3.9: Summary of Key Concepts
Summary of Key Concepts
Positive and Negative Numbers
A number is denoted as
positive
if it is directly preceded by a "+" sign or no sign at all. A number is denoted as negative if it is directly preceded by a "−" sign.
Opposites
Opposites are numbers that are the same distance from zero on the number line but have opposite signs.
Double-Negative Property
\(−(−a)=a\)
Absolute Value (Geometric)
The absolute value of a number a, denoted |a|, is the distance from a to 0 on the number line.
Absolute Value (Algebraic)
\(|a|=\left\{\begin{array}{cl}
a & \text { if } a \geq 0 \\
-a & \text { if } a<0
\end{array}\right.\)
Addition of Signed Numbers
To add two numbers with
like signs
, add the absolute values of the numbers and associate the common sign with the sum.
unlike signs
, subtract the smaller absolute value from the larger absolute value and associate the sign of the larger absolute value with the difference.
Addition with 0
0+any number=that particular number, that is, 0+a=a for any real number a.
Additive Identity
Since adding 0 to a real number leaves that number unchanged, 0 is called the additive identity.
Definition of Subtraction
\(a−b=a+(−b)\)
Subtraction of Signed Numbers
To perform the subtraction \(a−b\), add the opposite of \(b\) to \(a\), that is, change the sign of b and add.
Multiplication and Division of Signed Numbers
\(\begin{array}{l}
(+)(+)=+\dfrac{(+)}{(+)}=+\dfrac{(+)}{(-)}=- \\
(-)(-)=+ \\
(+)(-)=- \\
(-)(+)=-\dfrac{(-)}{(-)}=+\dfrac{(-)}{(+)}=-
\end{array}\)
Reciprocals
Two numbers are reciprocals of each other if their product is \(1\). The numbers \(4\) and \(\dfrac{1}{4}\) are reciprocals since \((4) (\dfrac{1}{4})=1\).
Negative Exponents
If \(n\) is any natural number and \(x\) is any nonzero real number, then \(x^{-n}=\dfrac{1}{x^n}\).
Writing a Number in Scientific Notation
To write a number in scientific notation:
- Move the decimal point so that there is one nonzero digit to its left.
- Multiply the result by a power of 10 using an exponent whose absolute value is the number of places the decimal point was moved. Make the exponent positive if the decimal point was moved to the left and negative if the decimal point was moved to the right.
Converting from Scientific Notation:
positive exponent
To convert a number written in scientific notation to a number in standard form when there is a
positive
exponent as the power of 10, move the decimal point to the
right
the number of places prescribed by the exponent on the 10.
Negative Exponent
To convert a number written in scientific notation to a number in standard form when there is a
negative
exponent as the power of 10, move the decimal point to the
left
the number of places prescribed by the exponent on the 10.


1.2: Putting It Together- Introduction to Management
At the beginning of this module, we presented examples of how two management teams reimagined their organizations in order to survive in a changing and competitive global environment. After its beginning as a paper mill in 1871, Nokia grew and combined with other firms to incorporate in 1967 as a diversified corporation producing paper, electronics, rubber, and cable. At every opportunity, Nokia took advantage of merging and entering new markets until, by 2000, it was a leader in the telecommunications industry. [1]
A slowdown in the mobile phone market plus a disastrous recall to correct faulty cell phone batteries contributed to a downturn in profit that eventually led to Microsoft’s partnering with and then acquiring Nokia in 2014. [2] In some ways, Nokia’s very success led to its demise. Its top managers failed to recognize the impact the smartphone would have on the mobile phone market. They failed to anticipate their customers’ needs.
Meanwhile, the top management at ThyssenKrupp AG has recognized that to save its organization, the time to change is now. From a major German steel manufacturer, the company leadership is transforming it into an “industrial goods company with a stake in a steel joint venture.” [3] If it succeeds (and the indications are good), then will management continue to adapt as market conditions change?
This module covered the primary management functions needed in all organizations: planning, leading, organizing, and controlling. We looked at four common levels of managers: top, middle, first-line, and team leaders. Finally, we discussed how the managers in an organization or business create a competitive advantage by managing people well.
- The rise, dominance, and epic fall - a brief look at Nokia's history. (n.d.). Retrieved July 28, 2017, from http://www.gsmarena.com/the_rise_dominance_and_epic_fall__a_brief_look_at_nokias_history-blog-13460.php ↵
- Ibid. ↵
- Andresen, T. (2017, June 19). Germany's Biggest Steelmaker Set to Unveil a Future Beyond Steel. Retrieved July 28, 2017, from https://www.bloomberg.com/news/articles/2017-06-19/germany-s-biggest-steelmaker-set-to-unveil-a-future-beyond-steel ↵
Contributors and Attributions
- Putting It Together: Introduction to Management. Authored by : John and Lynn Bruton. Published by Lumen Learning. License : CC BY: Attribution
- ThyssenKrupp wheelchair lift in Moscow. Authored by : Stolbovsky. Located at : https://commons.wikimedia.org/wiki/File:ThyssenKrupp_wheelchair_lift_in_Moscow_02.jpg . License : CC BY-SA: Attribution-ShareAlike


1.3: Use the Language of Algebra
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Learning Objectives
By the end of this section, you will be able to:
- Use variables and algebraic symbols
- Simplify expressions using the order of operations
- Evaluate an expression
- Identify and combine like terms
- Translate an English phrase to an algebraic expression
Use Variables and Algebraic Symbols
Suppose this year Greg is \(20\) years old and Alex is \(23\). You know that Alex is \(3\) years older than Greg. When Greg was \(12\), Alex was \(15\). When Greg is \(35\), Alex will be \(38\). No matter what Greg’s age is, Alex’s age will always be 3 years more, right? In the language of algebra, we say that Greg’s age and Alex’s age are variables and the \(3\) is a constant. The ages change (“vary”) but the \(3\) years between them always stays the same (“constant”). Since Greg’s age and Alex’s age will always differ by \(3\) years, \(3\) is the constant. In algebra, we use letters of the alphabet to represent variables. So if we call Greg’s age \(g\), then we could use \(g + 3g + 3\) to represent Alex’s age. See Table \(\PageIndex{1}\).
| Greg's age | Alex's age |
|---|---|
| \(12\) | \(15\) |
| \(20\) | \(23\) |
| \(35\) | \(38\) |
| \(g\) | \(g+3\) |
The letters used to represent these changing ages are called variables. The letters most commonly used for variables are \(x, y, a, b,\) and \(c\).
Definition: VARIABLE
A variable is a letter that represents a number whose value may change.
Definition: CONSTANT
A constant is a number whose value always stays the same.
To write algebraically, we need some operation symbols as well as numbers and variables. There are several types of symbols we will be using.
There are four basic arithmetic operations: addition, subtraction, multiplication, and division . We’ll list the symbols used to indicate these operations below (Table \(\PageIndex{2}\)). You’ll probably recognize some of them. \(\require{enclose}\)
| Operation | Notation | Say: | The result is... |
|---|---|---|---|
| Addition | \(a+b\) | \(a\) plus \(b\) | the sum of \(a\) and \(b\) |
| Subtraction | \(a−b\) | \(a\) minus \(b\) | the difference of \(a\) and \(b\) |
| Multiplication | \(a·b,ab,(a)(b),(a)b,a(b)\) | \(a\) times \(b\) | the product of \(a\) and \(b\) |
| Division | \(a\div{b}, a/b,\dfrac{a}{b}, b \enclose{longdiv}{a}\) | \(a\) divided by \(b\) | the quotient of \(a\) and \(b\), \(a\) is called the dividend, and \(b\) is called the divisor |
We perform these operations on two numbers. When translating from symbolic form to English, or from English to symbolic form, pay attention to the words “of” and “and.”
- The difference of \(9\) and \(2\) means subtract \(9\) and \(2\), in other words, \(9\) minus \(2\), which we write symbolically as \(9−2\).
- The product of \(4\) and \(8\) means multiply \(4\) and \(8\), in other words \(4\) times \(8\), which we write symbolically as \(4\cdot 8\).
In algebra, the cross symbol, \(\times\), is not used to show multiplication because that symbol may cause confusion. Does \(3xy\) mean \(3\times y\) (‘three times \(y\)’) or \(3\cdot x \cdot y\) (three times \(x\) times \(y\))? To make it clear, use \(\cdot\) or parentheses for multiplication.
When two quantities have the same value, we say they are equal and connect them with an equal sign .
EQUALITY SYMBOL
\(a = b\) is read “\(a\) is equal to \(b\)”
The symbol \(“=”\) is called the equal sign .
On the number line, the numbers get larger as they go from left to right. The number line can be used to explain the symbols \(“<”\) and \(“>"\).
INEQUALITY
\(a<b\) is read “\(a\) is less than \(b\)”
\(a\) is to the left of \(b\) on the number line
\(a>b\) is read "\(a\) is greater than \(b\)”
\(a\) is to the right of \(b\) on the number line
The expressions \(a < b\) or \(a > b\) can be read from left to right or right to left, though in English we usually read from left to right Table \(\PageIndex{3}\). In general, \(a < b\) is equivalent to \(b > a\). For example \(7 < 11\) is equivalent to \(11 > 7\). And \(a > b\) is equivalent to \(b < a\). For example \(17 > 4\) is equivalent to \(4 < 17\).
| Inequality Symbols | Words |
|---|---|
| \(a \neq b\) | \(a\) is not equal to \(b\) |
| \(a < b\) | \(a\) is less than \(b\) |
| \(a \leq b\) | \(a\) is less than or equal to \(b\) |
| \(a > b\) | \(a\) is greater than \(b\) |
| \(a \geq b\) | \(a\) is greater than or not equal to \(b\) |
Example \(\PageIndex{1}\)
Translate from algebra into English:
- \(17 \leq 26\)
- \(8 \neq 17 - 3\)
- \(12 > 27 \div 3\)
- \(y + 7 < 19\)
Solution
- \(17 \leq 26\), \(17\) is less than or equal to \(26\)
- \(8 \neq 17 - 3\), \(8\) is not equal to \(17\) minus \(3\)
- \(12 > 27 \div 3\), \(12\) is greater than \(27\) divided by \(3\)
- \(y + 7 < 19\), \(y\) plus \(7\) is less than \(19\)
Try It \(\PageIndex{2}\)
Translate from algebra into English:
- \(14 \leq 27\)
- \(19 - 2 \neq 8\)
- \(12 > 4 \div 2\)
- \(x - 7 < 1\)
- Answer
-
- \(14\) is less than or equal to \(27\)
- \(19\) minus \(2\) is not equal to \(8\)
- \(12\) is greater than \(4\) divided by \(2\)
- \(x\) minus \(7\) is less than \(1\)
Try It \(\PageIndex{3}\)
Translate from algebra into English:
- \(19 \leq 15\)
- \(7 = 12 - 5\)
- \(15 \div 3 < 8\)
- \(y + 3 < 6\)
- Answer
-
- \(19\) is greater then or equal to \(15\)
- \(7\) is equal to \(12\) minus \(5\)
- \(15\) divided by \(3\) is less than \(8\)
- \(y\) plus \(3\) is greater than \(6\)
Grouping symbols in algebra are much like the commas, colons, and other punctuation marks in English. They help to make clear which expressions are to be kept together and separate from other expressions. We will introduce three types now.
