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Lesson Plans and Worksheets for Grade 6

Lesson Plans and Worksheets for all Grades

More Lessons for Grade 6

Common Core For Grade 6

Videos and solutions to help Grade 6 students learn what are exponents and how to use exponents.

New York State Common Core Math Grade 6, Module 4, Lesson 5

Lesson 5 Student Outcomes

Students discover that 3x = x + x + x is not the same thing as x^{3} which is x • x • x.

Students understand that a base number can be represented with a positive whole number, positive fraction, or positive decimal and that for any number b, we define b^{m} to be m factors of b where b is the base and m is called the exponent or power of b.

Opening Exercise

As you evaluate these expressions, pay attention to how you arrive at your answers.

4 + 4 + 4 + 4 + 4 + 4 + 4 +4 + 4 + 4

9 + 9 + 9 + 9 + 9

10 + 10 + 10 + 10 + 10

Multiplication is a faster way to add numbers when the addends are the same.

When we add five groups of 10, we use an abbreviation and a different notation, called multiplication.

10 + 10 + 10 + 10 + 10 = 10 × 5 = 50.

If multiplication is a more efficient way to represent addition problems involving the repeated addition of the same addend, do you think there might be a more efficient way to represent the repeated multiplication of the same factor, as in 10 x 10 x 10 x 10 x 10.

When we add 5 groups of 10, we write 5 × 10, but when we multiply 5 copies of 10, we write 10^{5}. So, multiplication by 5 in the context of addition corresponds exactly to the exponent in the context of multiplication.

The repeated factor is called the **base** and the exponent is also called the **power**.

There is a special name for numbers raised to the second power. When a number is raised to the second power, it is called **squared**.

There is also a special name for numbers raised to the third power. When a number is raised to the third power, it is called **cubed**.

Examples

Write each expression in exponential form.

- 5 × 5 × 5 × 5 × 5
- 2 × 2 × 2 × 2

Write each expression in expanded form. - 8
^{3} - 10
^{6} - g
^{3}

What is the difference between 3g and g^{3}?

The base number can be written in decimal or fraction form.

(3.8)^{4}

(2/3)^{2}

Exercises

- Fill in the missing expressions for each row. For whole number and decimal bases, use a calculator to find the standard form of the number. For fraction bases, leave your answer as a fraction.
- Write “five cubed” in all three forms: exponential form, written as a series of products, standard form.
- Write “fourteen and seven tenths squared” in all three forms.
- One student thought two to the third power was equal to six. What mistake do you think they made and how would you help them fix their mistake?

Lesson Summary

**Exponential Notation for Whole Number Exponents**: Let be a non-zero whole number. For any number b, the expression b^{m} is the product of m factors of b.

The number b is called the base, and m is called the exponent or power of b. When m is 1, “the product of one factor of ” just means b, i.e. b^{1} = b. Raising any non-zero number to the power of 0 is defined to be 1, i.e., b^{0} = 1 for all b ≠ 0.

Lesson 5 Examples and Exercises

Examples 1 - 5

Go back to Examples 1ִ and use a calculator to evaluate the expressions.

Example 6

3.8^{4}

Example 7

2.1 × 2.1

Example 8

0.75 × 0.75 × 0.75

The base number can also be a fraction. Convert the decimals to fractions in Examples 7 and 8 and evaluate. Leave your answer as a fraction.

Example 9

1/2 × 1/2 × 1/2

Example 10

(2/3)^{2}

Exercises 1 - 4

Lesson 5 Exit Ticket

- What is the difference between 6z and z
^{6}? - Write 10
^{3}as a multiplication expression having repeated factors. - Write 8 × 8 × 8 × 8 using exponents.

**Problem Set**

- Complete the table by filling in the blank cells. Use a calculator when needed. Exponential Form, Expanded Form, Standard Form
- Why do whole numbers raised to an exponent get greater, while fractions raised to an exponent get smaller? 3. The powers of 2 that are in the range 2 through 1,000 are 2, 4, 8, 16, 32, 64, 128, 256, and 512. Find all the powers of 3 that are in the range 3 through 1,000.
- Find all the powers of 4 in the range 4 through 1,000.
- Write an equivalent expression for n × a using only addition.
- Write an equivalent expression for w
^{b}using only multiplication.

a. Explain what w is in this new expression.

b. Explain what b is in this new expression. - What is the advantage of using exponential notation?
- What is the difference between 4x and x
^{4}? Evaluate both of these expressions when x = 2.

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