21 st Century Standards

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1 ROCKAWAY TOWNSHIP PUBLIC SCHOOLS MATHEMATICS UNIT GUIDE GRADE 7 MATH PLUS Unit Title: Time Frame: First Marking Period Operations with Fractions and Rational Numbers Standard 7.NS Number System 8.NS Number System 21 st Century Standards 9.1.A Critical Thinking and Problem Solving 9.1.B Creativity and Innovation Enduring Understandings: Properties of whole numbers operations can be applied when solving operations with rational numbers. Strategies to represent and solve problems involving operations with rational numbers including fractions, decimals, and percents. Fluency with mental math and estimation facilitates efficient problem solving. Real numbers consist of both rational and irrational numbers. Essential Questions: Is there evidence of integers in our everyday lives? What kind of real-world experiences develop number sense? How is an understanding of rational and irrational numbers useful in problem solving? Cumulative Progress Indicator Number(s): 7.NS.1 Apply and extend previous understanding of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. a. Describe situations in which opposite quantities combine to make 0. b. Understand p + q as the number located a distance q from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts. c. Understand subtraction of rational numbers as adding the additive inverse, p q = p + (-q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts. d. Apply properties of operations as strategies to add and subtract rational numbers. 7.NS.2 Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.

2 a. Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts. b. Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers, then (p/q) = (- p)/q = p/(-q). Interpret quotients of rational numbers by describing real-world contexts. c. Apply properties of operations as strategies to multiply and divide rational numbers. d. Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats. 7.NS.3 Solve real-world and mathematical problems involving the four operations with rational numbers including complex fractions. 8.NS.1 Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats into a rational number. Unit Learning Targets: Suggested Activities: Vocabulary The student will be able to. Model addition of integers Add integers Model subtraction of integers Subtract integers Solve problems by looking for patterns Model multiplication and division of integers Multiply and divide integers Graph rational numbers on a number line Write fractions as terminating or repeating decimals and write decimals as fractions Compare and order rational numbers Add and subtract fractions with Including Differentiated Strategies (DI) Have students model an expression using positive and negative integers Use integer counters to represent zero pairs Use a common fraction such as ¾ when showing students how to write a fraction as a decimal using division (DI) Use visual problem solving models and shapes to model fractions Connect integers to real life situations in bank accounts, science, scuba diving, etc Use number cubes or cards to create various fractions to multiply and divide Have students work in groups to find decimal equivalents of rational numbers Zero pair Opposites Additive inverse Terminating decimal Repeating decimal Bar notation Rational numbers Irrational numbers Common denominators Least common denominator Like fractions Unlike fractions

3 like denominators Use models to add and subtract fractions with unlike denominators Add and subtract fractions with unlike denominators Add and subtract mixed numbers Model and perform multiplication of fractions and mixed numbers Solve problems by drawing a diagram Divide fractions and mixed numbers Resources and Technology Integration Illuminations.nctm.org/lessons.aspx Mathwire.com Easiteach Interactive Whiteboard Online text resources TWAIN Edhelper.com Kutasoftware Shodor.org Assessments Quizzes Formal assessments Renaissance assessments Participation Homework Rubric specific activities and projects Various math text assessments Teacher created assessments Open-Ended assessments Math journal Teacher observation during cooperative learning groups

4 ROCKAWAY TOWNSHIP PUBLIC SCHOOLS MATHEMATICS UNIT GUIDE GRADE 7 MATH PLUS Unit Title: Time Frame: First and Second Marking Periods Numerical and Algebraic Expressions and Equations Standard 7.EE Expressions and Equations 8.EE Expressions and Equations 21 st Century Standards 9.1.A Critical Thinking and Problem Solving 9.1.B Creativity and Innovation Enduring Understandings: Properties of Operations and Order of Operations are useful to simplify, evaluate, or find equivalent expressions. Inverse operations are used to solve equations and inequalities. Solutions to an equation/inequality are the values of the variables that make the equation/inequality true. Exponents can be used to represent numbers of varying magnitude. Essential Questions: What do the best problem solvers do to get answers? In what order should the operations be performed when simplifying an expression? How can algebraic symbols be used to efficiently express mathematical situations? How are equations and inequalities useful? How do you solve equations? Cumulative Progress Indicator Number(s): 7.EE.1 Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients. 7.EE.2 Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. 7.EE.3 Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. 7.EE.4 Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities. a. Solve word problems leading to equations in the form px + q = r and p(x+q) = r, where p, q, and r are specific rational

