Grade 7 Mathematics, Quarter 1, Unit 1.1
Properties and Relationships Among Numbers
Overview
Number of instructional days:
6
(1 day = 45–60 minutes)
Content to be learned
Mathematical Practices to be Integrated
•
Demonstrate a conceptual understanding of
scientific notation.
Attend to precision.
•
Demonstrate a conceptual understanding of
square of perfect squares and non-perfect
squares.
•
Calculate accurately and efficiently.
•
Express numerical answers with a degree of
precision appropriate for the problem context.
Look for and express regularity in repeated
reasoning.
•
Notice when calculations are repeated.
•
Look for both general methods and shortcuts
(scientific notation).
•
Continuously evaluate the reasonableness of
results.
•
What is the difference between a perfect and a
non-perfect square?
Essential questions
•
Why do we use scientific notation instead of
standard form?
•
What does it mean to find the square root of a
number?
Cumberland, Lincoln, and Woonsocket Public Schools
in collaboration with the Charles A. Dana Center at the University of Texas at Austin
C-1
Grade 7 Mathematics, Quarter 1, Unit 1.1
Final, July 2011
Properties and Relationships Among Numbers (6 days)
Written Curriculum
Grade-Level Expectations
M(N&O)–7–2 Demonstrates understanding of the relative magnitude of numbers by ordering,
comparing, or identifying equivalent rational numbers across number formats, numbers with whole
number bases and whole number exponents (e.g., 33, 43), integers, absolute values, or numbers
represented in scientific notation using number lines or equality and inequality symbols. (State)
M(N&O)–7–4 Accurately solves problems involving the addition or subtraction of integers, raising numbers
to whole number powers, and determining square roots of perfect square numbers and non-perfect square
numbers. (Local)
M(N&O)–7–6 Uses a variety of mental computation strategies to solve problems (e.g., using compatible
numbers, applying properties of operations, using mental imagery, using patterns) and to determine the
reasonableness of answers; and mentally calculates benchmark perfect squares and related square roots
(e.g., 12, 22 … 122, 152, 202, 252, 1002, 10002); determines the part of a number using benchmark percents and
1
2
1
33 %
66 %
33 %
3 , 50%,
3 , 75%, and 100%) (e.g., 25% of 16;
3 of 330).
related fractions (1%, 10%, 25%,
(Local)
(IMPORTANT: The intent of this GSE is to embed mental arithmetic throughout the instructional
program, not to teach it as a separate unit.)
Clarifying the Standards
Prior Learning
Students progressively applied properties of odd and even numbers, remainders, divisibility, and prime
factorization throughout grades 1–6. In grades 1 and 2, they used commutative, associative, and identity
properties to solve and simplify addition problems. Using these field properties to solve multiplication
problems began in grade 3. Beginning in grade 4, students were introduced to rational number formats
(fractions, decimals). Integers and benchmark percents were added in grades 5–6 as well as the distributive
property and using additive inverses for calculating rational numbers. Since kindergarten, students have used
mental computation strategies to solve problems with increasing complexity and to determine reasonableness
of answers.
Current Learning
Students have a conceptual understanding of scientific notation and calculating square roots of perfect squares
and non-perfect squares. Mentally calculating benchmark perfect squares and related square roots is new to
this grade. Students will continue to use mental computation strategies to solve problems throughout their
learning.
Future Learning
In grade 8, students will apply their understanding of absolute value and scientific notation in problemsolving formats. In grades 8–10, they will continue to use a variety of mental-computation strategies to
solve problems, and in grades 8–12 students will continue to apply the properties of numbers and field
properties to solve problems.
C-2
Cumberland, Lincoln, and Woonsocket Public Schools
in collaboration with the Charles A. Dana Center at the University of Texas at Austin
Properties and Relationships Among Numbers (6 days)
Grade 7 Mathematics, Quarter 1, Unit 1.1
Final, July 2011
Additional Research Findings
According to Principles and Standards for School Mathematics, students in the middle grades should
understand numbers, ways of representing numbers, relationships among numbers, and number systems.
Additionally, middle-schoolers should understand meanings of operations and how they relate to one
another. Computing fluently and making reasonable estimates is also an expectation at this grade level (p.
214).
