Summary of Skills

Age 10-12 - Concept 5: Math

Unit 1: Operations

Math

Apply properties of operations as strategies to multiply rational numbers
Compute fluently with multi-digit numbers
Compute fluently with multi-digit numbers and find common factors and multiples
Find the greatest common factor of two whole numbers
Find the least common multiple of two whole numbers
Fluently add and subtract multi-digit decimals using the standard algorithm for each operation
Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation
Fluently divide multi-digit decimals using the standard algorithm
Fluently divide multi-digit numbers using the standard algorithm
Fluently multiply and divide multi-digit decimals using the standard algorithm
Fluently multiply multi-digit decimals using the standard algorithm
Identify a natural number as prime or composite
Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations)
Solve real-world and mathematical problems involving the four operations with rational numbers
Use the distributive property to express a sum of two whole numbers 1-100 with a common factor as a multiple of a sum of two whole numbers with no common factor
Write and evaluate numerical expressions involving whole-number exponents

Unit 2: Integers and Rational Numbers

Math

Apply and extend previous understandings of multiplication and division to divide fractions by fractions
Apply and extend previous understandings of operations with fractions
Distinguish comparisons of absolute value from statements about order
Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates
Find and position pairs of integers and other rational numbers on a coordinate plane
Fluently add, subtract, multiply, and divide fractions and mixed numbers
Interpret and compute quotients of fractions
Plot rational numbers on number lines
Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line
Recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes
Show that a number and its opposite have a sum of 0 (are additive inverses)
Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane; include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate
Solve real-world and mathematical problems involving the four operations with rational numbers
Solve word problems involving division of fractions by fractions
Understand a rational number as a point on the number line
Understand and solve problems involving inequalities
Understand and solve problems involving ordering and absolute value of rational numbers
Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane
Understand that positive and negative numbers are used together to describe quantities having opposite directions or values
Use positive and negative numbers and number lines to describe real-world situations
Use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation
Use the coordinate plane to plot coordinates, draw simple figures, and calculate simple distances
Use visual models and equations to represent and solve fraction division word problems
Write, interpret, and explain statements of order for rational numbers in real-world contexts

Unit 3: Ratios and Percentages

Math

Calculate and compare unit prices
Find a part when given the whole amount and the percent of the whole the part represents
Find a percent of a quantity as a rate per 100
Find a whole amount when given a part and the percent of the whole the part represents
Find the percent that a part represents when given the whole amount and a part
Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane; use tables to compare ratios
Manipulate and transform units appropriately when multiplying or dividing quantities
Recognize and represent proportional relationships between quantities
Solve problems involving finding the whole, given a part and the percent
Solve real-world and mathematical problems involving percents
Solve unit rate problems including those involving unit pricing and constant speed
Understand and describe unit rates
Understand the concept of a ratio
Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship
Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations
Use ratio language to describe a ratio relationship between two quantities
Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities

Unit 4: Algebraic Expressions

Math

Apply properties of operations as strategies to add and subtract linear expressions with rational coefficients
Apply properties of operations as strategies to add and subtract rational numbers
Apply properties of operations as strategies to multiply and divide rational numbers
Apply rules for exponents to write and evaluate expressions
Apply the properties of operations to generate equivalent expressions
Evaluate expressions at specific values of their variables
Identify parts of an expression using mathematical terms
Identify when two expressions are equivalent
Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations)
Recognize parts of an expression as a single entity
Recognize that the opposite of the opposite of a number is the number itself, e.g., -(-3) = 3
Represent addition and subtraction on a horizontal or vertical number line diagram
Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts
Understand a rational number as a point on the number line
Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative
Understand subtraction of rational numbers as adding the additive inverse, p - q = p + (-q)
Understand that positive and negative numbers are used together to describe quantities having opposite directions or values
Understand what it means for two expressions to be equivalent
Use positive and negative numbers to represent quantities in real-world contexts
View one or more parts of an expression as a single entity
Write and evaluate numerical expressions involving whole-number exponents
Write expressions that record operations with numbers and with letters standing for numbers
Write, read, and evaluate expressions in which letters (i.e., variables) stand for numbers

Unit 5: Algebraic Equations

Math

Analyze the relationship between the dependent and independent variables using graphs and tables
Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients
Apply the properties of operations to generate equivalent expressions
Construct simple equations and inequalities to solve problems by reasoning about the quantities
List and graph ordered pairs that show the relationship between the independent and dependent variables
Recognize that inequalities of the form x > c or x < c have infinitely many solutions
Represent solutions of inequalities on number line diagrams
Solve multi-step, real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically
Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers
Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers; graph the solution set of the inequality, and interpret it in the context of the problem
Understand how to solve an inequality by determining which values from a specified set, if any, make the inequality true
Understand solving an equation as a process of answering a question: which values from a specified set, if any, make the equation or inequality true?
Understand that a variable can represent an unknown number
Understand that a variable can represent any number in a specified set
Use substitution to determine whether a given number in a specified set makes an equation or inequality true
Use substitution to determine whether a given number in a specified set makes an equation true
Use substitution to determine whether a given number in a specified set makes an inequality true
Use variables to represent numbers and write expressions when solving a real-world or mathematical problem
Use variables to represent quantities in a real-world or mathematical problem
Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems
Use variables to represent two quantities in a real-world problem that change in relationship to one another
Write an equation to express one quantity, thought of as the dependent variable, in terms of another quantity, thought of as the independent variable
Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem
Write expressions that record operations with numbers and with letters standing for numbers

