The absolute value of a number is its distance from 0 on the number line.
The absolute value of -7 is 7, because it is 7 units away from 0. The absolute value of 5 is 5, because it is 5 units away from 0.
Area is the number of square units that cover a two-dimensional region, without any gaps or overlaps.
For example, the area of region A is 8 square units. The area of the shaded region of B is \(\frac12\) square unit.
The average is another name for the mean of a data set.
For the data set 3, 5, 6, 8, 11, 12, the average is 7.5.
\(3+5+6+8+11+12=45\)
\(45 \div 6 = 7.5\)
We can choose any side of a parallelogram or triangle to be the shape’s base. Sometimes we use the word base to refer to the length of this side.
The word base can also refer to a face of a polyhedron.
A prism has two identical bases that are parallel. A pyramid has one 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. The data is divided into four sections. The sides of the box represent the first and third quartiles. A line inside the box represents the median. Lines outside the box connect to the minimum and maximum values.
For example, this box plot shows a data set with a minimum of 2 and a maximum of 15. The median is 6, the first quartile is 5, and the third quartile is 10.
The center of a set of numerical data is a value in the middle of the distribution. It represents a typical value for the data set.
For example, the center of this distribution of cat weights is between 4.5 and 5 kilograms.
A circle is made out of all the points that are the same distance from a given point.
For example, every point on this circle is 5 cm away from point \(A\), which is the center of the circle.
The circumference of a circle is the distance around the circle. If you imagine the circle as a piece of string, it is the length of the string. If the circle has radius \(r\) then the circumference is \(2\pi r\).
The circumference of a circle of radius 3 is \(2 \boldcdot \pi \boldcdot 3\), which is \(6\pi\), or about 18.85.
A coefficient is a number that is multiplied by a variable.
For example, in the expression \(3x+5\), the coefficient of \(x\) is 3. In the expression \(y+5\), the coefficient of \(y\) is 1, because \(y=1 \boldcdot y\).
Compose means “put together.” We use the word compose to describe putting more than one figure together to make a new shape.
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, because \(2 \boldcdot 3 = 6\), \(3 \boldcdot 3 = 9\), and \(5 \boldcdot 3 = 15\). This means that there are 3 apples for every 1 orange in the fruit salad.
| number of oranges | number of apples |
|---|---|
| 2 | 6 |
| 3 | 9 |
| 5 | 15 |
The coordinate plane is a system for telling where points are. For example. point \(R\) is located at \((3, 2)\) on the coordinate plane, because it is three units to the right and two units up.
We use the word cubed to mean “to the third power.” This is because a cube with side length \(s\) has a volume of \(s \boldcdot s \boldcdot s\), or \(s^3\).
Decompose means “take apart.” We use the word decompose to describe taking a figure apart to make more than one new shape.
The dependent variable is the result of a calculation.
For example, a boat travels at a constant speed of 25 miles per hour. The equation \(d=25t\) describes the relationship between the boat's distance and time. The dependent variable is the distance traveled, because \(d\) is the result of multiplying 25 by \(t\).
When you put money into an account, it is called a deposit.
For example, a person added $60 to their bank account. Before the deposit, they had $435. After the deposit, they had $495, because \(435+60=495\).
A diameter is a line segment that goes from one edge of a circle to the other and passes through the center. A diameter can go in any direction. Every diameter of the circle is the same length. We also use the word diameter to mean the length of this segment.
The distribution tells how many times each value occurs in a data set. For example, in the data set blue, blue, green, blue, orange, the distribution is 3 blues, 1 green, and 1 orange.
Here is a dot plot that shows the distribution for the data set 6, 10, 7, 35, 7, 36, 32, 10, 7, 35.
A double number line diagram uses a pair of parallel number lines to represent equivalent ratios. The locations of the tick marks match on both number lines. The tick marks labeled 0 line up, but the other numbers are usually different.
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 \(5x+2x\). No matter what value we use for \(x\), these expressions are always equal. When \(x\) is 3, both expressions equal 21. When \(x\) is 10, both expressions equal 70.
