Course Description In, instructional time should focus on four critical areas: (1) developing understanding of and applying proportional relationships; (2) developing understanding of operations with rational numbers and working with expressions and linear equations; (3) solving problems involving scale drawings and informal geometric constructions, and working with two-and three-dimensional shapes to solve problems involving area, surface area, and volume; and (4) drawing inferences about populations based on samples. Course Rationale Scope and Sequence Timeframe Unit Instructional Topics 5 Day(s) Statistics and Probability 1. Random Sampling 2. Comparative Inferences 3. Chance Processes and Probability Models 10 Week(s) The Number System 1. Rational Numbers 2. Order of Operations 7 Week(s) Expressions and Equations 1. Expressions 2. Equations 10 Day(s) Geometry 1. Geometric Measurement 2. Angle Measurement 3. Geometric Figures (scaling) 4 Week(s) Ratios and Proportional Relationships 1. Ratios and Proportions In alignment with Common Core State Standards, the School District's Mathematics courses provide students with a solid foundation in number sense while building to the application of more demanding math concepts and procedures. The courses focus on procedural skills and conceptual understandings to ensure coherence and depth in mathematical practices and application to real world issues and challenges. The number system, with focus on rational numbers, provides the foundation of number sense used to solve a variety of real-world problems. Expressions and equations are used to simplify and solve mathematical situations. Geometry is used to draw, construct, and model real-life problems. Statistics and probability are used to draw inferences and compare populations as well as to evaluate probability models. Ratios and proportional relationships are used to analyze and solve real-world problems. Key Resources Glencoe McGraw Hill Math Connects Concepts Skills and Problem Solving Course 2 UNIT: Statistics and Probability -- 5 Day(s) Course Details In this unit students explore data that enables them to make predictions about a population and learn about possible outcomes of an event. Statistics can be used to communicate information about a population by representing data. Probability can be used to model, understand, and make predictions about real life situations. Chance is expressed as a number between 0 and 1. How can I effectively communicate collected data? How do I know which measure of central tendency to use when analyzing data? Course Summary Page 1 of 7
How can statistics and probability help me make predictions and decisions? Given a set of data, explain which measure of central tendency is the best. Data display Relative frequency Probability model Tree diagram Observed frequency TOPIC: Random Sampling -- 0 Day(s) The student will understand that generalizations about a population from a sample are valid only if the sample is representative of that population. MACC.CC.SP.1 MACC.NETS. NETS-S:3.4 The student will understand that random sampling tends to produce representative samples and support valid inferences. MACC.CC.SP.1 The student will understand that statistics can be used to gain information about a population by examining a sample of the population. GF.CCMA.SP.1 MA.CC.SP.1 MACC.CC.SP.1 The student will 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. GF.CCMA.SP.2 MA.CC.SP.2 MACC.CC.SP.2 TOPIC: Comparative Inferences -- 0 Day(s) The student will assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability. GF.CCMA.SP.3 MA.CC.SP.3 MACC.CC.SP.3 The student will use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. GF.CCMA.SP.4 MA.CC.SP.4 MACC.CC.SP.4 Course Summary Page 2 of 7
TOPIC: Chance Processes and Probability Models -- 0 Day(s) The student will 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. GF.CCMA.SP.6 MA.CC.SP.6 MACC.CC.SP.6 MACC.NETS. NETS-S:1.4 The student will 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 of the discrepancy - Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. - Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. GF.CCMA.SP.7 MA.CC.SP.7 MACC.CC.SP.7 The student will find probabilities of compound events using organized lists, tables, tree diagrams, and simulation. - Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs. - 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. - Design and use a simulation to generate frequencies for compound events. GF.CCMA.SP.8 MA.CC.SP.8 MACC.CC.SP.8 The student will understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event. GF.CCMA.SP.5 MA.CC.SP.5 MACC.CC.SP.5 UNIT: The Number System -- 10 Week(s) During this unit the main focus in the classroom will be developing number sense and computation. Integers can be manipulated and applied to situations when numbers less than zero must exist. Numbers can represent quantity and position; symbols may be used to express these relationships. Calculating a series of operations requires a specific order to be followed. How do integers give me an understanding of situations in my life? How is the number system used in different situations? Why is it important to understand order of operations? Given a multi-step problem including integers, students find the solution by using order of operations. Order of Operations Course Summary Page 3 of 7
