International School of Curacao Curriculum Articulation ...

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International School of Curacao

Curriculum Articulation

August 2017

CURRICULUM AREA:

Mathematics 6 - 12

Standards & Indicators

(adapted from Union School of Haiti, Common Core State Standards, IB MYP and DP curricula)

Mathematics Grade 6 Overview

The Number System

● Apply and extend previous understandings of multiplication and division to divide fractions by

fractions.

● Compute fluently with multi-digit numbers and find common factors and multiples.

● Apply and extend previous understandings of numbers to the system of rational numbers.

Algebra

● Understand ratio concepts and use ratio reasoning to solve problems.

● Apply and extend previous understanding of arithmetic to algebraic expressions

● Reason about and solve one-variable equations and inequalities.

● Represent and analyze quantitative relationships between dependent and independent variables.

Geometry

● Solve real-world and mathematical problems involving area, surface area, and volume.

Statistics and Probability

● Develop understanding of statistical variability.

● Summarize and describe distributions.

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Mathematics Grade 6 Standards

The Number System

1. Apply and extend previous understandings of multiplication and division to divide fractions by

fractions.

1.1. Interpret and compute quotients of fractions, and solve word problems involving division of

fractions by fractions, e.g., by using visual fraction models and equations to represent the problem.

For example, create a story context for

and use a visual fraction model to show the

quotient; use the relationship between multiplication and division to explain that (2/3) ⎟

because

of

is

.

How much chocolate will each

person get if 3 people share

lb of chocolate equally? How many

cup servings are in

of a cup of

yogurt? How wide is a rectangular strip of land with length

mi and area

square mi?

2. Compute fluently with multi-digit numbers and find common factors and multiples.

2.1. Fluently divide multi-digit numbers using the standard algorithm.

2.2. Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for

each operation.

2.3. Find the greatest common factor of two whole numbers less than or equal to 100 and the least

common multiple of two whole numbers less than or equal to 12. 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. For example, express 36 + 8 as 4(9 + 2).

3. Apply and extend previous understandings of numbers to the system of rational numbers.

3.1. Understand that positive and negative numbers are used together to describe quantities having

opposite directions or values (e.g., temperature above/below zero, elevation above/below sea

level, credits/debits, positive/negative electric charge); use positive and negative numbers to

represent quantities in real-world contexts, explaining the meaning of 0 in each situation.

3.2. Understand a rational number as a point on the number line. 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.

3.2.1. Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the

number line; recognize that the opposite of the opposite of a number is the number itself, e.g.,

–(–3) = 3, and that 0 is its own opposite.

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3.2.2. Understand signs of numbers in ordered pairs as indicating locations in quadrants of the

coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of

the points are related by reflections across one or both axes.

3.2.3. Find and position integers and other rational numbers on a horizontal or vertical number line

diagram; find and position pairs of integers and other rational numbers on a coordinate plane.

3.3. Understand ordering and absolute value of rational numbers.

3.3.1. Interpret statements of inequality as statements about the relative positions of two numbers

on a number line diagram. For example, interpret –3 > –7 as a statement that –3 is located to

the right of –7 on a number line oriented from left to right.

3.3.2. Write, interpret, and explain statements of order for rational numbers in real-world contexts.

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

3.3.3. Understand the absolute value of a rational number as its distance from 0 on the number line;

interpret absolute value as magnitude for a positive or negative quantity in a real-world

situation. For example, for an account balance of –30 dollars, write |–30| = 30 to describe the

size of the debt in dollars.

3.3.4. Distinguish comparisons of absolute value from statements about order. For example,

recognize that an account balance less than –30 dollars represents a debt greater than

30 dollars.

3.4. 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.

Algebra

1. Understand ratio concepts and use ratio reasoning to solve problems.

1.1. Understand the concept of a ratio and use ratio language to describe a ratio relationship between

two quantities. For example, “The ratio of wings to beaks in the bird house at the zoo was 2:1,

because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C

received nearly three votes.”

1.2. 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. For example, “This recipe has a ratio of 3 cups of flour

to 4 cups of sugar, so there is ¾ cup of flour for each cup of sugar.” “We paid $75 for 15 hamburgers,

which is a rate of $5 per hamburger.”

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1.3. 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.

1.3.1. 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.

1.3.2. Solve unit rate problems, including those involving unit pricing and constant speed. For

example, if it took 7 hours to mow 4 lawns, then, at that rate, how many lawns could be mowed

in 35 hours? At what rate were lawns being mowed?

1.3.3. Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times

the quantity); solve problems involving finding the whole, given a part and the percent.

1.3.4. Use ratio reasoning to convert measurement units; manipulate and transform units

appropriately when multiplying or dividing quantities.

1.3.5. Solve problems that relate the mass of an object to its volume. Write and evaluate numerical

expressions involving whole-number exponents.

2. Apply and extend previous understandings of arithmetic to algebraic expressions.

2.1. Write, read, and evaluate expressions in which letters stand for numbers.

2.2. Write expressions that record operations with numbers and with letters standing for numbers. For

example, express the calculation “Subtract y from 5” as 5 – y.

2.3. 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. 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.

2.4. Evaluate expressions at specific values of their variables. Include expressions that arise from

formulas used in real-world problems. 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). For example, use the formulas V = s3 and A = 6s2 to find the

volume and surface area of a cube with sides of length s = ½ .

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2.5. Apply the properties of operations to generate equivalent expressions. For example, apply the

distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the

distributive property to the expression 24x + 18y to produce the equivalent expression 6(4x + 3y);

apply properties of operations to y + y + y to produce the equivalent expression 3y.

2.6. Identify when two expressions are equivalent (i.e., when the two expressions name the same

number regardless of which value is substituted into them). For example, the expressions y + y + y

and 3y are equivalent because they name the same number regardless of which number y stands for.

3. Reason about and solve one-variable equations and inequalities.

3.1. 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? Use substitution to determine

whether a given number in a specified set makes an equation or inequality true.

3.2. Use variables to represent numbers and write expressions when solving a real-world or

mathematical problem; understand that a variable can represent an unknown number, or,

depending on the purpose at hand, any number in a specified set.

3.3. 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.

3.4. Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world

or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many

solutions; represent solutions of such inequalities on number line diagrams.

4. Represent and analyze quantitative relationships between dependent and independent variables.

4.1. 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.

4.2. Analyze the relationship between the dependent and independent variables using graphs and

tables, and relate these to the equation. For example, in a problem involving motion at constant

speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent

the relationship between distance and time.

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Geometry

1. Solve real-world and mathematical problems involving area, surface area, and volume.

1.1. Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing

into rectangles or decomposing into triangles and other shapes; apply these techniques in the

context of solving real-world and mathematical problems.

1.2. Use the relationships among radius, diameter, and center of a circle to find its circumference and

area.

1.3. Solve real-world and mathematical problems involving the measurements of circles.

1.4. 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. Apply the formulas V = lwh and V = bh to

find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-

world and mathematical problems.

1.5. Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find

the length of a side joining points with the same first coordinate or the same second coordinate.

Apply these techniques in the context of solving real-world and mathematical problems.

1.6. Represent three-dimensional figures using nets made up of rectangles and triangles, and use the

nets to find the surface areas of these figures. Apply these techniques in the context of solving real-

world and mathematical problems.

2. Statistics and Probability

2.1. Develop understanding of statistical variability.

2.1.1. Recognize a statistical question as one that anticipates variability in the data related to the

question and accounts for it in the answers. For example, “How old am I?” is not a statistical

question, but “How old are the students in my school?” is a statistical question because one

anticipates variability in students’ ages.

2.1.2. 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.

2.1.3. Recognize that a measure of center for a numerical data set summarizes all of its values with a

single number, while a measure of variation describes how its values vary with a single

number.

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2.2. Summarize and describe distributions.

2.2.1. Display numerical data in plots on a number line, including dot plots, histograms, and box

plots. Read and interpret circle graphs.

2.2.2. Summarize numerical data sets in relation to their context, such as by:

2.2.3. Reporting the number of observations.

2.2.4. Describing the nature of the attribute under investigation, including how it was measured and

its units of measurement.

2.2.5. 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.

2.2.6. Relating the choice of measures of center and variability to the shape of the data distribution

and the context in which the data were gathered.

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The Number System

● Apply and extend previous understandings of operations with fractions to add, subtract, multiply,

and divide rational numbers.

Algebra

● Analyze proportional relationships and use them to solve real-world and mathematical problems.

● Use properties of operations to generate equivalent expressions.

● Solve real-life and mathematical problems using numerical and algebraic

Geometry

● Draw, construct and describe geometrical figures and describe the relationships between them.

● Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.


● Use random sampling to draw inferences about a population.

● Draw informal comparative inferences about two populations.

● Investigate chance processes and develop, use, and evaluate probability models.

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1. The Number System

1.1. Apply and extend previous understandings of operations with fractions to add, subtract, multiply,

and divide rational numbers.

1.2. Apply and extend previous understandings of addition and subtraction to add and subtract

rational numbers; represent addition and subtraction on a horizontal or vertical number line

diagram.

1.2.1. Describe situations in which opposite quantities combine to make 0. For example, a hydrogen

atom has 0 charge because its two constituents are oppositely charged.

1.2.2. 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. Show that a number and its opposite

have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-

world contexts.

1.2.3. Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (–q).

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.

1.2.4. Apply properties of operations as strategies to add and subtract rational numbers.

1.3. Apply and extend previous understandings of multiplication and division and of fractions to

multiply and divide rational numbers.

1.3.1. Understand that multiplication is extended from fractions to rational numbers by requiring

that operations continue to satisfy the properties of operations, particularly the distributive

property, leading to products such as (–1)(–1) = 1 and the rules for multiplying signed

numbers. Interpret products of rational numbers by describing real-world contexts.

1.3.2. Understand that integers can be divided, provided that the divisor is not zero, and every

quotient of integers (with non-zero divisor) is a rational number. If p and q are integers, then

–(p/q) = (–p)/q = p/(–q). Interpret quotients of rational numbers by describing real-world

contexts.

1.3.3. Apply properties of operations as strategies to multiply and divide rational numbers.

1.3.4. Convert a rational number to a decimal using long division; know that the decimal form of a

rational number terminates in 0s or eventually repeats.

1.4. Solve real-world and mathematical problems involving the four operations with rational numbers.

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2. Algebra

2.1. Use properties of operations to generate equivalent expressions.

2.2. Analyze proportional relationships and use them to solve real-world and mathematical problems.

2.3. Compute unit rates associated with ratios of fractions, including ratios of lengths, areas, and other

quantities measured in like or different units. For example, if a person walks ½ mile in each ¼ hour,

compute the unit rate as the complex fraction ½/¼ miles per hour, equivalently 2 miles per hour.

2.4. Recognize and represent proportional relationships between quantities.

2.4.1. Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent

ratios in a table, or graphing on a coordinate plane and observing whether the graph is a

straight line through the origin.

2.4.2. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and

verbal descriptions of proportional relationships.

2.4.3. Represent proportional relationships by equations. For example, if total cost t is proportional

to the number n of items purchased at a constant price p, the relationship between the total cost

and the number of items can be expressed as t = pn.

2.4.4. Explain what a point (x, y) on the graph of a proportional relationship means in terms of the

situation, with special attention to the points (0, 0) and (1, r ) where r is the unit rate.

2.5. Use proportional relationships to solve multi-step ratio and percent problems. Examples: simple

interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and

decrease, percent error.

2.6. Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions

with rational coefficients.

2.7. Understand that rewriting an expression in different forms in a problem context can shed light on

the problem and how the quantities in it are related. For example, a + 0.05a = 1.05a means that

“increase by 5%” is the same as “multiply by 1.05.”

2.8. Solve real-life and mathematical problems using numerical and algebraic expressions and

equations.

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2.9. 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. Apply

properties of operations to calculate with numbers in any form; convert between forms as

appropriate; and assess the reasonableness of answers using mental computation and estimation

strategies. For example: If a woman making $25 an hour gets a 10% raise, she will make an

additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a

towel bar 9¾ inches long in the center of a door that is 27½ inches wide, you will need to place the

bar about 9 inches from each edge; this estimate can be used as a check on the exact computation.

2.10. 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.

2.10.1. Solve word problems leading to equations of the form px + q = r and p(x ⎟ q) = r, where p, q,

and r are specific rational numbers. Solve equations of these forms fluently. Compare an

algebraic solution to an arithmetic solution, identifying the sequence of the operations used in

each approach. For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its

width?

2.10.2. 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. For example: As a salesperson, you are paid $50 per week plus $3 per

sale. This week you want your pay to be at least $100. Write an inequality for the number of sales

you need to make, and describe the solutions.

2.10.3. Extend analysis of patterns to include analyzing, extending, and determining an expression

for simple arithmetic and geometric sequences (e.g., compounding, increasing area), using

tables, graphs, words, and expressions.

3. Geometry

3.1. Draw, construct, and describe geometrical figures and describe the relationships between them.

3.2. Solve problems involving scale drawings of geometric figures, such as computing actual lengths

and areas from a scale drawing and reproducing a scale drawing at a different scale.

3.3. Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given

conditions. 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.

3.4. Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane

sections of right rectangular prisms and right rectangular pyramids.

3.5. Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.

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3.6. Know the formulas for the area and circumference of a circle and solve problems; give an informal

derivation of the relationship between the circumference and area of a circle.

3.7. Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step

problem to write and use them to solve simple equations for an unknown angle in a figure.

3.8. 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.


4.1. Use random sampling to draw inferences about a population.

4.2. Understand that statistics can be used to gain information about a population by examining a

sample of the population; 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.

4.3. 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. For example, estimate the mean word length in a

book by randomly sampling words from the book; predict the winner of a school election based on

randomly sampled survey data. Gauge how far off the estimate or prediction might be.

4.4. Draw informal comparative inferences about two populations.

4.5. Informally assess the degree of visual overlap of two numerical data distributions with similar

variabilities, measuring the difference between the centers by expressing it as a multiple of a

measure of variability. For example, the mean height of players on the basketball team is 10 cm

greater than the mean height of players on the soccer team, about twice the variability (mean

absolute deviation) on either team; on a dot plot, the separation between the two distributions of

heights is noticeable.

4.6. Use measures of center and measures of variability for numerical data from random samples to

draw informal comparative inferences about two populations. For example, decide whether the

words in a chapter of a seventh-grade science book are generally longer than the words in a chapter

of a fourth-grade science book.

4.7. Investigate chance processes and develop, use, and evaluate probability models.

4.8. Understand that the probability of a chance event is a number between 0 and 1 that expresses the

likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0

indicates an unlikely event, a probability around ½ indicates an event that is neither unlikely nor

likely, and a probability near 1 indicates a likely event.

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4.9. Approximate the probability of a chance event by collecting data on the chance process that

produces it and observing its long-run relative frequency, and predict the approximate relative

frequency given the probability. For example, when rolling a number cube 600 times, predict that a 3

or 6 would be rolled roughly 200 times, but probably not exactly 200 times.

