Two pages of visual notes for students to keep: the key words with pictures, perfect squares and cubes, how to estimate a root on a number line, four solved problems in the style of the Rooted to the Spot game, and where roots show up in real life. To make a PDF, click Print and choose "Save as PDF."
Visual Notes: Square Roots and Cube Roots
Grade 8 · CCSS 8.EE.A.2, 8.NS.A.2 · Pairs with the Rooted to the Spot game
Name
The big idea: squaring a number makes a square: a side of 7 makes an area of 49. A square root goes backward: from an area of 49 back to a side of 7. A cube root does the same for a cube: from a volume back to its edge.
Key vocabulary
Perfect square
A whole number times itself: 4 × 4 = 16.
Square root (√)
√49 = 7
The number that, times itself, gives the original: 7 × 7 = 49.
Perfect cube
A whole number used 3 times as a factor: 2 × 2 × 2 = 8.
Cube root (∛)
∛64 = 4
The number used 3 times to make it: 4 × 4 × 4 = 64.
Know these by heart
Perfect squares
n
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
n²
1
4
9
16
25
36
49
64
81
100
121
144
169
196
225
Perfect cubes
n
1
2
3
4
5
10
n³
1
8
27
64
125
1000
Estimating a square root that isn't whole
To estimate √30: find the perfect squares on each side. 25 < 30 < 36, so 5 < √30 < 6. 30 is closer to 25 than to 36, so √30 is closer to 5 (about 5.5).
Square root vs. half: √36 is 6 (because 6 × 6 = 36), not 18. Square roots of numbers between 0 and 100 are always between 0 and 10.
A square patio has an area of 144 ft². How long is each side?
The side is √144: what number times itself is 144?
12 × 12 = 144, so the side is 12 ft
✓ Check: 12 × 12 = 144
2. Edge of a cube
A cube-shaped box has a volume of 125 in³. How long is each edge?
The edge is ∛125: what number used 3 times makes 125?
5 × 5 × 5 = 125, so the edge is 5 in
✓ Check: 5 × 5 = 25, 25 × 5 = 125
3. Estimate √50
Perfect squares around it: 49 < 50 < 64
So 7 < √50 < 8
50 is just past 49, so √50 is just over 7 (about 7.1)
✓ Check: 7.1 × 7.1 = 50.41, close to 50
4. Rational or irrational?
√16 = 4: a whole number, rational
√2 = 1.41421… never ends or repeats: irrational
The square root of a whole number that isn't a perfect square is irrational.
✓ Irrational numbers still have a spot on the number line
Watch out
√36 is 6, not 18. A square root is not half.
∛27 is 3, not 9. A cube root uses the number three times as a factor.
To estimate, use the perfect squares on both sides, then decide which one it's closer to.
Where you see it in real life
Floors: 36 square tiles make a 6-by-6 square.Gardens: a square plot's side is the root of its area.Boxes and ice cubes: the edge is the cube root of the volume.Sports fields: square training areas are measured by area.