Evidence Outcomes:

1. 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 ¼ hour, compute the unit rate as the complex fraction1/2 over 1/4miles per hour, equivalently 2 miles per hour. (CCSS: 7.RP.A.1)

2. Identify and represent proportional relationships between quantities. (CCSS: 7.RP.A.2) Determine 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. (CCSS: 7.RP.A.2.a) Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. (CCSS: 7.RP.A.2.b) 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. (CCSS: 7.RP.A.2.c) 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. (CCSS: 7.RP.A.2.d)

3. Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error. (CCSS: 7.RP.A.3)

Transfer Goals: Based on the Evidence Outcomes, what will students transfer to new contexts/situations?

• Examine and apply a variety of strategies to accurately and efficiently solve problems

• Fluidly move between contextualizing and decontextualizing during the problem solving process

Essential Understandings: In order to meet these transfer goals, the essential ideas and core processes students must understand are...

• Decontextualize real-world problems by recognizing, representing, and solving problems that involve proportional relationships to make predictions and describe associations among variables

• Contextualize proportional relationship problems to build understanding of rates and their units

• Make meaning of real-world proportional relationship problems by analyzing constraints, determining what is known, identifying the relationships, and solving the problem

In order to meet these essential understandings, students must know...

• Rates and unit rates

• Proportional relationships

• Constant of proportionality (unit rate)

• Strategies for solving proportional relationships

In order to meet these essential understandings, students must be able to...

• Compute unit rates with ratios that involve fractions, use:

  • Lengths

  • Areas

  • Other quantities measured in like or different units

• Determine if two quantities are in a proportional relationship by:

  • Testing for equivalent ratios in a table

  • Graphing on a coordinate plane

• Find the constant of proportionality in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships

• Represent proportional relationships with an equation

• Analyze a proportional relationship represented as a graph and explain a point (x, y) in its context considering these questions:

  • What does the point (0,0) mean?

  • What does the point (1, r) represent?

  • What units go with which variable (independent vs dependent)?

Evidence Outcomes:

1. Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. (CCSS: 7.NS.A.1) Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged. (CCSS: 7.NS.A.1.a) 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. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts. (CCSS: 7.NS.A.1.b) Demonstrate subtraction of rational numbers as adding the additive inverse, p – q = p + (-q). 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. (CCSS: 7.NS.A.1.c) Apply properties of operations as strategies to add and subtract rational numbers. (CCSS: 7.NS.A.1.d)

2. Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. (CCSS: 7.NS.A.2) Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1) (-1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts. (CCSS: 7.NS.A.2.a) Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If p and q are integers, then – (p/q) = -p/q = p/-q. Interpret quotients of rational numbers by describing real-world contexts. (CCSS: 7.NS.A.2.b) Apply properties of operations as strategies to multiply and divide rational numbers. (CCSS: 7.NS.A.2.c) Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0 s or eventually repeats. (CCSS: 7.NS.A.2.d)

3. Solve real-world and mathematical problems involving the four operations with rational numbers. (Computations with rational numbers extend the rules for manipulating fractions to complex fractions.) (CCSS: 7.NS.A.3)

Transfer Goals: Based on the Evidence Outcomes, what will students transfer to new contexts/situations?

• Fluidly move between contextualizing and decontextualizing during the problem solving process

• Examine and apply a variety of strategies to accurately and efficiently solve problems

Essential Understandings: In order to meet these transfer goals, the essential ideas and core processes students must understand are...

• Contextualize problems involving the four operations and rational number to provide meaning and context to problems

• Decontextualize real-world problems in order to perform necessary operations with rational numbers

• Use previous understandings of operating with numbers and new knowledge to perform operations with rational numbers

In order to meet these essential understandings, students must know...

• Definition of rational numbers

• Algorithms and strategies for adding and subtracting rational numbers

• Properties of Operations

  • Associative property

  • Commutative Property

  • Distributive Property

• Multiplication Property of 0

• What happens when you multiply by -1

• Algorithms and strategies for multiplying with rational numbers

• All integers can be divided (as long as the divisor is not 0) and makes a rational number

• Every rational number either terminates in 0s or eventually repeats

• That -(pq)=-pq=p-q

• The various symbols used to represent the division and multiplication

In order to meet these essential understandings, students must be able to...

