Squares and Square Roots of Numbers for Primary 4

Squares and Square Roots of Numbers for Primary 4

Performance Objectives

By the end of the lesson, students should be able to:

1. calculate squares of 1- and 2-digit numbers.
2. find square roots of perfect squares up to 900;
3. solve quantitative aptitude problems involving square roots of perfect squares not greater than 900.

What is the Square of a Number?

The square of a number is the result of multiplying that number by itself.

When a number is multiplied by itself, the answer is called its square. For example, if you multiply 3 by 3, the answer is 9. So, 9 is the square of 3.

The symbol for squaring a number is a small ‘2’ written above and to the right of the number. This is called an exponent. For example, $3 \times 3$ can be written as $3^2$. This is read as “3 squared” or “3 to the power of 2”.

Calculating Squares of 1-Digit Numbers

To find the square of a 1-digit number, multiply the number by itself.

Examples of squares of 1-digit numbers include:

1. $1^2 = 1 \times 1 = 1$
2. $2^2 = 2 \times 2 = 4$
3. $3^2 = 3 \times 3 = 9$
4. $4^2 = 4 \times 4 = 16$
5. $5^2 = 5 \times 5 = 25$
6. $6^2 = 6 \times 6 = 36$
7. $7^2 = 7 \times 7 = 49$
8. $8^2 = 8 \times 8 = 64$
9. $9^2 = 9 \times 9 = 81$

Calculating Squares of 2-Digit Numbers

To find the square of a 2-digit number, multiply the 2-digit number by itself.

Let us calculate the squares of some 2-digit numbers:

1. Find the square of 10.

   Solution:

   $$
   10^2 = 10 \times 10 = 100
   $$

2. Find the square of 12.

   Solution:

   $$
   12^2 = 12 \times 12
   $$
   $$
   \begin{array}{r}
   12 \\
   \times 12 \\
   \hline
   24 \\
   +120 \\
   \hline
   144
   \end{array}
   $$

   So, $12^2 = 144$.

3. Find the square of 15.

   Solution:

   $$
   15^2 = 15 \times 15
   $$
   $$
   \begin{array}{r}
   15 \\
   \times 15 \\
   \hline
   75 \\
   +150 \\
   \hline
   225
   \end{array}
   $$

   So, $15^2 = 225$.

4. Find the square of 20.

   Solution:

   $$
   20^2 = 20 \times 20 = 400
   $$

5. Find the square of 25.

   Solution:

   $$
   25^2 = 25 \times 25
   $$
   $$
   \begin{array}{r}
   25 \\
   \times 25 \\
   \hline
   125 \\
   +500 \\
   \hline
   625
   \end{array}
   $$

   So, $25^2 = 625$.

6. Find the square of 30.

   Solution:

   $$
   30^2 = 30 \times 30 = 900
   $$

What is the Square Root of a Number?

The square root of a number is the number that, when multiplied by itself, gives the original number.

It is the opposite operation of squaring a number. For example, since $3 \times 3 = 9$, the square root of 9 is 3. The symbol for a square root is $\sqrt{\quad}$. This symbol is called a radical sign.

So, $\sqrt{9}$ is read as “the square root of 9”.

Perfect Square Numbers

A perfect square number is a number whose square root is a whole number.

For example, 4 is a perfect square because its square root is 2 (a whole number). 9 is a perfect square because its square root is 3. Numbers like 2, 3, 5, 6, 7, 8 are not perfect squares because their square roots are not whole numbers.

Examples of perfect square numbers up to 900 include:

1. 1 (since $1 \times 1 = 1$)
2. 4 (since $2 \times 2 = 4$)
3. 9 (since $3 \times 3 = 9$)
4. 16 (since $4 \times 4 = 16$)
5. 25 (since $5 \times 5 = 25$)
6. 36 (since $6 \times 6 = 36$)
7. 49 (since $7 \times 7 = 49$)
8. 64 (since $8 \times 8 = 64$)
9. 81 (since $9 \times 9 = 81$)
10. 100 (since $10 \times 10 = 100$)
11. 121 (since $11 \times 11 = 121$)
12. 144 (since $12 \times 12 = 144$)
13. 169 (since $13 \times 13 = 169$)
14. 196 (since $14 \times 14 = 196$)
15. 225 (since $15 \times 15 = 225$)
16. 256 (since $16 \times 16 = 256$)
17. 289 (since $17 \times 17 = 289$)
18. 324 (since $18 \times 18 = 324$)
19. 361 (since $19 \times 19 = 361$)
20. 400 (since $20 \times 20 = 400$)
21. 441 (since $21 \times 21 = 441$)
22. 484 (since $22 \times 22 = 484$)
23. 529 (since $23 \times 23 = 529$)
24. 576 (since $24 \times 24 = 576$)
25. 625 (since $25 \times 25 = 625$)
26. 676 (since $26 \times 26 = 676$)
27. 729 (since $27 \times 27 = 729$)
28. 784 (since $28 \times 28 = 784$)
29. 841 (since $29 \times 29 = 841$)
30. 900 (since $30 \times 30 = 900$)

