## Formative Assessment and Bridging activities

Grade 8

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Standards are considered a bridge when they: function as a bridge to which other content within the grade level/course is connected; serve as prerequisite knowledge for content to be addressed in future grade levels/courses; or possess endurance beyond a single unit of instruction within a grade level/course.**Note:** Links marked with ↗ will open in a new tab

## Standard 8.1

**S****tandard**** ****8.1 **Compare and order real numbers.

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## UNDERSTANDING THE LEARNING Trajectory

**Big Ideas:**

The density property states that between any two real numbers lies another real number. The set of real numbers is

**all the numbers that have a location on the number line**.The real numbers include natural numbers or counting numbers, whole numbers, integers, rational numbers (fractions and repeating or terminating decimals), and irrational numbers.

Patterns help us to order and compare real numbers.

In mathematics, we can compare and order real numbers by converting between different forms of numbers (ie. fractions → decimals or scientific notation → standard notation).

There are numbers that are not rational. We can approximate them using rational numbers (Achieve the Core)↗

**Important Assessment Look Fors:**

The student can differentiate between negative and positive integers and place them on the correct side of zero.

The students can convert between scientific notation and decimals.

The student can use a number line as a tool to correctly order numbers.

The student can approximate irrational numbers using rational numbers.

**Purposeful Questions: **

What does it mean if a number is negative or positive? How does that affect its placement on the number line?

What does a negative exponent mean? What does a positive exponent mean?

When a value isn’t listed on the number line, how do you determine its placement? Does it matter where it goes if there isn’t a given label?

**Student Strengths**

Students can represent and determine equivalencies among, and order fractions, mixed numbers, decimals, percents, exponents, perfect squares, and integers. Ordering may be in ascending or descending order.

**Bridging Concepts**

**Standard 8.1**

**Formative Assessments:**

**Formative Assessments:**

**Routine **

**Routine**

**Rich Tasks**

**Rich Tasks**

**Games/Tech**

**Games/Tech**

## Standard 8.4

**S****tandard**** ****8.4**** **Solve practical problems involving consumer applications.

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Understanding the Learning Trajectory

**Big Ideas:**

Students move from being able to reason proportionally with whole numbers to rational numbers. Students develop their proportional reasoning while working with ratios, rates, and unit rates representing them with expressions, tape diagrams, double number line diagrams, and tables. They may use these representations to reason about situations involving consumer application (unit price, discounts, tax and tips) as well as color mixtures, recipes, constant speed, and measurement conversions.

Context clues and vocabulary are used to describe real world situations that can be written using operations with rational numbers.

**Important Assessment Look Fors:**

Students understand how to use proportional reasoning to find a percent of a number.

Students use vocabulary to determine appropriate operations to perform to solve given consumer scenarios.

The student should answer the appropriate question.

**Purposeful Questions: **

Is your answer reasonable for the question?

Did you answer the question in its entirety?

How did you find the commission ( or sales tax, or tip, depending on the question)?

Would your total decrease or increase after including the sales tax, tip, or commission?

What is another way you could find the total?

**Student Strengths**

**Student Strengths**

Students can problem solve using rational numbers and proportional reasoning.

**Bridging Concepts**

**Bridging Concepts**

Students can find tax, tip and discount, solve problems with similar figures, and practical problems with proportional reasoning..

**Standard ****8.4**

**Standard**

**8.4**

Students can solve practical problems involving consumer applications.

**Full Module with Instructional Tips & Resources: **

**Full Module with Instructional Tips & Resources:**

**Formative Assessments: **** **

**Formative Assessments:**

**Routines:**

**Routines:**

**Rich Tasks: **** **

**Rich Tasks:**

**Games/Tech:**

**Games/Tech:**

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## Standard 8.5

**S****tandard**** ****8.5**** **Use relationships among pairs of angles that are vertical angles, adjacent angles, supplementary angles, and complementary angles to determine the measure of unknown angles.

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Understanding the Learning Trajectory

**Big Ideas:**

Relationships can be found within angles, and those relationships can be described and proved.

Relationships between angles can be used to solve problems for unknown angles.

**Important Assessment Look Fors:**

The student should locate and identify adjacent angles as two non-overlapping angles that share a common ray and a common vertex.