4.10. Develop a probability model and use it to find probabilities of events. Compare probabilities from

a model to observed frequencies; if the agreement is not good, explain possible sources of the

discrepancy.

4.10.1. Develop a uniform probability model by assigning equal probability to all outcomes, and use

the model to determine probabilities of events. For example, if a student is selected at

random from a class, find the probability that Jane will be selected and the probability that a

girl will be selected.

4.10.2. Develop a probability model (which may not be uniform) by observing frequencies in data

generated from a chance process. For example, find the approximate probability that a

spinning penny will land heads up or that a tossed paper cup will land open-end down. Do the

outcomes for the spinning penny appear to be equally likely based on the observed

frequencies?

4.10.3. Find probabilities of compound events using organized lists, tables, tree diagrams, and

simulation.

4.10.4. Understand that, just as with simple events, the probability of a compound event is the

fraction of outcomes in the sample space for which the compound event occurs.

4.10.5. Represent sample spaces for compound events using methods such as organized lists,

tables, and tree diagrams. For an event described in everyday language (e.g., “rolling double

sixes”), identify the outcomes in the sample space which compose the event.

4.10.6. Design and use a simulation to generate frequencies for compound events. For example, use

random digits as a simulation tool to approximate the answer to the question: If 40% of donors

have type A blood, what is the probability that it will take at least 4 donors to find one with type

A blood?

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The Number System

● Know that there are numbers that are not rational, and approximate them by rational numbers.

Algebra

● Work with radicals and integer exponents.

● Understand the connections between proportional relationships, lines, and linear equations.

● Analyze and solve linear equations and pairs of simultaneous linear equations.

● Define, evaluate, and compare functions.

● Use functions to model relationships between quantities.

Geometry

● Understand congruence and similarity using physical models or geometry software (Geogebra).

● Apply basic characteristics of polygons.

● Apply properties of circle geometry to calculate circumference and area.

● Solve real-world problems involving volume of cylinders, cones and spheres.


● Investigate patterns of association in bivariate data.

● Use random sampling to draw inferences about a population.

● Draw informal comparative inferences about two populations.

● Investigate chance processes and develop, use, and evaluate probability models.

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1. The Number System

1.1. Know that there are numbers that are not rational, and approximate them by rational numbers.

1.2. Know that numbers that are not rational are called irrational. Understand informally that every

number has a decimal expansion; for rational numbers show that the decimal expansion repeats

eventually, and convert a decimal expansion which repeats eventually into a rational number.

1.3. Use rational approximations of irrational numbers to compare the size of irrational numbers,

locate them approximately on a number line diagram, and estimate the value of expressions (e.g.,

€2). For example, by truncating the decimal expansion of show that is between 1 and 2, then

between 1.4 and 1.5, and explain how to continue on to get better approximations.

2. Algebra

2.1. Work with radicals and integer exponents.

2.2. Know and apply the properties of integer exponents to generate equivalent numerical expressions.

For example, 32 ⋅ 3–5 = 3–3 = 1/33 = 1/27.

2.3. Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3

= p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube

roots of small perfect cubes. Know that is irrational.

2.4. Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very

large or very small quantities, and to express how many times as much one is than the other. For

example, estimate the population of the United States as 3 ⋅ 108 and the population of the world as

7 ⋅ 109, and determine that the world population is more than 20 times larger.

2.5. Perform operations with numbers expressed in scientific notation, including problems where both

decimal and scientific notation are used. Use scientific notation and choose units of appropriate

size for measurements of very large or very small quantities (e.g., use millimeters per year for

seafloor spreading). Interpret scientific notation that has been generated by technology.

2.6. Understand the connections between proportional relationships, lines, and linear equations.

2.7. Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two

different proportional relationships represented in different ways. For example, compare a

distance-time graph to a distance-time equation to determine which of two moving objects has

greater speed.

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2.8. Use similar triangles to explain why the slope m is the same between any two distinct points on a

non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin

and the equation y = mx + b for a line intercepting the vertical axis at b.

2.9. Analyze and solve linear equations and pairs of simultaneous linear equations.

2.10. Solve linear equations in one variable.

2.10.1. Give examples of linear equations in one variable with one solution, infinitely many

solutions, or no solutions. Show which of these possibilities is the case by successively

transforming the given equation into simpler forms, until an equivalent equation of the form x

= a, a = a, or a = b results (where a and b are different numbers).

2.10.2. Solve linear equations with rational number coefficients, including equations whose

solutions require expanding expressions using the distributive property and collecting like

terms.

2.11. Analyze and solve pairs of simultaneous linear equations.

2.11.1. Understand that solutions to a system of two linear equations in two variables correspond

to points of intersection of their graphs, because points of intersection satisfy both equations

simultaneously.

2.11.2. Solve systems of two linear equations in two variables algebraically, and estimate solutions

by graphing the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x +

2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6.

2.11.3. Solve real-world and mathematical problems leading to two linear equations in two

variables. For example, given coordinates for two pairs of points, determine whether the line

through the first pair of points intersects the line through the second pair.

2.12. Functions

2.13. Define, evaluate, and compare functions.

2.14. Understand that a function is a rule that assigns to each input exactly one output. The graph

of a function is the set of ordered pairs consisting of an input and the corresponding output.

2.15. Compare properties of two functions each represented in a different way (algebraically,

graphically, numerically in tables, or by verbal descriptions). For example, given a linear function

represented by a table of values and a linear function represented by an algebraic expression,

determine which function has the greater rate of change.

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2.16. Interpret the equation y = mx + b as defining a linear function whose graph is a straight line;

give examples of functions that are not linear. For example, the function A = s2 giving the area of a

square as a function of its side length is not linear because its graph contains the points (1, 1), (2, 4)

and (3, 9), which are not on a straight line.

2.17. Use functions to model relationships between quantities.

2.18. Construct a function to model a linear relationship between two quantities. Determine the

rate of change and initial value of the function from a description of a relationship or from two (x,

y) values, including reading these from a table or from a graph. Interpret the rate of change and

initial value of a linear function in terms of the situation it models, and in terms of its graph or a

table of values.

2.19. Describe qualitatively the functional relationship between two quantities by analyzing a

graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that

exhibits the qualitative features of a function that has been described verbally.

3. Geometry

3.1. Understand congruence and similarity using physical models, transparencies, or geometry

software.

3.2. Verify experimentally the properties of rotations, reflections, and translations:

3.2.1. Lines are taken to lines, and line segments to line segments of the same length.

3.2.2. Angles are taken to angles of the same measure.

3.2.3. Parallel lines are taken to parallel lines.

3.3. Understand that a two-dimensional figure is congruent to another if the second can be obtained

from the first by a sequence of rotations, reflections, and translations; given two congruent figures,

describe a sequence that exhibits the congruence between them.

3.4. Describe the effects of dilations, translations, rotations, and reflections on two-dimensional figures

using coordinates.

3.5. Understand that a two-dimensional figure is similar to another if the second can be obtained from

the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-

dimensional figures, describe a sequence that exhibits the similarity between them.

3.6. Use informal arguments to establish facts about the angle sum and exterior angle of triangles,

about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion

for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of

the three angles appears to form a line, and give an argument in terms of transversals why this is so.

3.7. Understand and apply the Pythagorean Theorem.

18

3.8. Explain a proof of the Pythagorean Theorem and its converse.

3.9. Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-

world and mathematical problems in two and three dimensions.

3.10. Apply the Pythagorean Theorem to find the distance between two points in a coordinate

system.

3.11. Solve real-world and mathematical problems involving volume of cylinders, cones, and

spheres.

3.12. Know the formulas for the volumes of cones, cylinders, and spheres, and use them to solve

real-world and mathematical problems.


4.1. Investigate patterns of association in bivariate data.

4.2. Construct and interpret scatter plots for bivariate measurement data to investigate patterns of

association between two quantities. Describe patterns such as clustering, outliers, positive or

negative association, linear association, and nonlinear association.

4.3. Know that straight lines are widely used to model relationships between two quantitative

variables. For scatter plots that suggest a linear association, informally fit a straight line, and

informally assess the model fit by judging the closeness of the data points to the line.

4.4. Use the equation of a linear model to solve problems in the context of bivariate measurement data,

interpreting the slope and intercept. For example, in a linear model for a biology experiment,

interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated

with an additional 1.5 cm in mature plant height.

4.5. Understand that patterns of association can also 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 on two categorical variables collected from the same subjects. Use relative

frequencies calculated for rows or columns to describe possible association between the two

variables. For example, collect data from students in your class on whether or not they have a curfew

on school nights and whether or not they have assigned chores at home. Is there evidence that those

who have a curfew also tend to have chores?

19

Grade 8 Algebra Overview

Seeing Structure in Expressions

● Interpret the structure of expressions.

● Write expressions in equivalent forms to solve problems.

Arithmetic with Polynomials and Rational Expressions

● Perform arithmetic operations on polynomials.

● Understand the relationship between zeros and factors of polynomials.

● Use polynomial identities to solve problems.

● Rewrite rational expressions.

Creating Equations

● Create equations that describe numbers or relationships.

Reasoning with Equations and Inequalities

● Understand solving equations as a process of reasoning and explain the reasoning.

● Solve equations and inequalities in one variable.

● Solve systems of equations.

● Represent and solve equations and inequalities graphically.

20

Grade 8 Algebra Standards

1. Seeing Structure in Expressions

1.1. Interpret the structure of expressions.

1.1.1. Interpret expressions that represent a quantity in terms of its context.

1.1.1.1. Interpret parts of an expression, such as terms, factors, and coefficients.

1.1.1.2. Interpret complicated expressions by viewing one or more of their parts as a single

entity. For example, interpret P(1 + r)n as the product of P and a factor not depending on P.

1.1.2. Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as

(x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as

1.1.2.1. (x2 – y2)(x2 + y2).

1.2. Write expressions in equivalent forms to solve problems.

1.2.1. Choose and produce an equivalent form of an expression to reveal and explain properties of

the quantity represented by the expression.

1.2.1.1. Factor a quadratic expression to reveal the zeros of the function it defines.

1.2.1.2. Complete the square in a quadratic expression to reveal the maximum or minimum

value of the function it defines.

1.2.1.3. Use the properties of exponents to transform expressions for exponential functions.

For example, the expression 1.15t can be rewritten as (1.151/12)12t ≈ 1.01212t to reveal the

approximate equivalent monthly interest rate if the annual rate is 15%.

1.2.2. Derive the formula for the sum of a finite geometric series (when the common ratio is not 1),

and use the formula to solve problems. For example, calculate mortgage payments.

2. Arithmetic with Polynomials and Rational Expressions

2.1. Perform arithmetic operations on polynomials.

2.1.1. Understand that polynomials form a system analogous to the integers, namely, they are closed

under the operations of addition, subtraction, and multiplication; add, subtract, and multiply

polynomials.

2.1.2. Divide polynomials.

2.2. Understand the relationship between zeros and factors of polynomials.

2.2.1. Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the

remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).

2.2.2. Identify zeros of polynomials when suitable factorizations are available, and use the zeros to

construct a rough graph of the function defined by the polynomial.

21

2.3. Use polynomial identities to solve problems.

2.3.1. Prove polynomial identities and use them to describe numerical relationships. For example,

the polynomial identity (x2 + y2)2 = (x2 – y2)2 + (2xy)2 can be used to generate Pythagorean triples.

2.3.2. Know and apply the Binomial Theorem for the expansion of (x + y)n in powers of x and y for a

positive integer n, where x and y are any numbers, with coefficients determined for example

by Pascal’s Triangle.

2.4. Rewrite rational expressions.

2.4.1. Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form

q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less

than the degree of b(x), using inspection, long division, or, for the more complicated examples,

a computer algebra system.

2.4.2. Understand that rational expressions form a system analogous to the rational numbers, closed

under addition, subtraction, multiplication, and division by a nonzero rational expression;

add, subtract, multiply, and divide rational expressions.

3. Creating Equations

3.1. Create equations that describe numbers or relationships.

3.1.1. Create equations and inequalities in one variable and use them to solve problems. Include

equations arising from linear and quadratic functions, and simple rational and exponential

functions.

3.1.2. Create equations in two or more variables to represent relationships between quantities;

graph equations on coordinate axes with labels and scales. ★

3.1.3. Represent constraints by equations or inequalities, and by systems of equations and/or

inequalities, and interpret solutions as viable or non-viable options in a modeling context. For

example, represent inequalities describing nutritional and cost constraints on combinations of

different foods. ★

3.1.4. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving

equations. For example, rearrange Ohm’s law V = IR to highlight resistance R. ★

22

4. Reasoning with Equations and Inequalities

4.1. Understand solving equations as a process of reasoning and explain the reasoning.

4.1.1. Explain each step in solving a simple equation as following from the equality of numbers

asserted at the previous step, starting from the assumption that the original equation has a

solution. Construct a viable argument to justify a solution method.

4.1.2. Solve simple rational and radical equations in one variable, and give examples showing how

extraneous solutions may arise.

4.2. Solve equations and inequalities in one variable.

4.2.1. Solve linear equations and inequalities in one variable, including equations with coefficients

represented by letters.

4.2.2. Solve linear equations and inequalities in one variable involving absolute value.

4.2.3. Solve quadratic equations in one variable.

4.2.3.1. Use the method of completing the square to transform any quadratic equation in x

into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic

formula from this form.

4.2.3.2. Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots,

completing the square, the quadratic formula, and factoring, as appropriate to the initial

form of the equation. Recognize when the quadratic formula gives complex solutions and

write them as a ± bi for real numbers a and b.

4.2.4. Demonstrate an understanding of the equivalence of factoring, completing the square, or

using the quadratic formula to solve quadratic equations.

4.3. Solve systems of equations.

4.3.1. Prove that, given a system of two equations in two variables, replacing one equation by the

sum of that equation and a multiple of the other produces a system with the same solutions.

4.3.2. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on

pairs of linear equations in two variables.

4.3.3. Solve a simple system consisting of a linear equation and a quadratic equation in two variables

algebraically and graphically. For example, find the points of intersection between the line y = –

3x and the circle x2 + y2 = 3.

4.3.4. Represent a system of linear equations as a single matrix equation in a vector variable.

4.3.5. Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using

technology for matrices of dimension 3 ⋅ 3 or greater).

23

4.4. Represent and solve equations and inequalities graphically.

4.4.1. Understand that the graph of an equation in two variables is the set of all its solutions plotted

in the coordinate plane, often forming a curve (which could be a line).

4.4.2. Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and

y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately,

e.g., using technology to graph the functions, make tables of values, or find successive

approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute

value, exponential, and logarithmic functions.

4.4.3. Graph the solutions to a linear inequality in two variables as a half-plane (excluding the

boundary in the case of a strict inequality), and graph the solution set to a system of linear

inequalities in two variables as the intersection of the corresponding half-planes.

24

HIGH SCHOOL: GRADE 9 - 12

Algebra I Overview

Seeing Structure in Expressions

● Interpret the structure of expressions.

