• Course Description

In this course, 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.  Students engage with relevant mathematical contexts to make math meaningful.  As students encounter challenging problems, and practice asking for help and a willingness to learn from others, they build the social emotional learning competencies of self-management, self awareness, relationship skills, and social awareness.

Related Priority Standards (State &/or National): K-12 Mathematics Missouri Learning Standards

Essential Questions

• What do effective problem solvers do, and what do they do when they get stuck?
• How can I effectively explain my mathematical thinking and reasoning to others?
• What does it mean for two things to be proportionally related? How can you tell?
• How are percentages used to represent change?
• Which representations best help you make sense of certain mathematical scenarios?
• How can rational numbers be used to represent real world situations?
• Our world is really complex--how can we simulate parts of it to make better predictions?
• When is a sample not representative of a population?

Enduring Understandings/Big Ideas

• Any proportional relationship can be represented by an equation.
• A proportional relationship is a collection of equivalent ratios.
• Addition, subtraction, multiplication, and division can be extended to rational numbers.
• Circumference of a circle is proportional to its diameter.
• We can estimate probabilities of outcomes of chance experiments.

Course-Level Scope & Sequence (Units &/or Skills)

Unit 1: Scale Drawing

• Students will solve problems including scale drawings of real objects and geometric figures including actual lengths and areas from a scale drawing and reproducing the drawing at a different scale.
• Students will use visual models to build conceptual understanding of scale factor by examining the relationships among corresponding pints, sides, and angles of scaled copies.
• Students will measure precisely using rulers or informal measuring tools.
• Students will draw scaled copies, ensuring that angle measures are unchanged and side lengths are changed by a common factor.

Unit 2: Proportional Relationships

• Students will represent proportional relationships using equations.
• Students will write equations for the two ways a proportional relationship can be considered.
• Students will compare proportional and nonproportional relationships, focusing on the structure of the equation for each.
• Students will determine that proportional relationships are straight lines passing through the origin and categorize graphs as proportional or nonproportional.
• Students will interpret graphs of proportional relationships and reason that graphs can be used to compare constants of proportionality.
• Students will compute unit rates with complex fractions and with like or different units.
• Students will determine when two quantities are in a proportional relationship.
• Students will identify and or compute the constant of proportionality (unit rate).
• Students will explain what a point on the graph of a proportional relationship means in terms of the situation.
• Students will interpret the unit rate as the slope of the graph.
• Students will derive and write the equations y=mx + b (y-intercept, slope).

Unit 3: Measuring Circles

• Students will develop a formal definition of a circle and describe and measure a circle using the radius and diameter.
• Students will apply understanding of scaled figures and proportional relationships to explore the relationship between a square’s perimeter and diagonal length.
• Students will determine the perimeter of shapes composed of circular parts and solve for unknown lengths.
• Students will develop a formula for the area of a circle.
• Students will apply the area of a circle formula to solve problems involving the area of shapes composed by circular parts and polygons.

Unit 4: Proportional Relationships and Percentages

• Students will compute unit rates with complex fractions and with like or different units.
• Students will determine when two quantities are in a proportional relationship.
• Students will identify and or compute the constant of proportionality (unit rate).
• Students will explain what a point on the graph of a proportional relationship means in terms of the situation.
• Students will recognize that the graph of any proportional relationship will pass through the origin.
• Students will interpret the unit rate as the slope of the graph.
• Students will compare two different proportional relationships.
• Students will write the equations y=mx + b (y-intercept, slope).
• Students will understand that percentages are not whole numbers.
• Students will model problems involving percent increase and decrease with tape diagrams and write expressions to represent the scenarios.
• Students will use tape diagrams to make sense of problems involving percent increase and decrease to determine percent of change.
• Students will connect proportional relationships and percent change scenarios in order to write equations.
• Students will represent percent increase and decrease problems with equations, and use them to solve for various unknown values.
• Students will apply percent reasoning to contexts involving money - specifically sales tax, tips, and simple interest.
• Students will interpret and solve problems about real world situations involving proportional relationships and percent change.
• Students will determine missing measurements in proportional relationships involving fractional quantities or percentages.

