 ## 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.
Share Feedback for Grade 6 Modules

Standard 6.6a

Standard 6.6b

Standard 6.8a

Standard 6.8b

Standard 6.10b

Standard 6.11a

Standard 6.11b

Standard 6.12a

Standard 6.12b

Standard 6.12c

Standard 6.12d

Standard 6.13

Standard 6.14a

Standard 6.14b

## Standard 6.1

Standard 6.1 Represent relationship between quantities using ratio, and use appropriate notations such as a/b, a to b, and a:b.

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Understanding the Learning Trajectory

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Big Ideas:

• A ratio is a multiplicative comparison of quantities; there are different types of comparisons that can be represented as ratios (Charles, 2005).

• Ratios give the relative sizes of the quantities being compared, not necessarily the actual sizes (Charles, 2005).

• The student understands that relationships between quantities can be part to whole or part to part.

• The student understands that ratios can be written in multiple forms (Common Core Writing Team, 2019, Ratios and Proportional Relationships p. 3).

Important Assessment Look Fors:

• The student correctly determines if the scenario is a part to part or part to whole relationship.

• The student correctly expresses equivalent ratios for the given situation.

• The student expresses ratios in correct order (ex: 1:3 vs 3:1).

• The student expresses ratios using the correct format (a to b, a:b, or a/b).

Purposeful Questions:

• How do you know if you are looking for a part to part or a part to whole relationship?

• What strategies can you use to determine if ratios are equivalent?

• What are the different ways that you can represent a ratio? ### Student Strengths

Students can recognize part to whole relationships and use fractions to represent part to whole relationships.

### Bridging Concepts

Students can find equivalencies for part to whole relationships.

### Standard 6.1

Students can represent relationships between quantities using ratio, and use appropriate notations such as “a to b” a:b and a/b.     ### Routines:

• A vase holds red and white roses only. There are 1.5 times as many red roses as white roses. How many flowers might be in the vase? (Good Questions for Math Teaching, 2005, p. 71).

Games:

## Standard 6.2a

Standard 6.2a Represent and determine equivalencies among fractions, mixed numbers, decimals, and percents

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Understanding the Learning Trajectory

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Big Ideas:

• Fractions, decimals, and percents can be used to represent a part to whole relationship.

• Students should begin to identify equivalence among fractions and decimals starting with common fractions such as halves, thirds, fourths and eighths as decimal fractions. For example, using a decimal grid and shading ½ and 50/100= .5. A double number line, decimal grids, and rational numbers wheel (see chap. 16 of Van de Walle text) are useful models to connect decimals and fractions as one moves beyond common fractions to continue the development of fraction-decimal equivalence (Van de Walle et al., 2018).

• Decimal-fraction-percent equivalents can be named not just through procedures but through sense-making with our base-ten system (e.g. ⅖ = 4/10 =4/100= 0.4=40%). For example, although one can use division with the fraction to find the decimal equivalent, another way is to find an equivalent decimal fraction with 10 or 100 in the denominator to find the decimal and percent equivalent (e.g. 12/50=24/100=0.24=24%)

• Naming an equivalent fraction, decimal and percent means that the quantities are the same even though they are represented differently. ¾ is 0.75 and 75% written in a different form. Both ¾, 0.75, and 75% would appear at the same point on a number line, take up the same amount of space on an area model, and be shown similarly in a set or measurement model.

• Percents are used to solve practical problems including sales, data description, and data comparison.

Important Assessment Look Fors:

• The student recognizes, identifies, and names equivalent fractions, decimals and percents with concrete or pictorial models.

• The student recognizes, identifies, and names equivalent fractions, decimals and percents without concrete or pictorial models.

• The student demonstrates an understanding that fractional models (such as the one above showing ⅘) can also be written in many equivalent decimal and percent forms (0.8, 0.80, 80% etc.).

• The student can determine the equivalent value given a fraction, percent or decimal.

• The student can justify with reasoning why one form is equivalent to another form representing the same value.

Purposeful Questions:

• How do you know this model represents the equivalent values between fractions, decimals and percents?

• How does the model help you identify the equivalent values between fractions, decimals and percents?

• How can you compare two percents using pictorial representations?

• What strategy did you use to order your rational numbers?

• Which strategy is the most efficient for you and why? ### Student Strengths

Students can represent and identify equivalencies among fractions and decimals, with and without models.

### Bridging Concepts

Students can understand equivalent relationships and represent and identify equivalencies among fractions and decimals without models.

### Standard 6.2a

Students can represent and determine equivalencies among fractions, mixed numbers, decimals, and percents.    Games:

## Standard 6.2b

Standard 6.2b- Compare and order postive rational numbers

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Understanding the Learning Progression

Big Ideas:

• Any number can be represented in an infinite number of ways that have the same value and can be compared by their relative values (Charles, p.10, p.14). In order to use reasoning skills when comparing fractions, it is important to have students notice what happens to the size of fractions when the numerator increases and also when the denominator increases.

