Area is the number of square units that cover a two-dimensional region without gaps or overlaps.

• The area of region A is 8 square units.
• The area of the shaded region of B is \(\frac12\) square unit.

The area of a circle is the number of square units it covers. If a circle has radius \(r\) units, its area is \(\pi r^2\) square units.

For example, a circle has radius of 3 inches. Its area is \(\pi 3^2\), or \(9\pi\), square inches. This is about 28.3 square inches.

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 chance experiment is something that can be done over and over again, and what will happen each time is not known.

For example, each time the spinner is spun, it could land on red, yellow, blue, or green.

The circumference of a circle is the distance around the circle. If a circle has radius \(r\) units, its circumference is \(2\pi r\) units.

For example, a circle has a radius of 3 inches. Its circumference is \(2 \boldcdot \pi \boldcdot 3\), or \(6\pi\) inches. This is about 18.85 inches.

Complementary angles have measures that add up to 90\(^\circ\).

For example, a \(15^\circ\) angle and a \(75^\circ\) angle are complementary.

In a proportional relationship, the values for one quantity are each multiplied by the same number to get the values for the other quantity. This number is called the constant of proportionality.

In this example, the constant of proportionality is 3.

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 \((0,0)\).

Corresponding parts are the parts that match up between a figure and its scaled copy. They have the same relative position. Points, segments, angles, or distances can be corresponding.

Point \(B\) in the first triangle corresponds to point \(E\) in the second triangle. Segment \(AC\) corresponds to segment \(DF\).

A cross-section is the new face that is seen when a three-dimensional figure is sliced.

A rectangular pyramid is sliced parallel to the base. The cross-section is a smaller rectangle.

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A deposit is when money is put into an account.

For example, a person added \$60 to their bank account. Before the deposit, they had \$435. After the deposit, they had \$495, because \(435+60=495\).

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 \(6\boldcdot\frac12 = 3\).

An event is a set of 1 or more outcomes in a chance experiment.

For example, if a number cube is rolled, there are 6 possible outcomes.

Some events are “rolling a number less than 3,” “rolling an even number,” or “rolling a 5.”

To expand an expression, use the distributive property to rewrite a product as a sum. The new expression is equivalent to the original expression.

For example, the expression \(5(4x+7)\) can be expanded to get the equivalent expression \(20x + 35\).

To factor an expression, use the distributive property to rewrite a sum as a product. The new expression is equivalent to the original expression.

For example, the expression \(20x + 35\) can be factored to get the equivalent expression \(5(4x+7)\).

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 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.

Measurement error is the positive difference between a measured amount and the actual amount.

For example, Diego measures a line segment as 5.30 cm. The actual length is 5.32 cm. The measurement error is 0.02 cm, because \(5.32-5.30=0.02\).

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.

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.

An outcome of a chance experiment is one of the things that can happen.

For example, the possible outcomes of tossing a coin are heads and tails.

A percent decrease tells how much a quantity went down. It is written as a percentage of the starting amount.

For example, a store had 64 hats on Friday. It had 48 hats left on Saturday. The number of hats went down by 16.

This was a 25% decrease, because 16 is 25% of 64.

Percent error is a way to describe error as a percentage of the actual amount.

For example, a box is supposed to have 150 folders. Clare counts only 147 folders in the box. This is an error of 3 folders. The percent error is 2%, because 3 is 2% of 150.

A percent increase tells how much a quantity went up. It is written as a percentage of the starting amount.

For example, Elena had \$50 in the bank on Monday. She had \$56 on Tuesday. The amount went up by \$6.

This was a 12% increase, because 6 is 12% of 50. 

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 population is a set of people or things that is studied.

For example, someone wants to study the heights of people on different sports teams. The population is all the people on all of the teams.

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.

The probability of an event is a number that tells how likely it is to happen. A probability of 1 means the event will always happen. A probability of 0 means the event will never happen.

For example, the probability of selecting a moon block at random from this bag is \(\frac45\).

A proportion of a data set is the fraction of the data in a given category.

For example, a class has 20 students. There are 2 left-handed students and 18 right-handed students in the class. The proportion of students who are left-handed is \(\frac{2}{20}\), or 0.1.

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.

An outcome of a chance experiment is random if it cannot be predicted. For example, the result of rolling a standard number cube is random because it could result in 1, 2, 3, 4, 5, or 6, and nobody can know which value it will be until it is rolled.

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 repeating decimal has digits that keep going in the same pattern over and over. The repeating digits are marked with a line above them.