Two ratios are equivalent if you can multiply each of the numbers in the first ratio 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 \(6\boldcdot\frac12 = 3\).
A recipe for lemonade says to use 8 cups of water and 6 lemons. If we use 4 cups of water and 3 lemons, it will make half as much lemonade. Both recipes taste the same, because \(8:6\) and \(4:3\) are equivalent ratios.
| cups of water | number of lemons |
|---|---|
| 8 | 6 |
| 4 | 3 |
In expressions like \(5^3\) and \(8^2\), the 3 and the 2 are called exponents. They tell you how many factors to multiply. 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, and they are all squares.
The frequency of a data value is how many times it occurs in the data set.
For example, there were 20 dogs in a park. The table shows the frequency of each color.
| color | frequency |
|---|---|
| white | 4 |
| brown | 7 |
| black | 3 |
| multi-color | 6 |
The height is the shortest distance from the base of the shape to the opposite side (for a parallelogram) or opposite vertex (for a triangle).
We can show the height in more than one place, but it will always be perpendicular to the chosen base.
A histogram is a way to represent data on a number line. Data values are grouped by ranges. The height of the bar shows how many data values are in that group.
This histogram shows there were 10 people who earned 2 or 3 tickets. We can't tell how many of them earned 2 tickets or how many earned 3. Each bar includes the left-end value but not the right-end value. (There were 5 people who earned 0 or 1 tickets and 13 people who earned 6 or 7 tickets.)
The independent variable is used to calculate the value of another variable.
For example, a boat travels at a constant speed of 25 miles per hour. The equation \(d=25t\) describes the relationship between the boat's distance and time. The independent variable is time, because \(t\) is multiplied by 25 to get \(d\).
The interquartile range is one way to measure how spread out a data set is. We sometimes call this the IQR. To find the interquartile range we subtract the first quartile from the third quartile.
For example, the IQR of this data set is 20 because \(50-30=20\).
| 22 | 29 | 30 | 31 | 32 | 43 | 44 | 45 | 50 | 50 | 59 |
| Q1 | Q2 | Q3 |
Long division is a way to show the steps for dividing numbers in decimal form. It finds the quotient one digit at a time, from left to right.
For example, here is the long division for \(57 \div 4\).
\(\displaystyle \require{enclose} \begin{array}{r} 14.25 \\[-3pt] 4 \enclose{longdiv}{57.00}\kern-.2ex \\[-3pt] \underline{-4\phantom {0}}\phantom{.00} \\[-3pt] 17\phantom {.00} \\[-3pt]\underline{-16}\phantom {.00}\\[-3pt]{10\phantom{.0}} \\[-3pt]\underline{-8}\phantom{.0}\\ \phantom{0}20 \\[-3pt] \underline{-20} \\[-3pt] \phantom{00}0 \end{array} \)
Long division is an algorithm for finding the quotient of two numbers expressed in decimal form. It works by building up the quotient one digit at a time, from left to right. Each time you get a new digit, you multiply the divisor by the corresponding base ten value and subtract that from the dividend.
Using long division we see that \(513 \div 4 = 128 \frac14\). We can also write this as \(513 = 128 \times 4 + 1\).
\(\displaystyle \require{enclose} \begin{array}{r} 128 \\[-3pt] 4 \enclose{longdiv}{513}\kern-.2ex \\[-3pt] \underline{4{00}} \\[-3pt] 113 \\[-3pt] \underline{\phantom{0}8{0}} \\[-3pt] \phantom{0}33 \\[-3pt] \underline{\phantom{0}32} \\[-3pt] \phantom{00}1 \end{array} \)
The mean is one way to measure the center of a data set. We can think of it as a balance point. For example, for the data set 7, 9, 12, 13, 14, the mean is 11.
To find the mean, add up all the numbers in the data set. Then, divide by how many numbers there are. \(7+9+12+13+14=55\) and \(55 \div 5 = 11\).
