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Illustrative Mathematics Grade 7 Unit 5 ...

Illustrative Mathematics Grade 7 Unit 5 Review

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-25 + -20 will have a positive answer

36 + -12 will have a negative answers

-4 x -3 will have a positive answer

-5 - -3 will have a positive answer

-5c = 20 What is c?

-6 + v = 10 What is v?

g + (-8) = -10 What is g?

Tyler's bank account has a balance of -30 dollars. Tyler makes 3 withdrawals of $7 each. What is his new balance?

A hawk is perched in a tree 25 feet above sea level. Directly below the hawk, a seagull is flying 6 feet above sea level. Directly below the birds is a walleye, swimming 4 feet below sea level.

Select ALL of the true statements.

The difference in height between the seagull and the hawk is 19 feet.

The difference in height between the seagull and the hawk is -19 feet.

The difference in height between the seagull and the walleye is 10 feet.

The difference in height between the seagull and the walleye is –10 feet.

The distance in height between the hawk and the walleye is 29 feet.

Let A = -5/8 and B = 2/3 . Which expressions have a positive value? Select ALL that work.

A divided by B

What was the transaction amount?

What's the new account balance?

What is the new account balance?

What was the old account balance?

Select ALL of the FALSE statements

5.5 - (-5.5) is equal to zero

-34 + 34 = 68

-4 + 1/2 is negative

-16/8 - (-2.6) is positive

-3.9 + 3.8 is positive

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Unit 5 - Unit review including section questions and vocabulary

Unit 5: period 5: 1844–1877, ap u.s. history, ruben s. ayala high school - chino hills, recommended for you, students also viewed.

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Assignment information.

For this assignment, you will answer guiding questions that correspond to the AP® US History learning objectives for this time period. In addition, you will identify important terms, events, and people to ensure you have achieved the appropriate depth of knowledge for this era. In addition, you should review the AP US History thematic learning objectives and key concepts for this time period. You can find this information in the AP United States History Course and Exam Description, which you can download at the College Board website here: apstudent.collegeboard/apcourse/ap-united-states-history All multiple choice, short answer, and essay questions on the AP US History exam directly address one or more of the thematic learning objectives or key concepts provided in this document.

Step 1: As you progress through the video and reading lessons in this unit, take notes on the guiding questions and important terms. - Remember to save your work as you go. - Ask your teacher if you have questions about how to complete this assignment. - You will have the opportunity to turn this assignment in at the end of the unit.

Section I: Guiding questions

These guiding questions reflect the important concepts and thematic learning objectives for this time period. Take notes on these topics as you progress through the video and reading lessons in this unit, and then craft a thorough response. Be sure to include specific details, facts, and evidence to support your conclusions. - How did economic interests, cultural factors, and political ambitions contribute to an unprecedented territorial expansion from 1844–1860? Economic interests were a major driving force for territorial expansion from 1844-1860. The desire for resources such as gold and silver drove settlers to move westward and conquer new territories. Additionally, the cultural belief in Manifest Destiny and the idea that it was the