5 numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. b. Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. 8.EE.1 Know and apply the properties of integer exponents to generate equivalent numerical expressions. Unit Learning Targets: The student will be able to. Use powers and exponents Evaluate expression using the order of operations Evaluate simple algebraic expressions Use Commutative, Associative, Identity, and Distributive properties to solve problems Solve problems by looking for a pattern Describe the relationships and extend terms in arithmetic sequences Explore patterns in sequences of geometric figures Write and solve one-step equations using a variety of methods Write and solve two-step equations using a variety of methods Write and solve equations with variables on both sides using a variety of methods Model and solve inequalities using Suggested Activities: Including Differentiated Strategies (DI) Model algebraic expressions using classroom objects Write a multi-step numerical expression on the IWB, highlight the symbol of the operation to be performed first and then perform the operation, then continue until the problem is evaluated(di) Use balance scale to model and solve one step equations Use equations to model a story Play number cube games with equations or inequalities Number card games with equations of inequalities Card game 24 Students make posters displaying examples of Commutative, Associative, and Identity and Distributive Properties Place parentheses in correct location in an equation to make it true Use algebra tiles to model and solve multi-step equations Vocabulary Factors Exponent Base Powers Squared Cubed Evaluate Standard form Exponential form Numerical expression Order of Operations Variable Algebra Algebraic expression Coefficient Define a variable Equivalent expressions Properties Sequence Term Arithmetic sequence Geometric sequence Equation

6 a variety of methods Use percent diagrams to solve problems Find percent of a number Estimate percents by using fractions and decimals Solve problems involving percents Solve problems using percent proportions or the percent equation Solve problems by determining reasonable answers Find percent of increase or decrease Solve problems involving discount, sales tax and tip Solve problems involving simple interest and compound interest Solve real-world percent application word problems Resources and Technology Integration Illuminations.nctm.org/lessons.aspx Mathwire.com Easiteach Interactive Whiteboard Online text resources TWAIN Edhelper.com Kutasoftware Shodor.org Equivalent equation Coefficients Formula Multiplicative inverse Reciprocal Two-step equations Inequality Percent proportion Percent equation Discount Sales tax Tip Gratuity Percent of change Percent of increase Percent of decrease Principal Simple interest Compound interest Assessments Quizzes Formal assessments Renaissance assessments Participation Homework Rubric specific activities and projects Various math text assessments Teacher created assessments Open-Ended assessments Teacher observation during cooperative learning groups

7 Unit Title: Ratios and Proportional Relationships Standard 7.RP Ratios and Proportional Relationships 8.EE Expressions and Equations ROCKAWAY TOWNSHIP PUBLIC SCHOOLS MATHEMATICS UNIT GUIDE GRADE 7 MATH PLUS Time Frame: Second Marking Period 21 st Century Standards 9.1.A Critical Thinking and Problem Solving 9.1.B Creativity and Innovation 9.2.E Becoming a Critical Consumer Enduring Understandings: There are various strategies for deciding if a relationship is proportional (for example equivalent ratios in a table, observing points graphed on a coordinate plane, analyzing ratios for equivalence, etc). Proportional relationships can be represented symbolically (equation), graphically (coordinate plane), in a table, in diagrams, and verbal descriptions. Ratio and proportional reasoning strategies can be extended and applied to multi-step ratio and percent problems. Slope of a line is a constant rate of change. Essential Questions: How are ratios and proportions used in the real-world? What is the connection between a ratio and a fraction? What is the difference between a proportional and nonproportional relationship? When and why do I use proportional comparisons? How does comparing quantities describe the relationship between them? Cumulative Progress Indicator Number(s): 7.RP.1 Compute unit rates associated with ratios and fractions, including ratios of lengths, areas and other quantities measured in like or different units. 7.RP.2 Recognize and represent proportional relationships between quantities. a. Decide whether two quantities are in a proportional relationship by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. b. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. c. Represent proportional relationships by equations.

8 d. Explain what a point (x,y) on a graph of a proportional relationship means in terms of the situation, with special attention to the points (0,0) and (1,r) where r is the unit rate. 7.RP.3 Use proportional relationships to solve multistep ratio and percent problems. 8.EE.5 Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. Unit Learning Targets: Suggested Activities: Vocabulary The student will be able to. Model proportions using bar diagrams Determine unit rates Identify proportional and nonproportional relationships Use proportions to solve problems Use proportions to estimate populations Solve problems by drawing a diagram Solve problems involving scale drawings Use a spreadsheet to calculate measurements for scale drawings Solve problems by using a graph Compare and contrast proportional and nonproportional linear functions Use direct variation to solve problems Find slope of a line Including Differentiated Strategies (DI) Compare and contrast linear relationships that are proportional and nonproportional (DI) Model a unit rate using counters as the numerator and denominator and a pencil as a fraction bar (DI) Find the dimensions of their desktops and create scale drawings on grid paper using specified scales Collect, measure, and draw scale models of objects (DI) Use interactive web sites and virtual manipulatives Compare and contrast ratios of figures created by the students Use a tape measure or yardstick to find the height of a tree, flagpole, or other tall object Draw a diagram of the problem, write the proportion, and find the missing height Write a journal entry on how students use a measurable object to check a proportion Use clear sheet protectors or whiteboards to create a template to find the percent, part, and whole using is over of equals percent over one hundred (DI) Rate Unit rate Proportional Nonproportional Equivalent ratios Proportion Cross products Scale drawing Scale model Scale Scale factor Direct variation Constant of variation Slope-intercept form y-intercept

9 Resources and Technology Integration Illuminations.nctm.org/lessons.aspx Mathwire.com Easiteach Interactive Whiteboard Online text resources TWAIN Edhelper.com Kutasoftware Shodor.org Assessments Quizzes Formal assessments Renaissance assessments Participation Homework Rubric specific activities and projects Various math text assessments Teacher created assessments Open-Ended assessments Math journal Teacher observation during cooperative learning groups

10 Unit Title: Statistics and Probability Standard 7.SP Statistics and Probability 8.SP Statistics and Probability ROCKAWAY TOWNSHIP PUBLIC SCHOOLS MATHEMATICS UNIT GUIDE GRADE 7 MATH PLUS Time Frame: Third Marking Period 21 st Century Standards 9.1.A Critical Thinking and Problem Solving 9.1.B Creativity and Innovation 9.2.E Becoming a Critical Consumer Enduring Understandings: Probability is the likelihood of an event occurring. Theoretical and experimental probabilities and proportional reasoning are used to make predictions about a population. Multiplication of fractions can be used to find probabilities of compound events. What it means to have a valid, random sample representative of a population. Data displays are used to visually compare data sets and draw informal comparative inferences. The line of best fit represents the data set as a whole. Essential Questions: How can math help you make good decisions? How can counting the possible outcomes of compound events be used to determine their probability? How can predictions be made based on data or probability? How do people use data to influence others? What kind of relationships can be found when comparing two sets of data? Cumulative Progress Indicator Number(s): 7.SP.1 Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences. 7.SP.2 Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. For example, estimate the mean word length in a book by randomly sampling words from the book. 7.SP.3 Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between centers by expressing it as a multiple of a measure of variability.

11 7.SP.4 Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. 7.SP.5 Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Large numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around ½ indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event. 7.SP.6 Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. For example, when rolling a number cube 600 times, predict that 3 or 6 would be rolled roughly 200times, but probably not exactly 200 times. 7.SP.7 Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources for the discrepancy. a. Develop a uniform probability to all outcomes, and use the model to determine probabilities of events. b. Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. 7.SP.8 Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation. a. Understand that, just as with simple events, the probability of a compound is the fraction of outcomes in the sample space for which the compound event occurs. b. Represent sample spaces for compound events using methods such as organized lists, tables, and tree diagrams. For an event described in everyday language (e.g., rolling double sixes ). Identify the outcomes in the sample space which compose the event. c. Design and use a simulation to generate frequencies for compound events. 8.SP.2 Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line. Unit Learning Targets: Suggested Activities: Vocabulary The student will be able to. Find the probability of a simple event Find sample spaces and probabilities Use multiplication to count outcomes and find probabilities Find the number of permutations in a set of objects Explore and find the probabilities Including Differentiated Strategies (DI) Calculate the probability of events using number cubes and spinners (DI) Calculate experimental and theoretical probabilities using coins and number cubes Create data sets to calculate the mean, median, and mode Use graphing calculators and spreadsheet software to find the mean, median, and mode of a data set Outcome Simple event Probability Random Complementary event Geometric probability Sample space Tree diagram

12 of independent and dependent events Find and compare experimental and theoretical probabilities Solve problems by conducting simulations Use experimental and theoretical probabilities to decide whether a game is fair or unfair Predict actions of a larger group using a sample Compare data about two populations by using measures of center and measures of variability Gain an intuitive understanding of how to construct scatter plots and how to write an equation of the line of best fit Create foldables with the math vocabulary terms for mean, median, mode, and range (DI) Compare two populations using the measures of center and variation Create a scatter plot, graph and write an equation of best fit line and use that equation to make predictions Odds in favor Counting principle Permutation Combination Compound event Independent events Dependent events Disjoint events Theoretical probability Experimental probability Fair game Unfair game Survey Population Sample Unbiased sample Simple random sample Biased sample Interquartile range (IQR) Mean absolute deviation (MAD) Variability Scatter-plot

