1 Thinking Proportionally

Transcription

1 Thinking Proportionally Course, Module, Topic : Circles and Ratio Students learn formulas for the circumference and area of circles and use those formulas to solve mathematical and real-world problems. Students also learn that the irrational number pi (π) is the ratio of a circle s circumference to its diameter. Standard: 7.G. Pi: The Ultimate Ratio Exploring the Ratio of Circle Circumference to Diameter 7.G. Students explore the relationship between the distance around and the distance across various circles. They notice that for every circle the ratio of the circumference to diameter is pi. That's a Spicy Pizza! Area of Circles 7.G. Students explore the area of a circle in terms of its circumference. They derive the area for a circle and then solve problems using the formulas for the circumference and area for circles Circular Reasoning Solving Area and Circumference Problems 7.G. Students use the area of a circle formula and the circumference formula to solve for unknown measurements in real-world and mathematical problem. 7.G. Students practice solving problems involving area and circumference of circles. Accelerated Grade 7 Textbook:

2 Course, Module, Topic : Fractional Rates Students calculate and use unit rates from ratios of fractions. They review strategies for solving proportions and then use means and extremes to solve real-world proportion problems. Standards: 7.RP., 7.RP..c, 7.RP. Eggzactly! Solving Problems with Ratios of Fractions 7.RP. Students determine ratios and write rates, including complex ratios and rates. Students will write proportions and use rates to determine miles per hour. They will scale up and scale down to determine unknown quantities. Tagging Sharks Solving Proportions Using Means and Extremes 7.RP..c 7.RP. Students solve several proportions embedded in real world contexts. Several proportions that contain one variable are solved using one of three methods: the scaling method, the unit rate method, and the means and extremes method. Students learn to isolate a variable in an equation by using inverse operations. 7.RP. 7.RP..c Students determine and compare unit rates. They solve proportions using equivalent ratios and means and extremes. Accelerated Grade 7 Textbook:

3 Course, Module, Topic : Proportionality Students differentiate between proportional and non-proportional relationships, including linear relationships that are not proportional. They identify and use the constant of proportionality from tables, graphs, equations, and real-world situations; represent proportional relationships with equations; and explain the meaning of points on the graph of a proportional relationship. Standard: 7.RP. How Does Your Garden Grow? Proportional Relationships 7.RP..a Students explore graphs and tables of proportional and non-proportional relationships. They determine that the graphs of proportional relationships are straight lines that pass through the origin. They also learn that tables of proportional relationships have a constant ratio of corresponding values of the quantities. Students learn the term direct variation and relate it to proportional relationships. Complying with Title IX Constant of Proportionality 7.RP..b 7.RP..c Students explore equations of proportional relationships. They determine the constant of proportionality, the constant ratio of the outputs to the inputs in a proportional relationship. Students explore the reciprocal relationship of constants of proportionality in equations. They use the constant of proportionality to write and solve equations. Fish-Inches Identifying the Constant of Proportionality in Graphs 7.RP..b 7.RP..d Students analyze real world and mathematical situations, both proportional and non-proportional, represented on graphs and then identify the constant of proportionality when appropriate. Throughout the lesson, students interpret the meaning of points on graphs in terms of a proportional relationship, including the meaning of (, y) and (0, 0). Minding Your Ps and Qs Constant of Proportionality in Multiple Representations 7.RP. Students use proportional relationships to create equivalent multiple representations, such as diagrams, equations, tables, and graphs of the situation. A proportional relationship may initially be expressed using only words, or a table of values, or an equation, or a graph. 7.RP..a 7.RP..b 7.RP..c Students write ratios and determine the constant of proportionality in real-world problems. They practice determining the constant of proportionality, writing equations, and drawing a line to represent the direct variation equation to solve problems. Students are given graphs to determine if it represents a direct variation. Accelerated Grade 7 Textbook:

4 Course, Module, Topic : Proportional Relationships Students use proportions and percent equations to solve real-world problems about money and scale drawings. They use multiple representations to solve and compare percents. Then students use proportionality to solve problems with scale drawings and scale factors. Standards: 7.RP., 7.G.6, 7.G. Markups and Markdowns Introducing Proportions to Solve Percent Problems 7.RP. Students analyze strategies for determining the unknown value in a percent problem. Students use proportions to solve percent problems. They connect percent problems with direct variation and proportional relationships. Perks of Work Calculating Tips, Commission, and Simple Interest 7.RP. Students solve proportions and percent equations in the context of tipping and commissions. They analyze both strategies as they determine the amount of a tip or commission, the percent tip or commission, and the total sale when given the percent and the tip or commission amount. No Taxation Without Calculation Sales Tax, Income Tax, and Fees 7.RP. Students use percents to solve sales tax, income tax, and fee problems. They identify the percent relationship between two amounts as a proportional relationship, with a unit rate and constant of proportionality. More Ups and Downs Percent Increase and Percent Decrease 7.RP. 7.G.6 Students compute percent increase and percent decrease in several situations. They apply percent increase and decrease to solving problems involving geometric measurement. 5 Pound for Pound, Inch for Inch Scale and Scale Drawings 7.G. Students use scale models to calculate measurements and enlarge and reduce the size of models. They enlarge or reduce the size of objects and calculate relevant measurements, explore scale drawings, and describe the meaning of several different scales. Students then determine which scale will produce the largest and smallest drawing of an object when different units of measure are given. 7.RP. 7.G. Students practice converting between fractions, decimals, and percents. They solve percent problems for the part, the percent, or the whole, and solve percent change problems. Students use scale factors to determine unknown measures given real-life situations. Accelerated Grade 7 Textbook:

5 Operating with Signed Numbers Course, Module, Topic : Adding and Subtracting Rational Numbers Students use physical motion, number lines, and two-color counters to develop conceptual understanding of adding and subtracting integers. They develop rules for these operations and apply the rules to the set of rational numbers. Standards: 7.NS., 7.NS. Walk the Line Adding Integers, Part 7.NS..b Students explore patterns for adding two integers using a number line. They focus on the absolute values of the numbers being added and develop informal rules for adding integers. Two-Color Counters Adding Integers, Part 7.NS..a 7.NS..b Students use two-color counters to develop rules for adding integers. They model adding positive and negative integers with the two-color counters. Students use a graphic organizer to represent how to add additive inverses using a variety of representations. What's the Difference? Subtracting Integers 7.NS..c Students use number lines and two-color counters to model subtraction of signed numbers. They develop and apply rules for subtracting integers. 5 All Mixed Up Adding and Subtracting Rational Numbers 7.NS. Students apply their knowledge of adding and subtracting positive and negative integers to the set of rational numbers. 7.NS. Students practice adding and subtracting integers using a number line. Accelerated Grade 7 Textbook: 5

6 Course, Module, Topic : Multiplying and Dividing Rational Numbers Students use number lines and two-color counters to model and develop rules for the signs of the products and quotients of signed numbers. They convert rational numbers from fractional to decimal form. Then students apply rules and properties to the set of rational numbers. Standards: 7.NS..a, 7.NS..b, 7.NS..d, 7.NS., 7.RP. Equal Groups Multiplying and Dividing Integers 7.NS..a 7.NS. Students use number lines and two-color counters to model the product of two integers. They use the models to develop rules for multiplying integers. Using fact families, students apply rules for the multiplication of integers to the division of integers. Be Rational! Quotients of Integers 7.NS..b 7.NS..d Students write the quotients of integers as fractions and decimals. They learn that the decimal representation of rational numbers terminates or repeats. Building a Wright Brothers' Flyer Simplifying Expressions to Solve Problems 7.NS. 7.RP. Students solve real-world problems involving numeric expressions and signed rational numbers, including problems about percent error. Properties Schmoperties Using Number Properties to Interpret Expressions with Signed Numbers 7.NS..d 7.NS..c 7.NS. Students apply the distributive property to expanding and factoring with - and learn that subtraction is the same as adding the opposite. They identify properties in expressions with rational coefficients. 7.NS. 7.NS. Students use fact families to explore dividing integers. They then practice simplifying a variety of numeric expressions using order of operations. Accelerated Grade 7 Textbook: 6

7 Course, Module, Topic : Algebraic Expressions Students explore algebraic expressions with rational coefficients. They apply the Distributive Property as a strategy to write equivalent expressions and to factor linear expressions. Students combine like terms, including like linear terms, and use properties of operations to add and subtract expressions. Standards: 7.EE., 7.EE., 7.EE. No Substitution for Hard Work Evaluating Algebraic Expressions 7.EE. Students review variables, algebraic expressions, and evaluating algebraic expressions. They practice evaluating expressions with rational numbers. Mathematics Gymnastics Rewriting Expressions Using the Distributive Property 7.EE. Students rewrite algebraic expressions with rational coefficients using the Distributive Property. They then expand linear expressions. Students will factor linear expressions in a variety of ways, including by factoring out the greatest common factor and the coefficient of the variable. All My Xs Combining Like Terms 7.EE. 7.EE. Students simplify expressions by combining like terms, with integer, fraction, and decimal coefficients. They use properties to simplify the expressions. Students add and subtract algebraic expressions, using addition of the opposite to subtract. 7.EE. Students model the product of two factors and explore different factors of expressions. They then use the Distributive Property to factor and expand expressions. Students simplify variable expressions by combining like terms, by using number properties, and by using order of operations. Accelerated Grade 7 Textbook: 7

