Formative Assessment and Bridging activities

Grade 6

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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 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.

Rich Tasks:

• 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/Tech:

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/Tech:

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.

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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.

Rich Tasks:

• 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).

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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.

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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.

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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.

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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?

• Have you answered the question asked in 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.

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Standard 6.6a

Standard 6.6a Add, subtract, multiply, and divide 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.

• The real-world actions for operations with integers are the same for operations with whole numbers. (Charles, 2005).

• The sum of opposite integers is equal to zero. (Arizona, Number System, p. 7-9)

Important Assessment Look-fors:

• The student has demonstrated a clear understanding of positive and negative signs in their model.

• The student correctly simplified the expression.

• The student is able to accurately reflect the operation in their model.

• The student can justify the sign they used as part of their solution.

Purposeful questions:

• How do you know that your model represents the expression?

• How can you use the models to show how to simplify the expression?

• How do you know that the answer should be negative? Positive? Either?

• How can you use other manipulatives to prove your solution?

• How can you use the number line to show this number sentence?

Student Strengths

Students can add, subtract, multiply, and divide whole numbers fluently.Students can represent integers on a number line, as well as compare and order integers with and without number lines.

Bridging Concepts

Students can add, subtract, multiply and divide integers using concrete models.

Standard 6.6a

Students can add, subtract, multiply, and divide integers.

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Standard 6.6b

Standard 6.6b Solve practical problems involving operations with integers

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Understanding the Learning Progression

Big Ideas:

• Numbers can be represented in multiple ways, including scientific and exponential notation, negative numbers and irrational numbers.

• Integers are negative and positive counting numbers and 0. Positive and negative numbers describe quantities having both magnitude and direction (e.g. temperature above or below zero) (Van De Walle et al., 2018).

• Positive and negative numbers occur in a real world context (temperature, sea level)

• There are multiple meanings of the negative sign: a) a subtraction sign, b) a negative sign, c) an opposite sign

• Properties of operations can be used to make connections between integer operations and whole number operations (Common Core Standards Writing Team, 2019).

Important Assessment Look-fors:

• The student uses the correct operation to solve the practical situations.

• The student demonstrates understanding of positive and negative movements within the scenario.

• The student models the problem using pictures, words, and/or numbers.

• The student can justify why the solution makes sense within the context of the problem.

Purposeful questions:

• Can you explain what is happening in the problem?

• How might you draw a picture to represent what is happening in the problem?

• How can this scenario be modeled using a number line?

• How can you determine which operation is needed to solve this problem?

• Could you solve this problem in a different way?

Student Strengths

Students can solve single step practical problems using whole numbers and plot integers on a number line.

Bridging Concepts

Students can add, subtract, multiply, and divide integers and can recognize vocabulary related to positive or negative change..

Standard 6.6b

Students can solve practical problems involving operations with integers.

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Standard 6.8a

Standard 6.8a Identify the components of the coordinate plane

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Understanding the Learning Progression

Big Ideas:

• As students develop, locating objects move from spatial directions like under and over, to points on a map and the use of coordinate systems. The first quadrant of the coordinate plane is introduced in grade 5, then in grade 6 all quadrants are included. Later this will build on scale drawing and construction as well as coordinate axis to graph lines, perform transformation and explore distance (Van de Walle et al., 2018).

• The coordinate plane can be used to represent and solve practical problems, and model two-dimensional shapes using the coordinate plane system (Common Core Standards Writing Team, 2019).

• There are some conventions that students will need to develop, such as, the quadrants of a coordinate plane are the four regions created by the two intersecting perpendicular lines (x- and y-axes). Quadrants are named in counterclockwise order. The signs on the ordered pairs for quadrant I are (+,+); for quadrant II, (–,+); for quadrant III, (–, –); and for quadrant IV, (+,–).Apply the concept of parallel and perpendicular lines to number lines (Grade 6 Standard of Learning Curriculum Framework).

Important Assessment Look-fors:

• The student correctly identified the x and y axis.

• The student demonstrates understanding of positive and negative numbers within the coordinate plane.

• The student is familiar with and correctly uses coordinate plane vocabulary including: horizontal, vertical, quadrant, and origin.

• The student correctly identifies the x and y coordinate of an ordered pair.

Purposeful questions:

• How can you determine the quadrant where an ordered pair should be placed?

• What is some information you can include on your coordinate plane to help you locate the correct quadrants?

• What is one way you can help yourself remember the order of the quadrants in the coordinate plane?

• Where would “3” go on the x-axis? Where would “3” go on the y-axis? Where would -4 go on the y axis? Where would -4 go on the x axis?

• Why is (0,4) not in a quadrant? What about (4,0)? Where would you describe their location?

Student Strengths

Students can identify congruent figures and recognize parallel, perpendicular, and intersecting lines.

Bridging Concepts

Students can identify positive and negative numbers on a horizontal and vertical number line.

Standard 6.8a

Students can identify the components of the coordinate plane.

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Standard 6.8b

Standard 6.8b Identify the coordinates of a point and graph ordered pairs in a coordinate plane

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Understanding the Learning Progression

Big Ideas:

• As students develop, locating objects move from spatial directions like under and over, to points on a map and the use of coordinate systems. The first quadrant of the coordinate plane is introduced in grade 5, then in grade 6 all quadrants are included. Later this will build on scale drawing and construction as well as coordinate axis to graph lines, perform transformation and explore distance (Van de Walle et al., 2018).

• The coordinate plane can be used to represent and solve practical problems, and model two-dimensional shapes using the coordinate plane system (Common Core Standards Writing Team, 2019).

• There are some conventions that students will need to develop, such as, the quadrants of a coordinate plane are the four regions created by the two intersecting perpendicular lines (x- and y-axes). Quadrants are named in counterclockwise order. The signs on the ordered pairs for quadrant I are (+,+); for quadrant II, (–,+); for quadrant III, (–, –); and for quadrant IV, (+,–). Apply the concept of parallel and perpendicular lines to number lines (Grade 6 Standard of Learning Curriculum Framework).

• The concept of absolute value can be used to find the distance between two points on the same horizontal or vertical line.

Important Assessment Look-fors:

• The student has graphed points that fall on the x or y-axis correctly.

• The student has identified points accurately on the coordinate plane.

• The student can distinguish between the x and y-axis.

• The student can differentiate points that fall on an axis or in a quadrant.

• The student has correctly identified the distance between two points by using a number sentence and absolute values or by modeling the distance on the coordinate plane.

Purposeful questions:

• What technique did you use to help you plot your coordinates correctly?

• What does it mean when one of your coordinates is a 0?

• What method did you use to find the distance between the two points?

• What would happen to the point location if you switched the x and y value?

Student Strengths

Students can identify components of the coordinate plane, find the area and perimeter of a rectangle or triangle, and identify congruent figures.

Bridging Concepts

Students can identify integers on a horizontal and vertical number line.

Standard 6.8b

Students can identify points in the coordinate plane and graph ordered pairs in the coordinate plane.

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Standard 6.10b

Standard 6.10b Make observations and inferences about data represented in a circle graph

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Understanding the Learning Progression

Big Ideas:

• Circle graphs are used to represent a relationship of the parts of a circle to a whole.

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

• The information displayed in a circle graph may be examined to determine how data are or are not related, differences between characteristics (comparisons), trends that suggest what new data might be like (predictions), and/or “what could happen if” (inferences). (VDOE Curriculum Framework)

Important Assessment Look-fors:

• The student can calculate the fraction or percent of the total number.

• The student can successfully calculate a solution using multiple operations.

• The student can make predictions or inferences by analyzing a graph.

Purposeful questions:

• What does ____ (number or word) represent in the circle graph?

• How can you find the percent or fraction of the total number the circle graph represents?

• What percent equals 1 whole?

• How many ____ (people, objects, etc…) are represented in this graph? How many ____ does this piece represent? How do you know?

Student Strengths

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

Bridging Concepts

Students can find a percent or fraction of a whole number.

Standard 6.10b

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

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Standard 6.11a

Standard 6.11a Represent the mean of a data set graphically as the balance point

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Understanding the Learning Progression

Big Ideas:

• Measures of central tendency are used to describe the averages for all values in a data set.

• Students see the mean as a “leveling out” of the data in the sense of a unit rate. In this “leveling out” interpretation, the mean is often called the “average” and can be considered in terms of “fair share.” (Common Core Writing Team, 2019, p.6-8 statistics and probability). It can also be discussed as a balance point.

• Balance point requires that the sum of the distances of the points above and below the mean are equal.

• Graphical representations can be used to show statistical distribution.

Important Assessment Look-fors:

• The student accurately represents the given data on a line plot.

• The student finds the balance point of a data set.

• The student describes the balance point as the mean of a set of data.

• The student uses the distance of the points from the mean to verify the balance point.

• The student can describe how the balance point can be found.

Purposeful questions:

• How can you verify that the balance point of the data set is accurate?

• What does the balance point represent in the given scenario?

• Why might the balance point of a data set be useful?

• How did you find the balance point? What strategy did you use?

Student Strengths

Students can describe mean, median, mode as measures of central tendency and describe mean as a fair share.

Bridging Concepts

Students can determine the mean, median, mode and range for a set of data.

Standard 6.11a

Students can represent the mean of a set of data graphically as a balance point.

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Standard 6.11b

Standard 6.11b Determine the effect on measures of center when a single value of a data set is added, removed, or changed

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Understanding the Learning Progression

Big Ideas:

• Results and valid conclusions can be drawn from the data that relate to questions posed (Arizona Progression).

• Statistical variability can be changed when data is added, removed or changed (Arizona Progression).

• Different points of central tendency can be used to interpret a data set.

• Students need to learn more than how to identify the mean, median, mode, and range of a set of data. They need to build an understanding of what the measure tells them about the data, and see those values in the context of other characteristics of the data in order to best describe the results.

Important Assessment Look-fors:

• Student can correctly identify the mean, median and mode of a data set.

• The student can interpret the meaning of the mean, median or mode in relationship to a data set.

• Student can determine the effect on mean, median and mode when changing a data point in a data set.

• The student can determine the value adding or removing a single value in a data set with respect to measures of central tendency.

Purposeful questions:

• How does adding or removing a data point affect the mean, median or mode?

• What value could be added to the data set so the mean does not change? So the mode doesn’t change? So the median doesn’t change?

• What value would need to be added to increase the mean or decrease the mean?

• Which measure of center is affected the most by adding this data point? Or by taking away this data point?

Student Strengths

Students can describe mean, median, mode as measures of central tendency. Students can describe the mean as a fair share.

Bridging Concepts

Students can determine the mean, median, mode and range for a set of data.

Standard 6.11b

Students can determine the effect on measures of center when a value is added, removed or changed in a data set.

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Standard 6.12a

Standard 6.12a Represent a proportional relationship between two quantities, including those that arise from practical situations

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Understanding the Learning Progression

Big Ideas:

• Proportions are equations stating that two or more ratios are equivalent.

• Ratio tables are an effective way to show proportionality.

• When given one ratio, additional proportional relationships can be identified.

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

• Proportions and ratios are used in real world contexts. (Arizona, Ratios and Proportional Relationships, p.4)

Important Assessment Look-fors:

• The student can create a ratio table that shows proportional relationships.

• The student can use a multiplicative relationship when creating ratio tables.

• The student can express ratios using the correct order (ex: 1:2 vs 2:1).

• The student can understand real world context where ratios and proportions are used.

• The student can identify the multiplicative relationship in a proportional relationship.

Purposeful questions:

• What are additional ratios that have the same proportional relationship?

• What strategy did you use to determine additional proportional ratios?

• How can you model that two ratios have a proportional relationship?

• How do you know these ratios are proportional?

• What other strategy can you use to show the ratios are proportional?

Student Strengths

Students can represent relationships using ratios.

Bridging Concepts

Students can recognize equivalences between fractions, decimals, and percents. Students can recognize and express equivalent ratios.

Standard 6.12a

Students can represent a proportional relationship between two quantities, including those that arise from practical situations.

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Standard 6.12b

Standard 6.12b Determine the unit rate of a proportional relationship and use it to find a missing value in a ratio table

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Understanding the Learning Progression

Big Ideas:

• Proportional relationships can be expressed using verbal descriptions, tables, and graphs (VDOE, Curriculum Framework).

• Unit rates are used to determine additional proportional relationships. (Arizona, Ratios and Proportional Reasoning, p.5)

• Any ratio can be converted into a unit rate, which describes how many units of the first quantity of a ratio correspond to one unit of the second quantity (VDOE, Curriculum Framework).

• Ratio tables are used to represent proportional relationships that include pairs of values that represent equivalent rates and ratios (VDOE, Curriculum Framework).

Important Assessment Look-fors:

• The student can determine a unit rate when given a proportional relationship.

• The student can use proportional relationships in real world contexts.

• The student can use unit rates to compare two different ratios.

• The student can use multiplication/division to determine additional proportional relationships.

Purposeful questions:

• What strategy did you use to determine the unit rate?

• How can you use the unit rate to determine additional proportional relationships in this scenario?

• How can you use the unit rate to determine a better deal?

• Are these numbers proportional? How do you know?

• What is the relationship between these two numbers?

Student Strengths

Students can multiply and divide fluently.

Bridging Concepts

Students can recognize relationships that are proportional and determine equivalent ratios.

Standard 6.12b

Students can determine the unit rate of a proportional relationship and use it to find a missing value in a ratio table.

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Standard 6.12c

Standard 6.12c Determine whether a proportional relationship exists between two quantities

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Understanding the Learning Progression

Big Ideas:

• In order for there to be a proportional relationship between quantities, they must have a multiplicative relationship.

• Proportional relationships do not always exist between two quantities.

• Additive relationships are not proportional.

• Proportions can be represented using pictures, tables, and graphs.

• (Arizona, Ratios and Proportional Reasoning, p.5 and 8)

Important Assessment Look-fors:

• The student can analyze information to determine if proportional relationships exist.

• The student can find unit rate to justify thinking.

• The student can recognize differences between proportional and nonproportional relationships.

• The student can explain why a situation is proportional or nonproportional.

Purposeful questions:

• What differences did you notice between situations that were proportional and ones that were nonproportional?

• What strategies did you use to help determine if a situation was proportional or not?

• Can there be a relationship between quantities and still not be a proportional situation? Explain your thinking.

• How do you know the situation is proportional?

Student Strengths

Students can multiply and divide fluently.

Bridging Concepts

Students can determine equivalent ratios. Students can find a unit rate in a given proportional situation.

Standard 6.12c

Students can determine whether a proportional relationship exists between two quantities.