- Triangle inequality
- Relationships of vertical angles, complementary angles, supplementary angles
- Congruence of corresponding and alternate interior angles when parallel lines are cut by a transversal, and that such congruencies imply parallel lines.
- Locate interior/exterior angles of any triangle, and use the property that an exterior angle of a triangle is equal to the sum of the remote (opposite) interior angles
- Know that the sum of the exterior angles of a convex polygon is 360 degrees
- Understand that for polygons, congruence means corresponding sides and angles have equal measure
- Understand the basic rigid motions in the plane (reflections, rotations, translations) relate these to congruence, and apply them to solve problems
- Understand and use simple compositions of basic rigid transformations, e.g. a translations followed by a reflection
- Use paper folding to perform basic geometric constructions of perpendicular lines midpoints of line segments, and angle bisectors, justify informally.
Fractions, Ratios, Decimals, Percents, and Probability
- Add, subtract, multiply and divide positive rational numbers fluently.
- Understand division of fractions as the inverse of multiplication, e.g., if 4/5 ÷ 2/3 =, then 2/3 ∙=4/5, so =4/5 ∙3/2=12/10.
- Given an applied situation involving dividing fractions, write a mathematical statement to represent the situation.
- Solve for the unknown in equations such as:
- ¼ ÷=1, 3/4÷=1/4 and ½=1∙
- Multiply and divide any two fractions fluently.
- Order rational numbers and place them on a number line.
- Represent rational numbers as fractions or terminating decimals when possible, and translate between these representations.
- Understand that a fraction or a negative fraction is a quotient of two integers, e.g., -8/3 is-8 divided by 3.
- Find equivalent rations by scaling up or scaling down.
- Calculate part of a number given the percentage and the number.
- Express probabilities as fractions, decimals, or percentages, between 0 and 1: know that 0 probability means an event will not occur and that probability 1 means an event will occur.
- Compute probabilities of events from simple experiments with equally likely outcomes, e.g. tossing dice, flipping coins, spinning spinners, by listing all possibilities and finding the fractions meets given conditions.
- Solve word problems involving percentages in such contexts as sales taxes and tips, and involving positive rational numbers.
- For applied situations, estimate the answers to calculations involving operations with rational numbers.
- Solve applied problems that use the four operations with appropriate decimal numbers.
- Plot ordered pairs of integers and use ordered pairs of integers to identify points in all four quadrants of the coordinate plane.
- Solve applied problems involving rates including speed, e.g., if a car is going 50 mph, how far will it go in 3 ½ hours?
- Use letters with units to represent quantities in a variety of contexts, e.g., y lbs., k minutes, x cookies.
- Distinguish between an algebraic expression and an equation.
- Use standard conventions for writing algebraic expressions, e.g., 2x + 1 means “two time x, plus 1” and 2(x+1) means “two times the quantity (x+1).”
- Represent information given in words using algebraic expressions and equations.
- Simplify expressions of the first degree by combing like terms, and evaluate using specific values.
- Understand that relationships between quantities can be suggested by graphs and tables.
- Graph and write equations for linear functions of the form y=mx, and solve related problems, e.g., given n chairs, the ‘leg function’ is f(n)=4n; if you have 5 chairs, how many legs? If you have 12 legs how many chairs?
- Represent simple relationships between quantities using verbal descriptions, formulas or equations, tables, and graphs, e.g., perimeter-side relationship for a square, distance/time graphs, and conversions such as feet to inches.
- Relate simple linear equations with integer coefficients to particular contexts, and solve, e.g., 3x=8 or x+5=10.
- Understand that adding or subtracting the same number to both sides of an equation creates a new equation that has the same solution.
- Understand that multiplying or dividing both sides of an equation by the same non-zero number creates a new equation that has the same solutions.
- Solve equations of the form ax+b=c, e.g., 3x+8=15 by hand for positive integer coefficients less than 20, using calculators otherwise, and interpret the results.
- Understand integer subtraction as the inverse of integer addition; add and subtract integers using intergers from 10 to -10
- Add, subtract, multiply, and divide integers between -10 and 10; use number line and strip models for addition and subtraction.
- Locate negative rational numbers (including integers) on the number line; know that numbers and their negatives add to 0 and are on opposites sides and at equal distance from 0 on a number line.
- Understand that rational numbers are quotients of integers (non-zero denominators), e.g., a rational number is either a fraction or a negative fraction.
- Understand that 0 is an integer that is neither negative nor positive.
- Know that the absolute value of a number is the value of the number, ignoring the sign, or is the distance of the number from 0.