American's God-given duty to spread democracy and civilization led to the acquisition of new territories. Politically, the acquisition of new territories allowed for the expansion of slavery and the reinforcement of political power for southern states. - How did economic differences and political conflicts from 1848–1860 bring the nation into the Civil War? Economic differences and political conflicts from 1848-1860 created tension between the North and South. The North was primarily industrialized and had no need for slave labor, whereas the South's economy relied heavily on slave labor in agriculture. This led to disputes over states' rights and the spread of slavery into newly acquired territories. These conflicts ultimately led to the secession of Southern states and the start of the Civil War. - How did the leadership of Abraham Lincoln contribute to the Union victory in the Civil War? Abraham Lincoln's leadership was critical to the Union victory in the Civil War. He was able to effectively communicate his vision for the Union and rally support for the war effort. He also made strategic decisions regarding military tactics and personnel appointments that ultimately led to Union victories on the battlefield. Additionally, Lincoln's Emancipation Proclamation helped to shift the focus of the war towards ending slavery, which weakened the Confederacy's morale and support from European powers. - In what two ways was 1863 a turning point year in the Civil War? 1863 was a turning point year in the Civil War because of two significant events. Firstly, the Union army's victory at the Battle of Gettysburg in July marked the first major defeat for the Confederate army on northern soil and boosted Union morale. Secondly, President Lincoln's Emancipation Proclamation, which went into effect on January 1st, 1863, declared all slaves in Confederate- held territory to be free and shifted the war's focus toward the abolition of slavery. - What transformations were brought to the lives of African Americans as a result of the Civil War and Reconstruction? In what ways did their situation not see a major improvement? The Civil War and Reconstruction period brought significant transformation to the lives of African Americans. The abolition of slavery signaled the end of centuries-long oppression and segregation. Reconstruction ensured that African Americans had the right to vote, expanded their access to education, and created opportunities for economic advancement. Despite these advancements, the situation for African Americans did not see a major improvement. The end of Reconstruction led to the rise of Jim Crow laws and other discriminatory practices aimed at suppressing black political and economic power. Lynchings, segregation, and racial violence persisted, limiting the social and economic progress of African Americans for decades to come.

acquire parts of Arizona and New Mexico. The significance of the Gadsden Purchase was it completed the southern boundary of the continental United States and provided a future route for a transcontinental railroad. Ostend Manifesto A secret proposal in 1854 to acquire Cuba from Spain. The Ostend Manifesto was significant because it highlighted President Franklin Pierce's southern sympathies and added fuel to the sectional conflict over slavery and Westward expansion. Kansas-Nebraska Act A law passed in 1854 by Stephen Douglas that allowed settlers in these prospects to decide whether or not to allow slavery. The Kansas-Nebraska Act was significant because it repealed the Missouri Compromise and led to clashes between pro-slavery and anti- slavery advocates in Kansas known as "Bleeding Kansas." American, or Know-Nothing, Party A political party that emerged in the 1850s, which favored strict controls on immigration and anti-Catholic sentiments. The American, or Know-Nothing, Party is significant because it reflected the rising nativism, xenophobia, and anti-immigrant sentiments. “Bleeding Kansas” Refers to the violent conflict between pro-slavery and anti-slavery forces in Kansas in 1855-1857. The significance of "Bleeding Kansas" was that it symbolized the eruption of sectional conflict over the issue of slavery and white men's ability to determine the fate of African Americans. Dred Scott v. Sanford A legal case decided by the Supreme Court in 1857 that declared that African Americans, whether free or enslaved, could not be considered citizens under the U. Constitution. The significance of Dred Scott v. Sanford was that it greatly intensified sectional tensions and weakened the Court's legitimacy by revealing its pro-slavery bias. Freeport Doctrine A statement made by Stephen Douglas during a debate against Abraham Lincoln in 1858, emphasizing that territorial legislatures had the right to decide whether or not to allow slavery despite the Supreme Court's decision in Dred Scott v. Sanford. The significance of the Freeport Doctrine was that it alienated Southern Democrats who claimed universal protection for slaveholders. James K. Polk The 11th President of the United States who served from 1845 to 1849 and led the nation during the Mexican-American War. The significance of James K. Polk was that he effectively implemented Manifest Destiny, expanded the territorial holdings of the United States, and initiated the annexation of Texas. Zachary Taylor The 12th President of the United States who served from 1849 to 1850 and was a military hero in the Mexican-American War. The significance of Zachary Taylor was that he helped bring the national conversation about slavery to the forefront and Southern secessionists saw the possibility of being locked out of the process of westward expansion. Lewis Cass A Democratic Senator and unsuccessful presidential candidate in 1848 who advocated popular sovereignty. The significance of Lewis Cass was that his idea of popular sovereignty, which would allow territories to decide whether or not to allow slavery, became popular amongst politicians in the lead up to the Kansas-Nebraska Act. Stephen Douglas A Democratic Senator from Illinois who played a major role in the passage of the Kansas-Nebraska Act, proposed popular sovereignty for newly acquired territories, and was responsible for the Freeport Doctrine. The significance of Stephen Douglas was that his actions and beliefs made him the center of the increasingly contentious debates around slavery.