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-3
Grade 7 Mathematics, Quarter 1, Unit 1.1
Final, July 2011
C-4
Properties and Relationships Among Numbers (6 days)
Cumberland, Lincoln, and Woonsocket Public Schools
in collaboration with the Charles A. Dana Center at the University of Texas at Austin
Grade 7 Mathematics, Quarter 1, Unit 1.2
Comparing and Ordering
Overview
Number of instructional days:
8
(1 day = 45–60 minutes)
Content to be learned
Mathematical Practices to be Integrated
•
Understand and use number lines or equality
and inequality symbols to show the magnitude
of rational numbers across number formats
(fractions, decimals, percents, square roots, and
scientific notation).
Reason abstractly and quantitatively.
Apply properties of numbers.
•
•
Understand the meaning of quantities, not just
how to compute them.
•
Create a coherent representation of a problem.
•
Convert common denominators to the same
form using benchmark comparisons.
Use appropriate tools strategically.
•
Consider the available tools (paper and pencil,
calculator, number lines, concrete models)
when solving a problem.
•
Detect possible errors through strategic use of
estimation and other mathematical knowledge.
•
When given different forms of numbers, how
will you be able to compare and order them?
Essential questions
•
How would you compare two fractions with
different denominators?
•
When comparing, for example, 6.615 and 6.62,
explain how you know which number is
smaller.
Cumberland, Lincoln, and Woonsocket Public Schools
in collaboration with the Charles A. Dana Center at the University of Texas at Austin
C-5
Grade 7 Mathematics, Quarter 1, Unit 1.2
Final, July 2011
Comparing and Ordering (8 days)
Written Curriculum
Grade-Level Expectations
M(N&O)–7–2 Demonstrates understanding of the relative magnitude of numbers by ordering,
comparing, or identifying equivalent rational numbers across number formats, numbers with whole
number bases and whole number exponents (e.g., 33, 43), integers, absolute values, or numbers
represented in scientific notation using number lines or equality and inequality symbols. (State)
M(N&O)–7–6 Uses a variety of mental computation strategies to solve problems (e.g., using
compatible numbers, applying properties of operations, using mental imagery, using patterns) and to
determine the reasonableness of answers; and mentally calculates benchmark perfect squares and
related square roots (e.g., 12, 22 … 122, 152, 202, 252, 1002, 10002); determines the part of a number using
benchmark percents and related fractions (1%, 10%, 25%, 33 1 % , 50%, 66 2 % , 75%, and 100%) (e.g.,
3
3
25% of 16; 33 1 % of 330). (Local)
3
(IMPORTANT: The intent of this GSE is to embed mental arithmetic throughout the instructional
program, not to teach it as a separate unit.)
M(N&O)–7–8 Applies properties of numbers (odd, even, remainders, divisibility, and prime factorization)
and field properties (commutative, associative, identity, distributive, inverses) to solve problems and to
simplify computations, and demonstrates conceptual understanding of field properties as they apply to
subsets of the real numbers (e.g., the set of whole numbers does not have additive inverses, the set of integers
does not have multiplicative inverses). (Local)
Clarifying the Standards
Prior Learning
In grades K–2 students used “more” or “less” to compare whole numbers. Students continued wholenumber comparisons in grades 3–5 and went on to order and compare positive fractional numbers,
decimals, percents, and integers within each of the number formats. Comparisons with whole-number
exponents and comparisons across number formats were introduced in grade 6.
Current Learning
Using number lines or equality/inequality symbols, students order and compare rational numbers across
number formats (fractions, decimals, percents, square roots, and scientific notation). This unit is being
state-tested for this grade level.
Future Learning
Beginning in grade 8 and continuing through grade 12, students’ ordering and comparing will include
common irrational numbers, numbers with whole number or fractional bases, and square roots.
C-6
Cumberland, Lincoln, and Woonsocket Public Schools
in collaboration with the Charles A. Dana Center at the University of Texas at Austin
Comparing and Ordering (8 days)
Grade 7 Mathematics, Quarter 1, Unit 1.2
Final, July 2011
Additional Research Findings
According to Principles and Standards for School Mathematics, students in the middle grades should
understand numbers, ways of representing numbers, relationships among numbers, and number systems.
They should also compute fluently and make reasonable estimates (p. 214).
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-7
Grade 7 Mathematics, Quarter 1, Unit 1.2
Final, July 2011
C-8
Comparing and Ordering (8 days)
Cumberland, Lincoln, and Woonsocket Public Schools
in collaboration with the Charles A. Dana Center at the University of Texas at Austin
Grade 7 Mathematics, Quarter 1, Unit 1.3
Statistics
Overview
Number of instructional days:
8
(1 day = 45–60 minutes)
Content to be learned
Mathematical Practices to be Integrated
•
Develop conceptual understanding of the
effects of outliers.