The student should locate and identify vertical angles as a pair of nonadjacent angles formed by two intersecting lines. Vertical angles are congruent and share a common vertex.

The student describes complementary angles as any two angles such that the sum of their measures is 90°and explains that adjacent complementary angles form a right angle.

The student identifies and describes that supplementary angles are any two angles such that the sum of their measures is 180° and form a straight angle.

**Purposeful Questions: **

Are vertical angles equal?

How are complementary angles different from supplementary angles? How are they different from vertical angles?

Explain how you determined the measure of the missing angle. How can you check your answer? How do you know your answer is reasonable?

**Student Strengths**

**Student Strengths**

Students can classify and measure right, acute, obtuse, and straight angles.

**Bridging Concepts**

Students can use the symbol ≅ to represent congruence and can identify and denote congruent angles with the appropriate markings. **Bridging Concepts**

**Standard 8.****5**

**Standard 8.**

**5**

Use relationships among pairs of angles that are vertical angles, adjacent angles, supplementary angles, and complementary angles to determine the measure of unknown angles

**Full Module with Instructional Tips & Resources: **

**Full Module with Instructional Tips & Resources:**

**Formative Assessments: **

**Formative Assessments:**

**Routines:**

**Routines:**

**Rich Tasks: **** **

**Rich Tasks:**

**Games/Tech:**

**Games/Tech:**

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## Standard 8.9b

**S****tandard**** 8.****9b**** **Apply the Pythagorean Theorem.

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Understanding the Learning Trajectory

**Big Ideas:**

Lengths of triangle and square sides can be compared using ideas such as longer, shorter, and equal (Charles, 2005).

Behind every measurement formula lies a geometric result (NCTM)

Triangles and quadrilaterals can be described, categorized, and named based on the relative lengths of their sides and the sizes of their angles (Charles, 2005).

**Important Assessment Look Fors:**

The student can square numbers and find square roots.

The student can identify the hypotenuse of a triangle.

The student can identify right triangles given side lengths by using the Pythagorean Theorem.

The student can find the length of the hypotenuse or the legs using the Pythagorean theorem.

The student can describe how to use the Pythagorean Theorem when applied to real-world problems.

**Purposeful Questions: **

When thinking about the Pythagorean Theorem, does it matter which side length we substitute for each variable

(a2+b2 = c2)? Why or why not?

What is the hypotenuse? How is it relevant to real world applications?

What is distance? How do we measure it? Are there different ways of measuring distance when thinking about real-world applications?

How did you arrive at your length for the hypotenuse? Explain how you used the Pythagorean Theorem.

**Student Strengths**

**Student Strengths**

Students can identify the perfect squares from 0 to 400 and determine the positive square root of a perfect square from 0 to 400.

**Bridging Concepts**

Students can determine the positive or negative square root of a given perfect square from 1 to 400. They can verify the Pythagorean Theorem, using diagrams, concrete materials, and measurement.**Bridging Concepts**

**Standard 8.****9b**

**Standard 8.**

**9b**

Apply the Pythagorean Theorem

**Full Module with Instructional Tips & Resources: **

**Full Module with Instructional Tips & Resources:**

**Formative Assessments: **

**Formative Assessments:**

**Routines:**

**Routines:**

**Rich Tasks: **** **

**Rich Tasks:**

**Games:**

**Games:**

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## Standard 8.10

**S****tandard**** 8.1****0**** **Solve area and perimeter problems, including practical problems, involving composite plane figures.

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Understanding the Learning Trajectory

**Big Ideas:**

Polygons can be described uniquely by their sides and angles. (Charles, 2005)

Polygons can be constructed from or decomposed into other polygons. (Charles, 2005)

Behind every measurement formula lies a geometric result.

**Important Assessment Look Fors:**

The student can calculate the area and perimeter of triangles, rectangles, squares, trapezoids, parallelograms, and semicircles.

The student can use composition and/or decomposition to find the area or perimeter of a composite plane figure.

The student applies vocabulary associated with each shape (ie. parts of a circle, what makes a trapezoid, etc.) to correctly calculate the perimeter or area of the parts of the composite plane figures using the appropriate formulas.

**Purposeful Questions: **

How do we find the lengths of the unknown sides? Can we find the area or perimeter without finding these missing side lengths first?