8 Course, Module, Topic : Two-Step Equations and Inequalities Students use bar models and double number lines to reason about and solve two-step linear equations. Then they use inverse operations to fluently and efficiently solve two-step equations with rational coefficients. Finally, students investigate, solve, and graph two-step inequalities. Standard: 7.EE. Picture Algebra Modeling Equations by Equal Expressions 7.EE..a Students use bar models to write expressions and equations and to solve for unknown quantities. Expressions That Play Together Solving Equations on a Double Number Line 7.EE..a Students model contextual and mathematical situations using double number lines. Formally Yours Using Inverse Operations to Solve Equations 7.EE..a Students formalize the process and language of solving two-step equations. They refer to the Properties of Equality and use inverse operations flexibly to solve equations. Students consider efficient strategies for solving problems with different types of rational numbers. They practice solving two-step equations. Be Greater Than Solving Inequalities with Inverse Operations 7.EE..b Students solve inequalities. They compare solving equations with solving inequalities. Students informally develop the Properties of Inequalities before analyzing them formally. They then solve and graph inequalities. 7.EE. Students create visual models to represent real-world situations and solve using reasoning. They then write and solve expressions from a given real-world problem using tables. Students solve a variety of two-step equations and inequalities using formal strategies. Accelerated Grade 7 Textbook: 8

9 Course, Module, Topic : Multiple Representations of Equations Students use tables, graphs, verbal descriptions, and scenarios to write and analyze two-step linear equations and inequalities. They make connections across representations, interpreting graphs and equations in terms of the problem situation. Standards: 7.EE., 7.EE. Put It on the Plane Representing Equations with Tables and Graphs 7.EE..a Students analyze linear equations using tables and graphs. They write and solve equations, create tables of values, and create graphs of the situations. They use the graphs to answer questions about the situations. Students explain if the linear situations represent proportional relationships. Stretches, Stacks, and Structure Structure of Linear Equations 7.EE. 7.EE..a Students write and solve equations for more complicated contexts. They use tables to create equations that require the use of the expression (n ) to represent the quantity of the independent variable except for the initial value. They compare the two forms of the same equation and relate the equations to the graphs. Deep Flight I Building Inequalities and Equations to Solve Problems 7.EE. Students work with a negative rate of change. They use negative values to create a table and graph. Students then write and analyze an equations and inequalities with negative unit rates of change. Texas Tea and Temperature Using Multiple Representations to Solve Problems 7.EE. Students solve equations using tables of values, graphs, and equations. In each activity, a different representation is presented and students use that representation to solve problems. 7.EE. Students write and solve equations and inequalities to solve real-world problems. They model and analyze graphs of linear equations to solve and interpret real-world problems. Accelerated Grade 7 Textbook: 9

10 Constructing, Measuring, and Transforming Objects Course, Module 5, Topic : Angles and Triangles Students learn about formal constructions and use construction tools to duplicate segments and angles. Students explore and use different pairs of angles including supplementary angles, complementary angles, vertical angles, and adjacent angles. Finally, students use both patty paper and formal construction tools to determine if given information defines a unique triangle, multiple triangles, or no triangles. Standards: 7.G., 7.G.5 Here's Lookin' at Euclid Geometric Constructions 7.G. Students are introduced to geometry and geometric constructions. Measuring tools are distinguished from construction tools, and the concepts of sketch, draw, and construct are differentiated. Students will learn how to properly draw, sketch, and name each of the essential building blocks of geometry. They use a compass to construct circles and arcs and to duplicate line segments and angles using only construction tools. Special Delivery Special Angle Relationships 7.G.5 Students use protractors and patty paper to explore special angle pairs formed when two lines intersect. They use the definitions and write and solve equations about special angle pairs. Consider Every Side Constructing Triangles Given Sides 7.G. Students use pasta, patty paper, and construction tools to determine the number of segments and the conditions needed to construct a unique triangle, more than one triangle, or no triangles. Unique or Not? Constructing Triangles Given Angles 7.G. Students use patty paper and construction tools to determine what information is needed to construct a unique triangle, more than one triangle, or no triangles when given at least one angle. 7.G.5 Students measure angles and determine angle sums. They then identify complementary, supplementary, and vertical angles. Students write and solve equations to solve for unknown angle measures. Accelerated Grade 7 Textbook: 0

