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The following video will provide a step-by-step guide to note-taking using the Cornell Method. The video is 2 minutes and 18 seconds long.
Note-taking 101 using the Cornell Method (2:18)
Reference
YouTube. (2015). Note-taking 101 using the Cornell method. YouTube. https://www.youtube.com/watch?v=HJCnqj7j7rU
Watch the following video about Reading Comprehension:
Reading Comprehension (1:40)
Reading aloud integrates auditory learning of listening to the words and tactile-kinesthetic learning of the act of saying the words aloud, which intrinsically builds reading comprehension. It is easier to remember what you read when you have not only visually read the words but also hear them while reading out loud.
Making a conscious effort to identify the main idea of a passage can help improve your reading comprehension.
Identifying a couple of supporting details or information about the main idea helps you get a more complete back story of who or what the passage is about.
We love using graphic organizers to improve reading comprehension! Note-taking graphic organizers can help you organize the main idea and details of a passage in a visual way.
Can you quickly identify the structure of the reading selection?
Answering questions about what you have read can help you build your active recall skills.
Generating questions is a great way to stretch yourself as a reader. What can you ask that you could answer from reading the selection?
Put the reading selection into your own words and summarize what you have read. When you use your own words to describe what you have read, it makes it more relatable and more memorable to you.
Identifying and using the vocabulary you have read can help you further your understanding of the reading. Were there any words that you did not understand? Try to infer the meaning of the word from the context it was in.
Having background knowledge in various areas can help you make more connections to what you have read and more easily understand the reading selection.
For more information about how to improve reading comprehension, read 10 Best Practices to Improve Reading Comprehension
Research shows that you retain more when you actively engage and interact with texts, as opposed to simply reading and re-reading without a clear purpose.
Although many students don’t think about this step, engaging with a text before reading can crucially boost your understanding and retention. Below are some active reading strategies to use before you read.
Consider your purpose for reading and what you need to be able to understand, know, or do after reading. Keep this purpose in mind as you read.
Before previewing the text, determine what you already know about the material you are to read. Think about how the reading relates to other course topics, and ask why your professor might have assigned the text.
Give the text an initial glance, noting headings, diagrams, tables, pictures, bolded words, summaries, and key questions.
Plan to break your reading into manageable chunks
Taking breaks while reading improves focus, motivation, understanding, and retention.
Especially if you are taking courses online or studying remotely, some of your course materials may be in a digital format, such as online journal articles or electronic textbooks. Before you read, decide if your reading is something you could and would want to print out.
Keeping your brain active and engaged while you read decreases distractions, mind-wandering, and confusion. Try some of these strategies to keep yourself focused on the text and engaged in critical thinking about the text while you read.
As soon as you notice your mind drifting, STOP and consider your needs. Do you need a break? Before resuming, summarize the last chunk of text you remember to make sure that you know the appropriate starting point.
Overusing the highlighter? Put it down and try annotation. Develop a key/system to note the following in the text: key ideas/major points, unfamiliar words/unclear information, key words and phrases, important information, and connections.
After reading small sections of texts (a couple of paragraphs, a page, or a chunk of text separated by a heading or subheading), summarize the main points and two or three key details in your own words. These summaries can serve as the base for your notes while reading.
Think like a professor and ask yourself higher level, critical thinking questions.
Reading a text should not end at the end of the chapter. Using effective after reading strategies can help you better understand and remember the text long-term.
Whether you read a printed text or an online document, the most important thing to assess is how much you understood from your reading. Try “cross-referencing” the information you read with simpler writings on the same subject and discussing your takeaways with peers.
If any information remains unclear, locate other resources related to the topic such as a trusted video source or web-based study guide. Still have questions you can’t answer on your own? Make note of them to ask a professor, TA, or classmate.
For more details on these reading tips, go to Reading Comprehension Tips
Citations
Learning Skills, Reading Comprehension. (January 27, 2016). https://www.youtube.com/watch?v=3ajqvi9glvY
Reading Comprehension Tips. (2024). The Learning Center - University of North Carolina at Chapel Hill. https://learningcenter.unc.edu/tips-and-tools/the-study-cycle/
Terry, B. (April 3, 2024). 10 Best Practices to Improve Reading Comprehension. Scholars Within. https://scholarwithin.com/10-best-practices-to-improve-reading-comprehension
Watch the video below to get introduced to test-taking strategies:
Test Taking Strategies (4:04)
Understanding and remembering information for a test takes time, so developing good study habits long before test day is really important.
It might seem like a good idea to spend hours memorizing the material you need the night before the test.
Before going to bed (early, so you get a good night’s sleep), gather everything you need for the test and have it ready to go.
And speaking of sleep…showing up to your test well-rested is one of the best things you can do to succeed on test day.
Like sleeping, eating is an important part of self-care and test taking preparation. After all, it’s hard to think clearly if your stomach is grumbling.
Arriving early at a test location can help decrease stress. And it allows you to get into a positive state of mind before the test starts.
Don’t underestimate the importance of confidence and a positive mindset in test preparation.
Once the test is front of you, it’s tempting to block everything out so you can get started right away.
If possible, look over the entire test quickly before you get started. Doing so will help you understand the structure of the test and identify areas that may need more or less time.
For certain types of tests, remembering facts, data, or formulas is key. For these tests, it can be helpful to take a few minutes to write down all the information you need on a scrap paper before you get started.
When possible, do a first pass through the test to answer the “easy” questions or the ones you know right away. When you come to a question that you can’t answer (relatively) quickly, skip it on this first pass.
Once you’ve done a first pass, you now have to go back and answer the questions you skipped.
Once you think you’ve answered all the questions, double check to make sure you didn’t miss any. Check for additional questions on the back of the paper, for instance, or other places that you might have missed or not noticed during your initial read-through.
Finally, if you have time left, go back through the test and check your answers.
For more details on these tips, go to 14 Tips for Test Taking Success
To be ready for a test, you need to prepare. Check your study habits by reviewing the 5 steps of The Study Cycle
Citations
Avella, F. (April 30, 2019). Test Taking Strategies. YouTube. https://www.youtube.com/watch?v=uzZ3mDF1hns
Emerson, M. (September 29, 2022). 14 Tips for Test Taking Success. Harvard Summer School. https://summer.harvard.edu/blog/14-tips-for-test-taking-success/
The Study Cycle. (2024). The Learning Center - University of North Carolina at Chapel Hill. https://learningcenter.unc.edu/tips-and-tools/the-study-cycle/
The goal of balancing chemical equations is to ensure that the number of atoms of each element is the same on both sides of the equation. Let's take a look at what this means using the chemical equation for making water.
H2 + O2 ---> H2O
REACTANTS PRODUCT
There are 2 atoms of the element Hydrogen (H) and 2 atoms of the element Oxygen (O) on the reactant side. On the product side, there are 2 atoms of the element Hydrogen (H) and 1 atom of the element Oxygen (O) on the product side. Since the numbers of atoms of the elements are not the same on both sides of the chemical equation, this equation is not balanced.
To balance the equation we use coefficients (the numbers before the chemical formulas) but we cannot change the subscripts (the numbers within the chemical formulas). We can follow three steps to balance equations.
Step 1: Count each type of atom in reactants and products. Does the same number of each atom appear on both sides of the equation? If not, the equation is not balanced, and you need to go to Step 2.