Can we assume two side lengths are congruent if they look the same? Why or why not?

What is the formula for finding the area of this shape? Why does that formula make sense?

How did you arrive at your final area? Explain the process you used. Could you have found it a different way?

**Student Strengths**

**Student Strengths**

Students can derive an approximation for pi (3.14 or 22/7) by gathering data and comparing the circumference to the diameter of various circles, using concrete materials or computer models.

They can solve problems involving circumference and area of a circle when given the length of the diameter or radius.

They can solve problems involving area and perimeter of triangles and rectangles.

**Bridging Concepts**

Students can solve practical problems involving circumference and area of a circle when given the length of the diameter or radius. They can solve practical problems involving area and perimeter of triangles and rectangles.**Bridging Concepts**

**Standard 8.10**

**Standard 8.10**

Students can solve area and perimeter problems, including practical problems, involving composite plane figures.

**Full Module with Instructional Tips & Resources: **

**Full Module with Instructional Tips & Resources:**

**Formative Assessments: **

**Formative Assessments:**

**Routines:**

**Routines:**

**Rich Tasks: **** **

**Rich Tasks:**

**Games/Tech:**

**Games/Tech:**

- Composite Figures Putt Putt Golf↗
- Desmos 8.10 Area and Perimeter Review Sort 8.10 Area on the Grid: Reasoning to Find Area 8.10 Composite Figures Against a Grid 8.10 Practice Areas of Composite Shapes↗

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## Standard 8.11a

**S****tandard**** 8.1****1a**** **Compare and contrast the probability of independent and dependent event.

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Understanding the Learning Trajectory

**Big Ideas:**

Probability can provide a basis for making predictions. (Charles, 2005)

Some probabilities can be determined through experimental trials. (Charles, 2005)

The chance of an event occurring can be described numerically by a number between 0 and 1 inclusive and used to make predictions about other events. (Charles, 2005))

Two events are either dependent or independent. If the outcome of one event has an impact on the outcome of the other event, the events are called dependent. If events are dependent then the second event is considered only if the first event has already occurred. (VDOE Curriculum Framework)

**Important Assessment Look Fors:**

The student can determine if two events are independent or dependent and explain their reasoning.

The student can explain how the experimental probability changes in different scenarios (ie. if an item is replaced vs. if an item is not replaced).

The student can compare probabilities in the form of rational numbers in order to identify which event has a higher or lower probability.

**Purposeful Questions: **

What is the difference between an independent and dependent event?

Why are these two events independent? Dependent? Explain your reasoning.

How does the act of replacing or not replacing an object affect probability?

**Student Strengths**

**Student Strengths**

Students can determine the theoretical and experimental probabilities of an event.

**Bridging Concepts**

Students can describe changes in the experimental probability as the number of trials increases. **Bridging Concepts**

Students can investigate and describe the difference between the probability of an event found through experiment or simulation versus the theoretical probability of that same event.

**Standard 8.1****1a**

**Standard 8.1**

**1a**

Students can determine whether two events are independent or dependent. Students can compare and contrast the probability of independent and dependent events.

**Full Module with Instructional Tips & Resources: **

**Full Module with Instructional Tips & Resources:**

**Formative Assessments: **

**Formative Assessments:**

**Routines:**

**Routines:**

**Rich Tasks: **** **

**Rich Tasks:**

**Games/Tech:**

**Games/Tech:**

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## Standard 8.11b

**S****tandard**** 8.1****1b**** **Determine the probabilities for independent and dependent events.

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Understanding the Learning Trajectory

**Big Ideas:**

Probability can provide a basis for making predictions. (Charles, 2005)

Some probabilities can be determined through experimental trials. (Charles, 2005)

The chance of an event occurring can be described numerically by a number between 0 and 1 inclusive and used to make predictions about other events. (Charles, 2005))

Two events are either dependent or independent. If the outcome of one event has an impact on the outcome of the other event, the events are called dependent. If events are dependent then the second event is considered only if the first event has already occurred. (VDOE Curriculum Framework)

**Important Assessment Look Fors:**

The student can define the difference between independent and dependent events.

The student can explain how the theoretical and experimental probability changes based on if the events occuring are independent or dependent.

The student can compare probabilities in the form of rational numbers in order to identify which event has a higher or lower probability.