GROUPING SYMBOLS
\[\begin{align*} & \text{Parentheses} & & ( ) \\ & \text{Brackets} & & [ ] \\ & \text{Braces} & & \{ \} \end{align*}\]
Here are some examples of expressions that include grouping symbols . We will simplify expressions like these later in this section.
\[8(14−8) \qquad 21−3[2 + 4(9−8)] \qquad 24\div \{ 13−2[1(6−5)+4] \nonumber\}\]
What is the difference in English between a phrase and a sentence? A phrase expresses a single thought that is incomplete by itself, but a sentence makes a complete statement. “Running very fast” is a phrase, but “The football player was running very fast” is a sentence. A sentence has a subject and a verb. In algebra, we have expressions and equations .
EXPRESSION
An expression is a number, a variable, or a combination of numbers and variables using operation symbols.
An expression is like an English phrase. Here are some examples of expressions:
| Expression | Words | English Phrase |
|---|---|---|
| \(3 + 5\) | \(3\) plus \(5\) | the sum of three and five |
| \(n − 1\) | \(n\) minus one | the difference of \(n\) and one |
| \(6\cdot 7\) | \(6\) times \(7\) | the product of six and seven |
| \(\dfrac{x}{y}\) | \(x\) divided by \(y\) | the quotient of \(x\) and \(y\) |
Notice that the English phrases do not form a complete sentence because the phrase does not have a verb. An equation is two expressions linked with an equal sign . When you read the words the symbols represent in an equation, you have a complete sentence in English. The equal sign gives the verb.
Definition: EQUATION
An equation is two expressions connected by an equal sign.
Here are some examples of equations.
| Equation | English Sentence |
|---|---|
| \(3+5=8\) | the sum of three and five is equal to eight |
| \(n−1=14\) | \(n\) minus one equals fourteen |
| \(6 \cdot 7=42\) | The product of six ans seven is equal to forty-two |
| \(x=53\) | \(x\) is equal to fifty-three |
| \(y+9=2y−3\) | \(y\) plus nine is equal to two \(y\) minus three |
Example \(\PageIndex{4}\)
Determine if each is an expression or an equation:
- \(2(x + 3) = 10\)
- \(4(y - 1) + 1\)
- \(x \div 25\)
- \(y + 8 = 40\)
Solution
- \(2(x + 3) = 10\). This is an equation — two expressions are connected with an equal sign.
- \(4(y - 1) + 1\). This is an expression — no equal sign.
- \(x \div 25\). This is an expression — no equal sign.
- \(y + 8 = 40\). This is an equation — two expressions are connected with an equal sign.
Try It \(\PageIndex{5}\)
Determine if each is an expression or an equation:
- \(3(x - 7) = 27\)
- \(5(4y - 2) - 7\)
- Answer
-
- equation
- expression
Try It \(\PageIndex{6}\)
Determine if each is an expression or an equation:
- \(y^{3} \div 14\)
- \(4x - 6 = 22\)
- Answer
-
- expression
- equation
Suppose we need to multiply nine factors of \(2\). We could write this as \(2\cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2\). This is tedious and it can be hard to keep track of all those 2s, so we use exponents. We write \(2\cdot 2 \cdot 2\) as \(2^{3}\) and \(2\cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2\) as \(2^{9}\). In expressions such as \(2^{3}\), the \(2\) is called the base and the \(3\) is called the exponent . The exponent tells us how many times we need to multiply the base.
We read \(2^{3}\) as “two to the third power” or “two cubed.”
We say \(2^{3}\) is in exponential notation and \(2\cdot 2 \cdot 2\) is in expanded notation.
EXPONENTIAL NOTATION
\(a^{n}\) means the product of \(n\) factors of \(a\).
The expression \(a^{n}\) is read \(a\) to the \(n^{th}\) power.
While we read \(a^{n}\) as “\(a\) to the \(n^{th}\) power,” we usually read:
- \(a^{2}\) “a squared”
- \(a^{3}\) “a cubed”
We’ll see later why \(a^{2}\) and \(a^{3}\) have special names.
Table \(\PageIndex{6}\) shows how we read some expressions with exponents.
| Expression | In Words |
|---|---|
| \(7^{2}\) | \(7\) to the second power or \(7\) squared |
| \(5^{3}\) | \(5\) to the third power or \(5\) cubed |
| \(9^{4}\) | \(9\) to the fourth power |
| \(12^{5}\) | \(12\) to the fifth power |
Example \(\PageIndex{7}\)
Simplify: \(3^{4}\)
Solution
\[\quad 3^{4}\nonumber\]
\[\begin{align*} & \text{Expand the expression } & & 3 \cdot 3 \cdot 3 \cdot 3 \\[5pt]
& \text{Multiply left to right} & & 9 \cdot 3 \cdot 3 \\[5pt]
& \text{Multiply} & & 27 \cdot 3 \\[5pt]
& \text{Multiply} & & 81 \end{align*}\]
Try It \(\PageIndex{8}\)
Simplify:
- \(5^{3}\)
- \(1^{7}\)
- Answer
-
- \(125\)
- \(1\)
Try It \(\PageIndex{9}\)
- \(7^{2}\)
- \(0^{5}\)
- Answer
-
- \(49\)
- \(0\)
Simplify Expressions Using the Order of Operations
To simplify an expression means to do all the math possible. For example, to simplify \(4\cdot 2 + 1\) we’d first multiply \(4\cdot 2\) to get \(8\) and then add the \(1\) to get \(9\). A good habit to develop is to work down the page, writing each step of the process below the previous step. The example just described would look like this:
\[4\cdot 2 + 1\nonumber\]
\[8 + 1\nonumber\]
\[9\nonumber\]
By not using an equal sign when you simplify an expression, you may avoid confusing expressions with equations.
SIMPLIFY AN EXPRESSION
To simplify an expression , do all operations in the expression.
We’ve introduced most of the symbols and notation used in algebra, but now we need to clarify the order of operations. Otherwise, expressions may have different meanings, and they may result in different values. For example, consider the expression:
\[4 + 3\cdot 7\nonumber\]
If you simplify this expression, what do you get?
Some students say \(49\),
\[4 + 3\cdot 7\nonumber\]
Since \(4+3\) gives \(7\).
\[7 \cdot 7\nonumber\]
And \(7\cdot 7\) is \(49\) \[49\nonumber\]
Others say \(25\),
\[4 + 3\cdot 7\nonumber\]
Since \(3\cdot 7\) is \(21\).
\[4 + 21\nonumber\]
And \(21 + 4\) makes \(25\).
\[25\nonumber\]
Imagine the confusion in our banking system if every problem had several different correct answers!
The same expression should give the same result. So mathematicians early on established some guidelines that are called the Order of Operations .
PERFORM THE ORDER OF OPERATIONS.
-
Parentheses and Other Grouping Symbols
- Simplify all expressions inside the parentheses or other grouping symbols, working on the innermost parentheses first.
-
Exponents
- Simplify all expressions with exponents.
-
Multiplication and Division
- Perform all multiplication and division in order from left to right. These operations have equal priority.
-
Addition and Subtraction
- Perform all addition and subtraction in order from left to right. These operations have equal priority.
Note
Doing the Manipulative Mathematics activity “Game of 24” will give you practice using the order of operations.
Students often ask, “How will I remember the order?” Here is a way to help you remember: Take the first letter of each key word and substitute the silly phrase: “Please Excuse My Dear Aunt Sally.”
\[\begin{align*} &\textbf{P}\text{arentheses} & & \textbf{P}\text{lease} \\[5pt]
&\textbf{E}\text{xponents} & & \textbf{E}\text{xcuse} \\[5pt]
&\textbf{M}\text{ultiplication}\space\textbf{D}\text{ivision} & & \textbf{M}\text{y}\space\textbf{D}\text{ear} \\[5pt]
&\textbf{A}\text{ddition}\space\textbf{S}\text{ubtraction} & & \textbf{A}\text{unt}\space\textbf{S}\text{ally} \end{align*}\]
It’s good that “\(\textbf{M}\text{y}\space\textbf{D}\text{ear}\)” goes together, as this reminds us that m ultiplication and d ivision have equal priority. We do not always do multiplication before division or always do division before multiplication. We do them in order from left to right.
Similarly, “\(\textbf{A}\text{unt}\space\textbf{S}\text{ally}\)” goes together and so reminds us that a ddition and s ubtraction also have equal priority and we do them in order from left to right.
Let’s try an example.
Example \(\PageIndex{10}\)
Simplify:
- \(4 + 3\cdot 7\)
- \((4 + 3)\cdot 7\)
Solution
1.| \(4 + 3 \cdot 7\) | |
| Are there any p arentheses? No. | |
| Are there any e xponents? No. | |
| Is there any m ultiplication or d ivision? Yes. | |
| Multiply first. | \(4 + {\color{red}{3 \cdot 7}}\) |
| Add. | \(4+21\) |
| \(25\) |
2.
| \((4 + 3)\cdot 7\) | |
| Are there any p arentheses? Yes. | \({\color{red}{(4 + 3)}}\cdot 7\) |
| Simplify inside the parentheses. | \(({\color{red}{7}})7\) |
| Are there any e xponents? No. | |
| Is there any m ultiplication or d ivision? Yes. | |
| Multiply. | \(49\) |
Try It \(\PageIndex{11}\)
Simplify:
- \(12 - 5\cdot 2\)
- \((12 - 5)\cdot 2\)
- Answer
-
- \(2\)
- \(14\)
Try It \(\PageIndex{12}\)
Simplify:
- \(8 + 3\cdot 9\)
- \((8 + 3)\cdot 9\)
- Answer
-
- \(35\)
- \(99\)
Example \(\PageIndex{13}\)
Simplify: \(18\div 6 + 4(5 - 2)\)
Solution
| Parentheses? Yes, subtract first. |
\(18\div 6 + 4(5 - 2)\)
|
| Exponents? No. | |
| Multiplication or division? Yes. | \({\color{red}{18\div 6}} + {\color{red}{4(3)}}\) |
| Divide first because we multiply and divide left to right. | \(3+{\color{red}{4(3)}}\) |
| Any other multiplication or division? Yes. | |
| Multiply. | \(3 + 12\) |
| Any other multiplication or division? No. | |
| Any addition or subtraction? Yes. | \(15\) |
Try It \(\PageIndex{14}\)
Simplify: \(30\div 5 + 10(3 - 2)\)
- Answer
-
\(16\)
Try It \(\PageIndex{15}\)
Simplify: \(70\div 10 + 4(6 - 2)\)
- Answer
-
\(23\)
When there are multiple grouping symbols, we simplify the innermost parentheses first and work outward.
Example \(\PageIndex{16}\)
Simplify: \(5 + 2^{3} + 3[6 - 3(4 - 2)]\).
Solution
| \(5 + 2^{3} + 3[6 - 3(4 - 2)]\) | |
| Are there any parentheses (or other grouping symbol)? Yes. | |
| Focus on the parentheses that are inside the brackets. | \(5 + 2^{3} + 3[6 - 3{\color{red}{(4 - 2)}}]\) |
| Subtract. | \(5 + 2^{3} + 3[6 - {\color{red}{3(2)}}]\) |
| Continue inside the brackets and multiply. | \(5 + 2^{3} + 3[{\color{red}{6 - 6}}]\) |
| Continue inside the brackets and subtract. | \(5 + 2^{3} + 3[{\color{red}{0}}]\) |
| The expression inside the brackets requires no further simplification. | |
| Are there any exponents? Yes. | \(5 + {\color{red}{2^{3}}}+ 3[0]\) |
| Simplify exponents. | \(5 + 8 + {\color{red}{3[0]}}\) |
| Is there any multiplication or division? Yes. | |
| Multiply. | \({\color{red}{5 + 8}}+0\) |
| Is there any addition or subtraction? Yes. | |
| Add. | \({\color{red}{13 + 0}}\) |
| Add. | \(13\) |
Try It \(\PageIndex{17}\)
Simplify: \(9 + 5^{3} - [4(9 + 3)]\).
