Grade 7 Math

Contents Page

GRADE: 8

TOPIC: ALGEBRAIC REASONING: PATTERNS AND FUNCTIONS ALGEBRA 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: Algebraic methods (tables, graphs and equations) can be used to solve real-world problems. A relation is a mapping from one set of values to another, and a function is a relation in which there is only one value of the dependent variable that corresponds to each value of the independent value. Given one representation or function, other representations can be derived. A common solution to two linear equations is shown graphically by the intersection of their lines. National (NCTM) Standards: Represent, analyze, and generalize a variety of patterns with tables, graphs, words, and, when possible, symbolic rules. Relate and compare different forms of representation for a relationship. Identify functions as linear or nonlinear and contrast their properties from tables, graphs, or equations. Represent and analyze mathematical situations and structures using algebraic symbols. Develop an initial conceptual understanding of different uses of variables. Explore relationships between symbolic expressions and graphs of lines, paying particular attention to the meaning of intercept and slope. Use symbolic algebra to represent situations and to solve problems, especially those that involve linear relationships. Recognize and generate equivalent forms for simple algebraic expressions and solve linear equations. Use mathematical models to represent and understand quantitative relationships. Model and solve contextualized problems using various representations, such as graphs, tables, and equations. Analyze change in various contexts. Use graphs to analyze the nature of changes in quantities in linear relationships.
Enduring Understandings Insights earned from exploring generalizations via the essential questions (Students will understand THAT…)	Essential Questions Inquiry used to explore generalizations
1. Algebraic skills and concepts, including functions, can be used to describe real-world phenomena symbolically and graphically 2. Functions can used to solve many problems 3. Algebra and Geometry are intimately connected	How do functions function? What do dependent variables depend on? How variable is the relationship between Geometry and Algebra?
Knowledge and Skills What students are expected to know and be able to do
Students will know… How to define: function, slope, intersect, coordinate, origin, x-axis, y-axis, variable, dependent variable, independent variable, linear, non-linear, y-intercept, x-intercept, system of equations. That algebraic skills and concepts can used to model quantitative change. That patterns, relations and functions are all inter-connected.	Students will be able to… Use tables, graphs and equations to represent mathematical relationships and solve real-world problems. Identify functions as linear or non-linear, compare and contrast their graphs and equations, and use them to solve problems. Recognize the relationships between points of intersection, parallelism, perpendicularity, intercepts and systems of equations, both on the coordinate plane and in equation form. Represent and analyze mathematical and real-world situations using algebraic symbols. Recognize that on the coordinate plane, lines with the same slope are parallel and lines with different slopes intersect; and, lines either meet at one point, at no points or infinitely many points. Use a graphing calculator to represent and to describe a linear function with tables, patterns, graphs and equations. Explore solving problems involving direct variation.
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---decide which of three options is best (cheapest) for copying a math textbook to be piloted by your school Role---math coordinator for the school system Audience---board of education, budget subcommittee Situation---before purchasing new textbooks, school systems always test them out (called piloting). This is down by providing copies of the proposed book to a handful of teachers to try it out. The Board of Education wants to pilot a new 325 page math book, of which it has obtained one free copy in advance form the publisher. We are unsure at present just how many copies we need, so assume this amount is an unknown (i.e. a variable). There exist three choices for making copies: 1. A local printing company can do it for an initial cost of $5000 for typesetting and an additional $9.50 per book. 2. A local copy center can do it for $0.05 per page and an additional $2.00 per book, for binding. 3. The Town Hall's copy center can do it for $0.035 per page and an upfront fee of $3000. Product Performance and Purpose---Create a detailed plan explaining which option is the cheapest Standards and Criteria for Success---Report must include: 3 tables, one for each option, as well as an algebraic equation representing each; further, a graph of the three lines of these equations in the x-y coordinate system show slopes, y-intercepts and line intersections, and explain/interpret how they are related to our problem Finally, which option is best, and why? WHERETO is an acronym checklist for building and evaluating the final learning plan It is not a suggested sequence. W – where are we going/why are we learning this H – “hook” and “hold” question E – what prerequisite skills are needed R – how can students reflect on and refine their skills E – methods of evaluation T – how to include a variety of learning styles O – general sequence of the unit W – Understanding functions is critical to successful completion of any high school math course H – How do functions function? E – Algebraic substitution, concept of variable, computation skills, inverse operations R – Writing in journals and discussing with peers E – Homework, quizzes, writing assignments, group activities, in-class discussions, projects T – Use of manipulatives, open-ended questions, both visual and written examples O - Each algebraic equation in two variables can also be represented graphically on an x-y coordinate grid; geometric properties can be analyzed and understood algebraically; linear relationships, slope, intercepts and coordinate graphs are all useful in solving many types of real-world problems.	Homework assignments, daily practice problems, quizzes Weekly journal writing assignment Unit tests and quizzes District Assessments State Assessments