• Extend previous understandings of addition and subtraction to make meaning of the algorithm for adding and subtracting rational numbers

• Represent addition and subtraction on the number line

• Contextualize scenarios where opposites values combine to make 0

• Apply the concepts of absolute value to show addition of rational numbers as a distance

• Show the solution to combining a number with its opposite on the number line

• Provide a visual representation of the algorithm for subtracting rational numbers on the number line

• Apply properties of operations to add and subtract rational numbers

• Build on previous understandings of multiplying fractions to lead to the rules for multiplying signed (negative and positive) numbers

• Interpret the products of rational numbers by providing context to the problem

• Interpret quotients of rational numbers by providing context to the problem

• Apply properties of operations to multiply and divide rational numbers

• Use long division to convert a fraction to a decimal

• Solve real-world and mathematical problems with all four operations involving rational numbers

Evidence Outcomes:

1. Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients. (CCSS: 7.EE.A.1)

2. 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. For example, a + 0.05a = 1.05a means that “increase by 5%” is the same as “multiply by 1.05.” (CCSS: 7.EE.A.2)

Transfer Goals: Based on the Evidence Outcomes, what will students transfer to new contexts/situations?

• Articulate how mathematical concepts relate to one another in the context of a problem or abstract relationships.

Essential Understandings: In order to meet these transfer goals, the essential ideas and core processes students must understand are...

• Recognize that the structure of equivalent algebraic expressions provide different ways of seeing the same problem.

In order to meet these essential understandings, students must know...

• Properties of Operations

  • Associative Property

  • Commutative Property

  • Distributive Property

• Difference between expressions and equations

• Linear expressions

  • Coefficients

• The various symbols to represent multiplication and division

In order to meet these essential understandings, students must be able to...

• Apply properties of operations as strategies to simplify or expand linear expressions

  • Combine like terms

  • Factor

  • Expand

• Apply properties of operations to simplify or expand linear expression existing in the real-world in order to shed light on the problem and how the quantities in it are related

Evidence Outcomes:

1. 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. 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. For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation. (CCSS: 7.EE.B.3)

2. 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. (CCSS: 7.EE.B.4) 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. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width? (CCSS: 7.EE.B.4.a) Solve word problems leading to inequalities of the form px ± q > r and px ± q ≥ r, px ± q < r, 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. For example: As a salesperson, you are paid \ 50 per week plus \ 3 per sale. This week you want your pay to be at least \ 100. Write an inequality for the number of sales you need to make and describe the solutions. (CCSS: 7.EE.B.4.b)

Transfer Goals: Based on the Evidence Outcomes, what will students transfer to new contexts/situations?

• Based on an understanding of any problem, initiate a plan, execute it, and evaluate the reasonableness of the solution

• Use appropriate tools to deepen understanding of mathematical concepts

Essential Understandings: In order to meet these transfer goals, the essential ideas and core processes students must understand are...

• Adapt to different forms of equations and inequalities and reach solutions that make sense in context

• Make connections between the sequence of operations used in an algebraic approach and an arithmetic approach, understanding how simply reasoning about the numbers connects to writing and solving a corresponding algebraic equation

• Represent a situation symbolically and solve, attending to the meaning of quantities and variables

• Select an appropriate solution approach (calculator, mental math, drawing a diagram, etc.) based on the specific values and/or desired result of a problem

In order to meet these essential understandings, students must know...

• Linear Equations

• Linear Inequalities

• Properties of operations

  • Associative Property

  • Commutative Property

  • Distributive Property

• Properties of Equality

  • Reflexive Property

  • Symmetric Property

  • Transitive Property

  • Addition Property

  • Subtraction Property

  • Multiplication Property

  • Division Property

  • Substitution Property

  • Distributive Property

• Procedures for solving multi-step equations and inequalities

• Rules for solving multi-step inequalities

In order to meet these essential understandings, students must be able to...

