MADISON PUBLIC SCHOOL DISTRICT
Grade 6 Mathematics
Authored by: Mark Fisher
Updated by: Dana Collins and Jin Lee
Reviewed by: Lee Nittel,
Director of Curriculum and Instruction
Kathryn Lemerich,
Supervisor of Mathematics and Business Education
Updated with Common Core State Standards: Fall 2012
Members of the Board of Education:
Lisa Ellis, President
Patrick Rowe, Vice-President
David Arthur
Kevin Blair
Shade Grahling
Linda Gilbert
Thomas Haralampoudis
James Novotny
Superintendent : Dr. Michael Rossi
Madison Public Schools
359 Woodland Road, Madison, NJ 07940
www.madisonpublicschools.org
I.
OVERVIEW
The Grade 6 Mathematics course is designed to provide students with the opportunity to strengthen
content base in mathematics while previewing pre-algebra concepts and incorporating thought-provoking
enrichment activities in order to further develop clear, concise, and independent mathematical reasoning
skill. The course is directly correlated to the Common Core State Standards for Mathematics, designed to
cover such topics as number sense and numerical operations, geometry and measurement, patterns and
algebra, data analysis, probability, and discrete mathematics while promoting and instilling the skills of
problem solving, communication in mathematics, and making mathematical connections. Students will
utilize various tools and technology in the process, including manipulatives, calculators, software,
websites, and computers to better enhance a well-rounded understanding of course topics. A strong focus
of the program is on promoting high levels of mathematical thought through experiences which extend
beyond traditional computation.
II.
RATIONALE
The Grade 6 Mathematics course objectives promote effective content-specific skills, while developing
problem-solving skills and strategies, communication, and reasoning. Lessons are prepared and
implemented developmentally, sequentially, and with the understanding that the learner has the proper
pre-requisite skills necessary for success in higher level thinking. Students will be expected to sharpen
thinking skill by utilizing a strong set of problem-solving techniques and advanced reasoning processes.
The intent is to best prepare every learner for the challenges of future mathematical endeavors.
III.
STUDENT OUTCOMES (Linked to the Common Core State Standards for Mathematics)
Ratios and Proportional Relationships (6.RP)
Understand ratio concepts and use ratio reasoning to solve problems.
6.RP.1: Understand the concept of a ratio and use ratio language to describe a ratio relationship between
two quantities. For example, “The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2
wings there was 1 beak.” “For every vote candidate A received, candidate C received nearly three votes.”
6.RP.2: Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate
language in the context of a ratio relationship. For example, “This recipe has a ratio of 3 cups of flour to 4 cups of
sugar, so there is 3/4 cup of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per
hamburger.”1 1Expectations for unit rates in this grade are limited to non-complex fractions.
6.RP.3: Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning
about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
a. Make tables of equivalent ratios relating quantities with whole-number measurements, find
missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to
compare ratios.
b. Solve unit rate problems including those involving unit pricing and constant speed. For example,
if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what
rate were lawns being mowed?
c. Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the
quantity); solve problems involving finding the whole, given a part and the percent.
d. Use ratio reasoning to convert measurement units; manipulate and transform units appropriately
when multiplying or dividing quantities.
The Number System (6.NS)
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 length 3/4 mi
and area 1/2 square mi?
Compute 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.
6.NS.4: Find the greatest common factor of two whole numbers less than or equal to 100 and the least
common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a
sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers
with no common factor. For example, express 36 + 8 as 4 (9 + 2).
Apply and extend previous understandings of numbers to the system of rational numbers.
6.NS.5: Understand that positive and negative numbers are used together to describe quantities having
opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level,
credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities
in real-world contexts, explaining the meaning of 0 in each situation.
6.NS.6: Understand a rational number as a point on the number line. Extend number line diagrams and
coordinate axes familiar from previous grades to represent points on the line and in the plane with negative
number coordinates.
a. Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the
number line; recognize that the opposite of the opposite of a number is the number itself, e.g., –(–3)
= 3, and that 0 is its own opposite.
b. Understand signs of numbers in ordered pairs as indicating locations in quadrants of the
coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the
points are related by reflections across one or both axes.
c. Find and position integers and other rational numbers on a horizontal or vertical number line
diagram; find and position pairs of integers and other rational numbers on a coordinate plane.
