grade 7 mathematics, quarter 1, unit 1.1 properties and ... · • look for both general methods...

Cumberland, Lincoln, and Woonsocket Public Schools C-1 in collaboration with the Charles A. Dana Center at the University of Texas at Austin

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.

• Demonstrate a conceptual understanding of square of perfect squares and non-perfect squares.

Attend to precision.

• 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.

Essential questions • Why do we use scientific notation instead of

standard form?

• What does it mean to find the square root of a number?

• What is the difference between a perfect and a non-perfect square?

Grade 7 Mathematics, Quarter 1, Unit 1.1 Properties and Relationships Among Numbers (6 days) Final, July 2011

C-2 Cumberland, Lincoln, and Woonsocket Public Schools in collaboration with the Charles A. Dana Center at the University of Texas at Austin

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

related fractions (1%, 10%, 25%, 133 %3 , 50%,

266 %3 , 75%, and 100%) (e.g., 25% of 16;

133 %3 of 330).

(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 problem-solving 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.

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

Grade 7 Mathematics, Quarter 1, Unit 1.1 Properties and Relationships Among Numbers (6 days) Final, July 2011




Comparing and Ordering


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).

• Apply properties of numbers.

Reason abstractly and quantitatively.

• 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.

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.

• When given different forms of numbers, how will you be able to compare and order them?

Grade 7 Mathematics, Quarter 1, Unit 1.2 Comparing and Ordering (8 days) Final, July 2011


Written Curriculum


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%, 133 %

3, 50%, 266 %

3 , 75%, and 100%) (e.g.,

25% of 16; 133 %3

of 330). (Local)

(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)


Prior Learning

In grades K–2 students used “more” or “less” to compare whole numbers. Students continued whole-number 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.

Comparing and Ordering (8 days) Grade 7 Mathematics, Quarter 1, Unit 1.2 Final, July 2011



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).


Grade 7 Mathematics, Quarter 1, Unit 1.2 Comparing and Ordering (8 days) Final, July 2011




Statistics


Content to be learned Mathematical Practices to be Integrated • Develop conceptual understanding of the

effects of outliers.

• Demonstrate ability to use measures of central tendency in problem-solving situations.

• Evaluate a sample for bias and its ramifications.

Make sense of problems and persevere in solving them.

• Use central tendencies to draw conclusions regarding real-life situations.

• 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).

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?

• What is bias and how can bias be prevented in collecting data samples?

Grade 7 Mathematics, Quarter 1, Unit 1.3 Statistics (8 days) Final, July 2011


Written Curriculum


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)


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.


According to Principles and Standards for School Mathematics, middle-level students should select and use appropriate statistical methods to analyze data (p. 248).

Statistics (8 days) Grade 7 Mathematics, Quarter 1, Unit 1.3 Final, July 2011



Grade 7 Mathematics, Quarter 1, Unit 1.3 Statistics (8 days) Final, July 2011




Statistics: Collecting, Displaying, and Analyzing Data


Content to be learned Mathematical practices to be integrated • Develop the concepts and use of circle graphs,

scatter plots (discrete linear), and histograms.

• 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.

• Consider limitations of data collection that could affect results.

Use appropriate tools strategically.

• 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.

• 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?

• From histograms, circle graphs, and scatter plots (discrete linear), what types of predictions and conclusions can you make?

• What are the possible limitations to your data collection process?

• What must be considered when choosing a graph to display a set of data.

Grade 7 Mathematics, Quarter 1, Unit 1.4 Statistics: Collecting, Displaying, Final, July 2011 and Analyzing Data (18 days)


Written Curriculum


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.)


Prior Learning

In grades K–2 students interpreted a given representation using models, tally charts, pictographs with one-to-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.

Statistics: Collecting, Displaying, Grade 7 Mathematics, Quarter 1, Unit 1.4 and Analyzing Data (18 days) 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.


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).


Grade 7 Mathematics, Quarter 1, Unit 1.4 Statistics: Collecting, Displaying, Final, July 2011 and Analyzing Data (18 days)


grade 7 mathematics, quarter 1, unit 1.1 properties and ... · • look for both general methods...

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