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Grade 7 Math

GRADE: 6

TOPIC: ALGEBRAIC REASONING PATTERNS AND FUNCTIONS

STAGE 1: IDENTIFY DESIRED RESULTS
Content Standard(s) Generalizations about what students should know and be able to do
Established Goals: State Standards: Understand and model patterns and functional relationships. Model and analyze quantitative data. Represent and interpret data and physical phenomena using algebraic symbols. National (NCTM) Standards: Understand patterns, relations, and functions. Represent and analyze mathematical situations and structures using algebraic symbols. Use mathematical models to represent and understand quantitative relationships. Analyze change in various contexts.
Enduring Understandings Insights earned from exploring generalizations via the essential questions (Students will understand THAT…)	Essential Questions Inquiry used to explore generalizations
1. Algebraic models help to solve problems and allow us to reach conclusions or make predictions. 2. Many different kinds of problems can be solved using the same technique.	1. How do you know for sure that your solution is correct? 2. How can you convince others that your algebraic solution is correct by also using pictures, graphs, charts and/or simple word explanations? 3. What is algebra?
Knowledge and Skills What students are expected to know and be able to do
Students will know that… Relationships that are expressed in words may be translated into algebraic expressions, equations, or inequalities. Mathematical relationships may be represented and analyzed with the help of tables, graphs, equations, and inequalities. Simple one-step equations can be solved using informal methods. When there is a relationship between two variables, the rate of change may be constant or varying.	Students will be able to… Represent numerical situations with algebraic expressions, equations and inequalities. Explore using variables as placeholders, to denote a pattern, to write a formula and to represent a function or relation. Explore how codes are used to communicate information. Use substitution to evaluate algebraic expressions and formulas. Describe, extend and analyze numeric, geometric and statistical patterns and use them to identify trends and justify predictions. Identify linear functions from tables, graphs or equations and use graphs to analyze the nature of changes in linear relationships. Solve simple linear equations using materials that model equivalence such as balance or guess-and-check. Solve simple linear equations using order of operations and algebraic properties. Create and use tables, graphs, words, equations and inequalities to represent, analyze and describe relationships, with constant and varying rates of change.
STAGE 2: DETERMINE ACCEPTABLE EVIDENCE
Performance Task(s) Authentic application in new context to evaluate student achievement of desired results designed according to GRASPS (Goal, Role, Audience, Setting Performance, Standards)	Other Evidence Application that is functional in a classroom context only to evaluate student achievement of desired results
GOAL The goal is to write an article for the school newsletter showing how safe locker combinations are. ROLE You are responsible for writing the article. AUDIENCE The audience is every reader of Cougar Tales. SETTING You need to calculate how many possible combinations there are for a locker and to estimate how much time would be needed to try each possible combination. PRODUCT PERFORMANCE AND PURPOSE You need to give mathematical proof for your conjecture (For a locker with 30 numbers, 30x29x29 = 25,230 different possible combinations. If it’s possible to try a combination every 15 seconds, it would require about 105 hours to try every possible combination.) STANDARDS Your article should give your opinion and be backed by mathematical reasoning. Scoring Rubric: Score of 3: Student shows a correct and/or appropriate answer and shows work and/or an explanation that demonstrates full and complete understanding. Score of 2: Answer has minor flaws, but the work and/or explanation are acceptable and the reasoning is appropriate. Score of 1: Student does not write a reasonable explanation or show sufficient work, resulting in a demonstration of only limited understanding. Score of 0: Student shows no understanding of the problem or how to arrive at a solution.	Daily Homework Classroom Observations Quizzes Tests District Assessments State Assessments Projects Student Reflections Journal Entries