**Purposeful Questions: **

How many events are occurring? One or more? Are the events independent or dependent? How do you know?

How do we calculate the probability of dependent events? Independent events?

When calculating the probability of an event and writing it as a fraction - what value goes in the numerator? Why? What value goes in the denominator? Why?

How would the probability change if the item IS replaced? Or IS NOT replaced? Explain how you know.

**Student Strengths**

**Student Strengths**

Students can describe changes in the experimental probability as the number of trials increases. Investigate and describe the difference between the probability of an event found through experiment or simulation versus the theoretical probability of that same event.

**Bridging Concepts**

Students can determine whether two events are independent or dependent. Students can compare and contrast the probability of independent and dependent events.**Bridging Concepts**

**Standard 8.1****1b**

**Standard 8.1**

**1b**

Students can determine the probability of two independent or dependent events.

**Full Module with Instructional Tips & Resources: **

**Full Module with Instructional Tips & Resources:**

**Formative Assessments: **

**Formative Assessments:**

**Routines:**

**Routines:**

**Rich Tasks: **** **

**Rich Tasks:**

**Games:**

**Games:**

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## Standard 8.14A

**S****tandard**** 8.14a **Evaluate an algebraic expression for given replacement values of the variables.

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Understanding the Learning Trajectory

**Big Ideas:**

In mathematics, it is understood that a variable can be replaced by a value.

Letters are used in mathematics to represent generalized properties, unknowns in equations, and relationships between quantities (Charles, 2005).

In mathematics, following the order of operations is the correct way to simplify/evaluate an expression and there are specific notations to follow.

In mathematics, performing any operation involving rational numbers is necessary to simplify expressions.

**Important Assessment Look Fors:**

Students should be able to replace the variables with the assigned value with appropriate signs.

Students should start at the innermost grouping symbol in this expression.

Students should be able to follow the order of operations accurately with expressions including rational values.

**Purposeful Questions: **

What are the different types of grouping symbols you see in this expression?

How do you know where to start?

Should the answer be negative or positive? How do you know?

Explain how you arrived at your final expression.

**Student Strengths**

**Student Strengths**

Students can determine the square root of a perfect square, identify and describe absolute value, use the order of operations with limited exponents and limited grouping symbols.

**Bridging Concepts**

Students can use the order of operations with one set of parentheses or one grouping symbols.**Bridging Concepts**

**Standard 8.14A**

**Standard 8.14A**

Students can evaluate an algebraic expression for given replacement values of the variables.

**Full Module with Instructional Tips & Resources: **

**Full Module with Instructional Tips & Resources:**

**Formative Assessments: **** **

**Formative Assessments:**

**Routines:**

**Routines:**

**Rich Tasks: **** **

**Rich Tasks:**

**Games/Tech:**

**Games/Tech:**

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## Standard 8.14b

**S****tandard**** 8.14b **Simplify algebraic expressions in one variable.

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### Understanding the Learning Trajectory

**Big Ideas:**

Letters are used in mathematics to represent generalized properties, unknowns in equations, and relationships between quantities (Charles, 2005).

Algebraic expressions can be named in an infinite number of different but equivalent ways (Charles, 2005).

Properties of whole numbers apply to certain operations but not others (e.g., The commutative property applies to addition and multiplication but not subtraction and division) (Charles, 2005).

**Important Assessment Look Fors:**

Students should correctly use properties of real numbers to simplify expressions.

Students should properly distribute positive and negative signs, as well as when multiplying a value by an expression.

All like terms should be combined in the expression’s simplest form.

**Purposeful Questions: **

Can you identify all the like terms?

Can your expression be simplified further?

According to the order of operations and properties of real numbers, what should you do first?

**Student Strengths**

**Student Strengths**

Students can solve two step equations in one variable including practical problems.

**Bridging Concepts**

Students can apply properties of real numbers to find equivalent expressions. **Bridging Concepts**

**Standard 8.14B**

**Standard 8.14B**

Students can simplify algebraic expressions in one variable.