Unit 6: 2D Geometry

Math

Apply area formulas in the context of solving real-world and mathematical problems
Apply area formulas in the context of solving real-world and mathematical problems.
Compute actual lengths and areas from a scale drawing
Construct triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle
Draw geometric shapes with given conditions
Draw geometric shapes with given conditions using a ruler and protractor
Draw geometric shapes with given conditions using a ruler and protractor.
Draw polygons in the coordinate plane
Draw polygons in the coordinate plane and use this skill to solve real-world and mathematical problems
Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes
Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes.
Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle
Identify geometric elements in two-dimensional figures
Know the formulas for the area and circumference of a circle and use them to solve problems
Know the formulas for the area and circumference of a circle, and use them to solve problems
Reproduce a scale drawing at a different scale
Reproduce a scale drawing at a different scale.
Solve problems involving scale drawings of geometric figures
Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers
Understand how to classify, measure, and draw angles
Understand how to classify, measure, and draw angles.
Understand the relationship between the circumference and area of a circle
Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure
Use facts about supplementary, complementary, vertical, and adjacent angles to solve for an unknown angle
Use facts about supplementary, complementary, vertical, and adjacent angles to solve for an unknown angle.
Use facts about supplementary, complementary, vertical, and adjacent angles to write and solve simple equations for an unknown angle in a figure

Unit 7: 3D Geometry

Math

Apply the formulas V = l × w × h and V = b × h to find volumes of right rectangular prisms with fractional edge lengths
Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids
Find the area of right triangles, other triangles, special quadrilaterals, and polygons
Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism
Represent three-dimensional figures using nets made up of rectangles and triangles
Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms
Use nets to find the surface area of three-dimensional figures

Unit 8: Statistics

Math

Compare populations by measuring the difference between the centers and expressing it as a multiple of a measure of variability
Describe the nature of the attribute under investigation, including how it was measured and its units of measurement
Display numerical data in plots on a number line, including dot plots, histograms, and box plots
Find quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation) and describe any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered
Informally assess the degree of visual overlap of two numerical data distributions
Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers
Recognize that a measure of center for a numerical data set summarizes all of its values with a single number
Recognize that a measure of variation describes how a numerical data set's values vary with a single number
Relate the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered
Summarize numerical data sets by reporting the number of observations and describing the nature of the attribute under investigation, including how it was measured and its units of measurement
Summarize numerical data sets in relation to their context by giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered
Summarize numerical data sets in relation to their context, such as by reporting the number of observations
Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape
Understand that generalizations about a population from a sample are valid only if the sample is representative of that population
Understand that random sampling tends to produce representative samples and support valid inferences
Understand that statistics can be used to gain information about a population by examining a representative sample of the population
Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions
Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations

Unit 9: Skills Review

Math

Apply and extend previous understandings of multiplication and division to divide fractions by fractions
Apply area formulas in the context of solving real-world and mathematical problems
Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients
Apply the properties of operations to generate equivalent expressions
Evaluate expressions at specific values of their variables
Find a percent of a quantity as a rate per 100
Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes
Find the greatest common factor of two whole numbers
Find the least common multiple of two whole numbers
Find the volume of a right rectangular prism with fractional edge lengths
Fluently add, subtract, and multiply fractions with unlike denominators (including mixed numbers)
Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation
Fluently divide multi-digit numbers using the standard algorithm
Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity
Know the formulas for the area and circumference of a circle and use them to solve problems
Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams
Solve problems involving finding the whole, given a part and the percent
Solve problems involving scale drawings of geometric figures
Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane
Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms
Solve real-world and mathematical problems involving percents
Solve word problems involving division of fractions by fractions
Understand ordering and absolute value of rational numbers
Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true?
Understand the concept of a ratio
Use facts about supplementary, complementary, vertical, and adjacent angles to solve for an unknown angle
Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations
Use ratio language to describe a ratio relationship between two quantities
Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities
Use the coordinate plane to plot coordinates, draw simple figures, and calculate simple distances
Use variables to represent numbers and write expressions when solving a real-world or mathematical problem
Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem
Write expressions that record operations with numbers and with letters standing for numbers, and construct simple equations and inequalities to solve problems by reasoning about the quantities

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