6.NS.7: Understand ordering and absolute value of rational numbers.
a. Interpret statements of inequality as statements about the relative position of two numbers on a
number line diagram. For example, interpret –3 > –7 as a statement that –3 is located to the right of –7 on a
number line oriented from left to right.
b. Write, interpret, and explain statements of order for rational numbers in real-world contexts. For
example, write –3°C > –7°C to express the fact that –3°C is warmer than –7°C.
c. Understand the absolute value of a rational number as its distance from 0 on the number line;
interpret absolute value as magnitude for a positive or negative quantity in a real-world situation.
For example, for an account balance of –30 dollars, write |–30| = 30 to describe the size of the debt in dollars.
d. Distinguish comparisons of absolute value from statements about order. For example, recognize
that an account balance less than –30 dollars represents a debt greater than 30 dollars.
6.NS.8: Solve real-world and mathematical problems by graphing points in all four quadrants of the
coordinate plane. Include use of coordinates and absolute value to find distances between points with the
same first coordinate or the same second coordinate.
Expressions and Equations (6.EE)
Apply and extend previous understandings of arithmetic to algebraic expressions.
6.EE.1: Write and evaluate numerical expressions involving whole-number exponents.
6.EE.2: Write, read, and evaluate expressions in which letters stand for numbers.
a. Write expressions that record operations with numbers and with letters standing for numbers.
For example, express the calculation “Subtract y from 5” as 5 – y.
b. Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient,
coefficient); view one or more parts of an expression as a single entity. For example, describe the
expression 2 (8 + 7) as a product of two factors; view (8 + 7) as both a single entity and a sum of two terms.
c. Evaluate expressions at specific values of their variables. Include expressions that arise from
formulas used in real-world problems. Perform arithmetic operations, including those involving
whole number exponents, in the conventional order when there are no parentheses to specify a
particular order (Order of Operations). For example, use the formulas V = s3 and A = 6 s2 to find the
volume and surface area of a cube with sides of length s = 1/2.
6.EE.3: Apply the properties of operations to generate equivalent expressions. For example, apply the
distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive
property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y
+ y + y to produce the equivalent expression 3y.
6.EE.4: Identify when two expressions are equivalent (i.e., when the two expressions name the same
number regardless of which value is substituted into them). For example, the expressions y + y + y and 3y are
equivalent because they name the same number regardless of which number y stands for.
Reason about and solve one-variable equations and inequalities.
6.EE.5: Understand solving an equation or inequality as a process of answering a question: which values
from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a
given number in a specified set makes an equation or inequality true.
6.EE.6: Use variables to represent numbers and write expressions when solving a real-world or
mathematical problem; understand that a variable can represent an unknown number, or, depending on
the purpose at hand, any number in a specified set.
6.EE.7: Solve real-world and mathematical problems by writing and solving equations of the form x + p =
q and px = q for cases in which p, q and x are all nonnegative rational numbers.
6.EE.8: Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world
or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many
solutions; represent solutions of such inequalities on number line diagrams.
Represent and analyze quantitative relationships between dependent and independent variables.
6.EE.9: Use variables to represent two quantities in a real-world problem that change in relationship to
one another; write an equation to express one quantity, thought of as the dependent variable, in terms of
the other quantity, thought of as the independent variable. Analyze the relationship between the dependent
and independent variables using graphs and tables, and relate these to the equation. For example, in a
problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation
d = 65t to represent the relationship between distance and time.
Geometry (6.G)
Solve real-world and mathematical problems involving area, surface area, and volume.
6.G.1: Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing
into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of
solving real-world and mathematical problems.
6.G.2: Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit
cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be
found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find
volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and
mathematical problems.
6.G.3: Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find
the length of a side joining points with the same first coordinate or the same second coordinate. Apply
these techniques in the context of solving real-world and mathematical problems.
6.G.4: Represent three-dimensional figures using nets made up of rectangles and triangles, and use the
nets to find the surface area of these figures. Apply these techniques in the context of solving real-world
and mathematical problems.