Harriet Beecher Stowe An American author who wrote the novel Uncle Tom's Cabin in 1852, which emphasized the horrors of slavery and had a significant impact on public opinion. The significance of Harriet Beecher Stowe was that her novel helped galvanize popular sentiment against slavery and intensify sectional tensions. John Brown An American abolitionist who advocated for the use of violence against slave owners and orchestrated the raid on Harpers Ferry in 1859. The significance of John Brown was that his actions helped accelerate the onset of the Civil War and his ideas would influence many of the abolitionists during the war. Abraham Lincoln The 16th President of the United States, serving from 1861 to 1865, and was instrumental in the Union's victory over the Confederate States in the Civil War. The significance of Abraham Lincoln was that he played a critical role in the preservation of the Union and the abolition of slavery. Frederick Douglass A prominent African American abolitionist, orator, and writer who fought for the emancipation of all enslaved people. The significance of Frederick Douglass was that his work helped change public opinion and was instrumental in the abolition movement. Crittenden Compromise A proposal in 1860 to prohibit slavery north of 36° 30' while allowing it south of that line. The significance of the Crittenden Compromise was that it revealed the deepening sectional divides and demonstrated the extreme difficulty of achieving compromise over sla habeas corpus A legal principle that guarantees individuals the right to be brought before a court or judge before being detained. The significance of habeas corpus was that President Abraham Lincoln suspended it during the Civil War in order to suppress opposition to the war effort. King Cotton A slogan used by pro-slavery advocates in the South to argue that the cotton industry was critical to the economic prosperity of the nation. The significance of King Cotton was that it highlighted the Southerners' reliance on slavery and its position in international trade. Radical Republicans A group of Republicans that advocated for the emancipation of enslaved people, civil rights legislation, and military reconstruction of the South. The significance of Radical Republicans was that they helped shape the agenda of Reconstruction and ensured that Union victory in the Civil War would lead to significant changes in American society. Emancipation Proclamation A presidential proclamation issued by Abraham Lincoln in 1863 that declared that all enslaved people living in Confederate-held territories were to be declared free. The significance of the Emancipation Proclamation was that it helped transform the Civil War from a struggle to preserve the Union to a war for the abolition of slavery. War and Peace Democrats A group within the Democratic Party that opposed the Civil War and believed in a negotiated peace with the Confederacy. The significance of War and Peace Democrats was that they helped divide the nation and illustrate the difficulty of maintaining unity when there were deep ideological divides. March to the Sea A military campaign conducted by General William T. Sherman that targeted infrastructure and civilian property in Georgia. The significance of the March to the Sea was that it broke the South's ability to sustain its armies and helped end the Civil War. Jefferson Davis The first and only president of the Confederate States of America, serving from 1861 to 1865. The significance of Jefferson Davis was that he played a critical role in the

