1: Number Sense, Properties, and Operations

1.1: Make both relative (multiplicative) and absolute (arithmetic) comparisons between quantities. Multiplicative thinking underlies proportional reasoning

1.1: Proportional reasoning involves comparisons and multiplicative relationships among ratios

1.1.a: Students can: Analyze proportional relationships and use them to solve real-world and mathematical problems.

Beam to Moon (Ratios and Proportions)
Direct and Inverse Variation
Estimating Population Size
Geometric Probability
Part-to-part and Part-to-whole Ratios
Proportions and Common Multipliers

1.1.b: Students can: Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units.

Beam to Moon (Ratios and Proportions)
Household Energy Usage
Road Trip (Problem Solving)
Unit Conversions

1.1.c: Students can: Identify and represent proportional relationships between quantities.

1.1.c.i: Determine whether two quantities are in a proportional relationship.

Beam to Moon (Ratios and Proportions)
Direct and Inverse Variation
Estimating Population Size
Geometric Probability
Part-to-part and Part-to-whole Ratios
Percents and Proportions
Proportions and Common Multipliers

1.1.c.ii: Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.

Beam to Moon (Ratios and Proportions)
Dilations
Direct and Inverse Variation
Perimeters and Areas of Similar Figures
Similar Figures

1.1.c.iii: Represent proportional relationships by equations.

Beam to Moon (Ratios and Proportions)
Direct and Inverse Variation
Geometric Probability
Part-to-part and Part-to-whole Ratios
Polling: Neighborhood
Proportions and Common Multipliers
Theoretical and Experimental Probability

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.

Direct and Inverse Variation

1.1.d: Students can: Use proportional relationships to solve multistep ratio and percent problems.

Beam to Moon (Ratios and Proportions)
Estimating Population Size
Part-to-part and Part-to-whole Ratios
Percent of Change
Percents and Proportions
Percents, Fractions, and Decimals
Polling: Neighborhood
Proportions and Common Multipliers

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.

Household Energy Usage

1.1.d.ii: Solve problems involving percent of a number, discounts, taxes, simple interest, percent increase, and percent decrease.

Compound Interest
Percent of Change
Percents and Proportions
Polling: Neighborhood

1.2: Are fluent with basic numerical and symbolic facts and algorithms, and are able to select and use appropriate (mental math, paper and pencil, and technology) methods based on an understanding of their efficiency, precision, and transparency

1.2: Formulate, represent, and use algorithms with rational numbers flexibly, accurately, and efficiently

1.2.a: Students can: Apply understandings of addition and subtraction to add and subtract rational numbers including integers.

1.2.a.i: Represent addition and subtraction on a horizontal or vertical number line diagram.

Adding and Subtracting Integers
Adding on the Number Line
Fractions Greater than One (Fraction Tiles)

1.2.a.ii: Describe situations in which opposite quantities combine to make 0.

Adding and Subtracting Integers
Adding and Subtracting Integers with Chips
Integers, Opposites, and Absolute Values
Rational Numbers, Opposites, and Absolute Values

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.

Adding and Subtracting Integers
Adding on the Number Line
Improper Fractions and Mixed Numbers
Integers, Opposites, and Absolute Values
Rational Numbers, Opposites, and Absolute Values
Simplifying Algebraic Expressions I
Simplifying Algebraic Expressions II
Solving Algebraic Equations I
Sums and Differences with Decimals

1.2.a.iv: Show that a number and its opposite have a sum of 0 (are additive inverses).

Adding and Subtracting Integers
Adding on the Number Line
Improper Fractions and Mixed Numbers
Integers, Opposites, and Absolute Values
Rational Numbers, Opposites, and Absolute Values
Simplifying Algebraic Expressions I
Simplifying Algebraic Expressions II
Solving Algebraic Equations I
Sums and Differences with Decimals

1.2.a.v: Interpret sums of rational numbers by describing real-world contexts.

Adding and Subtracting Integers
Adding on the Number Line
Improper Fractions and Mixed Numbers
Integers, Opposites, and Absolute Values
Rational Numbers, Opposites, and Absolute Values
Simplifying Algebraic Expressions I
Simplifying Algebraic Expressions II
Solving Algebraic Equations I
Sums and Differences with Decimals

1.2.a.vi: Demonstrate subtraction of rational numbers as adding the additive inverse, p – q = p + (–q).

Adding and Subtracting Integers
Adding and Subtracting Integers with Chips
Adding on the Number Line
Equivalent Algebraic Expressions I
Simplifying Algebraic Expressions I
Simplifying Algebraic Expressions II
Solving Algebraic Equations I
Sums and Differences with Decimals

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.

Adding and Subtracting Integers
Adding and Subtracting Integers with Chips
Adding on the Number Line
Equivalent Algebraic Expressions I
Simplifying Algebraic Expressions I
Simplifying Algebraic Expressions II
Solving Algebraic Equations I
Sums and Differences with Decimals

1.2.a.viii: Apply properties of operations as strategies to add and subtract rational numbers.