Statistics and Probability (6.SP)
Develop understanding of statistical variability.
6.SP.1: Recognize a statistical question as one that anticipates variability in the data related to the
question and accounts for it in the answers. For example, “How old am I?” is not a statistical question, but “How
old are the students in my school?” is a statistical question because one anticipates variability in students’ ages.
6.SP.2: Understand that a set of data collected to answer a statistical question has a distribution which can
be described by its center, spread, and overall shape.
6.SP.3: Recognize that a measure of center for a numerical data set summarizes all of its values with a
single number, while a measure of variation describes how its values vary with a single number.
Summarize and describe distributions.
6.SP.4: Display numerical data in plots on a number line, including dot plots, histograms, and box plots.
6.SP.5: Summarize numerical data sets in relation to their context, such as by:
a. Reporting the number of observations.
b. Describing the nature of the attribute under investigation, including how it was measured and its
units of measurement.
c. Giving quantitative measures of center (median and/or mean) and variability (interquartile
range and/or mean absolute deviation), as well as describing any overall pattern and any striking
deviations from the overall pattern with reference to the context in which the data were gathered.
d. Relating the choice of measures of center and variability to the shape of the data distribution and
the context in which the data were gathered.
IV.
ESSENTIAL QUESTIONS AND CONTENT
A. Number Sense and Algebraic Thinking
1. What methods of estimation can be used to problem solve?
2. How does changing the order of operations affect the result of a mathematical
expression?
3. How can algebraic equations be used to solve real-life problems?
4. What is the relationship between expressions and equations?
B. Measurement and Statistics
1. Which measure of central tendency(mean, median, or mode) best represents
given sets of data?
2. What factors contribute to drawing false conclusions from misleading graphs?
3. What type of display best represents data in given situations?
4. How can data displays be used to model real-life situations?
C. Decimal Addition, Subtraction, Multiplication, or Division
1. What previous knowledge allows for accurate comparing, rounding, and
ordering of decimals?
2. What effect does decimal place value have when performing operations on
decimals?
3. What would be the benefits and drawbacks if the United States of America
changed over to the metric system?
4. Compare and contrast attributes within the metric system.
D. Number Patterns and Fractions
1. How does prime factorization contribute to determining the greatest common
factor of two or more numbers?
2. How does prime factorization contribute to determining the least common
multiple of two or more numbers?
3. How are visual or physical models helpful when comparing and ordering
fractions or mixed numbers?
4. How can improper fractions and/or mixed numbers be used to model real-life
situations?
E. Addition, Subtraction, Multiplication, and Division of Fractions
1. What effect do the numerator and denominator have when performing
operations on fractions or mixed numbers?
2. How are visual or physical models helpful when performing operations on
fractions or mixed numbers?
3. In the United States of America, what are the benefits and drawbacks of using
the customary system of measurement over the metric system of
measurement?
4. What is the relationship between addition and subtraction when performing
operations on fractions or mixed numbers?
5. What is the relationship between multiplication and division when performing
operations on fractions or mixed numbers?
F. Ratio, Proportion, and Percent
1. What is the relationship between ratios and rates?
2. How can the rules for solving algebraic equations be used to solve
proportions?
3. How is scale factor connected to ratios and proportions?
4. How are visual or physical models helpful when representing percents?
5. How can estimation be used to check the accuracy of a solution in the percent
equation?
6. What is the relationship between percentage and angle measure when
constructing a circle graph?
7. How can we use percent change to explain real-life situations?
G. Geometric Figures
1. How can properties of angles influence the construction of polygons?
2. How are similarity and congruency related?
3. What effect does interior angle measurement in polygons have on the polygon’s
rotation or line symmetry?
4. How are the properties of polygons used in real-life situations?
5. What are effects of transformations on plane figures?
H. Geometry and Measurement
1. What effect does scale factor have when calculating area and perimeter of
similar figures?
2. How do we appropriately label units when calculating perimeter and area of
geometric figures?
3. How can we use a two-dimensional grid to find perimeter and area of
geometric figures?
4. What method(s) can be used to calculate perimeter and area of convex
polygons?
5. Compare and contrast attributes of solid figures.
6. What role do units play in analyzing the calculated result of a solid figure’s
surface area and/or volume?
I. Integers
1. How can integers be used to model real-life situations?
2. What patterns can be seen when performing operations on integers?
3. How is the reflection of an object similar to the rotation of an object? How do
these transformations differ?
4. What effect does the order of the x and y values in any given coordinate have
when graphing in the coordinate plane?
J. Equations and Functions
1. What role does order of operations play when translating verbal phrases into
algebraic expressions?
2. What method(s) can be best used to solve algebraic equations?
3. What role does order of operations play when evaluating algebraic expressions
or solving algebraic equations?
4. What underlying principles connect equations and functions?
5. How does the study of data displays relate to the graphing of linear equations
in the coordinate plane?
K. Probability and Statistics
1. How do we use probability to explain real-life situations?
2. How do we create, test, and validate a projected probability?
3. What is the relationship between experimental and theoretical probability?
4. How can we apply the techniques of tree diagrams in a variety of contexts?
5. How can we apply the techniques of the counting principle in a variety of
contexts?
6. How are visual models helpful when finding permutations of ordered
situations?
7. How do the techniques of systematic listing relate to finding total
combinations?
8. What visual models can be used to represent disjoint events?
9. What role do independent and dependent events have in making predictions
based on experimental or theoretical probabilities?
L. Individual and/or Group Projects
1. What skills are needed in order to make group communication and problemsolving successful?
2. What are the most appropriate problem solving strategies for this given
project?
3. What previous learning helps us understand this given project?
4. What technological tools or other resources are available to complete this
project?
5. How do we determine the reasonableness of our results for this project?
6. What are the benefits and limitations of our final product?
V.
STRATEGIES
Students will be actively involved in daily lessons by means of guided and independent practice,
cooperative learning activities, as well as group project participation. The Grade 6 Mathematics
curriculum is a transactional process, utilizing instruction that incorporates problem-based learning,
including hands-on activities, manipulatives, projects, class discussions, and other approaches determined
by the teacher.
Students will be prompted either in group or individual problem-solving situations to use a variety of
mathematical reasoning strategies to deduce solutions. Students will evidence the ability to make
connections between past and present learning. Students will utilize discovery in order to develop cognitive
skills intended to promote independence of thought and analysis.
Students will also be given sets of oral or written problems to solve individually. Students will understand
the mathematical concepts in each problem and choose the appropriate path in order to calculate correct
solutions.
Students will also utilize the systems and tools of technology in order to solve problems appropriate to
each unit of study.
VI.
EVALUATION
Assessments may include but are not limited to:
Teacher Observations
Class Participation
Homework Assignments
Notebooks
Student Projects
Unit Tests and Quizzes
Anecdotal Records
District Math Assessments: Open-Ended Problem Solving Assessment (October and May),
Cumulative Math Assessment (May)
Standardized Test Scores
Final Exam
VII.
REQUIRED RESOURCES
Textbook for course:
Boswell, L., Kanold, T.D., Larson, R., & Stiff, L. (2007). McDougal Littell Math Course 1 –
Practice Workbook. Evanston, IL: McDougal Littell.
Boswell, L., Kanold, T.D., Larson, R., & Stiff, L. (2007). McDougal Littell Math Course 1 –
Student Edition. Evanston, IL: McDougal Littell.
Suggested supplemental resources for course:
Aaberg, N.H., Lindsey, J.F., & Overholt, J.L. (2008). Math Stories for Problem Solving
Success: Ready-to-Use Activities Based on Real-Life Situations, Grades 6 – 12, Second
Edition. San Francisco: Jossey-Bass.
Greenes, Carole. (2002) Houghton Mifflin Mathematics Teacher’s Edition 6. Boston, MA:
Houghton Mifflin.
Marx, P. (2006). Math Amazements: Astounding Investigations Uncover Math in Your World.
Tuscon, AZ: Good Year Books.
Pearce, Q.L. (2007). Math Logic. Quincy, IL: Mark Twain Media, Inc., Publishers
VIII. SCOPE AND SEQUENCE
McDougal Littell – Math Course 1 – Mathematics 6
Chapter 1 – Number Sense and Algebraic Thinking (12 days)
A. Whole number operations
B. Whole number estimation
C. Powers and exponents
D. Evaluate expressions using order of operations
E. Evaluate expressions that involve variables
F. Solve equations using mental math
G. Four step problems solving plan
Chapter 2 – Measurement and Statistics (12 days)
A. Measuring lengths using customary and metric units
B. Use formulas to find perimeter and area
C. Use scale drawings to find actual lengths
D. Frequency tables and line plots
E. Display data using bar graphs
F. Plot points on coordinate grids and make line graphs
G. Interpret circle graphs and make predictions
H. Mean, median mode and range
Chapter 3 – Decimals and Place Value (8 days)
A. Read and write decimals
B. Use decimals to express metric measurements
C. Compare and order decimals
D. Round decimals
E. Estimate decimal sums and differences
F. Add and subtract decimals
Chapter 4 – Decimal Multiplication and Division (14 days)
A. Multiply decimals and whole numbers
B. Use distributive property to evaluate expressions
C. Multiply decimals by decimals
D. Divide decimals by whole numbers
E. Multiply and divide by powers of ten
F. Divide decimals by decimals
G. Metric units of mass and capacity
H. Change one metric unit of measure to another
Chapter 5 – Number Patterns and Fractions (12 days)
A. Prime factorization
B. Divisibility rules
C. Primes and composites
D. Greatest common factor
E. Equivalent fractions
F. Least common multiple
G. Ordering fractions
H. Rewrite mixed numbers and improper fractions
I. Changing decimals to fractions
J. Changing fractions to decimals
Chapter 6 – Addition and Subtraction of Fractions (12 days)
A. Estimate with fractions and whole numbers
B. Find actual fraction sums and differences
C. Add and subtract fractions with different denominations
D. Add and subtract mixed numbers
E. Subtract mixed numbers by renaming
F. Add and subtract measures of time
Chapter 7 – Multiplication and Division of Fractions (10 days)
A. Multiply fractions and whole numbers
B.
C.
D.
E.
F.
G.
Multiply fractions
Multiply mixed numbers
Use reciprocals to divide fractions
Dividing mixed numbers
Use customary units of weights and capacity
Change customary units of measure
Chapter 8 – Ratio Proportion and Percent (14 days)
A. Write ratios and equivalent rations
B. Write rates, equivalent rates and unit rates
C. Write and solve proportions
D. Use proportions to find measures of objects (scale drawings)
E. Write percents as decimals and fractions
F. Write fractions and decimals as percents
G. Multiply to find a percent of a number (sales tax, simple interest)
Chapter 9 – Geometric Figures (14 days)
A. Identify lines, rays and segments
B. Name, measure and draw angles
C. Classify angles and find angle measures
D. Classify triangles by their angles and their sides
E. Classify quadrilaterals by their angles and sides
F. Classify polygons by their sides
G. Identify congruent and similar figures
H. Identify lines of symmetry
Chapter 10 – Geometry and Measurement (12 days)
A. Area of a parallelogram
B. Area of a triangle
C. Find circumference of a circle
D. Find area of a circle
E. Classify solids
F. Find surface area of a prism
G. Volume of a rectangular prism
H. Measuring mass and capacity in metric and customary units
Chapter 11 – Integers (14 days)
A. Compare and order integers
B. Add integers
C. Subtract integers
D. Multiply integers
E. Divide integers
F. Graph points with negative coordinates (translate a figure)
G. Recognize reflections and rotations
H. Identify and construct tessellations
Chapter 12 – Equations and Functions (14 days)
A. Write variable expressions and equations
B. Solve one-step addition equations
C. Solve one-step subtractions equations
D. Solve multiplication and division equations
E. Solving one-step inequalities
F. Find an expression for an input-output table
G. Evaluate functions and write function rules
H. Graph linear functions in a coordinate plane
Chapter 13 – Probability and Statistics (12 days)
A. Write probabilities
B. Use diagrams, tables and lists to find outcomes
C. Find the probability of 2 independent events
D. Recognize how statistics can be misleading
E. Organizing data using stem-and-leaf plots
F. Represent data using box-and-whisker plots
G. Choose appropriate data displays