## Formative Assessment and Bridging activities

These materials are part of an iterative design process and will continue to be refined during the 2021-2022 school year. Feedback is being accepted at the link below.
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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.
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## Standard 7.1b

Standard 7.1b Compare and order numbers greater than zero written in scientific notation.

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## Understanding the Learning Trajectory

Big Ideas:

• Students develop the understanding that scientific notation should be used whenever the situation calls for use of very large or very small numbers.

• A number written in scientific notation is the product of two factors — a decimal greater than or equal to 1 but less than 10, and a power of 10.

• Smaller numbers always lie to the left of larger numbers on the number line

Important Assessment Look Fors:

• The student correctly compared numbers written in scientific notation that had the same exponents.

• The student correctly compared numbers written in scientific notation that had negative exponents.

• The student was able to justify their reasoning.

Purposeful Questions:

• How are numbers written in scientific notation with positive powers of ten different than those with negative powers of ten?

• How did you know which was the smallest number?

• How did you know which was the largest number?

• Are powers of ten with a negative exponent negative or positive numbers? Explain how you know.

Student Strengths

Students can compare and order positive rational numbers.

Bridging Concepts

Students can convert between numbers in scientific notation and standard form.

Standard 7.1b

Students can compare and order numbers greater than zero written in scientific notation.

## Standard 7.1c

Standard 7.1c Compare and order rational numbers.

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## Understanding the Learning Trajectory

Big Ideas:

• Rational numbers may be expressed as positive and negative fractions or mixed numbers, positive and negative decimals, integers and percents.

• Equivalent relationships among fractions, decimals, and percents may be determined by using concrete materials and pictorial representations.

• Since fractions, decimals and percents are essentially the same numbers in different forms, they can be compared and ordered. Fractions, decimals, and percents can be compared and ordered using a variety of strategies including using benchmarks (0, halves, wholes), naming equivalencies, and other reasoning strategies.

• Negative numbers lie to the left of zero and positive numbers lie to the right of zero on a number line. Smaller numbers always lie to the left of larger numbers on the number line. Rational numbers can be compared using greater than, less than, or equal.

Important Assessment Look Fors:

• The student is able to compare rational numbers using greater than, less than, or equal to symbols.

• The student is able to compare negative rational numbers.

• The student can use a model to help show the relationship between equivalent rational numbers.

• The student can determine the relationship between rational numbers and their location on the number line.

Purposeful Questions:

• How is it possible to compare/order numbers that are represented in different formats?

• How can you determine whether a number is equivalent to, greater than, or less than another number?

• How can understanding benchmark numbers, such as ½, help when comparing rational numbers?

• Where might you place these numbers on the number line?

Student Strengths

Students can compare and order fractions, mixed numbers, and/or decimals, in a given set from least to greatest and greatest to least.

Bridging Concepts

Students can compare and order positive rational numbers.

Standard 7.1c

Students can compare and order rational numbers without the use of a calculator.

## Standard 7.1D

Standard 7.1d Determine square roots of perfect squares.

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## Understanding the Learning Trajectory

Big Ideas:

• A perfect square is a whole number whose square root is an integer.

• A square root of a number is a number which, when multiplied by itself, produces the given number.

• The square root of a number can be represented geometrically as the length of a side of a square.

Important Assessment Look Fors:

• Students correctly determined the values that are perfect squares.

• Students correctly determined the values that are not perfect squares.

• Students justified their reasoning for determining whether the values were perfect squares or not.

• Students justify their reasoning using connections to the area of a geometric square and its square roots.

Purposeful Questions:

• How did you determine the missing value for the perfect square or square root?

• Can you justify using a picture?

Student Strengths

Students can recognize and represent patterns with perfect squares.

Bridging Concepts

Students can understand the connection between a geometric square, a perfect square, and a square root.

Standard 7.1d

Students can determine square roots of perfect squares.

## Standard 7.2

Standard 7.2 Solve practical problems involving operations with rational numbers.

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## Understanding the Learning Trajectory

Big Ideas:

• Just as it is important to expose students to a variety of problem situations with whole numbers, it is important to expose students to a variety of problem situations with rational numbers. To develop students’ problem solving abilities with rational numbers, use familiar contextual problems like using fractions with baking or measuring fabric, using percent for discounts or tips, and using decimals for budgeting money (Cramer & Whitney, 2010; Lewis, Gibbons, Kazemi, & Lind, 2015).

• Estimations can help students keep the focus on the meanings of rational numbers and the operation. Benchmarks can help mentally add or subtract rational numbers.

