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### 7.1 Number Sense, Properties, and Operations

- 7.1.1 Proportional reasoning involves comparisons and multiplicative relationships among ratios
- 7.1.1.a Analyze proportional relationships and use them to solve real-world and mathematical problems.
- 7.1.1.b Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units.
- 7.1.1.c Identify and represent proportional relationships between quantities.
- 7.1.1.c.i Determine whether two quantities are in a proportional relationship.
- 7.1.1.c.ii Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.
- 7.1.1.c.iii Represent proportional relationships by equations.
- 7.1.1.c.iv Explain what a point (x, y) on the 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.

- 7.1.1.d Use proportional relationships to solve multistep ratio and percent problems.
- 7.1.1.d.i Estimate and compute unit cost of consumables (to include unit conversions if necessary) sold in quantity to make purchase decisions based on cost and practicality.
- 7.1.1.d.ii Solve problems involving percent of a number, discounts, taxes, simple interest, percent increase, and percent decrease.

- 7.1.2 Formulate, represent, and use algorithms with rational numbers flexibly, accurately, and efficiently
- 7.1.2.a Apply understandings of addition and subtraction to add and subtract rational numbers including integers.
- 7.1.2.a.i Represent addition and subtraction on a horizontal or vertical number line diagram.
- 7.1.2.a.ii Describe situations in which opposite quantities combine to make 0.
- 7.1.2.a.iii Demonstrate 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.
- 7.1.2.a.iv Show that a number and its opposite have a sum of 0 (are additive inverses).
- 7.1.2.a.v Interpret sums of rational numbers by describing real-world contexts.
- 7.1.2.a.vi Demonstrate subtraction of rational numbers as adding the additive inverse, p – q = p + (-q).
- 7.1.2.a.vii 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.
- 7.1.2.a.viii Apply properties of operations as strategies to add and subtract rational numbers.

- 7.1.2.b Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers including integers.
- 7.1.2.b.i Apply properties of operations to multiplication of rational numbers.
- 7.1.2.b.ii Interpret products of rational numbers by describing real-world contexts.
- 7.1.2.b.iii Apply properties of operations to divide integers.
- 7.1.2.b.iv Apply properties of operations as strategies to multiply and divide rational numbers.
- 7.1.2.b.v Convert a rational number to a decimal using long division.
- 7.1.2.b.vi Show that the decimal form of a rational number terminates in 0s or eventually repeats.

- 7.1.2.c Solve real-world and mathematical problems involving the four operations with rational numbers.

- 7.1.2.a Apply understandings of addition and subtraction to add and subtract rational numbers including integers.

### 7.2 Patterns, Functions, and Algebraic Structures

- 7.2.1 Properties of arithmetic can be used to generate equivalent expressions
- 7.2.1.a Use properties of operations to generate equivalent expressions.
- 7.2.1.a.i Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.
- 7.2.1.a.ii Demonstrate 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.

- 7.2.1.a Use properties of operations to generate equivalent expressions.
- 7.2.2 Equations and expressions model quantitative relationships and phenomena
- 7.2.2.a Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form, using tools strategically.
- 7.2.2.b 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.
- 7.2.2.c 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.
- 7.2.2.c.i Fluently solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers.
- 7.2.2.c.ii Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach.
- 7.2.2.c.iii 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.
- 7.2.2.c.iv Graph the solution set of the inequality and interpret it in the context of the problem.

### 7.3 Data Analysis, Statistics, and Probability

- 7.3.1 Statistics can be used to gain information about populations by examining samples
- 7.3.1.a Use random sampling to draw inferences about a population.
- 7.3.1.a.i Explain that generalizations about a population from a sample are valid only if the sample is representative of that population.
- 7.3.1.a.ii Explain that random sampling tends to produce representative samples and support valid inferences.
- 7.3.1.a.iii Use data from a random sample to draw inferences about a population with an unknown characteristic of interest.
- 7.3.1.a.iv Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions.

- 7.3.1.b Draw informal comparative inferences about two populations.
- 7.3.1.b.i Informally 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.
- 7.3.1.b.ii Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations.

- 7.3.1.a Use random sampling to draw inferences about a population.
- 7.3.2 Mathematical models are used to determine probability
- 7.3.2.a Explain that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring.
- 7.3.2.b 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.
- 7.3.2.c Develop a probability model and use it to find probabilities of events.
- 7.3.2.c.i Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy.
- 7.3.2.c.ii Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events.
- 7.3.2.c.iii Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process.

- 7.3.2.d Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.
- 7.3.2.d.i Explain that the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs.
- 7.3.2.d.ii Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams.
- 7.3.2.d.iii For an event described in everyday language identify the outcomes in the sample space which compose the event.
- 7.3.2.d.iv Design and use a simulation to generate frequencies for compound events.

### 7.4 Shape, Dimension, and Geometric Relationships

- 7.4.1 Modeling geometric figures and relationships leads to informal spatial reasoning and proof
- 7.4.1.a Draw, construct, and describe geometrical figures and describe the relationships between them.
- 7.4.1.a.i 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.
- 7.4.1.a.ii Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions.
- 7.4.1.a.iii Construct triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.
- 7.4.1.a.iv Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.

- 7.4.1.a Draw, construct, and describe geometrical figures and describe the relationships between them.
- 7.4.2 Linear measure, angle measure, area, and volume are fundamentally different and require different units of measure
- 7.4.2.a State the formulas for the area and circumference of a circle and use them to solve problems.
- 7.4.2.b Give an informal derivation of the relationship between the circumference and area of a circle.
- 7.4.2.c Use properties of supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.
- 7.4.2.d 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.