GRADE: 6

TOPIC: GEOMETRY AND MEASUREMENT

STAGE 1: IDENTIFY DESIRED RESULTS
Content Standard(s) Generalizations about what students should know and be able to do
Established Goals: State Standards: Develop and apply units, systems, formulas and appropriate tools to determine and approximate measurements. Use spatial reasoning, location and geometric relationships to solve problems. Use attributes of two and three dimensional relationships and geometric theorems to describe relationships, communicate ideas and solve problems. National ( NCTM) Standards: Analyze characteristics and properties of two and three dimensional geometric shapes and develop mathematical arguments about geometric relationships. Specify locations and describe spatial relationships using coordinate geometry and other representational systems. Apply transformations, and use symmetry to analyze mathematical situations. Use visualization, spatial reasoning, and geometric modeling to solve problems. Understand measurable attributes of objects and the units, systems, and processes of measurement. Apply appropriate techniques, tools, and formulas to determine measurements.
Enduring Understandings Insights learned from exploring generalizations via the essential questions (Students will understand THAT…)	Essential Questions Inquiry used to explore generalizations
1. Both metric and customary systems of measurement are used in everyday life. 2. There are relationships among units within the same system. 3. There are appropriate units for measuring angles, perimeter, area, surface area, and volume. 4. Geometry enables us to describe, analyze, and understand our physical world. 5. Geometry introduces an experience of mathematics that complements and supports the study of other aspects of mathematics such as number and measurement. 6. Geometry offers powerful tools for representing and solving problems in all areas of mathematics, in other school subjects, and in everyday applications.	1. How would our lives be different if we couldn’t measure? 2. How does what we measure change how we measure? 3. When is it better to use an exact measurement rather than an estimate? 4. How far is far? 5. How would our classroom, school, homes, etc be different without geometrical shapes/angles? 6. What’s your position in the world? 7. How are size and distance related?
Knowledge and Skills What students are expected to know and be able to do
Students will know that… The metric system of measurement is based on powers of ten and ratios with multiples of ten underlying unit conversions. Angle measurement is based on rotation. Problems involving measurement can be solved through the use of appropriate tools, techniques and strategies. Triangles and some combinations of polygons are more stable than other polygons under stress. The formulas for the area of triangles, parallelograms, trapezoids and circles are based on the rectangle.	Students will be able to… Explore the different ratios used to convert between units of length, area and volume in the customary system. Recognize and use powers of ten as conversion ratios in the metric system. Estimate and measure angles based on rotation about a point. Locate points on a circular coordinate system. Build and use angle measurement tools such as circular protractor and angle ruler. Choose among nonstandard and standard referents to estimate length, area, volume, mass and angle measures. Select and use appropriate units, strategies and tools to estimate, measure and solve problems involving length, perimeter, area, volume, capacity, weight, mass, temperature and angles. Explore similarity of polygons and the effect of dilations (a reduction or enlargement) on changes of perimeter and area. Use the relationships of sides and angles to classify sets and subsets of polygons. Use the rectangle as a basic shape to model and develop formulas for the area of triangles, parallelograms, trapezoids and circles. Explore symmetry on rectangular and circular coordinate grids. Explore the relationship of radius, diameter, circumference and area of the circle. Approximate the value of pi.
STAGE 2: DETERMINE ACCEPTABLE EVIDENCE
Performance Task(s) Authentic application in new context to evaluate student achievement of desired results designed according to GRASPS (Goal, Role, Audience, Setting, Performance, Standards)	Other Evidence Application that is functional in a classroom context only to evaluate student achievement of desired results
Task 1 GOAL The goal is to create a feasible plan for an addition to your house for a room of your choice. ROLE You are the “architect” for the addition. AUDIENCE The audience is your family. SETTING You need to convince your family that the addition is consistent with the rest of the house. PRODUCT PERFORMANCE AND PURPOSE You first need to measure all of the rooms in your house. You will need to draw your house to scale (i.e. 1 inch = 3 feet). You then have to create an addition that will fit with the rest of the house. STANDARDS Your plan should include a detailed drawing of your house with the addition you are proposing. You also need to include a plan for the room’s usage. Scoring rubric: Score of 3: Student shows a correct and/or appropriate answer and shows work and/or an explanation that demonstrates full and complete understanding. Score of 2: Answer has minor flaws, but the work and/or explanation are acceptable and the reasoning is appropriate. Score of 1: Student does not write a reasonable explanation or show sufficient work, resulting in a demonstration of only limited understanding. Score of 0: Student shows no understanding of the problem or how to arrive at a solution. Task 2 GOAL The goal is to enclose a playground area with fencing for a daycare agency. ROLE You are responsible for creating the plan for the playground area. AUDIENCE The audience is the manager of the facility. SETTING You need to explore the different possibilities and find the largest possible area with the given fencing. PRODUCT PERFORMANCE AND PURPOSE You have been provided with 60 feet of fencing in 4 foot sections and a 4-foot gate. You need to try different shapes for the design and find the design with the biggest possible area. You need to drawn a plan for the fenced in area. STANDARDS Your plan should include a written summary of the different plans, mathematical proof to show the biggest possible area, and a drawn plan which shows the design. Scoring Rubric: Score of 3: Student shows a correct and/or appropriate answer and shows work and/or an explanation that demonstrates full and complete understanding. Score of 2: Student work has minor flaws, but the work and/or explanation are acceptable and the reasoning is appropriate. Score of 1: Student does not write a reasonable explanation or show sufficient work, resulting in a demonstration of only limited understanding. Score of 0: Student shows no understanding of the problem or how to arrive at a solution.	Daily Homework Classroom Observations Quizzes Tests District Assessments State Assessments Projects Student Reflections Journal Entries