11 Course, Module, Topic : Rigid Motion Transformations Students use patty paper and the coordinate plane to investigate the creation of congruent figures with translations, reflections, and rotations. They use patty paper to develop intuition about the properties of the transformations, and then make generalizations about the coordinates of figures after transformations. Students determine sequences of transformations that map congruent figures to each other. Standards: 8.G., 8.G., 8.G. Patty Paper, Patty Paper Introduction to Congruent Figures 8.G..a 8.G..b 8.G. Students use patty paper to investigate congruent figures. They make conjectures about congruence, investigate their conjectures, and justify their conjectures using informal transformation language. Slides, Flips, and Spins Introduction to Rigid Motions 8.G..a 8.G..b 8.G..c 8.G. Students develop a formal understanding of translations, rotations, and reflections in the plane. They use patty paper to investigate each transformation, create images from pre-images, and determine the properties of each transformation. They learn that each rigid motion transformation preserves the size and shape of the original figure, and that translations and rotations also preserve the orientation of the figure. Lateral Moves Translations of Figures on the Coordinate Plane 8.G. 8.G. Students build on their patty paper exploration of slides, or translations. They investigate translations on the coordinate plane, conjecturing about the coordinates of images having been translated. At the end, students reason that figures that undergo translations (pre-images) are congruent to the resulting images. Mirror, Mirror Reflections of Figures on the Coordinate Plane 8.G. 8.G. Students build on their patty paper exploration of flips, or reflections. They investigate reflections on the coordinate plane, conjecturing about the coordinates of images having been reflected. Students use reflections to show that two figures are congruent. 5 Half Turns and Quarter Turns Rotations of Figures on the Coordinate Plane 8.G. 8.G. Students build on their patty paper exploration of turns, or rotations. They investigate rotations on the coordinate plane, conjecturing about the coordinates of images having been rotated. Students use rotations of 90 and 80 degrees to show that two figures are congruent. 6 Every Which Way Combining Rigid Motions 8.G. 8.G. Students use coordinates to determine the rigid motion or rigid motions used to map from one congruent figure to another. They write congruence statements for congruent triangles. Students generalize the effects of rigid motions on the coordinates of figures. 8.G. 8.G. Students describe the translations, rotations, and reflections needed to map a pre-image onto a congruent image. Accelerated Grade 7 Textbook:

12 Course, Module, Topic : Similarity Students investigate dilations and similarity. They make connections between scale factors and dilation factors and define similar figures. Students dilate figures on the coordinate plane using different locations for the center of dilation, and generalize the coordinates of images formed from a dilation with a center at the origin. Then they identify a sequence of transformations that map from a figure to a similar figure. Standards: 8.G., 8.G. Pinch-Zoom Geometry Dilations of Figures 8.G. Students review terms associated with scale factors and relate them to dilations and similarity. They learn that dilations create similar figures, with congruent corresponding angles and equal ratios of their corresponding side lengths. Rising, Running, Stepping, Scaling Dilating Figures on the Coordinate Plane 8.G. 8.G. Students investigate dilations on the coordinate plane, using different locations of the center of dilation. They create and modify conjectures about the effect of dilations from different centers on the coordinates of a figure. Students transformations to verify that two figures are similar. From Here to There Mapping Similar Figures using Transformations 8.G. Students determine similarity using a single dilation, and verify similarity through a sequence of transformations. They explore the relationship between images of a common pre-image under different conditions and the relationship between figures similar to congruent figures. 8.G. 8.G. 8.G. Students dilate triangle given a center of dilation and a scale factor and describe the dilations needed to map a preimage onto an image. Students describe the transformations needed to map a pre-image onto an image. Accelerated Grade 7 Textbook:

13 Course, Module, Topic : Line and Angle Relationships Students use their knowledge of transformations, congruence, and similarity to establish the triangle sum theorem, the exterior angle theorem, relationships between angles formed when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. They use hands-on tools to make and justify conjectures. Once results are justified, students solve related problems, including ones with complex diagrams. Standard: 8.G.5 Pulling a One-Eighty! Triangle Sum and Exterior Angle Theorems 8.G.5 Students explore and justify the Triangle Sum Theorem and the Exterior Angle Theorem. They also investigate the relationship between interior angle measures and the side lengths of a triangle. Then students practice applying the theorems. Crisscross Applesauce Angle Relationships Formed by Lines Intersected by a Transversal 8.G.5 Students identify the angles formed when two lines are intersected by a transversal. They determine that when the two lines are parallel, special angle pairs are either congruent or supplementary. Then they solve problems using the parallel line and angle relationships. The Vanishing Point The Angle-Angle Similarity Theorem 8.G.5 Students establish and use the Angle-Angle Theorem to show two triangles are similar. Students then determine if triangles in complex diagrams are similar using the AA Theorem. 8.G.5 Students identify and classify angle pairs in a given figures containing lines cut by transversals. Students use the Angle-Angle Similarity Theorem to verify that images are similar. Accelerated Grade 7 Textbook:

14 Course, Module, Topic : From Proportions to Linear Relationships Students connect proportional relationships, lines, and linear equations. They compare proportional relationships, review the constant of proportionality, k, as a rate of change, and assign the new term slope to the rate of change for any line. They connect the equation for proportional relationships to the equation, y = mx, where m is the slope of the line. From there, students derive the equation y = mx + b for a line that passes through the point 0, b rather than the origin. Standards: 8.EE.5, 8.EE.6, 8.G..a, 8.G..c Post-Secondary Proportions Representations of Proportional Relationships 8.EE.5 Students use ratios to analyze proportional relationship representations. They write equivalent ratios, write equations, create a table of values, and graph proportional relationships. Then, students analyze two scenarios involving proportional relationships represented in different forms. Jack and Jill Went Up the Hill Using Similar Triangles to Describe the Steepness of a Line 8.EE.5 8.EE.6 Students connect unit rate, constant of proportionality, and scale factor with slope, which is introduced as the rate of change of the dependent quantity compared to the independent quantity. Students derive the equations for a proportional linear relationship, y = mx, and for a non-proportional linear relationship, y = mx + b. Slippery Slopes Exploring Slopes using Similar Triangles 8.EE.6 Students use similar triangles to explain why the slope of any line is the same between any two distinct points on a non-vertical line. Students explore the slope of y = x and y = x 5 by drawing right triangles on the lines, verifying that the triangles are similar, and show the slope of any two points on the non-vertical line is the same. Up, Down, and All Around Transformations of Lines 8.EE.6 8.G..a 8.G..c Students apply geometric transformations to the basic function, y = x. They recognize y = mx as a dilation of y = x and y = mx + b as a translation of y = mx. Students learn that lines with the same slope are parallel and that parallel lines remain parallel after a reflection or rotation. 8.F. Students determine linear expressions that represent real-world context. They use these expressions to solve problems. Accelerated Grade 7 Textbook:

15 Modeling Linear Equations Course, Module, Topic : Solving Linear Equations Students solve equations with rational coefficients and variables on both sides of the equals sign. They learn strategies to efficiently solve equations with rational number coefficients. Students write and solve equations to represent real-world situations. Students learn to recognize, algebraically, when equations have one solution, no solution, or infinite solutions. Then they use given expressions to form equations with specific solution sets. Standard: 8.EE.7 Strategic Solving Equations with Variables on Both Sides 8.EE.7.b Students solve equations with the same variable on both sides of the equals sign. They use combining like terms, the Properties of Equality, the additive inverse, and the Distributive Property to solve. Students explore strategies to convert fractions and decimals in equations to whole numbers. MPs and DVDs Analyzing and Solving Linear Equations 8.EE.7.a Students write algebraic expressions within the context of different situations. They will then use the expressions to write equations and solve the equations for unknown values. Students interpret solutions and determine when equations have one solution, no solutions, or infinitely many solutions. Tic-Tac-Bingo Creating Linear Equations 8.EE.7.a 8.EE.7.b Students use their knowledge of solving linear equations to create linear equations with one solution, no solution, or infinite solutions. They play Tic-Tac-Bingo as they work together to create equations with given solution types from assigned expressions. Then students summarize the strategies they used to create the equations. 8.EE.7.a 8.EE.7.b Students practice solving equations with variables on both sides, including equations with one solution, no solutions, and infinitely many solutions. Accelerated Grade 7 Textbook: 5