Step 2: Place coefficients, as needed, in front of the formulas to increase the number of atoms or molecules of the substances. Use the smallest coefficients possible. Never change the subscripts in chemical formulas. Changing subscripts changes the substances in the reaction. Change only the coefficients.
Step 3: Repeat steps 1 and 2 until the equation is balanced. Check your work.
Example 1. Balancing Chemical Equations
Let's apply these steps to balancing the equation for water.
H2 + O2 ---> H2O
Step 1. Count each type of atom in reactants and products. The number in parentheses indicates the number of atoms of each element.
Reactants Products
H (2) H (2)
O (2) O (1)
Step 2. We can place a coefficient of 2 in the front of H2O to balance the oxygen. Next, place a 2 in front of H2 to balance the hydrogen.
__2 H2 + ____O2 -----> __2 H2O
Step 3. Check your work.
Reactants Products
H (4) H (4)
O (2) O (2)
Take a look at another example of how to balance a chemical equation.
Video Tutorial - Balancing Chemical Equations (5:02)
Example 2. Balancing Hard Chemical Equations
Now let's consider a more challenging equation. Combustion (that is, burning) of the natural gas ethane (C2H6) in oxygen or air, yields carbon dioxide (CO2) and water.
____ C2H6 + ____O2 ----> ____CO2 + ___ H2O
Step 1. Count each type of atom in reactants and products.
Reactants Products
C (2) C (1)
H (6) H (2)
O (2) O (3)
Step 2. Let's focus on balancing the C atoms first. We place a coefficient of 2 in front of CO2:
____ C2H6 + ____O2 ----> __2 CO2 + ___ H2O
To balance the H atoms, we place a 3 in front of H2O:
____ C2H6 + ____O2 ----> __2 CO2 + _3 H2O
At this stage, the C and H atoms are balanced, but the O atoms are not balanced, because there are seven O atoms on the right side and only two O atoms on the left side of the equation. Since there are two O atoms on the left side, we need a fraction that when multiplied by two will result in seven O atoms. We balance the O atoms by writing in front of the O2 on the left side of the equation.
____ C2H6 + O2 ----> __2 CO2 + __3 H2O
Step 3. Check your work.
Reactants Products
C (2) C (2)
H (6) H (6)
O (7) O (7)
However, whole number coefficients are often preferred to fractional coefficients. To change the fraction to a whole number, multiply the entire equation by the denominator of the fraction. See the balanced equation with whole number coefficients below:
__2 C2H6 + _7_O2 ----> __4_CO2 + __6 H2O
Reactants Products
C (4) C (4)
H (12) H (12)
O (14) O (14)
Check out this tutorial on how to balance a similar chemical equation with fractional coefficients.
Video Tutorial - Balancing Hard Chemical Equations (4:36)
Example 3. Balancing Very Hard Chemical Equations
Sometimes you may have to balance chemical equations with formulas that contain a number of different types of atoms on both sides of an equation. We can use the same steps to balance these equations.
Let's consider the following example:
A neutralization reaction (that is when an acid and base react) between Phosphoric acid (H3PO4) and Potassium hydroxide (KOH) yields Potassium phosphate (K3PO4) and water (H2O).
___H3PO4 + ___KOH ----> ___K3PO4 + ___H2O
Step 1. Count each type of atom in reactants and products.
Reactants Products
H (4) H (2)
P (1) P (1)
O (5) O (5)
K (1) K (3)
Step 2.
We need to balance the H and K atoms on both sides of the equation. Let's start with K because the one K atom on the left side may be easier to balance. We place a coefficient of 3 in front of KOH:
___H3PO4 + _3_KOH ----> ___K3PO4 + ___H2O
To balance the H atoms, we place a 3 in front of H2O:
___H3PO4 + _3_KOH ----> ___K3PO4 + __3_H2O
Step 3. Check your work.
Reactants Products
H (6) H (6)
P (1) P (1)
O (7) O (7)
K (3) K (3)
Checkout the tutorial on balancing very hard chemical equations.
Video Tutorial - Balancing Very Hard Chemical Equations (7:48)
1. Measuring GDP:
How is GDP measured, and who measures it?
Government economists at the Bureau of Economic Analysis (BEA), a division of the U.S. Department of Commerce, compile GDP estimates from a wide range of sources. Measuring GDP involves counting the production of millions of different goods and services, such as automobiles, appliances, furniture, smartphones, cars, music downloads, computers, steel, bananas, college educations, and all other new goods and services that a country produced in the current year. These quantities are then summed into a total dollar value, which gives us the GDP. The process is straightforward: take the quantity (Q) of each item produced, multiply it by the price (P) at which each item is sold, and sum these amounts to calculate the GDP.
Before we delve into the calculation of GDP, let's first examine some items that are excluded from GDP calculations.
2. Exclusions from GDP calculations:
i. Second-Hand sales (transactions): GDP includes only current transactions.
Examples: A person sells their used car to another individual. The sale of the used car is excluded because it does not represent new production. The car's value was already included in GDP when it was first sold as a new car. Pre-Owned Furniture: Someone sells their old sofa through an online marketplace. The transaction is excluded from GDP because the sofa was counted in GDP when it was originally purchased new. Clothes are sold at a thrift store. The sale of second-hand clothes is excluded because these items were already included in GDP when they were first sold as new
ii. Non-Productive transactions: GDP does not count purely private or public financial transactions.) Examples: Financial transactions: Buying and selling stocks and bonds. These transactions represent transfers of ownership of financial assets rather than the production of new goods or services. Private Transfers: Gifts of money between individuals. These are private financial transfers without any corresponding production of goods or services, so they are not included in GDP. Government Transfer Payments: Social security payments, unemployment benefits, and welfare. Exclusion Explanation: These payments are redistributions of income and do not correspond to the production of new goods or services. They are therefore excluded from GDP calculations. Inheritances: Receiving an inheritance from a deceased relative. Exclusion Explanation: Inheritances represent transfers of existing wealth and not new production, so they are not included in GDP.
iii. Intermediate goods: Intermediate goods are excluded from GDP calculations to prevent double counting. Examples: Steel purchased by an automobile manufacturer. Exclusion Explanation: The value of steel is excluded because it is incorporated into the final product, a car. Only the value of the car, the final good, is included in GDP to avoid double counting. Flour bought by a bakery to make bread. Exclusion Explanation: The flour is excluded as an intermediate good. The value of the bread, a final product sold to consumers, is included in GDP. Cotton fabric used by a clothing manufacturer to make shirts. Exclusion Explanation: The value of the cotton fabric is not counted separately. Instead, the value of the finished shirts, the final goods, is included in GDP. Wood used by a furniture manufacturer to make chairs. Exclusion Explanation: The wood is not counted separately. The value of the finished chairs, the final products, is included in GDP.
Generally, there are three methods to calculate GDP: (i) the production/value-added approach, (ii) the expenditure approach, and (iii) the income approach. However, in this note, we will focus only on the widely used expenditure and income approaches.
i. The expenditure approaches
Every time consumers purchase final goods and services, the cost is recorded, akin to a cash register transaction. The total expenditure on these items over a given year constitutes the GDP. Economists often categorize this expenditure into four main groups, reflecting different sources of demand: (i)Personal Consumption Expenditures (C) this includes spending on durable goods (items expected to last more than three years), non-durable goods (items with a shorter lifespan), and services (such as healthcare, education, and entertainment). (ii) Gross Private Domestic Investments (I), this category includes both fixed investment (such as business expenditures on equipment and structures) and changes in business inventories (goods produced but not yet sold). (iii) Government Consumption Expenditures and Gross Investment (G), This includes spending by federal, state, and local governments on goods and services, as well as investment in infrastructure and other public projects. (iv) Net Exports (X-M), This is calculated as the value of a country's exports (X) minus its imports (M). Exports are goods and services produced domestically in the United States and sold abroad, while imports are those purchased from other countries. Net exports can be positive, negative, or zero, depending on the balance between a country's exports and imports. Positive Net Exports: When a country exports more goods and services than it imports, it has a trade surplus. Negative Net Exports: When a country imports more goods and services than it exports, it has a trade deficit. Zero Net Exports: If a country's exports and imports are equal, net exports will be zero, indicating a balanced trade situation with no net impact on the GDP.
