The average is another name for the mean of a data set. To find the average, add all the numbers in the data set. Then divide by how many numbers there are.

• Data set: 3, 5, 6, 8, 11, 12
• \(3+5+6+8+11+12=45\)
• \(45 \div 6 = 7.5\)

The average is 7.5.

Any side of a parallelogram or triangle can be chosen its base. The length of this side is also called the base.

A base is a specific face of a prism or pyramid.

A prism has 2 identical bases that are parallel. A pyramid has 1 base.

A prism or pyramid is named for the shape of its base.

A box plot is a way to represent data on a number line with a box and some lines. The data is divided into four sections by 5 values. Those values are the minimum, first quartile, median, third quartile, and maximum.

A set of categorical data has values that are words instead of numbers.

For example, Han asks 5 friends to each name their favorite color. Their answers are “blue,” “blue,” “green,” “blue,” and “orange.”

The center of a data set is a value in the middle. It represents a typical value for the data set.

The center of this data set is between 4.5 and 5 kilograms. So a typical cat in this group weighs between 4.5 and 5 kilograms.

A coefficient is a number that is multiplied by a variable.

• In the expression \(3x+5\), the coefficient of \(x\) is 3.
• In the expression \(y+5\), the coefficient of \(y\) is 1, because \(y=1 \boldcdot y\).

A common factor of two numbers is a number that evenly divides both numbers. 

• The factors of 15 are 1, 3, 5, and 15.
• The factors of 20 are 1, 2, 4, 5, 10, and 20.

The common factors of 15 and 20 are 1 and 5.

A common multiple of two numbers is a product that results from multiplying each of the two numbers by some whole number. 

• The multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, . . . .
• The multiples of 5 are 5, 10, 15, 20, 25, 30, 35, 40, . . . .

The common multiples of 3 and 5 are 15, 30, 45, 60, . . . .

The word compose means “put together.” To compose a shape, put figures together to make a new shape.

When an object moves at a constant speed, it moves at the same speed at all times. It does not move faster or slower at any time. So, the ratios of distance traveled to elapsed time are always equivalent.

For example, a car moves at a constant speed of 50 miles per hour. This means it travels 50 miles in 1 hour, 100 miles in 2 hours, and 150 miles in 3 hours. The ratios of distance in miles to time in hours are equivalent.

The coordinate plane is one way to represent pairs of numbers. The plane is made of a horizontal number line and a vertical number line that cross at 0.

Pairs of numbers can be used to describe the location of a point in the coordinate plane.

Point \(R\) is located at \((3,\text-2)\). This means \(R\) is 3 units to the right and 2 units down from \((0,0)\).

The word cubed means “to the third power.” This is because a cube with side length \(s\) has a volume of \(s \boldcdot s \boldcdot s\), or \(s^3\).

The word decompose means “take apart.” To decompose a shape, take it apart to make more than one new shape.

The distribution of a data set tells how many times each value occurs.

• In the data set blue, blue, green, blue, orange, the distribution is 3 blues, 1 green, and 1 orange.

• This dot plot shows the distribution for the data set 6, 10, 7, 35, 7, 36, 32, 10, 7, 35 kilograms.

  

A dot plot is a way to represent data with dots. Each dot above a number shows one time the value occurs in the set.

A double number line diagram uses a pair of parallel number lines to represent equivalent ratios. There is one number line for each quantity in the ratio. The tick marks for equivalent ratios line up.

This double number line diagram shows that \(3:5\) and \(6:10\) are equivalent ratios.

Equivalent expressions are always equal to each other. If the expressions have variables, they are equal whenever the same value is used for the variable in each expression.

For example, \(3x+4x\) is equivalent to \(5x+2x\).

• When \(x\) is 3, both expressions equal 21.
• When \(x\) is 10, both expressions equal 70.
• When \(x\) is any other number, both expressions still have equal value.

Two ratios are equivalent if each of the numbers in the first ratio can be multiplied by the same factor to get the numbers in the second ratio. For example, \(8:6\) is equivalent to \(4:3\), because \(8\boldcdot\frac12 = 4\) and \(6\boldcdot\frac12 = 3\).

In expressions like \(5^3\) and \(8^2\), the numbers 3 and the 2 are called exponents. They tell how many times a number is used as a factor.

For example, \(5^3\) = \(5 \boldcdot 5 \boldcdot 5\), and \(8^2 = 8 \boldcdot 8\).

Each flat side of a polyhedron is called a face.

For example, a cube has 6 faces that are all squares.

The greatest common factor of two numbers is the largest number that evenly divides both numbers. 

• The factors of 45 are 1, 3, 5, 9, 15, and 45.
• The factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60.