16 Course, Module, Topic : Systems of Linear Equations Students analyze and solve pairs of simultaneous linear equations by graphing, through substitution, and by inspection of the coefficients of the equations in the systems. Students write systems of equations to represent problem situations, solve the systems, interpret the meaning of the solution in the context of the situation. They solve systems with no solutions and infinite solutions graphically and algebraically, making connections between the equations, graphs, and final solution. To build fluency with solving systems of linear equations, students write and solve additional systems, using the structure of the equations in the system to determine the most efficient solution strategy. Standard: 8.EE.8 Crossing Paths Point of Intersection of Linear Graphs 8.EE.8.a Students compare and analyze cost and income equations graphically and algebraically. They graph cost and income equations on the same graph to determine a point of intersection and interpret the point of intersection as the solution to two equations. Finally, students compare determining a point of intersection from a table alone with doing so using equations and a graph. The Road Less Traveled Systems of Linear Equations 8.EE.8.b Students write and analyze linear systems of equations. They informally calculate the solutions to systems of linear equations and then graph the systems of equations. Students conclude when parallel lines comprise the system, the lines will never intersect, so there is no solution to the system. They then determine whether the graphs of linear systems of equations are parallel, perpendicular, or neither through analysis of their slopes. The County Fair Using Substitution to Solve Linear Systems 8.EE.8.a 8.EE.8.b 8.EE.8.c Students learn to use the substitution method to solve systems of linear equations. They use substitution to solve systems of linear equations including systems with no solution or with infinite solutions. They define variables, write systems of equations, solve systems, and interpret the meaning of the solution in terms of the problem context. Rockin' Roller Rinks Choosing a Method to Solve a Linear System 8.EE.8.a 8.EE.8.b 8.EE.8.c Students compare the cost of holding a middle school skating event at three different locations. They write equations for each location and compare the cost for different numbers of skaters, by solving systems of equations, completing tables of values, and creating graphs. 8.EE.8.a 8.EE.8.b 8.EE.8.c Students practice graphing and solving systems of linear equations. Accelerated Grade 7 Textbook: 6

17 5 Analyzing Populations and Probabilities Course, Module, Topic : Introduction to Probability Students conduct probability experiments with familiar objects and determine theoretical and experimental probabilities of simple events. They learn about using uniform and non-uniform probability models to organize the probabilities of the outcomes in a sample space. Students use proportional reasoning to predict expected frequencies of favorable outcomes in larger samples and to calculate the percent error between theoretical and experimental probabilities. They also use simulation with a variety of tools to simulate the results of experiments. Standards: 7.RP., 7.SP.5, 7.SP.6, 7.SP.7 Rolling, Rolling, Rolling Defining and Representing Probability 7.SP.5 Students conduct an experiment that involves rolling one six-sided number cube. They calculate probabilities by rolling number cubes, using spinners, and drawing marbles from a bag. Give the Models a Chance Probability Models 7.SP.7 Using tables and dot plots, students construct and interpret uniform and non-uniform probability models comparing experimental probabilities to theoretical probabilities. Toss the Cup Determining Experimental Probability of Simple Events 7.SP.6 7.SP.7.b 7.RP. Students conduct probability experiments and compute experimental probabilities. They compare the experimental and theoretical probabilities and use proportional reasoning to predict frequencies and calculate percent error. A Simulating Conversation Simulating Simple Experiments 7.SP.6 7.SP.7.a Students use simulations to explore the relationship between the theoretical probability and the experimental probability as the number of trials increases. 7.SP.5 7.SP.6 7.SP.7 Students build probability models and determine probabilities of simple and disjoint events. They then use proportions to make predictions based on samples and theoretical probabilities. Students use results of probability experiments to make conjectures about theoretical probabilities. Accelerated Grade 7 Textbook: 7

18 Course, Module, Topic : Compound Probability Students use arrays, lists, and tree diagrams to organize the possible outcomes of an experiment. They create probability models, calculate experimental and theoretical probabilities of events, and use proportional reasoning to determine percent error and to make predictions of expected numbers of outcomes. Then students learn about compound events that use the conjunctions and and or. Students then design and conduct simulations for three compound probability problems. Standards: 7.SP.6, 7.SP.7, 7.SP.8 Evans or Odds? Using Arrays to Organize Outcomes 7.SP.6 7.SP.7.b 7.SP.8.a 7.SP.8.b Students use organized lists and arrays to organize outcomes of simple events requiring data from two sources (i.e., sum of numbers on number cubes, product of two spins of a numbered spinner). They use the arrays to determine probabilities and expected values. Three Girls and No Boys? Using Tree Diagrams 7.SP.7 7.SP.8.b Students use tree diagrams to illustrate a sample space and to create a probability model. They use tree diagrams to determine probabilities. Pet Shop Probability Determining Compound Probability 7.SP.7.b 7.SP.8 7.SP.8.a 7.SP.8.b Students use lists and tree diagrams to list outcomes of compound events. They then determine probabilities of compound events, contrasting "and" and "or" compound events. On a Hot Streak Simulating Probability of Compound Events 7.SP.6 7.SP.8.c Students design and conduct simulations of compound events. They choose the simulation tool of their choice in two activities, and they use a random number table in another activity. 7.SP.8 Students use simulation, tree diagrams, organized lists, and tables to determine compound probabilities. Accelerated Grade 7 Textbook: 8