● Write expressions in equivalent forms to solve problems.

Arithmetic with Polynomials and Rational Expressions

● Perform arithmetic operations on polynomials.

● Understand the relationship between zeros and factors of polynomials.

● Use polynomial identities to solve problems.

● Rewrite rational expressions.

Creating Equations

● Create equations that describe numbers or relationships.

Reasoning with Equations and Inequalities

● Understand solving equations as a process of reasoning and explain the reasoning.

● Solve equations and inequalities in one variable.

● Solve systems of equations.

● Represent and solve equations and inequalities graphically.

25

Algebra I Standards

1. Seeing Structure in Expressions

1.1. Interpret the structure of expressions.

1.1.1. Interpret expressions that represent a quantity in terms of its context.

1.1.1.1. Interpret parts of an expression, such as terms, factors, and coefficients.

1.1.1.2. Interpret complicated expressions by viewing one or more of their parts as a single

entity. For example, interpret P(1 + r)n as the product of P and a factor not depending on P.

1.1.2. Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as

(x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as

1.1.2.1. (x2 – y2)(x2 + y2).

1.2. Write expressions in equivalent forms to solve problems.

1.2.1. Choose and produce an equivalent form of an expression to reveal and explain properties of

the quantity represented by the expression.

1.2.1.1. Factor a quadratic expression to reveal the zeros of the function it defines.

1.2.1.2. Complete the square in a quadratic expression to reveal the maximum or minimum

value of the function it defines.

1.2.1.3. Use the properties of exponents to transform expressions for exponential functions.

For example, the expression 1.15t can be rewritten as (1.151/12)12t ≈ 1.01212t to reveal the

approximate equivalent monthly interest rate if the annual rate is 15%.

1.2.2. Derive the formula for the sum of a finite geometric series (when the common ratio is not 1),

and use the formula to solve problems. For example, calculate mortgage payments.

2. Arithmetic with Polynomials and Rational Expressions

2.1. Perform arithmetic operations on polynomials.

2.1.1. Understand that polynomials form a system analogous to the integers, namely, they are closed

under the operations of addition, subtraction, and multiplication; add, subtract, and multiply

polynomials.

2.1.2. Divide polynomials.

2.2. Understand the relationship between zeros and factors of polynomials.

2.2.1. Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the


2.2.2. Identify zeros of polynomials when suitable factorizations are available, and use the zeros to

construct a rough graph of the function defined by the polynomial.

26

2.3. Use polynomial identities to solve problems.

2.3.1. Prove polynomial identities and use them to describe numerical relationships. For example,

the polynomial identity (x2 + y2)2 = (x2 – y2)2 + (2xy)2 can be used to generate Pythagorean triples.

2.3.2. Know and apply the Binomial Theorem for the expansion of (x + y)n in powers of x and y for a

positive integer n, where x and y are any numbers, with coefficients determined for example

by Pascal’s Triangle.

2.4. Rewrite rational expressions.

2.4.1. Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form

q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less

than the degree of b(x), using inspection, long division, or, for the more complicated examples,

a computer algebra system.

2.4.2. Understand that rational expressions form a system analogous to the rational numbers, closed

under addition, subtraction, multiplication, and division by a nonzero rational expression;

add, subtract, multiply, and divide rational expressions.

3. Creating Equations

3.1. Create equations that describe numbers or relationships.

3.1.1. Create equations and inequalities in one variable and use them to solve problems. Include

equations arising from linear and quadratic functions, and simple rational and exponential

functions.

3.1.2. Create equations in two or more variables to represent relationships between quantities;

graph equations on coordinate axes with labels and scales. ★

3.1.3. Represent constraints by equations or inequalities, and by systems of equations and/or

inequalities, and interpret solutions as viable or non-viable options in a modeling context. For

example, represent inequalities describing nutritional and cost constraints on combinations of

different foods. ★

3.1.4. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving

equations. For example, rearrange Ohm’s law V = IR to highlight resistance R. ★

4. Reasoning with Equations and Inequalities

4.1. Understand solving equations as a process of reasoning and explain the reasoning.

4.1.1. Explain each step in solving a simple equation as following from the equality of numbers

asserted at the previous step, starting from the assumption that the original equation has a

solution. Construct a viable argument to justify a solution method.

27

4.1.2. Solve simple rational and radical equations in one variable, and give examples showing how

extraneous solutions may arise.

4.2. Solve equations and inequalities in one variable.

4.2.1. Solve linear equations and inequalities in one variable, including equations with coefficients

represented by letters.

4.2.2. Solve linear equations and inequalities in one variable involving absolute value.

4.2.3. Solve quadratic equations in one variable.

4.2.3.1. Use the method of completing the square to transform any quadratic equation in x

into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic

formula from this form.

4.2.3.2. Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots,

completing the square, the quadratic formula, and factoring, as appropriate to the initial

form of the equation. Recognize when the quadratic formula gives complex solutions and

write them as a ± bi for real numbers a and b.

4.2.4. Demonstrate an understanding of the equivalence of factoring, completing the square, or

using the quadratic formula to solve quadratic equations.

4.3. Solve systems of equations.

4.3.1. Prove that, given a system of two equations in two variables, replacing one equation by the

sum of that equation and a multiple of the other produces a system with the same solutions.

4.3.2. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on

pairs of linear equations in two variables.

4.3.3. Solve a simple system consisting of a linear equation and a quadratic equation in two variables

algebraically and graphically. For example, find the points of intersection between the line y = –

3x and the circle x2 + y2 = 3.

4.3.4. Represent a system of linear equations as a single matrix equation in a vector variable.

4.3.5. Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using

technology for matrices of dimension 3 ⋅ 3 or greater).

4.4. Represent and solve equations and inequalities graphically.

4.4.1. Understand that the graph of an equation in two variables is the set of all its solutions plotted

in the coordinate plane, often forming a curve (which could be a line).

4.4.2. Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and

y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately,

e.g., using technology to graph the functions, make tables of values, or find successive

28

approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute

value, exponential, and logarithmic functions.

4.4.3. Graph the solutions to a linear inequality in two variables as a half-plane (excluding the

boundary in the case of a strict inequality), and graph the solution set to a system of linear

inequalities in two variables as the intersection of the corresponding half-planes.

29

Geometry Overview

Congruence

● Experiment with transformations in the plane. Understand congruence in terms of rigid motions.

● Prove geometric theorems.

● Make geometric constructions.

Similarity, Right Triangles, and Trigonometry

● Understand similarity in terms of similarity transformations.

● Prove theorems involving similarity.

● Define trigonometric ratios and solve problems involving right triangles.

● Apply trigonometry to general triangles.

Circles

● Understand and apply theorems about circles.

● Find arc lengths and areas of sectors of circles.

Expressing Geometric Properties with Equations

● Translate between the geometric description and the equation for a conic section.

● Use coordinates to prove simple geometric theorems algebraically.

Geometric Measurement and Dimension

● Explain volume formulas and use them to solve problems.

● Visualize relationships between two-dimensional and three-dimensional objects.

Modeling with Geometry

● Apply geometric concepts in modeling situations.

30

Geometry Standards

1. Congruence

1.1. Experiment with transformations in the plane.

1.1.1. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment,

based on the undefined notions of point, line, distance along a line, and distance around a

circular arc.

1.1.2. Represent transformations in the plane using, e.g., transparencies and geometry software;

describe transformations as functions that take points in the plane as inputs and give other

points as outputs. Compare transformations that preserve distance and angle to those that do

not (e.g., translation versus horizontal stretch).

1.1.3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and

reflections that carry it onto itself.

1.1.4. Develop definitions of rotations, reflections, and translations in terms of angles, circles,

perpendicular lines, parallel lines, and line segments.

1.1.5. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure

using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of

transformations that will carry a given figure onto another.

1.2. Understand congruence in terms of rigid motions.

1.2.1. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a

given rigid motion on a given figure; given two figures, use the definition of congruence in

terms of rigid motions to decide if they are congruent.

1.2.2. Use the definition of congruence in terms of rigid motions to show that two triangles are

congruent if and only if corresponding pairs of sides and corresponding pairs of angles are

congruent.

1.2.3. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition

of congruence in terms of rigid motions.

31

1.3. Prove geometric theorems.

1.3.1. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when

a transversal crosses parallel lines, alternate interior angles are congruent and corresponding

angles are congruent; points on a perpendicular bisector of a line segment are exactly those

equidistant from the segment’s endpoints.

1.3.2. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle

sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of

two sides of a triangle is parallel to the third side and half the length; the medians of a triangle

meet at a point.

1.3.3. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite

angles are congruent, the diagonals of a parallelogram bisect each other, and conversely,

rectangles are parallelograms with congruent diagonals.

1.4. Make geometric constructions.

1.4.1. Make formal geometric constructions with a variety of tools and methods (compass and

straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.).

Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing

perpendicular lines, including the perpendicular bisector of a line segment; and constructing a

line parallel to a given line through a point not on the line.

1.4.2. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

2. Similarity, Right Triangles, and Trigonometry

2.1. Understand similarity in terms of similarity transformations.

2.1.1. Verify experimentally the properties of dilations given by a center and a scale factor:

2.1.1.1. A dilation takes a line not passing through the center of the dilation to a parallel line,

and leaves a line passing through the center unchanged.

2.1.1.2. The dilation of a line segment is longer or shorter in the ratio given by the scale

factor.

2.1.2. Given two figures, use the definition of similarity in terms of similarity transformations to

decide if they are similar; explain using similarity transformations the meaning of similarity

for triangles as the equality of all corresponding pairs of angles and the proportionality of all

corresponding pairs of sides.

2.1.3. Use the properties of similarity transformations to establish the Angle-Angle (AA) criterion for

two triangles to be similar.

32

2.2. Prove theorems involving similarity.

2.2.1. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle

divides the other two proportionally, and conversely; the Pythagorean Theorem proved using

triangle similarity.

2.2.2. Use congruence and similarity criteria for triangles to solve problems and to prove

relationships in geometric figures.

2.3. Define trigonometric ratios and solve problems involving right triangles.

2.3.1. Understand that by similarity, side ratios in right triangles are properties of the angles in the

triangle, leading to definitions of trigonometric ratios for acute angles.

2.3.2. Explain and use the relationship between the sine and cosine of complementary angles.

2.3.3. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied

problems.

2.4. Apply trigonometry to general triangles.

2.4.1. Derive the formula A = ½ab sin(C) for the area of a triangle by drawing an auxiliary line from a

vertex perpendicular to the opposite side.

2.4.2. Prove the Laws of Sines and Cosines and use them to solve problems.

2.4.3. Understand and apply the Law of Sines and the Law of Cosines to find unknown

measurements in right and non-right triangles (e.g., surveying problems, resultant forces).

3. Circles

3.1. Understand and apply theorems about circles.

3.1.1. Prove that all circles are similar.

3.1.2. Identify and describe relationships among inscribed angles, radii, and chords. Include the

relationship between central, inscribed, and circumscribed angles; inscribed angles on a

diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius

intersects the circle.

3.1.3. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles

for a quadrilateral inscribed in a circle.

3.1.4. Derive the formula for the relationship between the number of sides and sums of the interior

and sums of the exterior angles of polygons and apply to the solutions of mathematical and

contextual problems.

3.1.5. Construct a tangent line from a point outside a given circle to the circle.

33

3.2. Find arc lengths and areas of sectors of circles.

3.2.1. Derive using similarity the fact that the length of the arc intercepted by an angle is

proportional to the radius

3.2.2. define the radian measure of the angle as the constant of proportionality

3.2.3. derive the formula for the area of a sector.

4. Expressing Geometric Properties with Equations

4.1. Translate between the geometric description and the equation for a conic section.

4.1.1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem;

complete the square to find the center and radius of a circle given by an equation.

4.1.2. Derive the equation of a parabola given a focus and directrix.

4.1.3. Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or

difference of distances from the foci is constant.

4.2. Use coordinates to prove simple geometric theorems algebraically.

4.2.1. Use coordinates to prove simple geometric theorems algebraically. For example, prove or

disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or

disprove that the point (1, ) lies on the circle centered at the origin and containing the point

(0, 2).

4.2.2. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric

problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes

through a given point).

4.2.3. Find the point on a directed line segment between two given points that partitions the

segment in a given ratio.

4.2.4. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g.,

using the distance formula.

34

6. Geometric Measurement and Dimension

6.1. Explain volume formulas and use them to solve problems.

6.1.1. Give an informal argument for the formulas for the circumference of a circle, area of a circle,

volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and

informal limit arguments.

6.1.2. Give an informal argument using Cavalieri’s principle for the formulas for the volume of a

sphere and other solid figures.

6.1.3. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

6.2. Visualize relationships between two-dimensional and three-dimensional objects.

6.2.1. Identify the shapes of two-dimensional cross-sections of three-dimensional objects

6.2.2. identify three-dimensional objects generated by rotations of two-dimensional objects.

6.3. Modeling with Geometry

6.3.1. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling

a tree trunk or a human torso as a cylinder).

6.3.2. Apply concepts of density based on area and volume in modeling situations (e.g., persons per

square mile, BTUs per cubic foot).

6.3.3. Apply geometric methods to solve design problems (e.g., designing an object or structure to

satisfy physical constraints or minimize cost; working with typographic grid systems based on

ratios).

35

Algebra II Overview

Number and Quantity Review

The Real Number System

● Extend the properties of exponents to rational exponents.

● Use properties of rational and irrational numbers.

Reason Quantitatively and Use Units to Solve Problems.

The Complex Number System

● Perform arithmetic operations with complex numbers.

● Represent complex numbers and their operations on the complex plane.

● Use complex numbers in polynomial identities and equations.

Matrix Quantities

● Perform operations on matrices and use matrices in applications.

Algebra I Standards Extended

Interpreting Functions

● Understand the concept of a function and use function notation.

● Interpret functions that arise in applications in terms of the context.

● Analyze functions using different representations.

Building Functions

● Build a function that models a relationship between two quantities.

● Build new functions from existing functions.

Linear, Quadratic, and Exponential Models

● Construct and compare linear, quadratic, and exponential models and solve problems.

● Interpret expressions for functions in terms of the situation they model.

Trigonometric Functions

● Extend the domain of trigonometric functions using the unit circle.

● Model periodic phenomena with trigonometric functions.

● Prove and apply trigonometric identities.

Congruence

● Experiment with transformations in the plane. Understand congruence in terms of rigid motions.

● Prove geometric theorems.

● Make geometric constructions.

36

Similarity, Right Triangles, and Trigonometry

● Understand similarity in terms of similarity transformations.

● Prove theorems involving similarity.

● Define trigonometric ratios and solve problems involving right triangles.

● Apply trigonometry to general triangles.

Circles

● Understand and apply theorems about circles.

● Find arc lengths and areas of sectors of circles.

Expressing Geometric Properties with Equations

● Translate between the geometric description and the equation for a conic section.

● Use coordinates to prove simple geometric theorems algebraically.

Geometric Measurement and Dimension

● Explain volume formulas and use them to solve problems.

● Visualize relationships between two-dimensional and three-dimensional objects.

Modeling with Geometry

● Apply geometric concepts in modeling situations.

Interpreting Categorical and Quantitative Data

● Summarize, represent, and interpret data on a single count or measurement variable.

● Summarize, represent, and interpret data on two categorical and quantitative variables.

● Interpret linear models.

Making Inferences and Justifying Conclusions

● Understand and evaluate random processes underlying statistical experiments.

● Make inferences and justify conclusions from sample surveys, experiments and observational

studies.

Conditional Probability and the Rules of Probability

● Understand independence and conditional probability and use them to interpret data.

● Use the rules of probability to compute probabilities of compound events in a uniform probability

model.

Using Probability to Make Decisions

● Calculate expected values and use them to solve problems.

● Use probability to evaluate outcomes of decisions.

37

Algebra II Standards

1. Number and Quantity

2. The Real Number System

2.1. Extend the properties of exponents to rational exponents.

2.1.1. Explain how the definition of the meaning of rational exponents follows from extending the

properties of integer exponents to those values, allowing for a notation for radicals in terms of

rational exponents. For example, we define 51/3 to be the cube root of 5 because we want

(51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5.

2.1.2. Rewrite expressions involving radicals and rational exponents using the properties of

exponents.

2.2. Use properties of rational and irrational numbers.

2.2.1. Explain why the sum or product of two rational numbers is rational;

2.2.2. Explain that the sum of a rational number and an irrational number is irrational

2.2.3. Explain that the product of a nonzero rational number and an irrational number is irrational

2.3. Reason quantitatively and use units to solve problems.

2.3.1. Use units as a way to understand problems and to guide the solution of multi-step problems;

choose and interpret units consistently in formulas; choose and interpret the scale and the

origin in graphs and data displays.

2.3.2. Define appropriate quantities for the purpose of descriptive modeling.

2.3.3. Choose a level of accuracy appropriate to limitations on measurement when reporting

quantities.

2.4. The Complex Number System

2.4.1. Perform arithmetic operations with complex numbers.

2.4.1.1. Know there is a complex number i such that , and every complex number has

the form a + bi with a and b real.

2.4.1.2. Use the relation and the commutative, associative, and distributive

properties to add, subtract, and multiply complex numbers.

2.4.1.3. Find the conjugate of a complex number; use conjugates to find moduli and

quotients of complex numbers.

38

2.4.2. Represent complex numbers and their operations on the complex plane.

2.4.2.1. Represent complex numbers on the complex plane in rectangular and polar form

(including real and imaginary numbers), and explain why the rectangular and polar

forms of a given complex number represent the same number.

2.4.2.2. Represent addition, subtraction, multiplication, and conjugation of complex

numbers geometrically on the complex plane; use properties of this representation for

computation. For example, because has modulus 2 and argument

120°.

2.4.2.3. Calculate the distance between numbers in the complex plane as the modulus of the

difference, and the midpoint of a segment as the average of the numbers at its endpoints.

2.4.3. Use complex numbers in polynomial identities and equations.

2.4.3.1. Solve quadratic equations with real coefficients that have complex solutions.

2.4.3.2. Extend polynomial identities to the complex numbers. For example, rewrite x2 + 4 as

(x + 2i)(x – 2i).

2.4.3.3. Know the Fundamental Theorem of Algebra; show that it is true for quadratic

polynomials.

3. Matrix Quantities

3.1. Perform operations on matrices and use matrices in applications.

3.2. Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships

in a network.

3.3. Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are

doubled.

3.4. Add, subtract, and multiply matrices of appropriate dimensions.

3.5. Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not

a commutative operation, but still satisfies the associative and distributive properties.

3.6. Understand that the zero and identity matrices play a role in matrix addition and multiplication

similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if

and only if the matrix has a multiplicative inverse.

3.7. Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to

produce another vector. Work with matrices as transformations of vectors.

3.8. Work with 2 ⋅ 2 matrices as transformations of the plane, and interpret the absolute value of the

determinant in terms of area.

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4. Algebra I Standards Extended

4.1. Seeing Structure in Expressions

4.1.1. Interpret the structure of expressions.

4.1.1.1. Interpret expressions that represent a quantity in terms of its context.

4.1.1.1.1. Interpret parts of an expression, such as terms, factors, and coefficients.

4.1.1.1.2. Interpret complicated expressions by viewing one or more of their parts as a

single entity. For example, interpret P(1 + r)n as the product of P and a factor not

depending on P.

4.1.1.2. Use the structure of an expression to identify ways to rewrite it. For example, see x4 –

y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as

4.1.1.2.1. (x2 – y2)(x2 + y2).

4.1.2. Write expressions in equivalent forms to solve problems.

4.1.2.1. Choose and produce an equivalent form of an expression to reveal and explain

properties of the quantity represented by the expression.

4.1.2.1.1. Factor a quadratic expression to reveal the zeros of the function it defines.

4.1.2.1.2. Complete the square in a quadratic expression to reveal the maximum or

minimum value of the function it defines.

4.1.2.1.3. Use the properties of exponents to transform expressions for exponential

functions. For example, the expression 1.15t can be rewritten as (1.151/12)12t ≈ 1.01212t

to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.

4.1.2.2. Derive the formula for the sum of a finite geometric series (when the common ratio

is not 1), and use the formula to solve problems. For example, calculate mortgage

payments.

40

4.2. Arithmetic with Polynomials and Rational Expressions

4.2.1. Perform arithmetic operations on polynomials.

4.2.1.1. Understand that polynomials form a system analogous to the integers, namely, they

are closed under the operations of addition, subtraction, and multiplication; add,

subtract, and multiply polynomials.

4.2.1.2. Divide polynomials.

4.2.2. Understand the relationship between zeros and factors of polynomials.

4.2.2.1. Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the


4.2.2.2. Identify zeros of polynomials when suitable factorizations are available, and use the

zeros to construct a rough graph of the function defined by the polynomial.

4.2.3. Use polynomial identities to solve problems.

4.2.3.1. Prove polynomial identities and use them to describe numerical relationships. For

example, the polynomial identity (x2 + y2)2 = (x2 – y2)2 + (2xy)2 can be used to generate

Pythagorean triples.

4.2.3.2. Know and apply the Binomial Theorem for the expansion of (x + y)n in powers of x

and y for a positive integer n, where x and y are any numbers, with coefficients

determined for example by Pascal’s Triangle.

4.2.4. Rewrite rational expressions.

4.2.4.1. Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form

q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x)

less than the degree of b(x), using inspection, long division, or, for the more complicated

examples, a computer algebra system.

4.2.4.2. Understand that rational expressions form a system analogous to the rational

numbers, closed under addition, subtraction, multiplication, and division by a nonzero

rational expression; add, subtract, multiply, and divide rational expressions.

41

4.3. Creating Equations

4.3.1. Create equations that describe numbers or relationships.

4.3.1.1. Create equations and inequalities in one variable and use them to solve problems.

Include equations arising from linear and quadratic functions, and simple rational and

exponential functions.

4.3.1.2. Create equations in two or more variables to represent relationships between

quantities; graph equations on coordinate axes with labels and scales. ★

4.3.1.3. Represent constraints by equations or inequalities, and by systems of equations

and/or inequalities, and interpret solutions as viable or non-viable options in a modeling

context. For example, represent inequalities describing nutritional and cost constraints on

combinations of different foods. ★

4.3.1.4. Rearrange formulas to highlight a quantity of interest, using the same reasoning as

in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R. ★

4.4. Reasoning with Equations and Inequalities

4.4.1. Understand solving equations as a process of reasoning and explain the reasoning.

4.4.1.1. Explain each step in solving a simple equation as following from the equality of

numbers asserted at the previous step, starting from the assumption that the original

equation has a solution. Construct a viable argument to justify a solution method.

4.4.1.2. Solve simple rational and radical equations in one variable, and give examples

showing how extraneous solutions may arise.

4.4.2. Solve equations and inequalities in one variable.

4.4.2.1. Solve linear equations and inequalities in one variable, including equations with

coefficients represented by letters.

4.4.2.2. Solve linear equations and inequalities in one variable involving absolute value.

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4.4.2.3. Solve quadratic equations in one variable.

4.4.2.3.1. Use the method of completing the square to transform any quadratic

equation in x into an equation of the form (x – p)2 = q that has the same solutions.

Derive the quadratic formula from this form.

4.4.2.3.2. Solve quadratic equations by inspection (e.g., for x2 = 49), taking square

roots, completing the square, the quadratic formula, and factoring, as appropriate

to the initial form of the equation. Recognize when the quadratic formula gives

complex solutions and write them as a ± bi for real numbers a and b.

4.4.2.4. Demonstrate an understanding of the equivalence of factoring, completing the

square, or using the quadratic formula to solve quadratic equations.

4.4.3. Solve systems of equations.

4.4.3.1. Prove that, given a system of two equations in two variables, replacing one equation

by the sum of that equation and a multiple of the other produces a system with the same

solutions.

4.4.3.2. Solve systems of linear equations exactly and approximately (e.g., with graphs),

focusing on pairs of linear equations in two variables.

4.4.3.3. Solve a simple system consisting of a linear equation and a quadratic equation in

two variables algebraically and graphically. For example, find the points of intersection

between the line y = –3x and the circle x2 + y2 = 3.

4.4.3.4. Represent a system of linear equations as a single matrix equation in a vector

variable.

4.4.3.5. Find the inverse of a matrix if it exists and use it to solve systems of linear equations

(using technology for matrices of dimension 3 ⋅ 3 or greater).

4.4.4. Represent and solve equations and inequalities graphically.

4.4.4.1. Understand that the graph of an equation in two variables is the set of all its

solutions plotted in the coordinate plane, often forming a curve (which could be a line).

43

4.4.4.2. Explain why the x-coordinates of the points where the graphs of the equations y =

f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions

approximately, e.g., using technology to graph the functions, make tables of values, or

find successive approximations. Include cases where f(x) and/or g(x) are linear,

polynomial, rational, absolute value, exponential, and logarithmic functions.

4.4.4.3. Graph the solutions to a linear inequality in two variables as a half-plane (excluding

the boundary in the case of a strict inequality), and graph the solution set to a system of

linear inequalities in two variables as the intersection of the corresponding half-planes.

5. Interpreting Functions

5.1. Understand the concept of a function and use function notation.

5.1.1. Understand that a function from one set (called the domain) to another set (called the range)

assigns to each element of the domain exactly one element of the range. If f is a function and x

is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The

graph of f is the graph of the equation y = f(x).

5.1.2. Use function notation, evaluate functions for inputs in their domains, and interpret statements

that use function notation in terms of a context.

5.1.3. Recognize that sequences are functions, sometimes defined recursively, whose domain is a

subset of the integers. For example, the Fibonacci sequence is defined recursively by

5.1.3.1. f(0) = f(1) = 1, f(n + 1) = f(n) + f(n − 1) for n ≥ 1.

5.2. Interpret functions that arise in applications in terms of the context.

5.2.1. For a function that models a relationship between two quantities, interpret key features of

graphs and tables in terms of the quantities, and sketch graphs showing key features given a

verbal description of the relationship. Key features include: intercepts; intervals where the

function is increasing, decreasing, positive, or negative; relative maximums and minimums;

symmetries; end behavior; and periodicity.

5.2.2. Relate the domain of a function to its graph and, where applicable, to the quantitative

relationship it describes. For example, if the function h(n) gives the number of person-hours it

takes to assemble n engines in a factory, then the positive integers would be an appropriate

domain for the function.

5.2.3. Calculate and interpret the average rate of change of a function (presented symbolically or as

a table) over a specified interval. Estimate the rate of change from a graph.

5.3. Analyze functions using different representations.

44

5.3.1. Graph functions expressed symbolically and show key features of the graph, by hand in simple

cases and using technology for more complicated cases. ★

5.3.2. Graph linear and quadratic functions and show intercepts, maxima, and minima.

5.3.3. Graph square root, cube root, and piecewise-defined functions, including step functions and

absolute value functions.

5.3.4. Graph polynomial functions, identifying zeros when suitable factorizations are available, and

showing end behavior.

5.3.5. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are

available, and showing end behavior.

5.3.6. Graph exponential and logarithmic functions, showing intercepts and end behavior, and

trigonometric functions, showing period, midline, and amplitude.

5.4. Write a function defined by an expression in different but equivalent forms to reveal and explain

different properties of the function.

5.4.1. Use the process of factoring and completing the square in a quadratic function to show zeros,

extreme values, and symmetry of the graph, and interpret these in terms of a context.

5.4.2. Use the properties of exponents to interpret expressions for exponential functions. For

example, identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t,

and y = (1.2)t/10, and classify them as representing exponential growth or decay.

5.4.3. Translate among different representations of functions and relations: graphs, equations, point

sets, and tables.

5.5. Compare properties of two functions each represented in a different way (algebraically,

graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one

quadratic function and an algebraic expression for another, say which has the larger maximum.

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7. Building Functions

7.1. Build a function that models a relationship between two quantities.

7.1.1. Write a function that describes a relationship between two quantities.

7.1.1.1. Determine an explicit expression, a recursive process, or steps for calculation from a

context.

7.1.1.2. Combine standard function types using arithmetic operations. For example, build a

function that models the temperature of a cooling body by adding a constant function to a

decaying exponential, and relate these functions to the model.

7.1.1.3. Compose functions. For example, if T(y) is the temperature in the atmosphere as a

function of height, and h(t) is the height of a weather balloon as a function of time, then

T(h(t)) is the temperature at the location of the weather balloon as a function of time.

7.1.2. Write arithmetic and geometric sequences both recursively and with an explicit formula, use

them to model situations, and translate between the two forms.

7.2. Build new functions from existing functions.

7.2.1. Identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx), and f(x + k) for specific

values of k (both positive and negative); find the value of k given the graphs. Experiment with

cases and illustrate an explanation of the effects on the graph using technology. Include

recognizing even and odd functions from their graphs and algebraic expressions for them.

7.2.2. Find inverse functions.

7.2.2.1. Solve an equation of the form f(x) = c for a simple function f that has an inverse and

write an expression for the inverse. For example, f(x) =2x3 or f(x) = (x + 1)/(x − 1) for x ≠ 1.

7.2.2.2. Verify by composition that one function is the inverse of another.

7.2.2.3. cRead values of an inverse function from a graph or a table, given that the function

has an inverse.

7.2.2.4. Produce an invertible function from a non-invertible function by restricting the

domain.

7.2.2.5. Understand the inverse relationship between exponents and logarithms and use this

relationship to solve problems involving logarithms and exponents.

46

9. Linear, Quadratic, and Exponential Models

9.1. Construct and compare linear, quadratic, and exponential models and solve problems.

9.1.1. Distinguish between situations that can be modeled with linear functions and with

exponential functions.

9.1.1.1. Prove that linear functions grow by equal differences over equal intervals, and that

exponential functions grow by equal factors over equal intervals.

9.1.1.2. Recognize situations in which one quantity changes at a constant rate per unit

interval relative to another.

9.1.1.3. Recognize situations in which a quantity grows or decays by a constant percent rate

per unit interval relative to another.

9.1.2. Construct linear and exponential functions, including arithmetic and geometric sequences,

given a graph, a description of a relationship, or two input-output pairs (include reading these

from a table).

9.1.3. Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a

quantity increasing linearly, quadratically, or (more generally) as a polynomial function.

9.1.4. For exponential models, express as a logarithm the solution to abct = d where a, c, and d are

numbers and the base b is 2, 10, or e; evaluate the logarithm using technology. ★

9.2. Interpret expressions for functions in terms of the situation they model.

9.3. Interpret the parameters in a linear or exponential function in terms of a context. ★

10. Trigonometric Functions

10.1. Extend the domain of trigonometric functions using the unit circle.

10.1.1. Understand radian measure of an angle as the length of the arc on the unit circle subtended

by the angle.

10.1.2. Explain how the unit circle in the coordinate plane enables the extension of trigonometric

functions to all real numbers, interpreted as radian measures of angles traversed

counterclockwise around the unit circle.

10.1.3. Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3,

π/4 and π/6, and use the unit circle to express the values of sine, cosine, and tangent for π − x,

π + x, and 2π − x in terms of their values for x, where x is any real number.

10.1.4. Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric

functions.

47

10.2. Model periodic phenomena with trigonometric functions.

10.2.1. Choose trigonometric functions to model periodic phenomena with specified amplitude,

frequency, and midline.

10.2.2. Understand that restricting a trigonometric function to a domain on which it is always

increasing or always decreasing allows its inverse to be constructed.

10.2.3. Use inverse functions to solve trigonometric equations that arise in modeling contexts;

evaluate the solutions using technology, and interpret them in terms of the context. ★

10.3. Prove and apply trigonometric identities.

10.3.1. Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and use it to find sin(θ), cos(θ), or

tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant.

10.3.2. Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to

solve problems.

11. Congruence

11.1. Experiment with transformations in the plane.

11.1.1. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment,

based on the undefined notions of point, line, distance along a line, and distance around a

circular arc.

11.1.2. Represent transformations in the plane using, e.g., transparencies and geometry software;

describe transformations as functions that take points in the plane as inputs and give other

points as outputs. Compare transformations that preserve distance and angle to those that do

not (e.g., translation versus horizontal stretch).

11.1.3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and

reflections that carry it onto itself.

11.1.4. Develop definitions of rotations, reflections, and translations in terms of angles, circles,

perpendicular lines, parallel lines, and line segments.

11.1.5. Given a geometric figure and a rotation, reflection, or translation, draw the transformed

figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of

transformations that will carry a given figure onto another.

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11.2. Understand congruence in terms of rigid motions.

11.2.1. Use geometric descriptions of rigid motions to transform figures and to predict the effect of

a given rigid motion on a given figure; given two figures, use the definition of congruence in

terms of rigid motions to decide if they are congruent.

11.2.2. Use the definition of congruence in terms of rigid motions to show that two triangles are

congruent if and only if corresponding pairs of sides and corresponding pairs of angles are

congruent.

11.2.3. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the

definition of congruence in terms of rigid motions.

11.3. Prove geometric theorems.

11.3.1. Prove theorems about lines and angles. Theorems include: vertical angles are congruent;

when a transversal crosses parallel lines, alternate interior angles are congruent and

corresponding angles are congruent; points on a perpendicular bisector of a line segment are

exactly those equidistant from the segment’s endpoints.

11.3.2. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle

sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of

two sides of a triangle is parallel to the third side and half the length; the medians of a triangle

meet at a point.

11.3.3. Prove theorems about parallelograms. Theorems include: opposite sides are congruent,

opposite angles are congruent, the diagonals of a parallelogram bisect each other, and

conversely, rectangles are parallelograms with congruent diagonals.

11.4. Make geometric constructions.

11.4.1. Make formal geometric constructions with a variety of tools and methods (compass and

straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.).

Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing

perpendicular lines, including the perpendicular bisector of a line segment; and constructing a

line parallel to a given line through a point not on the line.

11.4.2. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

49

13. Similarity, Right Triangles, and Trigonometry

13.1. Understand similarity in terms of similarity transformations.

13.1.1. Verify experimentally the properties of dilations given by a center and a scale factor:

13.1.1.1. A dilation takes a line not passing through the center of the dilation to a parallel line,

and leaves a line passing through the center unchanged.

13.1.1.2. The dilation of a line segment is longer or shorter in the ratio given by the scale

factor.

13.1.2. Given two figures, use the definition of similarity in terms of similarity transformations to

decide if they are similar; explain using similarity transformations the meaning of similarity

for triangles as the equality of all corresponding pairs of angles and the proportionality of all

corresponding pairs of sides.

13.1.3. Use the properties of similarity transformations to establish the Angle-Angle (AA) criterion

for two triangles to be similar.

13.2. Prove theorems involving similarity.

13.2.1. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle

divides the other two proportionally, and conversely; the Pythagorean Theorem proved using

triangle similarity.

13.2.2. Use congruence and similarity criteria for triangles to solve problems and to prove

relationships in geometric figures.

13.3. Define trigonometric ratios and solve problems involving right triangles.

13.3.1. Understand that by similarity, side ratios in right triangles are properties of the angles in

the triangle, leading to definitions of trigonometric ratios for acute angles.

13.3.2. Explain and use the relationship between the sine and cosine of complementary angles.

13.3.3. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied

problems.

13.4. Apply trigonometry to general triangles.

13.4.1. Derive the formula A = ½ab sin(C) for the area of a triangle by drawing an auxiliary line

from a vertex perpendicular to the opposite side.

13.4.2. Prove the Laws of Sines and Cosines and use them to solve problems.

13.4.3. Understand and apply the Law of Sines and the Law of Cosines to find unknown

measurements in right and non-right triangles (e.g., surveying problems, resultant forces).

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14. Circles

14.1. Understand and apply theorems about circles.

14.1.1. Prove that all circles are similar.

14.1.2. Identify and describe relationships among inscribed angles, radii, and chords. Include the

relationship between central, inscribed, and circumscribed angles; inscribed angles on a

diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius

intersects the circle.

14.1.3. Construct the inscribed and circumscribed circles of a triangle, and prove properties of

angles for a quadrilateral inscribed in a circle.

14.1.4. Derive the formula for the relationship between the number of sides and sums of the

interior and sums of the exterior angles of polygons and apply to the solutions of

mathematical and contextual problems.

14.1.5. Construct a tangent line from a point outside a given circle to the circle.

14.2. Find arc lengths and areas of sectors of circles.

14.2.1. Derive using similarity the fact that the length of the arc intercepted by an angle is

proportional to the radius

14.2.2. define the radian measure of the angle as the constant of proportionality

14.2.3. derive the formula for the area of a sector.

15. Expressing Geometric Properties with Equations

15.1. Translate between the geometric description and the equation for a conic section.

15.1.1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem;

complete the square to find the center and radius of a circle given by an equation.

15.1.2. Derive the equation of a parabola given a focus and directrix.

15.1.3. Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or

difference of distances from the foci is constant.

51

15.2. Use coordinates to prove simple geometric theorems algebraically.

15.2.1. Use coordinates to prove simple geometric theorems algebraically. For example, prove or

disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or

disprove that the point (1, ) lies on the circle centered at the origin and containing the point

(0, 2).

15.2.2. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric

problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes

through a given point).

15.2.3. Find the point on a directed line segment between two given points that partitions the

segment in a given ratio.

15.2.4. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles,

e.g., using the distance formula.

16. Geometric Measurement and Dimension

16.1. Explain volume formulas and use them to solve problems.

16.1.1. Give an informal argument for the formulas for the circumference of a circle, area of a circle,

volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and

informal limit arguments.

16.1.2. Give an informal argument using Cavalieri’s principle for the formulas for the volume of a

sphere and other solid figures.

16.1.3. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

16.2. Visualize relationships between two-dimensional and three-dimensional objects.

16.2.1. Identify the shapes of two-dimensional cross-sections of three-dimensional objects

16.2.2. identify three-dimensional objects generated by rotations of two-dimensional objects.

16.3. Modeling with Geometry

16.3.1. Use geometric shapes, their measures, and their properties to describe objects (e.g.,

modeling a tree trunk or a human torso as a cylinder).

16.3.2. Apply concepts of density based on area and volume in modeling situations (e.g., persons

per square mile, BTUs per cubic foot).

16.3.3. Apply geometric methods to solve design problems (e.g., designing an object or structure to

satisfy physical constraints or minimize cost; working with typographic grid systems based on

ratios).

17. Interpreting Categorical and Quantitative Data

52

17.1. Summarize, represent, and interpret data on a single count or measurement variable.

17.1.1. Represent data with plots on the real number line (dot plots, histograms, and box plots).

17.1.2. Use statistics appropriate to the shape of the data distribution to compare center (median,

mean) and spread (interquartile range, standard deviation) of two or more different data sets.

17.1.3. Interpret differences in shape, center, and spread in the context of the data sets, accounting

for possible effects of extreme data points (outliers).

17.1.4. Use the mean and standard deviation of a data set to fit it to a normal distribution and to

estimate population percentages. Recognize that there are data sets for which such a

procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas

under the normal curve.

17.2. Summarize, represent, and interpret data on two categorical and quantitative

variables.

17.2.1. Summarize categorical data for two categories in two-way frequency tables. Interpret

relative frequencies in the context of the data (including joint, marginal, and conditional

relative frequencies). Recognize possible associations and trends in the data.

17.2.2. Represent data on two quantitative variables on a scatter plot, and describe how the

variables are related.

17.2.2.1. Fit a function to the data; use functions fitted to data to solve problems in the

context of the data. Use given functions or choose a function suggested by the context.

Emphasize linear, quadratic, and exponential models.

17.2.2.2. Informally assess the fit of a function by plotting and analyzing residuals.

17.2.2.3. Fit a linear function for a scatter plot that suggests a linear association.

17.3. Interpret linear models.

17.3.1. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in

the context of the data.

17.3.2. Compute (using technology) and interpret the correlation coefficient of a linear fit.

17.3.3. Distinguish between correlation and causation.

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18. Making Inferences and Justifying Conclusions

18.1. Understand and evaluate random processes underlying statistical experiments.

18.1.1. Understand statistics as a process for making inferences to be made about population

parameters based on a random sample from that population.

18.1.2. Decide if a specified model is consistent with results from a given data-generating process,

e.g., using simulation. For example, a model says a spinning coin falls heads up with probability

0.5. Would a result of 5 tails in a row cause you to question the model?

18.2. Make inferences and justify conclusions from sample surveys, experiments, and

observational studies.

18.2.1. Recognize the purposes of and differences among sample surveys, experiments, and

observational studies; explain how randomization relates to each.

18.2.2. Use data from a sample survey to estimate a population mean or proportion; develop a

margin of error through the use of simulation models for random sampling.

18.2.3. Use data from a randomized experiment to compare two treatments; use simulations to

decide if differences between parameters are significant.

18.2.4. Evaluate reports based on data.

19. Conditional Probability and the Rules of Probability

19.1. Understand independence and conditional probability and use them to interpret

data.

19.1.1. Describe events as subsets of a sample space (the set of outcomes) using characteristics (or

categories) of the outcomes, or as unions, intersections, or complements of other events (“or,”

“and,” “not”).

19.1.2. Understand that two events A and B are independent if the probability of A and B occurring

together is the product of their probabilities, and use this characterization to determine if they

are independent.

19.1.3. Understand the conditional probability of A given B as P(A and B)/P(B), and interpret

independence of A and B as saying that the conditional probability of A given B is the same as

the probability of A, and the conditional probability of B given A is the same as the probability

of B.

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19.1.4. Construct and interpret two-way frequency tables of data when two categories are

associated with each object being classified. Use the two-way table as a sample space to decide

if events are independent and to approximate conditional probabilities. For example, collect

data from a random sample of students in your school on their favorite subject among math,

science, and English. Estimate the probability that a randomly selected student from your school

will favor science given that the student is in tenth grade. Do the same for other subjects and

compare the results. ★

19.1.5. Recognize and explain the concepts of conditional probability and independence in

everyday language and everyday situations. For example, compare the chance of having lung

cancer if you are a smoker with the chance of being a smoker if you have lung cancer.

19.2. Use the rules of probability to compute probabilities of compound events in a

uniform probability model.

19.2.1. Find the conditional probability of A given B as the fraction of B’s outcomes that also belong

to A, and interpret the answer in terms of the model.

19.2.2. Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in

terms of the model.

19.2.3. Apply the general Multiplication Rule in a uniform probability model,

P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model.

19.2.4. Use permutations and combinations to compute probabilities of compound events and

solve problems.

20. Using Probability to Make Decisions

20.1. Calculate expected values and use them to solve problems.

20.1.1. Define a random variable for a quantity of interest by assigning a numerical value to each

event in a sample space; graph the corresponding probability distribution using the same

graphical displays as for data distributions.

20.1.2. Calculate the expected value of a random variable; interpret it as the mean of the

probability distribution.

55

20.1.3. Develop a probability distribution for a random variable defined for a sample space in

which theoretical probabilities can be calculated; find the expected value. For example, find the

theoretical probability distribution for the number of correct answers obtained by guessing on

all five questions of a multiple-choice test where each question has four choices, and find the

expected grade under various grading schemes.

20.1.4. Develop a probability distribution for a random variable defined for a sample space in

which probabilities are assigned empirically; find the expected value. For example, find a

current data distribution on the number of TV sets per household in the United States, and

calculate the expected number of sets per household. How many TV sets would you expect to find

in 100 randomly selected households?

20.2. Use probability to evaluate outcomes of decisions.

20.2.1. Weigh the possible outcomes of a decision by assigning probabilities to payoff values and

finding expected values

20.2.1.1. Find the expected payoff for a game of chance. For example, find the expected

winnings from a state lottery ticket or a game at a fast-food restaurant.

20.2.1.2. Evaluate and compare strategies on the basis of expected values. For example,

compare a high-deductible versus a low-deductible automobile insurance policy using

various, but reasonable, chances of having a minor or a major accident.

20.2.2. Use probabilities to make fair decisions (e.g., drawing by lots, using a random number

generator).

20.2.3. Analyze decisions and strategies using probability concepts (e.g., product testing, medical

testing, pulling a hockey goalie at the end of a game).

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Consumer Math Overview

Pre-writing

Vocabulary Development

Relations and functions

Simple and compound interest

Loans and Financing

Individual Financial and Investment Planning

Summarizing Data (Descriptive Statistics)

Mathematical Reasoning and Problem Solving

Consumer Math Standards

1. Pre-writing

1.1. Prewrite by using organizational strategies and tools to develop a personal organizational style

1.2. Uses technology, spreadsheet, outline, chart, table, graph, Venn Diagram, web, story map, plot

pyramid

2. Vocabulary Development

2.1. Use new vocabulary that is introduced and taught directly

2.2. Uses multiple strategies to develop grade appropriate vocabulary

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3. Relations and functions

3.1. Draw and interpret graphs of relations. Understand the notation and concept of a function, find

domains and ranges, and link equations to functions

3.2. Create a graph to represent a real-world situation

3.3. Interpret a graph representing a real-world situation.

3.4. Linear equations and Inequalities

3.5. Solve linear equations and inequalities

3.6. Symbolically represent and solve multi-step and real-world applications that involve linear

equations and inequalities.

3.7. Solve real-world problems involving systems of linear equations and inequalities in two and three

variables.

4. Simple and compound interest

4.1. Understand simple and compound interest

4.2. Explain the difference between simple and compound interest

4.3. Solve problems involving compound interest.

4.4. Demonstrate the relationship between simple interest and linear growth.

4.5. Demonstrate the relationship between compound interest and exponential growth.

4.6. net Present and net Future Value (NPV and NFV)

4.7. net Present and net Future Value (NPV and NFV)

4.8. Calculate the future value of a given amount of money with and without technology.

4.9. Calculate the present value of a certain amount of money for a given length of time in the future

with and without technology.

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5. Loans and Financing

5.1. Become familiar with and describe the advantages and disadvantages of short-term purchases,

long-term purchases, and mortgages

5.2. Compare the advantages and disadvantages of using cash versus a credit card.

5.3. Analyze credit scores and reports.

5.4. Calculate the finance charges and total amount due on a credit card bill.

5.5. Compare the advantages and disadvantages of deferred payments.

5.6. Calculate deferred payments

5.7. Calculate total cost of purchasing consumer durables over time given different down payments,

financing options, and fees.

5.8. Calculate the following fees associated with a mortgage:

5.9. discount points

5.10. origination fee

5.11. maximum brokerage fee on a net or gross loan

5.12. documentary stamps

5.13. prorated expenses (interest, county and/or city property taxes, and mortgage on an

assumed mortgage)

5.14. Calculate the total amount to be paid over the life of a fixed rate loan.

5.15. Calculate the effects on the monthly payment in the change of interest rate based on an

adjustable rate mortgage.

5.16. Calculate the final pay out amount for a balloon mortgage.

5.17. Compare the cost of paying a higher interest rate and lower points versus a lower interest

rate and more points.

5.18. Calculate the total amount paid for the life of a loan for a house including the down

payment, points, fees, and interest.

5.19. Compare the total cost for a set purchase price using a fixed rate, adjustable rate, and a

balloon mortgage.

5.20. Compare interest rate calculations and annual percentage rate calculations to distinguish

between the two rates.

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6. Individual Financial and Investment Planning

6.1. Develop personal budgets that fit within various income brackets.

6.2. Explain cash management strategies including debit accounts, checking accounts, and savings

accounts.

6.3. Calculate net worth.

6.4. Establish a plan to pay off debt.

6.5. Develop and apply a variety of strategies to use tax tables, and to determine, calculate, and

complete yearly federal income tax.

6.6. Compare different insurance options and fees

6.7. Compare and contrast the role of insurance as a device to mitigate risk and calculate expenses of

various options.

6.8. Collect, organize, and interpret data to determine an effective retirement savings plan to meet

personal financial goals.

6.9. Calculate, compare, and contrast different types of retirement plans, including IRAs, ROTH

accounts, and annuities.

6.10. Analyze diversification in investments

6.11. Purchase stock with a set amount of money, and follow the process through gains, losses,

and selling.

6.12. Compare and contrast income from purchase of common stock, preferred stock, and bonds.

6.13. Given current exchange rates be able to convert from one form of currency to another.

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7. Summarizing Data (Descriptive Statistics)

7.1. Learn to work with summary measures of sets of data, including measures of the center, spread,

and strength of relationship between variables. Learn to distinguish between different types of

data and to select the appropriate visual form to present different types of data.

7.2. Read and interpret data presented in various formats. Determine whether data is presented in

appropriate format, and identify possible corrections. Formats to include:

7.3. bar graphs

7.4. line graphs

7.5. stem and leaf plots

7.6. circle graphs

7.7. histograms

7.8. box and whiskers plots

7.9. scatter plots

7.10. cumulative frequency (ogive) graphs

7.11. Calculate and interpret measures of the center of a set of data, including mean, median, and

weighted mean, and use these measures to make comparisons among sets of data.

8. Mathematical Reasoning and Problem Solving

8.1. Use a variety of strategies to solve problems. Develop and evaluate mathematical arguments and

proofs

8.2. Use a variety of problem-solving strategies, such as drawing a diagram, guess-and-check, solving a

simpler problem, examining simpler problems, and working backwards, using technology when

appropriate.

8.3. Decide whether a solution is reasonable in the context of the original situation.

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IB Curricula have been directly copied from official IB documents

and follow a different formatting than previous standards.

IB Math Studies

Aims

The aims of all mathematics courses in group 5 are to enable students to:

§ 1. enjoy mathematics, and develop an appreciation of the elegance and power of mathematics

§ 2. develop an understanding of the principles and nature of mathematics

§ 3. communicate clearly and confidently in a variety of contexts

§ 4. develop logical, critical and creative thinking, and patience and persistence in problem-solving

§ 5. employ and refine their powers of abstraction and generalization

§ 6. apply and transfer skills to alternative situations, to other areas of knowledge and to future

developments

§ 7. appreciate how developments in technology and mathematics have influenced each other

§ 8. appreciate the moral, social and ethical implications arising from the work of mathematicians

and the applications of mathematics

§ 9. appreciate the international dimension in mathematics through an awareness of the

universality of mathematics and its multicultural and historical perspectives

§ 10. appreciate the contribution of mathematics to other disciplines, and as a particular “area of

knowledge” in the TOK course.

Assessment objectives

Problem-solving is central to learning mathematics and involves the acquisition of mathematical

skills and concepts in a wide range of situations, including non-routine, open-ended and real-

world problems. Having followed a DP mathematical studies SL course, students will be expected

to demonstrate the following.

§ 1. Knowledge and understanding: recall, select and use their knowledge of mathematical facts,

concepts and techniques in a variety of familiar and unfamiliar

contexts.

§ 2. Problem-solving: recall, select and use their knowledge of mathematical skills, results and

models in both real and abstract contexts to solve problems.

§ 3. Communication and interpretation: transform common realistic contexts into mathematics;

62

comment on the context; sketch or draw mathematical diagrams, graphs or constructions both on

paper and using technology; record methods, solutions and conclusions using standardized

notation.

§ 4. Technology: use technology, accurately, appropriately and efficiently both to explore new ideas

and to solve problems.

§ 5. Reasoning: construct mathematical arguments through use of precise statements, logical

deduction and inference, and by the manipulation of mathematical expressions.

§ 6. Investigative approaches: investigate unfamiliar situations involving organizing and analysing

information or measurements, drawing conclusions, testing their validity, and considering their

scope and limitations.

Syllabus – Topic 1—Number and algebra (20 hours)

The aims of this topic are to introduce some basic elements and concepts of mathematics, and to

link these to financial and other applications.

§ 1.1 Natural numbers, N ; integers, Z ; rational numbers, Q ; and real numbers, R .

§ 1.2 Approximation: decimal places, significant figures.

§ Percentage errors.

§ Estimation.

§ 1.3 Expressing numbers in the form a ×10k , where 1≤ a

§ Operations with numbers in this form.

§ 1.4 SI (Système International) and other basic units of measurement: for example, kilogram (kg),

metre (m), second (s), litre (l), metre per second (m s–1), Celsius scale.

§ 1.5 Currency conversions.

§ 1.6 Use of a GDC to solve: pairs of linear equations in two variables; quadratic equations.

§ 1.7 Arithmetic sequences and series, and their applications.

§ Use of the formulae for the nth term and the sum of the first n terms of the sequence.

§ 1.8 Geometric sequences and series.

§ Use of the formulae for the nth term and the sum of the first n terms of the sequence.

§ 1.9 Financial applications of geometric sequences and series: compound interest; annual

depreciation.

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Syllabus - Topic 2—Descriptive statistics (12 hours)

The aim of this topic is to develop techniques to describe and interpret sets of data, in preparation

for further statistical applications.

§ 2.1 Classification of data as discrete or continuous.

§ 2.2 Simple discrete data: frequency tables.

§ 2.3 Grouped discrete or continuous data: frequency tables; mid-interval values; upper and lower

boundaries.

§ Frequency histograms.

§ 2.4 Cumulative frequency tables for grouped discrete data and for grouped continuous data;

cumulative frequency curves, median and quartiles.

§ Box-and-whisker diagram.

§ 2.5 Measures of central tendency.

§ For simple discrete data: mean; median; mode.

§ For grouped discrete and continuous data: estimate of a mean; modal class.

§ 2.6 Measures of dispersion: range, interquartile range, standard deviation.

Syllabus - Topic 3—Logic, sets and probability (20 hours)

The aims of this topic are to introduce the principles of logic, to use set theory to introduce

probability, and to determine the likelihood of random events using a variety of techniques.

§ 3.1 Basic concepts of symbolic logic: definition of a proposition; symbolic notation of

propositions.

§ 3.2 Compound statements: implication, ⇒ ; equivalence, ⇔ ; negation, ¬ ; conjunction, ∧ ;

disjunction, ; exclusive disjunction, v .

§ Translation between verbal statements and symbolic form.

§ 3.3 Truth tables: concepts of logical contradiction and tautology.

§ 3.4 Converse, inverse, contrapositive.

§ Logical equivalence.

§ Testing the validity of simple arguments through the use of truth tables.

§ 3.5 Basic concepts of set theory: elements x∈ A, subsets A ⊂ B ; intersection A∩B ; union A∪B ;

complement A′ § Venn diagrams and simple applications.

§ 3.6 Sample space; event A; complementary event, A′ .

§ Probability of an event.

§ Probability of a complementary event.

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§ Expected value.

§ 3.7 Probability of combined events, mutually exclusive events, independent events.

§ Use of tree diagrams, Venn diagrams, sample space diagrams and tables of outcomes.

§ Probability using “with replacement” and “without replacement”.

§ Conditional probability.

Syllabus - Topic 4—Statistical applications (17 hours)

The aims of this topic are to develop techniques in inferential statistics in order to analyse sets of

data, draw conclusions and interpret these.

§ 4.1 The normal distribution.

§ The concept of a random variable; of the parameters µ and σ ; of the bell shape; the symmetry

about x = µ .

§ Diagrammatic representation.

§ Normal probability calculations.

§ Expected value.

§ Inverse normal calculations.

§ 4.2 Bivariate data: the concept of correlation.

§ Scatter diagrams; line of best fit, by eye, passing through the mean point.

§ Pearson’s product–moment correlation coefficient, r.

§ Interpretation of positive, zero and negative, strong or weak correlations.

§ 4.3 The regression line for y on x.

§ Use of the regression line for prediction purposes.

§ 4.4 The χ² test for independence: formulation of null and alternative hypotheses; significance

levels; contingency tables; expected frequencies; degrees of freedom; pvalues.

Syllabus - Topic 5—Geometry and trigonometry (18 hours)

The aims of this topic are to develop the ability to draw clear diagrams in two dimensions, and to

apply appropriate geometric and trigonometric techniques to problem-solving in two and three

dimensions.

§ 5.1 Equation of a line in two dimensions: the forms y = mx + c and ax + by + d = 0 .

§ Gradient; intercepts.

§ Points of intersection of lines.

§ Lines with gradients, m1 and m2 .

§ Parallel lines m1 = m2 .

65

§ Perpendicular lines, m1 ×m2 = −1 .

§ 5.2 Use of sine, cosine and tangent ratios to find the sides and angles of right-angled triangles.

§ Angles of elevation and depression.

§ 5.3 Use of the sine rule: a/sinA = b/sinB = c/sinC

§ Use of the cosine rule a² = b² + c² − 2bc cos A; cosA = (b² + c² – a²) / (2bc)

§ Use of area of a triangle = ½ab sinC.

§ Construction of labelled diagrams from verbal statements.

§ 5.4 Geometry of three-dimensional solids: cuboid; right prism; right pyramid; right cone;

cylinder; sphere; hemisphere; and combinations of these solids.

§ The distance between two points; eg between two vertices or vertices with midpoints or

midpoints with midpoints.

§ The size of an angle between two lines or between a line and a plane.

§ 5.5 Volume and surface areas of the three-dimensional solids defined in 5.4.

Syllabus - Topic 6—Mathematical models (20 hours)

The aim of this topic is to develop understanding of some mathematical functions that can be used

to model practical situations. Extensive use of a GDC is to be encouraged in this topic.

§ 6.1 Concept of a function, domain, range and graph.

§ Function notation, eg f (x), v(t), C(n) .

§ Concept of a function as a mathematical model.

§ 6.2 Linear models.

§ Linear functions and their graphs, f (x) = mx + c .

§ 6.3 Quadratic models.

§ Quadratic functions and their graphs (parabolas): f (x) = ax² + bx + c ; a ≠ 0

§ Properties of a parabola: symmetry; vertex; intercepts on the x-axis and y-axis.

§ Equation of the axis of symmetry, x= -b/2a.

§ 6.4 Exponential models.

§ Exponential functions and graphs: f (x) = kax + c; a∈Q+ , a ≠1, k ≠ 0 ; f (x) = ka-x + c; a∈Q + , a ≠1, k

≠ 0 .

§ Concept and equation of a horizontal asymptote.

§ 6.5 Models using functions of the form f (x) = axm+ bxn +…; m,n∈ Z .

§ Functions of this type and their graphs.

§ The y-axis as a vertical asymptote.

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§ 6.6 Drawing accurate graphs.

§ Creating a sketch from information given.

§ Transferring a graph from GDC to paper.

§ Reading, interpreting and making predictions using graphs.

§ Included all the functions above and additions and subtractions.

§ 6.7 Use of a GDC to solve equations involving combinations of the functions above.

Syllabus - Topic 7—Introduction to differential calculus (18 hours)

The aim of this topic is to introduce the concept of the derivative of a function and to apply it to

optimization and other problems.

§ 7.1 Concept of the derivative as a rate of change.

§ Tangent to a curve.

§ 7.2 The principle that f (x) = axn ⇒ f ′(x) = anxn-1 .

§ The derivative of functions of the form f (x) = axn + bxn-1 + …, where all exponents are integers.

§ 7.3 Gradients of curves for given values of x.

§ Values of x where f ′(x) is given.

§ Equation of the tangent at a given point.

§ Equation of the line perpendicular to the tangent at a given point (normal).

§ 7.4 Increasing and decreasing functions.

§ Graphical interpretation of f ′(x) > 0 , f ′(x) = 0 and f ′(x) < 0 .

§ 7.5 Values of x where the gradient of a curve is zero.

§ Solution of f ′(x) = 0 .

§ Stationary points.

§ Local maximum and minimum points.

§ 7.6 Optimization problems.

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IB Math SL

Aims

The aims of all mathematics courses in group 5 are to enable students to:







developments








Assessment objectives


skills and concepts in a wide range of situations, including non-routine, open-ended and real-world

problems. Having followed a DP mathematics SL course, students will be expected to demonstrate

the following.


concepts and techniques in a variety of familiar and unfamiliar contexts.




comment on the context; sketch or draw mathematical diagrams, graphs or constructions both

on paper and using technology; record methods, solutions and conclusions using standardized

notation.

68





§ 6. Inquiry approaches: investigate unfamiliar situations, both abstract and real-world, involving

organizing and analysing information, making conjectures, drawing conclusions and testing their

validity.

Syllabus - Topic 1—Algebra (9 hours)

The aim of this topic is to introduce students to some basic algebraic concepts and applications.

§ 1.1 Arithmetic sequences and series; sum of finite arithmetic series; geometric sequences and

series; sum of finite and infinite geometric series.

§ Sigma notation.

§ Applications.

§ 1.2 Elementary treatment of exponents and logarithms.

§ Laws of exponents; laws of logarithms.

§ Change of base.

§ 1.3 The binomial theorem: expansion of (a + b)ⁿ , n∈N .

§ Calculation of binomial coefficients using Pascal’s triangle and (n r).

Syllabus - Topic 2—Functions and equations (24 hours)

The aims of this topic are to explore the notion of a function as a unifying theme in mathematics,

and to apply functional methods to a variety of mathematical situations. It is expected that

extensive use will be made of technology in both the development and the application of this topic,

rather than elaborate analytical techniques. On examination papers, questions may be set requiring

the graphing of functions that do not explicitly appear on the syllabus, and students may need to

choose the appropriate viewing window. For those functions explicitly mentioned, questions may

also be set on composition of these functions with the linear function y = ax + b .

§ 2.1 Concept of function f: x→ f (x) .

§ Domain, range; image (value).

§ Composite functions.

§ Identity function. Inverse function f-1 .

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§ 2.2 The graph of a function; its equation y = f (x) .

§ Function graphing skills.

§ Investigation of key features of graphs, such as maximum and minimum values, intercepts,

horizontal and vertical asymptotes, symmetry, and consideration of domain and range.

§ Use of technology to graph a variety of functions, including ones not specifically mentioned.

§ The graph of y = f -1

(x) as the reflection in the line y = x of the graph of y = f (x) .

§ 2.3 Transformations of graphs.

§ Translations: y = f (x) + b ; y = f (x − a) .

§ Reflections (in both axes): y = − f (x) ; y = f (−x) .

§ Vertical stretch with scale factor p: y = pf (x) .

§ Stretch in the x-direction with scale factor (1/q): y= f (qx)

§ Composite transformations.

§ 2.4 The quadratic function x → ax² + bx + c : its graph, y-intercept (0, c) . Axis of symmetry.

§ The form x → a(x − p)(x − q), x-intercepts (p, 0) and (q, 0) .

§ The form x → a(x − h)² + k , vertex (h, k).

§ 2.5 The reciprocal function x → (1/x), x ≠ 0 : its graph and self-inverse nature.

§ The rational function x → (ax+b)/(cx+d) and its graph.

§ Vertical and horizontal asymptotes.

§ 2.6 Exponential functions and their graphs: x → ax , a > 0 , x → ex .

§ Logarithmic functions and their graphs: logax → x , x > 0 , x → ln x , x > 0 .

§ Relationships between these functions: ax = e x ln a ; loga ax = x ; alog a x = x , x > 0 .

§ 2.7 Solving equations, both graphically and analytically.

§ Use of technology to solve a variety of equations, including those where there is no appropriate

analytic approach.

§ Solving ax² + bx + c = 0 , a ≠ 0 . The quadratic formula.

§ The discriminant Δ = b² − 4ac and the nature of the roots, that is, two distinct real roots, two equal

real roots, no real roots.

§ Solving exponential equations.

§ 2.8 Applications of graphing skills and solving equations that relate to real-life situations.

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Syllabus - Topic 3—Circular functions and trigonometry (16 hours)

The aims of this topic are to explore the circular functions and to solve problems using

trigonometry. On examination papers, radian measure should be assumed unless otherwise

indicated.

§ 3.1 The circle: radian measure of angles; length of an arc; area of a sector.

§ 3.2 Definition of cosθ and sinθ in terms of the unit circle.

§ Definition of tanθ as sin θ / cosθ.

§ Exact values of trigonometric ratios of 0, π/6, π/4, π/3, π/2 and their multiples.

§ 3.3 The Pythagorean identity cos²θ + sin²θ =1.

§ Double angle identities for sine and cosine.

§ Relationship between trigonometric ratios.

§ 3.4 The circular functions sin x , cos x and tan x : their domains and ranges; amplitude, their

periodic nature; and their graphs.

§ Composite functions of the form f (x) = a sin (b(x + c)) + d .

§ Transformations.

§ Applications.

§ 3.5 Solving trigonometric equations in a finite interval, both graphically and analytically.

§ Equations leading to quadratic equations in sin x, cos x or tan x .

§ 3.6 Solution of triangles.

§ The cosine rule.

§ The sine rule, including the ambiguous case.

§ Area of a triangle, ½ ab sinC

§ Applications.

Syllabus - Topic 4—Vectors (16 hours)

The aim of this topic is to provide an elementary introduction to vectors, including both algebraic

and geometric approaches. The use of dynamic geometry software is extremely helpful to visualize

situations in three dimensions.

§ 4.1 Vectors as displacements in the plane and in three dimensions.

§ Components of a vector; column representation; v= v1 v2 v3 = v1i + v2j + v3k.

§ Algebraic and geometric approaches to the following: the sum and difference of two vectors; the

zero vector, the vector −v ; multiplication by a scalar, kv ; parallel vectors; magnitude of a vector,

│v│ ; unit vectors; base vectors; i, j and k; position vectors OA = a ; AB = OB – OA = b − a .

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§ 4.2 The scalar product of two vectors.

§ Perpendicular vectors; parallel vectors.

§ The angle between two vectors.

§ 4.3 Vector equation of a line in two and three dimensions: r = a + tb .

§ The angle between two lines.

§ 4.4 Distinguishing between coincident and parallel lines.

§ Finding the point of intersection of two lines.

§ Determining whether two lines intersect.

Syllabus - Topic 5—Statistics and probability (35 hours)

The aim of this topic is to introduce basic concepts. It is expected that most of the calculations

required will be done using technology, but explanations of calculations by hand may enhance

understanding. The emphasis is on understanding and interpreting the results obtained, in context.

Statistical tables will no longer be allowed in examinations. While many of the calculations required

in examinations are estimates, it is likely that the command terms “write down”, “find” and

“calculate” will be used.

§ 5.1 Concepts of population, sample, random sample, discrete and continuous data.

§ Presentation of data: frequency distributions (tables); frequency histograms with equal class

intervals; box-and-whisker plots; outliers.

§ Grouped data: use of mid-interval values for calculations; interval width; upper and lower

interval boundaries; modal class.

§ 5.2 Statistical measures and their interpretations.

§ Central tendency: mean, median, mode.

§ Quartiles, percentiles.

§ Dispersion: range, interquartile range, variance, standard deviation.

§ Effect of constant changes to the original data.

§ Applications

§ 5.3 Cumulative frequency; cumulative frequency graphs; use to find median, quartiles,

percentiles.

§ 5.4 Linear correlation of bivariate data.

§ Pearson’s product–moment correlation coefficient r.

§ Scatter diagrams; lines of best fit.

§ Equation of the regression line of y on x.

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§ Use of the equation for prediction purposes.

§ Mathematical and contextual interpretation.

§ 5.5 Concepts of trial, outcome, equally likely outcomes, sample space (U) and event.

§ The probability of an event A is P(A) = n(A) / n(U).

§ The complementary events A and A′ (not A).

§ Use of Venn diagrams, tree diagrams and tables of outcomes.

§ 5.6 Combined events, P(A ∪ B) .

§ Mutually exclusive events: P(A ∩ B) = 0 .

§ Conditional probability; the definition P(A | B ) = P(A∩B) / P(B).

§ Independent events; the definition P(A | B) = P(A) = P(A | B’) .

§ Probabilities with and without replacement.

§ 5.7 Concept of discrete random variables and their probability distributions.

§ Expected value (mean), E(X ) for discrete data.

§ Applications.

§ 5.8 Binomial distribution.

§ Mean and variance of the binomial distribution.

§ 5.9 Normal distributions and curves.

§ Standardization of normal variables (z-values, z-scores).

§ Properties of the normal distribution.

Syllabus - Topic 6—Calculus (40 hours)

The aim of this topic is to introduce students to the basic concepts and techniques of differential

and integral calculus and their applications.

§ 6.1 Informal ideas of limit and convergence.

§ Limit notation.

§ Definition of derivative from first principles as f’(x) = lim h → 0 ( f(x+h) – f(x) / h ).

§ Derivative interpreted as gradient function and as rate of change.

§ Tangents and normals, and their equations.

§ 6.2 Derivative of xⁿ (n∈Q) , sin x , cos x , tan x , e and ln x .

§ Differentiation of a sum and a real multiple of these functions.

§ The chain rule for composite functions.

§ The product and quotient rules.

§ The second derivative.

§ Extension to higher derivatives.

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§ 6.3 Local maximum and minimum points.

§ Testing for maximum or minimum.

§ Points of inflexion with zero and non-zero gradients.

§ Graphical behaviour of functions, including the relationship between the graphs of f , f' and f" .

§ Optimization.

§ Applications.

§ 6.4 Indefinite integration as anti-differentiation.

§ Indefinite integral of xⁿ (n∈Q) , sin x, cos x, (1/x) and e .

§ The composites of any of these with the linear function ax + b.

§ Integration by inspection, or substitution of the form ∫ f (g(x))g '(x)dx .

§ 6.5 Anti-differentiation with a boundary condition to determine the constant term.

§ Definite integrals, both analytically and using technology.

§ Areas under curves (between the curve and the x-axis).

§ Areas between curves.

§ Volumes of revolution about the x-axis.

§ 6.6 Kinematic problems involving displacement s, velocity v and acceleration a.

§ Total distance travelled.

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IB Math HL

Aims







developments








Objectives


skills and concepts in a wide range of situations, including non-routine, open-ended and real-world

problems. Having followed a DP mathematics HL course, students will be expected to demonstrate

the following.


concepts and techniques in a variety of familiar and unfamiliar contexts.




comment on the context; sketch or draw mathematical diagrams, graphs or constructions both on

paper and using technology; record methods, solutions and conclusions using standardized

notation.



75



§ 6. Inquiry approaches: investigate unfamiliar situations, both abstract and real-world, involving

organizing and analysing information, making conjectures, drawing conclusions and testing their

validity.

Syllabus - Topic 1 - Core: Algebra (30 hours)

The aim of this topic is to introduce students to some basic algebraic concepts and applications.

§ 1.1 Arithmetic sequences and series; sum of finite arithmetic series; geometric sequences and

series; sum of finite and infinite geometric series.

§ Sigma notation.

§ Applications.

§ 1.2 Exponents and logarithms.

§ Laws of exponents; laws of logarithms.

§ Change of base.

§ 1.3 Counting principles, including permutations and combinations.

§ The binomial theorem: expansion of (a+b)n , nEN.

§ Not required:

§ Permutations where some objects are identical.

§ Circular arrangements.

§ Proof of binomial theorem.

§ 1.4 Proof by mathematical induction.

§ 1.5 Complex numbers: the number

§ i=√-1 ; the terms real part, imaginary part, conjugate, modulus and argument.

§ Cartesian form z=a+ib.

§ Sums, products and quotients of complex numbers.

§ 1.6 Modulus–argument (polar) form z=r(cosΘ+i sin Θ) = r cis Θ = reiΘ.

§ The complex plane.

§ 1.7 Powers of complex numbers: de Moivre’s theorem.

§ nth roots of a complex number.

§ 1.8 Conjugate roots of polynomial equations with real coefficients.

§ 1.9 Solutions of systems of linear equations (a maximum of three equations in three unknowns),

including cases where there is a unique solution, an infinity of solutions or no solution.

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Syllabus - Topic 2 - Core: Functions and equations (22 hours)

The aims of this topic are to explore the notion of function as a unifying theme in mathematics, and

to apply functional methods to a variety of mathematical situations. It is expected that extensive use

will be made of technology in both the development and the application of this topic.

§ 2.1 Concept of function ƒ : x ↦ ƒ(x) domain, range; image (value).

§ Odd and even functions.

§ Composite functions ƒ ⋄ g.

§ Identity function.

§ One-to-one and many-to-one functions.

§ Inverse function ƒ-1, including domain restriction. Self-inverse functions.

§ 2.2 The graph of a function; its equation y= ƒ(x).

§ Investigation of key features of graphs, such as maximum and minimum values, intercepts,

horizontal and vertical asymptotes and symmetry, and consideration of domain and range.

§ The graphs of the functions y= | ƒ(x)| and y= ƒ(|x|).

§ The graph of y=1/(ƒ(x)) given the graph of y= ƒ(x).

§ 2.3 Transformations of graphs: translations; stretches; reflections in the axes.

§ The graph of the inverse function as a reflection in y= x

§ 2.4 The rational function x ↦ ((ax+b)/(cx+d)), and its graph.

§ The function x ↦ ax, a > 0, and its graph.

§ The function x ↦ logax,x>0, and its graph.

§ 2.5 Polynomial functions and their graphs.

§ The factor and remainder theorems.

§ The fundamental theorem of algebra.

§ 2.6 Solving quadratic equations using the quadratic formula.

§ Use of the discriminant Δ=b ²-4ac to determine the nature of the roots.

§ Solving polynomial equations both graphically and algebraically.

§ Sum and product of the roots of polynomial equations.

§ Solutions of ax=b using logarithms

§ Use of technology to solve a variety of equations, including those where there is no appropriate

analytic approach.

§ 2.7 Solutions of g(x) ≥ƒ(x).

§ Graphical or algebraic methods, for simple polynomials up to degree 3.

§ Use of technology for these and other functions.

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Syllabus - Topic 3 - Core: Circular functions and trigonometry (22 hours)

The aims of this topic are to explore the circular functions, to introduce some important

trigonometric identities and to solve triangles using trigonometry. On examination papers, radian

measure should be assumed unless otherwise indicated, for example, by x ↦sin x ↦.

§ 3.1 The circle: radian measure of angles.

§ Length of an arc; area of a sector.

§ 3.2 Definition of cosΘ, sinΘ, tanΘ in terms of the unit circle.

§ Exact values of sin, cos and tan of 0, Π/6, Π/4, Π/3, Π/2 and their multiples.

§ Definition of the reciprocal trigonometric ratios secΘ, cscΘ and cotΘ.

§ Pythagorean identities: cos²Θ + sin²Θ=1;

§ 1+tan²Θ =sec²; 1+cot²Θ=csc²Θ.

§ 3.3 Compound angle identities.

§ Double angle identities.

