Math 6 Appeals Study Guide
Students should know and be able to solve problems related to these benchmarks
at mastery level.
6.1.1.1 and 6.1.1.2: Locate and Compare Numbers
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Use a number line to locate and compare rational numbers, including fractions, decimals, and percents;
Convert between fractions, decimals, and percents;
Represent the relationship between two positive rational numbers using <, =, or >;
Identify the origin, horizontal (x), and vertical (y) axes on a coordinate grid;
Plot ordered pairs of rational numbers on a coordinate grid.
6.1.1.3, 6.1.1.4 and 6.1.1.7: Equivalence and Representation
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Demonstrate knowledge of percent as representing part per 100 or ratios to 100;
Understand that 100% is equivalent to 1;
Determine equivalencies between fractions and decimals, decimals and percents, and fractions and percents;
Recognize common benchmarks; e.g., 1/4 = 0.25 = 25%
Understand that percents must be converted to fractions or decimals in order to multiply or divide;
Choose among representations of rational numbers (fractions, decimals, and percents) to solve problems.
6.1.1.5 and 6.1.1.6: Factors and Primes/ GCF and LCM
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Use divisibility tests to factor whole numbers;
Express whole numbers as the product of prime factors, including the use of exponential notation;
Understand that each whole number has a unique prime factorization;
Understand relationships among factors, multiples, divisors, products and quotients;
Use a variety of strategies to determine factors, multiples, least common multiples, and greatest common
factors, including prime factorization;
Use common factors to simplify fractions;
Understand that dividing the numerator and denominator of a fraction by their greatest common factor results
in a fraction in simplest form. Example: the greatest common factor of 12 and 16 is 4.
12dividedby 4 3
12
= , which is
in simplest form;
16
16dividedby 4 4
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Use common multiples as common denominators and find equivalent fractions to perform calculations;
Understand that the lowest common denominators used to add or subtract fractions are their least common
multiples. Example: the least common multiple of 12 and 16 is 48. The lowest common denominator for adding
fractions such as
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5
and 5
is 48;
16
12
Solve a variety of problems involving factors, greatest common factors, multiples, and least common multiples.
6.1.2.1 thru 6.1.2.4: Ratios and Rates
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Understand that ratios can express part-to-part, part-to-whole, or whole-to-part relationships;
Identify ratios in various contexts and represent them in multiple ways;
Use reasoning about multiplication and division to determine equivalent ratios;
Determine unit rates;
Use the relationships between ratios, equivalent fractions, and percents to solve a variety of problems,
including those involving mixtures and concentrations.
6.1.3.1 and 6.1.3.2: Multiplication and Division
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Model multiplication with fractions and connect models of multiplication with fractions to procedures for
multiplying fractions;
Model division with fractions and connect models of division of fractions to procedures for dividing fractions;
Use fractions, mixed numbers, and decimals to represent quotients in division of whole numbers;
Recognize and use the place-value patterns in multiplying and dividing finite decimals by powers of 10;
Use place value and their understanding of multiplication of fractions to justify procedures for multiplying
finite decimals;
Use place value and their understanding of representing quotients as fractions to justify procedures for
dividing decimals;
Recognize fractions, decimals, and percents as ways of representing rational numbers;
Convert among fractions, decimals, and percents;
Develop efficient, accurate, and generalizable methods for multiplying and dividing fractions and decimals;
Estimate product and quotients of problems involving decimals, fractions, mixed numbers, and improper
fractions;
Solve problems requiring arithmetic with decimals, fractions, and mixed numbers.
6.1.3.1: Percents
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Understand fractions, percents, and decimals as ways of representing rational numbers;
Convert among fractions, percents, and decimals and decide which form to use in a given situation;
Know fraction and decimal equivalents for common percents; e.g.,1%, 10%, 25%;
Use fractions and decimals to calculate the percent of a number; e.g., 20% of 50 can be found with 0.2 x 50 or
1
x 50;
5
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Use the concept of percent as parts out of 100 and find equivalencies to determine what percent one number
is of another; e.g., What percent 12 is of 15 can be found with
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12
= 0.8 = 80%;
15
Solve problems in various contexts that require determining the percent of a number;
Solve problems in various contexts that require determining what percent one number is of another;
Estimate answers to percent problems using common fraction and decimal benchmarks and use the results to
assess reasonableness.
6.1.3.4 and 6.1.3.5: Problem Solving & Estimation
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Students will be able to choose the correct operation to solve real-world problems involving arithmetic of
decimals, fractions, and mixed numbers;
Students will be able to solve real-world and mathematical problems involving computation of positive rational
numbers;
Students will be able to convert among fractions, decimals, and percents, and decide which form to use in a
given situation;
Students will be able to decide how to interpret the answer based on the context. For example, should the
answer be written as a decimal, a fraction, or a whole number with a remainder? Is the remainder the answer
to the problem? Does the remainder affect the answer to the question? Should the remainder be disregarded
when answering the question?
Students will be able to determine reasonable answers to real-world and mathematical problems.
6.2.1.1 and 6.2.1.2: Variables and Representations of Relationships
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Use variables to represent quantities that can change;
Recognize and use variables to represent additive (y = x + 2) and multiplicative (y = 2x) relationships between
two varying quantities;
Distinguish between independent and dependent variables in relationships;
Represent relationships between two changing quantities with tables, graphs, or function rules;
Translate between tables, graphs, and rules;
Use tables, graphs, and function rules to solve real-world and mathematical problems.