- Answer
-
\(86\)
Try It \(\PageIndex{18}\)
Simplify: \(7^{2} - 2[4(5 + 1)]\).
- Answer
-
\(1\)
Evaluate an Expression
In the last few examples, we simplified expressions using the order of operations. Now we’ll evaluate some expressions—again following the order of operations. To evaluate an expression means to find the value of the expression when the variable is replaced by a given number.
EVALUATE AN EXPRESSION
To evaluate an expression means to find the value of the expression when the variable is replaced by a given number.
To evaluate an expression, substitute that number for the variable in the expression and then simplify the expression.
Example \(\PageIndex{19}\)
Evaluate \(7x - 4\), when
- \(x = 5\)
- \(x = 1\)
Solution
1.
| when \(x = {\color{red}{5}}\) | \(7x - 4\) |
| \(7({\color{red}{5}}) - 4\) | |
| Multiply. | \(35 - 4\) |
| Subtract. | \(31\) |
2.
| when \(x = {\color{red}{1}}\) | \(7x - 4\) |
| \(7({\color{red}{1}}) - 4\) | |
| Multiply. | \(7 - 4\) |
| Subtract. | \(3\) |
Try It \(\PageIndex{20}\)
Evaluate \(8x - 3\), when
- \(x = 2\)
- \(x = 1\)
- Answer
-
- \(13\)
- \(5\)
Try It \(\PageIndex{21}\)
Evaluate \(4y - 4\), when
- \(y = 3\)
- \(y = 5\)
- Answer
-
- \(8\)
- \(16\)
Example \(\PageIndex{22}\)
Evaluate \(x = 4\), when
- \(x^{2}\)
- \(3^{x}\)
Solution
1.
| \(x^{2}\) | |
| Replace \(x\) with \({\color{red}{4}}\). | \(({\color{red}{4}})^{2}\) |
| Use definition of exponent. | \(4\cdot 4\) |
| Simplify. | \(16\) |
2.
| \(3^{x}\) | |
| Replace \(x\) with \({\color{red}{4}}\). | \(3^{\color{red}{4}}\) |
| Use definition of exponent. | \(3\cdot3\cdot3\cdot3\) |
| Simplify. | \(81\) |
Try It \(\PageIndex{23}\)
Evaluate \(x = 3\), when
- \(x^{2}\)
- \(4^{x}\)
- Answer
-
- \(9\)
- \(64\)
Try It \(\PageIndex{24}\)
Evaluate \(x = 6\), when
- \(x^{3}\)
- \(2^{x}\)
- Answer
-
- \(216\)
- \(64\)
Example \(\PageIndex{25}\)
Evaluate \(2x^{2} + 3x + 8\) when \(x = 4\).
Solution
| \(2x^{2} + 3x + 8\) | |
| Substitute \(x = {\color{red}{4}}\). |
\(\small{2x^{2} + 3x + 8}\)
\(2({\color{red}{4}})^{2} + 3({\color{red}{4}}) + 8\) |
| Follow the order of operations. | \(2(16)+3(4)+8\) |
| \(32+12+8\) | |
| \(52\) |
Try It \(\PageIndex{26}\)
Evaluate \(3x^{2} + 4x + 1\) when \(x = 3\).
- Answer
-
\(40\)
Try It \(\PageIndex{27}\)
Evaluate \(6x^{2} - 4x - 7\) when \(x = 2\).
- Answer
-
\(9\)
Identify and Combine Like Terms
Algebraic expressions are made up of terms. A term is a constant, or the product of a constant and one or more variables.
TERM
A term is a constant, or the product of a constant and one or more variables.
Examples of terms are \(7, y, 5x^{2}, 9a\), and \(b^{5}\).
The constant that multiplies the variable is called the coefficient .
COEFFICIENT
The coefficient of a term is the constant that multiplies the variable in a term.
Think of the coefficient as the number in front of the variable. The coefficient of the term \(3x\) is \(3\). When we write \(x\), the coefficient is \(1\), since \(x=1\cdot x\).
Example \(\PageIndex{28}\)
Identify the coefficient of each term:
- \(14y\)
- \(15x^{2}\)
- \(a\)
Solution
- The coefficient of \(14y\) is \(14\)
- The coefficient of \(15x^{2}\) is \(15\)
- The coefficient of \(a\) is \(1\) since \(a=1a\).
Try It \(\PageIndex{29}\)
Identify the coefficient of each term:
- \(17x\)
- \(41b^{2}\)
- \(z\)
- Answer
-
- \(14\)
- \(41\)
- \(1\)
Try It \(\PageIndex{30}\)
Identify the coefficient of each term:
- \(9p\)
- \(13a^{2}\)
- \(y^{3}\)
- Answer
-
- \(9\)
- \(13\)
- \(1\)
Some terms share common traits. Look at the following 6 terms. Which ones seem to have traits in common?
\[5x \qquad 7 \qquad n^{2} \qquad 4 \qquad 3x \qquad 9n^{2}\nonumber\]
The \(7\) and the \(4\) are both constant terms.
The \(5x\) and the \(3x\) are both terms with \(x\).
The \(n^{2}\) and the \(9n^{2}\) are both terms with \(n^{2}\).
When two terms are constants or have the same variable and exponent, we say they are like terms .
- \(7\) and \(4\) are like terms.
- \(5x\) and \(3x\) are like terms.
- \(x^{2}\) and \(9x^{2}\) are like terms.
LIKE TERMS
Terms that are either constants or have the same variables raised to the same powers are called like terms .
Example \(\PageIndex{31}\)
Identify the like terms: \(y^{3},7x^{2}, 14, 23, 4y^{3}, 9x, 5x^{2}\).
Solution
\(y^{3}\) and \(4y^{3}\) are like terms because both have \(y^{3}\); the variable and the exponent match.
\(7x^{2}\) and \(5x^{2}\) are like terms because both have \(x^{2}\); the variable and the exponent match.
\(14\) and \(23\) are like terms because both are constants.
There is no other term like \(9x\).
Try It \(\PageIndex{32}\)
Identify the like terms: \(9, 2x^{3},y^{2}, 8x^{3}, 15, 9y, 11y^{2}\).
- Answer
-
\(9\) and \(15\), \(y^{2}\) and \(11y^{2}\), \(2x^{3}\) and \(8x^{3}\)
Try It \(\PageIndex{33}\)
Identify the like terms: \(4x^{3},8x^{2}, 19, 3x^{3}, 24, 6x^{3}\).
- Answer
-
\(19\) and \(24\), \(8x^{2}\) and \(3x^{2}\), \(4x^{3}\) and \(6x^{3}\)
Adding or subtracting terms forms an expression. In the expression \(2x^{2} + 3x + 8\), from Example , the three terms are \(2x^{2}\), \(3x\), and \(8\).
Example \(\PageIndex{34}\)
Identify the terms in each expression.
- \(9x^{2}+7x+12\)
- \(8x+3y\)
Solution
- The terms of \(9x^{2}+7x+12\) are \(9x^{2}, 7x\), and \(12\).
- The terms of \(8x+3y\) are \(8x\) and \(3y\).
Try It \(\PageIndex{35}\)
Identify the terms in the expression \(4x^{2}+5x+17\).
- Answer
-
\(4x^{2}, 5x, 17\)
Try It \(\PageIndex{36}\)
Identify the terms in the expression \(5x+2y\).
- Answer
-
\(5x, 2y\)
If there are like terms in an expression, you can simplify the expression by combining the like terms. What do you think \(4x+7x+x\) would simplify to? If you thought \(12x\), you would be right!
\[\begin{array} { c } { 4 x + 7 x + x } \\ { x + x + x + x \quad + x + x + x + x + x + x + x \quad+ x } \\ { 12 x } \end{array}\]
Add the coefficients and keep the same variable. It doesn’t matter what x is—if you have 4 of something and add 7 more of the same thing and then add 1 more, the result is 12 of them. For example, 4 oranges plus 7 oranges plus 1 orange is 12 oranges. We will discuss the mathematical properties behind this later.
Simplify: \(4x+7x+x\)
Add the coefficients. \(12x\)
Example \(\PageIndex{37}\): How To Combine Like Terms
Simplify: \(2x^{2} + 3x + 7 + x^{2} + 4x + 5\)
Solution
Try It \(\PageIndex{38}\)
Simplify: \(3x^{2} + 7x + 9 + 7x^{2} + 9x + 8\).
- Answer
-
\(10x^{2}+16x+17\)
Try It \(\PageIndex{39}\)
Simplify: \(4y^{2} + 5y + 2 + 8y^{2} + 4y + 5\).
- Answer
-
\(12y^{2}+9y+7\)
COMBINE LIKE TERMS.
- Identify like terms.
- Rearrange the expression so like terms are together.
- Add or subtract the coefficients and keep the same variable for each group of like terms.
Translate an English Phrase to an Algebraic Expression
In the last section, we listed many operation symbols that are used in algebra, then we translated expressions and equations into English phrases and sentences. Now we’ll reverse the process. We’ll translate English phrases into algebraic expressions. The symbols and variables we’ve talked about will help us do that. Table \(\PageIndex{7}\) summarizes them.
| Operation | Phrase | Expression |
|---|---|---|
| Addition |
\(a\) plus \(b\)
the sum of \(a\) and \(b\) \(a\) increased by \(b\) \(b\) more than \(a\) the total of \(a\) and \(b\) \(b\) added to \(a\) |
\[a+b\] |
| Subtraction |
\(a\) minus \(b\)
the difference of \(a\) and \(b\) \(a\) decreased by \(b\) \(b\) less than \(a\) \(b\) subtracted from \(a\) |
\[a−b\] |
| Multiplication |
\(a\) times \(b\)
the product of \(a\) and \(b\) twice \(a\) |
\[a\cdot b, ab, a(b), (a)(b)\]
\[2a\] |
| Division |
\(a\) divided by \(b\)
the quotient of \(a\) and \(b\) the ratio of \(a\) and \(b\) \(b\) divided into \(a\) |
\[a\div b, a/b, \frac{a}{b}, b \enclose{longdiv}{a}\] |
Look closely at these phrases using the four operations:
Each phrase tells us to operate on two numbers. Look for the words of and and to find the numbers.
Example \(\PageIndex{40}\)
Translate each English phrase into an algebraic expression:
- the difference of \(17x\) and \(5\)
- the quotient of \(10x^{2}\) and \(7\).
Solution
-
The key word is
difference
, which tells us the operation is subtraction. Look for the words
of
and
and t
o find the numbers to subtract.
- The key word is “quotient,” which tells us the operation is division.
This can also be written \(10x^{2}/7\) or \(\dfrac{10x^{2}}{7}\).