By summing up these categories, we get the GDP using the expenditure approach. As of 2024, personal consumption expenditure constitutes the largest component of the US GDP, with services being the largest category within this component. Detailed data can be accessed at https://www.bea.gov/data/gdp/gross-domestic-product.
Formula of GDP using expenditure approaches:
ii. The income Approach
The primary method for estimating GDP is the Expenditure Approach, but there's also the Income Approach. Instead of summing up all the expenditures by consumers, businesses, and the government on goods and services, government economists can aggregate all the incomes received by economic agents, such as households, businesses, and governments. This involves totaling all incomes generated within a specific period, including compensation of employees, rents, net interest, and profits.
For example, when a firm sells its production, it generates revenue. This revenue is then used to pay wages and salaries for labor, interest and dividends for capital, rent for land, and profit to the entrepreneur. By summing up all the income produced in a year, we obtain an alternative way to measure GDP.
Both the expenditure approach and the income approach yield the same GDP calculations, demonstrating that the total value of a nation’s output equals its total income. This equivalence is because what is spent by one economic agent constitutes income received by others.
Components of the income approach to GDP measurement include: (i) Compensation of employees: This includes income earned from wages, salaries, and certain supplements paid by firms and government to labor suppliers. (ii) Rental incomes: This category covers rent and royalties received by property owners who allow others to use their assets over a period. (iii) Profits: This encompasses both proprietor’s income (income of sole proprietors and partnerships) and corporate profits (income earned by corporations). (iv) Taxes: This includes various taxes such as sales tax, excise tax, and customs duties, which are part of the income generated in the economy. (v) Depreciation: This represents an allowance for the wear and tear or reduction in value of capital goods used in production, which is deducted to reflect the true value added to GDP.
Formula of GDP using the income approach:
Where: W: Wages – Compensation of employees. R: Rent – Income from property. i: Interest – Income from investments. P: Profits – Corporate profits. Indirect Taxes – Taxes on production and imports (e.g., sales tax, VAT). Subsidies – Government payments to businesses. Depreciation – Capital consumption allowance, accounting for wear and tear on capital assets.
1. Marginal Cost Formula:
Mathematically, is the change in total cost (
) divided by the change in output/quantity (
): In other words,
is the ratio of the change in total cost to a 1-unit change in
. Written as a formula:
: is called delt and represents change
: represents change in total cost
: represents is change in quantity produced
can also be obtained by taking the first derivative of the
function, expressed as:
The total cost can also be expressed in functional form as:
where: TC(Q) is the total cost as a function of Q, TFC is the fixed cost, TVC is the variable cost, which vary with quantity, and Q represents the quantity of goods produced.
Marginal cost is derived from the total cost function by taking the first derivatives with respect to the quantity produced (Q).
Mathematically it is expressed as:
Where: is the first derivatives of the total cost function with respect to Q.
Since the total cost function is: , we can rewrite this derivative as:
Given TFC is constant, so
Given that TFC is constant, . Therefore, the equation simplifies to:
Since fixed costs do not change as output increases, depends only on the variable cost.
It’s important to note that changes in production costs are not necessarily linear. Many companies encounter certain threshold points where costs change significantly. In between these points, changing the output volume may have little to no effect on the overall cost. This nonlinearity can be due to factors like economies of scale, changes in production technology, or shifts in the availability and cost of inputs.
Examples:
2. Suppose that a company has a cost function of C(Q)=100+10Q, where C(Q) is the total cost and Q is the number of units produced. What is the marginal cost of producing an additional unit?
Solution: The MC is the derivative of the TC function with respect to Q.
3. Suppose a farmer grows vegetables with a cost function , where
is the number of kilograms harvested. What is the marginal cost of harvesting an additional kilogram of vegetables?
Solutions: = Rate of change of
with respect to
4. Suppose a scenario where a company experiences a significant increase in production efficiency once production exceeds 20 units. For quantities below this threshold, the cost function might be:
. However, for quantities above 20 units, the cost function might change to
, reflecting the improved efficiency. Calculate at different thresholds at
and
respectively.
For
For
For
For
This example demonstrates how can change significantly at different production levels due to nonlinear cost functions. Understanding these thresholds and their implications helps companies optimize production and manage costs effectively.
Check out this short video explaining marginal costs: Marginal Cost : Formula and How to Calculate It (3:05)
2. Types of cost Curves
A cost curve represents the relationship between output and the different cost measures involved in producing the output. Cost curves are visual descriptions of the various costs of production. In order to maximize profits, firms need to know how costs vary with output, so cost curves are vital to the profit maximization decisions of firms.
We can decompose costs into fixed and variable costs.
1. Fixed costs are the costs of the fixed inputs (e.g. capital). Because fixed inputs do not change in the short run, fixed costs are expenditures that do not change regardless of the level of production. Whether you produce a great deal or a little, the fixed costs are the same. One example is the rent on a factory or a retail space. Once you sign the lease, the rent is the same regardless of how much you produce, at least until the lease expires. Fixed costs can take many other forms: for example, the cost of machinery or equipment to produce the product, research and development costs to develop new products, even an expense like advertising to popularize a brand name. The amount of fixed costs varies according to the specific line of business: for instance, manufacturing computer chips requires an expensive factory, but a local moving and hauling business can get by with almost no fixed costs at all if it rents trucks by the day when needed.
From the total fixed cost (), we can calculate the average fixed cost (
), which represents the fixed cost per unit of output. The formula for the
is:
The declines as output increases. This happens because fixed costs, which do not change with the level of production, are spread over an increasing number of units. As a result, the per-unit share of the fixed cost decreases, leading to a downward-sloping
curve.
Example: Consider a company with fixed costs of $1000. Here's how the changes as output increases:
At
At
At
Figure 1: The graph of Average fixed cost
2. Variable costs are the costs of the variable inputs (e.g. labor). The only way to increase or decrease output is by increasing or decreasing the variable inputs. Therefore, variable costs increase or decrease with output. We treat labor as a variable cost, since producing a greater quantity of a good or service typically requires more workers or more work hours. Variable costs would also include raw materials.
From the total variable costs (), we can derive the average variable cost (
) by dividing the
by the total output. This
represents the cost incurred per unit of output produced. By calculating this metric, we gain insights into the cost efficiency of production and can better understand how changes in output levels impact overall variable costs.
Formula:
Example: Give the function.
At
Average cost () refers to the
incurred per unit of output produced. It is calculated by dividing the
by the total quantity of output. The
is composed of two key components:
and
.
Formula: , Since,
,
Since, and
Figure 2: The graph of MC, AVC and AC
1. Steps to Solve Market Equilibrium
Step I: Identify the Demand and Supply Equations:
The demand equation typically has the form of: , where
is the quantity demanded,
is the price,
is the constant intercept, and
is the slope of the demand equation.