• Relate problems like multiplying a whole number by a fraction(5 x ½) with equal group problems where one iterates the fractional part. Then relate problems multiplying a fraction by a whole number (½ of 12) as partitioning or as an operator (Lamon, 2012). Relate the different meanings for operation like partitive and measurement models for division when modeling division problems with fractions.

Important Assessment Look Fors:

• The student displayed evidence of using a problem-solving strategy.

• The student understood the necessary operation(s) to solve the problem.

• The student’s implementation of their chosen strategy resulted in the correct solution.

• The student justified their answer through mathematical reasoning or visual representation.

Purposeful Questions:

• How do you know which operation is needed given the information in the problem?

• How can you use estimation to gauge if your answer (i.e., sum, difference, product or quotient) is reasonable?

• What happens to products when multiplying by a fraction? What happens to quotients when dividing by a fraction?

• What situations have you experienced where you compute fractions, decimals and/or percent in your everyday life?

Student Strengths

Students can solve single-step practical problems involving rational numbers.

Bridging Concepts

Students can solve multi-step practical problems using integers, fractions, or decimals.

Standard 7.2

Students can solve practical problems involving operations with rational numbers.

## Standard 7.3

Standard 7.3 Solve single-step and multistep practical problems, using proportional reasoning

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## Understanding the Learning Trajectory

Big Ideas:

• A proportion is a statement of equality between two ratios.

• Equivalent ratios arise by multiplying each value in a ratio by the same constant value.

• Ratio tables are an effective way to show proportionality.

• In a proportional relationship, two quantities increase multiplicatively. One quantity is a constant multiple of the other.

• A proportion is an equation which states that two ratios are equal. When solving a proportion, the ratios may first be written as fractions.

Important Assessment Look Fors:

• The student created a correct proportion using the given ratio and the additional value.

• The student is able to determine if there is only one or multiple steps to finding a solution.

• The student showed a direct relationship between the ratios and their answer.

• The student justified their response in writing/proportions/ratio table/picture.

• The student was able to estimate to determine reasonableness of the solution.

Purposeful Questions:

• How did you determine the length/width of the actual object? Of the object in the scale drawing?

• How does the scale help you determine the actual length/width of the object?

• What is the direct relationship between the picture and the actual object’s length?

• What is the factor that you are multiplying by in the task?

• What is another way to solve this proportion? How can you check your solution?

Student Strengths

Students can find equivalent fractions. Students can write equal ratios creating a proportion.

Bridging Concepts

Students can solve single-step ratio and proportion problems.

Standard 7.3

Students can solve single and multi-step practical problems using proportional reasoning.

## Standard 7.4a

Standard: 7.4a Describe and determine the volume and surface area of rectangular prisms and cylinders

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## Understanding the Learning Trajectory

Big Ideas:

• The volume of a three-dimensional figure is a measure of capacity and is measured in cubic units. The development of the volume of a rectangular prism formula is parallel to the development of the area of a rectangle. The area of the base determines how many cubes can cover the base, forming a single unit, a layer of cubes. The height of the box then determines how many of the layers will fit in the box (Van de walle et al., 2019)

• The volume of rectangular prisms and cylinders are analogous in that both are multiplied by the height but different because of the type of shape on the base. Students should be encouraged to describe and connect the volume and surface area using concrete objects, diagrams, and formulas.

• To determine the surface area of rectangular prisms and cylinders, one can use concrete objects, nets, diagrams, and formulas. One can draw a net and calculate the area of each face then add up the area of all the faces.

Important Assessment Look Fors:

• The student is able to utilize the properties of a rectangle to determine the side lengths where the amount isn’t given.

• The student uses the correct formula to determine the area of the given shape.

• The student is able to compute the surface area by either adding each two-dimensional component, or by using the distributive property to account for the opposing faces of the shape, and then adding sums of the opposing faces.

• The student uses three dimensions, l x w x h, to find the volume of the prism or finds the area (A) of the base then multiplies the height, A x h = Volume.

Purposeful Questions:

• How do you know the measurement for each side length?

• How many smaller shapes did you visualize separating the rectangular prism into to find your answer?

• Why might your answer be in a different dimensional unit when finding an area versus a surface area?

• How can you use the area of the base to help you find the volume?

• How is the volume of the prism and the cylinder similar or different?

• One friend says the surface area for a cylinder is A=2πrh+2πr² and another friend says, it is 2πr(h + r). Who is correct? How can you prove your answer?