19 Course, Module, Topic : Drawing Inferences Students use random samples to collect representative data from a specified population. They use the results of the sample and proportional reasoning to estimate population parameters. Then students use data displays and measures of center and variation to compare populations and random samples to draw inferences about populations or to compare two populations. Standards: 7.SP., 7.SP., 7.SP., 7.SP. We Want to Hear From You! Collecting Random Samples 7.SP. Students review the statistical process and begin learning elements of rigorous data collection, include generating representative and random samples. Tiles, Gumballs, and Pumpkins Using Random Samples to Draw Inferences 7.SP. 7.SP. Students use statistical information gathered from a sample along with proportional reasoning to determine a parameter for a population. They learn that statistics obtained from random samples are more likely to represent the parameter of the population than non-random samples. Dark or Spicy? Comparing Two Populations 7.SP. Students calculate the measures of center and measures of variability for two different populations. They plot the data and compare the measures of center with respect to the measure of variation. Finding Your Spot to Live Using Random Samples from Two Populations to Draw Conclusions 7.SP. 7.SP. Students use random samples to draw conclusions about two populations. They use means and mean absolute deviations and medians and interquartile ranges. 7.SP. Students compare the characteristics of data displays, specifying which numerical characteristics can be determined from each display. They then use data displays to compare populations by determining the visual overlap and describing the difference between the measures of centers in terms of measures of variability. Accelerated Grade 7 Textbook: 9

20 6 Applying Powers and Moving to the Third Dimension Course, Module, Topic : The Real Number System Students build onto their knowledge of number systems to include the set of irrational numbers, specially square roots and cube roots of non-perfect squares and cubes. They review natural numbers, whole numbers, integers, and rational numbers and determine which systems have an additive identity, additive inverse, multiplicative identity, or multiplicative inverse and which are closed under the four basic arithmetic operations. They learn to convert rational numbers from repeating decimals to fractional form. Students then determine square roots and cube roots of perfect squares and perfect cubes and use rational approximations to estimate the value of irrational numbers, including solutions to equations. Standards: 8.NS., 8.NS., 8.EE. Rational Decisions Rational and Irrational Numbers 8.NS. Students learn formal definitions for rational and irrational numbers. They conclude the set of rational numbers is closed and includes the set of whole numbers, the set of integers, some fractions, and some decimals. Then students write fractions as repeating decimals and convert terminating and repeating decimals to fractions. What Are Those?! The Real Numbers 8.NS. 8.EE. Students calculate square roots of perfect square and cube roots of perfect cubes. They locate irrational numbers on a number line between two rational numbers. Then they summarize the relationships among the sets of numbers in the real number system. 8.NS. 8.NS. 8.EE. Students determine perfect squares and their square root. They determine decimal approximations of square roots of non-perfect squares. Students classify real numbers as rational or irrational. Students practice comparing and ordering real numbers on a number line. Accelerated Grade 7 Textbook: 0