• In terms of decimal reasoning, students need to develop the notion that there is what we call decimal density where in between any two decimals there are an infinite number of other decimals (Widjaja et al., 2008).

• 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.

• Decimal to fraction-to percent equivalents can be named not just through procedures but through sense-making with our base-ten system (e.g. ⅖ = 4/10 = 40/100=40%= 0.4). For example, although one can use division with the fraction to find the decimal equivalent, another way is to find an equivalent decimal fraction with 10 or 100 in the denominator to find the decimal and percent equivalent (e.g. 12/50=24/100=0.24=24%).

Important Assessment Look Fors:

• Students can use a variety of strategies to order and compare rational numbers (close to 0, 1 or ½, converting to a different representation, using a model, using benchmarks, using equivalencies, etc…)

• The student demonstrates an understanding of percents that include a decimal point.

• The student demonstrates understanding of symbols repeating decimals.

• The student can explain the strategy and reasoning used to determine the order of the rational numbers.

• The student can determine a fraction, decimal, or percent that can fit a series of given criteria (less than, greater than, or between two quantities).

Purposeful Questions:

• Can you explain to me how you were able to determine that quantity a is less than/greater than/equal to quantity b?

• What strategy/strategies did you use in order to compare/order your numbers? Why was this an effective strategy?

• How are the strategies you use to compare and order fraction similar or different to the strategies you use to compare decimals and percents?

• How do you know ___ is greater/less than ___? (fill in rational numbers in the problem)

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

• What other way could you represent this rational number? ### Student Strengths

Students can compare and order fractions and/or decimals in a given set using models or a computational strategy.

### Bridging Concepts

Students can use multiple strategies to compare and order fractions, mixed numbers, and decimals.

### Standard 6.2b

Students can compare and order positive rational numbers.    ## Standard 6.3a

Standard 6.3a Identify and represent integers.

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Understanding the Learning Progression

Big Ideas:

• Integers are the whole numbers and their opposites on the number line, where zero is its own opposite.

• Each integer can be associated with a unique point on the number line, but there are many points on the number line that cannot be named by integers.

• An integer and its opposite are the same distance from zero on the number line. (Charles, 2005)

Important Assessment Look Fors:

• Students can correctly identify a value of a positive or negative integer in relation to 0 on a number line.

• Student has conceptual knowledge or real life situations, such as temperature (above/below zero), deposits/withdrawals in a checking account, golf (above/below par), time lines, football yardage, positive and negative electrical charges, and altitude (above/below sea level), students may not know what some of these are based on their cultural situation.

• Students should be able to model with a number line or with manipulatives to represent the situation.

• Students should be able to determine the opposite number of a given value.

Purposeful Questions:

• What is the difference in values of negative numbers in relation to 0 on a number line?

• What are some examples of practical situations where an integer can be used?

• What are opposites? ### Student Strengths

Students can create and solve single-step and multistep practical problems involving addition, subtraction, multiplication, and division of whole numbers.

### Bridging Concepts

Students can understand values on a number line and modeling with different manipulatives.

### Standard 6.3a

Students can identify and represent integers.    ### Routines:

• Josie is working on a homework assignment from class. She is asked to find the “opposites” of integers such as +12 and -48. Josie wonders, “Does every integer have an opposite? Since zero is an integer, does it have an opposite?” What do you think about Josie’s question? How would you answer it? (Good Questions for Math, 2005, pg. 39-41).

## Standard 6.3b

Standard 6.3b Compare and order integers.

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Understanding the Learning Progression

Big Ideas:

• An integer and its opposite are the same distance from zero on the number line.

• There is no greatest or least integer on the number line.

• A number to the right of another on the number line is the greater number.

• Numbers can be compared using greater than, less than, or equal.(Charles, 2005)

• Statements of inequality can be interpreted as statements about the relative position of two numbers on a number line diagram. (Common Core Writing Team, 2019, p. 8).

Important Assessment Look-fors:

• The student understands the meaning of ascending and descending.

• The student understands mathematical symbols <, > and cen conceptualize them from left to right, and right to left.

• The student understands that the further away from zero a negative number is, the smaller its value.

• The student understands the interval on the number line.

Purposeful questions:

• What intervals are measured on the number line?

• Where do you see positive and negative values?

• What does it mean when a negative number has a greater distance away from 0 on a number line?

• How do you know this integer is larger? How do you know this integer is smaller? ### Student Strengths

Students can model integers, correctly label a number line, and understand mathematical symbols and use them correctly.

### Bridging Concepts

Students can compare and order integers in ascending or descending order and identify integers represented by a point on a number line.

### Standard 6.3b

Students can compare and order integers with a number line; and compare integers using mathematical symbols.     ## Standard 6.5a

Standard 6.5a Multiply and divide fractions and mixed numbers.