Student Strengths

Students can solve practical problems that involve perimeter, area, and volume in standard units of measure.

Bridging Concepts

Students can solve problems, including practical problems, involving circumference, area of a circle; area and perimeter of rectangles.

Standard 7.4a

Students can describe and determine the volume and surface area of rectangular prisms and cylinders.

## Standard 7.5

Standard: 7.5 Solve problems, including practical problems, involving the relationship between corresponding sides and corresponding angles of similar quadrilaterals and triangles.

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## Understanding the Learning Trajectory

Big Ideas:

• Similar polygons have corresponding sides that are proportional and corresponding interior angles that are congruent.

• Similarity has practical applications in a variety of areas, including art, architecture, and the sciences.

• Similarity does not depend on the position or orientation of the figures.

• Congruent polygons have the same size and shape. Corresponding angles and sides are congruent.

Important Assessment Look Fors:

• The student identified corresponding sides and corresponding congruent angles of similar quadrilaterals and triangles.

• The student was able to write similarity statements given two similar quadrilaterals or triangles.

• The student wrote proportions to express the relationships between the lengths of corresponding sides of similar quadrilaterals and triangles.

• The student solved a proportion to determine a missing side length of similar quadrilaterals or triangles.

• The student determined the unknown angle measures in a similar quadrilateral or triangle.

Purposeful Questions:

• How did you determine which angles correspond with angles M and R?

• Does the orientation of the shapes matter when identifying corresponding parts?

• How can understanding the concept of congruence help with this task?

• Did you change the orientation of the figures to find the missing angle measures?

Student Strengths

Students can distinguish between similar and congruent polygons.

Bridging Concepts

Students can identify corresponding parts of similar figures.

Students can set up proportions written as 𝑎/𝑏 = 𝑐 /𝑑 .

Standard 7.5

Solve problems, including practical problems, involving the relationship between corresponding sides and corresponding angles of similar quadrilaterals and triangles.

## Standard 7.7

Standard: 7.7 Apply translations and reflections of right triangles or rectangles in the coordinate plane

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## Understanding the Learning Trajectory

Big Ideas:

• A transformation of a figure called the preimage changes the size, shape, or position of the figure to a new figure called the image.

• Translations and reflections do not change the size or shape of a figure (e.g., the preimage and image are congruent figures).

• Translations and reflections change the position of a figure. A translation is a transformation in which an image is formed by moving every point on the preimage the same distance in the same direction.

• A reflection is a transformation in which an image is formed by reflecting the preimage over a line called the line of reflection. All corresponding points in the image and preimage are equidistant from the line of reflection.

Important Assessment Look Fors:

• The student can identify the coordinates of the image of a right triangle or rectangle that has been translated or reflected when given a preimage.

• The student can sketch the image of a right triangle or rectangle that has been translated vertically, horizontally, or a combination of both.

• The student can sketch the image of a right triangle or rectangle that has been reflected over the x- or y-axis.

• The student can sketch the image of a right triangle or rectangle that has been translated and reflected over the x- or y-axis or reflected over the x- or y-axis and then translated.

Purposeful Questions:

• What is different between the coordinates of the preimage and new image when you reflect an image across the X-axis?

• When you translate a point to the right 5 units, what happens to the ordered pair?

• When you translate a point to down 4 units, what happens to the ordered pair?

• What do you notice about the coordinates of a point when it is reflected over the x-axis?

• What do you notice about the coordinates of a point when it is reflected across the y-axis?

Student Strengths

Students can identify parts of the coordinate plane and plot points in all four quadrants.

Bridging Concepts

Students can identify the coordinates of the image of a right triangle or rectangle that has been translated or reflected when given a preimage.

Standard 7.7

Students can apply translations and reflections of right triangles or rectangles in the coordinate plane.

## Standard 7.8a

Standard: 7.8a Determine the theoretical and experimental probabilities of an event

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## Understanding the Learning Trajectory

Big Ideas:

• CHANCE: 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.

• Probability can provide a basis for making predictions. Some probabilities can only be determined through experimental trials. An event that is certain to happen will always happen (the probability is 1.) and an event that is impossible will never happen (the probability is 0.)

• The theoretical probability is what SHOULD happen and experimental probability is the actual RESULT (outcome) of a trial and probability of an event can be represented as a ratio or equivalent fraction, decimal, or percent.

Important Assessment Look Fors:

• The student can determine the theoretical probability of an event.

• The student can determine the experimental probability of an event.