Americans. Fifteenth Amendment An amendment to the U. Constitution that guaranteed voting rights to African American men. The significance of the Fifteenth Amendment was that it was the first constitutional amendment to directly address race and voting rights. American Woman Suffrage Association A group founded in 1869 by Susan B. Anthony and Elizabeth Cady Stanton that sought to secure voting rights for women. The significance of the American Woman Suffrage Association was that it led to the eventual passage of the Nineteenth Amendment and demonstrated the growing political power of women. sharecropping A labor system used in the Southern United States after slavery was abolished that allowed landowners to rent or lease land to farmers in exchange for a share of the crops. The significance of sharecropping was that it perpetuated a system of peonage and debt and confined African Americans to a status similar to slavery. Union League A political organization established in the South during Reconstruction that encouraged African Americans to participate in politics and civic life. The significance of the Union League was that it provided a base for African American political power and helped shift the balance of power in Southern politics. scalawags Southern whites who supported Reconstruction and worked with the Republican Party. The significance of scalawags was that they helped create a biracial political coalition in the South and helped shape the early years of Reconstruction. carpetbaggers Northern whites who moved to the South during Reconstruction and worked with the Republican Party. The significance of carpetbaggers was that they helped shape the early years of Reconstruction and were often resented by Southern whites Civil Rights Act of 1875 A federal law that guaranteed equal treatment under the law regardless of race in public accommodations and transportation. The significance of the Civil Rights Act of 1875 was that it represented the last major piece of civil rights legislation passed before the end of Reconstruction and illustrated the challenges and limitations of Reconstruction-era reforms. Crédit Mobilier A financial scandal that involved the construction of the Union Pacific Railroad and the distribution of fraudulent stock. The significance of Crédit Mobilier was that it demonstrated the corruption and political cronyism associated with American expansion in the West. Ku Klux Klan A white supremacist organization formed in the South after the Civil War that aimed to intimidate and terrorize African Americans and their white allies. The significance of the Ku Klux Klan was that it represented a reactionary response to Reconstruction and the growing political power of African Americans, as well as the failure of the federal government to protect the civil rights of those groups. enforcement laws A series of federal laws passed in 1870 and 1871 to protect the civil rights of African Americans and to combat the Ku Klux Klan and other white supremacist groups. The significance of the enforcement laws was that they demonstrated the federal government's commitment to protecting the rights of African Americans during Reconstruction. Slaughter-House Cases A Supreme Court case that limited the scope of the Fourteenth Amendment by restricting its application to national citizenship rather than individual rights. The significance of the Slaughter-House Cases was that it weakened Reconstruction-era reforms and made it easier for Southern states to restrict the civil rights of African Americans

United States v. Cruikshank A Supreme Court case that declared that the federal government could not prosecute individuals for violating the civil rights of others. The significance of United States v. Cruikshank was that it weakened protections for African Americans and illustrated the limits of federal power to protect civil rights. civil rights cases A series of Supreme Court cases that invalidated the Civil Rights Act of 1875 and severely limited federal power to intervene in cases of racial discrimination. The significance of the civil rights cases was that they undermined gains made during Reconstruction and demonstrated the difficulty of achieving lasting civil rights reforms. Andrew Johnson The 17th President of the United States who served from 1865 to 1869 and oversaw the early years of Reconstruction. The significance of Andrew Johnson was that he favored a lenient approach to Reconstruction and frequently clashed with the Radical Republicans who sought a more transformative approach. Charles Sumner A U. Senator from Massachusetts who was a leading abolitionist and played a key role in drafting and passing the Thirteenth and Fourteenth Amendments. The significance of Charles Sumner was that he played a vital role in shaping the nation's policy towards Reconstruction, voting rights, and civil rights Thaddeus Stevens A U. Representative from Pennsylvania who was a leading Radical Republican and played a key role in shaping Reconstruction policy. The significance of Thaddeus Stevens was that he was a key architect of the Reconstruction Amendments and sought to ensure that Reconstruction transformed American society. Nathan Bedford Forrest A Confederate general and early leader of the Ku Klux Klan who later became a significant businessman. The significance of Nathan Bedford Forrest was that he represents the continuity of racial politics in the South from the Civil War to modern times.

• Multiple Choice

Topic : Unit 5: Period 5: 1844–1877

Subject : ap u.s. history.

• Discover more from: AP U.S. History AP (Advanced Placement) 999+   Documents Go to course
• More from: Unit 5: Period 5: 1844–1877 368   Documents Go to topic
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What AP Calculus Students Should Know for their Unit 7 Review

Updated: Jul 15

Need extra practice for your AP Calculus Unit 7 review? Outlined are the topics and differential equations practice problems aligned with College Board’s curriculum to study for a Unit 7 test.

