______² = ________ < 50 < ______² = ________, so √50 is between ______ and ______

______² = ________ < 10 < ______² = ________, so √10 is between ______ and ______

______² = ________ < 70 < ______² = ________, so √70 is between ______ and ______

D. Stretch

A square has an area of 40 cm². Is its side more or less than 6.5 cm? Show how you know. ____________________

Put in order, smallest first: 7.2, √52, 7. ______________________

Between which two whole numbers is √1000? ______________________

Squares and roots at work

The skill of the day is choosing: a side given and an area wanted means square it; an area given and a side wanted means take the root. Every answer is checked by working backwards.

We are learning to

• decide whether a problem needs a square or a square root,
• solve area and arrangement problems with the right units, checking by working backwards.

Success criteria

• I can find the side and the perimeter of a square from its area.
• I can explain why tripling each side of a shape gives nine times the area.

You need

The worksheet, calculators for checking only, and the unit’s self-check quiz on the board.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after problem 3; start the second with problem 4.

Watch for these ideas

• Dividing the area by 4 to get the side. The side of a square of 196 m² is √196, not 196 ÷ 4.
• Mixing units: a floor in metres and tiles in centimetres. Change to one unit first.
• Thinking sides 3 times as long give 3 times the area. Both ways grow: 3 × 3 = 9 times.

Answers with working

1. side = √196 = 14 m (check: 14² = 196); edging = 4 × 14 = 56 m.
2. 3.6 m = 360 cm; 360 ÷ 30 = 12 tiles along each side; 12² = 144 tiles.
3. 12² = 144 ≤ 150 < 13² = 169, so 12 rows of 12; 150 − 144 = 6 left over.
4. 3² = 9 times; 150 cm² becomes 30 × 45 = 1350 cm² = 9 × 150.
5. side = 52 ÷ 4 = 13 cm; area = 13² = 169 cm².
6. 17² = 289 < 300 < 18² = 324, so no; 24 more make 18 by 18.
7. 8 + 6 = 14 cm, not √100 = 10 cm.
8. 100 ÷ 4 = 25 cm² each, so each side is √25 = 5 cm, half of √100.

Squares and roots at work

For each problem: square it or root it? Show your working and a check.

Problems (7 and 8 are the stretch)

1. A square courtyard has an area of 196 m². How long is each side, and how much edging goes all the way round?
2. A square floor is 3.6 m along each side. It is covered with square tiles 30 cm along each side. How many tiles are needed?
3. 150 chairs are set out in the biggest square block possible. How many rows are there, and how many chairs are left over?
4. A photo is enlarged so that each side is 3 times as long. How many times as much area does it cover? Check with a 10 cm by 15 cm photo.
5. A square has a perimeter of 52 cm. What is its area?
6. Can 300 square tiles make one solid square? If not, how many more tiles would make the next square?
7. Stretch. Squares of 64 cm² and 36 cm² sit side by side, bottoms level. How wide is the shape? Is it √(64 + 36)?
8. Stretch. A square of 100 cm² is cut into four equal smaller squares. How long is each small side?

Square-builder pieces

For Day 3, one set per pair

Cut out the six L-shaped pieces along their outside edges, keeping each piece whole. On Day 3, pairs count the tiles in each piece and fit the pieces together, smallest in the corner, to build bigger and bigger squares.

Teacher note: the pieces hold A 5, B 11, C 1, D 7, E 3, F 9 tiles, and they nest in the order C, E, A, D, F, B to make a 6 by 6 square of 36 tiles.

Root match and line-up cards

Root match (Day 2): twelve areas

Root match (Day 2): twelve sides

Root line-up (Day 4)

Groups lay the twelve cards in a line, smallest on the left, squaring to decide each close call.

Teacher note: the matches are 16 cm² with 4 cm, 49 m² with 7 m, 81 cm² with 9 cm, 121 m² with 11 m, 144 cm² with 12 cm, 169 m² with 13 m, 196 cm² with 14 cm, 225 m² with 15 m, 256 cm² with 16 cm, 289 m² with 17 m, 324 cm² with 18 cm, 400 m² with 20 m. The line-up runs 2.5, √8, √12, 4, √20, √30, 6, √40, √50, 7.5, √70, 9.

What we know: Square numbers and square roots

Work on your own, without a calculator. Show your working.

1. Work out 8² = ______    13² = ______    20² = ______
2. Which is bigger, 9² or 2 × 9? By how much? ______________________
3. Find √36 = ______    √289 = ______    √1600 = ______
4. Circle the perfect squares and show how you know: 72    361    625    200
5. A square courtyard has an area of 256 m². How long is each side, and what is its perimeter? ______________________
6. Between which two whole numbers is √75? Which one is it closer to, and how do you know? ______________________
7. Add the first 9 odd numbers: 1 + 3 + 5 + ... + 17 = ______. How can you tell without adding them one by one?
8. Is √(25 + 144) the same as √25 + √144? Show why. ______________________

Answers and marking guide

Answers with working

1. 8² = 8 × 8 = 64; 13² = 13 × 13 = 169; 20² = 20 × 20 = 400.
2. 9² = 81 and 2 × 9 = 18, so 9² is bigger by 63.
3. √36 = 6 (6² = 36); √289 = 17 (17² = 289); √1600 = 40 (40² = 1600).
4. 72 is between 8² = 64 and 9² = 81, not a square; 361 = 19², circled; 625 = 25², circled; 200 is between 14² = 196 and 15² = 225, not a square.
5. Side = √256 = 16 m (check: 16² = 256); perimeter = 4 × 16 = 64 m.
6. 64 < 75 < 81, so √75 is between 8 and 9; 8.5² = 72.25, which is less than 75, so it is closer to 9.
7. 81: the first 9 odd numbers add to 9² = 81: the L-shaped layers of a 9 by 9 square.
8. No: √(25 + 144) = √169 = 13, but √25 + √144 = 5 + 12 = 17.

A quick three-level guide

Eight questions, four ideas. A student at standard answers most questions with working and can check each root by squaring it back.

Weekly class record

Jot a tick as you move around the room; the mini-check fills any gaps. A tick a day is plenty.

The five columns are the five days: squares as areas, roots, patterns in the squares, trapping a root, and problems.