The supply equation typically has the form of: , where
is the quantity supplied,
the constant intercept, and
is the slop of the supply function.
Step II: Set the Demand Equation Equal to the Supply Equation:
To find the equilibrium, set equal to
:
=
Step III: Solve for the Equilibrium and rearrange the equation to solve for
Step IV: Find the Equilibrium Quantity
Substitute back into either the demand or supply equation to find the equilibrium quantity.
Given the demand equation:
And substitute the equilibrium price
Examples:
Demand:
Supply: , Then, find equilibrium price (
) and quantity (
).
Answer: To find the equilibrium, set :
=
Combine like terms: =
Solve for :
Now, substitute back into either the demand or supply equation to find
:
So, the equilibrium price is 8 and the equilibrium quantity
is 34.
Video: Solving The Equilibrium: https://study.com/learn/lesson/video/equilibrium-price-economics-calculation.html
Video Showing Graphical Analysis of Market Equilibrium
The Equilibrium Price and Quantity (4:50)
In academic writing, it is essential to understand how to effectively use paraphrasing and direct quotes, along with proper MLA in-text citations.
Paraphrasing is to restate the information in your own words, whereas direct quotes use the original words of a source. Paraphrasing allows you to convey ideas from your sources in a clear and concise manner. When paraphrasing, it is absolutely necessary to rephrase the information without changing the original meaning.
For example:
Original text: "Climate change is a pressing global issue that requires immediate action" (Keovongsa 79).
Paraphrased text: Addressing climate change is an urgent matter on a worldwide scale that demands prompt attention (Keovongsa 79).
Direct quotes keep the exact wording of a source to emphasize a point and/or provide evidence. When incorporating direct quotes, it is important to integrate them into your writing in a smooth manner.
For example:
Original text: "Climate change is a pressing global issue that requires immediate action" (Keovongsa 79).
Direct quote in your writing: According to The American Journal of Climate Change, "Climate change is a pressing global issue that requires immediate action" (Keovongsa 79).
When incorporating direct quotes into your writing, it's essential to know how to effectively shorten or modify them to cater to your own writing while maintaining the original meaning. There are various scenarios where you might want to use parts of a quote but not the entire quote, or where modifying a quote using brackets becomes necessary.
Sometimes, a direct quote may contain excessive information that is not relevant to your argument or point. In such cases, you can selectively choose the essential parts of the quote that support your idea. By omitting these irrelevant sections, you can focus on the key pieces of the quote that strengthen your own writing. Remember to indicate the omission of text by using an ellipsis ( … ) within the quote.
For example:
Direct quote:
"Climate change is a pressing global issue that requires immediate action to protect the environment, human health, and future generations' well-being" (Keovongsa 79).
Shortened quote with ellipsis:
"Climate change is a pressing global issue that requires immediate action to protect the environment ..." (Keovongsa 79).
When a direct quote needs slight alterations to fit into your writing or to clarify a pronoun reference, you can use brackets to make these modifications without changing the original meaning. For example, if the original quote refers to a specific person, you can replace the pronoun with the person's name enclosed in brackets to provide clarity.
For example:
Direct quote:
"Climate change is a pressing global issue that requires immediate action to protect the environment, human health, and future generations' well-being" (Keovongsa 79).
Modified quote with brackets:
"[Addressing climate change] is a pressing global issue that requires immediate action to protect the environment, human health, and future generations' well-being." (Keovongsa 79).
It's important to note that brackets should not be used to alter the meaning of a quote. The main purpose of modifying a quote using brackets is to ensure that the quote seamlessly integrates into your writing while maintaining the original meaning.
It is necessary to give credit to the original sources you use in your paper. MLA in-text citations help readers locate the full citation on the Works Cited page and give insight to your research. MLA In-text citations are formatted the same for paraphrased information and direct quotes.
Formatting MLA in-text citations:
Example of MLA in-text citations:
Addressing climate change is an urgent matter on a worldwide scale that demands prompt attention (Keovongsa 79).
The article states, "Climate change is a pressing global issue that requires immediate action" (Keovongsa 79).
In academic writing, it is essential to understand how to effectively use paraphrasing and direct quotes, along with proper APA in-text citations.
Paraphrasing is to restate the information in your own words, whereas direct quotes use the original words of a source. Paraphrasing allows you to convey ideas from your sources in a clear and concise manner. When paraphrasing, it is absolutely necessary to rephrase the information without changing the original meaning.
For example:
Original text: "Climate change is a pressing global issue that requires immediate action" (Keovongsa, 2009, p. 79).
Paraphrased text: Addressing climate change is an urgent matter on a worldwide scale that demands prompt attention (Keovongsa, 2009, p.79).
Direct quotes keep the exact wording of a source to emphasize a point and/or provide evidence. When incorporating direct quotes, it is important to integrate them into your writing in a smooth manner.
For example:
Original text: "Climate change is a pressing global issue that requires immediate action" (Keovongsa, 2009, p.79).
Direct quote in your writing: According The American Journal of Climate Change, "Climate change is a pressing global issue that requires immediate action" (Keovongsa, 2009, p.79).
When incorporating direct quotes into your writing, it's essential to know how to effectively shorten or modify them to cater to your own writing while maintaining the original meaning. There are various scenarios where you might want to use parts of a quote but not the entire quote, or where modifying a quote using brackets becomes necessary.
Sometimes, a direct quote may contain excessive information that is not relevant to your argument or point. In such cases, you can selectively choose the essential parts of the quote that support your idea. By omitting these irrelevant sections, you can focus on the key pieces of the quote that strengthen your own writing. Remember to indicate the omission of text by using an ellipsis ( … ) within the quote.
For example:
Direct quote:
"Climate change is a pressing global issue that requires immediate action to protect the environment, human health, and future generations' well-being" (Keovongsa, 2009, p.79).
Shortened quote with ellipsis:
"Climate change is a pressing global issue that requires immediate action to protect the environment ..." (Keovongsa, 2009, p.79).
When a direct quote needs slight alterations to fit into your writing or to clarify a pronoun reference, you can use brackets to make these modifications without changing the original meaning. For example, if the original quote refers to a specific person, you can replace the pronoun with the person's name enclosed in brackets to provide clarity.
For example:
Direct quote:
"Climate change is a pressing global issue that requires immediate action to protect the environment, human health, and future generations' well-being" (Keovongsa, 2009, p.79).
Modified quote with brackets:
"[Addressing climate change] is a pressing global issue that requires immediate action to protect the environment, human health, and future generations' well-being." (Keovongsa, 2009, p.79).
When not to use brackets:
Note that brackets should not be used to alter the meaning of a quote. The main purpose of modifying a quote using brackets is to ensure that the quote seamlessly integrates into your writing while maintaining the original meaning.
In academic writing, it is essential to give credit to the original sources you use in your paper. APA in-text citations help readers locate the full citation on the Reference page and give insight to your research. The APA In-text citation format is the same for paraphrased information and direct quotes.
Formatting APA in-text citations:
Example of APA in-text citations:
Addressing climate change is an urgent matter on a worldwide scale that demands prompt attention (Keovongsa, 2009, p. 79).
The article states, "Climate change is a pressing global issue that requires immediate action" (Keovongsa, 2009, p. 79).
For more information on APA in-text citations, please see In-text Citations - APA - Learning Lesson
MLA (Modern Language Association) is used in various disciplines for documenting sources in a research paper or student work. In-text citations are a basic component of MLA formatting.
