Whole Number Division

Whole Number Division

Compare Mia’s Methods

Opening

Compare Mia’s Methods

Emma runs on a 14-mile track to train for a marathon. Yesterday Emma ran a total of 3 miles. How many times did she run around the track?

Mia decides to solve this problem using a model similar to the one she used to find 89÷4 in the previous lesson.

She reasons:

“I need to find the number of 14 miles in 3 miles. So, I need to find 3 ÷ 14.

“To make a model of this situation, I can draw a bar for each whole mile and then divide each bar into fourths.

“There are 12 fourths in all, so 3 ÷ 14 = 12. Emma ran around the track 12 times.”

  • Discuss Mia’s solution. How is her approach similar to the one she used to find 89÷4? How is it different?

Divide 3 Wholes

Math Mission

Opening

Explore strategies for dividing a whole number by a fraction.

Explore Dividing Whole Numbers by Fractions

Work Time

Explore Dividing Whole Numbers by Fractions

Find 3 ÷ 34.

  • Use Mia’s method of using a diagram or a number line.
  • Check your answer using multiplication.
  • How does your answer compare to Mia’s answer of 12 for 3 ÷ 14?
  • Given Mia’s answer for 3 ÷ 14, why does your answer make sense?
  • In your model, draw three rectangles to represent the whole number 3. How many equal parts should you divide each rectangle into?

Diagrams for Division

Hint:

  • In your model, draw three rectangles to represent the whole number 3. How many equal parts should you divide each rectangle into?
  • How can you use your model to show how many three-fourths ( 34 ) are in 3?

Jan’s Method

Work Time

Jan’s Method

Jan wants to find 2 ÷ 35.

She makes the model pictured below. She reasons:

“To find 2 ÷ 35, I made a model with 2 wholes and divided each whole into fifths. Then I shaded groups of 3 fifths. There are 3 groups of 3 fifths and 13 of a group of 3 fifths left over.

So, 2 ÷ 35 = 313.”

  • Discuss Jan’s method with your partner. Then try to explain it in your own words.
  • Carlos thinks the answer is 315 because 15 of a whole is left over. Explain why his reasoning is incorrect.

Ask yourself:

  • What value does each equal group in the model represent?
  • What fraction of an equal group is left over?

Carlos’s Method

Work Time

Carlos’s Method

Carlos wants to find 3 ÷ 23. He decides to use the same method he used to find 89 ÷ 4 in the previous lesson: multiply by the reciprocal.

At first, Carlos isn’t sure what to do with 23. What is the reciprocal of 23? He thinks back to a rule he learned: the multiplicative inverse, or reciprocal, of ab is ba. For example, the reciprocal of 4 is 14; the reciprocal of 15 is 5. So, Carlos determines that the reciprocal of 23 is 32. He then solves the problem as follows: 3 ÷ 23 = 3 • 32 = 92.

Thus, 3 ÷ 23 = 92.

  • Use Carlos’s method to find 6 ÷ 25.
  • Describe and explain each step of your solution.

Ask yourself:

Will the answer be greater than or less than 3? How can you use the reciprocal of 25 to solve the problem?

Prepare a Presentation

Work Time

Prepare a Presentation

  • Explain how you solved each problem and why your answer makes sense.
  • Explain Jan’s and Carlos’s methods.

Challenge Problem

Denzel uses the following shortcut to divide a whole number by a fraction:

“To divide a whole number by a fraction, I multiply the whole number by the denominator of the fraction, and then divide this product by the numerator of the fraction.”

  • Show that Denzel’s method works for the problems from today’s lesson: 3÷14, 2÷35, and 3÷23.
  • Why does Denzel’s shortcut work? Explain the mathematics of his method.

Make Connections

Performance Task

Ways of Thinking: Make Connections

Take notes to clarify your understanding of how to divide a whole number by a fraction.

As your classmates present, ask questions such as:

  • How did you use Mia’s solution to 3 ÷ 1 4 to show that your solution to 3 ÷ 3 4 makes sense?
  • How is the whole number (the dividend) represented in Jan’s model?
  • How is the fraction (the divisor) represented in Jan’s model?
  • What does “amount left over” in Jan’s problem mean?
  • Why does Carlos’s method work?

Divide Whole Numbers by Fractions

Work Time

Apply the Learning: Divide Whole Numbers by Fractions

Solve the problems using the method of your choice.

  1.  5 ÷ 56
  2.  6 ÷ 37 
  3.  8 ÷ 58 
  4. ​​​​​​​ 5 ÷ 158 

Divide a Whole Number by a Fraction

Formative Assessment

Summary of the Math: Divide a Whole Number by a Fraction

Write a summary about how to divide a whole number by a fraction.

Check your summary.

  • Do you describe at least one way that you can use a model to divide a whole number by a fraction?
  • Do you explain how to use the inverse relationship between multiplication and division to divide a whole number by a fraction?

Reflect On Your Work

Work Time

Reflection

Write a reflection about the ideas discussed in class today. Use the sentence starter below if you find it to be helpful.

A strategy that I found useful for dividing a whole number by a fraction is ….