Grade 6 Mathematics, Quarter 2, Unit 2.1
Foundations of Fractions (LCM, GCF, Mixed
Numbers, Improper Fractions)
Overview
Number of instructional days:
12
(1 day = 45–60 minutes)
Content to be learned
Mathematical practices to be integrated
•
Find the greatest common factor (GCF) of two or
more numbers.
Construct viable arguments and critique the
reasoning of others.
•
Find the least common multiple (LCM) of two or
more numbers.
•
Solve real-world problems involving GCF and
LCM.
•
Convert between and solve problems involving
proper fractions, improper fractions, and mixed
numbers.
•
Order and compare fractions (proper, improper,
mixed).
•
Justify conclusions (by showing
thinking/processes used to find LCM, GCF,
and comparing/ordering fractions).
•
Analyze situations (to determine when to use
GCF vs. LCM).
Attend to precision.
•
Use clear definitions in discussion (by using
GCF and LCM in discussion with others and in
one’s own reasoning).
•
Calculate accurately and efficiently (the GCF
and LCM in real-world problems).
Make sense of problems and persevere in solving
them.
•
Check solutions with another method (by using
multiple strategies/representations for GCF,
LCM, and comparing/ordering fractions).
•
Ask, “Does my solution make sense?” (when
comparing and ordering fractions or when
finding the GCF/LCM of two or more
numbers.
Reason abstractly and quantitatively.
•
Make sense of quantities and their
relationships in problem situations.
Cumberland, Lincoln, and Woonsocket Public Schools
in collaboration with the Charles A. Dana Center at the University of Texas at Austin
C-21
Grade 6 Mathematics, Quarter 2, Unit 2.1
Final, July 2011
Foundations of Fractions (LCM, GCF,
Mixed Numbers, Improper Fractions) (8 days)
Essential questions
•
How is a factor different from a multiple?
•
•
How do you find the common multiples and the
least common multiple of two numbers?
How can you decide if finding common multiples
or common factors is helpful in solving
problems?
•
What are the differences between a proper and
improper fraction?
•
How do you find the common factors and
greatest common factor of two numbers?
•
What is the relationship between improper
fractions and mixed numbers?
•
What strategies can you use to order and compare
fractions (proper, improper, mixed)?
•
Given two unequal fractions with different
denominators, which fraction is greater? How do
you know?
Written Curriculum
Grade-Level Expectations
M(N&O)–6–4 Accurately solves problems involving single or multiple operations on fractions (proper,
improper, and mixed), or decimals; and addition or subtraction of integers; percent of a whole; or
problems involving greatest common factor or least common multiple. (State)
(IMPORTANT: Applies the conventions of order of operations with and without parentheses.)
M(N&O)–6–2 Demonstrates understanding of the relative magnitude of numbers by ordering or
comparing numbers with whole number bases and whole number exponents (e.g.,33, 43), integers, or
rational numbers within and across number formats (fractions, decimals, or whole number percents from
1–100) using number lines or equality and inequality symbols. (State)
Clarifying the Standards
Prior Learning
In grades 3–4, students compared and ordered fractions using models, number lines, and explanations.
Fourth-grade students explored the properties of factors and multiples. Then in grade 5, students further
investigated multiples and factors, including prime and composite numbers.
Current Learning
In grade 6, students convert between and solve problems involving proper fractions, improper fractions,
and mixed numbers. These processes are introduced, reinforced, and mastered at this level. Students also
use their prior knowledge of factors and multiples to find and solve problems involving greatest common
factor and least common multiple. Students also compare and order fractions (proper, improper, and
mixed).
Future Learning
In grades 7–8, students will solve problems involving proportional reasoning. In grades 9–10, students
will solve problems involving ratios, rates, and proportional relationships.
C-22
Cumberland, Lincoln, and Woonsocket Public Schools
in collaboration with the Charles A. Dana Center at the University of Texas at Austin
Grade 6 Mathematics, Quarter 2, Unit 2.2
Adding and Subtracting Fractions
Overview
Number of instructional days:
9
(1 day = 45–60 minutes)
Content to be learned
Mathematical practices to be integrated
•
Demonstrate a conceptual understanding of
addition and subtraction of positive fractions
(proper, improper, mixed).