MLA in-text citations give credit to the person(s) whose ideas or words are being used in your writing.
MLA in-text citations:
• help writers to avoid plagiarism
• provide credibility within your own writing
• point the reader to the correct source on the works cited page which gives information to your research and writing process.
In MLA style, in-text citations can include the author's last name and the page number(s) where the information was found. If the author's name is mentioned in the sentence, only the page number(s) or indicator need to be included in the parentheses.
(Author Last Name, indicator).
A page number (no abbreviation), line number (line), section number (sec.), chapter number (ch.), paragraph number (par.), or time stamp. An indicator is used when source is quoted directly, or a specific passage is paraphrased.
Format: Examples:
| (Author's last name page number). | (Smith 22). |
| (Author's last name, Line). | (Smith, line 12). |
| (Author's last name, section). | (Smith, sec. 15). |
| (Author's last name, chapter). | (Smith, ch. 3). |
| (Author's last name, paragraph). | (Smith, par. 15). |
| (Author's last name, time stamp). | (Smith, 28:12). |
Parenthetical citation- when the author's name is inside the parentheses
Format: (author's name not in sentence)
Example:
•"Direct quotation will go here" (Wallace 37).
Narrative citation- when the author's name is mentioned in the sentence prior to the quotation
Format: (with author's name in sentence)
Example (page number):
• According to Wallace, "direct quotation will go here" (37).
When citing multiple works by the same author, differentiate between them by including a shortened version of the title in the citation and the page number and mention the author in the sentence(s) prior to the quotation(s).
Format:
Example:
• According to Smith, "direct quote will go here" (Rhetoric Matters 23). In addition to this, Smith states, "direct quote will go here" (Write Today 4).
When citing a source that has two authors, include both names in the citation. If a source has three or more authors, include only the first author's name followed by "et al."
Format: (two authors)
Example:
• (Wallace and Clemm 47).
Format: (three or more authors)
Example:
• (Wallace et al. 49).
When a source has no known author, use a shortened title of the work instead of an author name. Place the title in quotation marks if it's a short work (such as an article) or italicize it if it's a longer work (websites) and include an indicator if applicable.
Format: (no author)
• (Title or shortened name of work).
Example:
• (Rhetoric Matters).
Format: (online article with known author)
Example:
• (Wallace).
Format: (website/online article with no known author)
• ("Title of Website").
Example:
When citing two or more authors that have the same last name, provide both authors' first initials in your citation. If both authors share the same first initial, provide the first name. It may also be helpful to refer to each specific author by their first and last name in your writing prior to the in-text citations.
Format: (authors with the same last name)
Example:
• (D. Wallace).
An indirect source is a source that's cited within another source. For indirect quotations, use "qtd. in" to indicate the source you used in your writing.
Format: (indirect source quoted)
Example:
• (qtd. in Wallace 4).
For other scenarios and further explanation, please refer to MLA 9th Edition Guide
APA (American Psychological Association) is used in various disciplines for documenting sources in a research paper or student work. In-text citations are a basic component of APA formatting.
APA in-text citations give credit to the person(s) whose ideas or words are being used in your writing.
APA in-text citations:
• help writers to avoid plagiarism
• provide credibility within your own writing
• point the reader to the correct source on the reference page which gives information to your research and writing process.
APA in-text citations usually include the author's last name and the year the source was published. If the author's name is mentioned in the sentence, only the publication year needs to be included in parentheses.
(Author last name, year published, indicator).
Page number(s) (p. or pp.), no date (n.d.), volume(s) (Vol. or Vols., part (Pt.), paragraph number (para.), or time stamp. An indicator is used when source is quoted directly, or a specific passage is paraphrased.
Format: Examples:
| (Author's last name, year published, page number). | (Smith, 2005, p. 22). or (Smith, 2005, pp. 22-24). |
| (Author's last name, no date). | (Smith, n.d.). |
| (Author's last name, year published, volume). | (Smith, 2005, Vol. 2). or (Smith, 2005, Vols. 2-3). |
| (Author's last name, year published, part). | (Smith, 2005, Pt. 2). |
| (Author's last name, paragraph). | (Smith, 2005, para. 15). |
| (Author's last name, time stamp). | (Smith, 2005, 28:12). |
Parenthetical citation- when the author's name is inside the parentheses
Format & example: (with author's name not in sentence)
Narrative citation- when the author's name is mentioned in the sentence prior to the quotation
Format & example: (author's name in sentence)
In order to cite multiple works by the same author, arrange them in chronological order and separate the years with commas.
Format:
Example:
In order to cite two authors contributing to the same source, name both authors and use the word "and" between the authors' names. For more than two authors, use et al. after the name of the first author followed by et al. Lastly, for no known author, use the title or shortened title of the work being referenced to.
Format: (two authors)
Example:
Format: (three or more authors)
Example:
Format: (no known author)
Example:
In order to cite an organization as the author of a source, place the organization's title in parenthesis. If the organization has a well-known abbreviation, include the abbreviation in brackets after the organization name the first time the source is cited and then use only the abbreviation in subsequent citations.
Format:
Example:
When citing two or more authors that have the same last name, provide both authors' first initials in your citations. It may also be helpful to refer to each specific author by their first and last name in your writing prior to the in-text citations.
Format: (authors with the same last name)
Example:
• (D. Smith, 2005; F. Smith, 2008).
For other scenarios and further explanation, please refer to APA Style Guide
Watch the following video, which describes the similarities and differences between MLA and APA citations.
MLA vs APA: Works Cited and References (2:57)
Source: Isccyfairlibrary. (December 17, 2018). MLA vs APA: Works Cited and References. https://www.youtube.com/watch?v=qPODFCNIJeA
Use the following document as a quick reference to MLA/APA citations.
Examples
Roadmap Thesis
A thesis statement does not always have to include sub-claims (often called a “roadmap thesis”). For longer essays, it can be helpful.
Topic + Claim + Reasons/Subclaims = Thesis Statement
Examples
These are some strategies to help you create a thesis statement:
#1: Asking Questions
Writers often think that the thesis statement should be a question, but it is more accurate to say that the thesis is an answer to a question the writer has asked about the topic. Asking questions about a general topic usually assists writers in discovering what they want to say about the topic. For example, in the very broad question of “Should women be allowed in combat in the military?,” it would be helpful to ask a few more specific questions:
By answering these questions, it becomes easier to form a working thesis, such as: “The United States Armed Forces should change their policy against women in combat in order to quash the misguided stereotypes of women’s capabilities, which keep women from serving their country to the best of their abilities.”
#2: Understanding the Purpose of Your Essay
Having a clear purpose in mind when you sit down to write will be extremely useful in forming a thesis. It may help to recall that the essay is supposed to inform readers about a topic, or to express a position and persuade others to accept that particular view as correct. One way to form a thesis is to write down the purpose of the paper, and then rewrite it until it more closely resembles an argument. For example, the general topic is animal testing:
Purpose: To convince readers that cosmetics companies should not test their products on animals
Tentative Thesis: Cosmetics companies should not test their products on animals because animal testing is cruel and unnecessary.
Here are examples for the three most common essay purposes:
Informative: Presents facts and research on a subject; a less complex thesis
Example Thesis Statement: The life of the typical college student is characterized by time spent studying, attending class, and socializing with peers.