• The student can show that probability can be expressed as fractions/decimals/percents.

• The student shows understanding that frequency directly impacts experimental probability.

Purposeful Questions:

• How do you determine a sample space in a given probability situation?

• How do you determine if an event is impossible, less likely, more likely, certain?

• How can you represent the probability of an event using a fraction/decimal/percent?

• How is the experimental probability different from the theoretical probability?

Student Strengths

Students can determine the probability of a given sample space.

Bridging Concepts

Students can predict the probability of an event and also run trials to compare their guesses with actual trial results.

Standard 7.8a

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

## Standard 7.8b

Standard: 7.8b Investigate and describe the difference between the experimental probability and theoretical probability of an event

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## Understanding the Learning Trajectory

Big Ideas:

• CHANCE: 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.

• Probability can provide a basis for making predictions. Some probabilities can only be determined through experimental trials. An event that is certain to happen will always happen (the probability is 1.) and an event that is impossible will never happen (the probability is 0.)

• The theoretical probability is what SHOULD happen and experimental probability is the actual RESULT (outcome) of a trial and probability of an event can be represented as a ratio or equivalent fraction, decimal, or percent.

• In experimental probability, as the number of trials increases, the experimental probability gets closer to the theoretical probability (Law of Large Numbers).(VDOE, Curriculum Framework)

Important Assessment Look Fors:

• The student can determine the difference between theoretical and experimental probability given an event.

• The student can describe changes in the experimental probability as the number of trials increases.

• The student 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.

Purposeful Questions:

• What is the theoretical probability for this event to occur?

• What is the experimental probability for this event to occur?

• What do you think would happen to the experimental probability if you increased your number of trials? How would it compare to the theoretical probability?

• How is experimental probability different from theoretical probability? Provide an example.

Student Strengths

Students can predict the probability of an event and also run trials to compare their guesses with actual trial results.

Bridging Concepts

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

Standard 7.8b

Students can investigate and describe the difference between the experimental probability and theoretical probability of an event.

## Standard 7.9b

Standard: 7.9b Make observations and inferences about data represented in a histogram

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## Understanding the Learning Trajectory

Big Ideas:

• A histogram is used to deduce information and analyze data when presented in consecutive intervals.

• Comparisons, predictions and inferences are made by examining characteristics of a data set displayed in a variety of graphical representations to draw conclusions.

• Data analysis helps describe data, recognize patterns or trends, and make predictions.

• Inferential statistics uses data in a sample selected from a population to describe features of the population (Kader & Jacobbe, 2013. NCTM, Essential Understandings Statistics 6-8).

• The sampling distribution of a statistic describes the sample-to-sample variability and values of the statistic from the multiple samples of the same size selected from the same population.

• Selecting a simple random sample from a population is a fair way to select a sample.

• The predictable pattern for the sampling distribution of a statistic based on random sampling provides a way for making inferences about a population.

Important Assessment Look Fors:

• The student shows understanding of the verbiage of “no fewer than” means “equal to or more.”

• The student recognizes that there is more than one interval that serves a solution.

• The student computes the combination of the occurrences in each appropriate interval, and not the amount represented by the interval.

• The student is able to create a proper representation of the number of successful solutions to the scenario in comparison to the total number of cities represented.

Purposeful Questions:

• Do you know all of the exact data values? Why or why not?

• Which measures of center could be computed from a histogram? Explain your reasoning.

• Which intervals would be included in fewer than___? At least_____? Greater than ____? How do you know?

• How will you convert the data from the histogram to a percentage?

Student Strengths

Students can interpret data represented in line plots, circle graphs, and stem-and-leaf plots.

Bridging Concepts

Students can represent data in a histogram.

Standard 7.9b

Students can make observations and inferences about data represented in a histogram.

## Standard 7.10a

Standard: 7.10a 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

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## Understanding the Learning Trajectory

Big Ideas:

• When two quantities, x and y, vary in such a way that one of them is a constant multiple of the other, the two quantities are “proportional.”

• The slope of a proportional relationship can be determined by finding the unit rate.

• The slope of a line is a rate of change, a ratio describing the vertical change to horizontal change of the line.

• Linear equations can be solved by symbolic, graphical, and numerical methods. On some occasions and in some contexts, one solution method may be more elegant, efficient, or informative than another (Lloyd, Herbel Eisenmann, Star & Zbiek, 2011, NCTM’s Essential Understanding of Expressions, Equations and Functions 6-8)

Important Assessment Look Fors:

• The student is able to determine the slope of a proportional relationship by finding the unit rate.