6.2.2.1: Properties and Equivalent Expressions
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Understand that algebraic expressions behave in the same way as numerical expressions;
Apply the order of operations to generate equivalent numeric expressions involving rational numbers;
Examples: 12.6 − (5.3 + 4.2) = 12.6 − 9.3 = 3.3 ;
Apply commutative, associative, and distributive properties to generate equivalent expressions;
Examples: 9 52=9 (50+2)=(9 50)+(9 2)=450+18=468 ;
12x+2x=2x+12x =14x ;
5x 3=3(5x)=15x ;
(x+2) 5=5(x+2)=5x+10 ;
Identify commutative, associative, and distributive properties used to generate equivalent numeric and
algebraic expressions;
Examples: 3 (x+5)=(x+5) 3 ; Commutative Property of Multiplication;
Evaluate algebraic expressions when given positive rational numbers as values for variables.
6.2.3.1 and 6.2.3.2: Represent and Solve Equations
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Understand that a variable represents any value that makes a statement true, and may represent one value,
many values, or no value;
Generalize additive and multiplicative numerical relationships and express them using algebraic
representations (e.g., x + 2, 2x);
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Use the idea of maintaining equality on both sides to solve equations and inequalities that involve positive
rational numbers;
Understand and apply inverse operations (addition-subtraction and multiplication-division) when solving
equations and inequalities;
Represent and solve real-world and mathematical situations using equations and inequalities;
Interpret solutions to real-world problems represented by equations and inequalities in its original context.
6.3.1.1. 6.3.1.2 and 6.3.1.3: Measurement of Polygons and Prisms
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Find areas of squares, rectangles, rhombuses, parallelograms, trapezoids, and kites, including the use of
formulas;
Justify why formulas for areas of squares, rectangles, parallelograms, triangles, and trapezoids are valid;
Estimate perimeter and area of irregular figures on a grid and use appropriate units;
Distinguish between the concepts of surface area and volume;
Calculate surface area and volume of prisms using appropriate units;
Use two-dimensional representations of three-dimensional figures to visualize and solve problems involving
surface area and volume;
Use nets to analyze figures, then calculate surface area and volume;
Develop an understanding of formulas used to determine surface areas and volumes of three-dimensional
figures;
Use formulas to determine surface area and volume;
Justify formulas for surface area and volume using nets, decomposition, or other models;
Find lengths from areas or volumes;
Solve simple equations involving formulas by using number sense, properties of operations, and the idea of
maintaining equality on both sides of an equation.
6.3.2.1: Angles and Intersecting Lines
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Identify supplementary angles and know that the sum of the measures of these angles is 180o;
Recognize straight angles and know that these angles measure 180o;
Identify complementary angles and know that the sum of the measures of these angles is 90o;
Know that right angles measure 90º;
Recognize vertical (opposite) angles in a pair of intersecting lines and know that these angles are congruent;
Determine the measure of all angles in a pair of intersecting lines when given the measure of one angle;
Use the relationships formed by angles of intersecting lines to solve problems.
6.3.2.2 and 6.3.2.3: Angles in Geometric Figures
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Know that the sum of the interior angles of any triangle is 180o;
Find missing angle measures in triangles using the fact that the sum of the interior angles of any triangle is
180o;
Decompose polygons into triangles;
Find sums of interior angles of polygons by decomposing them into triangles and using the fact the sum of the
interior angles of any triangle is 180o;
Develop and use a formula to find sums of interior angles of polygons.
6.3.3.1 and 6.3.3.2: Converting and Estimating Measurement
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Know what is meant by length, capacity and weight;
Select appropriate size and type of unit for a given measurement situation;
Understand the size of each benchmark unit and the relationship among benchmark units within a
measurement system;
Identify equivalent ratios within a measurement system;
Construct and analyze tables to show quantities that are in equivalent ratios;
Use ratios and reasoning about multiplication and division to solve problems that require conversion of
measurement units within a measurement system;
Use common benchmarks to make reasonable estimates of measurements;
Recognize new equivalences, such as 1 square yard equals 9 square feet and 1 cubic meter equals 1,000,000
cubic centimeters;
Know useful approximate equivalents between the customary and metric systems, such as a quart is a little less
than a liter, and a yard is a little less than a meter.
6.4.1.1: Sample Space
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Determine the set of all possible outcomes (sample space) for an experiment;
Represent the set of possible outcomes using a variety of strategies, such as tree diagrams, organized lists,
tables, and pictures;
Use tree diagrams, tables, and pictures to determine the size of the sample space (number of possible
outcomes);
Use representations of a sample space to determine which members are related to certain events.
6.4.1.2, 6.4.1.3 and 6.4.1.4: Probability and Experiments
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Perform experiments and collect data;
Determine probabilities;
Express probabilities as fractions, decimals, and percents;
Use probabilities to make predictions;
Use words such as impossible, unlikely, equally likely, likely, or certain to describe the likelihood of an event
occurring;
Compare relative frequencies (experimental results) to known probabilities and know that there may be
differences;
Understand that relative frequencies (experimental probabilities) don't always fit with probabilities based on
mathematical models;
Understand that relative frequencies (experimental probabilities) come closer to probabilities based on
mathematical models as the number of trials increase.