Analytical: Breaks down and evaluates an issue or a text
Example Thesis Statement: Although praised for its realism, Saving Private Ryan glorifies American patriotism and heroism, excluding alternative perspectives.
Argumentative: Take a position on a topic and defends this claim
Example Thesis Statement: High school graduates should be required to take a year off to pursue community service projects before entering college in order to increase their maturity and global awareness.
#3: Finding a Thesis through Prewriting
It is not uncommon for writers to discover a brilliant thesis randomly hidden in a mess of prewriting or in a rough draft. Many times, writers sit down to write a rough draft and suddenly discover their true thesis statement on page four. You should not be discouraged by this; it simply means that you needed a bit of a warm-up before your true purpose could find its way to the page. If this happens, it is time for the working thesis to move toward the front of the essay where it can better serve its purpose of directing the reader and organizing the writing.
Outlines you create for yourself do not have to be formal and can use numbered points or bullet points. And many professors won't require a formal outline.
But, if you are required to complete a formal outline, this is the format:
The Four Rules of an Effective Outline
Each heading and subheading should preserve parallel structure. If the first heading is a verb, the second heading should be a verb. Example:
("Choose" and "Prepare" are both verbs. The present tense of the verb is usually the preferred form for an outline.)
All the information contained in Heading 1 should have the same significance as the information contained in Heading 2. The same goes for the subheadings (which should be less significant than the headings). Example:
(Campus and website visits are equally significant. They are part of the main tasks you would need to do. Finding statistics and classes found on college websites are parts of the process involved in carrying out the main heading topics.)
The information in the headings should be more general, while the information in the subheadings should be more specific. Example:
(A favorite teacher and grandparent are specific examples from the generalized category of influential people in your life.)
Each heading should be divided into 2 or more parts. Example:
(The heading "Compile Résumé" is divided into 3 parts.)
Examples
In the image below, you will see an informal outline and what should be included:
In the image below, you will see a formal outline and what should be included:
Reverse outlining is different in a few ways from traditional (topic and sentence) outlines. First, it happens later in the process, after a draft is written rather than before. Second, it gives you an opportunity to review and assess the ideas and connections that are actually present in the completed draft.
This is a different approach from traditional outlining, as the traditional pre-writing outline considers an initial set of ideas, which might shift as the draft is being written and new ideas are added or existing ones are moved, changed, or removed entirely.
A reverse outline can help you improve the structure and organization of your already-written draft, letting you see where support is missing for a specific point or where ideas don’t quite connect on the page as clearly as you wanted them to.
How To Create a Reverse Outline
To complete a reverse outline, you re-read your draft; extract the main idea from each paragraph, summarizing it in one phrase or sentence; determine what steps need to be taken to present the ideas more logically; and reorganize more effectively during the revision stage.
Another strategy is to read and outline each paragraph, starting with the last one then the previous one until you get to the introduction.
Once you have completed these steps, you have a reverse outline!
For one more reverse outline example, check out https://writing.wisc.edu/handbook/reverseoutlines/
Characteristics of a Strong Body Paragraph
#1 Development: Development means that the main idea is discussed in enough depth and supported by enough evidence and detail that it is clear and convincing to the reader. A paragraph that is 2-3 sentences will not be developed enough for an academic assignment.
#2 Unity: When you revise a paragraph, you want to make sure it has paragraph unity. If each sentence supports the paragraph’s idea or point (topic sentence), then you will promote paragraph unity. Remember this writing mantra: “One point per paragraph!”
Here’s an example:
This paragraph has a few sentences that do not relate to the topic sentence and main idea of the body paragraph. Can you point them out?
One of the most important aspects of ice hockey is speed because players must skate around defensemen and get to open areas of the rink. The fastest players are able to sneak around an opposing defense and go on breakaways, creating scoring chances. Being big is also important because size allows players to hit hard. One of the fastest players in the NHL is Teemu Selanne of the Anaheim Ducks. Because of his iconic speed and Finish heritage, Teemu is known as the “Finnish Flash.” He has used his speed to score more than 600 goals during his career. He can usually be seen streaking down the boards, flying by helpless defensemen, and crashing the net to score goals. Teemu Selanne’s impressive career resulted in 10 All Star Game appearances. Ultimately, such speed is what makes a player extraordinary, even though it is just one of many attributes an ice hockey player must have to succeed.
The following two sentences do not relate to the topic sentence and main idea of the body paragraph:
Therefore, they need to be taken out.
Our new paragraph is now unified:
One of the most important aspects of ice hockey is speed because players must skate around defensemen and get to open areas of the rink. The fastest players are able to sneak around an opposing defense and go on breakaways, creating scoring chances. One of the fastest players in the NHL is Teemu Selanne of the Anaheim Ducks. Because of his iconic speed and Finish heritage, Teemu is known as the “Finnish Flash.” He has used his speed to score more than 600 goals during his career. He can usually be seen streaking down the boards, flying by helpless defensemen, and crashing the net to score goals. Ultimately, such speed is what makes a player extraordinary, even though it is just one of many attributes an ice hockey player must have to succeed.
When you revise your body paragraphs, strive to improve paragraph unity!
#3 Coherence: Coherence helps the paragraph logically flow from sentence to sentence. This enables the reader to easily follow from one idea to the next in the paragraph. Does the idea in Sentence #2 logically make sense after Sentence #1, does Sentence #3 logically make sense after Sentence #2, etc? Without coherence, your reader will be really confused and unable to understand what you’re trying to say.
Here's a sample paragraph with the sentences out of order—therefore, it is not coherent yet. Does it make sense to you? Probably not.
Many of the most common toys children play with today are descendants of those used by children centuries ago. Then, in 1200 B.C. Chinese youngsters played with yo-yos and marbles. Today's popular board games, with the exception of chess, checkers, and a few other such games, are products of twentieth-century thinking. Not long after, children living in the Mediterranean basin began enjoying themselves by spinning ovals around their waists, much in the same way that a contemporary child might use a hula hoop. For example, the popular toy dolls probably originated in prehistoric times.
Here is the paragraph in the correct order. Because the writer is discussing toys in different parts of history, it makes sense that the writer discusses the oldest first and ends with the most recent. Now it achieves coherence!
Many of the most common toys children play with today are descendants of those used by children centuries ago. For example, the popular toy dolls probably originated in prehistoric times. Then, in 1200 B.C. Chinese youngsters played with yo-yos and marbles. Not long after, children living in the Mediterranean basin began enjoying themselves by spinning ovals around their waists, much in the same way that a contemporary child might use a hula hoop. Today's popular board games, with the exception of chess, checkers, and a few other such games, are products of twentieth-century thinking.
Strive to have well-developed, unified, and coherent paragraphs!
Here are five strategies to help you write a strong introduction paragraph:
#1. Start by thinking about the question (or questions) you are trying to answer. Your entire essay will be a response to this question, and your introduction is the first step toward that end. Your direct answer to the assigned question will be your thesis, and your thesis will likely be included in your introduction, so it is a good idea to use the question as a jumping off point. Imagine that you are assigned the following question:
Drawing on the Narrative of the Life of Frederick Douglass, discuss the relationship between education and slavery in 19th-century America. Consider the following: How did white control of education reinforce slavery? How did Douglass and other enslaved African Americans view education while they endured slavery? And what role did education play in the acquisition of freedom? Most importantly, consider the degree to which education was or was not a major force for social change with regard to slavery.