**Full Module with Instructional Tips & Resources: **

**Full Module with Instructional Tips & Resources:**

**Formative Assessments: **** **

**Formative Assessments:**

**Routines:**

**Routines:**

**Rich Tasks: **** **

**Rich Tasks:**

**Games/Tech:**

**Games/Tech:**

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## Standard 8.15a

**S****tandard**** 8.1****5a**** **Determine whether a given relation is a function.

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### Understanding the Learning Trajectory

**Big Ideas:**

Ordered pairs, a table, or a graph of discrete points is a function with the property that each input is related to exactly one output.

Mathematical rules (relations) can be used to assign members of one set to members of another set. (Charles, 2005)

A special rule (function) assigns each member of one set to a unique member of the other set. (Charles, 2005)

**Important Assessment Look Fors:**

The student can rule out tables and ordered pairs that repeat the x or input value for two separate y or output values.

The student can identify functions even if the y values are repeated as long as the x value is not.

The student can recognize that a graph of a function cannot have more than one coordinate plotted on the same vertical line and explain why this “vertical line” test helps to identify functions.

**Purposeful Questions: **

Is this a function? Explain your reasoning.

Do any of the x-values repeat in your table or ordered pairs?

Why can the x-values not repeat?

Why can the y-values not repeat?

Is there only one coordinate on each vertical line representing the x-values for your function?

How could you change this table, set of ordered pairs, or graph so that it becomes a function?

**Student Strengths**

**Student Strengths**

Students can identify ordered pairs represented by points in the four quadrants and on the axes of the coordinate plane.

Students can find the rule in input-output tables.

**Bridging Concepts**

Students can identify ordered pair coordinates expressed as rational numbers, in a table, or on a graph. **Bridging Concepts**

**Standard 8.1****5a**

**Standard 8.1**

**5a**

Students can determine whether a relation, represented by a set of ordered pairs, a table, or a graph of discrete points is a function.

**Full Module with Instructional Tips & Resources: **

**Full Module with Instructional Tips & Resources:**

**Formative Assessments: **** **

**Formative Assessments:**

**Routines:**

**Routines:**

**Rich Tasks: **** **

**Rich Tasks:**

**Games/Tech:**

**Games/Tech:**

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## Standard 8.15b

**S****tandard**** 8.1****5****b **Determine the domain and range of a function.

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### Understanding the Learning Trajectory

**Big Ideas:**

A special rule (function) assigns each member of one set to a unique member of the other set. (Charles, 2005)

The domain of a function represented as a set of ordered pairs, a table, or a graph of discrete points is the input or x value.

The range of a function represented as a set of ordered pairs, a table, or a graph of discrete points is the output or y value.

If a function consists of a discrete set of ordered pairs, then the domain is the set of all the x-coordinates, and the range is the set of all the y-coordinates.

**Important Assessment Look Fors:**

The student identifies the x-value in a coordinate as the domain.

The student identifies the y-value in a coordinate as the range.

The student can identify the domain using the x-axis and the range using the y=axis of a coordinate plane.

**Purposeful Questions: **

What are the values of the domain? How do you know?

What are the values of the range? How do you know?

Could two relations have the same domains, but different ranges? Explain or show an example for your reasoning.

**Student Strengths**

**Student Strengths**

Students can identify input and output from rules of one operation.

**Bridging Concepts**

Students can identify and graph ordered pairs in a coordinate plane.Students can identify functions.**Bridging Concepts**

**Standard 8.1****5b**

**Standard 8.1**

**5b**

Students can identify the domain and range of a function represented as a set of ordered pairs, a table, or a graph of discrete points.

**Full Module with Instructional Tips & Resources: **

**Full Module with Instructional Tips & Resources:**

**Formative Assessments: **** **

**Formative Assessments:**

**Routines:**

**Routines:**

**Rich Tasks: **** **

**Rich Tasks:**

**Games/Tech:**

**Games/Tech:**

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## Standard 8.16a

**S****tandard**** 8.1****6a**** **Recognize and describe the graph of a linear function with a slope that is positive, negative, or zero.

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### Understanding the Learning Trajectory

**Big Ideas:**

Mathematical rules (relations) can be used to assign members of one set to members of another set. A special rule (function) assigns each member of one set to a unique member of the other set. (Charles, 2005)

Mathematical relationships can be represented and analyzed using words, tables, graphs, and equations. (Charles, 2005)

The graph of a relationship can be analyzed with regard to the change in one quantity relative to the change in the other quantity. (Charles, 2005)

In a linear function of the form y = mx, m is the constant of variation and it represents the rate of change of y with respect to x, otherwise known as slope. (Charles, 2005)

**Important Assessment Look Fors:**

The student describes the line as having a negative slope when the line is decreasing from left to right.

