Document1
Connections to Previous Learning:
Time Frame: Approximately 3-4 weeks
In Grades 4 and 5, students learned models for operations on multi-digit whole numbers to develop fluency with these operations. In grade 5, students used
these models to add, subtract, multiply and divide decimals within the hundredths place. Fifth graders also developed the understanding that fractions
represent division and learned models for operations on fractions. Students added, subtracted and multiplied fractions using number lines and area models.
They investigated and understood the structure of fraction multiplication (scaling) by considering how the resulting product increases when multiplied by a value
greater than 1, decreases when multiplied by a value less than 1, or remains the same as the value of the original number when it is multiplied by a value equal
to 1. They also learned to divide a whole number by a unit fraction and a unit fraction by a whole number. They solved real world problems involving all 4
operations in accordance with these expectations.
Focus within the Grade Level:
Sixth graders will extend their understanding of fraction operations to divide a fraction by a fraction. In addition, they use their understanding of place value
with whole numbers and decimals to fluently add, subtract, multiply and divide whole numbers and decimals. There is not one single standard recording
method; however, the standard algorithm is represented by any method that uses place value understanding and the distributive property to consistently result
in the correct solution. Some well-known methods include partial sums, differences, products and quotients, column (traditional) methods for each operation.
Students develop algorithms that are efficient, allow for fluent calculation and demonstrate understanding of both the value and the operations. If students can
justify their reasoning for an algorithm, it may be categorized as a “standard algorithm.”
Connections to Subsequent Learning:
Students can add, subtract, multiply and divide using decimals and fractions to solve complex application problems. In seventh grade students will use percents
and scale factors to determine percent of increase or percent of decrease, discounts and markups. They will also use the understanding of area and surface to
solve problems involving area, volume and surface area of 2-D and 3-D objects in 7th grade.
2/8/2016 2:02:37 PM
Major Standards
Adapted from UbD framework
Supporting Standards
Page 1
Additional Standards
Document1
Desired Outcomes
Standard(s):
Apply and extend previous understandings of multiplication and division to divide fractions by fractions
 6.NS.1 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 length3/4 mi and area 1/2 square mi?
Compare fluently with multi-digit numbers and find common factors and multiples
 6.NS.2 Fluently divide multi-digit numbers using the standard algorithm.
 6.NS.3 Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.
Transfer: Students will apply…
Fraction and decimal concepts and procedures to interpret, solve, and create real-world problem scenarios that involve operations with fractions and/or
decimals.
Ex. Create a story context for 3 ¸ 2 and use a visual fraction model to show the quotient.
8 9
Understandings: Students will understand that …





The two types of division – quotative (partitive) and measurement are applied to fractions and decimals as well as to whole numbers.
Multiplication and division are inverse operations for whole numbers, fractions and decimals.
The relationship of the location of the digits and the value of the digits is part of understanding multi-digit operations.
Division of fractions by fractions can be represented using multiple formats (manipulatives, diagrams, real-life situations, equations).
Operations on decimals and whole numbers are based upon place value relationships.
Essential Questions:



How is division related to realistic situations and to other operations?
What role does place value play in multi-digit operations?
How can division be represented and interpreted?
2/8/2016 2:02:37 PM
Major Standards
Adapted from UbD framework
Supporting Standards
Page 2
Additional Standards
Document1
Mathematical Practices: (Practices to be explicitly emphasized are indicated with an *.)
* 1. Make sense of problems and persevere in solving them. Students make sense of real-world fraction and decimal problem situations by representing
the context in tactile and/or virtual manipulatives, visual, or algebraic models.
* 2. Reason abstractly and quantitatively. Students will reason about the value of numbers as they perform operations. Students use their understanding
3.
* 4.
* 5.
6.
* 7.
8.
of multiplication of fractions as scaling to reason about the effects of multiplying or dividing fractions and decimals and the values of the resulting
products or quotients.
Construct viable arguments and critique the reasoning of others. Students construct and critique arguments regarding the portion of a whole as
represented in the context of real-world situations.
Model with mathematics. Students will model real-world situations to show multiplication and division of fractions and decimals.
Use appropriate tools strategically. Students will use visual or concrete tools for division of fractions with understanding. (Such as fraction square or
circle pieces, fraction equivalence towers, bar models, and number line diagrams.
Attend to precision. Students attend to the language of problems to determine appropriate representations and operations for solving real-world
problems. In addition, students attend to the units of measure used in real-world problems.
Look for and make use of structure. Students examine the relationship of rational numbers to the number line and the place value structure as related
to multi-digit operations. They also use their knowledge of problem solving structures to make sense of word problems.
Look for and express regularity in repeated reasoning. Students demonstrate repeated reasoning when dividing fractions by fractions by fractions and
see the inverse relationship to multiplication.
Prerequisite Skills/Concepts:
Students should already be able to:





Add, subtract and multiply fractions.
Divide fractions by whole numbers and whole numbers by
fractions.
Use area models for fraction or decimal computation situations.
Fluently add, subtract, multiply and divide whole numbers.
Use concepts of area, perimeter and volume to solve problems
with whole numbers.
2/8/2016 2:02:37 PM
Major Standards
Advanced Skills/Concepts:
Some students may be ready to:

Represent and solve multi-step problems involving positive and negative rational
numbers with tape diagrams, double number lines, equations and expressions.
Adapted from UbD framework
Supporting Standards
Page 3
Additional Standards
Document1
Knowledge:
Students will know…
Skills:
Students will be able to…



Standard algorithms for addition, subtraction, multiplication
and division of multi-digit decimals


Compute quotients of fractions divided by fractions. (6.NS.1)
Explain the meaning of a quotient determined by division of fractions, using visual
fraction models, equations, real-life situations, and language. (6.NS.1)
Divide multi-digit numbers fluently using the standard algorithm. (6.NS.2)
Fluently add, subtract, multiply and divide decimals to solve problems. (6.NS.3)
WIDA Standard: (English Language Learners)
English language learners communicate information, ideas and concepts necessary for academic success in the content area of Mathematics.
English language learners benefit from:
 the opportunity to use visual and concrete models in order to understand and apply fraction and decimal concepts and language.
 explicit vocabulary instruction regarding fractions and decimals.
Academic Vocabulary:
Critical Terms:
Reciprocal
Inverse operation
Pre-assessments
6.NS.1 Pre-assessment
2/8/2016 2:02:37 PM
Major Standards
Supplemental Terms:
Compose
Decompose
Quotient
Dividend
Assessment
Formative Assessments
Summative Assessment
Division and Fractions Assessment
Matching Representations
Division Models ppt
Forming a Division Algorithm
Fraction Division Exit Slip
Reasoning with Reciprocals Practice
Context Stories
Task Cards
6.NS.1 Summative Assessment
Unit 1 Fractions and Decimals
Assessment
Reflections Letter
Playground assignment
Adapted from UbD framework
Supporting Standards
Divisor
Remainder
Self-Assessment
Gallery Walk
Skeleton 6.NS.1 Assessment
Skeleton Unit 1 Fractions and
Decimals Assessment
Page 4
Additional Standards
Document1
Sample Lesson Sequence
1. 6.NS.2 & 6.NS.3 – Division of multi-digit whole numbers, and all operations with decimals
2. 6.NS.1 – Interpreting and Computing Quotients of Fractions (model lesson)
3. 6.NS.1, 6.NS.2 & 6.NS.3 – Problem Solving
2/8/2016 2:02:37 PM
Major Standards
Adapted from UbD framework
Supporting Standards
Page 5
Additional Standards