• The decimal representation for \(\frac13\) is \(0.\overline{3}\), which means 0.3333333 . . . .
• The decimal representation for \(\frac{25}{22}\) is \(1.1\overline{36}\), which means 1.136363636 . . . .

A sample is representative of a population if they match closely. They are very alike in shape, center, and spread.

This dot plot shows the distribution for a population.

A dot plot labeled dollars per pound of catfish. The numbers 1.6 through 2.8 in increments of zero point 2 are indicated. The data are as follows: 1 point 6 dollars per pound, 4 dots. 1 point 7 dollars per pound, 20 dots. 1 point 8 dollars per pound, 24 dots. 1 point 9 dollars per pound, 2 dots. 2 dollars per pound, 10 dots. 2 point 1 dollars per pound, 2 dots. 2 point 2 dollars per pound, 6 dots. 2 point 3 dollars per pound, 4 dots. 2 point 4 dollars per pound, 4 dots. 2 point 5 dollars per pound, 6 dots. 2 point 6 dollars per pound, 10 dots. 2 point 7 dollars per pound, 4 dots. 2 point 8 dollars per pound, 4 dots.

This dot plot shows the distribution of a sample that is representative of the population.​​​​​

A dot plot labeled dollars per pound of catfish. The numbers 1.6 through 2.8 in increments of zero point 2 are indicated. The data, titled Sample 1, are as follows: 1 point 6 dollars per pound, 1 dots. 1 point 7 dollars per pound, 4 dots. 1 point 8 dollars per pound, 5 dots. 2 dollars per pound, 2 dots. 2 point 2 dollars per pound, 2 dots. 2 point 4 dollars per pound, 1 dot. 2 point 5 dollars per pound, 2 dots. 2 point 6 dollars per pound, 2 dots. 2 point 7 dollars per pound, 1 dot. 2 point 8 dollars per pound, 1 dot

A right angle is half of a straight angle. It measures 90 degrees.

A sample is part of a population. For example, a population could be all the seventh grade students at one school. One sample of that population is all the seventh grade students who are in band.

The sample space is the list of every possible outcome for a chance experiment.

For example, the sample space for tossing two coins is:

A scale tells how the measurements in a scale drawing represent the actual measurements.

The scale on this floor plan tells us that 1 inch on the drawing represents 8 feet in the actual room. This means that 2 inches represent 16 feet, and \(\frac12\) inch represents 4 feet.

A scale drawing represents an actual place or object. All the measurements in the drawing correspond to the measurements of the actual object by the same scale.

To create a scaled copy of a figure, all the side lengths in the original figure are multiplied by the same number. This number is called the scale factor.

In this example, the scale factor is 1.5, because \(4 \boldcdot (1.5) = 6\), \(5 \boldcdot (1.5)=7.5\), and \(6 \boldcdot (1.5)=9\).

A scaled copy is a copy of a figure where every side length in the original figure is multiplied by the same number.

Triangle \(DEF\) is a scaled copy of triangle \(ABC\). Each side length on triangle \(ABC\) is multiplied by 1.5 to get the corresponding side length on triangle \(DEF\).

A simulation is an experiment that is used to estimate the probability of a real-world event.

Suppose the weather forecast says there is a 25% chance of rain. This situation can be simulated with a spinner that has four equal sections. If the spinner stops on red, it represents rain. If the spinner stops on any other color, it represents no rain.

A circular spinner divided into 4 equal parts. The top left section is red and labeled “R.” The top right section is blue and labeled “B.” The bottom right section is green and labeled “G.” The bottom left section is yellow and labeled “Y.” The spinner dial points to the section labeled "G."

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.

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 straight angle is an angle that forms a straight line. It measures 180 degrees.

Supplementary angles have measures that add up to 180\(^\circ\).

For example, a \(15^\circ\) angle and a \(165^\circ\) angle are supplementary.

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.

Terms are the parts of an expression that are added together. They 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 \(5x+3-18\) has 3 terms: \(5x\), 3, and -18.

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 \(2 \div 12 = \frac16\).

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 \(10-6=4\).

Vertical angles are opposite angles that share the same vertex. They are formed when two lines cross each other. Their angle measures are equal.

Angles \(AEC\) and \(DEB\) are vertical angles. If angle \(AEC\) measures \(120^\circ\), then angle \(DEB\) must also measure \(120^\circ\).

Angles \(AED\) and \(BEC\) are another pair of vertical angles.

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.

A withdrawal is when money is taken out of an account.

For example, a person took out \$25 from their bank account. Before the withdrawal, they had \$350. After the withdrawal, they had \$325, because \(350-25=325\).