You will probably refer back to your assignment extensively as you prepare your complete essay, and the prompt itself can also give you some clues about how to approach the introduction. Notice that it starts with a broad statement and then narrows to focus on specific questions from the book. One strategy might be to use a similar model in your own introduction—start off with a big picture sentence or two and then focus in on the details of your argument about Douglass. Of course, a different approach could also be very successful, but looking at the way the professor set up the question can sometimes give you some ideas for how you might answer it.
#2. Decide how general or broad your opening should be. Keep in mind that even a “big picture” opening needs to be clearly related to your topic; an opening sentence that said “Human beings, more than any other creatures on earth, are capable of learning” would be too broad for our sample assignment about slavery and education. If you have ever used Google Maps or similar programs, that experience can provide a helpful way of thinking about how broad your opening should be. Imagine that you’re researching Chapel Hill. If what you want to find out is whether Chapel Hill is at roughly the same latitude as Rome, it might make sense to hit that little “minus” sign on the online map until it has zoomed all the way out and you can see the whole globe. If you’re trying to figure out how to get from Chapel Hill to Wrightsville Beach, it might make more sense to zoom in to the level where you can see most of North Carolina (but not the rest of the world, or even the rest of the United States). And if you are looking for the intersection of Ridge Road and Manning Drive so that you can find the Writing Center’s main office, you may need to zoom all the way in. The question you are asking determines how “broad” your view should be. In the sample assignment above, the questions are probably at the “state” or “city” level of generality. When writing, you need to place your ideas in context—but that context doesn’t generally have to be as big as the whole galaxy!
#3. Try writing your introduction last. You may think that you have to write your introduction first, but that isn’t necessarily true, and it isn’t always the most effective way to craft a good introduction. You may find that you don’t know precisely what you are going to argue at the beginning of the writing process. It is perfectly fine to start out thinking that you want to argue a particular point but wind up arguing something slightly or even dramatically different by the time you’ve written most of the paper. The writing process can be an important way to organize your ideas, think through complicated issues, refine your thoughts, and develop a sophisticated argument. However, an introduction written at the beginning of that discovery process will not necessarily reflect what you wind up with at the end. You will need to revise your paper to make sure that the introduction, all of the evidence, and the conclusion reflect the argument you intend. Sometimes it’s easiest to just write up all of your evidence first and then write the introduction last—that way you can be sure that the introduction will match the body of the paper.
#4. Don’t be afraid to write a tentative introduction first and then change it later. Some people find that they need to write some kind of introduction in order to get the writing process started. That’s fine, but if you are one of those people, be sure to return to your initial introduction later and rewrite if necessary.
#5. Open with something that will draw readers in. Consider these options (remembering that they may not be suitable for all kinds of papers):
Pay special attention to your first sentence. Start off on the right foot with your readers by making sure that the first sentence actually says something useful and that it does so in an interesting and polished way.
One or more of the following strategies may help you write an effective conclusion:
Identify Variables and Constants
Vocabulary
Examples
In the expression ,
is the variable and 4 is the constant.
In the expression ,
is the variable and 2 is the constant.
Identify Expressions and Equations.
Vocabulary
Examples
is an expression (no equal sign).
is an equation (notice the equal sign).
Evaluate Algebraic Expressions
To evaluate an algebraic expression, means to substitute a specific number in for a specific variable. Once you replace the variable with a number, be sure to follow the Order of Operations to correctly evaluate the expression. Also, be careful with negative numbers.
Examples
#1) Evaluate the expression when
.
Solution: 4
Steps: (notice the variable
was replaced with the number five)
#2) Evaluate the expression when
.
Solution: 0
Steps: (notice the variable
was replaced with the number negative two)
Instructional Video
To learn more about evaluating algebraic expressions, watch
Identify Like Terms
Vocabulary
Examples
In the expression ,
and
are like terms (notice the variables match).
In the expression ,
and
are like terms and
and
are like terms (notice
is not a like term because the variable has an exponent).
Simplify Algebraic Expressions by Combining Like Terms
To simplify an algebraic expression, means to combine like terms.
Examples
#1) Simplify the expression .
Solution:
Steps:
Identify the like terms: and
Combine the like terms: (notice we added the numbers in front of the variable, these are called coefficients)
Write the simplified expression: (because there are no other
terms or constants, they are brought down unchanged)
#2) Simplify the expression .
Solution:
Steps:
Identify the like terms: and
are like terms and
and
are like terms
Combine the like terms: and
(be careful with signs!)
Write the simplified expression:
Instructional Video
To learn more about simplifying algebraic expressions, watch
For further study, read OpenStax - PreAlgebra, 2e sections 2.1 and 2.2.
Need a little extra help with algebra? Check out these options:
Free interactive Pre-Algebra textbook.
Instructional videos organized by subject and topic.
What's not on Khan Academy? They even have an AI tutor, but you do have to subscribe for that feature.
Instructional videos organized by subject and topic.
Instructional videos organized by subject and topic.
Instructional videos organized by subject and topic.
Instructional videos organized by subject and topic.
Need a little extra help with your math? Check out these options:
Free interactive Pre-Algebra textbook.
Instructional videos organized by subject and topic.
What's not on Khan Academy? They even have an AI tutor, but you do have to subscribe for that feature.
Instructional videos organized by subject and topic.
Instructional videos organized by subject and topic.
Instructional videos organized by subject and topic.
Instructional videos organized by subject and topic.
Vocabulary
Examples
The function is an exponential function (the variable is in the exponent position).
The function is NOT an exponential function (the variable is the base, not the exponent).
To evaluate an exponential function, means to substitute a specific number in for the input variable. Once you replace the variable with a number, be sure to follow the Order of Operations to correctly evaluate the expression. Also, be careful with negative numbers.
Examples
#1) Evaluate the function when
.
Solution:
Steps: (notice the variable
was replaced with the number five)
#2) Given the function , find
.
Solution:
Steps: (notice the variable
was replaced with the number zero)
Instructional Video
To learn more about evaluating exponential functions, watch the video Evaluate Exponential Functions (3:41)
Vocabulary
Given , then
.
** The expression is a logarithmic expression with base 3 (this is understood as 3 raised to what power gives 27).
** The expression is a logarithmic expression with base
(this is understood as
raised to what power gives 42).
Examples
#1) Write the following exponential equation in its equivalent logarithmic form.
Solution: (notice 3 raised to what power gives 9)
#2) Write the following exponential equation in its equivalent logarithmic form.
Solution: (notice
raised to what power gives
)
Instructional Video
To learn more about writing exponential equations as a logarithmic equation, watch Write Exponential Equations as Logarithmic Equations - Variables (3:02)
Vocabulary
Given , then
.
** The expression is a logarithmic expression with base 3 (this is understood as 3 raised to what power gives 27).
** The expression is a logarithmic expression with base
(this is understood as
raised to what power gives 42).
Examples
#1) Write the following logarithmic equation in its equivalent exponential form.
Solution: (notice 5 is the base, 3 is the exponent, and 125 is the result)
#2) Write the following logarithmic equation in its equivalent exponential form.
Solution: (notice
is the understood base,
is the exponent, and
is the result)
Instructional Video
To learn more about writing logarithmic equations as an exponential equation, watch Write Logarithmic Equations As Exponential Equations - Variables (2:30)
To evaluate a simple logarithmic expression, write the logarithmic expression as an equivalent exponential equation to determine the result.
Examples
#1) Evaluate without the use of a calculator.
Solution: 4
Steps: (notice 2 raised to the 4th power is 16)
#2) Evaluate without the use of a calculator.
