# Common Core Unit Summary Grades 6 to 8

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1 Common Core Unit Summary Grades 6 to 8 Grade 8: Unit 1: Congruence and Similarity- 8G1-8G5 rotations reflections and translations,( RRT=congruence) understand congruence of 2 d figures after RRT Dilations rotations translations and reflections on coordinate plane (dilations=similarity) Understand similarity of 2 d figures after RRT and dilations Angle sums and external angles of triangles, parallel lines cut by transversal, angle criterion for similar triangles Unit 2: Real Numbers -NS1 NS2 Know the difference between rational and irrational numbers and decimal expansions of numbers, convert a decimal expansion that repeats into a rational number Rational approximations of irrational numbers, where do they go on a number line with rational numbers Unit 3: Exponential Expressions -EE1 EE4 Know and apply properties of integer exponents to generate equivalent expressions Use square and cube root to represent solutions to equations of the form x²= p and x³= p evaluate perfect square and small cube roots Use numbers expressed in the form of a single digit times a whole number power of ten to estimate very large or very small quantities (how many times bigger or smaller) Scientific notation Unit 4: Proportional Relationships and Linear Relationships -EE5 EE8 Graph proportional relations and interpret unit rate as the slope of the graph. Compare two different proportional relationships in two different ways (algebraically, graphically, table, verbally) Use similar triangles to demonstrate why slope is the same between any two distinct points on a non-vertical line on a coordinate plane. Derive an equation for y=mx for a line through the origin and y=mx+b as a line intercepting the vertical axis at b Solve linear equations in one variable with one solution, no solution and many solutions. Solve linear equations with rational number coefficients including distributive property and combining like terms Solve systems of linear equations, solution lies in the point of intersection, solve real world math problems leading to two linear equations in two variables

2 Unit 5: Pythagorean Theorem- EE2 G6-G8 Use square and cube root to represent solutions to equations of the form x²= p and x³= p evaluate perfect square and small cube roots Explain a proof of the Pythagorean theorem and its converse Apply the Pythagorean theorem to determine missing side lengths of a right triangle in real world applications in two and three dimensions Apply the Pythagorean theorem to find the distance between two points in the coordinate system Unit6: Functions- F1-F5 G9 Is it a function Compare properties of functions are presented in a variety of ways verbal, table, equations, graph Interpret y=mx+b as a linear function whose graph is a straight line Construct a function to model a linear relationship between two quantities. Determine rate of change and initial value of the function by looking at coordinate pairs Describe qualitatively the functional relationship between two quantities by analyzing the graph (increasing or decreasing linear or nonlinear) sketch a graph that exhibits the qualitative features of a function expressed verbally. Unit 7: Patterns and Data- SP1 SP4 Construct and interpret scatter plots for bivariate data measurement to investigate patterns of association between two quantities. Describe patterns clustering outliers positive negative linear or nonlinear association Straight lines are widely used to model relationships between two quantitative variables for scatterplots that suggest a linear association, assess the model fit by how close the data points are to the line, line of best fit Use the equation of a linear model to solve problems in bivariate measurement data. Determine slope and intercept Patterns of association can be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two way table. Construct and interpret a two way table summarizing data. Grade 7: Unit 1: Operating with Rational Numbers (add/subtract) NS 1 and NS3 Apply and extend previous understanding of addition subtraction to include rational numbers. Represent operations in a horizontal or vertical number line diagram. Opposites combine to equal 0. Absolute value. Interpret sums of rational numbers by describing real world contexts.

3 Subtraction is adding the additive inverse. Apply properties of operations as strategies to add and subtract rational numbers Solve real world problems involving the four operations with rational numbers. Computations with rational numbers extend the rules for manipulating fractions to complex fractions Unit 2: Operating with Rational Numbers (multiply/divide) NS2 and NS3; EE2 and EE3 Apply and extend previous understanding of multiplication and division and fractions to multiply and divide rational numbers. Rules for multiplying and dividing integers. Integers can be divided as long as the divisor is not 0. Every quotient of integers is a rational number. Apply properties of operations as strategies to multiply and divide rational numbers. Convert a rational number to a decimal using long division. The decimal form of a rational number ends in 0 or eventually repeats Solve real world problems involving the four operations with rational numbers. Computations with rational numbers extend the rules for manipulating fractions to complex fractions Understand that rewriting an expressions in different forms in a problem context can shed light on the problem and how the quantities in it are related (a+0.05a = 1.05a means that increase by 5% is the same as multiply by 1.05 ) Solve multistep real life math problems posed with positive and negative rational numbers in any form using tools strategically. Apply properties of operations to calculate with numbers in any form, convert between forms as appropriate. Assess the reasonableness of answers using mental computation and estimation strategies. Unit 3: Algebraic Reasoning EE1, EE2 and EE4 Apply properties of operations as strategies to add subtract factor and expand linear expressions with rational coefficients Understand that rewriting an expressions in different forms in a problem context can shed light on the problem and how the quantities in it are related (a+0.05a = 1.05a means that increase by 5% is the same as multiply by 1.05 ) Use variables to represent quantities in a real world or mathematical problem and construct simple equations and inequalities to solve problems by reasoning about the quantities Unit 4: Proportional Relationships RP1 RP3, G1 Compute unit rates associated with ratios and fractions including ratios of lengths areas and other quantities measured in like or different units Recognize and represent proportional relationships between quantities. Decide when two quantities are equal through a table or coordinate plane noticing the line is through the origin. Determine unit rate or constant or proportionality in table s graphs equations diagrams and verbal description. Represent proportional relationships by equations. Explain what a point on the graph of a proportional relationship means in terms of the situation

