# 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