GRADE: 6

TOPIC: NUMERICAL AND PROPORTIONAL REASONING ESTIMATION AND APPROXIMATION

STAGE 1: IDENTIFY DESIRED RESULTS
Content Standard(s) Generalizations about what students should know and be able to do
Established Goals: State Standards: Use a variety of numerical representations in the base ten system to describe quantitative relationships. Use numbers and their properties to compute flexibly and fluently, and to reasonably estimate measures and quantities. National (NCTM) Standards: Understand numbers, ways of representing numbers, relationships among numbers, and number systems. Understand meanings of operations and how they relate to one another. Compute fluently and make reasonable estimates.
Enduring Understandings Insights learned from exploring generalizations via the essential questions (Students will understand THAT…)	Essential Questions Inquiry used to explore generalizations
1. Estimation serves as an important companion to computation. 2. Estimation provides an effective tool for judging the reasonableness of calculator, mental, and paper and pencil computations. 3. There are appropriate times to use estimation rather than computing an exact answer.	Why is estimation important? When does estimating make more sense than finding an exact answer? How will estimating make solving a problem easier? What is the difference between rounding and estimating? What is reasonable?
Knowledge and Skills What students are expected to know and be able to do
Students will know that… Appropriate computational strategies facilitate problem solving. With fractions and decimals, products or quotients may be larger or smaller than either factor.	Students will be able to… Explain orally and in writing when a situation requires an exact answer or when an estimate is sufficient. Use estimation to predict reasonable answers and recognize and explain when an estimate will be more or less than an exact answer. Estimate and use a variety of computational strategies (mental computation, paper and pencil, and calculator) to add, subtract, multiply and divide multi-digit numbers in the context of multi-step word and practical problems. Use factors of composite numbers, multiples of 10, 100 and 1000 and divisibility rules to estimate products and missing factors.
	Choose, construct and use a variety of models and pictures to estimate and demonstrate addition and subtraction of fractions, decimals and mixed numbers, and relate the models to the use of equivalent forms and common denominators. Estimate and use calculators to add, subtract and multiply fractions and decimals. Write and round division problems in fraction form to estimate an answer to a division problem. Construct and use models and the distributive property to estimate reasonable answers and multiply fractions, decimals, mixed numbers and percents. Develop, describe and use a variety of ways to estimate and calculate with large numbers and connect the strategies to powers of ten.
STAGE 2: DETERMINE ACCEPTABLE EVIDENCE
Performance Task(s) Authentic application in new context to evaluate student achievement of desired results designed according to GRASPS (Goal, Role, Audience, Setting, Performance, Standards)	Other Evidence Application that is functional in a classroom context only to evaluate student achievement of desired results
GOAL The goal is to win the contest, “Estimate the amount of popcorn kernels that are in the jar.” ROLE You need to devise a plan to estimate a large number. AUDIENCE Your audience is the class. SETTING You need to convince the class that your estimation is a good one. PRODUCT PERFORMANCE AND PURPOSE You need to explain your strategy and your reasoning. (Students can use graph paper, scales, cups, rulers, calculators.) STANDARDS Students need to show they used a systematic and organized approach. Their explanation has to be very clear and their thinking process easy to follow. Scoring Rubric: Score of 3: Student shows a correct and/or appropriate answer and shows work and/or an explanation that demonstrates full and complete understanding. Score of 2: Answer has minor flaws, but the work and/or explanation are acceptable and the reasoning is appropriate. Score of 1: Student does not write a reasonable explanation or show sufficient work, resulting in a demonstration of only limited understanding. Score of 0: Student shows no understanding of the problem or how to arrive at a solution.	Daily Homework Classroom Observations Quizzes Tests District Assessments State Assessments Projects Student Reflections Journal Entries