The student describes the line as having a positive slope when the line increases from left to right.

The student can describe how slope and the “steepness” of the line is related.

**Purposeful Questions: **

Can you think of an example of a real life scenario that would have a positive slope? What about a negative slope? What about a slope of zero?

Is the line increasing from left to right, or decreasing from left to right?

Is it possible to have all coordinate values positive and have a negative slope? Or all negative coordinate values have a negative slope?

**Student Strengths**

**Student Strengths**

Students can determine the proportional relationships given ordered pairs, tables, graphs, and verbal descriptions.

**Bridging Concepts**

Students can determine the slope, m, as rate of change in a proportional relationship between two quantities and write an equation in the form 𝒚 = 𝒎𝒙 to represent the relationship. **Bridging Concepts**

**Standard 8.1****6a**

**Standard 8.1**

**6a**

Students can recognize and describe the graph of a linear function with a slope that is positive, negative, or zero.

**Full Module with Instructional Tips & Resources: **

**Full Module with Instructional Tips & Resources:**

**Formative Assessments: **** **

**Formative Assessments:**

**Routines:**

**Routines:**

**Rich Tasks: **** **

**Rich Tasks:**

**Games/Tech:**

**Games/Tech:**

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## Standard 8.16b

**S****tandard**** 8.1****6****b **Identify the slope and y-intercept of a linear function, given a table of values, a graph, or an equation in y = mx + b.

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### Understanding the Learning Trajectory

**Big Ideas:**

Mathematical rules (relations) can be used to assign members of one set to members of another set. A special rule (function) assigns each member of one set to a unique member of the other set.

Mathematical relationships can be represented and analyzed using words, tables, graphs, and equations.

Mathematical situations and structures can be translated and represented abstractly using variables, expressions, and equations.

The parameters in an equation representing a function affect the graph of the function in predictable ways (Charles, 2005).

**Important Assessment Look Fors:**

Given y=mx+b, the student identifies that m is slope, and b is the y-intercept.

The student describes slope as the ratio of the change in y to the change in x.

The student can describe where the slope and the y intercept can be found in a graph, table, or equation.

**Purposeful Questions: **

Can you identify the slope and y-intercept in the equations?

What determines if the slope is positive, negative or zero?

What determines the y-intercept?

Why is the y-intercept considered the constant?

How do you know the slope is positive or negative when you look at each representation?

**Student Strengths**

**Student Strengths**

Students can determine the slope, m, as rate of change in a proportional relationship between two quantities and write an equation in the

form y = mx to represent the relationship and the student can determine the y-intercept, b, in an additive relationship between two quantities and write an equation in the form y = x + b to

represent the relationship.

**Bridging Concepts**

Students can, when using real world context, make connections between proportional and additive relationships using tables, equations, and graphs. **Bridging Concepts**

**Standard 8.1****6****b**

**Standard 8.1**

**6**

**b**

Students can identify the slope and y-intercept of a linear function given a table of values, a graph, or an equation in y = mx + b form.

**Full Module with Instructional Tips & Resources: **

**Full Module with Instructional Tips & Resources:**

**Formative Assessments: **** **

**Formative Assessments:**

**Routines:**

**Routines:**

**Rich Tasks: **** **

**Rich Tasks:**

**Games:**

**Games:**

Desmos 8.16b Put the Point on the Line 8.16bd Match My Picture 8.16bd Match My Line 8.16bd Land the Plane 8.16bde Investigating T-Shirt Offers 8.16abde Linear Slalom↗

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## Standard 8.16c

**S****tandard**** 8.16****c**** **Determine the independent and dependent variable, given a practical situation modeled by a linear function.