Adding Fractions (Fraction Tiles)
Adding and Subtracting Integers
Adding on the Number Line
Equivalent Algebraic Expressions I
Estimating Sums and Differences
Fractions Greater than One (Fraction Tiles)
Improper Fractions and Mixed Numbers
Sums and Differences with Decimals

1.2.b: Students can: Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers including integers.

1.2.b.i: Apply properties of operations to multiplication of rational numbers.

Adding and Subtracting Integers
Multiplying Fractions
Multiplying Mixed Numbers
Multiplying with Decimals

1.2.b.ii: Interpret products of rational numbers by describing real-world contexts.

Multiplying Fractions
Multiplying Mixed Numbers
Multiplying with Decimals

1.2.b.iii: Apply properties of operations to divide integers.

Dividing Mixed Numbers

1.2.b.iv: Apply properties of operations as strategies to multiply and divide rational numbers.

Adding and Subtracting Integers
Dividing Fractions
Dividing Mixed Numbers
Multiplying Fractions
Multiplying Mixed Numbers
Multiplying with Decimals

1.2.b.v: Convert a rational number to a decimal using long division.

Percents, Fractions, and Decimals

1.2.b.vi: Show that the decimal form of a rational number terminates in 0s or eventually repeats.

Percents, Fractions, and Decimals

1.2.c: Students can: Solve real-world and mathematical problems involving the four operations with rational numbers.

Adding Fractions (Fraction Tiles)
Adding and Subtracting Integers
Adding on the Number Line
Dividing Fractions
Dividing Mixed Numbers
Estimating Population Size
Estimating Sums and Differences
Fractions Greater than One (Fraction Tiles)
Improper Fractions and Mixed Numbers
Multiplying Fractions
Multiplying Mixed Numbers
Multiplying with Decimals
Sums and Differences with Decimals

2: Patterns, Functions, and Algebraic Structures

2.1: Understand that equivalence is a foundation of mathematics represented in numbers, shapes, measures, expressions, and equations

2.1: Properties of arithmetic can be used to generate equivalent expressions

2.1.a: Students can: Use properties of operations to generate equivalent expressions.

2.1.a.i: Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.

Equivalent Algebraic Expressions I
Equivalent Algebraic Expressions II
Simplifying Algebraic Expressions I
Simplifying Algebraic Expressions II
Solving Algebraic Equations I
Solving Algebraic Equations II

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.

Exponents and Power Rules
Modeling the Factorization of ax²+bx+c
Modeling the Factorization of x²+bx+c

2.2: Use critical thinking to recognize problematic aspects of situations, create mathematical models, and present and defend solutions

2.2: Equations and expressions model quantitative relationships and phenomena

2.2.a: Students can: Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form, using tools strategically.

Adding Fractions (Fraction Tiles)
Adding Whole Numbers and Decimals (Base-10 Blocks)
Adding and Subtracting Integers
Adding on the Number Line
Dividing Fractions
Dividing Mixed Numbers
Estimating Sums and Differences
Fraction Garden (Comparing Fractions)
Fractions Greater than One (Fraction Tiles)
Fractions with Unlike Denominators
Improper Fractions and Mixed Numbers
Multiplying Fractions
Multiplying Mixed Numbers
Multiplying with Decimals
Part-to-part and Part-to-whole Ratios
Percent of Change
Percents and Proportions
Percents, Fractions, and Decimals
Rational Numbers, Opposites, and Absolute Values
Subtracting Whole Numbers and Decimals (Base-10 Blocks)
Sums and Differences with Decimals
Toy Factory (Set Models of Fractions)

2.2.b: Students can: 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.

Adding Fractions (Fraction Tiles)
Adding Whole Numbers and Decimals (Base-10 Blocks)
Adding and Subtracting Integers
Adding on the Number Line
Dividing Fractions
Dividing Mixed Numbers
Estimating Sums and Differences
Fraction Garden (Comparing Fractions)
Fractions Greater than One (Fraction Tiles)
Fractions with Unlike Denominators
Improper Fractions and Mixed Numbers
Multiplying Fractions
Multiplying Mixed Numbers
Multiplying with Decimals
Part-to-part and Part-to-whole Ratios
Percent of Change
Percents and Proportions
Percents, Fractions, and Decimals
Subtracting Whole Numbers and Decimals (Base-10 Blocks)
Sums and Differences with Decimals
Toy Factory (Set Models of Fractions)

2.2.c: Students can: 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.

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.

Absolute Value Equations and Inequalities
Circles
Modeling One-Step Equations
Modeling and Solving Two-Step Equations
Order of Operations
Solving Algebraic Equations I
Solving Algebraic Equations II
Solving Equations on the Number Line
Solving Two-Step Equations

2.2.c.ii: Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach.