• Solve multi-step real-life and mathematical problems by:

  • Applying properties of operations to compute with numbers in any form

  • Convert between different forms as appropriate

  • Assess the reasonableness of an answers using mental computation and estimation strategies

• Decontextualize real-world problems to construct simple equations and inequalities

• Contextualize mathematical problems to provide context to simple equations and inequalities

• Apply procedures to accurately, efficiently, and flexibly solve word problems leading to the form pxq=r and p(xq)=r where p, q, and r are rational numbers

• Compare an algebraic solution to an arithmetic solution and identify the sequence of the operations used in each

• Apply procedures and rules to accurately, efficiently, and flexibility solve word problems leading to inequalities of the form pxq>r, pxqr, pxq<r, or pxqr where p, q, and r are rational numbers

• Graph the solution set of the inequality and interpret it in the context of the problem

Evidence Outcomes:

1. Understand that statistics can be used to gain information about a population by examining a sample of the population; explain that generalizations about a population from a sample are valid only if the sample is representative of that population. Explain that random sampling tends to produce representative samples and support valid inferences. (CCSS: 7.SP.A.1)

2. 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. For example, estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be. (CCSS: 7.SP.A.2)

Transfer Goals: Based on the Evidence Outcomes, what will students transfer to new contexts/situations?

• Communicate effectively based on purpose, task, and audience.

Essential Understandings: In order to meet these transfer goals, the essential ideas and core processes students must understand are...

• Communicate inferences made about a population using a random sample

• Make conjectures about population parameters and support arguments with sample data

• Use a statistical model to informally assess the variability of sample statistics (like mean)

In order to meet these essential understandings, students must know...

• Statistics can be used to gain information about a population

• Characteristics of a representative sample

• That generalizations about a population are only valid if the sample is representative of the population

• Basic understanding of various sampling methods

  • Simple random sampling

  • Systematic sampling

  • Stratified sampling

  • Cluster sampling

  • Random sampling

• Random sampling produces the most representative sample

In order to meet these essential understandings, students must be able to...

• 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

Evidence Outcomes:

1. 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. For example, the mean height of players on the basketball team is 10 cm greater than the mean height of players on the soccer team, about twice the variability (mean absolute deviation) on either team; on a dot plot, the separation between the two distributions of heights is noticeable. (CCSS: 7.SP.B.3)

2. Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. For example, decide whether the words in a chapter of a seventh-grade science book are generally longer than the words in a chapter of a fourth-grade science book. (CCSS: 7.SP.B.4)

Transfer Goals: Based on the Evidence Outcomes, what will students transfer to new contexts/situations?

• Draw informal comparative inferences.

Essential Understandings: In order to meet these transfer goals, the essential ideas and core processes students must understand are...

• Interpret variability in statistical distributions and draw conclusions

• Produce arguments about the difference between two distributions by analyzing the relative variability of the distributions

In order to meet these essential understandings, students must know...

• The distributions of data

  • Normal

  • Uniform

  • Skewed

• Measures of Central Tendencies

  • Mean

  • Median

  • Mode

• Measures of Variability

  • Range

  • Interquartile Range

  • Variance

  • Standard Deviation

In order to meet these essential understandings, students must be able to...

• Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities by measuring the difference between the centers and express it as a multiple of the variability

• Draw informal comparative inferences about two populations using measures of central tendency and measures of variability

• Model real-world populations with statistical distributions and compare the distributions using measures of central tendency

Evidence Outcomes:

1. Explain 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. (CCSS: 7.SP.C.5)

2. 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. For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times. (CCSS: 7.SP.C.6)

3. 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. (CCSS: 7.SP.C.7) Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. For example, if a student is selected at random from a class, find the probability that Jane will be selected and the probability that a girl will be selected. (CCSS: 7.SP.C.7.a) Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. For example, find the approximate probability that a spinning penny will land heads up or that a tossed paper cup will land open-end down. Do the outcomes for the spinning penny appear to be equally likely based on the observed frequencies? (CCSS: 7.SP.C.7.b)

4. Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation. (CCSS: 7.SP.C.8)Explain 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. (CCSS: 7.SP.C.8.a) 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. (CCSS: 7.SP.C.8.b) Design and use a simulation to generate frequencies for compound events. For example, use random digits as a simulation tool to approximate the answer to the question: If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood? (CCSS: 7.SP.C.8.c)

Transfer Goals: Based on the Evidence Outcomes, what will students transfer to new contexts/situations?

• Articulate how mathematical concepts relate to one another in the context of a problem or abstract relationships

Essential Understandings: In order to meet these transfer goals, the essential ideas and core processes students must understand are...