4 Use proportional relationships to solve multistep ratio and percent problems. Simple interest, tax markups, markdowns commissions fees percent increase and decrease percent error Solve problems involving scale drawings of geometric figures including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale Unit 5: Two and Three Dimensional Geometry G2-G6 Draw geometric shapes (by hand, ruler, protractor, technology geometers sketchpad) with given conditions. Focus on constructing triangles from three measures of angles or sides noticing when the conditions determine a unique triangle or no triangle Describe the two dimensional figures that result from slicing three dimensional figures, as in plane sections of right triangular prisms and right triangular pyramids Know that formulas for the area and circumference of a circle and use them to solve problems, give an informal derivation of the relationship between the circumference and area of a circle Use facts about supplementary, complimentary, vertical and adjacent angles in a multistep problem to write and solve simple equations for an unknown angle in a figure. Solve real world and math problems involving area, volume and surface area of two and three dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms Unit 6: Probability SP5 SP8 Probability is between 0 and 1 unlikely and likely events Probability of a chance event by collecting data Develop a probability model and use it to find probability of events. Compare probability from a model to observed frequencies Find probabilities of compound events using organized lists tables tree diagrams and simulation Unit 7: Inferences about Populations SP1-SP4 Samples and populations, random sampling Use data from a random sample Compare two numerical data distributions with similar variabilities Use measures of center and measures of variability for numerical data from random samples Grade 6: Unit 1: Operating with Positive Rational Numbers NS1-NS4 Divide fractions using multiplicative inverse Fluently divide multi-digit numbers using the standard algorithm Fluently add subtract multiply and divide multi-digit decimals using the standard algorithm for each operation Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two numbers less than or equal to 12. Use distributive property to express

5 a sum of two whole numbers with a common factors a multiple of a sum of two whole numbers with no common factor (*express 36+8 as 4(9+2) Unit 2: Ratios and Rates RP1 RP3 Understand the concept of a ratio and use ratio language to describe a ratio relationship between 2 quantities Understand the concept of a unit rate a/b associated with a ratio a:b with b 0 Use rates and ratios to solve real world problems by reasoning about tables of equivalent ratios, tape diagrams, double line diagrams or equations Unit 3: Using Expressions EE1-EE4 Write and evaluate numerical expressions involving whole number exponents Write read and evaluate expressions in which letters stand for numbers (order of operations) Apply the properties of operations to generate equivalent expressions (distributive property) Identify when two expressions are equivalent regardless what value is substituted in for the variable (3y = y+y+y) Unit 4: Understanding Positive and Negative Numbers NS5 to NS8 Understand that positive and negative numbers are used together to describe quantities having opposite directions or values. Use positive and negative quantities to represent quantities in real world contexts, explaining the meaning of zero in each situation Understand a rational number as a point on the number line. Extend number line and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates Understand ordering and absolute value of rational numbers 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 same second coordinate Unit 5: Using Equations EE5 to EE9 Understand solving an equation or inequality as a process of answering a question: which values if any, from a specified set make the equation or inequality true. Use substitution to determine whether a given number in a specified set makes an equation or inequality true. Use variables to represent numbers and write expressions when solving a real world or mathematical problem; understand that a variable can represent and unknown number, or depending on the purpose at hand any number in a specified set. Solve real world and mathematical problems by writing and solving equations in the form of x + p = q and px =q for cases in which p, q, and x are all nonnegative numbers

6 Write an inequality of the form x < c or x < c to represent a constraint or condition in a real world or mathematical problem. Realize they have infinitely many solutions and represent such quantities on number line diagrams 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 the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variable using graphs and tables and relate these to the equation. (motion at a constant speed, distance and time) Unit 6: Applications of Geometry G1 to G4 Find the area of right triangles, other triangles and special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques to solve real world math problems Find the volume of a right rectangular prism with fractional edge lengths by packing it with 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. Apply v=lwh and v=bh to find volumes of right rectangular prisms in real world problems. Draw polygons in the coordinate plane given coordinates of the vertices, use coordinates to find the length of the sides joining points with the same first coordinate or the same second coordinate. Represent three dimensional figures using nets made up of rectangles and triangles and use the nets to find surface area of these figures Unit 7: Statistics and Distributions SP1 to SP5 Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answer, example: how old am I?, Is not a statistical question but how old are the students in my school?, is a statistical question because it anticipates variability in students age 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 Recognize that a measure of center for a numerical data set summarizes all of its values with a single number and a measure of variation describes how its values vary with a single number Display numerical data in plots on a number line including dot plots, histograms and box plots Summarize numerical data sets in relation to their context such as by reporting number of observations, describing the nature of the attribute under investigation, including how it was measured and its units of measurement, (quantitive,measures of center mean median interquartile range, absolute deviation, patterns) relating the choice of measures of center and variability to the shape of the data distribution

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