• The student is able to determine the slope of proportional relationship by using a ratio that describes the vertical change to horizontal change of the line.

• The student is able to describe a proportional relationship from a table.

• The student is able to describe a proportional relationship from a verbal description.

Purposeful Questions:

• How could you represent a proportional relationship?

• How did you determine the slope from the table?

• How did you determine the equation from the table?

• How can you represent this problem situation using algebraic symbols, numbers, graphs, verbal descriptions, and tables?

Student Strengths

Students can represent a proportional relationship between two quantities.

Bridging Concepts

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

Standard 7.10a

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

## Standard 7.10b

Standard: 7.10b Graph a line representing a proportional relationship between two quantities given the slope and an ordered pair, or given the equation in y = mx form where m represents the slope as rate of change

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## Understanding the Learning Trajectory

Big Ideas:

• When two quantities, x and y, vary in such a way that one of them is a constant multiple of the other, the two quantities are “proportional.”

• The slope of a proportional relationship can be determined by finding the unit rate.

• The slope of a line is a rate of change, a ratio describing the vertical change to horizontal change of the line.

Important Assessment Look Fors:

• The student utilizes the origin and the given slope to locate another point in a given graph.

• The student is able to insert a chosen input value into the equation of the line, and determine an output value.

• The student can consistently determine the x-value given the y-value, or find the y-value given the x-value in a linear equation.

Purposeful Questions:

• How will knowing the slope of the line help you to figure out an initial set of points?

• What strategy do you prefer to use when selecting three other points on the line? Explain your strategy.

• Is the incline of the line between all of your points identical? Why is this important?

Student Strengths

Students can identify, describe, create, express, and extend number patterns found in objects, pictures, numbers, and tables.

Bridging Concepts

Students can identify the coordinates of a point and graph ordered pairs in a coordinate plane. Students can identify proportional relationships given a table or verbal description.

Standard 7.10b

Students can graph a line representing a proportional relationship between two quantities given the slope and an ordered pair, or given the equation in y = mx form.

## Standard 7.10c

Standard: 7.10c 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

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## Understanding the Learning Trajectory

Big Ideas:

• The y-intercept of a line represents the initial value in a relationship between two variables.

• Two quantities, x and y, have an additive relationship when a constant value, b, exists where y = x + b, where b does not equal 0.

• The y-intercept of an additive relationship can be determined by adding a value to the other quantity.

Important Assessment Look Fors:

• The student is able to determine the y-intercept of an additive relationship from a table, graph, and equation.

• The student is able to write the equation of a line of an additive relationship from a table, graph, and equation.

• The student is able to identify the y-intercept of an additive relationship by adding a value to the other quantity in a table.

Purposeful Questions:

• How could you represent an additive relationship?

• What is the difference between an additive relationship and a proportional relationship?

• How did you determine the y-intercept from the table?

• How did you determine the equation of the line?

Student Strengths

Students can determine the slope as a 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

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

Standard 7.10c

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

## Standard 7.10d

Standard: 7.10d 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

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## Understanding the Learning Trajectory

Big Ideas:

• The y-intercept and an ordered pair, or an equation given in slope-intercept form, provide a pathway to determining the slope of a line.

• The y-intercept of a line represents the initial value in a relationship between two variables.

• Two quantities, x and y, have an additive relationship when a constant value, b, exists where y = x + b, where b does not equal 0.

• The y-intercept of an additive relationship can be determined by adding a value to the other quantity.

Important Assessment Look Fors:

• The student can plot an ordered pair when provided with an input-output combination, and when given an intercept.

• The student can describe the relationship between slope of a line, and every ordered pair on the line.

• The student can graph a line given an ordered pair and y-intercept.

• The student can locate additional points to be included on the graph of a line, working in accordance with the original constraints.

Purposeful Questions:

• How do you know that the two additional points you have chosen indicate the same slope as the two given points?

• What is the y-intercept of the graph of this line? How do you know?

• What is the difference between an additive and proportional relationship? Can you provide an example of each?

Student Strengths

Students can identify, describe, create, express, and extend number patterns found in objects, pictures, numbers, and tables.

Bridging Concepts

Students can identify the coordinates of a point and graph ordered pairs in a coordinate plane. The students can identify additive and proportional relationships.

Standard 7.10d

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.