Finding Square Roots of Perfect Squares up to 900

To find the square root of a perfect square, think of which number, when multiplied by itself, gives the original number.

Let us find the square roots of some perfect squares:

1. Find $\sqrt{4}$.

   Solution:

   We need to find a number that, when multiplied by itself, gives 4.

   We know that $2 \times 2 = 4$.

   So, $\sqrt{4} = 2$.

2. Find $\sqrt{25}$.

   Solution:

   We need to find a number that, when multiplied by itself, gives 25.

   We know that $5 \times 5 = 25$.

   So, $\sqrt{25} = 5$.

3. Find $\sqrt{100}$.

   Solution:

   We need to find a number that, when multiplied by itself, gives 100.

   We know that $10 \times 10 = 100$.

   So, $\sqrt{100} = 10$.

4. Find $\sqrt{400}$.

   Solution:

   We need to find a number that, when multiplied by itself, gives 400.

   We know that $20 \times 20 = 400$.

   So, $\sqrt{400} = 20$.

5. Find $\sqrt{625}$.

   Solution:

   We need to find a number that, when multiplied by itself, gives 625.

   We know that $25 \times 25 = 625$.

   So, $\sqrt{625} = 25$.

6. Find $\sqrt{900}$.

   Solution:

   We need to find a number that, when multiplied by itself, gives 900.

   We know that $30 \times 30 = 900$.

   So, $\sqrt{900} = 30$.

Solving Quantitative Aptitude Problems

Quantitative aptitude problems involving square roots test your understanding of squares and square roots in different situations. These problems often involve finding missing numbers or solving simple word problems.

Let us solve some quantitative aptitude problems:

1. What is the value of $5^2 + \sqrt{16}$?

   Solution:

   First, find the square of 5:

   $$
   5^2 = 5 \times 5 = 25
   $$

   Next, find the square root of 16:

   We know that $4 \times 4 = 16$, so $\sqrt{16} = 4$.

   Now, add the results:

   $$
   25 + 4 = 29
   $$

   So, $5^2 + \sqrt{16} = 29$.

2. A square garden has an area of 81 square metres. What is the length of one side of the garden?

   Solution:

   The area of a square is found by multiplying the side length by itself (side $\times$ side = area).

   To find the length of one side, we need to find the square root of the area.

   Length of side = $\sqrt{81}$

   We know that $9 \times 9 = 81$.

   So, $\sqrt{81} = 9$.

   The length of one side of the garden is 9 metres.

3. If a number is squared, the result is 144. What is the number?

   Solution:

   Let the number be $x$.

   When the number is squared, it means $x \times x = x^2$.

   We are given that $x^2 = 144$.

   To find $x$, we need to find the square root of 144.

   $x = \sqrt{144}$

   We know that $12 \times 12 = 144$.

   So, $x = 12$.

   The number is 12.

4. Calculate $(\sqrt{49} \times \sqrt{9}) – 10$.

   Solution:

   First, find the square root of 49:

   We know that $7 \times 7 = 49$, so $\sqrt{49} = 7$.

   Next, find the square root of 9:

   We know that $3 \times 3 = 9$, so $\sqrt{9} = 3$.

   Now, multiply the results:

   $$
   7 \times 3 = 21
   $$

   Finally, subtract 10:

   $$
   21 – 10 = 11
   $$

   So, $(\sqrt{49} \times \sqrt{9}) – 10 = 11$.

Using Math Tools for Square Roots

In mathematics, there are various tools and applications that can help in solving problems related to square roots. These tools can be physical charts or digital applications.

A square root chart lists numbers and their corresponding squares and square roots. This allows for quick identification of perfect squares and their roots without manual calculation.

Digital tools, such as math apps on phones or computers, can also calculate squares and square roots very quickly. These applications are useful for checking answers or for finding square roots of larger numbers efficiently. While using such tools, it is important to understand the basic mathematical concepts of squares and square roots first.