Try It \(\PageIndex{41}\)
Translate each English phrase into an algebraic expression:
- the difference of \(14x^{2}\) and \(13\)
- the quotient of \(12x\) and \(2\).
- Answer
-
- \(14x^{2} - 13\)
- \(12x \div 2\)
Try It \(\PageIndex{42}\)
Translate each English phrase into an algebraic expression:
- the sum of \(17y^{2}\) and \(19\)
- the product of \(7\) and \(y\).
- Answer
-
- \(17y^{2} + 19\)
- \(7y\)
How old will you be in eight years? What age is eight more years than your age now? Did you add 8 to your present age? Eight “more than” means 8 added to your present age. How old were you seven years ago? This is 7 years less than your age now. You subtract 7 from your present age. Seven “less than” means 7 subtracted from your present age.
Example \(\PageIndex{43}\)
Translate the English phrase into an algebraic expression:
- Seventeen more than \(y\)
- Nine less than \(9x^{2}\).
Solution
-
The key words are
more than.
They tell us the operation is addition.
More than
means “added to.”
\(\begin{array} { c } { \text { Seventeen more than } y } \\ { \text { Seventeen added to } y } \\ { y + 17 } \end{array}\)
-
The key words are
less than
. They tell us to subtract.
Less than
means “subtracted from.”
\(\begin{array} { c } { \text { Nine less than } 9 x ^ { 2 } } \\ { \text { Nine subtracted from } 9 x ^ { 2 } } \\ { 9 x ^ { 2 } - 9 } \end{array}\)
Try It \(\PageIndex{44}\)
Translate the English phrase into an algebraic expression:
- Eleven more than x
- Fourteen less than \(11a\).
- Answer
-
- \(x+11\)
- \(11a−14\)
Try It \(\PageIndex{45}\)
Translate the English phrase into an algebraic expression:
- \(13\) more than \(z\)
- \(18\) less than \(8x\).
- Answer
-
1. \(z+13\)
2. \(8x−18\)
Example \(\PageIndex{46}\)
Translate the English phrase into an algebraic expression:
- five times the sum of \(m\) and \(n\)
- the sum of five times \(m\) and \(n\).
Solution
There are two operation words—
times
tells us to multiply and
sum
tells us to add.
1. Because we are multiplying \(5\) times the sum we need parentheses around the sum of \(m\) and \(n\), \((m+n)\). This forces us to determine the sum first. (Remember the order of operations.)
\[\begin{array} { c } { \text { five times the sum of } m \text { and } n } \\ { 5 ( m + n ) } \end{array}\]
2. To take a sum, we look for the words “of” and “and” to see what is being added. Here we are taking the sum of five times \(m\) and \(n\.)\[\begin{array} { c } { \text { the sum of five times } m \text { and } n } \\ { 5 m + n } \end{array}\]
Try It \(\PageIndex{47}\)
Translate the English phrase into an algebraic expression:
- four times the sum of \(p\) and \(q\)
- the sum of four times \(p\) and \(q\).
- Answer
-
- \(4(p+q)\)
- \(4p+q\)
Try It \(\PageIndex{48}\)
Translate the English phrase into an algebraic expression:
- the difference of two times x and \(8\),
- two times the difference of x and \(8\).
- Answer
-
- \(2x−8\)
- \(2(x−8)\)
Later in this course, we’ll apply our skills in algebra to solving applications. The first step will be to translate an English phrase to an algebraic expression. We’ll see how to do this in the next two examples.
Example \(\PageIndex{49}\)
The length of a rectangle is \(6\) less than the width. Let \(w\) represent the width of the rectangle. Write an expression for the length of the rectangle.
Solution
\[\begin{array} { l l } { \text { Write a phrase about the length of the rectangle. } } &{ 6 \text { less than the width } } \\ { \text { Substitute } w \text { for "the width." } } &{\text{6 less then w}} \\ { \text { Rewrite "less than" as "subtracted from." } } &{\text{6 subtracted from w}} \\ { \text { Translate the phrase into algebra. } } &{w - 6} \end{array}\]
Try It \(\PageIndex{50}\)
The length of a rectangle is \(7\) less than the width. Let \(w\) represent the width of the rectangle. Write an expression for the length of the rectangle.
- Answer
-
\(w - 7\)
Try It \(\PageIndex{51}\)
The width of a rectangle is \(6\) less than the length. Let \(l\) represent the length of the rectangle. Write an expression for the width of the rectangle.
- Answer
-
\(l - 6\)
Example \(\PageIndex{52}\)
June has dimes and quarters in her purse. The number of dimes is three less than four times the number of quarters. Let \(q\) represent the number of quarters. Write an expression for the number of dimes.
Solution
\[\begin{array} { ll } { \text { Write the phrase about the number of dimes. } } &{\text{three less than four times the number of quarters}} \\ { \text { Substitute } q \text { for the number of quarters. } } &{\text{3 less than 4 times q}} \\ { \text { Translate "4 times } q \text { ." } } &{\text{3 less than 4q}} \\ { \text { Translate the phrase into algebra. } } &{\text{4q - 3}} \end{array}\]
Try It \(\PageIndex{53}\)
Geoffrey has dimes and quarters in his pocket. The number of dimes is eight less than four times the number of quarters. Let \(q\) represent the number of quarters. Write an expression for the number of dimes.
- Answer
-
\(4q - 8\)
Try It \(\PageIndex{54}\)
Lauren has dimes and nickels in her purse. The number of dimes is three more than seven times the number of nickels. Let \(n\) represent the number of nickels. Write an expression for the number of dimes.
- Answer
-
\(7n + 3\)
Key Concepts
-
Notation The result is…
\(\begin{array} { l l } {\bullet \space a + b } &{ \text { the sum of } a \text { and } b } \\ { \bullet \space a - b } &{ \text { the difference of } a \text { and } b } \\ {\bullet\space a \cdot b , a b , ( a ) ( b ) ( a ) b , a ( b ) } &{ \text { the product of } a \text { and } b } \\ {\bullet\space a \div b , a / b , \frac { a } { b } , b ) \overline{a} } &{ \text { the quotient of } a \text { and } b } \end{array}\) -
Inequality
\(\begin{array} { l l } { \bullet \space a < b \text { is read "a is less than } b ^ { \prime \prime } } &{a \text { is to the left of } b \text { on the number line } } \\ { \bullet \space a > b \text { is read "a is greater than } b ^ { \prime \prime } } & { a \text { is to the right of } b \text { on the number line } } \end{array}\) -
Inequality Symbols Words
\(\begin{array} {ll} { \bullet a \neq b } &{ a \text { is not equal to } b } \\ { \bullet a < b } &{ a \text { is less than } b } \\ { \bullet a \leq b } &{ a \text { is less than or equal to } b } \\ { \bullet a > b } & { a \text { is greater than } b } \\ { \bullet a \geq b } & { a \text { is greater than or equal to } b } \end{array}\) -
Grouping Symbols
- Parentheses ()
- Brackets []
- Braces {}
-
Exponential Notation
- \(a^{n}\) means the product of \(n\) factors of \(a\). The expression \(a^{n}\) is read \(a\) to the \(n^{th}\) power.
-
Order of Operations:
When simplifying mathematical expressions perform the operations in the following order:
- Parentheses and other Grouping Symbols: Simplify all expressions inside the parentheses or other grouping symbols, working on the innermost parentheses first.
- Exponents: Simplify all expressions with exponents.
- Multiplication and Division: Perform all multiplication and division in order from left to right. These operations have equal priority.
- Addition and Subtraction: Perform all addition and subtraction in order from left to right. These operations have equal priority.
-
Combine Like Terms
- Identify like terms.
- Rearrange the expression so like terms are together.
- Add or subtract the coefficients and keep the same variable for each group of like terms.
Glossary
- coefficient
- The coefficient of a term is the constant that multiplies the variable in a term.
- constant
- A constant is a number whose value always stays the same.
- equality symbol
- The symbol “\(=\)” is called the equal sign. We read \(a=b\) as “\(a\) is equal to \(b\).”
- equation
- An equation is two expressions connected by an equal sign.
- evaluate an expression
- To evaluate an expression means to find the value of the expression when the variable is replaced by a given number.
- expression
- An expression is a number, a variable, or a combination of numbers and variables using operation symbols.
- like terms
- Terms that are either constants or have the same variables raised to the same powers are called like terms.
- simplify an expression
- To simplify an expression, do all operations in the expression.
- term
- A term is a constant or the product of a constant and one or more variables.
- variable
- A variable is a letter that represents a number whose value may change.


5.3: Developmental Domains
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- Angela Blums & Sally Nyblad Holloway
While development is complex, four domains have been identified: physical, cognitive, social, and emotional. This section will introduce each of these developmental domains and explain some important features of each domain. keep in mind that developmental domains are interrelated, so consider as you read the next sections how one aspect of development could affect another.
Physical Development
Physical development includes all things related to the body. Sometimes this domain is called biological development because it also encompasses processes that occur inside the body at the biological level.
Brain Development
Brain development is an important part of physical development and was thoroughly covered in Chapter 4. As a reminder, infant brains begin to develop while still in utero. Because of this, it is important to support fetal brain development through proper nutrition and care of the mother. Once born, infant brains develop rapidly. Brain connections, called synapses, develop rapidly during the first few months to the first 3 years of life, making this period critical for healthy brain development. During infancy, babies require responsive interactions with trusted caregivers that include verbal, expressive, face-to-face engagement. Additionally, and above all, they require a safe and secure home environment for healthy brain development.
Motor Skills
Children’s bodies are designed to wiggle and move. Motor skills refer to a child’s ability to move and coordinate motion using their bodies. Motor skills are generally organized into two categories: gross motor skills and fine motor skills. Gross motor skills are related to the large muscle groups in the body: walking, running, climbing, balancing, and jumping are examples of gross motor skills. Fine motor skills are related to small muscle groups in the body. Grasping a spoon, holding a pencil, stringing beads, and stacking blocks are examples of fine motor skills.
In order for gross motor skills to develop successfully, infants and children must be given opportunities to move their bodies freely, safely and without restriction, starting from birth. This is why most pediatric physical therapists recommend against the use of so-called “baby containers” – chairs, bouncy seats, swings, and jumpers that place an infant in an upright position before their body is ready to do so on its own. These devices can hinder development. Instead, babies should play on a blanket or mat and have a few minutes of “tummy time” each day starting at birth. This helps to strengthen neck and shoulder muscles that will later be used in crawling. As children begin to crawl and walk, soft climbers can help to develop gross motor skills.
Once solid on their feet, children need opportunities to practice running, jumping, balancing, and climbing. This is why daily outdoor play is vital. Playgrounds can be great places for children to experiment with gross motor skills. Outdoor play in natural areas is also important. Consider the difference between climbing stairs or monkey bars on a playground vs climbing the branches of a strong, low, tree. Both offer opportunities for using arms and legs, but tree climbing allows children to use muscle groups in new positions and learn to balance in an irregular position. If no safe climbing trees are available, learning to walk or run on uneven ground in a field or meadow can also be beneficial. It is important for children to move their bodies in many different ways.
Playing outdoors on playgrounds and in nature provides children opportunities to take risks. When children climb, they not only build muscle and coordination, but also learn to overcome fears. This can lead to greater self-confidence that carries over into other areas as well. This is one way in which outdoor activity supports not only physical but also emotional development.