Absolute Value Equations and Inequalities
Circles
Modeling One-Step Equations
Modeling and Solving Two-Step Equations
Order of Operations
Solving Algebraic Equations I
Solving Algebraic Equations II
Solving Equations on the Number Line
Solving Two-Step Equations

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.

Absolute Value Equations and Inequalities
Rational Numbers, Opposites, and Absolute Values
Solving Linear Inequalities in One Variable

2.2.c.iv: Graph the solution set of the inequality and interpret it in the context of the problem.

Absolute Value Equations and Inequalities
Rational Numbers, Opposites, and Absolute Values
Solving Linear Inequalities in One Variable

3: Data Analysis, Statistics, and Probability

3.1: Use critical thinking to recognize problematic aspects of situations, create mathematical models, and present and defend solutions

3.1: Statistics can be used to gain information about populations by examining samples

Polling: City
Polling: Neighborhood
Populations and Samples

3.1.a: Students can: Use random sampling to draw inferences about a population.

3.1.a.i: Explain that generalizations about a population from a sample are valid only if the sample is representative of that population.

Polling: City
Polling: Neighborhood
Populations and Samples

3.1.a.ii: Explain that random sampling tends to produce representative samples and support valid inferences.

Polling: City
Polling: Neighborhood
Populations and Samples

3.1.a.iii: Use data from a random sample to draw inferences about a population with an unknown characteristic of interest.

Polling: City
Polling: Neighborhood
Populations and Samples

3.1.a.iv: Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions.

Polling: City
Polling: Neighborhood
Populations and Samples

3.1.b: Students can: Draw informal comparative inferences about two populations.

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.

Box-and-Whisker Plots
Describing Data Using Statistics
Mean, Median, and Mode
Movie Reviewer (Mean and Median)
Reaction Time 1 (Graphs and Statistics)
Reaction Time 2 (Graphs and Statistics)
Real-Time Histogram

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.

Box-and-Whisker Plots
Polling: City
Populations and Samples
Reaction Time 1 (Graphs and Statistics)

3.2: Recognize and make sense of the many ways that variability, chance, and randomness appear in a variety of contexts

3.2: Mathematical models are used to determine probability

3.2.a: Students can: Explain that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring.

Geometric Probability
Independent and Dependent Events
Probability Simulations
Spin the Big Wheel! (Probability)
Theoretical and Experimental Probability

3.2.b: Students can: 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.

Probability Simulations
Theoretical and Experimental Probability

3.2.c: Students can: Develop a probability model and use it to find probabilities of events.

3.2.c.i: Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy.

Geometric Probability
Independent and Dependent Events
Probability Simulations
Spin the Big Wheel! (Probability)
Theoretical and Experimental Probability

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.

Independent and Dependent Events
Probability Simulations
Spin the Big Wheel! (Probability)
Theoretical and Experimental Probability

3.2.c.iii: Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process.

Spin the Big Wheel! (Probability)
Theoretical and Experimental Probability

3.2.d: Students can: Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.

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.

Independent and Dependent Events
Theoretical and Experimental Probability

3.2.d.ii: Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams.

Independent and Dependent Events
Permutations and Combinations

3.2.d.iii: For an event described in everyday language identify the outcomes in the sample space which compose the event.

Independent and Dependent Events
Permutations and Combinations

3.2.d.iv: Design and use a simulation to generate frequencies for compound events.

Independent and Dependent Events

4: Shape, Dimension, and Geometric Relationships

4.1: Apply transformation to numbers, shapes, functional representations, and data

4.1: Modeling geometric figures and relationships leads to informal spatial reasoning and proof

4.1.a: Students can: Draw, construct, and describe geometrical figures and describe the relationships between them.

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.

Dilations
Perimeters and Areas of Similar Figures
Similar Figures

4.1.a.ii: Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions.

3D and Orthographic Views
Concurrent Lines, Medians, and Altitudes
Triangle Inequalities

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.

3D and Orthographic Views
Concurrent Lines, Medians, and Altitudes
Triangle Inequalities

4.2: Understand quantity through estimation, precision, order of magnitude, and comparison. The reasonableness of answers relies on the ability to judge appropriateness, compare, estimate, and analyze error

4.2: Linear measure, angle measure, area, and volume are fundamentally different and require different units of measure

4.2.a: Students can: State the formulas for the area and circumference of a circle and use them to solve problems.

Circumference and Area of Circles

4.2.b: Students can: Give an informal derivation of the relationship between the circumference and area of a circle.

Circumference and Area of Circles

4.2.c: Students can: 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.

Investigating Angle Theorems
Triangle Angle Sum

4.2.d: Students can: 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.

Area of Parallelograms
Area of Triangles
Balancing Blocks (Volume)
Chocomatic (Multiplication, Arrays, and Area)
Fido's Flower Bed (Perimeter and Area)
Perimeter and Area of Rectangles
Prisms and Cylinders
Pyramids and Cones
Surface and Lateral Areas of Prisms and Cylinders
Surface and Lateral Areas of Pyramids and Cones