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### Understanding the Learning Trajectory

**Big Ideas:**

Functions can be used to model relationships between quantities. (Achieve the Core)

In mathematical relationships, the value for one quantity depends on the value of the other quantity. (Charles, 2005)

The nature of the quantities in a relationship determines what values of the input and output quantities are reasonable. (Charles, 2005)

In a prediction context, the 𝑥-variable is referred to as the independent variable, explanatory variable, or predictor variable. The 𝑦-variable is referred to as the dependent variable, response variable, or predicted variable. Students should become equally comfortable with using the pairings (independent, dependent), (explanatory, response), and (predictor, predicted). Statistics builds on data, and in this lesson, students investigate bivariate data that are linearly related. Students examine how the dependent variable relates to the independent variable or how the predicted variable relates to the predictor variable (From Engage NY 8th grade lesson 10)

**Important Assessment Look Fors:**

The student describes independent and dependent variables as the input and output of the function and can provide real life examples of each.

The student can identify the independent and dependent variable given a function modeled by words, tables, graphs, and equations.

**Purposeful Questions: **

What is an independent variable? What is a dependent variable? What is a real life example of each?

How can you use an input/output table to help you identify the independent and dependent variables?

How can we identify the independent and dependent variables of a function in the following scenarios: given a word problem? A table? A graph? An equation?

Does being able to identify a variable being independent or dependent matter? Why or why not?

**Student Strengths**

**Student Strengths**

Students can represent a proportional situation between two quantities from practical situations.

**Bridging Concepts**

Students can identify the domain and range of a function represented as a set of ordered pairs, a table, or a graph of discrete points. **Bridging Concepts**

Students can identify inputs and outputs in a table and graph.

Students can make sense of which variables belong on the x- and y-axis when graphing.

**Standard 8.16****c**

**Standard 8.16**

**c**

Students can identify the dependent and independent variable, given a practical situation modeled by a linear function.

**Full Module with Instructional Tips & Resources: **

**Full Module with Instructional Tips & Resources:**

**Formative Assessments: **** **

**Formative Assessments:**

**Routines:**

**Routines:**

**Rich Tasks: **** **

**Rich Tasks:**

**Games/Tech:**

**Games/Tech:**

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## Standard 8.16d

**S****tandard**** 8.1****6d**** **Graph a linear function given the equation in y = mx + b form.

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### Understanding the Learning Trajectory

**Big Ideas:**

Mathematical rules (relations) can be used to assign members of one set to members of another set. A special rule (function) assigns each member of one set to a unique member of the other set.

Mathematical relationships can be represented and analyzed using words, tables, graphs, and equations.

In a linear function of the form y = mx, m is the constant of variation and it represents the rate of change of y with respect to x, otherwise known as slope (Charles, 2005)

Functions of the form y = mx + b are proportional relationships exactly when b = 0, so that y is proportional to x. (Achieve the Core)

The parameters in an equation representing a function affect the graph of the function in predictable ways. (Charles, 2005)

**Important Assessment Look Fors:**

The student can identify the slope and y-intercept in the equation.

The student identifies the value of the y-intercept as the point on the line where the line crosses the y- axis.

The student can apply rise over run (or change in y over change in x) for the slope starting at the y-intercept to find another point on the line.

The student can create a table of values to find other points that lie on the line and graph the points.

**Purposeful Questions: **

What is the y-intercept of the line? Where would the point be on the graph?

What is the slope of the line? How do you use the slope to find another point on the line?

If you create a table of values for the equation, how can you use that to graph the line?

**Student Strengths**

**Student Strengths**

Students can determine the difference between a proportional linear relationship and an additive linear relationship.

**Bridging Concepts**

Students can graph a line representing a proportional relationship between two quantities given the slope and an ordered pair, or given theequation in y = mx form where m represents the slope as rate of change.**Bridging Concepts**

Students can graph a line representing an additive relationship between two quantities given the y-intercept and an ordered pair, or given the equation in the form y = x + b, where b represents the y-intercept.

**Standard 8.1****6d**

**Standard 8.1**

**6d**

Students can, when given the equation of a linear function in the form y = mx + b, graph the function.

**Full Module with Instructional Tips & Resources: **

**Full Module with Instructional Tips & Resources:**

**Formative Assessments: **** **

**Formative Assessments:**

**Routines:**

**Routines:**

**Rich Tasks: **** **

**Rich Tasks:**

**Games/Tech:**

**Games/Tech:**

Desmos 8.16bd Match My Picture 8.16bd Match My Line 8.16bd Land the Plane 8.16bde Investigating T-Shirt Offers 8.16abde Linear Slalom↗

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## Standard 8.16e

**S****tandard**** 8.1****6e**** **Make connections between and among representations of a linear function using verbal descriptions, tables, equations, and graphs.