§ Not required:

§ Proof of compound angle identities.

§ 3.4 Composite functions of the form ƒ(x)= a sin(b(x+c))+d.

§ Applications.

§ 3.5 The inverse functions x ↦ arcsin x, x ↦ arcos x, x ↦artan x; their domains and ranges; their

graphs.

§ 3.6 Algebraic and graphical methods of solving trigonometric equations in a finite interval,

including the use of trigonometric identities and factorization.

§ Not required:

§ The general solution of trigonometric equations.

§ 3.7 The cosine rule

§ The sine rule including the ambiguous case.

§ Area of a triangle as 1/2 absinC.

§ Applications.

Syllabus - Topic 4 - Core: Vectors (24 hours)

The aim of this topic is to introduce the use of vectors in two and three dimensions, and to facilitate

solving problems involving points, lines and planes.

§ 4.1 Concept of a vector.

§ Representation of vectors using directed line segments.

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§ Unit vectors; base vectors i, j, k.

§ Components of a vector: *equation not displayed

§ Algebraic and geometric approaches to the following:

§ • the sum and difference of two vectors;

§ • the zero vector 0, the vector –v;

§ • multiplication by a scalar, kv;

§ • magnitude of a vector, |v|;

§ • position vectors OA=a.

§ AB= b-a

§ 4.2 The definition of the scalar product of two vectors.

§ Properties of the scalar product:

§ v⋅w= w⋅v:

§ u⋅(v+w)=u⋅v+u⋅w;

§ (kv) ⋅w=k(v⋅w);

§ v⋅v=|v|².

§ The angle between two vectors.

§ Perpendicular vectors; parallel vectors.

§ 4.3 Vector equation of a line in two and three dimensions: r= a+λb.

§ Simple applications to kinematics.

§ The angle between two lines.

§ 4.4 Coincident, parallel, intersecting and skew lines; distinguishing between these cases.

§ Points of intersection.

§ 4.5 The definition of the vector product of two vectors.

§ Properties of the vector product:

§ v x w = -w x v;

§ u x (v+w) = u x v +u x w;

§ (kv) x w = k(v x w);

§ v x v = 0.

§ Geometric interpretation of |v x w|.

§ 4.6 Vector equation of a plane r = a + λb + µc.

§ Use of normal vector to obtain the form r⋅n=a⋅n.

§ Cartesian equation of a plane ax+by+cz=d.

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§ 4.7 Intersections of: a line with a plane; two planes; three planes.

§ Angle between: a line and a plane; two planes.

Syllabus - Topic 5 - Core: Statistics and probability (36 hours)

The aim of this topic is to introduce basic concepts. It may be considered as three parts:

manipulation and presentation of statistical data (5.1), the laws of probability (5.2–5.4), and

random variables and their probability distributions (5.5–5.7). It is expected that most of the

calculations required will be done on a GDC. The emphasis is on understanding and interpreting the

results obtained. Statistical tables will no longer be allowed in

examinations.

§ 5.1 Concepts of population, sample, random sample and frequency distribution of discrete and

continuous data.

§ Grouped data: mid-interval values, interval width, upper and lower interval boundaries.

§ Mean, variance, standard deviation.

§ Not required:

§ Estimation of mean and variance of a population from a sample.

§ 5.2 Concepts of trial, outcome, equally likely outcomes, sample space (U) and event.

§ The probability of an event A as P(A) = (n(A))/(n(U)).

§ The complementary events A and A' (not A).

§ Use of Venn diagrams, tree diagrams, counting principles and tables of outcomes to solve

problems.

§ 5.3 Combined events; the formula for P(A⌣B).

§ Mutually exclusive events.

§ 5.4 Conditional probability; the definition P(A|B)= (P(A⌢B)/P(B)).

§ Independent events; the definition P(A|B)=P(A)=P(A|B’).

§ Use of Bayes’ theorem for a maximum of three events.

§ 5.5 Concept of discrete and continuous random variables and their probability distributions.

§ Definition and use of probability density functions.

§ Expected value (mean), mode, median, variance and standard deviation.

§ Applications.

§ 5.6 Binomial distribution, its mean and variance.

§ Poisson distribution, its mean and variance.

§ Not required:

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§ Formal proof of means and variances.

§ 5.7 Normal distribution.

§ Properties of the normal distribution.

§ Standardization of normal variables.

Syllabus - Topic 6 - Core: Calculus (48 hours)

The aim of this topic is to introduce students to the basic concepts and techniques of differential

and integral calculus and their application.

§ 6.1 Informal ideas of limit, continuity and convergence.

§ Definition of derivative from first principles. *Equation not displayed

§ The derivative interpreted as a gradient function and as a rate of change.

§ Finding equations of tangents and normals.

§ Identifying increasing and decreasing functions.

§ The second derivative.

§ Higher derivatives.

§ 6.2 Derivatives of xn, sin x, cos x, tan x, ex and ln x.

§ Differentiation of sums and multiples of functions.

§ The product and quotient rules.

§ The chain rule for composite functions.

§ Related rates of change.

§ Implicit differentiation.

§ Derivatives of sec x, csc x, cot x, ax, logax, arcsin x, arccos x and arctan x.

§ 6.3 Local maximum and minimum values.

§ Optimization problems.

§ Points of inflexion with zero and non-zero gradients.

§ Graphical behaviour of functions, including the relationship between the graphs of ƒ, ƒ’, ƒ”.

§ Not required:

§ Points of inflexion, where ƒ”(x) is not defined, for example, y =x1/3 at (0,0).

§ 6.4 Indefinite integration as anti-differentiation.

§ Indefinite integral of xn, sin x, cos x and ex.

§ Other indefinite integrals using the results from 6.2.

§ The composites of any of these with a linear function.

§ 6.5 Anti-differentiation with a boundary condition to determine the constant of integration.

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§ Definite integrals.

§ Area of the region enclosed by a curve and the x-axis or y-axis in a given interval; areas of regions

enclosed by curves.

§ Volumes of revolution about the x-axis or y-axis.

§ 6.6 Kinematic problems involving displacement s, velocity v and acceleration a.

§ Total distance travelled.

§ 6.7 Integration by substitution

§ Integration by parts.

Syllabus - Topic 7 - Option: Statistics and probability (48 hours)

The aims of this option are to allow students the opportunity to approach statistics in a practical

way; to demonstrate a good level of statistical understanding; and to understand which situations

apply and to interpret the given results. It is expected that GDCs will be used throughout this option,

and that the minimum requirement of a GDC will be to find probability distribution function (pdf),

cumulative distribution function (cdf), inverse cumulative distribution function, p-values and test

statistics, including calculations for the following distributions: binomial, Poisson, normal and t.

Students are expected to set up the problem mathematically and then read the answers from the

GDC, indicating this within their written answers. Calculator-specific or brand-specific language

should not be used within these explanations.

§ 7.1 Cumulative distribution functions for both discrete and continuous distributions.

§ Geometric distribution.

§ Negative binomial distribution.

§ Probability generating functions for discrete random variables.

§ Using probability generating functions to find mean, variance and the distribution of the sum of n

independent random variables.

§ 7.2 Linear transformation of a single random variable.

§ Mean of linear combinations of n random variables.

§ Variance of linear combinations of n independent random variables.

§ Expectation of the product of independent random variables.

§ 7.3 Unbiased estimators and estimates.

§ Comparison of unbiased estimators based on variances.

§ X-Bar as an unbiased estimator for µ.

§ S² as an unbiased estimator for σ².

§ 7.4 A linear combination of independent normal random variables is normally distributed. In

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particular, N(µ, σ²) ⇒ N(µ, (σ²/ n)).

§ The central limit theorem.

§ 7.5 Confidence intervals for the mean of a normal population.

§ 7.6 Null and alternative hypotheses, Null and alternative hypotheses, H₀ and H₁ .

§ Significance level.

§ Critical regions, critical values, p-values, one-tailed and two-tailed tests.

§ Type I and II errors, including calculations of their probabilities.

§ Testing hypotheses for the mean of a normal population.

§ 7.7 Introduction to bivariate distributions.

§ Covariance and (population) product moment correlation coefficient p.

§ Proof that p= 0 in the case of independence and ±1 in the case of a linear relationship between X

and Y.

§ Definition of the (sample) product moment correlation coefficient R in terms of n paired

observations on X and Y. Its application to the estimation of p.

§ Informal interpretation of r, the observed value of R. Scatter diagrams.

§ The following topics are based on the assumption of bivariate normality.

§ Use of the t-statistic to test the null hypothesis p = 0.

§ Knowledge of the facts that the regression of X on Y(E(X)|Y=y) and Y on X (E(Y)|X=x) are linear.

§ Least-squares estimates of these regression lines (proof not required).

§ The use of these regression lines to predict the value of one of the variables given the value of the

other.

Syllabus - Topic 8 - Option: Sets, relations and groups (48 hours).The aims of this option are to

provide the opportunity to study some important mathematical concepts, and introduce the

principles of proof through abstract algebra.

§ 8.1 Finite and infinite sets. Subsets.

§ Operations on sets: union; intersection; complement; set difference; symmetric difference.

§ De Morgan’s laws: distributive, associative and commutative laws (for union and intersection).

§ 8.2 Ordered pairs: the Cartesian product of two sets.

§ Relations: equivalence relations; equivalence classes.

§ 8.3 Functions: injections; surjections; bijections.

§ Composition of functions and inverse functions.

§ 8.4 Binary operations.

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§ Operation tables (Cayley tables).

§ 8.5 Binary operations: associative, distributive and commutative properties.

§ 8.6 The identity element e.

§ The inverse a-1 of an element a.

§ Proof that left-cancellation and right-cancellation by an element a hold, provided that a has an

inverse.

§ Proofs of the uniqueness of the identity and inverse elements.

§ 8.7 The definition of a group {G ,*}.

§ The operation table of a group is a Latin square, but the converse is false.

§ Abelian groups.

§ 8.8 Examples of groups:

§ • ℝ, ℚ , ℤ and ℂ under addition;

§ • integers under addition modulo n;

§ • non-zero integers under multiplication, modulo p, where p is prime;

§ symmetries of plane figures, including equilateral triangles and rectangles;

§ invertible functions under composition of functions.

§ 8.9 The order of a group.

§ The order of a group element.

§ Cyclic groups.

§ Generators.

§ Proof that all cyclic groups are Abelian.

§ 8.10 Permutations under composition of permutations.

§ Cycle notation for permutations.

§ Result that every permutation can be written as a composition of disjoint cycles.

§ The order of a combination of cycles.

§ 8.11 Subgroups, proper subgroups.

§ Use and proof of subgroup tests.

§ Definition and examples of left and right cosets of a subgroup of a group.

§ Lagrange’s theorem.

§ Use and proof of the result that the order of a finite group is divisible by the order of any

element. (Corollary to Lagrange’s theorem.)

§ 8.12 Definition of a group homomorphism.

§ Definition of the kernel of a homomorphism. Proof that the kernel and range of a

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homomorphism are subgroups.

§ Proof of homomorphism properties for identities and inverses.

§ Isomorphism of groups.

§ The order of an element is unchanged by an isomorphism.

Syllabus - Topic 9 - Option: Calculus (48 hours). The aims of this option are to introduce limit

theorems and convergence of series, and to use calculus results to solve differential equations.

§ 9.1 Infinite sequences of real numbers and their convergence or divergence.

§ 9.2 Convergence of infinite series.

§ Tests for convergence: comparison test; limit comparison test; ratio test; integral test.

§ The p-series, ∑(1/np).

§ Series that converge absolutely. Series that converge conditionally. Alternating, series.

§ Power series: radius of convergence and interval of convergence. Determination of the radius of

convergence by the ratio test.

§ 9.3 Continuity and differentiability of a function at a point.

§ Continuous functions and differentiable functions.

§ 9.4 The integral as a limit of a sum; lower and upper Riemann sums.

§ Fundamental theorem of calculus.

§ Improper integrals of the type *equation not displayed

§ 9.5 First-order differential equations.

§ Geometric interpretation using slope fields, including identification of isoclines.

§ Numerical solution of Dy/dx= ƒ( x, y) using Euler’s method.

§ Variables separable.

§ Homogeneous differential equation *equation not displayed using the substitution y = vx.

§ Solution of y′ + P(x)y = Q(x), using the integrating factor.

§ 9.6 Rolle’s theorem.

§ Mean value theorem.

§ Taylor polynomials; the Lagrange form of the error term.

§ Maclaurin series for ex , sin x , cos x , ln(1+x) , (1+x) P, p εℚ.

§ Use of substitution, products, integration and differentiation to obtain other series.

§ Taylor series developed from differential equations.

§ 9.7 The evaluation of limits of the form *equation not displayed

§ Using l’Hôpital’s rule or the Taylor series.

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Syllabus - Topic 10 - Option: Discrete mathematics (48 hours) The aim of this option is to provide

the opportunity for students to engage in logical reasoning, algorithmic thinking and applications.

§ 10.1 Strong induction.

§ Pigeon-hole principle.

§ 10.2 a|b ⇒ b =na for some N ε ℤ

§ The theorem a|b and a|c ⇒ a | (bx±cy) where x,y ε ℤ.

§ Division and Euclidean algorithms.

§ The greatest common divisor, gcd(a, b) , and the least common multiple, lcm(a, b) , of integers a

and b.

§ Prime numbers; relatively prime numbers and the fundamental theorem of arithmetic.

§ 10.3 Linear Diophantine equations ax + by = c .

§ 10.4 Modular arithmetic.

§ The solution of linear congruences.

§ Solution of simultaneous linear congruences. (Chinese remainder theorem).

§ 10.5 Representation of integers in different bases.

§ 10.6 Fermat’s little theorem.

§ 10.7 Graphs, vertices, edges, faces. Adjacent vertices, adjacent edges.

§ Degree of a vertex, degree sequence.

§ Handshaking lemma.

§ Simple graphs; connected graphs; complete graphs; bipartite graphs; planar graphs; trees;

weighted graphs, including tabular representation.

§ Subgraphs; complements of graphs.

§ Euler’s relation: v – e + ƒ = 2 ; theorems for planar graphs including ℯ ≤3v-6, ℯ ≤2v-4, leading to

the results that K₅ and K3,3 are not planar.

§ 10.8 Walks, trails, paths, circuits, cycles.

§ Eulerian trails and circuits.

§ Hamiltonian paths and cycles.

§ 10.9 Graph algorithms: Kruskal’s; Dijkstra’s.

§ 10.10 Chinese postman problem.

§ Not required:

§ Graphs with more than four vertices of odd degree.

§ Travelling salesman problem.

§ Nearest-neighbour algorithm for determining an upper bound.

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§ Deleted vertex algorithm for determining a lower bound.

§ 10.11 Recurrence relations. Initial conditions, recursive definition of a sequence.

§ Solution of first- and second-degree linear homogeneous recurrence relations with constant

coefficients.

§ The first-degree linear recurrence relation un =aun-1+b.

§ Modelling with recurrence relations.

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