Fine motor skills begin to develop in infancy when the child first learns to grasp a toy such as a rattle or a cloth. Later, they learn to throw a ball, hold a crayon, and use scissors. Children can be given opportunities to use their hands in a variety of ways using age-appropriate materials. Young children love to practice fine motor skills using real-life items, such as bucking the clasp of a highchair strap or zipping a zipper. This has the added benefit of fostering independence which supports emotional development. Children can coordinate their finger movements while learning a zipper and they can also feel proud when they can zip their own jacket to go outside. Opportunities for fine motor development are everywhere and can be as simple as picking up a leaf and ripping it into tiny pieces. Teachers can foster fine motor skills by making sure to create challenging activities based on each child’s developmental level.
Cognitive Development
Cognitive development includes all things related to thinking and learning. In the early childhood stage, the focus is on problem solving, memory, and language acquisition.
Executive Function
One key area of cognitive development is called executive function . Executive function is the collection of processes that encompasses attention, working memory, and inhibition, and it develops between the ages of about 3 and 6 years. Attention is the ability of a child to focus on something, like carefully concentrating on a picture in a book. Working memory refers to the ability of a child to maintain several pieces of information in the mind for a short period of time. This could be recalling some items present in the picture in the book, like a tree, a bird, and a house. Inhibition is the ability to block out distractions, such as other background designs on the book or even the sounds of other children playing in the classroom. Development of executive function is key to building social relationships, acquiring, and maintaining learning, and eventual academic success. For example, if a child is going to learn how to add using blocks, she might have to first pay attention to the blocks in front of her, remember that she had three blocks on one side and one block on the other side, and inhibit the distraction of other toys on the table. All of these components come together with executive function. It is important to remember that because executive function is developing between the ages of about 3 and 5 years, young children do not have the ability to pay attention to things for long periods of time. Nor do they have the ability to remember multiple pieces of information or block out distractions. This is important when planning activities for young children. Setting the expectation that young children should sit at a table and focus on one activity will result in frustration of both children and teachers, as children will communicate their lack of executive function with a lot of wiggles and movement! This is healthy, and teachers should keep this in mind when planning activities that require concentration.
Children with Attention Deficit Hyperactivity Disorder (ADHD) need extra support to develop executive function, as concentration and inhibition are particularly difficult. Children who are in chaotic environments or those who have experienced toxic stress and trauma may struggle with the development of healthy executive function. Strategies that have been successful in promoting executive function are games and activities that practice concentration and task-switching. Such games need to be fun, not tied to punishment or reward. Deep breathing and mediation have also been shown to aid in the development of executive function for children who are struggling.
Problem Solving
When children encounter a challenge, they learn to practice problem solving. Problem solving involves assessing the problem, devising a plan, carrying out the plan, and reflecting on the outcome. Children may do this without actively knowing that they are carrying out these four steps. For instance, if a child is building a block tower on a soft carpet, it might be unstable and fall down. The child might think of a solution: get a tray or a large book to set on the carpet to create a flat surface. The child then gets the tray, builds the block tower, and then checks to see if it is wobbly or stable. Problem solved! Children solve many such problems each day during play. It is helpful for teachers to facilitate problem solving by asking open-ended questions , or questions that do not have a yes or no answer. This type of question helps the child see the main features and problems in the situation, encourages thinking about solutions, and suggests solutions, all without giving the answer. An exchange might look like the following:
Teacher: “It looks like your block tower is falling down a lot. I wonder why?”
Child “Because the carpet is bumpy and soft”
Teacher: “How can you make a more stable base for your tower?”
Child: “I could put a tray underneath it!”
Teacher: “Want to try that and see what happens?”
This exchange of open-ended questions helps the child learn to problem solve on their own. The use of the phrase “see what happens” encourages the child to engage in the fourth step of problem solving, assessing the outcome. After all, a problem-solving strategy may not succeed, and the child might have to try something else. This is all part of the process of learning.
Theory of Mind
As young as 2 ½ healthy children start to figure out how other people are thinking and feeling. The study of this ability is referred to as theory of mind, a cognitive ability supporting social development. One of the hallmarks of theory of mind is ability to pick up cues and understand the mental states of others. For example, if a child sees a friend crying, he might infer that the friend is sad. It is difficult for a young child to get out of their own head. Children see things from their own perspective and have a difficult time understanding the perspectives of others. So, if a child is happily playing and sees another child crying, it takes some time before they can understand that what they are thinking and feeling is not the same as what others are thinking and feeling. It is clear to see the connection between this cognitive ability and building social relationships. Children with Autism have difficulty with theory of mind, and many children with Autism do not develop it at all. For this reason, Autism interventions include instruction on how to read the mental stats of others.
Self-reflection
Self-reflection is the ability to think about one’s own thoughts. This is sometimes referred to as metacognition , which develops between middle childhood and adolescence. However, during early childhood there are signs of self-reflection, or thinking back on one’s thoughts, Self-reflection is helpful in problem solving and emotional development. If a child has solved a problem such as mixing the colors just right for a painting, the teacher might ask “Why did you decide to do it this way?” or “Tell me about why you chose to mix those colors?”. This helps children to think through their process and helps them remember the strategy for next time. Self-reflection in early childhood is best facilitated by teachers using probing questions such as the examples above, but older children can engage in self-reflection through journaling or planned exchanges with peers.
Language
Language development is a very broad and rich area of development that encompasses much more than the scope of this section. Nevertheless, this section will highlight the key points of language development in early childhood.
Children learn to understand language in infancy through responsive interactions with adults. Adults should speak with and listen to infants starting at birth, making eye contact and using a gentle tone. When infants begin to babble, adults can take turns talking and letting baby babble. This sets up the format for later turn taking in conversation. Throughout childhood, children will usually understand more words than they can speak. As children begin to understand language, they can answer simple questions with a yes or no response by shaking their heads or using simple sign language. It has become a common practice to teach hearing infants a few common signs to facilitate communication earlier than when they begin speaking. Infants with hearing impairments can be taught sign language as soon as challenges are detected.
Books should be read and stories told to children starting in infancy, as this helps develop new vocabulary and creates a connection to literature that will later be important for learning. It also helps strengthen the relationship of the child and adult. Joint attention is the action of a child and a caregiver focusing on the same object or concept at the same time. This shared experience helps to form new vocabulary. Children learn new words much more quickly and efficiently with joint attention than they ever could by viewing a video. In fact, there is evidence that suggests that television can actually hinder language development rather than help. Here is an example of joint attention in action.
Setting: Teacher Tanisha and 18-month-old Sofia are looking out the window. Teacher Tanisha sees some autumn leaves falling from the tree.
Teacher Tanisha: “Look Sofia, leaves are falling from the tree.”
[Sofia looks at the leaves]
[Teacher Tanisha and Sofia look at each other and make eye contact]
Sofia: Smiles and says “leaves!”
This action has three steps: adult and child focus on the same object, adult and child make eye contact, adult names the object. In this way, children add many new words to their vocabulary, the words are presented in a real-life context, and they strengthen social relationships with their caregivers.
Once children begin to speak, language use should be encouraged through open-ended questions and topics that interest the child. Songs, rhymes, and books are all effective and fun ways to encourage language, but everyday conversation is also helpful. Children like to be involved in the real-life daily activities of adults, so engaging a child while doing tasks around the classroom is a good way to support language. Mealtimes are another opportunity for language development, as eye contact is easy while sitting at a table.
When learning to speak, children will inevitably make errors with grammar and syntax. The best way to support language development is to model the correct grammar and word order rather than correcting the child. This avoids the embarrassment that might come with being called out on making a mistake, and language modeling is simply a better way for the brain to remember how to use grammar. An exchange might look something like this:
Child: “After breakfast, I goed outside.”
Teacher: “After breakfast you went outside? How fun! What did you do outside?”
Child: “I played with Weihua out there”
By using language modeling, the teacher provides an example of when and how to use the word in a certain situation, thus helping the child remember it for next time.
Social Development
Social development includes all things related to interacting with others. Parents, family, teachers, and peers all have special relationships with children, and those relationships are also related to one another. Teachers form relationships with parents, which benefits the child, and adults can facilitate communication between children. At the center of this is the child, who is driven to form relationships with adults and children alike.
Relationships with Caregivers
Infants begin to form relationships with caregivers at birth. Forming a strong relationship with a primary caregiver gives a baby a feeling of safety and security. The primary caregiver is typically the mother, but can also be the father, relative or other main person in the infant’s life. Eventually, infants form relationships with secondary caregivers like relatives and teachers. Healthy relationships are characterized by things like reciprocal interactions, sensitivity to emotions, a warm, calm, voice, and lots of positive language. These are good ways to interact with children of all ages but are particularly important with infants as toddlers. Children may form preferences to particular caregivers and be upset when one is not available. Similarly, children around from about 9 to 18 months of age may go through separation anxiety , or a fear of being separated from their primary caregiver. This is normal, and children’s emotions should be respected during this time. They do not understand that their mother will return later, which can be upsetting. Teachers can provide support by recognizing the child’s emotions and proving physical comfort (if the child wants it). Using phrases like “I see you are sad that mommy has left” “You are feeling upset right now” are helpful in supporting the child through the tough time.
Temperament
Temperament is a developmental characteristic that intersects social, emotional, and physical development, and because it has genetic underpinnings, it is thought to be “inborn”. When most people think of genetics, they think of physical traits such as eye color or hair texture. While these are genetically inherited characteristics, genetics are also involved in a child’s disposition, which can later become a key part of their personality. Temperament is an infant’s regular way of reacting with their environment, and is categorized as either easy, difficult, or slow to warm up. This is measured by several factors including: smiling and laughter; regularity in eating and sleep habits; approach or withdrawal; adaptability to new situations; intensity of responsiveness; general cheerfulness or unpleasantness; distractibility or persistence, and soothability. Parents of multiple children often report how their first child’s temperament differed from their second: “Jorge was so quiet and peaceful as a baby. He slept all day and never cried! Carlo on the other hand, fussed and cried all day. We thought he might never grow out of it!” It is important to know that babies of all temperaments can grow up to be happy, healthy, and balanced individuals.
Children’s temperament does indeed impact their behavior as an infant and this is a good example of how children’s genetic disposition can interact with their environment to help them develop. This is also an example of how different developmental domains can overlap. What is biological can also be social and emotional. The relationship between a child’s temperament and a caregiver’s personality is sometimes referred to goodness of fit . If an infant has a difficult temperament, she may have frequent periods of intense crying, be difficult to soothe, and may not fall asleep easily. If she has a primary caregiver who is ready for a challenge and sees this baby as an individual who needs love and understanding, then they have a good “goodness of fit”. On the other hand, if an infant has an easy temperament and a caregiver who does not share her sunny disposition, they may not have a “goodness of fit”.
That is why it is important for caregivers of multiple children to be adaptable to multiple infant temperaments. Each child is an individual, and no child is better or worse than another. Further, there is no perfect temperament. Children are who they are, and it is up to the important adults in their lives to respect that and treat all children with love and care.
Relationships with Peers
As children grow and mature, they begin to show interest in other children. At first, it may be just a 10-month-old watching other children playing on a playground. This may not seem like much but observing older children at play lays the groundwork for later social interactions. Later, children will play with toys alongside one another, but not yet interact. Adults may be eager for children to form friendships with others, but this time of play is important for children to experience before they move into play involving rules and negotiation.
