7th Grade Math · Unit 2
Explain how a perfect square and its positive square root go together.
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A perfect square
A perfect square is a whole number times itself.
6 × 6 = 36, so 36 is a perfect square.
The positive square root undoes the square
Because 6 × 6 = 36, = 6.
Squaring the root returns the square: 6² = 36.
For a whole number 0 or greater, these two steps undo each other.
Not every whole number is a perfect square
4 × 4 = 16 and 5 × 5 = 25.
20 is between 16 and 25, so 20 is not a perfect square.
Two numbers can square to the same result
6 × 6 = 36 and (−6) × (−6) = 36.
The positive square root is the positive one: = 6.
A square and its positive square root name the same fact.
A perfect square is a whole number times itself.
The positive square root is that whole number.
6 × 6 = 36.
= 6
Answer: = 6
A perfect square is a whole number times itself.
The positive square root is that whole number.
5 × 5 = 25.
= 5
Yes. 25 is 5 squared.
Answer: Yes. = 5
4 × 4 = 16.
5 × 5 = 25.
20 is between 16 and 25.
No whole number times itself is 20.
Answer: No. 20 is between 16 and 25.
Go both ways. Squaring a positive square root returns the perfect square.
n is the perfect square of the root.
15 × 15 = 225.
n = 225
Answer: 225
n is the perfect square of the root.
13 × 13 = 169.
n = 169
Answer: 169
A perfect square is a whole number times itself.
The positive square root is that whole number.
14 × 14 = 196.
= 14
Answer: 14
7 × 7 = 49.
8 × 8 = 64.
50 is between 49 and 64.
There is no whole number whose square is 50.
Answer: 50 is between 49 and 64.
0 × 0 = 0.
So 0 is a perfect square, and = 0.
Answer: Because 0 × 0 = 0
Take the positive square root first.
= 12 because 12 × 12 = 144.
Then square that result.
12 × 12 = 144.
Answer: 144
Take the positive square root first.
= 3 because 3 × 3 = 9.
Then square that result.
3 × 3 = 9.
Answer: 9
Take the positive square root first.
= 4 because 4 × 4 = 16.
Then square that result.
4 × 4 = 16.
Answer: 16
Take the positive square root first.
= 7 because 7 × 7 = 49.
Then square that result.
7 × 7 = 49.
Answer: 49
Take the positive square root first.
= 9 because 9 × 9 = 81.
Then square that result.
9 × 9 = 81.
Answer: 81
Explain a mistake, or connect the area of a square to its side length.
A perfect square is a whole number times itself.
The positive square root is that whole number.
15 × 15 = 225.
= 15
Answer: = 15
50 × 50 = 2,500, not 100.
10 × 10 = 100.
= 10.
The square root is not half of the number.
Answer: = 10
The area of a square is the side length times itself.
The side length is the positive square root of the area.
13 × 13 = 169.
= 13.
Answer: 13 feet
Check each number. A perfect square is a whole number times itself.
A perfect square is a whole number times itself.
3 × 3 = 9.
4 × 4 = 16.
4 × 4 = 16.
5 × 5 = 25.
24 is between 16 and 25.
6 × 6 = 36.
24 is not a perfect square.
Answer: 24
A perfect square is a whole number times itself.
4 × 4 = 16.
6 × 6 = 36.
6 × 6 = 36.
7 × 7 = 49.
48 is between 36 and 49.
8 × 8 = 64.
48 is not a perfect square.
Answer: 48
A perfect square is a whole number times itself.
7 × 7 = 49.
9 × 9 = 81.
9 × 9 = 81.
10 × 10 = 100.
90 is between 81 and 100.
10 × 10 = 100.
90 is not a perfect square.
Answer: 90
A perfect square is a whole number times itself.
11 × 11 = 121.
12 × 12 = 144.
12 × 12 = 144.
13 × 13 = 169.
150 is between 144 and 169.
13 × 13 = 169.
150 is not a perfect square.
Answer: 150
A perfect square is a whole number times itself.
1 × 1 = 1.
2 × 2 = 4.
2 × 2 = 4.
3 × 3 = 9.
8 is between 4 and 9.
3 × 3 = 9.
8 is not a perfect square.
Answer: 8
A perfect square is a whole number times itself.
14 × 14 = 196.
15 × 15 = 225.
15 × 15 = 225.
16 × 16 = 256.
240 is between 225 and 256.
16 × 16 = 256.
240 is not a perfect square.
Answer: 240
A perfect square is a whole number times itself.
0 × 0 = 0.
1 × 1 = 1.
1 × 1 = 1.
2 × 2 = 4.
2 is between 1 and 4.
2 × 2 = 4.
2 is not a perfect square.
Answer: 2
If the positive square root is given, square it.
The number under the radical is the root times itself.
8 × 8 = 64.
= 8
Answer: 64
The number under the radical is the root times itself.
12 × 12 = 144.
= 12
Answer: 144
The number under the radical is the root times itself.
7 × 7 = 49.
= 7
Answer: 49
The number under the radical is the root times itself.
18 × 18 = 324.
= 18
Answer: 324
The number under the radical is the root times itself.
11 × 11 = 121.
= 11
Answer: 121
The number under the radical is the root times itself.
19 × 19 = 361.
= 19
Answer: 361
The number under the radical is the root times itself.
6 × 6 = 36.
= 6
Answer: 36
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