PURPOSE Seated and Silent by 7:53. Have forms ready to turn in. Have Student Survey ready to turn in. Complete All About Me Assignment or Time Capsule Activity if not completed yesterday. All About Me and Time Capsule are on the back counter. Today we will be discussing TCAP scores.

Advisory – Silent Reading If you do not have your own reading material, take a book, magazine, comic, or graphic novel from my shelf.

Procedures Review Think of at least 3 procedures we talked about yesterday. Be prepared to share one of those procedures if your UNO card is drawn.

Square Roots, Cube Roots, and Irrational Numbers 8.EE.2: Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.

Words to know… Square root Perfect square Irrational Cube root

Individual Think Time: How can you turn two blocks into a square?

2 x 2 = 4 2 2

3 x 3 = 9 3 3

4 x 4 = 16 4 4

5 x 5 = 25 5 5

2 The square root of 4 is

3 The square root of 9 is

4 The square root of 16 is

5 The square root of 25 is

Square numbers Here are the first 10 square numbers: 12 = 1 × 1 = 1 22 = 2 × 2 = 4 32 = 3 × 3 = 9 42 = 4 × 4 = 16 52 = 5 × 5 = 25 62 = 6 × 6 = 36 Introduce the notation 2 to mean ‘squared’ or ‘to the power of two’. Pupils sometimes confuse ‘to the power of two’ with ‘times by two’. Point out that raising to a power can be thought of as repeated multiplication. Squaring means that the number is multiplied by itself, while multiplying by two means that the number is added to itself. For example, 42 means 4 × 4 and 4 × 2 means 4 + 4. Point out that these square numbers form a sequence. Look at the difference between consecutive terms and conclude that we could write a term-to-term rule for this sequence as add consecutive odd numbers. Ask pupils to tell you the position-to-term rule. Conclude that each square number can be made, not only by multiplying a whole number by itself (raising it to the power of two) but also by adding together consecutive odd numbers starting with one. 72 = 7 × 7 = 49 82 = 8 × 8 = 64 92 = 9 × 9 = 81 102 = 10 × 10 = 100

Adding consecutive odd numbers The seventh square number is 49. The eighth square number is 64. The ninth square number is 81. The sixth square number is 36. The tenth square number is 100. The fifth square number is 25. The second square number is 4. The first square number is 1. The fourth square number is 16. The third square number is 9. By using a different colour for the counters that are added on between consecutive square numbers we can see how square numbers can be made by adding consecutive odd numbers starting from 1. 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 = 64 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 = 81 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 = 100 1 + 3 + 5 + 7 + 9 + 11 + 13 = 49 1 + 3 + 5 + 7 = 16 1 + 3 = 4 1 + 3 + 5 = 9 1 + 3 + 5 + 7 + 9 = 25 1 + 3 + 5 + 7 + 9 + 11 = 36

Think about this….. Could you figure out what the 11th square number would be?

Think about this….. What about the 15th square number?

Square roots Finding the square root is the inverse of finding the square: squared 8 64 square rooted Talk through the slide and introduce the square root symbol. Explain that to work out the square root we need to ask ourselves ‘what number multiplied by itself will give this answer’. Does any other number multiply by itself to give 64? Obtain the answer –8. Remember that when you multiply two negative numbers together the answer is positive. Point out that when we use the  symbol we are usually referring to the positive square root. Ask verbally for the square root of some known square numbers. For example, What is the square root of 81? We write 64 = 8 The square root of 64 is 8.

Square roots We can easily find the square root of a square number. 1 = 1 36 = 6 4 = 2 49 = 7 9 = 3 64 = 8 16 = 4 81 = 9 25 = 5 100 = 10

Square numbers When we multiply a number by itself we say that we are squaring the number. To square a number we can write a small 2 after it. For example, the number 3 multiplied by itself can be written as Three squared 3 × 3 or 32 Introduce the notation 2 to mean ‘squared’ or ‘to the power of two’. Pupils sometimes confuse ‘to the power of two’ with ‘times by two’. Point out that raising to a power can be thought of as repeated multiplication. Squaring means that the number is multiplied by itself, while multiplying by two means that the number is added to itself. For example, 42 means 4 × 4 and 4 × 2 means 4 + 4. Point out that these square numbers form a sequence. Look at the difference between consecutive terms and conclude that we could write a term-to-term rule for this sequence as add consecutive odd numbers. Ask pupils to tell you the position-to-term rule. Conclude that each square number can be made, not only by multiplying a whole number by itself (raising it to the power of two) but also by adding together consecutive odd numbers starting with one. The value of three squared is 9. The result of any whole number multiplied by itself is called a perfect square.

Think about this……

Think about this……

Approximating Square Roots Square roots that are not perfect squares are called irrational. An irrational number in a non-repeating, non-terminating decimal. This means the decimal does not repeat, but it also doesn’t end. For example, is not a perfect square and so it’s irrational. Remember

Categorize the following square roots as rational or irrational. For example, the is rational because 22 or 2 x 2 = 4. The is irrational because there is no whole number that can be multiplied by itself to result in 5.

Think about this…. What could be a square root that is irrational? How do you know?

Cubes 5 6 7 8 1 2 3 4

2 x 2 x 2 = 8 2 2 2

3 x 3 x 3 = 27

Cube Roots The index of a cube root is always 3. The cube root of 64 is written as .

The cube root of a number is… What does cube root mean? The cube root of a number is… …the value when multiplied by itself three times gives the original number.

Cube Root Vocabulary radical sign index radicand

Perfect Cubes If a number is a perfect cube, then you can find its exact cube root. A perfect cube is a number that can be written as the cube (raised to third power) of another number.

What would the next perfect cube be? What are Perfect Cubes? 13 = 1 x 1 x 1 = 1 23 = 2 x 2 x 2 = 8 33 = 3 x 3 x 3 = 27 43 = 4 x 4 x 4 = 64 53 = 5 x 5 x 5 = 125 What would the next perfect cube be?

Examples: because

Examples: