6.NS.1 - About the Math, Learning Targets, and Rigor

Grade 4 G/T The Number System

6.NS.1

About the Math

Full Standard

Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi?

 

Learning Targets (I can)

  • Explain the meaning of the quotient relative to a division of fractions expression (i.e. explain that 3/4 ÷ 1/2 asks how many halves are in three-fourths; there is one half and a half of a half in three-fourths).
  • Explain division of a fraction by a fraction using representations.
  • Estimate the quotient of an expression that divides a fraction by a fraction (i.e. 3/4 ÷ 1/2 will be more than 1 because 1/2 is less than 3/4 therefore there will be 1 whole half and some more in 3/4. Conversely, 1/5 ÷ 1/2 will yield a quotient less than 1 because 1/2 is greater than 1/5 so there cannot be one whole half in 1/5 of something.)
  • Connect a representation of division of a fraction by a fraction to an equation.
  • Divide a fraction by a fraction.
  • Solve word problems that divide a fraction by a fraction.

 

About the Math

Multiplication and division have an inverse relationship. So we say 10 ÷ 2 = 5 is the same as 2 x 5 = 10. Prior to developing the procedure for division of fractions, students need to experience visual representations of the meaning of division of fractions. Eventually, we get to the procedure for division of fractions which is to multiply by the reciprocal of the divisor. In other words, 10 ÷ 2 is the same as 10 x 1/2. They both result in the answer of 5. So just as 10 ÷ 2 means how many 2's are in 10, a division problem like 7/8 ÷ 1/8 is asking how many one-eighths are in 7/8? There are 7 one-eighths in 7/8. When dividing fractions, you need to think about what the problem is asking. When a greater fraction is divided by a smaller fraction, the quotient is greater than one. For example, 2/3 ÷ 1/3 = 2 because there are two one-thirds in two thirds. When a smaller fraction is divided by a greater fraction, the quotient is less than one. For example 1/3 ÷ 2/3 is 1/2 because there is only part of two-thirds in one-third. Students need to experiment with lots of examples to find this generalization. Essential vocabulary for this standard includes: divisor, dividend, inverse operation, quotient, and reciprocal. Visit the online dictionary Links to an external site. or visual math dictionary Links to an external site. for vocabulary support.

 

Progression of Standard within Grade 4 G/T

This progression informs how to develop the standard within the grade level. This progression is provided by HCPSS Elementary Mathematics.

Progression Throughout Year
Quarter 1 Quarter 2 Quarter 3 Quarter 4
  • Explain the meaning of the quotient relative to a division of fractions expression (i.e. explain that 3/4 ÷ 1/2 asks how many halves are in three-fourths; there is one half and a half of a half in three-fourths).
  • Explain division of a fraction by a fraction using representations.
  • Estimate the quotient of an expression that divides a fraction by a fraction (i.e. 3/4 ÷ 1/2 will be more than 1 because 1/2 is less than 3/4 therefore there will be 1 whole half and some more in 3/4. Conversely, 1/5 ÷ 1/2 will yield a quotient less than 1 because 1/2 is greater than 1/5 so there cannot be one whole half in 1/5 of something.)
  • Connect a representation of division of a fraction by a fraction to an equation.
  • Solve word problems that divide a fraction by a fraction.

 

 

 

Progression of this Standard Across Grades

This progression is informed by the Achieve the Core Coherence Map Links to an external site.. Information is not the complete standard.

Progression Across Grades
Grade 5 Grade 7
  • Divide unit fractions by whole numbers and whole numbers by unit fractions (5.NF.7 )
  • Multiply and divide rational numbers (7.NS.2)
  • Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities (7.SP.3)

 

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Instructional Tasks

TASKS

These tasks can be used with small group or whole group instruction.

  • You divide two fractions, and the numerator of the quotient is a 4. What could the fractions be?
  • Compare and contrast the following, using visual models/drawings: ½ ÷ ¾ and ¾ ÷ ½.
  • Larry ordered 5 ½ subs. He wants to share the subs with 4 friends and himself. Larry thinks they will each receive 1 ¼ subs. Is Larry correct? Why or why not?
  • Can you use the fractions (2/3), (8/9), and (3/4) to create a true equation? (one answer: 8/9 x ¾ = ⅔ )
  • Alana wants to run 1 ⅞ miles and the track is ¼ mile around. How many laps will she need to run to make her goal?

 

SLIDE-BASED TASKS 

 These links are HCPSS created instructional tasks. These tasks are provided in Google slides. These tasks should be used for inspiration and resources, but instruction should start with students having the opportunity to engage with the math first (often involving physical and/or visual models) followed by discussion and explicit instruction to ensure student understanding.

Module 5 • Dividing Fractions

Solve Problems by Dividing Fractions by Fractions:

 

 

Additional Tasks 

These links provide instructional ideas connected to this standard. 

 

 

Tasks From Print Resources

These publications have been provided for each school. They are typically stored in team closets or the media center. Check with your team leader if you cannot find them. 

Print Resources
Book Thumbnail Book Title Grade Pages
Teaching Student-Centered Mathematics 6-8 How Much For One?, Activity 8.22, Page 137
Expanded Lesson:  Division-of-Fractions Stories, Pages 141- 142

Nimble With Numbers

 

 5-6 85-87 (Fraction Arrangements)
90 (Digits to Fractions II)

Nimble with Numbers

 6-7 56-58 (Fraction Formulation)
59-60 (Seeking Fractions)
62 (Finding Fractions Practice)
64 (Creating and Computing Fractions)

Hands-On Standards Common Core

 6 22 (Lesson 1)

 

 

 

 

 

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Independent Work

Centers

These print resources can be used during independent or center time. These resources could also be used as lesson seeds.

 

 

 

INDEPENDENT PRACTICE/HOMEWORK/ASSESSMENT

 These resource sheets can be used for independent practice, homework, or assessment. They are intended to reinforce procedures and concepts. They should not be used as a source of direct instruction or whole-group practice.

 

 

 

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Assessment

Learning Targets

rubric

  • Explain the meaning of the quotient relative to a division of fractions expression (i.e. explain that 3/4 ÷ 1/2 asks how many halves are in three-fourths; there is one half and a half of a half in three-fourths).
  • Explain division of a fraction by a fraction using representations.
  • Estimate the quotient of an expression that divides a fraction by a fraction (i.e. 3/4 ÷ 1/2 will be more than 1 because 1/2 is less than 3/4 therefore there will be 1 whole half and some more in 3/4. Conversely, 1/5 ÷ 1/2 will yield a quotient less than 1 because 1/2 is greater than 1/5 so there cannot be one whole half in 1/5 of something.)
  • Connect a representation of division of a fraction by a fraction to an equation.
  • Solve word problems that divide a fraction by a fraction.

 

Learning targets identify what students should be able to do. This rubric can be applied to tasks and observations for assessment and/or grading. 

 

 

 

 

 

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