21 Course, Module 5, Topic : Exponents and Scientific Notation Students learn and apply properties of integer exponents. They explore exponent rules with positive and negative bases, including products of powers, quotients of powers, and powers of powers, and develop rules for each operation. Then they determine rules for 0 and negative exponents. Students then explore scientific notation. They learn to express numbers in standard form in scientific notation and those in scientific notation in standard form. Students multiply, divide, add, and subtract numbers expressed in scientific notation, making connections to the exponent rules. Then they compare the relative sizes of and operate on numbers expressed in scientific notation and standard form. Standards: 8.EE., 8.EE., 8.EE. It's a Generational Thing Properties of Powers with Integer Exponents 8.EE. Students write and evaluate expressions with positive integer exponents. They begin with a context using the power with a base of. Students then investigate positive and negative integer bases where the negative sign may or may not be raised to a power depending on the placement of parentheses. Show What You Know Analyzing Properties of Powers 8.EE. Students organize and discuss properties of powers. They write mathematical justifications for each step within a solution path using the properties of powers. Students solve additional practice problems and examine student work for correctness. The Big and Small of It Scientific Notation 8.EE. 8.EE. Students are introduced to scientific notation. They convert from standard form to scientific notation and from scientific notation to standard form and begin comparing numbers written in scientific notation. How Much Larger? Operations with Scientific Notation 8.EE. 8.EE. Students perform operations on numbers represented in scientific notation with and without context. They connect the Product Rule of Powers and the Quotient of a Power Rule with scientific notation. Students perform operations with numbers written in scientific notation. They also compare and operate on numbers expressed in standard form or scientific notation. 8.EE. 8.EE. 8.EE. Students practice simplifying mathematical expressions using the rules of exponents. Students convert between and compare numbers written in standard form and scientific notation. Accelerated Grade 7 Textbook:

22 Course, Module 5, Topic : Three-Dimensional Figures Students create and describe cross-sections of right rectangular prisms and right rectangular pyramids. They determine the areas of regular polygons through decomposition. Then, they calculate the volumes and surface areas of right prisms and pyramids. Standards: 7.G., 7.G.6 Slicing and Dicing Cross-Sections of Rectangular Prisms 7.G. Students explore cross-sections of cubes and general right rectangular prisms. They determine how to make each of six different cross-section results with each solid. Dissecting a Pyramid Cross-Sections of Rectangular Pyramids 7.G. Students explore cross-sections of right rectangular pyramids. They determine how to make three different crosssection results with the pyramids. Hey, Mister, Got Some Bird Seed? Volume of Pyramids 7.G.6 Students use nets of an open rectangular prism and an open rectangular pyramid with congruent bases and heights to investigate the volume of pyramids. They use the volume formulas to solve problems involving prisms and pyramids. Students then investigate the effect that doubling and tripling dimensions of prisms and pyramids has on their volumes. The Sound of Surface Area Surface Area of Pyramids 7.G.6 Students compare two different pieces of acoustical foam one that is made up of square pyramids and one that is made up of triangular prisms and determine their surface areas. They use formulas to calculate surface area and volumes of rectangular and triangular prisms and pyramids. 5 More Than Four Sides of the Story Volume and Surface Area of Prisms and Pyramids 7.G.6 Students use the strategies for calculating the volumes and surface areas of right rectangular prisms and pyramids to calculate the volumes and surface areas of prisms and pyramids with non-rectangular bases. They also develop a strategy to calculate the areas of regular polygons. 7.G.6 Students calculate the volume of pyramids in mathematical and real-world contexts. Accelerated Grade 7 Textbook:

23 Course, Module 5, Topic : Volume of Curved Figures Students derive formulas for the volumes of cylinders, cones, and spheres. They apply each formula to mathematical and real-world problems. In some cases, students must use the Pythagorean Theorem to determine needed measures. Finally, students compute the volumes of composite figures and compare the volumes of cones and spheres, of two cylinders, and of cones and cylinders as they solve problems. Standard: 8.G.9 Drum Roll, Please! Volume of a Cylinder 8.G.9 Students explore how to determine the volume of a cylinder. They identify characteristics of a cylinder such as the radius, diameter, and height. Using circular discs, they calculate the volume of the cylinder. Students then answer several questions related to the formula used to compute the volume of right and oblique cylinders. Cone of Silence Volume of a Cone 8.G.9 Students learn how to calculate the volume of a cone and how to solve problems involving cones. They use the formulas for the volume of a cylinder and the volume of a pyramid to write formulas for the volume of a cone. They solve problems when given different dimensions of a cone, including the slant height. Pulled in All Directions Volume of a Sphere 8.G.9 Students investigate the volume of a sphere. They derive and then use the formula for the volume of a sphere to calculate the volume of various spheres. Silos, Frozen Yogurt, and Popcorn Volume Problems with Cylinders, Cones, and Spheres 8.G.9 Students review all of the formulas they have learned up to this point and use them to solve real-world and mathematical problems. Students determine the volume of grain needed to fill a silo, the volume of a cone and a melted scoop of yogurt, and the volume of cylindrical and rectangular-prism shaped popcorn containers. 8.G.9 Students apply the formulas for the volume of a cylinder, cone, and sphere to solve a variety of problems. Accelerated Grade 7 Textbook: