Standard 8-2: The student will demonstrate through the...

Numbers and Operations Eighth Grade

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Indicator 8-2.1

Apply an algorithm to add, subtract, multiply, and divide integers.

Continuum of Knowledge

In seventh grade, students generated their own strategies to add, subtract, multiply, and divide integers (7-2.8)

In eighth grade, students apply an algorithm to add, subtract, multiply and divide integers (8-2.1).

Taxonomy Level

Cognitive Dimension: Apply Knowledge Dimension: Procedural Key Concepts

• Additive inverse Vocabulary/Symbols

• Multiplicative inverse • Zero pairs • Sum • Product • Quotient

Instructional Guidelines For this indicator, it is essential

• Connect concrete and/or pictorial models to an algorithm (numbers only) for students to:

• Understand the meaning of additive inverse • Understand that addition means combining. Subtracting an integer is the

same as adding the opposite, a “double negative.” • Gain computational fluency

Standard 8-2: The student will demonstrate through the mathematical processes an understanding of operations with integers, the effects of multiplying and dividing with rational numbers, the comparative magnitude of rational and irrational numbers, the approximation of cube and square roots, and the application of proportional reasoning.


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For this indicator, it is not essential

for students to:

None noted Student Misconceptions/Errors

Although rules and fun saying are easy for students to remember, many students still struggle with operations with integers because they lack a conceptual understanding of these operations. Rules such as “like signs give positive products” and “unlike signs give negative products” are deceptively simple. Correct answers and meaningful justification of these answers are equally important.

Instructional Resources

• Use arrows to represent all integers and don’t refer to number line coordinates as “numbers.” The arrows help students think of the integers as directed distances.

• Allow students to use colored pencils (yellow for positive and red for negative) to connect the concept of the models they learned in previous grade levels.

• It is important to use appropriate mathematical terminology when discussing strategies. Using words like flip may be easier for students to understand but they decrease the student’s conceptual understanding.

• A strategy for moving students through operations with integers is to ensure that students have a thorough understanding of addition and subtraction prior to instructing multiplication and division. Progress from combining (addition) of positive integers, to combining negative integers, and finally to combining unlike signs, a mixture of positive and negative integers that include zero pairs.

• Although students have generated strategies in 7th

• When using colored counters it may be helpful to relate them to the process of using the number line to help students make additional understanding.

grade and explored problems in context, the use of these problems is still essential to helpful students deepen their understanding of procedures.

Assessment Guidelines The objective of this indicator is to apply which is in the “apply procedural” cell of Revised Taxonomy. Although the focus is to gain computational fluency with operations with integers, the learning progression should integrate strategies to enhance both conceptual and procedural knowledge. The learning progression to apply requires students to recall and understand the concept of integers. Students explore these operations in context as opposed to rote memorization of rules. They


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apply their conceptual knowledge of integers to transfer their understanding of concrete and/or pictorial representations to symbolic representations (numbers only) by generalizing connections among these representational forms and real world situations (8-1.7). Students use correct and clearly written or spoken words to communicate their understanding (8-1.6). Students engage in repeated practice using pictorial models, if needed, to support learning. Lastly, students should evaluate

the reasonableness of their answers using appropriate strategies.


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Indicator 8-2.2

Understand the effect of multiplying and dividing a rational number by another rational number.


In seventh grade, students learned to apply an algorithm to multiply and divide fractions and decimals (7-2.9).

In eighth grade, students gain an understanding of the effect of multiplying and dividing a rational number by another rational number.

Taxonomy Level

Cognitive Dimension: Understand Knowledge Dimension: Conceptual Key Concepts

• Effect Vocabulary

• Rational number • Product • Quotient


• Multiply and divide rational numbers for students to:

• Understand that division does not always result in a smaller answer • Understand that multiplication does not always result in a larger answer • Recognize fractional forms of one • Understand the concept of equivalent fractions (same value; different form) • Understand that zero is a rational number • Understand why division by zero is not possible



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for students to:


• Students may experience difficulty understanding why division by zero is impossible.


• The focus of the indicator is an emphasis on understanding relationship not on computational fluency. Understanding these relationships will also be beneficial to students as their perform operations with rational numbers.

• Provide contextual opportunities for students to focus on the effects to the products and quotients of operations with rational numbers.

• One strategy for helping students understand why division by zero is not possible is to examine the relationship between rational numbers (fractions) and their decimal equivalent. The denominator x quotient = the numerator. For example, ½ = 0.5; therefore, 2 x 0.5 must equal 1. Students sometimes state the #/0 = 0 so by their logic 4/0 =0 but if that were true then 0 x 0 should equal 4.

Assessment Guidelines The objective of this indicator is to understand which is in the “understand conceptual” knowledge cell of the Revised Taxonomy. To understand is to construct meaning; therefore, students develop a conceptual understanding of these effects. The learning progression to understand requires students to recall the meaning of rational numbers. Students understand how to multiply and divide rational numbers using an appropriate algorithm. Given examples, students generate and evaluate conjectures (8-1.2) about the effect on the product or quotient. They use inductive reasoning (specific to general) to generalize

mathematical statements about the effect using correct and clearly written or spoken words and notation (8-1.6). They then generate and solve complex problems such as division by zero.


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Indicator 8-2.3

Represent the approximate location of irrational numbers on a number line.


In seventh grade, students work involved approximating the location of perfect squares on the number line (7-2.2).

In eighth grade, students represent the approximate location of irrational numbers on a number line. This is the first time students are introduced to the concept of irrational numbers and their approximate location on a number line.

Taxonomy Level


Vocabulary

• irrational number • terminating decimal • non-terminating decimal


for students to:

• Understand that irrational numbers can be approximated as a decimal • Recall the value of pi (It is used a benchmark irrational number by National

Association for Educational Progress (NAEP). • Understand that every number on a number line is either rational or

irrational. • Understand how to find the cube root of a number • Determine the value between which an irrational number occurs using

perfect squares or cubes • Recall the perfect squares and their values



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• Approximate for square roots and cubes roots less than 1000 (8-2.6) For this indicator, it is not essential

for students to:

• Compute the exact value of irrational numbers Student Misconceptions/Errors

• Students using calculators need to realize that the display may only have 8 digits visible. This could be a very long repeating decimal and not be easily identified.

• When students see a radical sign, the mistakenly think that the number is irrational


• Discuss with students why a number is labeled as rational or irrational. • Provide practice with rounding and estimation for students to perfect their

skills. Knowledge that every number on a number line is either rational or irrational and solid connections with rounding will better enable students to place numbers in the appropriate place on a number line.

• Although students do not have to compute the exact value of irrational numbers without a calculator, they should use a strategy to determine between which two numbers the irrational number lies. For example, to locate square root of 79, student should reason that 79 is between 64 and 81; therefore, the square root of 79 is between 8 and 9 and probably closer to nine because 79 is closer to 81. The answer is ≈ 8.9.

Assessment Guidelines The objective of this indicator is to represent which is in the “understand conceptual” knowledge of Bloom’s Taxonomy. To represent is to translate from one form to another; therefore, students develop an understanding of irrational number by translate them from numerical form (number) to graphical form (number line). The learning progression to represent requires students to recall the concept of rational number. Students understand the characteristics of irrational numbers and make the connection to their prior knowledge of rational numbers. Students explore how to represent these types of numbers in the correct location on a number line as their relate to rational numbers and justify

their placement of a rational number on a number line using general mathematical statements (8-1.5) based on inductive and deductive reasoning (8-1.3).


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Indicator 8-2.4

Compare rational and irrational numbers by using the symbols <, >, < , >, and =.


In sixth grade, students compare rational numbers and whole number percentages through 100 by using symbols <, >, < , >, and = (6-2.3).

In seventh grade, students focused on comparing rational numbers and square roots of perfect squares (7-2.3).

In eighth grade, student compare rational and irrational numbers by using the symbols <, >, < , >, and =.

Taxonomy Level


• Real numbers Vocabulary/Symbols

• (Square/Cube) root ( ) • Pi (π )


• Understand the meaning of rational numbers for students to:

• Find the square root and cube root • Understand the difference between ≤ and ≥ • Translate numbers to same form, where appropriate before comparing

numbers • Translate between the fraction and percents



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• Recall the benchmark fractions and common fraction – decimal equivalents. For this indicator, it is not essential

for students to:

• Find the exact value of an irrational number Student Misconceptions/Errors

• Students using calculators need to realize that the display may only have 8 digits visible. This could be a very long repeating decimal and not be easily identified.

• When students see a radical sign, the mistakenly think that the number is irrational


• In indicator 8-2.3, students approximate the value of irrational numbers; therefore, when students compare numbers the value should not close. For example, compare the square root of 3 and 1.752. Students only know that the square root of 3 is between 1 and 2 not that it equals 1.732. So this would be a difficult comparison. A more appropriate comparison may be square root of 3 and 2.5. They know for sure what the relationship is between the two numbers.

• To alleviate the misconception that a number in a radical sign is irrational, consider roots from the opposite direction. Imagine that every number is a square root, a cube root, a fourth root, etc so that students realize that a root is just a way of expressing a relationship between two numbers. For example, rather than ask, “What is the square root of 64 or the cube root of 27”?, we might suggest that every number is the square (second) root, the third root, the fourth root and so on, of some number. Three is the second root of 9 and the third root of 27, etc. From this vantage point students can see that “square root” is just a way of indicating a relationship between two numbers. That the cube root of 27 is 3 indicates special relationship between 3 and 27.

• Calculators should be allowed unless students are estimating between square roots and cube roots.

Assessment Guidelines The objective of this indicator is to understand which is in the “understand conceptual” knowledge cell of the Revised Taxonomy. Conceptual knowledge is not bound by specific examples; therefore, the student’s conceptual knowledge should include numerous examples. The learning progression to compare requires students to recall common fraction/decimal equivalents and identify perfect square roots. They understand how to find the approximate value of irrational numbers. Students understand equivalent symbolic expressions (8-1.4) and translate


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numbers to a common form, if necessary. Students use their conceptual understanding to determine the location of irrational numbers. They compare without dependence on a traditional algorithm and use concrete models to support understanding where appropriate. Students recognize mathematical symbols <, >, >, < and = and their meanings. As students analyze the relationships to compare percentages and rational numbers, they evaluate conjectures and explain and justify their answer to classmates and their teacher. Students should use

correct and clearly written or spoken words, variables and notation to communicate their reasoning (8-1.6).


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Indicator 8-2.5

Apply the concept of absolute value.


Seventh grade was the first experience with absolute value as a distance away from zero (7-2.4).

In eighth grade, students apply the concept of absolute value (8-2.5).

Taxonomy Level


• absolute value symbol | | Vocabulary/Symbols


for students to:

• Understand that absolute value is a distance from zero, not a direction. • Understand that distance is always a positive value; therefore, the absolute

value is always positive For this indicator, it is not essential

for students to:


• Students may think that distance from zero can be a negative number. • When students only evaluate numerical expressions they learn to do the

computation and then ignore or “remove” the minus sign. This causes difficulty in later grades when values are unknown or variables are involved.



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• To deepen conceptual understanding, students need opportunities to compare absolute values using order symbols. For example, |-5 | -|4 |

• Students can work examples that involve finding the absolute value such as finding the difference between below and above sea level, freezing temperatures which are above and below zero, being in the red as a deficit balance and then in the black for having a positive bank balance, deposits and withdrawals, cars that are going different directions, yards gained or lost in a football game, etc.

Assessment Guidelines The objective of this indicator is to understand, which is in the “understand conceptual” knowledge cell of the Revised Taxonomy. To understand is to construct meaning; therefore, the student’s conceptual knowledge of absolute value should include numerous real world examples and non-examples. The learning progression to understand requires students to recall the concept of integers and their position in relation to zero. When directed to determine the absolute value of a number, students should understand that absolute value is a distance away from zero and does not include which direction (positive or negative) away from zero. They illustrate this understanding by representing absolute value relationship on the number line. Students use their understanding to generate and solve complex problems to deepen conceptual knowledge. They use

correct and clearly spoken words, variables and notation to communicate their understanding (8-1.6).


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Indicator 8-2.6

Apply strategies and procedures to approximate between two whole numbers the square roots or cube roots of numbers less than 1,000.


In seventh grade, students gained an understanding of the inverse relationship between squaring and finding the square roots of perfect squares (7-2.10).

In eighth grade, students apply strategies and procedures to approximate between two whole numbers the square roots or cube roots of numbers less than 1,000 (8-2.6). They also represent the approximate location of irrational numbers on a number line.

Taxonomy Level


• Vocabulary/Symbols

3 (cube root) • Square root • Approximation


• Understand the meaning of square roots for students to:

• Understand the meaning of cube roots • Understand the relationship between area and square root and volume and

cube roots.



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for students to:


• Students often mistakenly compute 32

• Students may think that the square root is dividing by 2 and the cube root is dividing by 3

as if the exponent were a factor, so they come up with 6 instead of 9. They do the same when the exponent is 3. If students gain an understanding of the relationships between multiplication, exponent, and radical, they’ll be much less likely to make this common error. It will also lay a good foundation for working with other roots.


• When asked to find roots, answers should be given as whole numbers and/or decimals rounded to the hundredths place.

• The focus of the indicator is to approximate values; therefore, students should use their conceptual understanding of square roots and cube roots to approximate these values.

Assessment Guidelines The objective of this indicator is to apply which is in the “apply procedural” knowledge cell of the Revised Taxonomy. Although the focus of the indicator is on computational fluency with strategies for approximating cube and square roots values, the essence of the strategy is rooted in conceptual understanding. The learning progression to apply requires students to recall and understand the meaning of square roots and cube roots. Students explore concrete and/or pictorial models of square roots and cube roots using area and volume. Students generalize connection among these representation and finding the approximate value of cube and square roots. They generalize mathematical statements (8-1.5) about the connection between these representations to develop a strategy for determining approximations. As students explore these representations, they use

correct and clearly written or spoken word to communicate their reasoning (8-1.6).


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Indicator 8-2.7 Apply ratios, rate and proportions Continuum of Knowledge In sixth grade, students understand the relationship between ratio/rate and multiplication/division (6-2.6). In seventh grade, students apply ratios, rates, and proportions to discounts, taxes, tips, interest, unit costs and similar shapes (7-2.5). In eighth grade, students apply ratios, rates and proportions (8-2.7) Taxonomy Level Cognitive Dimension: Apply Knowledge Dimension: Procedural Key Concepts

• Ratios • Rates • Proportions


for students to:

• Understand the meaning of ratio, rate and proportions • Develop strategies for determining discounts • Connect the similar ideas computing discounts, taxes, tips and interest in

order to develop strategies • Understand that with discounts, they pay less and with taxes, tips and

interest they pay more • Understand the concept of percents • Multiply by a decimal • Understand the relationship between unit cost and ratio and rate • Understand the proportionality of similar shapes • Solve proportions using an appropriate strategy • Explore real world examples that goes beyond discounts, taxes, tips,etc..

that were explored in 7th grade


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for students to:

None noted Student Misconceptions/Errors Students may still struggle with converting decimals. Instructional Resources and Strategies Although students are not required to compute the final cost, it may helpful for them to understand that discounts (subtracting from original amount) and taxes, tips and interest (add to the original amount). Representing these concepts as pictorial or concrete models may help students deepen their understanding these procedures. For example, using a hundreds charts to represent $100 dollars. Shade appropriate blocks to represent discounts like 20% off mean subtracting 20 blocks or interest of 30% means adding 30 more blocks. The cross multiplying algorithm for solving proportions is a valid strategy that should be taught with understanding. Using proportional reasoning is another powerful strategy that focuses more on a conceptual understanding of proportional relationship as opposed to a traditional algorithm. Proportional reasoning always allows students make estimates. Assessment Guidelines The objective of this indicator is to apply which is in the “apply procedural” knowledge cell of the Revised Taxonomy. Although the focus of the indicator is procedural, students also use their conceptual understanding of ratio, rate and proportion to solve problems. The learning progression to apply requires students to recall and understand the meaning of the concepts rate, ratio and proportions. . Students explore a variety of problems (beyond discounts, taxes, tips, interest, unit cost and similar shapes) in context to generalize connections (8-1.7). They analyze pictorial and/or concrete models, where appropriate, to support conceptual understanding of these procedures. They generate mathematical statements (8-1.5) related to how to use ratios, rates and proportions to solve problems and use correct and clearly written and spoken words, variable and notation to communicate their understanding (8-1.6). Students should use their understanding to generate

and solve more complex problems using ratio, rate and proportion (8-1.1).

Algebra Eight Grade

1

Indicator 8.3-1

Translate among verbal, graphic, tabular, and algebraic representations of linear functions.


In fifth grade, students match tables, graphs, expressions, equations, and verbal descriptions of the same problem situation (5-3.3). In seventh grade, students represent proportional relationships with graphs, tables and two-step equations (7-3.6)

In eighth grade, students translate among verbal, graphic, tabular and algebraic representations of linear functions (8-3.1) and classify relationships between two variables in graphs, tables and/or equations as either linear or nonlinear (8-3.5). and applying abstract reasoning (8-1.1) in regards to the various representations.

In Elementary Algebra, students will carry out a procedure to write an equation of a line (EA-4.1, EA-4.2, EA-4.3), carry out a procedure to graph a line (EA-5.1), and carry out a procedure to determine the slope of a line from data given tabularly, graphically, symbolically and verbally (EA-5.6).

Taxonomy Level


• Nonlinear Vocabulary

• Constant • Coefficient • Multiple representations • Functions


for students to:

• Recall and understand the slope intercept form of a linear function • Construct a table of values given a linear equation

Standard 8-3: The student will demonstrate through the mathematical processes an understanding of equations, inequalities, and linear functions.

Algebra Eight Grade

2

• Analyze verbal descriptions for key words that relate to linear concepts such as slope, rate of change, initial value, y-intercept, etc..

• Produce the other two representation when given one of them (produce a table given the graph or the equation)

• Identify the slope and y-intercept when given a linear equation in slope-intercept form.

• Identify increasing and decreasing linear patterns from the graph, table and equation

• Connect numeric patterns in tables and graphs in coordinate planes.


for students to:

None noted Student Misconceptions/Errors Although students have worked with the multiple representations in prior grades, many of them may not recognize that each form is a representation of the same relationship. Explicit connections between and among these forms will support students conceptual and procedural understanding of concepts related to linear functions.

Instructional Resources and Strategies

• The emphasis is students not only being able to translate among the form but also for students to make connections among the forms.

• It is important for students to be able to easily translate between the different representations and know when to use the appropriate representation.

• Although the slope and y-intercept are easily recognized in the slope intercept form, students need opportunities to investigate how these values impact the graph and table.

• Building Bridges: Extending Problems from Arithmetic to Algebraic Thinking Illuminations: http://illuminations.nctm.org/LessonDetail.aspx?id=L247

• Navigating through Algebra pg 19-29 o Walking Stories o From Stories to Graphs o From Graphs to Stories

Assessment Guidelines The objective of this indicator is to translate, which is in the “understand conceptual” knowledge cell of the Revised Bloom’s Taxonomy. To understand is to construct meaning. Conceptual knowledge is not bound by specific examples;

http://illuminations.nctm.org/LessonDetail.aspx?id=L247�

Algebra Eight Grade

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therefore, the student’s conceptual knowledge of translating among verbal, graphic, tabular, and algebraic representations of linear functions should include a variety of examples. The learning progression to translate requires students to recall and understand the multiple methods of expressing real world functional relationships (words, graphs, equations and tables). Students analyze these representations simultaneously to generalize connections among these forms (8-1.7) such as the location of the y-intercept on the graph, in the table and in the equation. They use correct and clearly written and spoken words, variables and notation to communicate their understanding of these generalizations. Students then use these generalizations to translate among representations. They explain and justify

their understanding to their classmates and teacher.

Algebra Eight Grade

4

Indicator 8.3-2

Indicator 8.3-2

Represent algebraic relationships with equations and inequalities.


In sixth grade, students represent algebraic relationships with variables in expressions and simple equations and inequalities (6-3-3). In the seventh grade, the learning extends to include representing the solution of a two-step inequality on a number line (7-3.5).

In eighth grade, represent algebraic relationships with equations and inequalities (8.3-2).

In Elementary Algebra, students analyze given information to write a linear function that models a given problem situation( EA-5.9) and analyze given information to write a linear inequality in one variable that models a given problem situation (EA-5.12).

Taxonomy Level


• Equations Vocabulary

• Inequalities • Less than < • Less than or equal to ≤ • Greater than > • Greater than or equal to ≥ • Multiple representations


for students to:

• Understand pattern generalizations (example: Less than is the same as <) • Identify patterns as linear and nonlinear • Translate from one representation to another (graph, table, word)


Algebra Eight Grade

5

• Write an equation/inequality that represents a situation (words, graph, table)

• Read inequalities and equations using appropriate terminology. For example, -23 < x (-23 is less than x) -23 < x < 100 (-23 is less than x which is less than or equal to 100)

OR

(x is between -23 and 100 and equal to 100)

• Understand that a closed circle is inclusive of the number it represents and therefore is associated with ≤ and ≥.

• Understand that an open circle is not inclusive of the number it represents and therefore is associated with < and >.


• Solve the equation or inequality for students to:

• Graph the equation or inequality Students Misconceptions/Errors

• Students may think that the inequality symbols indicate the direction of the shading of the graph


• Teachers should connect numeric patterns in tables to equations and inequalities to support conceptual understanding.

• Students may give examples of numbers that would satisfy the inequality

and develop a rule or a verbal description. For example, Describe the solutions of the inequality x < 2 or x > 1.

Answer: All real numbers between 1 and 2 but not including 1 or 2

• Once students are able to write inequalities, have them apply what they know by posing contextual situations. For example,

Water can take on 3 forms: solid, liquid or gas. Under ordinary conditions, water is a solid at temperatures of 32o

temperatures of 212F or lower and a gas at

o

What inequality describes when water is F or higher.

NOTAnswer: t

a liquid? ≤ 32 or t ≥ 212

What inequality describes when water IS

Answer: 32 < t < 212 a liquid?

Algebra Eight Grade

6

Assessment Guidelines The objective of this indicator is to represent which is in the “understand conceptual” knowledge cell of the Revised Taxonomy. To understand means to construct meaning; therefore, the students’ focus is on building conceptual knowledge of the relationships between the forms. The learning progression to represent requires students to understand the concepts of equivalency and inequalities. Students analyze algebraic relationships (words, tables and graphs) to determine known and unknown values and the operations involved. They generate descriptions of the observed relationship and generalize the connection (8-1.7) between their description and structure of algebraic equations or inequalities. Students explain and justify their ideas with their classmates and teachers using correct and clearly written or spoken words, variables and notation to communicate their ideas (8-1.6). Students then compare

the relationships (words, tables and graphs) to their equation, inequality or expression to verify that each form conveys the same meaning.

Algebra Eight Grade

7

Indicator 8.3-3

Indicator 8.3-3

Use commutative, associative, and distributive properties to examine the equivalence of a variety of algebraic expressions.


In fifth grade, students identified applications of commutative, associative and distributive properties (5-3.4). In sixth grade, students learned to show that two expressions (numbers only) are equivalent by applying the commutative, associative, and distributive properties (6-3.4) and to apply inverse operations to solve simple one step equations (6-3.5).

In eighth grade, students use commutative, associative, and distributive properties to examine the equivalence of a variety of algebraic expressions.

In Algebra I, students will carry out a procedure using the real properties of numbers (including commutative, associative and distributive) to simplify expressions (EA-2.5)

Taxonomy Level


Vocabulary

• Commutative Property • Associative Property • Distributive Property • Numerical expressions • Algebraic expressions • Variable • Constant • Coefficient • Equivalency


Algebra Eight Grade

8


for students to:

• Recall the commutative, associative and distributive properties • Simplify an expression by collecting like terms • Verify equivalency by substituting values


for students to:

• Distribute variables when simplifying expressions Student Misconceptions/Errors

• Students may collect like terms that are not alike. By building the concept properly, students are less likely to make this error. Hands-on activities using manipulatives to represent variables are helpful in support their understanding of like terms. For example, use popsicle sticks to represent “a” and straws to represent “b”. Have students represent the expression 5a + 2b + a + 3b using the manipulatives. Then tell them to organize them into like groups. Ask them what did they get using manipulatives? 6 popsicle sticks and 5 straws. Then they write an expression: 6a + 5b. Ask them can they add the two groups? No because they can’t add sticks and straws; they are not alike. So 6a + 5b ≠ 11ab.


• Although students have worked with these properties in previous grades, they may still need to support their understanding through the use of concrete and pictorial models. Connecting what the students know about finding the area of rectangles to the distributive property is one strategy. Students will need practice with several examples in order to understand this concept. For example, prove the following by using an area model:

6 x 53 = 6(40) + 6(13)

• Use a model to multiply 3(2x + 5). How can the commutative property of addition be used to prove the distributive property? Answer: 3 groups of “2x + 5” commutative property allows you to group the 3 groups of 2x and the 3 groups of 5. Resulting in 6x + 15.

• Use the commutative, associative, and/or distributive property to get from

expression A to expression B:

Algebra Eight Grade

9

Answer: Expression A: 5(7x – 8) – 6(7x – 8) = (5-6)(7x – 8) = (-1)(7x – 8) = -7x + 8 OR Expression A: 5(7x – 8) – 6(7x – 8) = 35x – 40 – 42x + 48 = 35x – 42x – 40 + 48 = -7x + 8 Assessment Guidelines The objective of this indicator is to use which is in the “apply procedural” knowledge cell of the Revised Taxonomy. Although the focus of the indicator is to use which is a knowledge of specific steps and details, learning experiences should integrate both memorization and concept building strategies to support retention. The learning progression to use requires student to explore a variety of examples of these number properties using a various types of numbers. They analyze these examples and generalize connections (8-1.7) about what they observe using correct and clearly written or spoken language (8-1.6) to communicate their understanding. Students translate these connections into mathematical statements using variables. Students connect these statements to the terms commutative, associative and distributive. Students then develop

meaningful and personal strategies that enable them to recall these relationships.

Expression A Expression B

5(7x – 8) – 6(7x – 8) -7x + 8

Algebra Eight Grade

10

Indicator 8.3-4

Apply procedures to solve multi-step equations.


In sixth grade, students begin to build the foundation by using inverse operations to solve simple one-step equations with whole numbers solutions and variables with whole number coefficients (6-3.5). In 7th

In 8

grade, students use inverse operations to solve two-step equations (with rational numbers) and two-step inequalities (7-3.4).

th

In Elementary Algebra, students will demonstrate through the mathematical processes an understanding of the real number system and operations involving exponents, matrices and algebraic expressions (EA2).

grade, the process continues as students apply procedures to solve multi-step equations (8-3.4).

Taxonomy Level


• Multi-step equations Vocabulary

• Properties of Equalities • Properties of Real Numbers


• Understand the concept of equivalency for students to:

• Solve all types of equations (one-step, two-step, and multi-step) • Apply properties to solve equations with rational numbers • Solve equations incorporating all grouping symbols • Check their solutions


• Apply the law of exponents and roots for students to:

• Perform operations with numbers written in scientific notation • Simplify expressions that require distributing a variable i.e. x(2x -5)


Algebra Eight Grade

11

Student Misconceptions/Errors

• Students may forget to reverse the order of operations when solving. Instructional Resources and Strategies

• Although students have solved equations in the past, proficiency with procedures is support by understanding of concepts. Students may still use concrete and pictorial models to support understanding.

• The types of problems students encounter should move beyond basic procedural knowledge. Sample problems:

o Explain how to solve 3 ( 9 + 4a) – 19 = 32 o Together, Donald, Yolanda, and Iris made 27 birdhouses for a school

fair. Yolanda made the fewest number of birdhouses. Donald made one more than Yolanda, and Iris made one more than Donald. Write and solve and equation to find out how many birdhouses each of them made.

Let n = the number of birdhouses Yolanda made • Comparing multiple strategies for solving the same problems builds both

conceptual and procedural knowledge. • Making a connection to using the order of operations in reverse order may be

helpful to students when solving equations or inequalities (particularly two-step)

Assessment Guidelines The objective of this indicator is use which is in the “apply procedural” of the Revised Taxonomy. Procedural knowledge is knowledge of specific steps or strategies that can be used to solve a problem or problem situation. Although the focus is to gain computational fluency with solving multi-step equations, the learning progression should integrate strategies to enhance both conceptual and procedural knowledge. The learning progression to use requires students to explore and understand the concepts of equivalency, inequality and variables using concrete models such as balance scales. Student use this understanding of balance to analyze a variety of examples of simple multi-step equations. Students use inductive reasoning (8-1.3) to generalize connections (8-1.7) among types of equations/inequalities and generate mathematical statements (8-1.5) related to how these equations can be solved. They use concrete manipulatives and pictorial to support their conceptual understanding. Student engage in meaningful practice to support retention of these processed and check

their answers.


Algebra Eight Grade

12

Indicator 8.3-5

Classify relationships between two variables in graphs, tables, and/or equations as either linear or nonlinear.


In 7th

In 8

grade, students analyze tables and graphs to describe the rate of change among quantities and to determine if the rate of change is constant (7-3.2). They gained an understanding of slope as constant rate of change (7-3.3).

th

grade, students gain the ability to classify relationships between two variables in graphs, tables, and/or equations as linear or nonlinear (8-3.5).

In Elementary Algebra, students use this understanding to classify a variation as direct (linear) or inverse (EA-3.6). Taxonomy Level


Vocabulary

• Linear • Nonlinear • Constant • Coefficient • Variable • Multiple representations


for students to:

• Understand the structure of a linear equation i.e. constant multiplied by a variable (+ or –) a constant or constant (+ or –) a constant multiplied by a variable

• Understand that linear equations do not have variables in the denominator • Understand that a linear equation cannot contain variables with exponents

other than one.

Algebra Eight Grade

13

• Recognize that if a relationship does not fit these criteria, the relationship is nonlinear.

• Determine if a given set of data or equation is linear. • Make connections between numeric, pictorial, and algebraic models.


for students to:


• Students may struggles with the exponent of the variable being one because it is not visible.


• Whether using a two variable table, a coordinate plane graph, or an equation, the goal for 8th graders is to be able to classify data as linear or nonlinear. Each model should be seen as a tool to support the conclusions drawn from another model. The relationship between variables determined by students with one format should be confirmed easily by using the other two methods for representing the same data.

• It is in the 8th

grade that students are first exposed to the slope-intercept form of a linear equation (y=mx+b), with m being the slope and b the y-intercept. They learn that a linear function is a function whose ordered pairs satisfy a linear equation and that any linear function can be written in slope-intercept form. Because equations are commonly written in this form, students should be able to recognize anything that can be written in slope-intercept form as linear.

Assessment Guidelines The objective of this indicator is to classify which is in the “understand conceptual” knowledge cell of the Revised Taxonomy. To understand is to construct meaning and conceptual knowledge is not bound by specific examples; therefore, students should gain a conceptual understanding of linear and nonlinear in order to classify relationships. The learning progression to classify requires students to recall and understand the relationship between the multiple forms of linear functions. They understand the linear characteristics of each form. Students analyze these representations and compare them against known characteristics. As students analyze these relationships, they categorize them as linear or nonlinear. Students

Algebra Eight Grade

14

explain and justify

their answers using correct and clearly written and spoken words, variables and notation to communicate their understanding (8-1.6).

Algebra Eight Grade

15

Indicator 8.3-6

Indicator 8.3-6

Identify the coordinates of the x- and y- intercepts of a linear equation from a graph, equation and/or table.


Eighth grade is the first time that students are first introduced to slope-intercept form (y=mx + b) of a linear equation (linear function).

In Elementary Algebra, students analyze the effects of changes in the slope, m, and the y-intercept, b, on the graph of y=mx +b (EA-5.2). Students also carry out a procedure to determine the x-intercept and y-intercept of lines from data given tabularly, graphically, symbolically, and verbally (EA-5.5)

Taxonomy Level

Cognitive Dimension: Remember Knowledge Dimension: Procedural Key Concepts

• X-intercept Vocabulary

• Y-intercept • Multiple representations


for students to:

• Understand that the x-intercept is in the form (x,0) • Understand that the y-intercept is in the form (0,y) • Identify slope (m) and y-intercept (b) of a linear equation • Understand that the x-intercept is where the graph crosses the x axis • Understand that the y-intercept is where the graph crosses the y axis


• Identify x- and y-intercepts from a graph that are not integers for students to:


Algebra Eight Grade

16


• Students confuse which variable as a value of zero for each intercepts. Instructional Resources and Strategies Not only should students identify x and y intercepts from each form but they should also make connections among the forms; therefore, each time students examine one representation, they should examine how it looks in the other forms. For example, once the y-intercept is identified from slope-intercept form, illustrate what it looks like on the graph and in a table. Assessment Guidelines The objective of this indicator is to remember, which is in the “remember procedural” knowledge cell of the Revised Bloom’s Taxonomy. Procedural knowledge is knowledge of specific steps or strategies that can be used to solve a problem or problem situation. Although the focus is on remembering procedures to identify the coordinates of the x- and y- intercepts of a linear equation from a graph, equation and/or table, the learning progression should integrate strategies to enhance both conceptual and procedural knowledge. The learning progression to remember requires students to recall that the x intercept is where the graph crosses the x axis and the y-intercept is where the graph crosses the y axis. By generalizing connections (8-1.7) among forms, students use this understanding to create a coordinate representation of intercepts using zero appropriately. They graph these points to examine the graphical form. They analyze the equation to determine how these coordinates connect i.e. the y-intercept is the value of b. They generate other values in the table and observe how the intercept fit into the table. Students generalize

mathematical statements (8-1.5) about the relationships between and among the representations using correct and clearly written or spoken words, variables, and notation (8-1.6).

Algebra Eight Grade

17

Indicator 8.3-7

Identify the slope of a linear equation from a graph, equation and/or table.


In seventh grade, students analyze tables and graphs to describe the rate of change between and among quantities (7-3.2). Students gain an understanding slope as a constant rate of change (7-3.3)

In eighth grade, students identify the slope of a linear equation from a graph, equation, and/or table (8-3.7). Students did not relate slope to linear equations in seventh grade.

In Elementary Algebra, students carry out a procedure to determine the slope of a line from data given tabularly, graphically, symbolically, and verbally (EA-5.6).

Taxonomy Level

Cognitive Dimension: Remember Knowledge Dimension: Procedural Key Concepts

Vocabulary

• Rise • Run • Slope • Coefficient • Rate of change


for students to:

• Have a conceptual understanding of slope (rate of change) • Use the process of rise over run to find the slope from graph • Understand that slope is the steepness and direction of the line. • Recall that the slope is the coefficient of x in y=mx + b and y = b + mx • Understand the effect the m has on the equation and the graph • Review the vocabulary related to the coordinate plane.


Algebra Eight Grade

18

For this indicator, it is not essential• Identify the slope for horizontal and vertical lines

for students to:


• Students often reverse the “change in y/change in x” when finding slope • When given a linear equation in the form y = b + mx, students often identify

b as the slope. Instructional Resources and Strategies Students have had experience with rate of change in tables and graphs. Eighth grade is the first time that explore slope from an equations; therefore, it is important that students understand the relationship between the coefficient of x and rate of change. Below is a sample inquiry activity: (chart on next page)

a. Determine if the line is above the origin, below the origin, or through the origin. What variable determines the position on the y-axis?

b. Determine if the line is increasing or decreasing. What determine the direction of the line?

c. Determine if the line is steeper or flatter than the line y = x. What determine the steepness of the line?

Assessment Guidelines The objective of this indicator is to remember which is in the “remember procedural” knowledge cell of the Revised Bloom’s Taxonomy. Procedural knowledge is knowledge of specific steps or strategies that can be used to solve a problem or problem situation. Although the focus is on remembering procedures to identify, the learning progression should integrate strategies to enhance both conceptual and procedural knowledge. The learning progression to remember requires students to recall and understand the connection between the rate of change and slope of a line. Students use their understanding of slope from tables and graphs to explore slope with equations. Students analyze a variety of linear equations and their graphs and generalize connections (8-1.7) between the coefficient of x, the direction of the line and the steepness of the line. Students generalize

mathematical statements (8-1.5) about these connections using correct and clearly written or spoken words, variables, and notation (8-1.6). After having sufficient experiences with slope from equations, students use this understanding to identify slope from a graph, table and equation.

Algebra Eight Grade

19

y = mx + b above the origin

below the origin

through the origin

increasing decreasing steeper flatter

y = 3x + 1

y = - 2x – 4

y = 741

+x

y = 253

−x

y = 5x

y = -4x – 9

y = 7x – 1

y = x31

−

y = -5x + 23

Geometry Grade Eight

1

Standard 8-4: The student will demonstrate through the mathematical processes an understanding of the Pythagorean theorem; the use of ordered pairs, equations, intercepts, and intersections to locate points and lines in a coordinate plane; and the effect of a dilation in a coordinate plane.

Indicator 8-4.1

Apply the Pythagorean Theorem.

Continuum of Knowledge:

In seventh grade the students learned about the inverse relationship between perfect squares and square roots (7-2.10)

In eighth grade, students apply the Pythagorean Theorem (8-4.1).

This topic is also addressed in high school Geometry.

Taxonomy Level

Cognitive Dimension: Apply Knowledge Dimension: Conceptual

Key Concepts

• Hypotenuse

Vocabulary

• Legs • Right triangle • Square root • Radical

• a

Mathematical Formula

2+b2=c

Instructional Guidelines

2

For this indicator, it is essential

• Recall the formula

for students to:

• Understand the relationship among the sides beyond memorization of the formula

• Know that the hypotenuse is always the longest side and that the legs are connected to the right angle.


2

• Know how to use the Pythagorean Theorem in situations involving both perfect and non-perfect squares.

• Recognize when the Pythagorean Theorem is being described in a story problem


None noted

for students to:


• The students often forget to take the square root of the sum to find the length of the missing side.

• Students will often misuse the formula thinking all values go in the place of a and b because they do not have a sound understanding of the difference between the legs and the hypotenuse.


Although the focus of the indicator is procedural, students need to have an understanding of the “concept” of the Pythagorean Theorem. Conceptual knowledge is based on relationships; therefore, students should gain of an understanding of how these values connect beyond the reciting the formula.

The Pythagorean relationship states that if a square is constructed on each side of a right triangle, the sum of the areas of the two smaller squares equals the area of the largest square. The Pythagorean Theorem is most frequently represented as a2 + b2 = c2

, with "a" and "b" being the legs (the two sides that create the right angle) and "c" being the hypotenuse (the side directly across from the right angle) of a right triangle.

It is extremely important that the students understand (not just memorize) the different applications of the Pythagorean Theorem. The students should be given a variety of story problems that can be solved by using the Pythagorean Theorem. An example of this: A diver dove off of the dock and swam 50 feet to the where the buoy was attached to the bottom of the lake. The buoy is 25 feet above his head; how far is it from the dock to the buoy? This question requires the students to draw


3

a picture and to realize that they are looking for a missing leg; the information that was given is the length of the hypotenuse and one leg.

• Children’s book: “What’s your angle Pythagoras” By: Julie Ellis

Assessment Guidelines

The objective of this indicator is to apply which is in the “apply procedural” knowledge cell of the Revised Taxonomy. Although the focus of the indicator is to gain computational fluency with using the Pythagorean Theorem, the learning progression should also build the student’s conceptual knowledge in order to support retention. The learning progression to apply requires students to recall explore the Pythagorean relationship using a variety of examples. Students use their observations to generalize connections (8-1.7) among the areas of the squares. They then generalize a mathematical statement (8-1.5) summarizing this connection using correct and clearly written or spoken words (8.1.6). Students translate this verbal description to mathematical notation understanding that both are equivalent symbolic expressions of the same relationship (8-1.4). They use

their conceptual understanding of the Pythagorean Theorem as they engage in problem solving and repeated practice to gain computational fluency.


4


Indicator 8-4.2

Use ordered pairs, equations, intercepts, and intersections to locate points and lines in a coordinate plane.


In the sixth grade students represented with ordered pairs of integers the location of a point in the coordinate grid (6-4.1). In seventh grade, students analyzed tables and graphs to determine the rate of change between and among quantities (7-3.2). Students gained an understanding of slope as a constant rate of change (7-3.3).

In eighth grade, students use ordered pairs, equations, intercepts, and intersections to locate points and lines in a coordinate system (8-4.2). They also the coordinates of the x- and y- intercepts of a linear equation from a graph, equation, and/or graph (8-3.6)

Taxonomy Level

Cognitive Dimension: Apply Knowledge Dimension: Procedural

Key Concepts

• y-intercept

Vocabulary

• x-intercept • slope • function table • function rule • coordinate system • ordered pair • y-axis • x-axis • linear function


5



• Use an equation to locate points by creating a function table

for students to:

• Use points from a table to graph a line • Use (plot) intercepts to graph a line • Determine the point of intersection after graphing two equations or from a

given graph. • Understand that the point of intersection is where both equations have the

same value for x and y. • Understand that a line consists of infinitely many points • Understand that the x-intercept is the result of the intersection between a

line and the x-axis line. It is in the form (x, 0) • Understand that the y-intercept is the result of the intersection between a

line and the y-axis line. It is in the form (0, y).


• Know that the point of intersection is the solution to a system of equations.

for students to:

• Solve for the x- and y- intercept without the graph.


None noted


• To clarify the intent of this indicator, replace the word locate with graph. For example, o Students use ordered pairs to grapho Students use equations to

points and lines. graph

o Students use intercepts to points and line.

grapho Students use intersections to

points and line. graph

points and line.

• The intent of the indicator as it relates to intersections of lines is not to solve a system of linear equations. Real world story problems can be used to explore the concept of intersection. For example, at Kira’s Video, they charge $12 a month and $0.50 for each video. At Arika’s video there is no fee to join but charges $1 for each video. For how many movies will the cost at each video store by the same?


6

• Below is an example that may be used to illustrate how students use points to determine lines.

o Which line contains the following points (4,4) and (0,0)? Answer: c o What two lines intersect at (4,4)? Answer: c & d o What is the line that has an x-intercept of 4? Answer: d o What line has a y-intercept of 4? Answer: a o What line has the point (0,-1) on it? Is there another way to name

this point? Answer b, yes it’s the y intercept of -1


The objective of this indicator is to use which is in the “apply procedural” cell of the Revised Taxonomy. Procedural knowledge is not only knowledge of steps and techniques but also knowing when to use appropriate those steps. The learning progression to use requires students to recall how to plot points and how to write points given the graph. Students use their understanding of ordered pairs, equations, intercepts and intersections to create point and lines on the coordinate plane. They generalize connections (7-1.7) among these concepts and understand that each is a distinct symbolic form that represents the same linear relationship (7-1.4). Students translate from equation to ordered pairs to graph, from intercepts to graph and explore

real world problems to gain a conceptual understanding of intersection.


7


Indicator 8-4.3

Apply a dilation to a square, rectangle, or a right triangle in a coordinate plane.


In sixth grade the students applied strategies to find the missing vertices of various polygons (6-4.2).

In eighth grade, students apply a dilation to a square, rectangle, or a right triangle in a coordinate plane (8-4.3)

Taxonomy Level

Cognitive Dimension: Apply Knowledge Dimension: Procedural

Key Concepts

• Dilation

Vocabulary

• Scale Factor • Ordered Pairs • Image • Center of dilation

• A’- A prime and denotes the image of the original or pre-image.

Notation



• Understand the meaning of dilation

for students to:

• Recall the characteristics of a square, rectangle and right triangle • Apply a dilation factor to enlarge or reduce squares, rectangles, and right

triangles. • Multiply the ordered pairs by the dilation factor. • Use the new coordinates to graph the image of the pre-image.


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• Understand that if the scale factor of the dilation is greater than 1, the image will be bigger than the original.

• Understand if the scale factor of the dilation is less than 1 but greater than 0, the image will be smaller than the original.

• Understand that the image and pre-mage may overlap or one may be inside the other.

• Use appropriate notation to denote the image and pre-image


• Apply a dilation factor to enlarge or reduce irregular shapes.

for students to:


• Students sometimes have difficulty plotting ordered pairs, by wanting to reverse the x and y axis “moves.” If this is the case, location activities should precede or be used in a differentiated lesson for those students having difficulty.

• Students often think that all dilation factors will always enlarge the pre- image. They do not connect the idea that multiplying by a fraction or decimal less than one but greater that zero reduces the size and that multiplying by a whole number 1 and greater enlarges the image.


Students should be given several opportunities to investigate the process of applying dilation to various polygons. A dilation is defined as a transformation that copies, enlarges, or reduces an image or polygon. One strategy would be to have students construct a square, rectangle or a right triangle in a coordinate plane and record the original coordinate points next to the vertices of the shape. Students should then multiply these coordinate pairs by a number (this number is called the scale factor) to create a similar figure. The new shape should then be constructed using the new coordinate pairs and the new coordinates recorded next to the vertices of the new shape. By doing so, students have performed a dilation on the figure. This provides the opportunity to discuss similarities and differences between the old and new ordered pairs and the shapes they produced to assist in the development of mental models. Once students have a deeper understanding of the effect of a dilation, the teacher may want to move to more abstract examples by giving them only the coordinates of the vertices of a polygon and having them find the coordinates of a dilation of the given polygon, given the scale factor.


The objective of this indicator is to apply which is in the “apply procedural knowledge” cell of the Revised Taxonomy. To apply is to carry out or use a


9

procedure in various situations. The learning progression to apply requires students to recall and understand the meaning of dilation. They also recall the characteristics of squares, rectangles and right triangles. Students explore examples where they multiply by several different scale factors. They generalize connections (8-1.7) among examples where the factors are between zero and one or greater than one. They generalize mathematical statements (8-1.5) about these relationships using correct and clearly written or spoken words (8-1.6). They use their understanding of the relationships to apply dilations to other examples and explain and justify

their answers to their classmates and teacher.


10

Standard 8-4: The student will demonstrate through the mathematical processes an understanding of the Pythagorean theorem; the use of ordered pairs, equations, intercepts, and intersections to locate points and lines in a coordinate plane; and the effect of a dilation in a coordinate plane

Indicator 8-4.4

Analyze the effect of a dilation on a square, rectangle, or right triangle in a coordinate plane.


In sixth grade the students applied strategies to find the missing vertices of various polygons (6-4.2).

In eighth grade, students apply a dilation to a square, rectangle, or a right triangle in a coordinate plane (8-4.3)

Taxonomy Level

Cognitive Dimension: Analyze Knowledge Dimension: Conceptual

Key Concepts

• Dilation

Vocabulary

• Scale Factor • Ordered Pairs • Image • center of dilation • Proportional reasoning

• A’- A prime and denotes the image of the original or pre-image.

Notation



• Name the new coordinates of a polygon when given the coordinates and the scale factor.

for students to:


11

• Understand that a fractional scale factor will reduce the pre-image and a scale factor greater that 1 will enlarge the pre-image.

• Know that a scale factor of 1 will copy the image. • Analyze the impact the scale factor has on the area of the polygon. • Understand that the image and pre-image are similar shapes. • Use proportional reasoning to analyze relationships


• Apply a dilation to irregular shapes.

for students to:


Students often think that dilations will always enlarge the pre- image. They do not connect the idea that multiplying by a fraction or decimal less than one but greater that zero reduces the size and that multiplying by a whole number 1 and greater enlarges the image.

Student may not understand that if the anchor point or the center of dilation is a point within the pre-image the dilation factor would not be applied to that point.


This is an extension of 8-4.3. When students are provided with examples of dilations, they should be able to identify them as either being a dilation or as not being a dilation. As part of this indicator, students also explore the effect of a dilation on the area of the figures.


The objective of this indicator is to analyze which is in the “analyze conceptual” knowledge cell of the Revised Taxonomy. To analyze means to break material down into its constituent parts and determine how the parts relate to each other and the overall purpose; therefore, students examine the coordinates and area of figures (parts) and determine how their relate to dilations (purpose). The learning progression to analyze requires students to recall the meaning of dilations and understand how to perform dilations. They recognize the relationships among scale factor and proportional relationship between the pre-image and image. They explore a variety of examples and generalize connections (8-1.7) among those examples in order to generalize mathematical statements (8-1.5) related to the effect of dilations. Students use deductive reasoning (8-1.3) to move from generalized statements to specific relationships in order to describe the effects.

Measurement Eight Grade

1

Indicator 8-5.1

Use proportional reasoning and the properties of similar shapes to determine the length of a missing side.


In sixth grade, students use proportional reasoning to determine unit rates (6-5.6). They also use scale to determine distance (6-5.7). They gained an understanding of the relationship between ratio/rate and multiplication/division (6-2.6). They also classify geometric shapes as being similar (6-4.8). In seventh grade, students must apply proportional reasoning to find missing attributes of similar shapes (7-4.8).

In eighth grade, students use proportional reasoning and the properties of similar shapes to determine the length of a missing side (8-5.1).

The concepts of proportional reasoning and similar figures will be used throughout high school Geometry.

Taxonomy Level


• Ratio • Proportional • Congruent • Corresponding • Similar Figures • Ratio of similitude - The ratio of the corresponding sides of similar figures is

called the ratio of similitude. This ratio of similitude is also the ratio of the corresponding diagonals, altitudes, medians, angle bisectors, apothems and perimeters.

Standard 8-5: The student will demonstrate through the mathematical processes an understanding of the proportionality of similar figures; the necessary levels of accuracy and precision in measurement; the use of formulas to determine circumference, perimeter, area, and volume; and the use of conversions within and between the U.S. Customary System and the metric system.


2


for students to:

• Recognize similar polygons as shapes that have the same shape, but different sizes

• Recognize that corresponding angles are congruent in similar shapes • Recognize corresponding sides as proportional • Find corresponding sides of any pair of similar polygons (by observation and

by angle measurement) • Solve proportions • Find the missing side length going from the smaller shape to the larger and

from the larger to the smaller shape • Apply proportional reasoning to similar shapes in different geometric

“families”


for students to:

• Find the missing angle measures in similar figures • Use the ratio of similitude to calculate measurements related to the

diagonals, altitudes, medians, angle bisectors, apothems and perimeters of similar figures.

Student Misconceptions/Errors Students may have difficulty determining which sides are the corresponding sides.

Instructional Resources and Strategies Students should begin with explorations of drawings for various types of geometric figures. This knowledge should be extended to problems in different contexts.

Example: Given that rectangle ABCD is similar to rectangle A'B'C'D' where A corresponds to A' and so on, if AB = 12, A'B' = 4, AD = 9, find A'D'.

Problems that involve creating and analyzing scale drawings also give students good experience with similarity and proportionality.

To use proportional reasoning and properties of similar shape, have students start with rectangles. They are easily sorted into “families” or groups by examining ratios of corresponding sides. When a family of rectangles is nested (rectangles share the lower left corner) on top of each other, the diagonals will line up and corresponding vertices will form a straight line.


3

Students should develop a deep understanding of the method of cross-multiplying through meaningful experiences. Cross multiplication is a powerful technique but needs to be taught with understanding. Expressing the terms as equivalent ratios is one way of developing cross multiplication. They need to be able to analyze and explain how the method works and know when it is reasonable to make use of this method. One way to develop cross multiplying is to create equivalent ratios by multiplying each term by 1 so that the denominators are equal then students find the unknown value in the numerator:

372 a=

377

7323

xxa

xx

= xax 723 =

• http://illuminations.nctm.org/LessonDetail.aspx?ID=L515 lesson 6

Resources

Assessment Guidelines The objective of this indicator is to use proportional reasoning, which is in the “apply procedural” knowledge cell of the Revised Taxonomy table. Procedural knowledge is tied to knowledge of criteria for determining when to use appropriate procedures. The learning progression to use proportional reasoning requires students to recall the characteristics of similar polygons. After students determine that given polygons are similar figures, they must compare the characteristics of the polygons in order to locate the corresponding parts. Students will use proportions fluently in order to find a missing side length. Students should understand similarity as well as proportionality at a physical, numeric, graphic, and symbolic level. By generalizing the connections (8-1.7) among each of the representations, the students will gain a more in-depth understanding than if these representations were viewed independently. Students use correct and clearly written or spoken words in order to communicate their ideas about the proportionality of these similar figures (8-1.6). They use

proportional reasoning to determine if their answer is reasonable.


4

Indicator 8-5.2

Explain the effect on the area of two-dimensional shapes and on the volume of three dimensional shapes when one or more of the dimension are changed.


In fourth grade, students generated strategies to determine the area of rectangles and triangles (4-5.5). They extend this knowledge to apply formulas to determine area of rectangles, triangles and parallelograms in fifth grade (5-5.4). Also in fifth grade, students apply strategies and formulas to determine volume of rectangular prisms (5-5.5). In seventh grade, the students apply strategies and formulas to determine volume of prism, pyramid, and cylinder (7-5.2).

In eighth grade, the students will explain the effect on the area of two-dimensional shapes and on the volume of three dimensional shapes when one or more of the dimension are changed (8-5.2).

In Geometry, the students will analyze how changes in dimensions affect the perimeter and extend the list of geometric figures to include quadrilaterals and other regular polygons (G-4.4). Spheres are added when analyzing the changes in dimensions and how it affects the volume of objects (G-7.3).

Taxonomy Level


• Area Vocabulary

• Volume • Geometric figures (rectangles, triangles, parallelograms, rectangular prism

and pyramid, triangular prism and pyramid, cylinder) Instructional Guidelines For this indicator, it is essential for students to:



5

• Recall area and volume formulas for geometric figures • Calculate area and volume of geometric figures • Understand the concept of volume and cubic units as well as area and square

units. • Examine patterns and draw conclusions based on the patterns that exist


for students to:

• Use spheres when changing the dimensions and analyzing the effect on volume

• Analyze how changes in dimensions effect perimeter Student Misconceptions/Errors Students may forget that volume is measured in cubic units and area is measured in square units. Students may not understand the meaning of doubling or tripling a dimension.

Instructional Resources and Strategies Give students a rectangle with a width of 2 and a length of 5 drawn on a grid. Have the students determine the area of the given shape. Then have them draw a rectangle whose length is twice as long and determine the area. Have them complete a table or chart to record the measurements (width, length, and area). With a different or the “new” drawn rectangle, change both dimensions and have the students determine the effect on the area. Students should be given many opportunities to explore the effects on the area of two-dimensional shapes.

An example of a table you can use is:

Sketch of Original Shape

Original Dimensions of

the Edges

Original Area New Dimensions of

the Edges

New Area


6

This same type of teaching strategy can be used when explaining the effects on the volume of a three-dimensional shape. Start with a cube then use other three-dimensional shapes. A table similar to the one above can be used to record information being sure to replace area with volume and square units with appropriate cubic units.

Share the data in a class discussion. Lead the students in formulating ideas about how the lengths of the edges affect areas of these two-dimensional shapes.

Assessment Guidelines The objective of this indicator is to explain, which is in the “understand conceptual” knowledge cell of the Revised Taxonomy table. Conceptual knowledge is not bound by specific examples. For this reason, the student’s conceptual knowledge of how the area of two-dimensional shapes and the volume of three dimensional shapes when one or more of the dimension are changed should be evaluated using a variety of examples. To explain is to construct a cause and effect model; therefore, explanation should include the cause (doubling, tripling, etc..) and the effect (area doubles, etc..). The learning progression to explain requires that students recall the formulas for area and volume as well as calculate these answers with fluency. Students should use their previous knowledge of area and volume and apply it to a variety of examples in order to examine the patterns that exist. Students draw conclusions based on these patterns and use questioning techniques to either prove or disprove their conclusions (8-1.2). This process should be completed separately for two-dimensional and three-dimensional figures. Throughout this process students should use

correct terminology to communicate their thoughts clearly through either speaking or writing (8-1.6).


7

Indicator 8-5.3

Apply strategies and formulas to determine the volume of the three-dimensional shapes cone and sphere.


In fifth grade, students applied strategies and formulas to determine the volume of a rectangular prism (5-5.5). In seventh grade, students applied formulas to find the volume of prisms, pyramids, and cylinders (7-5.2).

In 8th grade, students apply strategies and formulas to determine the volume of the three-dimensional shapes cone and sphere (8-5.3).

In Geometry, students extend their knowledge of volume to include hemispheres and composite objects (G-7.2). Students will also analyze how changes in dimensions affect the volume of a geometric object (G-7.3).

Taxonomy Level


• Volume Vocabulary

• Three-Dimensional Figures • Base • Height • Sphere and Cone (review characteristics) • Area of Base


for students to:

• Understand the concept of height of a solid • Be able to fluently calculate with fractions, decimals, and exponents



8

• Know the estimated equivalent of pi as 3.14 and 722

• Recall the formula to find area of a circle • Understand the concept of volume and cubic units • Be able calculate volume of spheres and cones • Identify characteristics of cones and sphere from verbal and pictorial

representations • Estimate answers by rounding


for students to:

• Use Pythagorean Theorem to find the height of a cone • Find exact solutions (in terms of pi) • Calculate with measurements that are irrational


• It is difficult for students to understand conceptually that volume of the cone is 1/3 the volume of a cylinder.

• Students may not understand that multiplying by 1/3 is the same as dividing

by 3.

• Students confuse the height and lateral height of a pyramid.

• Students may not use order of operations when simplifying the formula.

• Students have difficulty remembering volume is cubic units.

• Students may confuse the diameter and the radius.

Instructional Resources Below is one strategy that may be used to build a conceptual understanding of the volume formula for a cone.

Example:

Divide your class into groups. Give each group a different size cylinder and cone (each have the same size base and height) and use either a container of water, sand, rice, etc. Have each group predict the relationship between the volume of the cylinder to the volume of the cone. Then have them fill the cone with the water, sand or rice and pour it into the cylinder until the cylinder is filled. Students will see that the


9

cone needs to be filled three times in order to fill the cylinder. Therefore, the volume of the cylinder is 3 time the volume of the cone (3:1) or the volume of the cone is 1/3 the volume of the cylinder with the same base and height regardless of the shape of the base or the position of the vertex (V = ⅓(A x h, where A is area of the base)).

Below is one strategy that may be used to build conceptual knowledge of volume of a sphere.

A comparison used for finding the volume of a sphere can be made with the volume of a cylinder, V = πr2h or the area of its base times its height which is (πr2) (2r) or 2 πr3. The sphere does not fill the entire cylinder; but its volume is 2/3 of the volume of a cylinder. So, 2/3(2 πr2) = 4/3 πr3. For example, use a sphere and cylinder that have the same height and diameter.

Either fill the sphere with water or sand and pour into cylinder or make the sphere out of playdough or clay and flatten the sphere so that it tightly fits the bottom of the cylinder. Be sure that all the air and space is out of the dough.

Using water or sand is easier to see the relationship. Students should observe that the volume of the sphere is 2/3 the volume of the cylinder.

• Resources

Connections to Literature:

Counting on Frank by Rod Clement is a great picture book that offers scenarios for students to explore and apply their knowledge and understanding of volume.



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The objective of this indicator is to apply, which is in the “apply procedural” knowledge cell of the Revised Taxonomy. Procedural knowledge is knowledge of specific steps or strategies to solve a problem or problem situation. Although the focus is to gain computational fluency with calculating volume of spheres and cones, the learning progression should integrate strategies to enhance both conceptual and procedural knowledge. The learning progression to apply requires students to recall the formula used to find the area of a circle. They understand the concept of volume being the number of cubic units to fill a space. It is essential for students to model these geometric figures to discover the relationships among the solids and their connections to the formulas used (8-1.1). Students also need to generalize and communicate the formula to see relationships among the solids and connections to the symbols in the formulas (8-1.4). Students should also use correct and clearly written or spoken words to explain their reasoning for their answers (8-1.6). Students apply these procedures in context as opposed to only rote computational exercises and generalize

connections among representational forms and real world situations (8-1.7). They engage in meaningful practice to support retention.



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Indicator 8-5.4

Apply formulas to determine the exact (pi) circumference and area of a circle.


In sixth grade, students applied strategies to find the circumference and area of a circle using an approximation of pi (3.14 and 22/7) (6-5.2). They also learned the relationship among circumference, diameter, and radius of a circle (6-5.1). In seventh grade, the students applied strategies and formulas to determine the surface area and volume of the three dimensional shapes prisms, pyramid and cylinder (7-5.2).

In eighth grade, students should apply formulas to determine the exact (pi) circumference and area of a circle.

In Geometry, students will carry out a procedure to compute the circumference and area of circles (G-5.1 and G-5.2). The students will also analyze how a change in the radius affects the circumference or area of a circle (G-5.3).

Taxonomy Level


• Circumference Vocabulary

• Area • Diameter • Radius • Pi



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for students to:

• Recall the formula for area and circumference • Find the diameter when given the radius and vice-versa • Understand the relationship between an estimate of pi and the exact value • Calculate the exact area and circumference when the diameter and radius is

a rational number. For example, find the exact area of a circle with a radius of 4 in. The area is 16π .

• Leave answers in π in the answer • Find the area or circumference of a fractional part of a circle (half, three-

fourths, etc..


for students to:

• Calculate the exact area and circumference when the diameter or radius is irrational.

• Give an approximation for the area or circumference of a circle • Find the length of the diameter or radius when given the circumference or

area in a real-world situation. Student Misconceptions/Errors

• Students may not understand that leaving pi in the answer is the exact value of circumference and area.

• Students may use the diameter instead of the radius to find area of a circle. • Students may interchange the area and circumference formula. • Students may forget that area is measured in square units and circumference

is a linear measure. Instructional Resources For this indicator, students are asked to leave pi in the answer to determine the exact value of the circumference and area. A discussion about the difference between leaving pi and using 3.14 or (22/7) may be needed. Assessment Guidelines The objective of this indicator is to applyknowledge cell of the Revised Taxonomy table. Procedural knowledge is knowledge of specific steps or strategies that can be used to solve a problem. Procedural knowledge is also tied to knowledge of criteria for determining when to use appropriate procedures. The learning progression to apply requires students to

, which is in the “apply procedural”

recall the formulas for both circumference and area. The students must be able to


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determine which formula would be used with respect to the context of the problem presented. A variety of real-world situations should be presented so that the students are comfortable differentiating between the concepts of circumference and area (8-1.7). Once the correct formula is selected, students will be required to simplify the equation and also examine the reasonableness of their answer. Since an approximation for area and circumference were found in a previous grade, students will understand

the distinct symbolic forms that represent the same relationship (8-1.4) and will use correct and clearly written or spoken words to communicate their answer (8-1.6).



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Indicator 8-5.5

Apply formulas to determine the perimeters and areas of trapezoids.


In fifth grade, students applied formulas to determine the perimeters and areas of triangles, rectangles and parallelograms (5-5.4). In sixth grade, students found the perimeter and area of irregular shapes (6-5.5). In seventh grade, students generate strategies to determine the perimeters and areas of trapezoids (7-5.3).

In eighth grade, students apply formulas to determine the perimeters and areas of trapezoids (8-5.5).

In Geometry, students will carry out a procedure to compute both the perimeter and area of quadrilaterals, regular polygons, and composite figures (G-4.1 and G-4.2). They will also use their knowledge of quadrilaterals to analyze how changes in dimensions affect the perimeter and area (G-4.4).

Taxonomy Level


• Area Vocabulary

• Perimeter • Base • Height


for students to:

• Recall the formula for area and perimeter of trapezoids • Recall the attributes of a trapezoid • Understand how to locate the height and bases of a trapezoid given in pictorial and verbal form • Substitute values correctly into the formula • Simplify


15


for students to:

• Calculate the area and perimeter when the height or bases are irrational. Student Misconceptions/Errors

• Students may think that the base of a trapezoid always means the bottom. • Students may forget to add the lengths of the bases together first. • Students may select the length of one of the legs of the trapezoids instead of

the height. • Students may interchange the area and perimeter formula. • Students may forget that area is measured in square units and perimeter is a

linear measure. Instructional Resources In seventh grade, students generated strategies to find area and perimeter of trapezoid. Students will need opportunities to connect their generated strategies with the formulas. Assessment Guidelines The objective of this indicator is to applyknowledge cell of the Revised Taxonomy table. Procedural knowledge is knowledge of specific steps or strategies that can be used to solve a problem. Procedural knowledge is also tied to knowledge of criteria for determining when to use appropriate procedures. The learning progression to apply requires students to

, which is in the “apply procedural”

recall the formulas for both perimeter and area. The students determine which formula would be used with respect to the context of the problem presented. A variety of real-world situations should be presented so that the students are comfortable differentiating between the concepts of perimeter and area (8-1.7). Once the correct formula is selected, students are required to simplify the equation using correct mathematical procedures. Students should examine the reasonableness of their answer and will use

correct and clearly written or spoken words to communicate their answer (8-1.6).



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Indicator 8-5.6

Analyze a variety of measurement situations to determine the necessary level of accuracy and precision.


Students have been taught to use units of measurement since the 1st

Levels of precision and accuracy are new skills to be introduced to 8

grade.

th

Taxonomy Level

grade students for the first time.

Cognitive Dimension: Analyze Knowledge Dimension: Conceptual Key Concepts

• Precision Vocabulary

• Accuracy • Review of the different units of measurement (Metric and U.S. Customary)


for students to:

• Recall the different types of measurement in both U.S. Customary and Metric units

• Determine what type of accuracy is needed for a given real-life situation • Distinguish between the concepts of accuracy and precision • Understand that a measurement is precise only to one-half of the smallest

unit used in a measurement • Identify situations where accuracy or precision is more appropriate • Understand that there is really no limit to how precisely an object can be

measured. It depends on the measurement tool being used For this indicator, it is not essential

for students to:


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• Measure attributes to a certain accuracy or precision Student Misconceptions/Errors Students may confuse the concepts of precision and accuracy.

Instructional Resources Students must distinguish between "accuracy," which means the closeness of a measurement to the exact value, and "precision," which means the claimed or implied closeness. For example, if I said my desk was 2 meters wide, and you said it was 2.345 meters wide, your answer would be more precise (claiming that you know it down to the millimeter); but if the desk is really 2.123 meters wide, then my answer is more accurate! Accuracy is the difference between our own calculation and the accepted real value. For example, suppose your math textbook tells you that the value of Pi is 3.14. You do a careful measurement by drawing a circle and measuring the circumference and diameter, and then you divide the circumference by the diameter to get a value for Pi of 3.16. The accuracy of your answer is how much it differs from the accepted value. In this case, the accuracy is 3.16 - 3.14 = 0.02. The precision with which the accepted value has been measured is not important. All that matters is how different your measurement is from that value; the bigger the difference, the less accurate your measurement. Also, accuracy depends on the instrument you are measuring with. But as a general rule: The degree of accuracy is half a unit each side of the unit of measure. The precision of a measurement describes the units

you used to measure something. For example, you might describe your height as 'about 6 feet'. That wouldn't be very precise. If however you said that you were '74 inches tall', that would be more precise. The smaller the unit you use to measure with, the more precise the measurement is. In mathematics, it is often necessary to make measurements that are as precise as you can make them. This requires that you use measuring instruments with smaller units. For example, measure the pencil below.

How long is the pencil? Students might say it’s about 9 centimeters. Some might guess and say about 9.5 centimeters, but the decimal place is just a guess.


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Because the smallest unit on the ruler you are using is one centimeter, the precision of your measurement is to the nearest centimeter. Now look at the picture below, where we are using a different ruler to measure the pencil.

How long is the pencil? Students might say about 9.5 centimeters. Again, some might guess and say about 9.51 centimeters, but the second decimal place is just a guess. Because the smallest unit on the ruler you are now using is one millimeter (one tenth of a centimeter), the precision of your measurement is to the nearest millimeter, or tenth of a centimeter. This second measurement is more precise, because you used a smaller unit

with which to measure. Discuss with the students this statement; the smaller the unit, the more precise the measurement. Discuss with the students why it is impossible to make a perfectly precise measurement. (There is really no limit to how precisely an object can be measured. It depends on the measurement tool being used.)

Assessment Guidelines The objective of this indicator is to analyze, which is in “analyze conceptual” knowledge cell of the Revised Taxonomy table. Analyze requires students to break material (measurement) into its constituent parts (accuracy and precision) and to determine how the parts relate to one another and to an overall structure or purpose. Conceptual knowledge is not bound by specific examples; therefore, the student’s conceptual knowledge of precision and accuracy should be assessed using a variety of real-life examples (8-1.7). The learning progression to analyze requires students to recall basic units of measurements and to apply them in real-world situations. When given these authentic situations, students should be able to differentiate between the concepts of precision and accuracy when asked specific questions. They also generate problems (8-1.1) that relate to accuracy and precision. Students should be able to explain there answers using correct and clearly written or spoken words to communicate

their answers to both teachers and their classmates (8-1.6).



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Indicator 8-5.7

Use multistep unit analysis to convert between and within the U.S. Customary System and the metric system.


In fourth grade, students recalled basic conversion facts within the U.S. Customary System for length, time, weight and capacity (4-5.8). They used equivalencies to convert units of length, weight, and capacity within the Customary System (4-5.3). In fifth grade, students recalled equivalencies associated with length, liquid volume and mass in the metric system (5-5.8) and converted within the metric system (5-5.3). In seventh grade, students recall equivalencies (7-5.4) and used one step unit analysis to convert within and between the U.S. Customary System and metric system (7-5.5)

In eighth grade, students use multi-step unit analysis problems to convert within the U.S. Customary System and the metric system, as well as, between the systems (8-5.7).

In Elementary Algebra, students will use dimensional analysis to convert units of measure within a system (EA-2.4)

Taxonomy Level


• Unit analysis (dimensional analysis)

Vocabulary

• System (Customary or Metric) • Unit cancellation

Mathematical Notation/Symbols:


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Abbreviations for length, weight and mass, and liquid volume for U.S. Customary and metric system units


for students to:

• Understand proportional reasoning • Multiplying fractions • Simplify expression • Understand that each equivalency when written as a fraction is a form of one • Use equivalencies to convert between systems • Set up ratios to convert using unit analysis:

For example, how many inches are in 10 yards? Students should know that 1 yd = 3 ft and 1 ft = 12 inches. The set up might look like this:

10 yd x 3 ft x 12 in = 10 x 3 x 12 in

1 1 yd 1 ft 1 x 1 x 1

= 360 inches.


• Use unit analysis for conversions within the metric system (although it works if they choose to use it). Students can rely on their knowledge of computing metric conversions (place value equivalencies or multiplying/dividing by powers of 10).

for student to:

Student Misconceptions/Errors Students may have trouble with placing their units in the correct position of the fraction to get their units to “cancel”.

Students need to know ALL conversion factors. Without knowing conversion factors, they cannot and will not accurately use unit analysis even if they know the correct procedure.


• A review of fractions or ratios that simplify to 1 and the process of cross simplification (cancelling) may need to be reviewed.

• Although the focus of the indicator is procedural, explore unit analysis in context builds conceptual understanding and supports retention. For example,


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You kept track of the number of miles you drove in your car and the amount of gasoline used for two months.

Number of Miles 290 242 196 237 184

Number of Gallons 12.1 9.8 8.2 9.5 7.8

a. What was the average mileage for a gallon of gasoline? (Round the answer to the nearest tenth)

b. Estimate the distance you can drive on 18 gallons.


The objective of this indicator is use, which is the “apply procedural” knowledge cell of the Revised Taxonomy. Procedural knowledge is knowledge of specific steps or strategies that can be used to solve a problem. The focus is to gain computational fluency with conversions within and between the US Customary and metric system. The learning progression to use requires students to recall the conversion equivalencies. They understand the relationship between the equivalencies and a form of one. Students review teacher generated examples of one step unit analysis problems and generalize mathematical statements (8-1.5) summarizing how that strategy may be applied to two step problems. They explain and justify their strategy using correct and clearly written or spoken words (8-1.6). They apply a strategy to other examples and real world situations (8-1.7) and use

their understanding of proportional reasoning to justify their answers.

Data Analysis and Probability Eighth Grade

1

Standard 8-6: The student will demonstrate through the mathematical processes an understanding of the relationships between two variables within one population or sample. Indicator 8-6.1 Generalize the relationship between two sets of data by using scatterplots and lines of best fit. Continuum of Knowledge In seventh grade, students were introduced to slope as a constant rate of change. [7-3.3] Seventh grade students predicted the characteristics of two populations. In eighth grade, students identify the slope of a linear equation from a graph, equation, and/or table [8-3.7]. They make scatterplots for two sets of data, find the line of best fit for each and compare these lines of best fit. In Elementary Algebra, students carry out a procedure to write an equation of a trend line from a given scatterplot (EA-4.4) and they analyze a scattterplot to make predications (EA-4.5). Taxonomy Level Cognitive Dimension: Understand Knowledge Dimension: Conceptual Key Concepts

• Scatterplot Vocabulary

• Line of best fit • Data • Inference • Data sets • Trend line • Scale • Outlier • Correlation (positive, negative or no)


for students to:

• Understand the relationship between correlation and slope • Recognize data that is increasing (positive correlation – positive slope) • Recognize data that is decreasing (negative correlation – negative slope)


2

• Recognize data that is varying or no change (no correlation) • Understand the structure and purpose of a scatterplot • Understand the purpose of a line of best fit • Interpret a trend in data • Understand the limitations of a trend line when interpreting data • Determine if a given line drawn with the scatterplot is a line of best fit based

on the relationship between the points and the line


for students:

• Understand how to determine slope as a numerical value. • Find the line of best fit.


• In the process of finding the line of best fit many students want to make sure that their line goes through the origin even though this is not in line with the data points.

• Students may try to connect all of the data points.

• Some students may include an outlier in their estimate of the line of best fit

and have their line be far from the expected slope. Instructional Resources and Strategies Some generalizations to consider when determining the relationship between two sets of data:

The more the points tend to cluster around a straight line, the stronger the linear relationship between the two variables (the higher the correlation).

If the line around which the points tends to cluster runs from lower left to upper right, the relationship between the two variables is positive (direct).

If the line around which the points tends to cluster runs from upper left to lower right, the relationship between the two variables is negative (inverse).

If there exists a random scatter of points, there is no relationship between the two variables (very low or zero correlation).

Very low or zero correlation could result from a non-linear relationship between the variables. If the relationship is in fact non-linear (points clustering around a curve, not a straight line), the correlation coefficient will not be a good measure of the strength.

Scatterplots give students the opportunity to interpolate and predict values not present in a display by estimating where the values will fall. Students need to realize that predictions from data have limitations.

Also it is important for students to be able to distinguish between variables that covary “by accident” and variables that illustrate a causal relationship.


3

Below is an activity that may be used to engage students in a discussion about these relationships. Since students have created scatterplots in 8-6.2, they can now begin to discuss the relationships and draw conclusions from the scatterplots. Have students work in groups. Use example 1 to model your thinking about how to interpret the scatterplot by referring the generalizations, a – e, in the “Content Overview “ section. (for example: trends in the data – positive, draw line of best fit, etc.)Have the groups discuss the remaining graphs and record the findings. Have each group report their findings. Be sure to discuss each graph for their relationships.

3.2.1.

654

987

10

1211


4

Assessment Guidelines The objective of this indicator is to generalize, which is the “understand conceptual” knowledge cell of the Revised Bloom’s Taxonomy. To generalize to abstract a general theme or major points; therefore, students are summarizing general themes as it related to the relationship between two set of data. The learning progression to generalize requires students to recall and understand the meaning of a scatterplot and a line of best fit. Students analyze graphical data to determine trends and the relationship between the points and the line of best fit. They use inductive reasoning (8-1.3) to formulate mathematical arguments about their observations. Students explain and justify their answers to their classmates and teacher using correct and clearly written or spoken words (8-1.6). If necessary, they revise their thinking based on the classroom discussion then generalize

mathematical statements (8-1.5) about the relationships between two sets of data.

Standard 8-6: The student will demonstrate through the mathematical processes an understanding of the relationships between two variables within one population or sample. Indicator 8-6.2


5

Organize data in matrices or scatterplots as appropriate. Continuum of Knowledge In fourth grade, students organized data in tables, line graphs, and bar graphs whose scale increments are greater than or equal to 1 (4-6.3). In sixth grade, they organized data in frequency tables, histograms, and stem-and-leaf plots (6-6.2). In seventh grade, they organized data in box plots and circle graphs (7-6.2). In eighth grade, students organize data in matrices or scatterplots as appropriate Students will also learn about matrices (8-6.2). In Elementary Algebra, students carry out a procedure to write an equation of a trend line from a given scatterplot (EA-4.4) and they analyze a scattterplot to make predications (EA-4.5). Taxonomy Level Cognitive Dimension: Analyze Knowledge Dimension: Conceptual Key Concepts

• Scatterplot Vocabulary:

• Matrix (matrices) • Row • Column • Data • Element • Data set • Scale • Range • Rectangular arrangement • Coordinates


for students to:

• Understand that a matrix is an array or another form of a table • Understand the purpose of creating a matrix • Identify an element in by row and column • Identify the size of a matrix • Understand the components of a matrix: rows, columns, and size of matrix. • Understand the purpose of a scatterplot • Set up the axes of the scatterplot appropriately • Plot the points


6


for students to:

• Perform mathematical operations on the matrices. Student Misconceptions/Errors

• The most common error in matrices is the interchanging of row and column. • The most common error in scatterplots is interchanging the axes components

when plotted. Instructional Resources and Strategies A matrix is a rectangular arrangement of values. Another way to describe a matrix is that it is a two-dimensional table in which the rows (horizontal groups of entries) and the columns (vertical groups of entries) are labeled to indicate the meaning of the data. The plural form of matrix is matrices. The size of the matrix is expressed by giving its dimensions (number of rows by the number of columns). For example, a matrix with 4 rows and 3 columns is a 4 by 3 or 4 x 3 matrix. An example of a matrix giving information about group of isosceles right triangles might look like this: Hypotenuse Leg Area √2 1 ½ √8 2 2 √18 3 9/2 √50 5 25/2 Students should be given information about the components of a matrix in order to use a matrix to organize data. Students should begin looking at a matrix as a table. Students need to become comfortable with matrices and see why they are useful. The students should begin to see the connection between a matrix and a spreadsheet.

A scatterplot is a useful summary of a set of bivariate, numerical data (two variables). It gives a good visual picture of the relationship between the two variables, and aids the interpretation of the correlation coefficient or regression model. Each unit contributes one point to the scatterplot, on which points are plotted but not joined. The resulting pattern indicates the type and strength of the relationship between the two variables. Below is an example of a scatterplot:


7

Assessment Guidelines The objective of this indicator is to organize, which is in the “analyze conceptual” knowledge cell of the Revised Taxonomy. To organize is to determine how elements (data points) fit or function together within a structure (scatterplot or matrix). The learning progression to organize requires students to understand the purpose of matrices and scatterplots. Students recall the components of each and understand how those components are used to organize the data. They analyze the given data and generalize connections (8-1.7) between the data and the components of each representation to determine how to organize their data. After the data has been organized, students explain and justify

their answer using correct and clearly written or spoken word (8-1.6).

Standard 8-6: The student will demonstrate through the mathematical processes an understanding of the relationships between two variables within one population or sample.


8

Indicator 8-6.3 Use theoretical and experimental probability to make inferences and convincing arguments about an event or events. Continuum of Knowledge In sixth grade, students used theoretical probability to determine the sample space and probability for one- and two-stage events using tree diagrams, models, lists, charts, and pictures (6-6.4). In seventh grade students will find and differentiate between the theoretical and experimental probability of the same events. (7-6.7)

In eighth grade students will use theoretical and experimental probability to make inferences and convincing arguments about an event or events (8-6.3) High school Data Analysis and Probability asks students to compare theoretical and experimental probability [DA-5.9]. Taxonomy Level Cognitive Dimension: Apply Knowledge Dimension: Conceptual Key Concepts

• Theoretical probability Vocabulary:

• Experimental probability • Event


for students to:

• Understand the characteristics of theoretical probability • Understand the characteristics of experimental probability • Understand the purpose of a simulation • Understand that experimental probability is the background for examining

theoretical probability • Understand that the experimental approach has its limitations because it is

impossible to perform an infinite number of trials • Understand the relationship between a simulation and experimental

probability • Understand that when all outcomes of an experiment are equally likely, the

theoretical probability of an event is the fraction of the outcomes in which the event occurs

• Write a probability using appropriate notation


9

• Interpret the probability For this indicator, it is not essential

for students to:

• Conduct the experiment to arrive at the experimental probability. Student Misconceptions/Errors

• Students may attempt to substitute theoretical probability for experimental results because they feel their answer is wrong.

• Students may omit a resultant experimental event because it is an evident outlier and would not fit in with the other data.

• Students may not understand the impact of having very few attempts can cause the results to be deceptive.


• The focus of the indicator is for students to apply their understanding of experimental and theoretical probability to make inference. Probability in the eighth grade engages students in inferential thinking skills and persuasive arguments based upon theoretical or experimental probability. After determining probability of both independent and dependent events, students should be able to use this information to make inferences, draw conclusions, and develop arguments to support their interpretation of the data.

• Use theoretical probability and experimental results to make predictions and decisions. Students should analyze the factors that cause experimental data to be poor predictors. For example, have students design games, try out the games, and decide whether they are fair or not using experimental results and theoretical probabilities.

Game 1: Toss 3 two-color counters. You get a point if there are at least two red counters showing. Otherwise, your partner gets a point. Is the game fair? Why or why not? Game 2: Roll two number cubes. If the product of the two numbers is even, you get a point. If it is odd, your partner gets a point. Is the game fair? Why or why not?

Students then use this information to predict outcomes. For example, if you played the game 100 times, who would win? 200 games?

Assessment Guidelines The objective of the indicator is to use, which is in the “apply conceptual” knowledge cell of the Revised Taxonomy. To apply conceptual knowledge means to use your understanding of interrelationships among concepts to solve problems as opposed to an algorithm. The learning progression to use requires students to recall and understand the meaning of theoretical and experimental probability.


10

They use that understand to make observations about the results of an event or events and explain and justifyBased on their discussions, students

their observations to their classmate and teacher. generalize mathematical statements (8-1.5)

about the data and use those statements to make inferences about the event or events. They use

correct and clearly written or spoken words to communicate their understanding (8-1.6).

Standard 8-6: The student will demonstrate through the mathematical processes an understanding of the relationships between two variables within one population or sample. Indicator 8-6.4 Apply procedures to calculate the probability of two dependent events. Continuum of Knowledge


11

In sixth grade, students applied procedures to calculate the probability of complementary events (6-6.5). They also used theoretical probability to determine the sample space and probability for one- and two-stage events such as tree diagrams, models, lists, charts, and pictures (6-6.4) and applied procedures to calculate the probability of complementary events (6-6.5). In seventh grade, students calculated (7-6.5) and interpreted (7-6.6) the probability of mutually exclusive simple or compound events. They also differentiated between experimental and theoretical probability of the same event (7-6.7) and used the fundamental counting principle to determine the number of possible outcomes for a multistage event (7-6.8) In the eighth grade, students will calculate the probability of two dependent events (8-6.4). In high school data analysis and probability, students classify events as either dependent or independent (DA-5.3) and find their probabilities. Taxonomy Level Cognitive Dimension: Apply Knowledge Dimension: Procedural Key Concepts

• Probability Vocabulary:

• Dependent events • Independent events • Event • Sample Space


for students to:

• Understand the characteristics of dependent events • Understand the difference between a dependent and independent event • Be able to multiply fractions to find the probability of a two-stage event. • Use appropriate strategies such as tree diagrams, area model, list, etc.. • Represent probabilities in fractional form • Understand probability notation. Example: P(blue, green) • Determine the reasonableness of their answers


for students to:

• Calculate complementary, mutually exclusive or compound events. Student Misconceptions/Errors


12

• Students sometimes have a difficult time understanding the meaning of the

term “dependent” in both probability and in algebra. • Sample Space is a term that is confused by many students as they think it is

related to a sample. Instructional Resources and Strategies Students not only calculate the probability of two dependent

events but also interpret the results (8-6.5). With regard to dependent events, the results are more complex since before multiplying probabilities, students must consider the change in the number of outcomes for each event. Students need to know or understand that when the outcome of one event affects the outcome of another event, the events are dependent. To find the probability for two dependent events, students can use the formula, P(A and B) = P(A)∙P(B given A), where P(B given A) means the likelihood that B will happen when A happens.

To find the probability of the dependent event have the students use multiplication. Explain that for getting 2 white marbles without replacement, you have to assume that the first event you are looking for occurred because that is the situation you are interested in. If a white marble was removed, there will be only 3 marbles left and only one of the 3 will be white. The sentence for 2 white marbles without replacement should be ¼ x ⅓= 1/12.

• Conduct probability exercises using marbles in a bag, candy or fake lotto tickets with student names. Have students use the software Inspiration, the drawing tools in MS Office or another graphic organizer to create tree diagrams to represent probability results visually.

Assessment Guidelines The objective of this indicator is to apply, which is in the “apply procedural” knowledge cell of the Revised Taxonomy. Procedural knowledge is tied to knowledge of criteria for determining when to use appropriate procedures and is applied to familiar and unfamiliar situations. The learning progression to apply requires students to recall and understand the meaning of dependent events. Students generate examples of dependent events to deepen conceptual understanding. They useetc..) to model the probability. They further

an appropriate strategy (area model, lists, tree diagrams, analyze these events using an

experimental approach where appropriate. They make conjectures about the probability and evaluate those conjectures to prove or disprove their reasoning (8-1.2). Students use inductive and deductive reasoning (8-1.3) as they calculate the


13

probability. Then explain and justify

their answers using correct and clearly written or spoken words (8-1.6).

Standard 8-6: The student will demonstrate through the mathematical processes an understanding of the relationships between two variables within one population or sample. Indicator 8-6.5 Interpret the probability of two dependent events. Continuum of Knowledge In sixth grade, students applied procedures to calculate the probability of complementary events (6-6.5). They also used theoretical probability to


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determine the sample space and probability for one- and two-stage events such as tree diagrams, models, lists, charts, and pictures (6-6.4) and applied procedures to calculate the probability of complementary events (6-6.5). In seventh grade, students calculated (7-6.5) and interpreted (7-6.6) the probability of mutually exclusive simple or compound events. They also differentiated between experimental and theoretical probability of the same event (7-6.7) and used the fundamental counting principle to determine the number of possible outcomes for a multistage event (7-6.8) In the eighth grade, students will calculate the probability of two dependent events (8-6.4). In high school data analysis and probability, students classify events as either dependent or independent (DA-5.3) and find their probabilities. Taxonomy Level Cognitive Dimension: Understand Knowledge Dimension: Conceptual Key Concepts

• Probability Vocabulary:

• Sample Space • Event • Dependent events • Independent events


for students to:

• Understand the characteristics of a dependent event • Understand the difference between an independent and dependent event • Interpret the meaning of a fractional probability • Understand probability notation


for students to:

• Calculate the probability • Calculate complementary, mutually exclusive or compound events.


• Students sometimes have a difficult time understanding the meaning of the term “dependent” in both probability and in algebra.

• Sample Space is a term that is confused by many students as they think it is related to a sample.


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• The focus of the indicator is for students to interpret. Although having students calculate probabilities first then interpret the results would be a natural and viable progression, it would also be reasonable for students to be given

the probability of events that they did not calculate and be asked to interpret the probability.

Assessment Guidelines The objective of this indicator is to interpret

requires students to

which is in the “understand conceptual” knowledge cell of the Revised Taxonomy. To interpret is to change from one form (numerical) to another form (verbal). The learning progression to interpret

understand the meaning of dependent events. Given a real world problem and a probability, students use inductive and deductive reasoning (8-1.3) and summarize observations. They use their understanding of the characteristics of dependent events to generalize mathematical statements (8-1.5) about their observations. Students use

correct and clearly written or spoken words to communicate their understanding.

Standard 8-6: The student will demonstrate through the mathematical processes an understanding of the relationships between two variables within one population or sample. Indicator 8-6.6 Apply procedures to compute the odds of a given event. Continuum of Knowledge In fifth grade, students represented the probability of a single-state event in words and fractions (5-6.5). In sixth grade, students applied procedures to calculate the probability of complementary events (6-6.5). They also used theoretical probability to determine the sample space and probability for one- and two-stage


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events such as tree diagrams, models, lists, charts, and pictures (6-6.4) and applied procedures to calculate the probability of complementary events (6-6.5). In seventh grade, students calculated (7-6.5) and interpreted (7-6.6) the probability of mutually exclusive simple or compound events. They also differentiated between experimental and theoretical probability of the same event (7-6.7) and used the fundamental counting principle to determine the number of possible outcomes for a multistage event (7-6.8) In eighth grade, students learn what odds are and how to compute the odds of a given event (8-6.6). Taxonomy Level Cognitive Dimension: Apply Knowledge Dimension: Procedural Key Concepts

• Odds Vocabulary:

• Probability • Event • Favorable • Unfavorable • Sample Space • Outcome


for students to:

• Understand the meaning of odd • Understand the difference between odds and probability • Understand the relationship between odds and probability • Understand the difference between favorable outcomes and unfavorable

outcomes (other phrases like desirable/undesirable, chances for/chances against, etc…)

• Write the odd in fractional form (3/2) or as a ratio 3:2 • Interpret the meaning of odds. For example, 3 to 2 means 3 favorable

outcomes to every 2 unfavorable outcomes, and we write 3:2

.


for students to:

• None noted Student Misconceptions/Errors


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Students may confuse finding probability with determining the odds. Instructional Resources and Strategies

• The discussion of odds should include that odds are another way to express probability; ratio involving favorable outcomes and unfavorable outcomes or Chances for versus Chances against.

Probability = # of possible outcomes

# of favorable outcomes

Odds = # of unfavorable outcomes

# of favorable outcomes

To go from probability to odds:

1. Students should note that the numerator is the same in both probability and odds, so the numerator of the answer will be the same.

2. To find the denominator (or the number of undesirable outcomes), we subtract the numerator from the denominator.

Example, if the probability of winning is 7/15, then the odds of winning is ___?

The numerator is the same which is 7. The denominator is: 15-7 = 8 (possible – favorable = unfavorable). The answer is 7/8

• Students may also explore going from

.

odds to probability: If the odds for an event are 3: 2, what is the probability of the event happening?” Discuss with the students how to solve this problem. If the favorable outcomes = 3 then the possible outcomes = favorable outcomes + unfavorable outcomes = 3 + 2 = 5. Thus, the probability of the event happening is .)

Assessment Guidelines The objective of this indicator is to apply which is in the “apply procedural” knowledge cell of the Revised Taxonomy. Although the focus of the indicator is to apply, students also use their conceptual understanding of the relationship between probability and odds to compute the odds. The learning progression to apply requires students to understand the meaning of and how to compute probability. Students explore real world examples involving probability and odds in order to generalize connections (8-1.7) among these concepts and real world situations. Students use these connections to generalize mathematical statements (8-1.5) about the relationship between probability and odds. They use this understanding


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to solve problems and explain and justify their answers using correct and clearly written or spoken words (8-1.6). To deepen understanding, students generate and solve

more complex problems (8-1.1).

Standard 8-6: The student will demonstrate through the mathematical processes an understanding of the relationships between two variables within one population or sample. Indicator 8-6.7 Analyze probability using area models. Continuum of Knowledge In fifth grade, students applied formulas to determine the perimeters and areas of triangles, rectangles, and parallelograms (5-5.4). In the sixth grade, they applied strategies and procedures to estimate the perimeters and areas of irregular shapes (6-5.4). Seventh graders used ratio and proportion to solve problems involving scale factors and rates (7-5.1). As it relates to probability, seventh grade students calculated (7-6.5) and interpreted (7-6.6) the probability of mutually exclusive simple or compound events. They differentiated between experimental and theoretical probability of the same event (7-6.7) and used the fundamental


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counting principle to determine the number of possible outcomes for a multistage event (7-6.8) In eighth grade, students calculate (8-6.4) and interpret (8-6.5) probability for two dependent events. They also analyze probability using area models (8-6.7). In high school Geometry, student use geometric probability to solve problems [G-2.7]. Taxonomy Level Cognitive Dimension: Analyze Knowledge Dimension: Conceptual Key Concepts

• Area model • Probability • Ratio


for students to:

• Compute the area of polygons • Interpret probability notation • Understand how to represent probability using a rectangular area model • Understand that the probability is the ratio between the area of the shapes;

regardless of the shapes • Understand how to set up a ratio • Interpret area models


• When constructing the rectangular area models, students may not understand that the answer is the overlapping region.

• Students may struggle with the conceptual basis of constructing area models. Instructional Resources and Strategies

• An area model is drawn to represent a whole with partitions drawn to represent the data for first event and then partition the square in the other directions to represent the data for the second event. The area model is easy for students to use and understand. Students can make connections with pictorial models used to multiply fractions.

• Another aspect is to actually work with area itself. The dart board analogy usually interests students. Find the probability of hitting the center.

P (hitting the center) = area of center circle


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area of dart board circle

• Students’ understanding of the area model should extend beyond circle inside of circle. They should transfer their understanding to circle inside of rectangles, rectangles inside of circles, etc… (any combination of polygons their know).

Assessment Guidelines The objective of this indicator is to analyze which is in the “analyze conceptual” knowledge cell of the Revised Taxonomy. To analyze means break material (probability) into its constituent parts (area of shapes) and determine how the parts relate to one another and to the overall structure. The learning progression to analyze requires students to understand the meaning of probability. Students recall their knowledge of multiplying fractions using rectangular models and how to compute the area of polygons. They explore probability using these models and generalize the connections (8-1.7) between these concepts. They generalize mathematical statements (8-1.5) about these connections and use that understanding to solve problems. Students explore other types of area models and generalize the connection between (8-1.5) part/whole ratios and area of shape/areas of whole shape ratios. They solve a variety of problems and explain and justify


Standard 8-6: The student will demonstrate through the mathematical processes an understanding of the relationships between two variables within one population or sample. Indicator 8-6.8 Interpret graphic and tabular data representations by using range and the measures of central tendency (mean, median, and mode). Continuum of Knowledge In third grade, students applied a procedure to find the range of a data set (3-6.1). In fifth grader, students calculated (5-6.3) and interpreted (5-6.4) the measures of central tendency (mean, median, and mode). In the sixth grade, students analyzed which measure of central tendency was the most appropriate for a given purpose (6-6.3) and predicted the characteristics of one population based on the analysis of sample data (6-6.1). In eighth grade, students interpret graphic and tabular data representations by using range and the measures of central tendency (mean, median, and mode) (8-6.8).


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High school students will apply and interpret data via these concepts in Data Analysis and Probability. Taxonomy Level Cognitive Dimension: Understand Knowledge Dimension: Conceptual Key Concepts

• Range Vocabulary:

• Mean • Mode • Median • Data • Measures of Central Tendency • Outlier


for students to:

• Calculate range • Calculate the measures of central tendency • Understand the meaning of range

• Understanding what each measure of central tendency says about the data • Describe the data using appropriate mathematics language


for students to:

None noted Student Misconceptions/Errors The word average has been misinterpreted by students. The belief is that average refers to only the mean. Students need to realize that it can refer to the median or mode as well. Instructional Resources and Strategies Assessment Guidelines The objective of this indicator is to interpret, which is in the “understand conceptual” knowledge cell of the Revised Taxonomy. To interpret involves


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changing from one form of representation to another (e.g. from graphic or tabular to verbal). The learning progression to interpret requires students to recall and understand the meaning of range, mean, median and mode. Students use their understanding of these statistics to analyze data in graphical and tabular form. They use inductive and deductive reasoning (8-1.5) to reach a conclusion and explain how each statistic impacts the data. Students explain and justify


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