**(Pull down for more)**

### Understanding the Learning Trajectory

**Big Ideas:**

Mathematical rules (relations) can be used to assign members of one set to members of another set. A special rule (function) assigns each member of one set to a unique member of the other set. (Charles, 2005)

In a linear function of the form y = mx, m is the constant of variation and it represents the rate of change of y with respect to x, otherwise known as slope (Charles, 2005)

Functions of the form y = mx + b are proportional relationships exactly when b = 0, so that y is proportional to x. (Achieve the Core)

If two quantities vary proportionally, that relationship can be represented as a linear function. (Charles, 2005)

**Important Assessment Look Fors:**

The student identifies slope as the change in y over the change in x.

The student is able to represent the same situation in a table, graph, equation and words.

The student expresses that m represents the slope and b represents the y-intercept.

The student identifies the constant in the practical situation as the y-intercept on the graph.

The student corresponds the constant rate of change in the practical situation as the slope on the graph.

**Purposeful Questions: **

How can you find the slope from a table? An equation? A graph? From a verbal description?

How can you find the y-intercepts from a table? An equation? A graph? From a verbal description?

What is the constant in the practical situation? How do you know?

What parts of the practical situation match the table? The graph? The equation?

**Student Strengths**

**Student Strengths**

Students can make connections between and among representations of a proportional or additive relationship between two quantities using verbal descriptions, tables, equations, and graphs.

**Bridging Concepts**

Students can write the equation of a linear function in the form y = mx + b given values for the slope, m, and the y-intercept or given apractical situation in which the slope, m, and y-intercept aredescribed verbally.**Bridging Concepts**

**Standard 8.1****6e**

**Standard 8.1**

**6e**

Students can make connections between and among representations of a linear function using verbal descriptions, tables, equations, and graphs.

**Full Module with Instructional Tips & Resources: **

**Full Module with Instructional Tips & Resources:**

**Formative Assessments: **** **

**Formative Assessments:**

**Routines:**

**Routines:**

**Rich Tasks: **** **

**Rich Tasks:**

**Games/Tech:**

**Games/Tech:**

Desmos 8.16bde Investigating T-Shirt Offers 8.16abde Linear Slalom 8.16e Pool Border Problem 8.16e LEGO Prices 8.16e How are the Patterns Changing? 8.16e Linear Pattern Match 8.16e Charge!↗

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## Standard 8.17

**S****tandard**** 8.1****7**** **Solve multistep linear equations in one variable with the variable on one or both sides of the equation, including practical problems that require the solution of a multistep linear equation in one variable.

**(Pull down for more)**

### Understanding the Learning Trajectory

**Big Ideas:**

A given equation can be represented in an infinite number of different ways that have the same solution (Charles, 2005).

A variety of concrete materials such as colored chips, algebra tiles, or weights on a balance scale may be used to model solving equations in one variable.

Properties of real numbers and properties of equality can be applied when solving equations, and justifying solutions.

Some problem situations can be represented as algebraic expressions or algebraic equations.

**Important Assessment Look Fors:**

The student accurately distributes where necessary.

The student combines like terms as needed as well as uses the appropriate inverse operation to isolate the variable.

The student uses the order of operations correctly.

The student creates an equation that accurately represents the practical situation.

**Purposeful Questions: **

What is the important information in this problem?

How did you determine the terms and their placement within the equation?

How did you determine what is unknown?

Were all negatives distributed correctly?

Were only like terms combined?

How did you check to see if your solution is correct? Does your solution make sense in the context of the problem.

**Student Strengths**

**Student Strengths**

Students can solve one-step linear equations in one variable, including practical problems that require the solution of a one-step linear equation in one variable.

**Bridging Concepts**

Students can solve two-step linear equations in one variable, including practical problems that require the solution of a two-step linear equation in one variable.**Bridging Concepts**

**Standard 8.1****7**

**Standard 8.1**

**7**

Students can solve multistep linear equations in one variable with the variable on one or both sides of the equation, including practical problems that require the solution of a multistep linear equation in one variable.