Make sense of problems and persevere in solving
them.
•
Demonstrate ability to use measures of central
tendency in problem-solving situations.
•
Use central tendencies to draw conclusions
regarding real-life situations.
•
Evaluate a sample for bias and its
ramifications.
•
Analyze givens, constraints, relationships, and
goals.
•
Make conjectures about the form and meaning
of a solution.
•
Plan a solution pathway, rather than jumping
into the work.
•
Monitor and evaluate progress, changing
course if necessary.
•
Transform information into different
representations.
•
Check solutions with another method.
Construct viable arguments and critique the
reasoning of others.
•
Reason inductively about data, making
plausible arguments that take into account the
context from which the data arose (using
central tendencies to draw conclusions).
•
Identify flawed logic or reasoning, explaining
how it is flawed (i.e., bias, outliers).
•
What is bias and how can bias be prevented in
collecting data samples?
Essential questions
•
Why is it important to represent and analyze
data using measures of central tendency?
•
What effect do outliers have on measures of
central tendency?
Cumberland, Lincoln, and Woonsocket Public Schools
in collaboration with the Charles A. Dana Center at the University of Texas at Austin
C-9
Grade 7 Mathematics, Quarter 1, Unit 1.3
Final, July 2011
Statistics (8 days)
Written Curriculum
Grade-Level Expectations
M(DSP)–7–2 Analyzes patterns, trends, or distributions in data in a variety of contexts by solving
problems using measures of central tendency (mean, median, or mode), dispersion (range or variation),
or outliers to analyze situations to determine their effect on mean, median, or mode; and evaluates the
sample from which the statistics were developed (bias). (State)
Clarifying the Standards
Prior Learning
In grades K–2, students analyzed patterns, trends, or distributions of data using more, less, or equal in a
variety of contexts. They moved on to measures of central tendency and range in grades 3–4, as well as
the concept of largest or smallest. In grades 5–6, students began to analyze situations and solve problems.
The term dispersion for range or variation was introduced in grade 6.
Current Learning
In grade 7, the focus shifts to problem solving using measures of central tendency and dispersion.
Students consider outliers and their effect on mean, median, and mode. They also evaluate data samples
for indicators of bias. This material is new to seventh grade and will be state-tested.
Future Learning
Concepts of outliers and bias will be reinforced and expanded upon in grades 8–12. They are the
foundation for line of best fit, line of regression, correlation, calculating and analyzing measures of
dispersion, and a conceptual understanding of the sample from which the statistics are developed.
Additional Research Findings
According to Principles and Standards for School Mathematics, middle-level students should select and
use appropriate statistical methods to analyze data (p. 248).
C-10
Cumberland, Lincoln, and Woonsocket Public Schools
in collaboration with the Charles A. Dana Center at the University of Texas at Austin
Statistics (8 days)
Grade 7 Mathematics, Quarter 1, Unit 1.3
Final, July 2011
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-11
Grade 7 Mathematics, Quarter 1, Unit 1.3
Final, July 2011
C-12
Statistics (8 days)
Cumberland, Lincoln, and Woonsocket Public Schools
in collaboration with the Charles A. Dana Center at the University of Texas at Austin
Grade 7 Mathematics, Quarter 1, Unit 1.4
Statistics: Collecting, Displaying, and
Analyzing Data
Overview
Number of instructional days:
18
(1 day = 45–60 minutes)
Content to be learned
Mathematical practices to be integrated
•
Develop the concepts and use of circle graphs,
scatter plots (discrete linear), and histograms.
Use appropriate tools strategically.
•
Evaluate circle graphs, scatter plots (discrete
linear), and histograms in order to make
predictions and conclusions based on the types
of information presented in each
representation.
•
Identify the most appropriate representation for
a given situation or set of data.
•
Choose the most effective method to collect
data in response to a question or hypothesis.
•
•
Know which tool to use, what will be gained
by using the tool, and the limitations (types of
graphs).
•
Know which tools are available and
appropriate for the grade level (pencil and
paper, calculator, ruler, protractor, spreadsheet,
internet, statistics software).
Attend to precision.
•
Consider limitations of data collection that
could affect results.
Specify units of measure and label axes to
clarify the correspondence with quantities in a
problem (appropriately labeling graphs with
titles, scales, and keys).
Model with mathematics.
•
Identify important quantities in a practical
situation and map relationships using graphs
(histograms, scatter plots, circle graphs).
•
Analyze those relationships mathematically to
draw conclusions.
