Numbers and Beyond numbers: Praxis Math: the Basics

Praxis Math: the Basics, Length: 90 minutes, Format: 56 questions

Multiple choice questions that ask you to select one answer or one or more answers

On-screen calculator available (but not in Khan Academy's practice system)

The Core Academic Skills for Educators test in Mathematics measures academic skills in mathematics needed to prepare successfully for a career in education.

All skills assessed have been identified as needed for college and career readiness, in alignment with the Common Core State Standards for Mathematics.

The Praxis Math test covers four major content areas:

#1 Number and Quantity

Understanding of order among fractions
Representation of a number in more than one way
Place value
Properties of whole numbers
Counterexamples
Ratios, rates, and percents

#2 Algebra and functions

Manipulating expressions and equations
Solving linear and quadratic equations
Solving algebraic word problems
Creating expressions and equations from verbal descriptions

#3 Geometry

Characteristics and properties of geometric shapes
Congruence and similarity
Angle relationships
Circles

#4 Statistics and Probability

Reading and interpreting visual display of quantitative information
Understanding the correspondence between data and graph
Making inferences from a given data display
Determining mean, median, and mode
Assigning a probability to an outcome

Number and Quantity

#1 Rational number operations

What are rational number operations?

Rational numbers include integers, fractions, and decimals.

For integers and decimals, we can rely on our calculators to add, subtract, multiply, and divide them. However, basic calculators can't give us answers in fractions. Therefore, we need to be able to perform fraction operations by hand.

What skills are tested?
Writing equivalent fractions
Multiplying fractions
Dividing fractions
Adding and subtracting fractions
Performing operations on a mix of integers, fractions, and decimals

#2 What are ratios and proportions?

It is just equivalent fractions. It is about property of 1.

A ratio is a comparison of two quantities. The ratio of a to b can also be expressed as a:b or a/b

A proportion is an equality of two ratios. We write proportions to help us establish equivalent ratios and solve for unknown quantities.

Equivalent ratios are ratios that have the same value. Given a ratio, we can generate equivalent ratios by multiplying both parts of the ratio by the same value.

What skills are tested?

Identifying and writing equivalent ratios
Solving word problems involving ratios
Solving word problems using proportions

How do we write ratios?

Two common types of ratios we'll see are part to part and part to whole. part:whole =part:sum of all parts

To write a ratio:

Determine whether the ratio is part to part or part to whole.
Calculate the parts and the whole if needed.
Plug values into the ratio.
Simplify the ratio if needed. Integer-to-integer ratios are preferred.

I did algebraically 3 e = 5 t, 7 t = 3b, 21 e = 35 t, 35 t = 15b, 21e = 15b, 7e = 5 b

#3 Percentages

What are percentages?

A percentage is a ratio whose second term is 100. Percent means parts per hundred. Percentages are useful because we tend to have a better intuitive understanding of something out of 100 than fractions and decimals. That's good because in the real world, we're surrounded by percent calculations: a waiter's tip, income tax, results of surveys, etc.

#4 Rate

What's the connection between rates and proportional relationships?

Since rates are ratios, we can apply them to different quantities. To do so, we can either multiply the unit rate by the new quantity or find the desired quantity by solving a proportional relationship.

#5 Naming and ordering numbers

#6 Number concepts

Factors are whole numbers that divide evenly (no remainder) into another whole number.

Factoring a number means listing all its factors. The factors are usually presented in a list ordered from least to greatest.

What is prime factorization?

Prime numbers are whole numbers greater than 1 whose only factors are 1 and itself. For example, 17 is prime because it is only divisible by 1 and 17.

Prime factorization means writing a number as a product of factors that are all prime numbers. Prime factorization is helpful when finding the greatest common factor or least common multiple, but is not the only way to do it.

What are multiples?

A multiple is a number that results when we multiply a whole number by another non-zero whole number.

To find the first few multiples of a number, multiply the number by whole numbers starting with 1.

The greatest common factor (or greatest common divisor) of two numbers is the largest whole number that both numbers are divisible by.

The least common multiple of two numbers is the smallest whole number that is divisible by both numbers

#7 Counterexamples

What is a counterexample?

A mathematical statement is a sentence that is either true or false. Showing that a mathematical statement is true requires a formal proof. However, showing that a mathematical statement is false only requires finding one example where the statement isn't true. Such an example is called a counterexample because it's an example that counters, or goes against, the statement's conclusion.

#8 Pre-algebra word problems

Pre-algebra word problems are questions that require translating sentences to expressions, then evaluating those expressions.The expressions we need to write will involve numbers and operations, but no variables.

What is unit reasoning?

How do we convert between measurement units?

We can convert measurements from one unit to another using conversion factors.

How do we use scale factors?

Architects use blueprints to construct large buildings, and engineers use schematics to build electronics with tiny components. In both cases, the plans are not the same size as the objects, but rather scaled representations of the objects. A scale factor is a relationship we use to translate between representation and reality. For example, each inch on a map represents a certain number of miles in the real world.