Construct viable arguments and critique the
reasoning of others.
•
Solve problems involving single or multiple
operations on fractions (proper, improper, and
mixed) including order of operations and apply
it in real-world situations.
•
Make conjectures (by estimating sums and
differences).
•
Justify conclusions (by showing
thinking/processes used to solve problems).
•
Clarify and improve arguments (by using
different representations to clarify a solution).
Attend to precision.
•
Calculate accurately and efficiently (the sums
and differences of problems in real-world
situations and problems involving fractions).
•
Use clear definitions in discussion (to use math
vocabulary in discussion and in one’s own
reasoning).
•
Communicate precisely.
Model with mathematics.
•
Apply math to solve problems involving
addition and subtraction of fractions in
everyday life, society, and the workplace.
•
How are addition and subtraction of fractions
related?
•
In what real-world situation do you need to
apply addition and subtraction of fractions?
Explain.
•
When solving a real-world problem, how do
you know when to add or subtract to arrive at a
correct solution?
Essential questions
•
How do you add and subtract fractions with
like and unlike denominators?
•
How could you explain to a peer that
11 7
−
=
12 12
4
is not correct?
0
•
How do you add and subtract mixed numbers
with unlike denominators?
Cumberland, Lincoln, and Woonsocket Public Schools
in collaboration with the Charles A. Dana Center at the University of Texas at Austin
C-23
Grade 6 Mathematics, Quarter 2, Unit 2.2
Final, July 2011
Adding and Subtracting Fractions (9 days)
Written Curriculum
Grade-Level Expectations
M(N&O)–6–3 Demonstrates conceptual understanding of mathematical operations by adding and
subtracting positive fractions and integers; and multiplying and dividing fractions and decimals. (Local)
M(N&O)–6–4 Accurately solves problems involving single or multiple operations on fractions (proper,
improper, and mixed), or decimals; and addition or subtraction of integers; percent of a whole; or
problems involving greatest common factor or least common multiple. (State)
(IMPORTANT: Applies the conventions of order of operations with and without parentheses.)
Clarifying the Standards
Prior Learning
In grades K–2, students developed a conceptual understanding of addition and subtraction of whole
numbers through solving problems a variety of ways, including through part-part-whole relationships.
They also began formal addition and subtraction. In grade 3, students looked at the inverse relationship
between addition and subtraction. In grades 4 and 5, students learned how to accurately solve problems
involving the addition and subtraction of positive fractions with like and unlike denominators using
number lines, models, and explanations.
In grade 4, students began solving problems involving the addition and subtraction of positive proper
fractions with like denominators. In grade 5, students solved problems involving the addition and
subtraction of positive proper fractions with unlike denominators.
Current Learning
In grade 6, students continue to develop their conceptual understanding of the addition and subtraction of
positive fractions, including mixed numbers and improper fractions. Students also accurately solve
problems involving single or multiple operations on fractions (proper, improper, and mixed). Problem
types must include adding mixed numbers where an improper fractions results as part of the answer,
and—when subtracting, requires regrouping. These concepts should be mastered by the end of this unit.
Students continue to use their estimation and mental-computation skills to verify the accuracy of their
results. The order of operations, with and without parentheses, will be reinforced throughout the unit as
students solve problems.
Future Learning
In grades 7–12, the addition and subtraction of positive fractions will be expanded to the addition and
subtraction of negative fractions and will show up in probability (e.g., determining probabilities), algebra
(e.g., solving equations), and geometry (e.g., finding perimeter). In grades 9–10, students will solve
problems that involve proportional relationships, percents, ratios, and rates. In grades 11–12, students will
solve problems involving scientific notation and will interpret rational exponents and their relationship to
radicals.