As College Board outlines, the topics to review for Unit 7 are:

·         Modeling Situations with Differential Equations

This is essentially taking words and converting the relationship to an equation. Recall from past math classes that a directly proportional relationship is of the form kx, and an inversely proportional relationship is of the form k/x.

While this is not a common question on the AP exam, it leads us into the topics below, Exponential Models or Logistic Models.

1.      Write a differential equation to describe the relationship: The rate of change of a radioactive substance R with respect to t is proportional to the amount of the substance present at any given time.

·         Verifying Solutions for Differential Equations

For this section, you’re often given dy/dx and asked to find a particular solution through a given point. To go from dy/dx back to y, you must take the antiderivative. In taking the antiderivative, don’t forget to add C since it’s an indefinite integral. Use the given point to find C.

2.      Find the particular solution to dy/dx= sin(2x) through (0, 4).

·         Sketching Slope Fields

The two common ways this section is assessed on the AP exam is first by matching the differential equation to its slope field in the multiple choice section, or second, by drawing a slope field given a differential equation in the free response section.

To fill out a slope field, plug in each coordinate into dy/dx. The result will determine the slope of the small tangent line segment you draw at that point. To differentiate between positive slopes, for example, make sure a slope of 3 is steeper than a slope of 1. Remember a slope of 0 is a horizontal tangent, while a slope that’s undefined (from dividing by 0) would be a vertical tangent.

To match a differential equation to its slope field, you can follow the same idea as above, choosing a coordinate and plugging it into dy/dx and seeing if it makes sense for the slope field. This might take several iterations to determine the correct differential equation. A tip is to choose a coordinate that has a negative slope on the slope field. That often cancels out more of the multiple choice answers.

Another tip/method is to set all the differential equations equal to 0 and solve. For example, if you had dy/dx=x-y, I would set x-y=0 and solve for y: y=x. Along that diagonal line, if the slope field has horizontal tangents, that would be the correct solution. If dy/dx=2x, setting that to 0 would give x=0. Along the vertical line x=0, or the y-axis, the correct solution will have horizontal tangents.

For a practice problem of this section, see the free response question from AP Calculus AB 2010 Form B below. Part (a) is this section and part (b) is the next section.

·         Reasoning using Slope Fields

This section includes questions about slope fields like “describe all points in the xy-plane with negative slopes” or “explain why the slope field could not match the given differential equation.” For the first problem, I would see when dy/dx is negative (if it’s a complicated expression, you may need to factor and set each part to 0 first). For the second question, try to find a counterexample with conflicting information such as “The slope at (1,2) is positive on the slope field but plugging (1,2) into dy/dx equals 0”

Part C is from an upcoming section Finding Particular Solutions, but since that is arguably the most important section of this chapter, I included this for extra practice.

·         Approximating Solutions using Euler’s Method (BC only)

There are two ways to apply Euler's method, either using tangent line approximations or a table that organizes the point, slope (dy/dx), change in x, and change in y. If you look at scoring guidelines for an AP Calculus free response question, they’ll typically use the tangent line method. I admit, when I started teaching AP Calculus, I used the table method since my textbook used that method. Now, I recommend using tangent lines since it’s one less thing to memorize. We’ve been doing tangent line equations for months now so hopefully you feel confident in that!

4.      Let y=f(x) be the solution to the differential equation dy/dx= x+y with the initial condition f(1)=2. What is the approximation for f(2) using Euler’s method with 2 steps of equal length starting at x=1?

·         Finding General Solutions using Separation of Variables

Most differential equations we encounter in AP Calculus have dy/dx in terms of both x and y. To get back to the solution y=f(x), we must take an antiderivative. But we can’t take an antiderivative with x and y together. We must first separate the variables. Then integrate, or take the antiderivative, of both sides. Don’t forget the +C for indefinite integrals! The general solution will leave the C, as the solution is the family of curves with the same slope. The final step is to get y by itself.

5.      Find the general solution to the differential equation: dy/dx = (y+1)cosx

·         Finding Particular Solutions using Initial Conditions and Separation of Variables

This section is the same idea as the previous one, but we use the initial condition/coordinate to solve for C. I like to use the steps: separate, integrate, solve for C, solve for y. You can switch the last two steps, but I’ve found it’s usually more algebraic work if you solve for y first.

6.      Find the particular solution to the differential equation: dy/dx= xy^2 through the point (-1, 3).

·         Exponential Models with Differential Equations

Exponential differential equations are of the form dy/dt=ky. You can find the general solution by separating and integrating, and you’ll see that the general solution to exponential differential equations is y=y0e^(kt). You probably recognize this formula from previous math classes as continuous exponential growth or decay. Any time you see dy/dt=ky or the wording “a rate proportional to the number of y” (y varies from problem to problem), you can automatically use y=y0e^(kt).

You should be given y0, the initial amount, or if it’s not given, you can use 1 for simplicity. The first step is to solve for k (you’ll be given a y value and time to plug in). Once you get k, you can plug that into your y= equation. The last step is to plug in the remaining time or y-value and solve for the other variable.

7.      During the first month of the COVID pandemic, the number of people infected increased at a rate proportional to the number of people infected at that time. If 1000 people were infected when the pandemic was first discovered, and 4000 people were infected 2 days later, how many people were infected 14 days after it was first discovered?

·         Logistic Models with Differential Equations (BC only)

Logistic models are typically better models for population growth than exponential models because there are factors that limit growth, like food availability, so it doesn't make sense for a population to continue to grow without bound.

Logistic differential equations are of the form dy/dt=ky(M-y). M is the carrying capacity. If you set dy/dt = 0, that will give you horizontal tangents for y. In the case of logistic curves, that will give you the two horizontal asymptotes, at y=0 and y=M. Some teachers also give the general logistic curve formula, but you don’t need to know that for the AP exam.

There’s a variety of logistic questions that can be asked (this is ultimately the application of integration using partial fractions!), but the most common questions ask about the form of a logistic differential equation, the carrying capacity, the greatest rate of change, or a limit approaching infinity. I’ve provided one problem below, but you can get more practice on logistic models and Unit 7  at that link.

Notice on the logistic curve below that the green dot for the point of inflection, or the greatest rate of change, occurs in the middle of the two horizontal asymptotes. The easiest way to calculate this is to find half of the carrying capacity.

In this case, the carrying capacity is 6 so the greatest rate of change occurs when y=3.

8.      A population P changes at a rate modeled by the logistic differential equation dP/dt=0.1P(2000-P), where t is measured in years, and P(0)=300. A) At what population does the maximum rate of change occur? B) What is the limit as t approaches infinity of P(t)?

AP Calculus Unit 7 Review

For your AP Calculus Unit 7 review, make sure you are practicing differential equation problems from multiple perspectives: given functions, tables, graphs, and word problems. If you need extra practice, check out the 7 best resources to study for AP Calculus tests . These resources have questions that match the rigor of your tests.

If you need further explanation on how to approach some of these difficult Unit 7 review questions, especially when all of the topics are mixed together, consider individual tutoring with me. I answer any questions students have, then provide practice solving my past test questions and previous AP exam questions. Getting more practice with problems given graphs, tables, and word problems will help you be prepared for in class tests and the AP exam.

Sign up for individual tutoring  today!

Solutions to the above problems:

1.      dR/dt=kR

Going back to the horizontal asymptotes being y=0 and y=M, recall that a limit approaching infinity is essentially asking about horizontal asymptotes. So your answer for part B above will either be 0 or 2000. To determine which was correct, I checked dP/dt at the given point. Logistic curves are always increasing or always decreasing so if dP/dt is positive, P is increasing so 2000 is our answers. If dP/dt was negative, P would be decreasing so 0 would be the answer.

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