 ## 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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The Bridging Standards in bold below are currently live. Others are coming soon!

Standard 7.7

Standard 7.8a

Standard 7.8b

Standard 7.9b

Standard 7.10a

Standard 7.10b

Standard 7.10c

Standard 7.10d

Standard 7.10e

Standard 7.13

## 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 7.11

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.     ### Routines: 