Solution: -2
Steps: (notice 3 raised to the power of -2 is
)
Instructional Video
To learn more about evaluating logarithmic expressions, watch Evaluate Logarithms Without a Calculator - Whole Numbers (2:12)
Identify a Polynomial Function.
Vocabulary
Examples
The function is a polynomial function (all exponents are whole numbers).
The function is NOT a polynomial function (the exponent
is not a whole number).
Instructional Video
To learn more about identifying polynomial functions, watch this video available at Mathispower4u.com (3:54).
Evaluate a Polynomial fFunction for a Given Value
To evaluate a polynomial function, means to substitute a specific number in for the input variable. Once you replace the variable with a number, be sure to follow the Order of Operations to correctly evaluate the expression. Also, be careful with negative numbers.
Examples
#1) Evaluate the function when
.
Solution:
Steps: (notice the variable
was replaced with the number five)
#2) Given the function , find
.
Solution:
Steps: (notice the variable
was replaced with the number negative two)
Instructional Video
To learn more about evaluating polynomial functions, watch this video available at Mathispower4u.com (5:15).
Complete an Input/Output Table for a Polynomial Function
To complete an input/output table, sometimes referred to as a t-table, simply evaluate the function for each of the given input values. Notice that each input/output pair, also known as an ordered pair, corresponds to one point on the Cartesian Coordinate System.
Examples
Given the function , complete the following table.
| Input (x) | Output (y) |
| -2 | |
| -1 | |
| 0 | |
| 1 | |
| 2 |
Solution:
| Input (x) | Output (y) | Ordered Pair |
| -2 | 25 | (-2 , 25) |
| -1 | 14 | (-1 , 14) |
| 0 | 9 | (0 , 9) |
| 1 | 10 | (1 , 10) |
| 2 | 17 | (2 , 17) |
Steps:
(notice the variable was replaced with the number -2)
(notice the variable was replaced with the number -1)
Repeat this process for 0, 1, and 2.
Instructional Video
To learn more about input/output tables, watch this video available at Mathispower4u.com (3:19).
For further study, read OpenStax - College Algebra, 2e sections 3.1 and 5.2.
***
Need a little extra help with polynomial functions? Check out these options:
Free interactive College Algebra textbook
Instructional videos organized by subject and topic.
What's not on Khan Academy? They even have an AI tutor, but you do have to pay for that feature.
Instructional videos organized by subject and topic.
Instructional videos organized by subject and topic.
Instructional videos organized by subject and topic.
Instructional videos organized by subject and topic.
When we are evaluating algebraic expressions, we use one set of rules so that everyone arrives at the same correct answer.
These rules used for simplifying (or evaluating) algebraic expressions are called Order of Operations.
The rules are listed below.
How To:
PEMDAS, “Please Excuse My Dear Aunt Sally”
Example:
Evaluate the following expressions using the rules for order of operations.
Find the value of the following expression using the rules for order of operations.
a. (12*4÷2^3 ) - [(3*2^3 )÷(4*6)] we deal with the exponents within the grouping symbols
(12*4÷8)-[(3*8)÷(4*6)] we now work on multiplication before the division
(48÷8)-[(24)÷(24)] we now work on division
6-1 =5 The final step is subtraction
b. (82 – 25) ÷ (24 ÷ 6) + 32 we deal with the exponents within the grouping symbols
(64-32)÷(24÷6)+9 we still simplify what is inside the parentheses
(32)÷(4)+9 now we divide and add
8+9=17
c. (5(16-5)-1)/(4^2-7) we simplify the numerator and denominator separately
(5(11)-1)/(16-7)
(55-1)/9
54/9=6
When we are evaluating algebraic expressions, we use one set of rules so that everyone arrives at the same correct answer.
These rules used for simplifying (or evaluating) algebraic expressions are called Order of Operations.
The rules are listed below.
Example:
Evaluate the following expressions using the rules for order of operations.
Find the value of the following expression using the rules for order of operations.
a. (12*4÷2^3 ) - [(3*2^3 )÷(4*6)] we deal with the exponents within the grouping symbols
(12*4÷8)-[(3*8)÷(4*6)] we now work on multiplication before the division
(48÷8)-[(24)÷(24)] we now work on division
6-1 =5 The final step is subtraction
b. (82 – 25) ÷ (24 ÷ 6) + 32 we deal with the exponents within the grouping symbols
(64-32)÷(24÷6)+9 we still simplify what is inside the parentheses
(32)÷(4)+9 now we divide and add
8+9=17
c. (5(16-5)-1)/(4^2-7) we simplify the numerator and denominator separately
(5(11)-1)/(16-7)
(55-1)/9
54/9=6
Integer Exponent: This is a repeated multiplication by the same number (or variable) called the base.
is called the: coefficient.
is called the: base.
is called the: exponent.
Rational Exponent: Sometimes fractional exponents are used to represent power of numbers or variables. The numerator of the fraction (m) represents the power, the denominator (n) represents the root. The exponent in the denominator must always be positive.
Video for Exponential Notation (4:57)
Properties of Exponents
Video for Product and Quotient Properties (6:06)
Note:
Video of Fraction Raised to a Power (3:45)
Examples:
See attachment for examples Examples for Integer and Rational Exponents.pdf
Fractions are ways of writing parts of whole numbers. For example, if a pizza is divided into 6 equal pieces, each piece will be . Fraction can be written as
where a is the numerator, b is the denominator and b≠0.
Note: Fraction is also a way of writing division problem. That is, is the same as
.
Equivalent Fractions: These are fractions obtained by either multiplying or dividing the numerator and the denominator of a given fraction by the same factor (number).
Example: is equivalent to
and this is obtained by multiplying the numerator and denominator by 2.
Decimals:
Decimals are another way of representing fractions. We use the decimal point to separate the whole part of a number from the fraction part.
For example, is 13 wholes and one-half.
Fractions and Decimals Video (9:14)
Proper Fractions:
These are fractions that are less than 1, in other words the numerator is less than the denominator. Examples of proper fractions are: .
Improper Fractions:
These are fractions that are greater than 1, in other words the numerator is greater than the denominator. Examples of improper fractions are: . Improper fractions can be written as mixed numbers. Example,
Addition and Subtraction of Fractions:
We can add or subtract two or more fractions if and only if the denominators are the same. In cases where the denominators are not the same, we make them the same by finding the least common denominator (LCD). We find the LCD by listing the multiples of all the denominators and finding the least number common to all of them.
Example: Perform the indicated operations. list the multiples of 2, 4, and 5
2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30…
4, 8, 12, 16, 20, 24, 28, 32, 36, 40…
5, 10, 15, 20, 25, 30, 35, 40….
LCD = 20. We write equivalent fraction with a denominator of 20 for each fraction.
Multiplication of Fractions: We multiple the numerators and denominators and simplify the new fraction if possible.
Division of Fractions: We take the reciprocal (flip) of the fraction after the fraction after the division sign and multiply the outcomes.
Example: Perform the indicated operations.
Addition, Subtraction, Multiplication, and Division Video (7:15)
Conversion Between Fractions and Decimals:
How to Convert Decimal to Fraction and Vice Versa Video (5:02)
The goal of this lesson focused on "meiosis" is to educate learners about the process of cell division in human.
Chromosomes: Structure & Function (5:33)
Meiosis (0:51)
Complete the short quiz below. You may take it as many times as you wish to comprehend the material.
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