Unit 5: Rational Number Arithmetic

• Students will add, subtract, multiply, and divide rational numbers.
• Students will represent addition and subtraction on horizontal and vertical number lines.
• Students will describe situations that have a number and its opposite with a sum of zero.
• Students will understand subtraction of rational numbers as adding the additive inverse.
• Students will determine the distance between two rational numbers on the number line as the absolute value of their difference.
• Students will interpret sums, differences, products, and quotients of rational numbers by describing real world context.
• Students will generate equivalent representations of rational numbers.
• Students will explore adding rational numbers and generalize rules about the sign of the sum.
• Students will subtract rational numbers to compare differences and notice that the order of subtraction changes the sign of the difference.
• Students will formalize that the product of a positive and negative number is negative, by relating multiplication to repeated addition.
• Students will perform all four operations with positive and negative numbers using a variety of strategies.
• Students will reason about variable expressions involving adding, subtracting, multiplying, and dividing signed numbers.
• Students will generalize patterns for the value of variable expressions.
• Students will apply understanding of the distance formula, d = rt, to make observations about the rules for multiplying rational numbers.
• Students will identify and create equivalent expressions involving positive, negative, and zero exponents.
• Students will express and perform operations with very large or very small quantities using powers of ten and scientific notation.
• Students will synthesize understandings of rational number arithmetic and interpret negative quantities such as rates of change.
• Students will solve equations of the form p + x = q and px = q with rational values.

Unit 6: Expressions, Equations, and Inequalities

• Students will find unknown values on balanced hanger diagrams that model two-step equations.
• Students will connect the Distributive Property with solving equations of the form p(x + q) = r, using hanger diagrams to assist.
• Students will practice solving equations of the form p(x + q) = r, focusing on the structure of the equation to determine the most efficient solution method.
• Students will build fluency in algebraic manipulation by solving a variety of equations.
• Students will use tape diagrams and equations of the form px + q = r and p(x + q) = r to describe relationships in real-world scenarios.
• Students will create tape diagrams, write equations, and solve real-world problems, focusing on equations of the form p(x + q) = r.
• Students will decide which type of equation, px + q = r or p(x + q) = r, describes the relationships in a real-world story problem.
• Students will represent percent increase and decrease using tape diagrams and equations.
• Students will solve inequalities of the forms px + q < r and p(x + q) < r by first writing and solving a related equation.
• Students will interpret and solve inequalities that represent real-world situations, while making sense of quantities and their relationships in the problem.
• Students will use the Distributive Property to expand and factor expressions that include subtraction and negative values.
• Students will use the properties of operations to understand how like terms can be combined to write an equivalent expression with fewer terms.
• Students will combine like terms to write equivalent expressions with fewer terms, now including negative coefficients and parentheses.

Unit 7: 2D and 3D Geometry

• Students will compose, decompose, and measure angles.
• Students will find an unknown angle given the measure of a supplementary or complementary angle.
• Students will calculate the surface area and volume of three-dimensional figures.
• Students will find the measures of non-adjacent supplementary and complementary angles and draw conclusions about the angle relationships of polygons.
• Students will write and solve equations of the form px + q = r and p(x + q) = r to represent angle relationships shown in diagrams.
• Students will examine and compare sets of triangles that share three common angle measures or side lengths.
• Students will use various tools to draw triangles, noticing that certain conditions determine how many unique triangles can be drawn.
• Students will draw triangles given one angle and two side lengths, or three angles.
• Students will calculate the volume of any right prism by multiplying the area of its base by its height.
• Students will determine fractions and decimal approximations for rational and irrational numbers.

Unit 8: Data, Statistics, and Probability

• Students will use data from multiple samples to draw inferences about a population and investigate variability in estimates of the characteristic of interest.
• Students will analyze different data distributions of statistical measures.
• Students will compare the numerical measures of center, measures of frequency, and measures of variability from two random samples to draw inferences about the population.
• Students will investigate the probability of chance events between zero and one.
• Students will predict outcomes using theoretical probability.
• Students will perform experiments that model theoretical probability.
• Students will compare and explain theoretical and experimental probabilities.
• Students will find probabilities of compound events using organized lists, tables, tree diagrams, and simulations.
• Students will compare and explain differences between data displays.
• Students will calculate percents.
• Students will determine probability of unknown events by comparing the results of repeated experiments and the expected probability.
• Students will explain the purpose of sampling and which methods of obtaining a sample tend to produce representative samples.

Course Resources & Materials: DESMOS

Date Last Revised/Approved: 2022