The mean absolute deviation is one way to measure how spread out a data set is. Sometimes we call this the MAD. For example, for the data set 7, 9, 12, 13, 14, the MAD is 2.4. This tells us that these travel times are typically 2.4 minutes away from the mean, which is 11.
To find the MAD, add up the distance between each data point and the mean. Then, divide by how many numbers there are.
\(4+2+1+2+3=12\) and \(12 \div 5 = 2.4\)
Measurement error is the positive difference between a measured amount and the actual amount.
For example, Diego measures a line segment and gets 5.3 cm. The actual length of the segment is really 5.32 cm. The measurement error is 0.02 cm, because \(5.32-5.3=0.02\).
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.
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 two numbers in the middle. The median is the average of these two numbers. \(6+8=14\) and \(14 \div 2 = 7\).
A negative number is a number that is less than zero. On a horizontal number line, negative numbers are usually shown to the left of 0.
A net is a two-dimensional figure that can be folded to make a polyhedron.
Here is a net for a cube.
Two numbers are opposites if they are the same distance from 0 and on different sides of the number line.
For example, 4 is the opposite of -4, and -4 is the opposite of 4. They are both the same distance from 0. One is negative, and the other is positive.
For each side of a triangle, there is one vertex that is not on that side. This is the opposite vertex.
For example, point \(A\) is the opposite vertex to side \(BC\).
The origin is the point \((0,0)\) in the coordinate plane. This is where the horizontal axis and the vertical axis cross.
A parallelogram is a type of quadrilateral that has two pairs of parallel sides.
Here are two examples of parallelograms.
The word per means “for each.” For example, if the price is $5 per ticket, that means you will pay $5 for each ticket. Buying 4 tickets would cost $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. We can say that a quarter is worth 25% of a dollar.
A percentage is a rate per 100.
For example, a fish tank can hold 36 liters. Right now there is 27 liters of water in the tank. The percentage of the tank that is full is 75%.
A percentage decrease tells how much a quantity went down, expressed as a percentage of the starting amount.
For example, a store had 64 hats in stock on Friday. They had 48 hats left on Saturday. The amount went down by 16.
This was a 25% decrease, because 16 is 25% of 64.
A percentage increase tell how much a quantity went up, expressed 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.
Percent error is a way to describe error, expressed as a percentage of the actual amount.
For example, a box is supposed to have 150 folders in it. 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.
There is a proportional relationship between the diameter and circumference of any circle. The constant of proportionality is pi. The symbol for pi is \(\pi\).
We can represent this relationship with the equation \(C=\pi d\), where \(C\) represents the circumference and \(d\) represents the diameter.
Some approximations for \(\pi\) are \(\frac{22}{7}\), 3.14, and 3.14159.
A polygon is a closed, two-dimensional shape with straight sides that do not cross each other.
Figure \(ABCDE\) is an example of a polygon.
A polyhedron is a closed, three-dimensional shape with flat sides. When we have more than one polyhedron, we call them polyhedra.
Here are some drawings of polyhedra.
A population is a set of people or things that we want to study.
For example, if we want to study the heights of people on different sports teams, the population would be all the people on the teams.
A positive number is a number that is greater than zero. On a horizontal number line, positive numbers are usually shown to the right of 0.
A prism is a type of polyhedron that has two bases that are identical copies of each other. The bases are connected by rectangles or 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 \(\frac45\).
In a proportional relationship, the values for one quantity are each multiplied by the same number to get the values for the other quantity.
For example, in this table every value of \(p\) is equal to 4 times the value of \(s\) on the same row.
We can write this relationship as \(p = 4s\). This equation shows that \(s\) is proportional to \(p\).
| \(s\) | \(p\) |
|---|---|
| 2 | 8 |
| 3 | 12 |
| 5 | 20 |
| 10 | 40 |
A pyramid is a type of polyhedron that has one base. All the other faces are triangles, and they all meet at a single vertex.
Here are some drawings of pyramids.
The coordinate plane is divided into 4 regions called quadrants. The quadrants are numbered using Roman numerals, starting in the top right corner.