GRADE: 8

TOPIC: ALGEBRAIC REASONING: PATTERNS AND FUNCTIONS PATTERNS

STAGE 1: IDENTIFY DESIRED RESULTS
Content Standard(s) Generalizations about what students should know and be able to do
Established Goals: State Standards: A relation is a mapping from one set of values to another, and a function is a relation in which there is only one value of the dependent variable that corresponds to each value of the independent value. Given one representation or function, other representations can be derived. National (NCTM) Standards: Represent, analyze and generalize a variety of patterns with tables, graphs, words, and when possible, symbolic rules. Solve problems using scale factors, using ratio and proportion.
Enduring Understandings Insights earned from exploring generalizations via the essential questions (Students will understand THAT…)	Essential Questions Inquiry used to explore generalizations
1. Patterns can be used to describe, discover and analyze mathematical and real-world phenomena. 2. Patterns can be used to solve a variety of problems. 3. Patterns can be numerical and/or geometrical.	1. How are patterns described? 2. How are representations used to describe patterns?
Knowledge and Skills What students are expected to know and be able to do
Students will know… How to define and recognize and analyze: pattern, sequence, iteration, symbolic rules.	Students will be able to… Represent, analyze and generalize a variety of patterns with tables, graphs, words or symbolic rules.
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 - To solve the "chess board problem" as proposed to the King of Persia. Role - An advisor to the King. Audience - The King of Persia. Situation - A peasant farmer and servant in the ancient kingdom of Persia has created a game called "Chess" at the request of the King. The king is so enthralled with the game that he offers the servant/farmer any reward of his choosing. The servant asks for grain (the most valuable commodity in this ancient kingdom) from the Kings' warehouses, in the following amount: place one grain on the first square of the 8 x 8 chessboard, then place two grains on the second square, 4 on the 3^rd, double that amount on the 4^th, etc…keep doubling the amount for each subsequent square. As advisor to the King, your job is to decide if the King should honor this request. Product Performance and Purpose - Decide what the King should do, and create a graph or table of mathematical information that supports your view. Make sure to include any and all patterns you perceive, particularly any that involve exponents. Standards and Criteria for Success---A written answer for the King with clear information to backup your decision. WHERETO is an acronym checklist for building and evaluating the final learning plan It is not a suggested sequence. W – where are we going/why are we learning this H – “hook” and “hold” question E – what prerequisite skills are needed R – how can students reflect on and refine their skills E – methods of evaluation T – how to include a variety of learning styles O – general sequence of the unit W – Understanding patterns and relationships is an integral part of mathematics and is an important pretense to working with functions. Students will understand numerical and geometric patterns. H - Is there a pattern to everything? E – Methods of counting, math computations, ratio and proportion and knowledge of geometry are all important aspects to working with patterns. R – Writing about patterns in addition to working with them symbolically and computationally provides an avenue to further understanding and assessment. E – Class work, homework, quizzes and group activities. T – A variety of activities can be offered to students, including individual work, group work, writing assignments and projects. O – Students will be exposed to particular examples of patterns and functions that gradually increase in complexity and difficulty.	Homework assignments, daily practice problems, quizzes Weekly journal writing assignment Unit tests and quizzes District Assessments State Assessments

GRADE: 8

TOPIC: GEOMETRY AND MEASUREMENT

MEASUREMENT

STAGE 1: IDENTIFY DESIRED RESULTS
Content Standard(s) Generalizations about what students should know and be able to do
Established Goals: State Standards: Relationships exist among sides, angles, perimeters, areas, surface areas and volumes of congruent and similar polygons and solids. Indirect measures of volume for some solids can be found through the use of formulas. Direct measure of the volume of irregular solid objects can be accomplished through the use of displacement. The Pythagorean Theorem can be used to find an unknown length. Understand relationships among units and convert from one unit to another within the same system. National (NCTM) Standards: Understand both metric and customary systems of measurement. Understand, select, and use units of appropriate size and type to measure angles, perimeter, area, surface area, and volume. Use common benchmarks to select appropriate methods for estimating measurements. Develop and use formulas to determine the circumference of circles and the area of triangles, parallelograms, trapezoids, and circles and develop strategies to find the area of more complex shapes. Select and apply techniques and tools to accurately find length, area, volume and angle measure to appropriate levels of precision. Develop strategies to determine the surface area and volume of selected prisms, pyramids and cylinders.
Enduring Understandings Insights earned from exploring generalizations via the essential questions (Students will understand THAT…)	Essential Questions Inquiry used to explore generalizations
1. There exist different systems of measurement and that it is possible to convert from one system to another. 2. There is a difference between direct and indirect measurement and when each is most applicable. 1. 3. Various units of measurement exist and are appropriate for different categories of measurement (i.e. angles, areas, lengths, etc.).	1. How do we measure up? (Or down, or sideways, or backward, or forward) 2. Is equal always the same? 3. How does what we measure change how we measure? 4. When is it better to use an exact measurement rather than an estimate?
Knowledge and Skills What students are expected to know and be able to do