GRADE: 6

TOPIC: NUMERICAL AND PROPORTIONAL REASONING NUMBER SENSE

STAGE 1: IDENTIFY DESIRED RESULTS
Content Standard(s) Generalizations about what students should know and be able to do
Established Goals: State Standards: Use a variety of numerical representations in the base ten system to describe quantitative relationships. Use numbers and their properties to compute flexibly and fluently, and to reasonably estimate measures and quantities. National (NCTM) Standards: Understand numbers, ways of representing numbers, relationships among numbers, and number systems.
Enduring Understandings Insights learned from exploring generalizations via the essential questions (Students will understand THAT…)	Essential Questions Inquiry used to explore generalizations
1. Fractions can be represented in many ways. 2. Fractions, decimals and percents are useful in many types of problem solving. 3. Many things in life involve parts of a whole. Fractions, decimals and percents can be used to represent them.	How does multiplication bring you down? How do people use fractions, decimals and percents in everyday life? How would our lives be different if we did not have fractions, decimals and percents? Why do you need common denominators when adding and subtracting fractions? Why is it necessary to show fractions in different forms? When can a half be more than a whole?
Knowledge and Skills What students are expected to know and be able to do
Students will know that… Whole numbers, fractions, decimals and integers may be modeled on number lines, scales, and the coordinate plane and used to solve problems. Place value patterns may be expressed using exponents to write powers of ten.	Students will be able to… Choose appropriate linear, area, and set models and pictures of fractions, decimals, mixed numbers, and improper fractions to locate, label, order, compare, round, and estimate values on number lines, coordinate grids, scales and measuring tools. Explore magnitude of decimal values by comparing 0.1 and 0.01 more and less than a given number. Compare large numbers using expanded forms and powers of ten. Use models and common factors to identify equivalent fractions and decimals. Locate, order, and compare whole numbers and integers on number lines, scales and the coordinate grid. Use absolute value to represent distance between two points on a number line. Use models, number patterns and common factors to rewrite a rational number in its equivalent fraction, decimal, ratio and percent forms and as powers of ten.
STAGE 2: DETERMINE ACCEPTABLE EVIDENCE
Performance Task(s) Authentic application in new context to evaluate student achievement of desired results designed according to GRASPS (Goal, Role, Audience, Setting, Performance, Standards)	Other Evidence Application that is functional in a classroom context only to evaluate student achievement of desired results
GOAL The goal is to rewrite recipes for a healthy meal for the entire Cobalt/Yellow team. ROLE You are responsible for creating the menu and rewriting the recipes. AUDIENCE The audience is the Cobalt/Yellow team. SETTING You need to create a healthy menu to be shared with your team. The recipes will have to be rewritten so there is enough food for the entire team. PRODUCT PERFORMANCE AND PURPOSE You need to take all of the ingredients and multiply accordingly to make enough for the entire team. STANDARDS Your recipe should be detailed with the new amounts needed. You need to show why the meal you have selected is healthy and a good choice for your team. Scoring Rubric: Score of 3: Student shows a correct and/or appropriate answer and shows work and/or an explanation that demonstrates full and complete understanding. Score of 2: Student work has minor flaws, but the work and/or explanation are acceptable and the reasoning is appropriate. Score of 1: Student does not write a reasonable explanation or show sufficient work, resulting in a demonstration of only limited understanding. Score of 0: Student shows no understanding of the problem or how to arrive at a solution.	Daily Homework Classroom Observations Quizzes Tests District Assessments State Assessments Projects Student Reflections Journal Entries

GRADE: 6

TOPIC: NUMERICAL AND PROPORTIONAL REASONING OPERATIONS

STAGE 1: IDENTIFY DESIRED RESULTS
Content Standard(s) Generalizations about what students should know and be able to do
Established Goals: State Standards: Use a variety of numerical representations in the base ten system to describe quantitative relationships. Use numbers and their properties to compute flexibly and fluently, and to reasonably estimate measures and quantities. National (NCTM) Standards: Understand numbers, ways of representing numbers, relationships among numbers, and number systems. Understand meanings of operations and how they relate to one another. Compute fluently and make reasonable estimates.
Enduring Understandings Insights learned from exploring generalizations via the essential questions (Students will understand THAT…)	Essential Questions Inquiry used to explore generalizations
1. There are appropriate times to use the operations when problem solving with fractions, decimals, percents and integers. 2. There is an inverse relationship among the operations. 3. Many real world problems involve multi-step operations, with and without extraneous information.	1. How does multiplication bring you down? 2. Why does “of” mean multiplication? 3. When is the “correct” mathematical answer not the best solution? 4. How do you know for sure that your solution is correct?
Knowledge and Skills What students are expected to know and be able to do
Students will know… The division interpretation of fractions can be used to write equivalent decimal forms. With fractions and decimals, products or quotients may be larger or smaller than either factor.	Students will be able to… Use models to explore the definition of division with decimals, fractions and mixed numbers. Explore place value patterns when multiplying and dividing decimals by 10, 100, 1000 and multiples of 10. Interpret finding a fractional part of a set as a two-step division and multiplication problem. Use the order of operations and algebraic properties (associative, commutative, distributive, inverse operations and additive and multiplicative identities) to estimate and solve multi-step problems. Recognize that multiplication by a unit fraction is equivalent to dividing by the fraction’s denominator. Create and solve a variety of problems involving fractions, decimals, mixed numbers, money, and simple percents.