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Dividing Fractions CCSS.6.NS.1: Apply and extend previous understandings multiplication and division to divide fractions by fractions. You need your Journal, glue, ruler, highlighter and a pencil. Bar Models and Number Lines Using the problems for 12 ÷ 3 = 4 to create bar models in your Journal. Create a number line for 10 ÷ 5 = 2. Check/correct your work with the examples on the board. PART 1 Apply what you learned about dividing whole numbers to dividing whole numbers by fractions. 1. 2. Use a bar diagram to help you understand what it means to divide. This board is 1 yard (3 feet) long. Sam wants to divide it equally into 1 foot pieces. What do you know? What do you need to find? Draw a model that represents the length of the board. Draw lines to separate the board into thirds. Each third represent one foot. How many groups of 1 foot are in 3 feet? Juan is building a set for the school musical. He has a 3-foot board that he needs to equally divide into ½ foot pieces. How many pieces will he have after he cuts the board? What do you know? What do you need to find? 1. Draw a model that represents the length of the board. Draw lines to separate the board into thirds. Each third represent one foot. 2. Divide each foot into halves. 3. Determine how many groups of ½ are in 3. Circle the groups that are the size of the divisor ½. ½ ½ ½ ½ ½ ½ Juan is building a set for the school musical. He has a 3-foot board that he needs to equally divide into ½ foot pieces. How many pieces will he have after he cuts the board? 3. Determine how many groups of ½ are in 3. Circle the groups that are the size of the divisor ½. ½ ½ ½ ½ ½ ½ Conclusion: There are 6 groups of ½ . So, 3 ÷ ½ = 6. (And 3 x 2 = 6.) Check by multiplying: 6 x ½ = 3. 2 3 Find 4 ÷ . 1. Draw a model to represent 4. 1 1 1 1 2. Divide each whole into thirds. 3. Circle groups of on the model. Think: How many 2 3 2 3 groups of are in 4? 2 3 Find 4 ÷ . 3. 2 3 Circle groups of on the model. Think: How many 2 3 groups of are in 4? 4. 5. 2 3 2 3 There are 6 groups of . So, 4 ÷ = 6. Check by multiplying: 6 x 2 3 = 12 3 = 4. 1 You try it. Find 3 ÷ . 3 1. Draw a model to represent 3. 2. Divide each whole into _____. 3. Determine how many groups of _____ are in _____. Circle groups of _____ on the model. 4. There are _____ groups of ___. So, 1 3 ÷ = _____. 3 Work with a partner in your group. Draw a bar model or number line to find each quotient. 1. 2÷ 2. 6÷ 3. 4÷ 4. 3÷ 1 4 2 3 1 2 3 4 Check your answers 1 4 1. 2÷ =8 2. 6÷ =9 3. 1 2 4÷ =8 4. 3÷ =4 2 3 3 4 How can a bar diagram or a number line help you understand what it means to divide fractions? Discuss this idea in your groups. The diagrams show the relationship between the factors and the quotient. The model shows that a quotient can be greater than the dividend when the divisor is less than 1. HOMEWORK Use the same process for the bar diagram to create number lines for the following problems. 1 1. 2 ÷ = 3 1 2. 3÷ = 3 2 3. 4 ÷ = 3 Part 2 Developing the Algorithm. Vocabulary Reciprocal: any two numbers with a product of 1. Number 1 2 2 3 Product 1 2 x2=1 2 3 x2=1 3 Reciprocal 2 3 2 Describe the relationship between the numerator and denominator of a number and its reciprocal. Connect to the vocabulary. Another name for reciprocal is multiplicative inverse. What are some words in everyday language that are similar to reciprocal or inverse? Pilots can fly in an inverted position, or upside down. How can you use the everyday meaning of invert to help you remember the mathematical meaning of multiplicative inverse, or reciprocal? Find Reciprocals 1 2 Dividing 3 by gives the same result as multiplying 3 by 2, which is the reciprocal 1 of . Any two numbers with a product of 2 1 are called reciprocals. reciprocals 1 2 3÷ =6 3x2=6 same result Practice finding the reciprocal. 2 3 Find the reciprocal of . 2 3 3 2 2 3 3 2 Since x = 1, the reciprocal of is . Find the reciprocal of each number. 1. 2. 3. 1 8 3 5 2 9 4. 7 5. 1 1 2 Divide by a fraction: Words: To divide a whole number or a fraction by a fraction, multiply by its reciprocal (or multiplicative inverse) of the divisor. EX. 2 3 5 1 3 2 5÷ = x = 15 2 =7 1 2 Five divided by two thirds means you need to find how many two thirds are in 5. (part of a whole) 3 8 2 3 3 8 3 2 9 16 ÷ = x = Three eighths divided by two thirds means you need to find how many two thirds are in three eighths. (part of a part) Practice – find the quotients. 1. 5÷ 2. 4÷ 3 6 7 9 3. 10 ÷ 4. 12 ÷ 1 5. 8 3 6. 4 1 7. 3 ÷ ÷ 1 2 2 3 ÷8 5 6 3 8 2 8. 4 ÷1 3 3 2 9. 2 ÷ 4 3 Practice Complete the practice worksheet applying what you have learned about finding quotients to the contextual problems.