Unlike a conversion factor, which uses two equivalent measurements with different units, a scale factor can be chosen when connecting representation to reality.

Geometry

#1 Properties of shapes

Triangles are polygons with three sides and three interior angles.

Isosceles triangles have two sides with the same length. The two angles opposite these two sides have the same measure. 相同的角度

Equilateral triangles have three sides with the same length. Each interior angle of an equilateral triangle measures 60, degrees.

Quadrilaterals are polygons with four sides and four interior angles.

Parallelograms are quadrilaterals with two pairs of parallel sides and two pairs of angles with the same measure. The opposite sides have the same length, and adjacent angles are supplementary. (add up to 180)

Rectangles are parallelograms with four 90 degrees angles. The adjacent sides are perpendicular. While all rectangles are parallelograms, not all parallelograms are rectangles.

Squares are parallelograms with four sides of equal length and four degrees angles. While all squares are both rectangles and parallelograms, not all parallelograms are squares and not all rectangles are squares.

#2 angles=

An angle is formed by two lines, line segments, or rays diverging from a vertex.

Acute angles measure less than 90, degrees.

Right angles measure 90, degrees.

Obtuse angles measure greater than 90, degrees and less than 180, degrees.

Straight angles measure 180, degrees.

For a polygon with n sides, the sum of its interior angle measures is equal to:(n−2)×180. For example, for a four-sided polygon such as a square or a rectangle, the sum of interior angles is: (4−2)×180 =360

From the above diagram, the pair of corresponding angles are:

∠a and ∠e, ∠b and ∠g, ∠d and ∠f, ∠c and ∠h

对顶角（vertical angles, opposite angles) always equal 两直线相交，对顶角相等。

同位角, corresponding angles ∠3和 ∠7, ∠4和 ∠8, ∠1和 ∠5, ∠2和∠ 6,

内错角: alternate interior angles, ∠2 和 ∠8, ∠3 和 ∠5

外错角: alternate exterior angles, ∠4 和 ∠6, ∠1 和 ∠7

同旁内角 ∠3和 ∠8 ∠2和 ∠5

同旁外角 ∠1和 ∠6 ∠4和 ∠7

#3 Congruence and similarity 相等相似 any side is also congruent.

Congruent triangles have both the same shape and the same size. they have the same angle measures and the same side lengths.

Similar triangles have the same shape, but not necessarily the same size. they have the same angle measures, but not the same side lengths.

If two objects are congruent, then they are also similar.

Two triangles are congruent if they meet one of the following criteria.

SSS: All three pairs of corresponding sides are equal.
SAS: Two pairs of corresponding sides and the corresponding angles between them are equal.
ASA: Two pairs of corresponding angles and the corresponding sides between them are equal.
AAS: Two pairs of corresponding angles and one pair of corresponding sides (not between the angles) are equal.

Two triangles are similar if they meet one of the following criteria.

AA: Two pairs of corresponding angles are equal.
SSS: Three pairs of corresponding sides are proportional.
SAS: Two pairs of corresponding sides are proportional and the corresponding angles between them are equal.

NOT ASS or SSA

#4 Circles

Circles are round. They don't have any corners. All of the points on a circle are the same distance from the center.

That distance is called the radius, and it's usually represented by r. The diameter, represented by d, is twice as long as the radius.

The circumference of a circle is the distance around it. It's usually represented by C. The area of a circle is the amount of flat space inside the circle's circumference.

#5 Perimeter, area, and volume

The perimeter, P, of a polygon is the total length around the polygon's edges. When we add up all the side lengths of a polygon, we get its perimeter.

The area, A, of a polygon is the amount of flat space inside the polygon's edges. Area is measured in square units such as square miles.

The volume, V, of an object is the amount of space that object occupies. Volume is measured in cubic units such as cubic centimeters.

I only think as 1/6 of the whole volume. Never think of I can draw height 1/6, that is 4 inches tall.

Statistics and probability

#1 Data representations

We collect both qualitative data and quantitative data. However, long lists of data points can be difficult to interpret.

Data representations are graphics that display and summarize data and help us to understand the data's meaning.

Data representations can help us answer the following questions:

How much of the data falls within a specified category or range of values?
What is a typical value of the data?
How much spread is in the data?
Is there a trend in the data over time?
Is there a relationship between two variables?

Don't try to calculate mean (average) by adding them up and divide, try to detect a pattern.

#2 Center and spread

Center describes a typical value of a data point. Two measures of center are mean and median

Spread describes the variation of the data. Two measures of spread are range and standard deviation.

Range measures the total spread of the data. range=highest value−lowest value

It's stupid for me to write 1, 2.5, 2.5, 3, 3, 3, 3,5, 4.5, 6. Khan solution is smarter.

#3 Random sampling

A sample provides information about a population without having to survey the entire group.

To make valid conclusions about a population, we need a sample that recreates the characteristics of the entire population on a smaller scale.

A good sample is representative and random.

Representative means that the sample includes only members of the population being studied.
Random means that every member of the population being studied has an equal chance to be selected for the sample.

#4 Scatterplots

A scatterplot displays data about two variables as a set of points in the xy-plane. A scatterplot is a key tool to determine if there is a relationship between the values of two variables.

Positive correlation: As x increases, y tends to increase.

Negative correlation: As x increases, y tends to decrease.

No correlation: As x increases, y stays about the same or has no clear pattern.

Linearity describes whether or not the trend of the dots in a scatterplot can be approximated by a line.

In a linear relationship (as x increases, y changes at a constant rate) data points tend to fall along a line. The scatterplot below shows a linear relationship with a line of best fit illustrating how the data might be approximated.

In a nonlinear relationship (as x increases, y changes at varing rate), data points do not fall along a line. The scatterplot below shows a nonlinear relationship with a curve illustrating how the data might be approximated.

#5 Interpreting linear models

A line of best fit can be estimated by drawing a line so that the number of points above and below the line is about equal. While a line of best fit is not an exact representation of the actual data, it is a useful model that helps us interpret the data and make estimates.

How do we determine the equation of a line of best fit?

A line of best fit usually shows two key features.

The y-intercept, b, is the y-value when x=0.

The slope, m, is the change in y when xxx increases by 1.

#6 Correlation and Causation

Correlation means there is a relationship or pattern between the values of two variables. A scatterplot displays data about two variables as a set of points in the xy-plane and is a useful tool for determining if there is a correlation between the variables.

Causation means that one event causes another event to occur. Causation can only be determined from an appropriately designed experiment. In such experiments, similar groups receive different treatments, and the outcomes of each group are studied. We can only conclude that a treatment causes an effect if the groups have noticeably different outcomes.

//这种题我一般是英文差，读不懂题目。

Data from a certain city shows that the size of an individual's home is positively correlated with the individual's life expectancy. Which of the following factors would best explain why this correlation does not necessarily imply that the size of a individual's home is the main cause of increased life expectancy?

In this type of question, we are looking for a choice that explains why the relationship between home size and life expectancy isn't causal even though the variables are correlated

Let's consider each choice:

Choice (A): Larger homes have more safety features and amenities, which lead to increase in life expectancy.

This choice supports, rather than refutes, the claim that increased home size is the cause of increased life expectancy. This choice is not correct.

Choice (B): The ability to afford a larger home and better healthcare is a direct effect of having more wealth.

This choice says that wealth—not the size of a person's home— is the main cause of increased life expectancy.

We wouldn't expect someone to live longer simply because they moved to a larger house. Rather, wealth allows people to afford both a larger home and better health care and is the underlying cause of the correlation between these variables. This choice is correct.

Choice (C): The citizens were not selected at random for the study.

This choice provides information about the sampling method, but doesn't explain why increased home size isn't the cause of increased life expectancy.

This choice is not correct.

Choice (D): There are more people living in small homes than large homes in the city.

This choice provides information about the population but doesn't explain why increased home size isn't the cause of increased life expectancy.

This choice is not correct.

Choice (E): Some responses may have been counted twice during the data collection process.

This choice mentions an error in data handling but doesn't explain why increased home size isn't the cause of increased life expectancy.

This choice is not correct.

The factor that best explains why this correlation does not necessarily imply that the size of a individual's home is the main cause of increased life expectancy is:

The ability to afford a larger home and better healthcare is a direct effect of having more wealth.

#7 Probability

A probability indicates the chance that an event will happen.

A probability can be any number from 0 to 1.
A probability of 0 or 0%=means that an event will never happen.
A probability of 1/2 or 50% means that an event is equally likely to happen or not happen.

A probability of 1 or 100% means that an event will certainly happen.

The probability that event A will happen is written as P(A), left parenthesis, A, right parenthesis.

The probability that event A will not happen, P(notA) is equal to 1 - P(A).

Algebra

#1 Algebraic properties

#2 Solution procedures

Addition and subtraction are inverse operations (operations that "undo" each other)
Multiplication and division are inverse operations.

A flowchart is a visual representation of a sequence of operations. Each operation is applied to the result of the previous operation.

To find the result of a flowchart given the input, carry out all of the operations in the flowchart in order. To find the input of a flowchart given the result, find the inverse operation for each operation in the flowchart, then carry out all of the inverse operations in reverse order starting with the result.

What's a recursive sequence?

In a recursive sequence, the terms beyond the first one(s) are generated based on the values of the preceding terms. This means carrying out operations on the previous term to calculate the next term.

For each subsequent term, the operations remain the same, but the input changes, e.g., the 2nd term is calculated using the 1st term, the 3rd term is calculated using the 2nd term, etc.

#3 Equivalent expressions

Equivalent expressions are expressions that work the same even though they look different. If two algebraic expressions are equivalent, then the two expressions have the same value when we plug in the same value for the variable.

Just as we can add, subtract, multiply, and divide constants, we can do so with variables. To isolate a specific variable, perform the same operations on both sides of the equation until the variable is isolated. The new equation is equivalent to the original equation.

#4 Creating expressions and equations