 ## 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 8.4
Standard 8.5

Standard 8.9b

Standard 8.10

Standard 8.11a

Standard 8.11b

Standard 8.15a

Standard 8.15b

Standard 8.16a

Standard 8.16b

Standard 8.16c

Standard 8.16d

Standard 8.16e

Standard 8.17

Standard 8.18

## Standard 8.1

Standard 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

Students can represent and determine equivalencies square roots and numbers written in scientific notation. Ordering may be in ascending or descending order.

Standard 8.1

Students can compare and order real numbers.     ## Standard 8.4

Standard 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:

• 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

Students can problem solve using rational numbers and proportional reasoning.

### Bridging Concepts

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

### Standard 8.4

Students can solve practical problems involving consumer applications.      ## Standard 8.14A

Standard 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

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.

### Standard 8.14A

Students can evaluate an algebraic expression for given replacement values of the variables.       ## Standard 8.14b

Standard 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

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.

### Standard 8.14B

Students can simplify algebraic expressions in one variable.    ### Routines: 