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 that each have the same number of values.
For example, in this data set the first quartile is 30. The second quartile is the same thing as the median, which is 43. The third quartile is 50.
| 22 | 29 | 30 | 31 | 32 | 43 | 44 | 45 | 50 | 50 | 59 |
| Q1 | Q2 | Q3 |
A radius is a line segment that goes from the center to the edge of a circle. A radius can go in any direction. Every radius of the circle is the same length. We also use the word radius to mean the length of this segment.
For example, \(r\) is the radius of this circle with center \(O\).
Outcomes of a chance experiment are random if they are all equally likely to happen.
The range is the distance between the smallest and largest values in a data set. For example, for the data set 3, 5, 6, 8, 11, 12, the range is 9, because \(12-3=9\).
A ratio is an association between two or more quantities.
For example, the ratio \(3:2\) could describe a recipe that uses 3 cups of flour for every 2 eggs, or a boat that moves 3 meters every 2 seconds. One way to represent the ratio \(3:2\) is with a diagram that has 3 blue squares for every 2 green squares.
A rational number is a fraction or the opposite of a fraction.
For example, 8 and -8 are rational numbers because they can be written as \(\frac81\) and \(\text-\frac81\).
Also, 0.75 and -0.75 are rational numbers because they can be written as \(\frac{75}{100}\) and \(\text-\frac{75}{100}\).
Dividing 1 by a number gives the reciprocal of that number. For example, the reciprocal of 12 is \(\frac{1}{12}\), and the reciprocal of \(\frac25\) is \(\frac52\).
A region is the space inside of a shape. Some examples of two-dimensional regions are inside a circle or inside a polygon. Some examples of three-dimensional regions are the inside of a cube or the inside of a sphere.
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.
For example, 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 its distribution resembles the population's distribution in center, shape, and spread.
For example, this dot plot represents a population.
This dot plot shows a sample that is representative of the population.
We use the words same rate to describe two situations that have equivalent ratios.
For example, a sink is filling with water at a rate of 2 gallons per minute. If a tub is also filling with water at a rate of 2 gallons per minute, then the sink and the tub are filling at the same rate.
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:
| heads-heads | tails-heads |
| heads-tails | tails-tails |
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.
For example, 7 is the solution to the equation \(m+1=8\), because it is true that \(7+1=8\). The solution to \(m+1=8\) is not 9, because \(9+1 \ne 8\).
The spread of a set of numerical data tells how far apart the values are.
For example, the dot plots show that the travel times for students in South Africa are more spread out than for New Zealand.
We use the word squared to mean “to the second power.” This is because a square with side length \(s\) has an area of \(s \boldcdot s\), or \(s^2\).
The surface area of a polyhedron is the number of square units that covers all the faces of the polyhedron, without any gaps or overlaps.
For example, if the faces of a cube each have an area of 9 cm2, then the surface area of the cube is \(6 \boldcdot 9\), or 54 cm2.
A table organizes information into horizontal rows and vertical columns. The first row or column usually tells what the numbers represent.
For example, here is a table showing the tail lengths of three different pets. This table has four rows and two columns.
| pet | tail length (inches) |
|---|---|
| dog | 22 |
| cat | 12 |
| mouse | 2 |
A tape diagram is a group of rectangles put together to represent a relationship between quantities.
For example, this tape diagram shows a ratio of 30 gallons of yellow paint to 50 gallons of blue paint.
If each rectangle were labeled 5, instead of 10, then the same picture could represent 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. For example, the expression \(5x + 18\) has two terms. The first term is \(5x\) and the second term is 18.
The unit price is the cost for one item or for one unit of measure. For example, if 10 feet of chain link fencing cost $150, then 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 \(2 \div 12 = \frac16\).
A variable is a letter that represents a number. You can choose different numbers for the value of the variable.
For example, 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 \(10-6=4\).
When you take money out of an account, it is called a withdrawal.
For example, a person removed $25 from their bank account. Before the withdrawal, they had $350. After the withdrawal, they had $325, because \(350-25=325\).