6th Grade

1. Number Sense
2. Expressions/Equations
3. Rates/Proportions
4. Geometry
5. Statistics/Data

7th Grade

1. Number Sense
2. Expressions/Equations
3. Rates/Proportions
4. Geometry
5. Statistics/Data

8th Grade

1. Number Sense
2. Expressions/Equations
3. Functions
4. Geometry
5. Statistics/Data

1a. DIVIDE FRACTIONS
1b. Solve/Create word problems with Fractions

Understand that
(a/b)÷(c/d) = (ad/bc)

2. LONG DIVISION
Understand what division and its applications are, not just the algorithm

NO REMAINDERS
Convert remainders to decimal or fractional answer

3. DECIMAL OPERATIONS (+/-/x/÷)

4. NUMBER THEORY
4a. Find GCF of 2 or more numbers ≤ 100
4b. Find LCM of 2 or more numbers ≤ 12
4c. Use DISTRIBUTIVE PROPERTY to express a Product as the Sum of 2 numbers

         ex. 4*12 = 4(10 + 2) = (4*10 + 4*2) = (40 + 8)

5. POSITIVE/NEGATIVE INTEGERS
5a. Positive and Negative values are used to describe quantities with opposite directions. OPPOSITE = NEGATIVE

5b. Use POSITIVE/NEGATIVE numbers to represent quantities in the Real World, explaining the meaning of 0 in each context

6. Extend Number Lines and Coordinate Axes to include RATIONAL/NEGATIVE numbers on the Coordinate Plane:

6a. Recognize OPPOSITE SIGNS of numbers as indicating opposite sides of 0; and the opposite of the opposite of a number is the number itself
          ex. -(-8) = 8

6b. Associate signs in ordered pairs with quadrants

6c. FIND and GRAPH positive/negative rational numbers on the Coordinate Plane

1a. ADD/SUBTRACT FRACTIONS. Represent Fraction Operations on Vertical and Horizontal Number Lines.

1b. Understand (p + q) as a number that is q units away from p, and p units away from q, and that the sum of any number and its opposite, such as p + (-p), is 0.

     ex. The quantity (3 + 5) is located 5 units away from 3, and 3 units away from 5. And 3 + -3 = 0. 5 + -5 = 0.

1c. Understand what is meant by the Opposite of a number, and that subtracting a number is the same as adding its opposite.
ex. p – q = p + (-q)

2a. MULTIPLY/DIVIDE FRACTIONS in situations with DISTRIBUTIVE PROPERTY, NEGATIVES, and when the PRODUCT = 1.

2b.

Understand that
-(p/q) = (-p)/q = p/(-q)

and that

(-p)/(-q) = p/q

Number System

1a. DIVIDE FRACTIONS
1b. Solve/Create Word Problems with Fractions

Understand that
(a/b) ÷ (c/d) = (ad/bc)

1. IRRATIONAL numbers. Understand that every number has a decimal expansion; show that the decimal expansion of rational numbers repeat eventually.

CONVERT FRACTIONS TO REPEATING DECIMALS


CONVERT REPEATING DECIMALS INTO FRACTIONS/RATIONAL NUMBERS

2. LONG DIVISION

NO REMAINDERS

Convert remainders to decimals or fraction answers

2. MULTIPLY/DIVIDE FRACTIONS
2a. Multiply fractions; Use distributive property with rational numbers, with negatives, and when the product = 1.
2b. Interpret quotients; Understand Integers can be divided if divisor ≠ 0, and every quotient of integers is a rational number. Understand rules of dividing signed numbers. -(p/q) = (-p)/q = p/(-q)

2. Use rational approximations to compare the size of irrational numbers; locate irrational numbers on a number line; estimate the value of irrational expressions.
COMPARE/APPROXIMATE IRRATIONAL NUMBERS ON A NUMBER LINE

Number System

3. DECIMAL OPERATIONS
(+/-/*/÷)

2d. CONVERT FRACTIONS TO DECIMALS using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats

4a. Find GCF of 2 numbers <100

4b. Find LCM of 2 #s ≤ 12
4c. Use DISTRIBUTIVE PROPERTY to Express a Sum of 2 numbers (ex. 4*12 = 4(10+2) = (4*10 + 4*2)

3. Solve FRACTION WORD PROBLEMS involving all four operations with rational numbers

5a. Positive and negative values are used to describe quantities with opposite directions. OPPOSITE = NEGATIVE

5b. Use POSITIVE/NEGATIVE NUMBERS to represent quantities in Real World, explaining the meaning of 0 in each context

6. Extend Number Lines and Coordinate Axes to include RATIONAL/NEGATIVE #s ON THE COORDINATE PLANE. 

6a. Recognize OPPOSITE SIGNS of #s as indicating OPPOSITE SIDES of 0; and the opposite of the opposite of a # is the # itself
6b. Associate signs in ordered pairs with quadrants
6c. FIND/GRAPH INTEGERS/RATIONAL NUMBERS ON COORD. PLANE

7b. For example, write -3 ^oC > -7 ^oC to express the fact that -3 ^oC is warmer than -7 ^oC.

7c. For example, for an account balance of -30 dollars, write |-30| = 30 to describe the size of the debt in dollars.

7d. For example, recognize that an account balance less than -30 dollars represents a debt greater than 30 dollars.

7. ORDERING and ABSOLUTE VALUE OF RATIONAL NUMBERS
7a. INEQUALITY STATEMENTS indicate relative position of 2 #s on a line diagram
7b. Write/Interpret/Explain statements of order for Integer/Rational #s in contexts
7c. ABSOLUTE VALUE = Distance from 0; Magnitude

7d. Distinguish comparison of Absolute Value from statements about Order

8. GRAPH POINTS and FIND DISTANCE between points that share a coordinate (horizontal and vertical distance)

Expressions and Equations

1. CONSTANTS. Write/evaluate WHOLE-NUMBER EXPONENTS

1. ADD/SUBTRACT/FACTOR and EXPAND linear expressions with rational coefficients

1. INTEGER EXPONENTS. 0, negative exponents. Generate equivalent exponent expressions.

2b. For example, describe the expression 2 (8 + 7) as a product of two factors; view (8 + 7) as both a single entity and a sum of two terms.

2. VARIABLES. Write, read, and evaluate expression in which letters stand for numbers.
2a. Translate VERBAL EXPRESSIONS

2b. Identify parts of expressions using math terms (sum, term, product, factor, quotient, coefficient); view multiple parts of an expression as a single entity.
2c. EVALUATE EXPRESSIONS at specific values of their variables, formulas with exponents, subst.

1a. COMBINE LIKE TERMS with RATIONAL COEFFICIENTS

1b. Use DISTRIBUTIVE PROPERTY and FACTOR multi-term expressions a(x+y)

1c. TRANSLATE AND SIMPLIFY VERBAL EXPRESSIONS using OPERATIONS and DISTRIBUTION

2. Use SQUARE/CUBE ROOTS to represent solutions, solve equations in the form:

x²=p and x³=p

Evaluate square roots of perfect squares and cube roots of perfect cubes. Know √2 is IRRATIONAL.

Ex. 3(2+y)=6+3y; y+y+y=3y
24x+18y=6(4x+3y)

3. Generate EQUIVALENT EXPRESSIONS

2. REWRITE EXPRESSIONS to shed light on how the quantities are related

3. SCIENTIFIC NOTATION EXPRESSIONS for large and small quantities. 

4. IDENTIFY equivalent expressions

3. Solve MULTI-STEP world problems with positive and negative rational numbers in any form (whole numbers, fractions, decimals)

3a. Express how many times as much one quantity expressed in scientific notation is to another. 

5. EQUATIONS/INEQUALITIES; value(s that make them true

4. Solve MULTI-STEP EQUATIONS with RATIONAL COEFFICIENTS in the form
px+q=r and p(x+q)=r

WORD PROBLEMS++

4. SCIENTIFIC NOTATION OPERATIONS (+/-/*/÷). Determine appropriate measurements of very large/small quantities. Translate between SCIENTIFIC <> STANDARD notation

6. USE VARIABLE expressions to represent numbers in world problems

4a. Compare an algebraic solution to an arithmetic solution, identifying the sequence of operations used in each

5. GRAPH PROPORTIONAL RELATIONSHIPS. Interpret UNIT RATE as SLOPE. Compare proportional relationships expressed in different ways.

7. Solve world problems by WRITING/SOLVING 1-STEP EQUATIONS (x+p=q or px=q)

4b. Solve MULTI-STEP INEQUALITIES in the form
px+q>r or px+q<r

6. DERIVE EQUATIONS in the form of y=mx for line through origin, and y=mx+b for a line through the y-axis at b. Use Similar Triangles to explain why the slope is the same between any two points on a line.

8. WRITE INEQUALITIES to represent a constraint or condition; recognize when inequalities have infinitely many solutions, represent solutions on number lines

4bb. GRAPH SOLUTIONS of INEQUALITIES and INTERPRET CONTEXT

7. Solve 1-variable linear equations with rational coefficients, resulting in x=a, a=a, or a=b (i.e. 1 solution, infinite solutions, and no solution)

9. WRITE EQUATIONS that represent a RELATIONSHIP between two variables; d=rt

8. Solve Systems of Equations by Graphing, Substitution, and Elimination. Determine which method to use based on given equations. Determine when a system has “1 solution, no solution, or infinitely many.”

8b. Solve simple Systems by Inspection
8c. Solve Systems in Multi-step/Word problems, ex. given 2 pairs of points, determine intersection/solution

Ratios & Proportions

SET UP RATIOS to Compare numbers/relationships, a:b

1. Compute Unit Rates associated with Ratios of FRACTIONS

F1. Understand Functions as RULES: 1 output per input; graph of a function is the “set of ordered pairs consisting of input/outputs”

2a. Understand UNIT RATES associate with ratio a:b as a/b:1

2a. Recognize proportional relationships by testing ratios in a table, or graphing on a coordinate plane and observing the line

2. Compare properties of 2 functions represented in different ways: algebraically, graphically, in tables, or verbal description

3a. Find Missing Values in Proportion Tables; Use tables to compare ratios

2b. ID Constant of Proportionality (unit rate) in tables, graphs, equations, diagrams, verbal

3. Interpret equations y=mx+b as defining a linear function; give examples of non-linear functions

3b. Calculate UNIT PRICE and Average/Constant Speed (d=rt) 

3c. Find the percent of a quantity; solve problems involving finding the whole, given a part and the percent

2c. Represent proportional relationships by equations y=kx

2d. Explain what point (x, y) on a graph of a proportional relationship means, and what (0, 0) and (1, r)

4. Interpret Rate of Change and Initial Value of a linear function, in terms of the situation is models. Determine slope and intercept from 2 points.

5. Describe a relationship by analyzing graphs; Graph functions described verbally

3d. Convert measurement units; manipulate and transform units appropriately when Multiplying/Dividing

3. Use proportional relationships
to solve multistep ratio and percent problems (Simple interest, tax, markups, sales, commissions, feeds, percent increase/decrease)

Geometry

1. AREAS of TRIANGLES, QUADRILATERALS, and POLYGONS by COMPOSITION and DECOMPOSITION

1. Determine SCALE FACTORS by observing images; Compute Lengths/Areas from SCALE DRAWINGS; REPRODUCE an image at a different scale

1. Properties of ROTATIONS, REFLECTIONS, and TRANSLATIONS (lines, angles)

2. VOLUMES of RECTANGULAR PRISMS with FRACTIONAL EDGES. Apply V=lwh and V=Bh to solve real-world shapes

2. Draw geometric shapes with given conditions. Construct Triangles from given angles/sides.

2. PROVE CONGRUENCE with ISOMETRIES

3. Coordinate Plane. DRAW POLYGONS and find Horizontal/Vertical SIDE LENGTHS/DISTANCES given VERTICES. 

3. Describe 2-D figures resulting from SLICING 3-D figures in plane sections.

3. Describe effects of DILATIONS, TRANSLATIONS, ROTATIONS, and REFLECTIONS using coordinates

4. Use NETS to find the SURFACE AREA of figures; Represent figures using nets

4. AREA/CIRCUMFERENCE of circles; informally derive pi

4. Understand if figures are Similar based on Transformations and Dilations

5. Angle relationship facts: SUPPLEMENTARY, COMPLEMENTARY, VERTICAL, ADJACENT angles, solve for unknowns

5. ANGLE SUM and EXTERIOR ANGLES of Triangles; angles created when Parallel Lines are cut by transversal; angle-angle criterion for similarity of triangles

6. AREA, VOLUME, SURFACE AREA of 2- and 3-D objectives composed of triangles, quadrilaterals, polygons, cubes, and right prisms

Statistics/Probability

1. Recognize STATISTICAL QUESTIONS - ones that anticipate variability. Compose a statistical question

1. Understand that examining samples of a population can provide insight about that population, and that generalizations are only valid if the sample is REPRESENTATIVE of that population. RANDOM SAMPLING = REPRESENTATIVE SAMPLES

2. Describe Data DISTRIBUTIONS by Center, Spread, and Shape

2. Use data from a sample to draw INFERENCES about that population. Ex. Estimate the mean word length in a book by randomly sampling words from the book; or predict the winner of a school election based on randomly samples survey data.

Use random sampling to draw inferences about a population.

3. Recognize and calculate MEASURES OF CENTER and VARIABILITY

3. Draw informal comparative inferences about 2 populations. Assess the degree of visual overlap of two distributions with similar variabilities; measure the difference between the 2 centers, and express that difference as a multiple of a measure of variability (M.A.D. or ST.DEV.)

4. DISPLAY DATA on Number Lines, Dot Plots, Histograms, and Box Plots

5. Summarize data sets in relation to context, such as by
a) reporting # of observations,
b) describing attributes of measurements,
c) giving quantitative measures of center and variability (M.A.D.), describing patterns, deviations, or
d) relating choice of measures of center/variability to the shape of the data