The greatest common factor of 45 and 60 is 15.

The height is the shortest distance from the base of the shape to the opposite side (for a parallelogram) or to the opposite vertex (for a triangle).

The height can be shown in more than one place. It is always perpendicular to the chosen base.

An inequality is a statement that compares two values or expressions using symbols such as “\(>\)” or “\(<\)”. 

• The symbol “\(>\)” means “is greater than.”
• The symbol “\(<\)” means “is less than.”

For example, the inequality \(\text-3 > \text-7\) tells us that -3 is greater than -7.

The interquartile range is one way to measure how spread out a data set is. To find the IQR, subtract the first quartile (Q1) from the third quartile (Q3).

For example, the IQR of this data set is 20 because \(50-30=20\).

The least common multiple of two numbers is the smallest product that results from multiplying each of the two numbers by some whole number.

• The multiples of 6 are 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, . . . .
• The multiples of 10 are 10, 20, 30, 40, 50, 60, 70, 80, . . . .

The least common multiple of 6 and 10 is 30.

The mean is one way to measure the center of a data set. It can be thought of as a balance point. To find the mean, add all the numbers in the data set. Then divide by how many numbers there are.

• Data set: 7, 9, 12, 13, 14
• Add values: \(7+9+12+13+14=55\)
• Divide by number of values: \(55 \div 5 = 11\)

The mean is 11. So, the typical travel time is 11 minutes.

The mean absolute deviation (MAD) is one way to measure how spread out a data set is. To find the MAD, find the distance between each data value and the mean. Add all the distances. Then divide by how many distances there are.

• Data set: 7, 9, 12, 13, 14
• Add distances: \(4+2+1+2+3=12\)
• Divide by number of distances: \(12 \div 5 = 2.4\)

The MAD is 2.4. So, these travel times are typically 2.4 minutes away from the mean of 11 minutes.

A measure of center is a value that seems typical for a data distribution.

Mean and median are both measures of center.

The median is one way to measure the center of a data set. It is the middle number when the data set is listed in order of value.

• For the data set 7, 9, 12, 13, 14, the median is 12.
• For the data set 3, 5, 6, 8, 11, 12, there are 2 numbers in the middle. The median is the average of these 2 numbers. \(6+8=14\), and \(14 \div 2 = 7\). The median is 7.

A negative number is a number that is less than 0.

A set of numerical data has values that are numbers.

For example, Han lists the ages of people in his family: 7, 10, 12, 36, 40, 67.

Two numbers are opposites if they are the same distance from 0 on the number line, but on different sides. One is negative, and the other is positive.

Pace is one way to describe how fast something moves. Pace tells how much time it takes the object to travel a certain distance.

For example, Diego runs at a pace of 10 minutes per mile. Elena runs at a pace of 11 minutes per mile. Elena runs slower than Diego, because it takes her more time to travel the same distance.

A parallelogram is a type of quadrilateral that has 2 pairs of parallel sides.

Here are 2 examples of parallelograms.

The word per means “for each.”

For example, the price is \$5 per ticket. This means \$5 must be paid for each ticket. Buying 4 tickets costs $20, because \(4 \boldcdot 5 = 20\).

The word percent means “for each 100.” The symbol for percent is %.

For example, a quarter is worth 25 cents, and a dollar is worth 100 cents. A quarter is worth 25% of a dollar.

A diagram of two bars with different lengths. The top bar is labeled 1 Quarter and 25 cents is labeled inside the bar. The bottom bar is labeled 1 Dollar. It is 4 times longer than the top bar and 100 cents is labeled inside the bar.

A percentage is a rate per 100.

For example, a fish tank can hold 36 liters. There are 27 liters of water in the tank. The percentage of the tank that is full is 75%.

A polyhedron is a closed, three-dimensional shape with flat sides. When there is more than one polyhedron, they are called polyhedra.

Here are some drawings of polyhedra.

A positive number is a number that is greater than 0.

A prism is a type of polyhedron with 2 bases that are identical and parallel. The bases are connected by parallelograms.

Here are some drawings of prisms.

A pyramid is a type of polyhedron that has 1 base. All the other faces are triangles that meet at a single vertex.

Here are some drawings of pyramids.

The coordinate plane is divided into 4 sections called quadrants. The quadrants are numbered using Roman numerals, as shown.

A quadrilateral is a type of polygon that has 4 sides.

• A rectangle is an example of a quadrilateral.
• A pentagon is not a quadrilateral, because it has 5 sides.

Quartiles are the numbers that divide a data set into four sections. Each section has the same number of data values.

In this data set, the first quartile (Q1) is 30. The second quartile (Q2) is the median, 43. The third quartile (Q3) is 50.