• Use probability models and simulations to predict outcomes of real-world chance events both theoretically and experimentally

• Explain chance events using probabilities

In order to meet these essential understandings, students must know...

• That probability shows the likelihood of an event occurring and is a number between 0 and 1

• Relative frequency

• The meaning of a compound events as a fraction of outcomes in the sample space for which it occurs

• Methods for finding probability of compound events

  • Organized lists

  • Tables

  • Tree diagrams

  • Simulation

In order to meet these essential understandings, students must be able to...

• Explain the meaning of the probability of a chance event

• Calculate the approximate relative frequency of a chance event

• Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency

• Develop a uniform and not uniform probability model and use them to find probabilities

• Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy

• Represent compound events using organized lists, tables, tree diagrams, and simulations

• For an everyday event, identify the outcomes in the sample space which compose the event

• Design and use a simulation to generate frequencies for compound events.

Evidence Outcomes:

1. 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. (CCSS: 7.G.A.1)

2. 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. (CCSS: 7.G.A.2)

3. Describe the two-dimensional figures that result from slicing three-dimensional figures, as in cross sections of right rectangular prisms and right rectangular pyramids. (CCSS: 7.G.A.3)

Transfer Goals: Based on the Evidence Outcomes, what will students transfer to new contexts/situations?

• Use appropriate tools to deepen understanding of mathematical concepts.

Essential Understandings: In order to meet these transfer goals, the essential ideas and core processes students must understand are...

• Decontextualize real-world problems to answer questions involving scaling

• Use appropriate tools to contextualize geometric figures and their relationships to one another

In order to meet these essential understandings, students must know...

• Scale drawings of geometric figures

• Scale factor and how it is used to find side lengths

• Scale factors and how it is used to find areas

• How to use rulers, protractors, and technology to construct geometric shapes

• Geometric shapes

• Conditions for generating a unique triangle, more than one triangle, or no triangle (triangle inequality)

• What a cross-sections

• Right rectangular prisms

• Right rectangular pyramids

• Difference between three-dimensional figures and two-dimensional figures

In order to meet these essential understandings, students must be able to...

• Solve problems involving scale drawings of geometric figures

  • Compute actual lengths from a scale drawing

  • Compute areas from a scale drawing

  • Reproduce a scale drawing at a different scale

• Draw (free hand, with a ruler and protractor, and with technology) geometric shapes with given conditions

• Determine what conditions of sides and angles give one unique triangle, more than one triangle, or no triangles (triangle inequality)

• Describe two-dimensional figures that result from slicing three-dimensional figures

  • Right rectangular prisms

  • Right rectangular prisms

Evidence Outcomes:

1. State 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. (CCSS: 7.G.B.4)

2. Use facts about supplementary, complementary, vertical, and adjacent angles in a multistep problem to write and solve simple equations for an unknown angle in a figure. (CCSS: 7.G.B.5)

3. 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. (CCSS: 7.G.B.6)

Transfer Goals: Based on the Evidence Outcomes, what will students transfer to new contexts/situations?

• Based on an understanding of any problem, initiative a plan, execute it, and evaluate the reasonableness of the solution

• Examine and apply a variety of methods to accurately and efficiently solve problems

Essential Understandings: In order to meet these transfer goals, the essential ideas and core processes students must understand are...

• Represent, analyze and draw conclusions about area and volume of complex shapes that exist in the real world

• Apply geometric thinking, concepts, and formulas to solve real-world phenomena problems about geometric shapes

In order to meet these essential understandings, students must know...

• Formulas for area and circumference

• Supplementary angles

• Complementary angles

• Vertical angles

• Adjacent angles

• How to set up and solve equations to find unknown angles

• Area of two-dimensional objects composed of triangles, quadrilaterals, and polygons

• Volume and surface area of three-dimensional objects composed of cubes and right prisms

In order to meet these essential understandings, students must be able to...

• Use the formulas for area and circumference to solve problems

• Give an informal derivation of the relationship between circumference and area of a circle

• Set-up and solve simple equations to find unknown angle measurements using facts about supplementary, complementary, vertical, and adjacent angles

• Solve real-world and mathematical problems involving area of two-dimensional objects composed of triangles, quadrilaterals, and polygons

• Solve real-world and mathematical problems of three-dimensional objects objects composed of cubes and right prisms