•
Interpret mathematical results in the context of
the situation and reflect on whether the results
make sense.
Essential questions
•
What are the advantages of having a graphic
representation of data?
•
What are the possible limitations to your data
collection process?
•
From histograms, circle graphs, and scatter
plots (discrete linear), what types of predictions
and conclusions can you make?
•
What must be considered when choosing a
graph to display a set of data.
Cumberland, Lincoln, and Woonsocket Public Schools
in collaboration with the Charles A. Dana Center at the University of Texas at Austin
C-13
Grade 7 Mathematics, Quarter 1, Unit 1.4
Final, July 2011
Statistics: Collecting, Displaying,
and Analyzing Data (18 days)
Written Curriculum
Grade-Level Expectations
M(DSP)–7–1 Interprets a given representation (circle graphs, scatter plots that represent discrete linear
relationships, or histograms) to analyze the data to formulate or justify conclusions, to make predictions,
or to solve problems. (State)
(IMPORTANT: Analyzes data consistent with concepts and skills in M(DSP)–7–2.)
M(DSP)–7–3 Organizes and displays data using tables, line graphs, scatter plots, and circle graphs to
answer questions related to the data, to analyze the data to formulate or justify conclusions, to make
predictions, or to solve problems. (Local)
M(DSP)–7–3 Identifies or describes representations or elements of representations that best display
a given set of data or situation, consistent with the epresentations required in M(DSP)–7–1. (State)
(IMPORTANT: Analyzes data consistent with concepts and skills in M(DSP)–6–2.)
M(DSP)–7–4 Uses counting techniques to solve problems in context involving combinations or
permutations (e.g., How many different ways can eight students place first, second, and third in a race?)
using a variety of strategies (e.g., organized lists, tables, tree diagrams, models, Fundamental Counting
Principle, orsc others). (Local)
M(DSP)–7–6 In response to a teacher or student generated question or hypothesis decides the most
effective method (e.g., survey, observation, experimentation) to collect the data (numerical or categorical)
necessary to answer the question; collects, organizes, and appropriately displays the data; analyzes the
data to draw conclusions about the question or hypothesis being tested while considering the limitations
that could affect interpretations; and when appropriate makes predictions; and asks new questions and
makes connections to real world situations. (Local)
(IMPORTANT: Analyzes data consistent with concepts and skills in M(DSP)–7–2.)
Clarifying the Standards
Prior Learning
In grades K–2 students interpreted a given representation using models, tally charts, pictographs with oneto-one correspondence, tables, and line plots. Students used these models to formulate conclusions. In
grades 3–5, pictographs, bar, circle, and line graphs were used to analyze data and to formulate or justify
conclusions. In grade 6, stem-and-leaf plots were added. Students also used given models to make
predictions or to solve problems. They began to organize and display their own data using the various
models previously studied.
C-14
Cumberland, Lincoln, and Woonsocket Public Schools
in collaboration with the Charles A. Dana Center at the University of Texas at Austin
Statistics: Collecting, Displaying,
and Analyzing Data (18 days)
Grade 7 Mathematics, Quarter 1, Unit 1.4
Final, July 2011
Current Learning
Students apply central tendencies to given circle graphs, scatter plots (discrete linear—students should be
able to look at a scatter plot and determine if the data has a positive or negative correlation, or no
correlation), and histograms in order to make predictions, draw or justify conclusions, and solve
problems. Scatter plots and histograms are new to this grade level and will be state-tested.
Future Learning
In grade 8, students will analyze scatter plots and box-and-whisker plots. They will continue to make
predictions, draw or justify conclusions, and solve problems. In grades 9–10, students will critique
conclusions and expand use of analysis across other disciplines. In grades 8 and beyond students will also
choose appropriate representations to best display a given set of data or situation.
Additional Research Findings
According to Principles and Standards for School Mathematics, students in the middle grades should
build upon prior experience collecting, organizing, and representing sets of data. They should have had
experience in using some methods of analyzing information and answering questions about a single
population. In grades 6–8, teachers should build on this base of experience to help students answer more
complex questions about the data. Students should make observations, inferences, and conjectures, and
develop new questions (pp. 249–252).
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-15
Grade 7 Mathematics, Quarter 1, Unit 1.4
Final, July 2011
C-16
Statistics: Collecting, Displaying,
and Analyzing Data (18 days)
Cumberland, Lincoln, and Woonsocket Public Schools
in collaboration with the Charles A. Dana Center at the University of Texas at Austin