The range is the distance between the smallest and largest values in a data set.

In the data set 3, 5, 6, 8, 11, 12, the range is 9, because \(12-3=9\).

A ratio is a way to relate 2 or more quantities.

For example, the ratio \(3:2\) could describe:

• A recipe that uses 3 cups of flour for every 2 eggs.
• A boat that moves 3 meters every 2 seconds.
• A diagram that has 3 blue squares for every 2 green squares.

A rational number is a number that can be written as a positive fraction, a negative fraction, or zero.

• 8 and -8 are rational numbers because they can be written as \(\frac81\) and \(\text-\frac81\).
• 0.75 and -0.75 are rational numbers because they can be written as \(\frac{75}{100}\) and \(\text-\frac{75}{100}\).

Two numbers that multiply to equal 1 are reciprocals.

• 8 and \(\frac{1}{8}\) are reciprocals because \(8 \boldcdot \frac{1}{8} = 1\).
• \(\frac{2}{5}\) is a reciprocal of \(\frac{5}{2}\) because \(\frac{2}{5} \boldcdot \frac{5}{2} = 1\).

A region is the space inside of a shape.

• Examples of two-dimensional regions are the inside of a circle and the inside of a polygon.
• Examples of three-dimensional regions are the inside of a cube and the inside of a sphere.

The phrase same rate is used to describe two situations that have equivalent ratios.

For example, a sink is filled at a rate of 2 gallons per minute. A tub is also filled at a rate of 2 gallons per minute. So, the sink and the tub are filled at the same rate.

The sign of any number other than 0 is either positive or negative.

For example, the sign of 6 is positive. The sign of -6 is negative. Zero does not have a sign, because it is not positive or negative.

A solution to an equation is a number that can be used in place of the variable to make the equation true.

• The solution to the equation \(m+1=8\) is 7, because it is true that \(7+1=8\).
• The solution to \(m+1=8\) is not 9, because \(9+1\) does not equal 8. 

A solution to an inequality is a number that can be used in place of the variable to make the inequality true.

• A solution to the inequality \(c<10\) is 5, because it is true that \(5<10\).
• Some other solutions to this inequality are 9.9, 0, and -4.

Speed is one way to describe how fast something moves. Speed tells how much distance the object travels in a certain amount of time.

For example, Tyler walks at a speed of 4 miles per hour. Priya walks at a speed of 5 miles per hour. Priya walks faster than Tyler, because she travels more distance in the same amount of time.

The spread of a set of data tells how far apart the values are.

These dot plots show that the travel times for students in South Africa are more spread out than for students in New Zealand.

The word squared means “to the second power.” This is because a square with side length \(s\) has an area of \(s \boldcdot s\), or \(s^2\).

A statistical question can be answered by collecting data that has different values. Here are some examples of statistical questions:

• Who is the most popular musical artist at your school?
• When do students in your class typically eat dinner?
• Which classroom in your school has the most books?

The surface area of a polyhedron is the number of square units that covers all of its faces with no gaps or overlaps.

For example, the 6 faces of a cube each have an area of 9 cm². So, the surface area of the cube is \(6 \boldcdot 9\), or 54 cm².

A tape diagram is a group of rectangles put together to represent a relationship between quantities.

This tape diagram shows a ratio of 30 gallons of yellow paint to 50 gallons of blue paint.

This tape diagram represents the equivalent ratio of 15 gallons of yellow paint to 25 gallons of blue paint.

A term is a part of an expression. It can be a single number, a variable, or a number and a variable that are multiplied together.

• 7, \(y\), and \(9a\) are examples of terms.
• The expression \(5x+3\) has 2 terms: \(5x\) and \(3\).

The unit price is the cost for 1 item or for 1 unit of measure.

For example, 10 feet of fencing costs \$150. The unit price is \(150 \div 10\), or \$15 per foot.

A unit rate is a rate per 1.

For example, 12 people share 2 pies equally. One unit rate is 6 people per pie, because \(12 \div 2 = 6\). The other unit rate is \(\frac16\) of a pie per person, because \(2 \div 12 = \frac16\).

The variability of a data set describes how different the values are.

Data set B has many different values, while data set A has more of the same values. So, data set B has more variability.

A variable is a letter that represents a number. Different numbers can be chosen for the value of the variable.

In the expression \(10-x\), the variable is \(x\).

• If the value of \(x\) is 3, then \(10-x=7\), because \(10-3=7\).
• If the value of \(x\) is 6, then \(10-x=4\), because \(10-6=4\).

Volume is the number of cubic units that fill a three-dimensional region with no gaps or overlaps.

This rectangular prism has 3 layers that are each 20 units³. So, the volume of the prism is 60 units³.