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Understanding the Learning Progression

Big Ideas:

• Different real-world interpretations can be associated with division calculations involving fractions (decimals).

• The product of two positive fractions each less than one is less than either factor.

• A fraction division calculation can be changed to an equivalent multiplication calculation (i.e., a/b ÷ c/d = a/b x d/c, where b, c, and d = 0) (Charles, 2005).

Important Assessment Look-fors:

• The student can model multiplication and division with multiple representations.

• The student can use estimation to develop computational strategies.

• The student can demonstrate understanding/justify that when multiplying a fraction by a fraction, this results in the part of a part.

• The student can demonstrate understanding/justify that when multiplying a fraction by a whole number, this results in the part of a whole.

Purposeful questions:

• How can estimation be used to determine the reasonableness of a solution?

• When a whole number is divided by a fraction, why is the quotient larger than the dividend?

• What kind of product do you get when you multiply a whole number by a fraction?

• Can you show me how your multiplication/division model relates to your multiplication/division computation? ### Student Strengths

Students can solve single step practical problems involving multiplication of a whole number limited to twelve or less and a proper fraction with models.

### Bridging Concepts

Students can simplify fractions, understand how to change mixed fractions to improper fractions and vice versa, and multiply a whole number by a fraction.

### Standard 6.5a

Students can multiply and divide fractions and mixed numbers.      ## Standard 6.5B

Standard 6.5b Solve single-step and multistep practical problems involving addition, subtraction, multiplication, and division of fractions.

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Understanding the Learning Progression

Big Ideas:

• The real-world actions for addition and subtraction of whole numbers are the same for operations with fractions.

• Different real-world interpretations can be associated with the product of a whole number and fraction, a fraction and whole number, and a fraction and fraction.

• Different real-world interpretations can be associated with division calculations involving fractions (Charles, 2005).

• Fractional relationships can be used when solving practical problems involving addition, subtraction, multiplication and/or division (Common Core Writing Team, 2019, Number Systems p. 5-6)

• In mathematics, emphasis should be placed on representing the problem and applying reasoning to understand it rather than relying on keywords. (See Grade 4 VDOE Standards of Learning Document p.19).

• In mathematics, estimation should be used to determine if an answer is reasonable.

Important Assessment Look-fors:

• The student can model fraction operations using manipulatives or pictorial representations.

• The student can show their answer expressed in simplest form.

• The student can justify why the number sentence matches the model and situation presented in the problem.

• The student can justify why the answer makes sense.

• The student can give a complete answer, including units of measurement.

Purposeful questions:

• What are you trying to find in the problem?

• How can you begin to organize your thinking ? Will a picture or chart help you?

• What is happening in the problem? What does that tell you about which operation(s) you will need to use?

• How do you know your answer is reasonable and what does it mean? ### Student Strengths

Students can solve single step problems and add and subtract fractions.

### Bridging Concepts

Students can multiply a whole number and a proper fraction. Students can solve multistep problems using whole numbers and/or fractions.

### Standard 6.5b

Students can solve single-step and multistep practical problems involving addition, subtraction, multiplication, and division of fractions.     ## Standard 6.5C

Standard 6.5c Solve multistep practical problems involving addition, subtraction, multiplication and division of decimals.

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Understanding the Learning Progression

Big Ideas:

• The real-world actions for addition and subtraction of whole numbers are the same for operations with decimals.

• Different real-world interpretations can be associated with the product of a whole number and decimal, a decimal and whole number, and a decimal and decimal.

• Different real-world interpretations can be associated with division calculations involving decimals (Charles, 2005).

• In mathematics, emphasis should be placed on representing the problem and applying reasoning to understand it rather than relying on keywords (See Grade 4 VDOE Standards of Learning Document p.19).

• Examples of practical situations solved by using estimation strategies include shopping for groceries, buying school supplies, budgeting an allowance, and sharing the cost of a pizza or the prize money from a contest.

• Different strategies can be used to estimate the result of computations and judge the reasonableness of the result.

Important Assessment Look-fors:

• The student determines the correct operation or operations needed to solve the problem and can justify his/her choices.

• The student uses estimation to determine reasonableness of solution.

• The student can justify why the answer makes sense.

• The student uses a strategy to organize the information presented in the problem, such as a chart, diagram, list, or picture.

Purposeful questions:

• What are you trying to find in the problem?

• How can you begin to organize your thinking ? Will a picture or chart help you?

• How do you know your answer is reasonable and what does it mean?

• How can you determine the operation or operations that can be used to solve the problem? ### Student Strengths

Students can create and solve single-step and multistep practical problems involving addition, subtraction of decimals.

### Bridging Concepts

Students can solve multistep practical problems with multiplication of decimals and single-step practical problems involving division of decimals.

### Standard 6.5c

Students can solve multistep practical problems involving addition, subtraction, multiplication and division of decimals.    ### Routines: 