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Here you will learn about reciprocals, including the definition of reciprocal and how to find reciprocals.

Students will first learn about reciprocals in 6th grade math as part of their work in number and operations with fractions when they learn how to divide fractions and extend their knowledge as they work through number systems in middle school.

The **reciprocal **of a number is one over that number. When the reciprocal is multiplied by the original number, the product is 1. Reciprocals are also called the multiplicative inverse.

For example,

Number | Reciprocal | Number \textbf{×} Reciprocal |
---|---|---|

4 = \cfrac{4}{1} | The numerator of the original number, 4, becomes the denominator. The denominator of the original number , 1, becomes the numerator. Reciprocal: \cfrac{1}{4} | \cfrac{4}{1}\times \cfrac{1}{4}=\cfrac{4}{4}=1 |

\cfrac{5}{6} | The numerator of the original number, 5, becomes the denominator. The denominator of the original number , 6, becomes the numerator. Reciprocal: \cfrac{6}{5} | \cfrac{5}{6} \times \cfrac{6}{5}=\cfrac{30}{30}=1 |

0.9 = \cfrac{9}{10} | The numerator of the original number, 9, becomes the denominator. The denominator of the original number , 10, becomes the numerator. Reciprocal: \cfrac{10}{9} or 1.\overline{1} | \cfrac{9}{10}\times \cfrac{10}{9}=\cfrac{90}{90}=1 |

1\cfrac{2}{3}=\cfrac{5}{3} | The numerator of the original number, 5, becomes the denominator. The denominator of the original number , 3, becomes the numerator. Reciprocal: \cfrac{3}{5} | \cfrac{5}{3}\times \cfrac{3}{5}=\cfrac{15}{15}=1 |

To help make sense of a reciprocal, let’s look at visual models.

Do you see a pattern?

2 → \cfrac{1}{2} 3 → \cfrac{1}{3}Not yet? Do you see a pattern now?

\cfrac{2}{1} → \cfrac{1}{2} \cfrac{3}{1} → \cfrac{1}{3}You can think of a reciprocal as writing the number as a fraction and then flipping it so that the numerator becomes the denominator and the denominator becomes the numerator. The numbers are turned upside down.

How does this apply to 6th grade math?

**Grade 6: Number System (6.NS.A.1)**Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, for example, by using visual fraction models and equations to represent the problem.

In order to write the reciprocal of a number:

**If the number is a whole number, mixed number, or decimal, write it as an improper fraction.****“Flip” the fraction by switching the numerator and denominator.****Check to make sure the product is**\bf{1}**.**

Assess math progress for the end of grade 4 and grade 5 or prepare for state assessments with these mixed topic, multiple choice questions and extended response questions!

DOWNLOAD FREEAssess math progress for the end of grade 4 and grade 5 or prepare for state assessments with these mixed topic, multiple choice questions and extended response questions!

DOWNLOAD FREEFind the reciprocal of 7.

**If the number is a whole number, mixed number, or decimal, write it as an improper fraction.**

2**“Flip” the fraction by switching the numerator and denominator.**

7 becomes the denominator and 1 becomes the numerator.

3**Check to make sure the product is ** \bf{1} **.**

The reciprocal of 7 is \, \cfrac{1}{7}.

Find the reciprocal of \, \cfrac{3}{5}.

**If the number is a whole number, mixed number, or decimal, write it as an improper fraction.**

\cfrac{3}{5} \, is already a fraction.

**“Flip” the fraction by switching the numerator and denominator.**

3 becomes the denominator and 5 becomes the numerator.

**Check to make sure the product is ** \bf{1} **.**

\cfrac{3}{5}\times \cfrac{5}{3}=\cfrac{15}{15}=1

The reciprocal of \, \cfrac{3}{5} \, is \, \cfrac{5}{3}.

Find the reciprocal of \, 1\cfrac{1}{4}.

**If the number is a whole number, mixed number, or decimal, write it as an improper fraction.**

1\cfrac{1}{4} \, as an improper fraction is \, \cfrac{5}{4}.

**“Flip” the fraction by switching the numerator and denominator.**

5 becomes the denominator and 4 becomes the numerator.

**Check to make sure the product is ** \bf{1} **.**

\cfrac{5}{4}\times \cfrac{4}{5}=\cfrac{20}{20}=1

The reciprocal of \, 1\cfrac{1}{4} \, which is \, \cfrac{5}{4} \, is \, \cfrac{4}{5}.

Find the reciprocal of 0.3.

**If the number is a whole number, mixed number, or decimal, write it as an improper fraction.**

0.3 is three tenths which is \, \cfrac{3}{10}.

**“Flip” the fraction by switching the numerator and denominator.**

3 becomes the denominator and 10 becomes the numerator.

**Check to make sure the product is ** \bf{1} **.**

\cfrac{3}{10}\times \cfrac{10}{3}=\cfrac{30}{30}=1

The reciprocal of 0.3 which is \, \cfrac{3}{10} \, is \, \cfrac{10}{3}.

\cfrac{10}{3}= 3.\overline{3}

Find the reciprocal of \, 1.7.

**If the number is a whole number, mixed number, or decimal, write it as an improper fraction.**

1.7 is one and seven tenths which is \, 1\cfrac{7}{10}.

1\cfrac{7}{10} \, as an improper fraction is \, \cfrac{17}{10}.

**“Flip” the fraction by switching the numerator and denominator.**

17 becomes the denominator and 10 becomes the numerator.

**Check to make sure the product is ** \bf{1} **.**

\cfrac{17}{10}\times \cfrac{10}{17}=\cfrac{170}{170}=1

The reciprocal of 1.7 \, \Bigl(\cfrac{17}{10}\Bigr) \, is \, \cfrac{10}{17}.