TOPIC: Rational Numbers -- 0 Day(s) The student will apply and extend previous understandings of addition and subtraction to add and subtract rational numbers. GF.CCMA.NS.9 MA.CC.NS.9 MACC.CC.NS.9 The student will apply and extend previous understandings of multiplication, division, and of fractions to multiply and divide rational numbers. MACC.CC.NS.10 The student will represent addition and subtraction on a horizontal or vertical number line. MACC.CC.NS.9 TOPIC: Order of Operations -- 0 Day(s) The student will solve real-world and mathematical problems involving the four operations with rational numbers. MACC.CC.NS.11 UNIT: Expressions and Equations -- 7 Week(s) In this unit students write verbal phrases and sentences as mathematical expressions and equations. Expressions and equations can be represented in multiple ways. Real world situations are represented as expressions and equations using a combination of numbers, symbols, variables, and operations. The ability to assess the reasonableness of answers using mental computation and estimation strategies is essential. How can I make numbers more manageable? How does writing a situation as an expression or equation help me to solve a problem? Why is it important to consider the reasonableness of my answer and how do I do that? Variable TOPIC: Expressions -- 0 Day(s) The student will apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients. MACC.CC.EE.10 The student will 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. MACC.CC.EE.11 Course Summary Page 4 of 7
TOPIC: Equations -- 0 Day(s) The student will construct simple equations and inequalities to solve problems by reasoning about the quantities. MACC.CC.EE.13 The student will convert between forms as appropriate, assessing reasonableness of answers using mental computation and estimation strategies. MACC.CC.EE.12 The student will 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. GF.CCMA.EE.12 MA.CC.EE.12 MACC.CC.EE.12 The student will 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. MACC.CC.EE.11 The student will use variables to represent quantities in a real world or mathematical problem. - Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p and q, are r are specific rational numbers. Solve equations of these forms fluently. Compare and algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. -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. MACC.CC.EE.13 The student will apply properties of operations to calculate with numbers in any form. MACC.CC.EE.10 UNIT: Geometry -- 10 Day(s) MACC.CC.EE.12 In this unit students will create a scale model of different geometric shapes and know the formulas to calculate different geometric measurements. Figures can be composed and deconstructed into smaller, simpler figures. Attributes of shapes and solids can be uniquely measured in a variety of ways, using a variety of tools for a variety of purposes. Angle relationships can be used to solve simple equations where an angle measurement is unknown. What does a two-dimensional representation of a three-dimensional figure look like, and when would it be useful? How do I choose the appropriate formula and apply it? How can I describe and use the relationship between angles' measurements? Given a figure, students will draw a scale model. Circumference Complementary Angles Corresponding Angles Adjacent Angles Scaling Pi Scale Factor Similar Supplementary Angles Congruent Vertical Angles Course Summary Page 5 of 7
TOPIC: Geometric Measurement -- 0 Day(s) The student will 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. GF.CCMA.G.13 MA.CC.G.13 MACC.CC.G.13 MACC.NETS. NETS-S:6.1 The student will solve real-world and mathematical problems involving area, volume and surface area of two- and threedimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. GF.CCMA.G.15 MA.CC.G.15 MACC.CC.G.15 TOPIC: Angle Measurement -- 0 Day(s) The student will 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. GF.CCMA.G.14 MA.CC.G.14 MACC.CC.G.14 TOPIC: Geometric Figures (scaling) -- 0 Day(s) The student will describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids. GF.CCMA.G.12 MA.CC.G.12 MACC.CC.G.12 The student will 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. GF.CCMA.G.11 MA.CC.G.11 MACC.CC.G.11 MACC.NETS. NETS-S:1.3 MACC.NETS. NETS-S:4.2 The student will 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. GF.CCMA.G.10 MA.CC.G.10 MACC.CC.G.10 UNIT: Ratios and Proportional Relationships -- 4 Week(s) In in this unit students will use ratios and proportions to solve multi-step problems. Students apply proportional reasoning to percent problems to improve and enhance their ability to be wise consumers. Proportional reasoning can be used to model and understand relationships which exist throughout numerous aspects of the world. Proportions can be used as a strategy to solve for unknown quantities. How does proportional reasoning help me as a consumer? Why is it helpful for me to use proportions to solve real life problems? How can proportions be used to solve for unknown quantities? Given a real world problem involving a percent increase students will be able to use proportions to solve. Proportion Percent of Change Course Summary Page 6 of 7
TOPIC: Ratios and Proportions -- 0 Day(s) The student will compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction 1/2/1/4 miles per hour, equivalently 2 miles per hour. MACC.CC.RP.1 The student will recognize and represent proportional relationships between quantities. - Decide whether two quantities are in a proportional relationship, e.g., 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. - Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. - Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn. - Explain what a point (x,y) on the graph of a proportional relationship means in term of the situation, with special attention to the points (0,0) and (1,r) where r is the unit rate. MACC.CC.RP.2 The student will use proportional relationships to solve multi-step ratio and percent problems. GF.CCMA.RP.3 MA.CC.RP.3 MACC.CC.RP.3 Course Summary Page 7 of 7