C-24
Cumberland, Lincoln, and Woonsocket Public Schools
in collaboration with the Charles A. Dana Center at the University of Texas at Austin
Adding and Subtracting Fractions (9 days)
Grade 6 Mathematics, Quarter 2, Unit 2.2
Final, July 2011
Additional Research Findings
According to Principles and Standards for School Mathematics, students in the middle grades should
continue to refine their understanding of addition, subtraction, multiplication, and division as they use
these operations with fractions, decimals, percents, and integers. Students in grades 6–8 will continue to
build their understanding of the inverse relationship between addition and subtraction (as well as the
inverse relationship between multiplication and division) (p. 218).
The book also states, “teachers can help students add and subtract fractions correctly by helping them
develop meaning for numerator, denominator, and equivalence and by encouraging them to use
benchmarks and estimation. Students who have a solid conceptual foundation in fractions should be less
prone to committing computational errors than students who do not have such a foundation” (Principles,
p. 218).
Notes About Resources and Materials
Cumberland, Lincoln, and Woonsocket Public Schools
in collaboration with the Charles A. Dana Center at the University of Texas at Austin
C-25
Grade 6 Mathematics, Quarter 2, Unit 2.2
Final, July 2011
C-26
Adding and Subtracting Fractions (9 days)
Cumberland, Lincoln, and Woonsocket Public Schools
in collaboration with the Charles A. Dana Center at the University of Texas at Austin
Grade 6 Mathematics, Quarter 2, Unit 2.3
Multiplying and Dividing Fractions
Overview
Number of instructional days:
7
(1 day = 45–60 minutes)
Content to be learned
Mathematical practices to be integrated
•
Demonstrate a conceptual understanding of
multiplication and division of positive
fractions.
Construct viable arguments and critique the
reasoning of others.
•
Solve and apply problems involving single or
multiple operations on fractions (proper,
improper, and mixed) including order of
operations.
•
Make conjectures (by estimating products and
quotients).
•
Justify conclusions (by showing
thinking/processes used to solve problems).
•
Clarify and improve arguments (by using
different representations to clarify a solution).
Attend to precision.
•
Calculate accurately and efficiently (the
products and quotients of real-world problems
involving fractions).
•
Communicate precisely (use math vocabulary
in discussion and in one’s own reasoning).
Model with mathematics.
•
Apply math to solve problems (involving
multiplication and division of fractions) in
everyday life, society, and the workplace.
Essential questions
•
How are multiplication and division of
fractions related?
•
Why is the quotient of two positive fractions
larger than the dividend?
•
How would you represent 1/2 of 1/3 on a
number line?
•
•
How do you multiply and divide proper
fractions, improper fractions, and mixed
numbers?
Why is the product of two positive fractions
less than each of the two fractions you
multiplied?
•
•
In what real-world situation would you apply
multiplication and division of fractions?
Explain.
Describe and illustrate how to multiply
fractions by fractions, fractions by mixed
numbers, and fractions by whole numbers.
•
When solving real-world problems, how do
you know when to multiply or divide?
Cumberland, Lincoln, and Woonsocket Public Schools
in collaboration with the Charles A. Dana Center at the University of Texas at Austin
C-27
Grade 6 Mathematics, Quarter 2, Unit 2.3
Final, July 2011
Multiplying and Dividing Fractions (7 days)
Written Curriculum
Grade-Level Expectations
M(N&O)–6–3 Demonstrates conceptual understanding of mathematical operations by adding and
subtracting positive fractions and integers; and multiplying and dividing fractions and decimals. (Local)
M(N&O)–6–3 Demonstrates conceptual understanding of mathematical operations by describing or
illustrating the meaning of a power by representing the relationship between the base (whole number) and
the exponent (whole number) (e.g.,33, 43); and the effect on the magnitude of a whole number when
multiplying or dividing it by a whole number, decimal, or fraction. (State)
M(N&O)–6–4 Accurately solves problems involving single or multiple operations on fractions (proper,
improper, and mixed), or decimals; and addition or subtraction of integers; percent of a whole; or
problems involving greatest common factor or least common multiple. (State)
(IMPORTANT: Applies the conventions of order of operations with and without parentheses.)