Find the reciprocal of -8.

**If the number is a whole number, mixed number, or decimal, write it as an improper fraction.**

-8 as an improper fraction is \, -\cfrac{8}{1}.

**“Flip” the fraction by switching the numerator and denominator.**

8 becomes the denominator and 1 becomes the numerator.

The reciprocal is a negative number like the original number.

**Check to make sure the product is ** \bf{1} **.**

-\cfrac{8}{1}\times -\cfrac{1}{8}=\cfrac{8}{8}=1

The reciprocal of -8 is \, -\cfrac{1}{8}.

- Reinforce that the reciprocal is also the multiplicative inverse.

- Have students discover reciprocal numbers on their own by strategically asking questions such as, “what number multiplied to 5 will give a product of 1 ?”

- When working with reciprocals, have students see that the reciprocal of a unit fraction is a whole number.

- Although worksheets serve a purpose in the classroom and can help with skill practice and test prep practice, having students discover the mathematical concepts is more meaningful for building long lasting understanding.

**Trying to find the reciprocal of zero**

Zero does not have a reciprocal. This is because we can think about 0 as being \frac{0}{1}. When the numerator and the denominator are flipped, it becomes \frac{1}{0} or 1 \div 0 which is undefined (it does not exist).

**Forgetting that a whole number can be written as an improper fraction**

Whole numbers can be written as improper fractions. For example, 10 can be written as \frac{10}{1}.

- Fractions operations
- Multiplying and dividing fractions
- Adding fractions
- Subtracting fractions
- Adding and subtracting fractions
- Multiplying fractions
- Dividing fractions
- Adding and subtracting fraction word problems
- Multiplicative inverse
- Multiplicative inverse & reciprocals
- Interpret a fraction as division

1. Find the reciprocal of 9.

1.9

-\cfrac{1}{9}

\cfrac{1}{9}

0.9

Write 9 as an improper fraction.

9=\cfrac{9}{1}.

Flip the numerator and the denominator.

9 becomes the denominator and 1 becomes the numerator.

The reciprocal of \, \cfrac{9}{1} \, is \, \cfrac{1}{9}.

2. Find the reciprocal of \, \cfrac{6}{11}.

6.11

\cfrac{11}{6}

11.6

-\cfrac{11}{6}

\cfrac{6}{11} \, is already a fraction.

Flip the numerator and the denominator.

6 becomes the denominator and 11 becomes the numerator.

The reciprocal of \, \cfrac{6}{11} \, is \, \cfrac{11}{6}.

3. Find the reciprocal of \, \cfrac{8}{5}.

1\cfrac{3}{5}

1.6

\cfrac{8}{13}

\cfrac{5}{8}

\cfrac{8}{5} \, is already in fraction form.

Flip the numerator and the denominator.

8 becomes the denominator and 5 becomes the numerator.

The reciprocal of \, \cfrac{8}{5} \, is \, \cfrac{5}{8}.

4. Find the reciprocal of \, 4\cfrac{1}{5}.

4.15

\cfrac{41}{5}

\cfrac{5}{21}

\cfrac{21}{5}

Change 4\cfrac{1}{5} \, to an improper fraction.

4\cfrac{1}{5} \, is \, \cfrac{21}{5}.

Flip the numerator and the denominator.

21 becomes the denominator and 5 becomes the numerator.

The reciprocal of \, 4\cfrac{1}{5} \; \Bigl(\cfrac{21}{5}\Bigr) is \, \cfrac{5}{21}.

5. Find the reciprocal of 0.11.

1.1

\cfrac{100}{11}

1\cfrac{1}{10}

\cfrac{11}{100}

Change 0.11 to a fraction.

0.11 = \cfrac{11}{100}

Flip the numerator and the denominator.

11 becomes the denominator and 100 becomes the numerator.

The reciprocal of \, 0.11=\cfrac{11}{100} \, is \, \cfrac{100}{11}.

6. Find the reciprocal of 3.1.

\cfrac{10}{31}

\cfrac{31}{10}

1.3

3\cfrac{1}{10}

Change 3.1 to an improper fraction.

3.1 = 3\cfrac{1}{10}= \cfrac{31}{10}

Flip the numerator and the denominator.

31 becomes the denominator and 10 becomes the numerator.

The reciprocal of \, 3.1 = \cfrac{31}{10} \, is \, \cfrac{10}{31}.

Rational numbers and irrational numbers have reciprocals. The only number that does not have a reciprocal is 0. We can use the formal definition of reciprocals to prove that all numbers have a reciprocal.

By definition, the reciprocal of a number is 1 divided by that number. The product of a number and its reciprocal is 1.

For example, the reciprocal of 5 can be written as 1 \div 5 or \frac{1}{5}.

5 \times \frac{1}{5} = 1.
\; \pi is an irrational number.

The reciprocal of \pi is 1 \div \pi or \frac{1}{\pi}.

\pi \times \frac{1}{\pi} = 1.

The reciprocal of 0 is 1 \div 0 or \frac{1}{0} which does not exist because you cannot divide by 0.

Yes, if you take a reciprocal of a reciprocal, you will wind up back at the original given number.

For example, the reciprocal of 5 (written as \frac{5}{1} ) is \frac{1}{5}.

If you then take the reciprocal of \frac{1}{5}, it will be \frac{5}{1} or 5. \; 5 is the original given number.

Finding the reciprocal of a number is flipping the numerator and the denominator so it can be seen as turning the fraction upside down.

The inverse of a number is the reciprocal.

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