A great deal of peer relationship building is in the context of play. Play is, after all, the work of childhood, and this is what children spend most of their waking time doing. During play, children learn the rules of games, how to read a peer’s emotions, and how to engage in social problem solving. If a group of 5-year-old children are playing a game of hide and seek, younger children who are new to the game will quickly learn the rules from the others. If two children are playing with crayons and one takes the crayon from the other, the ensuing frustration will be evident. In this way, children learn to read the emotions of peers.
Perhaps the most interesting development is social problem solving. When children encounter a conflict, it can disrupt their play, and if there is one thing, we know to be true about children, it is that they do not care for their play to be interrupted. This motivates both children to solve the conflict as quickly as possible in order to continue play. Take an example of two children playing in the kitchen dramatic play area. Both children want to cook, but there are only enough materials for one to stir the soup. A conflict arises, and in order to navigate this, the children decide that one will stir the soup and the other will chop the vegetables. This is an example of sophisticated negotiations and problem solving. Both children need to regulate their emotions, come up with alternative activities, and decide who will do which activity. If they give up at any time during the process, then the game is over. This type of problem-solving behavior in the context of play is highly complex and takes a lot of practice. Teachers can facilitate this by stepping in to help regulate emotions, brainstorm alternative activities, or help decide who will do which activity, but only if it seems that the children need help.
Emotional Development
Emotional development includes the development of emotions, emotion regulation, and a sense self. Emotional development is closely related to social development. They are so closely related, in fact, that some frameworks of child development refer to them as socioemotional development or social-emotional development. In this section we will discuss some main areas of development that lean more toward the emotional side of things.
The Development of Emotions
When babies are born, they experience basic emotions such as contentment, interest, and distress. After a few months they begin to experience fear, anger, happiness, and surprise. Later, when a sense of self develops, more emotions come along such as guilt, pride, and embarrassment. It is important for caregivers to allow children to express emotions freely and teach them to express them in socially appropriate ways. One good way to do this is through modeling. Teachers can express mild frustration and model strategies to overcome it. For example, a teacher may demonstrate frustration with difficulty opening a jar, and then take some deep breaths before asking another teacher for help. This is an effective way to teach children how to manage emotions. Another way is to read books that depict children overcoming difficult emotions so that children have a benchmark that they can relate to. Yet another approach is to encourage the expression of emotions through dance and art. Children may also act out emotions through pretend play. Providing children space and time to work through feelings in a pretend setting can be very helpful for the development of emotions.
Emotion Regulation
Emotion regulation refers to a child’s ability to control or modify one’s own emotions. It is closely tied to brain development. The underpinnings for emotion regulation begin in infancy with the infant’s primary caregiver. When a baby cries for attention and is soothed by the caregiver, it sends signals to the brain which help it to calm down. With this repeated action of crying and receiving comfort from a caregiver, the baby’s brain slowly learns the process of how to regulate emotions. This process takes a very long time, and more sophisticated emotion regulation occurs alongside executive function. This means that between the ages of 3 and 5 years, children are able to shake off frustrations in a socially acceptable way.
Children with ADHD can have persistent difficulty with emotion regulation, making it hard to succeed in school environments designed for typically developing children. These children need special support and understanding in order to work toward developing emotion coping strategies.
Reflection
What are some activities that you could create to support child development? On which developmental domains will your activities focus?


4.4: Snapchat
If you have a smartphone you have probably heard of the app Snapchat: it’s an app that allows you to connect and share images with your friends. The application allows you to send pictures, text messages, and videos that disappear after being opened. At least, that is how the app started. Snapchat has since grown to include games, public “stories,” and news headlines. The Snapchat app has around 158 million users daily with around 85% of those users being between the age of 18 and 35. Snapchat (or Snap Inc.) is a publicly traded company on the New York Stock Exchange , and is still ran by the original founders Evan Spiegel and Bobby Murphy. Facebook in the past has made offers to buy out the company, but as of yet, Snapchat has remained independent. The status of being independently owned and not associated with other large corporations has definitely shaped the platform in terms of what news they show and how Snapchat makes revenue.
Components of Snapchat
Chat : Friends who have each others’ usernames are able to send text messages that disappear once exited. The chat may also be used to respond directly to a friend’s image or video post. Users can choose to “save” messages if they wish.
Lenses : Lenses allow users to supplement their images with various filters and effects, including sponsored content for different companies and organizations.
Product identification : By pressing on one’s screen for a few seconds, one can identify songs with Shazam or find products on Amazon that most closely resemble the product selected within the frame.
Snapchat’s Influence on the News
Over the years Snapchat has accumulated more platforms within the app that allowed for the expansion of public content. From more notable sources like The New York Times , National Geographic , and The Wall Street Journal , to more blog-like stories such as SEVENTEEN and Cosmo , Snapchat now contains a wide variety of “story” content. With access to these stories, Snapchat users get a snippet of what is happening in the other parts of the world with celebrities, politics, and “shocking” events in the past 24 hours. Because Snapchat is able to freely publicize user material, news stories sometimes include images and videos from other users present at the scene of the story. Because these stories don’t require much time or reading, a user can find out about a current event within seconds before deciding if they want to look further into or continue on to other headlines. These news stories are broken up by video advertisements, and users can choose to subscribe to news platforms. Once subscribed, Snapchat’s algorithm presents the user with other stories that may interest them.
As stated by The Center for Digital Ethics and Policy , Snapchat presents itself as a camera app, not a media platform, and similar to Facebook, Snapchat does not feel responsible for the content that is placed on the platform. The news stories that are posted on the app generally originate in live pictures and videos posted by users, due to the nature of the app. With crowd sourcing information, it is extremely difficult to monitor the content and to fact check the events. In regards to the Discovery section of Snapchat, which presents users with news stories from news sources, only groups that are partnered with Snapchat are allowed to post. Due to this, Snapchat can pick and choose which sources they would like to be partnered with, and are able to cancel partnerships if Snapchat feels the information is not what they want. This can both be good and bad, on one hand Snapchat can eliminate false news from the platform, but on the other hand Snapchat can censor sources that they do not agree with.
Snapchat Trends
Snapchat’s stated mission is “At Snap, we believe that reinventing the camera represents our greatest opportunity to improve the way people live and communicate. We contribute to human progress by empowering people to express themselves, live in the moment, learn about the world, and have fun together.” This social media platform has pushed the boundaries of how people communicate, especially for teenagers. The current state of Snapchat is very different from what it was in its inception, and the popularity of the platform is too. Snapchat is reported to be the highest activity for social media at 35% among teens, according to Pew Research Center .
Snapchat’s arguably most recognizable trend is that the things you send or post “disappear”, except that is not necessarily the case. Users have the capabilities on their devices to “screenshot” or “screen record” pictures and videos they are sent personally, as well as the content posted on their friends’ stories (public posts that all of the users’ friends can view). Snapchat was the first to implement stories, and this trend has spread and been copied by the other giant social media platforms like Instagram, Facebook, and now Youtube as well.
According to Amnesty International , Snapchat has not done enough to ensure privacy among its users, and they do not put enough effort into educating their users about the dangers possible on the site. One of the bigger issues is that Snapchat holds users’ pictures and videos for up to 30 days on a company server , leaving that data susceptible to hackers. However, the newest “trend” on Snapchat is its relatively new advertising strategy that makes it possible for smaller and bigger brands alike to put ads through Snapchat, and this is done through Snapchat sending user data to companies so they can target their respective audiences. Pushing the boundary further, Snapchat has a new partnership with Amazon for product identification that involves users being able to scan items or barcodes with their camera to then be provided with a list of options to buy from Amazon. Convenient? Sure. Good for privacy? We will just have to wait and find out for ourselves.
Loopholes and Community Concerns
- Ability to circumvent age restrictions: Snapchat, like most social media sites requires users to be at least 13 years old. While this is true, people simply lie.
- Terms of service say you can’t do anything illegal: Because there isn’t anyone going through and checking every message sent to look for illegal activity, it is very easy for people to send messages involving, for example, illegal substances.
- CAN regulate users by reviewing any and all materials posted, but regulation is very lax.
- Snap Inc., is not held liable for any illegal and/or harmful content posted; more concerned about copyright infringement.
- Product identification: partnering with Amazon to identify and market Amazon products: the camera can identify products and suggest similar products for the user to buy.
- Access to geolocations of users unless turned off in settings: Snapchat can see where you are
- Access to all contacts and photos on user devices.
- May mimic your face, voice, or other traits for the purposes of advertising without compensation.
What Should Snapchat do?
While many other social media sites struggle with fake news media, Snapchat remains relatively free of misleading news. According to Bloomberg Businessweek , Snapchat is one of few social media sites that does not have an issue with fake news. As was stated in previous sections, Snapchat only posts content from sources who they have a partnership with. Due to Snapchat picking and choosing who to partner with, they have been able to be clear of fake news and only partner with reputable news sources. Their content chief has also said “If an individual story gets hundreds of thousands of views, a team of our editors looks at it.” If the story is checked and found to be true, it is then posted by Snapchat to the “Our Stories” section on Snapchat, which can function as a type of news. Snapchat has also already been actively fighting back against fake news. The company has reportedly been helping to fund research in creating an algorithm to detect fake news, this way they can assist other platforms in riding the internet of the problem.
One method of regulation that Snapchat has is community regulation. Individual users can report other users and content by their discretion. This can allow for Snapchat to eliminate illegal activities, and to eliminate user abuse. Another option for closing loopholes could be to require verification of user identity. This will hold people more accountable and make sure that users are the appropriate age to be using the application. Snapchat already allows an optional two-factor identification for users, requiring new accounts to verify with a mobile phone number. However there are other ways to go about verifying users, for instance they could use Twitter’s model of verification. This can also help to reduce the amount of fake bot accounts on the platform. On the other side, requiring verification could have negative impacts for people that run multiple accounts. For instance, a person who has an account for personal use and an account to use as a public figure. Verification can also lead to the false assumption that because a person is verified, the company supports them, when in reality it just means that the person is who they say they are.
Now we will leave it to you, the reader, do you think that Snapchat is doing enough? What would you like to see change?
Recommended Reading
“Facebook is Being Eclipsed by its Youthful Rival Snapchat” by Mark Sweney, The Guardian, Aug. 27, 2018.
“How Snapchat is Shaping Social Media” by Katie Benner, The New York Times , Nov. 30, 2016.


11.7: Understand Slope of a Line (Part 1)
- Use geoboards to model slope
- Find the slope of a line from its graph
- Find the slope of horizontal and vertical lines
- Use the slope formula to find the slope of a line between two points
- Graph a line given a point and the slope
- Solve slope applications
Before you get started, take this readiness quiz.
- Simplify: \(\dfrac{1 − 4}{8 − 2}\). If you missed this problem, review Example 4.6.12 .
- Divide: \(\dfrac{0}{4}, \dfrac{4}{0}\). If you missed this problem, review Example 7.5.5 .
- Simplify: \(\dfrac{15}{-3}, \dfrac{−15}{3}, \dfrac{−15}{−3}\). If you missed this problem, review Example 4.6.11 .
As we’ve been graphing linear equations, we’ve seen that some lines slant up as they go from left to right and some lines slant down. Some lines are very steep and some lines are flatter. What determines whether a line slants up or down, and if its slant is steep or flat?
The steepness of the slant of a line is called the slope of the line . The concept of slope has many applications in the real world. The pitch of a roof and the grade of a highway or wheelchair ramp are just some examples in which you literally see slopes. And when you ride a bicycle, you feel the slope as you pump uphill or coast downhill.
Use Geoboards to Model Slope
In this section, we will explore the concepts of slope.
Using rubber bands on a geoboard gives a concrete way to model lines on a coordinate grid. By stretching a rubber band between two pegs on a geoboard, we can discover how to find the slope of a line. And when you ride a bicycle, you feel the slope as you pump uphill or coast downhill.
We’ll start by stretching a rubber band between two pegs to make a line as shown in Figure \(\PageIndex{1}\).
Figure \(\PageIndex{1}\)
Does it look like a line?
Now we stretch one part of the rubber band straight up from the left peg and around a third peg to make the sides of a right triangle as shown in Figure \(\PageIndex{2}\). We carefully make a 90° angle around the third peg, so that one side is vertical and the other is horizontal.
Figure \(\PageIndex{2}\)
To find the slope of the line, we measure the distance along the vertical and horizontal legs of the triangle. The vertical distance is called the rise and the horizontal distance is called the run , as shown in Figure \(\PageIndex{3}\).
Figure \(\PageIndex{3}\)
To help remember the terms, it may help to think of the images shown in Figure \(\PageIndex{4}\).
Figure \(\PageIndex{4}\)
On our geoboard, the rise is 2 units because the rubber band goes up 2 spaces on the vertical leg. See Figure \(\PageIndex{5}\).
What is the run? Be sure to count the spaces between the pegs rather than the pegs themselves! The rubber band goes across 3 spaces on the horizontal leg, so the run is 3 units.
Figure \(\PageIndex{5}\)
The slope of a line is the ratio of the rise to the run. So the slope of our line is \(\dfrac{2}{3}\). In mathematics, the slope is always represented by the letter m.
The slope of a line is m = \(\dfrac{rise}{run}\).
The rise measures the vertical change and the run measures the horizontal change.
What is the slope of the line on the geoboard in Figure \(\PageIndex{5}\)?
\[\begin{split} m &= \dfrac{rise}{run} \\ m &= \dfrac{2}{3} \\ The\; line\; &has\; slope\; \dfrac{2}{3} \ldotp \end{split}\]
When we work with geoboards, it is a good idea to get in the habit of starting at a peg on the left and connecting to a peg to the right. Then we stretch the rubber band to form a right triangle.
If we start by going up the rise is positive, and if we stretch it down the rise is negative. We will count the run from left to right, just like you read this paragraph, so the run will be positive.
Since the slope formula has rise over run, it may be easier to always count out the rise first and then the run.
What is the slope of the line on the geoboard shown?
Solution
Use the definition of slope.
\[m = \dfrac{rise}{run}\]
Start at the left peg and make a right triangle by stretching the rubber band up and to the right to reach the second peg. Count the rise and the run as shown.
The rise is 3 units. \[m = \dfrac{3}{run}\]
The run is 4 units. \[m = \dfrac{3}{4}\]
The slope is \(\dfrac{3}{4}\).
What is the slope of the line on the geoboard shown?
- Answer
-
\(\frac{4}{3}\)
What is the slope of the line on the geoboard shown?
- Answer
-
\(\frac{1}{4}\)
What is the slope of the line on the geoboard shown?
Solution
Use the definition of slope.
\[m = \dfrac{rise}{run}\]
Start at the left peg and make a right triangle by stretching the rubber band to the peg on the right. This time we need to stretch the rubber band down to make the vertical leg, so the rise is negative.
The rise is −1.
\[m = \dfrac{−1}{run}\]
The run is 3.
\[\begin{split} m &= \dfrac{−1}{3} \\ m &= − \dfrac{1}{3} \end{split}\]
The slope is \(− \dfrac{1}{3}\).
What is the slope of the line on the geoboard?
- Answer
-
\(-\frac{2}{3}\)
What is the slope of the line on the geoboard?
- Answer
-
\(-\frac{4}{3}\)
Notice that in the first example, the slope is positive and in the second example the slope is negative. Do you notice any difference in the two lines shown in Figure \(\PageIndex{6}\).
Figure \(\PageIndex{6}\)
As you read from left to right, the line in Figure A, is going up; it has positive slope. The line Figure B is going down; it has negative slope.
Figure \(\PageIndex{7}\)
Use a geoboard to model a line with slope \(\frac{1}{2}\).
Solution
To model a line with a specific slope on a geoboard, we need to know the rise and the run.
| Use the slope formula. | $$m = \dfrac{rise}{run}$$ |
| Replace m with \(\dfrac{1}{2}\). | $$\dfrac{1}{2} = \dfrac{rise}{run}$$ |
So, the rise is 1 unit and the run is 2 units.
Start at a peg in the lower left of the geoboard. Stretch the rubber band up 1 unit, and then right 2 units.
The hypotenuse of the right triangle formed by the rubber band represents a line with a slope of \(\dfrac{1}{2}\).
Use a geoboard to model a line with the given slope: m = \(\dfrac{1}{3}\).
- Answer
-
Use a geoboard to model a line with the given slope: m = \(\dfrac{3}{2}\).
- Answer
-
Use a geoboard to model a line with slope \(\dfrac{−1}{4}\).
Solution
| Use the slope formula. | $$m = \dfrac{rise}{run}$$ |
| Replace m with \(− \dfrac{1}{4}\). | $$- \dfrac{1}{4} = \dfrac{rise}{run}$$ |
So, the rise is −1 and the run is 4.
Since the rise is negative, we choose a starting peg on the upper left that will give us room to count down. We stretch the rubber band down 1 unit, then to the right 4 units.
The hypotenuse of the right triangle formed by the rubber band represents a line whose slope is \(− \dfrac{1}{4}\).
Use a geoboard to model a line with the given slope: m = \(\dfrac{−3}{2}\).
- Answer
-
Use a geoboard to model a line with the given slope: m = \(\dfrac{−1}{3}\).
- Answer
-
Find the Slope of a Line from its Graph
Now we’ll look at some graphs on a coordinate grid to find their slopes. The method will be very similar to what we just modeled on our geoboards.
To find the slope, we must count out the rise and the run. But where do we start?
We locate any two points on the line. We try to choose points with coordinates that are integers to make our calculations easier. We then start with the point on the left and sketch a right triangle, so we can count the rise and run.
Find the slope of the line shown:
Solution
Locate two points on the graph, choosing points whose coordinates are integers. We will use (0, −3) and (5, 1).
Starting with the point on the left, (0, −3), sketch a right triangle, going from the first point to the second point, (5, 1).
| Count the rise on the vertical leg of the triangle. | The rise is 4 units. |
| Count the run on the horizontal leg. | The run is 5 units. |
| Use the slope formula. | $$m = \dfrac{rise}{run}$$ |
| Substitute the values of the rise and run. | $$m = \dfrac{4}{5}$$ |
The slope of the line is \(\dfrac{4}{5}\).
Notice that the slope is positive since the line slants upward from left to right.
Find the slope of the line:
- Answer
-
\(\frac{2}{5}\)
Find the slope of the line:
- Answer
-
\(\frac{3}{4}\)
Step 1. Locate two points on the line whose coordinates are integers.
Step 2. Starting with the point on the left, sketch a right triangle, going from the first point to the second point.
Step 3. Count the rise and the run on the legs of the triangle.
Step 4. Take the ratio of rise to run to find the slope.
\[m = \dfrac{rise}{run}\]
Find the slope of the line shown:
Solution
Locate two points on the graph. Look for points with coordinates that are integers. We can choose any points, but we will use (0, 5) and (3, 3). Starting with the point on the left, sketch a right triangle, going from the first point to the second point.
| Count the rise – it is negative. | The rise is −2. |
| Count the run. | The run is 3. |
| Use the slope formula. | $$m = \dfrac{rise}{run}$$ |
| Substitute the values of the rise and run. | $$m = \dfrac{-2}{3}$$ |
| Simplify. | $$m = - \dfrac{2}{3}$$ |
The slope of the line is \(− \dfrac{2}{3}\).
Notice that the slope is negative since the line slants downward from left to right.
What if we had chosen different points? Let’s find the slope of the line again, this time using different points. We will use the points (−3, 7) and (6, 1).
Starting at (−3, 7), sketch a right triangle to (6, 1).
| Count the rise. | The rise is −6. |
| Count the run. | The run is 9. |
| Use the slope formula. | $$m = \dfrac{rise}{run}$$ |
| Substitute the values of the rise and run. | $$m = \dfrac{-6}{9}$$ |
| Simplify the fraction. | $$m = - \dfrac{2}{3}$$ |
The slope of the line is \(− \dfrac{2}{3}\).
It does not matter which points you use—the slope of the line is always the same. The slope of a line is constant!
Find the slope of the line:
- Answer
-
\(-\frac{4}{3}\)
Find the slope of the line:
- Answer
-
\(-\frac{3}{5}\)
The lines in the previous examples had y-intercepts with integer values, so it was convenient to use the y-intercept as one of the points we used to find the slope. In the next example, the y-intercept is a fraction. The calculations are easier if we use two points with integer coordinates.
Find the slope of the line shown:
Solution
| Locate two points on the graph whose coordinates are integers. | (2, 3) and (7, 6) |
| Which point is on the left? | (2, 3) |
Starting at (2, 3), sketch a right angle to (7, 6) as shown below.
| Count the rise. | The rise is 3. |
| Count the run. | The run is 5. |
| Use the slope formula. | $$m = \dfrac{rise}{run}$$ |
| Substitute the values of the rise and run. | $$m = \dfrac{3}{5}$$ |
The slope of the line is \(\dfrac{3}{5}\).
Find the slope of the line:
- Answer
-
\(\frac{5}{4}\)
Find the slope of the line:
- Answer
-
\(\frac{3}{2}\)
Find the Slope of Horizontal and Vertical Lines
Do you remember what was special about horizontal and vertical lines? Their equations had just one variable.
- horizontal line y = b; all the y -coordinates are the same.
- vertical line x = a; all the x -coordinates are the same.
So how do we find the slope of the horizontal line y = 4? One approach would be to graph the horizontal line, find two points on it, and count the rise and the run. Let’s see what happens in Figure \(\PageIndex{8}\). We’ll use the two points (0, 4) and (3, 4) to count the rise and run.
Figure \(\PageIndex{8}\)
| What is the rise? | The rise is 0. |
| What is the run? | The run is 3. |
| What is the slope? | $$m = \dfrac{rise}{run} \tag{11.4.21}$$ |
| $$m = \dfrac{0}{3} \tag{11.4.22}$$ | |
| $$m = 0 \tag{11.4.23}$$ |
The slope of the horizontal line y = 4 is 0.
All horizontal lines have slope 0. When the y-coordinates are the same, the rise is 0.
The slope of a horizontal line, y = b, is 0.
Now we’ll consider a vertical line, such as the line x = 3, shown in Figure \(\PageIndex{9}\). We’ll use the two points (3, 0) and (3, 2) to count the rise and run.
Figure \(\PageIndex{9}\)
| What is the rise? | The rise is 2. |
| What is the run? | The run is 0. |
| What is the slope? | $$m = \dfrac{rise}{run} \tag{11.4.24}$$ |
| $$m = \dfrac{2}{0} \tag{11.4.25}$$ |
But we can’t divide by 0. Division by 0 is undefined. So we say that the slope of the vertical line x = 3 is undefined. The slope of all vertical lines is undefined, because the run is 0.