Clarifying the Standards
Prior Learning
Multiplication was introduced in third grade. Students learned multiplication of whole numbers based on
repeated addition and used models, number lines, and explanations. In grade 4, students learned division
of whole numbers (with no remainders) based on repeated subtraction and were introduced to the inverse
relationship between multiplication and division of whole numbers. In grade 5, students continued their
work with division and the meaning of a remainder with respect to whole numbers using models,
explanations, or solving problems.
As early as grade 3, students began to solve problems involving multiplication of whole numbers. In
grade 4, they solved problems using division with single-digit divisors. Their work with solving division
problems was expanded to two-digit divisors in grade 5. Ongoing practice with multiplication continued
from grades 3–5.
Current Learning
In grade 6, students are introduced to multiplication and division of positive fractions, including mixed
numbers and improper fractions. Students also accurately solve problems involving single or multiple
operations on fractions (proper, improper, and mixed). These concepts should be mastered by the end of
this unit. Students continue to use their estimation and mental computation skills to verify the accuracy of
their results.
Future Learning
In grades 7–12, the multiplication and division of positive fractions will be expanded to the multiplication
and division of negative fractions and will show up in probability (e.g., determining probabilities),
algebra (e.g., solving equations), and geometry (e.g., finding area).
C-28
Cumberland, Lincoln, and Woonsocket Public Schools
in collaboration with the Charles A. Dana Center at the University of Texas at Austin
Multiplying and Dividing Fractions (7 days)
Grade 6 Mathematics, Quarter 2, Unit 2.3
Final, July 2011
Additional Research Findings
According to Principles and Standards for School Mathematics, “multiplying and dividing fractions …
can be challenging for many students because of problems that are primarily conceptual rather than
procedural. From their experience with whole numbers, many students appear to develop the belief that
‘multiplication makes bigger and division makes smaller’… a mistaken expectation about the magnitude
of a computational result is likely to interfere with students’ making sense of multiplication and division
of fractions” (p. 218). It is imperative that students master these concepts for further growth in
mathematics. The basic operations form the foundation for future math courses.
Notes About Resources and Materials
Cumberland, Lincoln, and Woonsocket Public Schools
in collaboration with the Charles A. Dana Center at the University of Texas at Austin
C-29
Grade 6 Mathematics, Quarter 2, Unit 2.3
Final, July 2011
C-30
Multiplying and Dividing Fractions (7 days)
Cumberland, Lincoln, and Woonsocket Public Schools
in collaboration with the Charles A. Dana Center at the University of Texas at Austin
Grade 6 Mathematics, Quarter 2, Unit 2.4
Ratios, Rates, and Percentages
Overview
Number of instructional days:
12
(1 day = 45–60 minutes)
Content to be learned
Mathematical practices to be integrated
•
Demonstrate conceptual understanding of
rational numbers with respect to ratios and
rates.
Make sense of problems and persevere in solving
them.
•
Demonstrate understanding of the relative
magnitude of numbers by ordering and
comparing across number formats (fractions,
decimals, and whole number percents from 1–
100).
•
Make conjectures (using benchmark percents
to estimate the percent of a whole number).
•
Decide on a solution pathway before jumping
into the work (when determining rates and unit
rates).
•
Solve problems involving percent of a whole
accurately.
•
Check solution with another method (when
determining rates and unit rates).
•
Convert between fractions, decimals, and
percents (not evident in GLEs).
•
Ask, “Does my answer make sense?” (when
finding the percent of a whole; when
converting between fractions, decimals, and
percents; and when finding rates and unit
rates).
Attend to precision.
•
Specify units of measure (when finding rates
and unit rates).
•
State the meaning of symbols (when writing
ratios, rates, and percents).
•
Use “=” consistently and appropriately (when
determining whether two ratios are
proportional; when converting between
fractions, decimals, and percents; and when
comparing rates and unit rates.
•
If 5% is $4.50 what is 100%?
•
Given a fraction, a percent, and a decimal of
different values, how would you order them
from least to greatest?
Essential questions
•
How are fractions and ratios similar? How are
they different?
•
How can you find a percent of a whole?
•
How is a percent a ratio?
•
What strategy do you use to order and compare
percents?
Cumberland, Lincoln, and Woonsocket Public Schools
in collaboration with the Charles A. Dana Center at the University of Texas at Austin
C-31
Grade 6 Mathematics, Quarter 2, Unit 2.4
Final, July 2011
Ratios, Rates, and Percentages (12 days)
Written Curriculum
Grade-Level Expectations
M(N&O)–6–1 Demonstrates conceptual understanding of rational numbers with respect to ratios
(comparison of two whole numbers by division a/b, a : b, and a ÷ b , where b ≠ 0); and rates (e.g., a out
of b, 25%) using models, explanations, or other representations. (State)
M(N&O)–6–2 Demonstrates understanding of the relative magnitude of numbers by ordering or
comparing numbers with whole number bases and whole number exponents (e.g.,33, 43), integers, or
rational numbers within and across number formats (fractions, decimals, or whole number percents from
1–100) using number lines or equality and inequality symbols. (State)
M(N&O)–6–4 Accurately solves problems involving single or multiple operations on fractions (proper,
improper, and mixed), or decimals; and addition or subtraction of integers; percent of a whole; or
problems involving greatest common factor or least common multiple. (State)
(IMPORTANT: Applies the conventions of order of operations with and without parentheses.)
Clarifying the Standards
Prior Learning
In grades K–5, students progressively learned about fractions—starting with the concept of (1/2) as a “fair
share” in kindergarten moving on to benchmark fractions such as fourths and thirds in grade 5—using
models, explanations, and other representations.
In grade 5, they continued these concepts and began to look at benchmark percents within number
formats (fractions to fractions, decimals to decimals, and percents to percents) using models or number
lines.
Current Learning
In grade 6, students are introduced to the concepts of “ratio” and “rate” using models, explanations, and
other representations. Students expand their work of understanding the relative magnitude of benchmark
percents to encompass all whole number percents. Students are introduced to problem solving involving
percent of a whole. Although not evident in the GLEs (grade 6 or prior), students must demonstrate an
understanding of converting between fractions, decimals, and percents in order to be successful in this
unit.
Future Learning
In grades 7 and 8, students will demonstrate conceptual understanding of rational numbers with respect to
percents as a means of comparing the same or different parts of the whole when the wholes vary in
magnitude; as a way of expressing multiples of a number using models, explanations, and other
representations; and as a way of describing change (percent increase and decrease). These concepts will
show up again in grades 11 and 12 when students are working with why certain real numbers are rational.
C-32
Cumberland, Lincoln, and Woonsocket Public Schools
in collaboration with the Charles A. Dana Center at the University of Texas at Austin
Ratios, Rates, and Percentages (12 days)
Grade 6 Mathematics, Quarter 2, Unit 2.4
Final, July 2011
Students will continue to develop their conceptual understanding of the relative magnitude of numbers
across number formats in grade 7 and will begin to look at irrational numbers in grade 8. In grades 10–12,
students will expand this understanding to real numbers.
In grades 7 and 8, students will take their knowledge of ratios and rates and begin to accurately solve
problems involving discounts, tax, tips, rates, and proportional reasoning. Grade 8 also will add percent
increase and decrease, interest rates, and markups. These concepts will continue in grades 9–12.
Additional Research Findings
According to Principles and Standards for School Mathematics, “by solving problems that require
multiplicative comparisons, students will gain extensive experience with ratios, rates, and percents, which
helps form a solid foundation for their understanding of, and their facility with, proportionality. The study
of rational numbers in the middle grades should build on students’ prior knowledge of whole-number
concepts and skills and their encounters with fractions, decimals, and percents in lower grades and in
everyday life … Students’ facility with rational numbers and proportionality can be developed in concert
with their study of many topics in the middle-grades curriculum” (p. 215).
Notes About Resources and Materials
Cumberland, Lincoln, and Woonsocket Public Schools
in collaboration with the Charles A. Dana Center at the University of Texas at Austin
C-33
Grade 6 Mathematics, Quarter 2, Unit 2.4
Final, July 2011
C-34
Ratios, Rates, and Percentages (12 days)
Cumberland, Lincoln, and Woonsocket Public Schools
in collaboration with the Charles A. Dana Center at the University of Texas at Austin