## Standard 7.10e

Standard: 7.10e Make connections between and among representations of a proportional or additive relationship between two quantities using verbal descriptions, tables, equations, and graphs

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## Understanding the Learning Trajectory

Big Ideas:

• Functions can be represented as ordered pairs, tables, graphs, equations, physical models, or in words.

• Any given relationship can be represented using multiple representations.

• Relationships can be described and generalizations made for mathematical situations that have numbers or objects that repeat in predictable ways.

Important Assessment Look Fors:

• Students can represent additive and proportional relationships as ordered pairs, tables, graphs, equations, and in words.

• Students can determine the relationship between the same function written as an ordered pairs, tables, graphs, equations, physical models and in words.

• Students can create one form of the additive or proportional relationship when given another form of the same relationship.

• Students can represent the same proportional or additive relationship in a table, graph, equation and words.

Purposeful Questions:

• How can you describe and understand linear relationships through various representations -in algebraic symbols, situations, graphs, verbal descriptions, tables? Give an example of how you can make connections between and among these representations.

• What are similarities and differences between a proportional relationship and an additive relationship? (or ask how is y=2x different from y=x+2?).

• How could you represent this relationship in a different format?

• Show the additive or proportional relationship in all of these representations. Explain how you know.

Student Strengths

Students can determine the slope as a 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

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

Standard 7.10e

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

## Standard 7.11

Standard: 7.11 Evaluate algebraic expressions 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:

• The student replaces the given values for the correct variable.

• The student follows the correct steps of order of operations.

• The student correctly evaluates each step of the expression.

Purposeful Questions:

• What does it mean when you see two variables right beside one another?

• What is the first step in evaluating this given expression? How do you know?

• What is the coefficient in this expression and what operation does a coefficient represent?

• What does the grouping symbol indicate?

• Explain how you arrived at your final expression.

Student Strengths

Students can evaluate numerical expressions with integers using order of operations with up to 3 steps.

Bridging Concepts

Students can evaluate numerical expressions with exponents up to 4, and grouping symbols such as absolute value and radicals.

Standard 7.11

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

## Standard 7.12

Standard: 7.12 Solve two-step linear equations in one variable, including practical problems that require the solution of a two-step linear equation in one variable.

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## 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.

• The inverse operation for addition is subtraction, and the inverse operation for multiplication is division.

• 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 can identify the inverse operation(s) required for solving the equation.

• The student can translate the equation between visual and concrete representations.

• The student can verify their solution.

• The student provided a response that was reasonable.

Purposeful Questions:

• What is the important information in this problem?

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

• What inverse operations will you use to solve your written equation?

• How did you determine what is unknown?

Student Strengths

Students can perform operations and simplify expressions with fractions, decimals, and integers.

Bridging Concepts

Students can solve one-step linear equations, including practical problems.

Standard 7.12

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

## Standard 7.13

Standard: 7.13 Solve one- and two-step linear inequalities in one variable, including practical problems, involving addition, subtraction, multiplication, and division, and graph the solution on a number line

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## Understanding the Learning Trajectory

Big Ideas:

• An inequality is “another way to describe a relationship between expressions but instead of showing that the values of two expressions are equal, inequalities indicate that the value of one expression is greater than ( or greater than or equal to) the value of the other expression” (NCTM, 2016, p. 37).

• Solving an inequality is similar to solving an equation in that “we need to add or subtract the same value on both sides of the expression in an inequality to preserve the particular relationship of inequality between the expressions” (NCTM, 2016, p. 39).

• When solving an inequality, multiplying and dividing both expressions by a negative number reverses the sign that indicates the relationship between the two expressions. For students to understand this mathematical behavior conceptually, they need to explore what happens when they multiply or divide by a -1 ( NCTM, 2016, p. 41).

Important Assessment Look Fors:

• The student can identify the inverse operation(s) required for solving the inequality.

• The student can translate inequalities between visual and concrete representations.

• The student can explain why inequalities have multiple values within its solution set.

• The student reverses the inequality sign when multiplying or dividing both sides of the inequality by a negative number, and can explain or show why this had to be done.

Purposeful Questions:

• What is the important information in this problem?

• How did you determine the terms and their placement within the inequality?

• What inverse operations will you use to solve your written inequality?

• What happens when you solve the inequality by multiplying or dividing by a negative? Why does this happen?

Student Strengths

Students can represent a practical situation with a linear inequality in one variable.

Bridging Concepts

Students can solve one-step linear inequalities in one variable, involving addition or subtraction, and graph the solution on a number line

Standard 7.13

Students can solve one and two-step linear inequalities, including practical problems, and graph the solution on a number line.