13 Resources and Technology Integration Illuminations.nctm.org/lessons.aspx Mathwire.com Easiteach Interactive Whiteboard Online text resources TWAIN Edhelper.com Kutasoftware Shodor.org Assessments Quizzes Formal assessments Renaissance assessments Participation Homework Rubric specific activities and projects Various math text assessments Teacher created assessments Open-Ended assessments Math journal Teacher observation during cooperative learning groups

14 ROCKAWAY TOWNSHIP PUBLIC SCHOOLS MATHEMATICS UNIT GUIDE GRADE 7 MATH PLUS Unit Title: Geometrical Figures Relationships Time Frame: Fourth Marking Period Geometrical Figures Relationships Standard 7.G Geometry 8.G Congruence, Similarity, Pythagorean Theorem, and Volume 21 st Century Standards 9.1.A Critical Thinking and Problem Solving 9.1.B Creativity and Innovation Enduring Understandings: Two and three dimensional objects can be described, classified, and analyzed by their attributes. The relationship between dimensions of a scale drawing and the original figure is proportional. Areas of polygons can be found by decomposing them into triangles and/or other shapes and rearranging or removing shapes. There is a relationship between the volumes of cylinders, cones, and spheres. Reflections, rotations, and translations are rigid transformations and maintain congruence. Essential Questions: How does geometry shape our lives? How do we describe, sort, and classify shapes? How is area related to circumference? How are shapes related to one another? How do geometric models describe spatial relationships? How can finding the area of a composite figure be helpful in real-world? What is the difference between volume and surface area? What are real-world situations in which you need to know volume or surface area? What is the connection between the volume of cylinders and cones? How can a coordinate grid be used to model and describe the results of various transformations? Cumulative Progress Indicator Number(s): 7.G.1 Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale. 7.G.2 Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.

15 7.G.3 Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids. 7.G.4 Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle. 7.G.5 Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure. 7.G.6 Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes and right prisms. 8.G.3 Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. 8.G.9 Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems. Unit Learning Targets: Suggested Activities: Vocabulary The student will be able to. Justify formulas for the volume of prisms Find the volume of rectangular and triangular prisms Find the volumes of cylinders Justify formulas for the volume of pyramids and cones Find the volume of pyramids Find the volume of cones Find the surface area of prisms and cylinders using models and nets Find the surface area of prisms Find the surface area of a cylinder Compare surface area and volume of rectangular prisms and cylinders Find the surface area of pyramids Justify the formula for the surface Including Differentiated Strategies (DI) Create a rectangular prism using centimeter cubes to determine the length, width, and height (DI) Have students use centimeter cubes to create at least two different prisms that have the same value Students can use an empty box to create a net to find the surface area of prisms (DI) Use centimeter cubes to create composite figures (DI) Groups create two or more composite figures with the same surface area and find the volume of each figure, then compare and contrast their findings Use graph paper to estimate the area of a composite figure Use an animated picture and graph paper to transform (translations, reflections, rotation, dilation) the picture in the coordinate plane Have students bring in a cylindrical object to determine its volume Scale drawing Scale model Scale Scale factor Prism Volume Rectangular prism Triangular prism Cylinder Precision Pyramid Lateral face Cone Cylinder Sphere Net Surface area Slant height

16 area of a cone by using a net Solve problems by solving a similar problem Explore and find volume and surface area of composite figures Determine how changes in dimensions affect area and volume Solve problems involving similar solids Classify and identify angles and find missing measures Demonstrate that the sum of the angles in a triangle is Identify and classify triangles and find missing angle measures Classify quadrilaterals and find missing angle measures Find the sum of the angle measures of a polygon and the measure of an interior angle of a regular polygon Graph transformations on the coordinate plane Find volume of cylinders, cones, and spheres Lateral surface area Composite figure Similar solids Angle Straight angle Vertical angles Adjacent angles Complementary angles Supplementary angles Alternate interior angles Alternate exterior angles Corresponding angles Parallel lines Transversal Triangle Congruent segments Acute triangle Right triangle Obtuse triangle Scalene triangle Isosceles triangle Equilateral triangle Quadrilateral Rectangle Square Parallelogram Rhombus Trapezoid Polygon Pentagon

17 Hexagon Heptagon Octagon Nonagon Decagon Equilateral Equiangular Regular polygon Translations Rotations Reflections Dilations Transformations Resources and Technology Integration Illuminations.nctm.org/lessons.aspx Mathwire.com Easiteach Interactive Whiteboard Online text resources TWAIN Edhelper.com Kutasoftware Shodor.org Geometers Sketchpad Assessments Quizzes Formal assessments Renaissance assessments Participation Homework Rubric specific activities and projects Various math text assessments Teacher created assessments Open-Ended assessments Math journal Teacher observation during cooperative learning groups

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