Lesson 1 – Determining Square Roots of Rational Numbers
Explore
1a) Explain how the diagram represents √16.
b) Draw a diagram that represents √25.
c) Explain how you could use a diagram like these to represent a rational number
with a square root that is between 4 and 5.
2. a) Explain how the shading on the hundred grid
represents √0.25.
b) Draw a diagram that represents √0.36.
Lesson 1 – Determining Square Roots of Rational Numbers
Connect
Whole number perfect squares and their square roots can be used as benchmark
values when trying to estimate the square root of a number. In other words, they
provide a lower point of reference and an upper point of reference. Estimation helps
confirm your answers are reasonable.
Consider a square sticky note that has a side
length of 7.6 cm.
You can estimate the area of this sticky note
by comparing it to the area of the nearest
whole number side length. A side length of 7.6
cm is between a side length of 7 cm and a side
length of 8 cm. The areas of the squares
formed by each of these whole numbers are
shown.
The area of the sticky note with a side length of 7.6 cm will be between 49 cm2 and
64 cm2. Since 7.6 is closer to 8, the area of the sticky note can be estimated to be
approximately 57 cm2.
Lesson 1 – Determining Square Roots of Rational Numbers
Example 1: Determine a Rational Number from its Square Root
A square trampoline has a side length of 2.6 m. Calculate the area of the trampoline.
Show you know
Estimate and calculate the area of a square photo with a side length of 7.1 cm.
We have seen that the whole number perfect squares can be used to estimate a
rational number when given its square root. But how can you determine whether a
rational number is a perfect square?
Consider the fraction
Remember that a perfect square can be expressed as the product of the same two
rational factors. For the fraction to be a perfect square, both the numerator and
the denominator must be perfect squares.
numerator: 81 = 92 = 9 × 9 The number 81 is a perfect square.
denominator: 121 = 112 = 11 × 11 The number 121 is a perfect square.
Since both the numerator and denominator are perfect squares, then the rational
number is also a perfect square.
But what about decimals?
Now consider the rational number 0.12. To determine whether 0.12 is a perfect
square, first write it in fraction form.
Next, determine whether both the
numerator and the denominator are
perfect squares.
numerator: 12 The number 12 is not a perfect square.
denominator: 100 = 102 = 10 × 10 The number 100 is a perfect square
So, 0.12 is not a perfect square.
Lesson 1 – Determining Square Roots of Rational Numbers
Wanna know a neat shortcut?
Well…you can use your calculator to find the square root of any value.
Take the square root of the following on your calculator and write down the
values…based on what we just learned, which ones are perfect squares?
a) √16
1
b) √9
1
c) √4
d) √0.36
e) √0.04
4
f) √9
g) √8
1
h) √5
Notice a pattern? If we want to know whether a value is a perfect square, we simply
use our calculator to find the value and look at …
Show you know
Determine which of the following numbers is a perfect square
121
a) 64
b) 1.2
c) 0.09
Suppose you are given the following square tile with an area of 0.16
m2. You are asked to determine the dimensions of the tile.
Lesson 1 – Determining Square Roots of Rational Numbers
The side length of the square tile will be a number that when multiplied by itself
results in a product of 0.16.
Is 0.16 a perfect square value?
Example 2: Determine the Square root of a perfect square number
Evaluate √1.44 (in other words the square root of 1.44 is _____)
Show you know
Evaluate
a)√2.25 (the square root of 2.25 is ______)
b) √0.16
Example 3: Determine the Square root of a non perfect square
For non perfect squares, it’s a good idea to estimate the value first, but in reality, the
fastest method to calculate is to simply stick it in your calculator.
Evaluate √0.73 to the nearest thousandth.
Show you know
Evaluate to the nearest hundredth.
√0.34
Assignment Worksheet 1
Skills today:
1. Find the square root of a non perfect square using benchmarks
2. Find the square root of a rational number – determine whether it is a perfect
square or not.
- Fractions – reduce! Are both the top and bottom perfect square
numbers? Then yes it is a perfect square value
- Decimal – take the square root on your calculator, then check – does
it terminate or repeat? Then yes it is a perfect square.
3. NON perfect squares will form non terminating and non repeating values.
Lesson 1 – Determining Square Roots of Rational Numbers
4. Connect rational numbers and square roots. Can you draw a diagram?
Area – represents the radical
Side length – represents the square root.
5. Follow the pattern:
The square root of 49 is _____
The square root of 25 is _____
The square root of 121 is _____
The square root of 16 is _____
The square root of 36 is _____
The square root of _____ is 4
The square root of _____ is 9
The square root of _____ is 3
The square root of _____ is 11
The square root of _____ is 5
Pay very close attention to the words used!!! IS versus OF tells us whether we’re
looking for the big number(radicand) or the square root small number (square root)