The slope of a vertical line, x = a, is undefined.
Find the slope of each line: (a) x = 8 (b) y = −5
Solution
(a) x = 8
This is a vertical line, so its slope is undefined.
(b) y = −5
This is a horizontal line, so its slope is 0.
Find the slope of the line: x = −4.
- Answer
-
undefined
Find the slope of the line: y = 7.
- Answer
-
0
Use the Slope Formula to find the Slope of a Line between Two Points
Sometimes we need to find the slope of a line between two points and we might not have a graph to count out the rise and the run. We could plot the points on grid paper, then count out the rise and the run, but there is a way to find the slope without graphing.
Before we get to it, we need to introduce some new algebraic notation. We have seen that an ordered pair (x, y) gives the coordinates of a point. But when we work with slopes, we use two points. How can the same symbol (x, y) be used to represent two different points?
Mathematicians use subscripts to distinguish between the points. A subscript is a small number written to the right of, and a little lower than, a variable.
- (x 1 , y 1 ) read x sub 1, y sub 1
- (x 2 , y 2 ) read x sub 2, y sub 2
We will use (x 1 , y 1 ) to identify the first point and (x 2 , y 2 ) to identify the second point. If we had more than two points, we could use (x 3 , y 3 ), (x 4 , y 4 ), and so on.
To see how the rise and run relate to the coordinates of the two points, let’s take another look at the slope of the line between the points (2, 3) and (7, 6) in Figure \(\PageIndex{10}\).
Figure \(\PageIndex{10}\)
Since we have two points, we will use subscript notation.
\[\begin{split} x_{1}, y_{1} \qquad &x_{2}, y_{2} \\ (2, 3) \qquad &(7, 6) \end{split}\]
On the graph, we counted the rise of 3. The rise can also be found by subtracting the y-coordinates of the points.
\[\begin{split} y_{2} &- y_{1} \\ 6 &- 3 \\ &\; 3 \end{split}\]
We counted a run of 5. The run can also be found by subtracting the x-coordinates.
\[\begin{split} x_{2} &- x_{1} \\ 7 &- 2 \\ &\; 5 \end{split}\]
| We know | $$m = \dfrac{rise}{run} \tag{11.4.26}$$ |
| So | $$m = \dfrac{3}{5} \tag{11.4.27}$$ |
| We rewrite the rise and run by putting in the coordinates. | $$m = \dfrac{6 - 3}{7 - 2} \tag{11.4.28}$$ |
| But 6 is the y-coordinate of the second point, y 2 and 3 is the y-coordinate of the first point y 1 . So we can rewrite the rise using subscript notation. | $$m = \dfrac{y_{2} - y_{1}}{7 - 2} \tag{11.4.29}$$ |
| Also 7 is the x-coordinate of the second point, x 2 and 2 is the x-coordinate of the first point x 2 . So we rewrite the run using subscript notation. | $$m = \dfrac{y_{2} - y_{1}}{x_{2} - x_{1}} \tag{11.4.30}$$ |
We’ve shown that m = \(\dfrac{y_{2} − y_{1}}{x_{2} − x_{1}}\) is really another version of m = \(\dfrac{rise}{run}\). We can use this formula to find the slope of a line when we have two points on the line.
The slope of the line between two points (x 1 , y 1 ) and (x 2 , y 2 ) is
\[m = \dfrac{y_{2} - y_{1}}{x_{2} - x_{1}} \tag{11.4.31}\]
Say the formula to yourself to help you remember it:
Slope is y of the second point minus y of the first point
over
x of the second point minus x of the first point.
Find the slope of the line between the points (1, 2) and (4, 5).
Solution
| We’ll call (1, 2) point #1 and (4, 5) point #2. | $$\begin{split} x_{1}, y_{1} \qquad &x_{2}, y_{2} \\ (1, 2) \qquad &(4, 5) \end{split}$$ |
| Use the slope formula. | $$m = \dfrac{y_{2} - y_{1}}{x_{2} - x_{1}} \tag{11.4.32}$$ |
| Substitute the values in the slope formula: | |
| y of the second point minus y of the first point | $$m = \dfrac{5 - 2}{x_{2} - x_{1}} \tag{11.4.33}$$ |
| x of the second point minus x of the first point | $$m = \dfrac{5 - 2}{4 - 1} \tag{11.4.34}$$ |
| Simplify the numerator and the denominator. | $$m = \dfrac{3}{3} \tag{11.4.35}$$ |
| m = 1 |
Let’s confirm this by counting out the slope on the graph.
The rise is 3 and the run is 3, so
\[\begin{split} m &= \dfrac{rise}{run} \\ m &= \dfrac{3}{3} \\ m &= 1 \end{split}\]
Find the slope of the line through the given points: (8, 5) and (6, 3).
- Answer
-
1
Find the slope of the line through the given points: (1, 5) and (5, 9).
- Answer
-
1
How do we know which point to call #1 and which to call #2? Let’s find the slope again, this time switching the names of the points to see what happens. Since we will now be counting the run from right to left, it will be negative.
| We’ll call (4, 5) point #1 and (1, 2) point #2. | $$\begin{split} x_{1}, y_{1} \qquad &x_{2}, y_{2} \\ (4, 5) \qquad &(1, 2) \end{split}$$ |
| Use the slope formula. | $$m = \dfrac{y_{2} - y_{1}}{x_{2} - x_{1}} \tag{11.4.36}$$ |
| Substitute the values in the slope formula: | |
| y of the second point minus y of the first point | $$m = \dfrac{2 - 5}{x_{2} - x_{1}} \tag{11.4.37}$$ |
| x of the second point minus x of the first point | $$m = \dfrac{2 - 5}{1 - 4} \tag{11.4.38}$$ |
| Simplify the numerator and the denominator. | $$m = \dfrac{-3}{-3} \tag{11.4.39}$$ |
| m = 1 |
The slope is the same no matter which order we use the points.
Find the slope of the line through the points (−2, −3) and (−7, 4).
Solution
| We’ll call (−2, −3) point #1 and (−7, 4) point #2. | $$\begin{split} x_{1}, y_{1} \qquad &x_{2}, y_{2} \\ (-2, -3) \qquad &(-7, 4) \end{split}$$ |
| Use the slope formula. | $$m = \dfrac{y_{2} - y_{1}}{x_{2} - x_{1}} \tag{11.4.40}$$ |
| Substitute the values. | |
| y of the second point minus y of the first point | $$m = \dfrac{4 - (-3)}{x_{2} - x_{1}} \tag{11.4.41}$$ |
| x of the second point minus x of the first point | $$m = \dfrac{4 - (-3)}{-7 - (-2)} \tag{11.4.42}$$ |
| Simplify. | $$m = \dfrac{7}{-5} \tag{11.4.43}$$ |
| m = \(- \dfrac{7}{5}\) |
Let’s confirm this on the graph shown.
\[\begin{split} m &= \dfrac{rise}{run} \\ m &= \dfrac{-7}{5} \\ m &= - \dfrac{7}{5} \end{split}\]
Find the slope of the line through the pair of points: (−3, 4) and (2, −1).
- Answer
-
-1
Find the slope of the line through the pair of points: (−2, 6) and (−3, −4).
- Answer
-
10
Contributors and Attributions
-
Lynn Marecek (Santa Ana College) and MaryAnne Anthony-Smith (Formerly of Santa Ana College). This content is licensed under Creative Commons Attribution License v4.0 "Download for free at<EMAIL_ADDRESS>."


24.7: Energy and Heat Balance
By the end of this section, you will be able to:
- Describe how the body regulates temperature
- Explain the significance of the metabolic rate
The body tightly regulates the body temperature through a process called thermoregulation , in which the body can maintain its temperature within certain boundaries, even when the surrounding temperature is very different. The core temperature of the body remains steady at around 36.5–37.5 °C (or 97.7–99.5 °F). In the process of ATP production by cells throughout the body, approximately 60 percent of the energy produced is in the form of heat used to maintain body temperature. Thermoregulation is an example of negative feedback.
The hypothalamus in the brain is the master switch that works as a thermostat to regulate the body’s core temperature (Figure \(\PageIndex{1}\)). If the temperature is too high, the hypothalamus can initiate several processes to lower it. These include increasing the circulation of the blood to the surface of the body to allow for the dissipation of heat through the skin and initiation of sweating to allow evaporation of water on the skin to cool its surface. Conversely, if the temperature falls below the set core temperature, the hypothalamus can initiate shivering to generate heat. The body uses more energy and generates more heat. In addition, thyroid hormone will stimulate more energy use and heat production by cells throughout the body. An environment is said to be thermoneutral when the body does not expend or release energy to maintain its core temperature. For a naked human, this is an ambient air temperature of around 84 °F. If the temperature is higher, for example, when wearing clothes, the body compensates with cooling mechanisms. The body loses heat through the mechanisms of heat exchange.
Mechanisms of Heat Exchange
When the environment is not thermoneutral, the body uses four mechanisms of heat exchange to maintain homeostasis: conduction, convection, radiation, and evaporation. Each of these mechanisms relies on the property of heat to flow from a higher concentration to a lower concentration; therefore, each of the mechanisms of heat exchange varies in rate according to the temperature and conditions of the environment.
Conduction is the transfer of heat by two objects that are in direct contact with one another. It occurs when the skin comes in contact with a cold or warm object. For example, when holding a glass of ice water, the heat from your skin will warm the glass and in turn melt the ice. Alternatively, on a cold day, you might warm up by wrapping your cold hands around a hot mug of coffee. Only about 3 percent of the body’s heat is lost through conduction.
Convection is the transfer of heat to the air surrounding the skin. The warmed air rises away from the body and is replaced by cooler air that is subsequently heated. Convection can also occur in water. When the water temperature is lower than the body’s temperature, the body loses heat by warming the water closest to the skin, which moves away to be replaced by cooler water. The convection currents created by the temperature changes continue to draw heat away from the body more quickly than the body can replace it, resulting in hyperthermia. About 15 percent of the body’s heat is lost through convection.
Radiation is the transfer of heat via infrared waves. This occurs between any two objects when their temperatures differ. A radiator can warm a room via radiant heat. On a sunny day, the radiation from the sun warms the skin. The same principle works from the body to the environment. About 60 percent of the heat lost by the body is lost through radiation.
Evaporation is the transfer of heat by the evaporation of water. Because it takes a great deal of energy for a water molecule to change from a liquid to a gas, evaporating water (in the form of sweat) takes with it a great deal of energy from the skin. However, the rate at which evaporation occurs depends on relative humidity—more sweat evaporates in lower humidity environments. Sweating is the primary means of cooling the body during exercise, whereas at rest, about 20 percent of the heat lost by the body occurs through evaporation.
Metabolic Rate
The metabolic rate is the amount of energy consumed minus the amount of energy expended by the body. The basal metabolic rate (BMR) describes the amount of daily energy expended by humans at rest, in a neutrally temperate environment, while in the postabsorptive state. It measures how much energy the body needs for normal, basic, daily activity. About 70 percent of all daily energy expenditure comes from the basic functions of the organs in the body. Another 20 percent comes from physical activity, and the remaining 10 percent is necessary for body thermoregulation or temperature control. This rate will be higher if a person is more active or has more lean body mass. As you age, the BMR generally decreases as the percentage of less lean muscle mass decreases.