B1.3 Estimate and calculate square roots, in various contexts.
Activity 1: Estimating Square Roots From Perfect Squares
Activation of prior knowledge: Review perfect squares by going back to its definition and also to the definition of the square root.
Example
49 is a perfect square, because \(7 \times 7 = \;{7^2}\).
\(\sqrt {49} \; = \;7\) (the square root of 49 is 7, because the positive real number 7 is multiplied by itself to get the result of 49)
Ask students to write down in ascending order all the perfect squares less than 150.
Distribute number cards randomly to students (number on cards from 0 to 150). The student should be able to estimate the value of the square root (put emphasis on the student's justification and reasoning).
Example
Estimate the value of the square root of the number on the card by explaining the steps in your reasoning.
Number on card \(\sqrt {40} \)
Example of steps in the reasoning:
- It is not a perfect square, because no two natural numbers multiplied by themselves give 40.
- The radicand (40, the number found under the square root symbol) lies between the perfect squares 36 and 49 so the square root lies between 6 and 7.
- 40 is closer to 36 \(\left( {36\; + \;4} \right)\) than to 49 \(\left( {49\; - \;9} \right)\) so the square root should be between 6 and 6.5
- Estimation:\(\sqrt {40} \; \approx \;6,3\)
Activity 2: Rational or Irrational Number (Exact or Approximate Root?)
Project 10 numbers written as radicands, using the square root symbol.
The student transcribes them into a table like this one.
Example
| Number | Exact or Approximate Value? (Why) | Rational or Irrational Number |
|---|---|---|
| \(\sqrt {40} \) | Approximate, because the radicand is not a perfect square (or by evaluating with the calculator the square root obtained is a number with non-terminating and non-repeating decimals) | Irrational |
| \(\sqrt {49} \) | Exact, because the radicand is a perfect square | Rational |
The student completes a chart with the 10 statements.