• List 3 things you learned, found interesting or found intriguing from watching this video.
• How can the ideas of the video be applied to your learning in this class?
• What are 3 things you learned from the video and/or the Tips for Studying Math that you plan to do to help you succeed in this course?

• Heliocentric vs. geocentric solar system, Kepler’s laws, Inductive reasoning and Galileo
• Analytic geometry and deductive reasoning: Descartes and Fermat—What purpose does coordinate
  geometry have for Descartes, and what relationship does it have with geometric construction and
  problem-solving? How does it relate to his Method?
• Indivisibles: Galileo’s paradoxes of the infinite, Cavalieri’s principle
• Was there any work before Newton and Leibniz about finding equations of tangent and normal
  lines to curves? or about quadrature (areas under curves)? If so, why do we credit Newton and
  Leibniz with the discovery of calculus instead of Isaac Barrow and some of these other folks?
• Isaac Newton’s method of fluxions, fluents, moments
• Newton’s Principia Mathematica—What is it about, why is it important, how does it differ from his
  earlier version of calculus with fluxions?
• G.W. Leibniz’s infinitesimals—How are these different from indivisibles? How are they different
  from Newton’s infinitesimals? What complaints do some contemporaries have about them?
• Who are some folks who applied calculus ideas in the early 1700s and what is an example of how
  they did?
• Why are power series important in the development of calc? Why was it important that Euler’s
  work with infinite series and analysis was so formal and symbolic? How does this influence
  Lagrange’s development of calculus?
• How does the idea of function change between 1700 and 1850?
• Why does each of the following examples of 19th-century mathematics reflect a crisis of the
  Enlightenment perspective of the previous century? How did things not work quite so smoothly as
  previously thought? Examples:

o Fourierseries
o Insolvabilityof5th-degreepolynomials,andofthe3toughGreekconstructionproblems
o Non-Euclideangeometry(whatisthis,whodowecreditwithitsdiscovery,andwhydoes

it reflect a change of perspective?)
o Infinitesimalsgetreplacedbylimits(when,bywho?)

• What is set theory and how does it relate to the axiomatization of mathematics? Who are the
  characters in that story?
• What are formalism, platonism, intuitionism, and logicism? More specifically:

o Whywereintuitionists(constructivists)opposedtoCantor’sideasaboutinfinitesets?
o Whatareacoupleimportantmathematicalideaswemightnothavehadifwewereall

platonists, and why?
o HowdoGødel’sincompletenesstheoremschallengetheformalistprogram?
o Asaformalist,whydoesitmakesensethatHilbertthoughttheContinuumHypothesis

was an important question?

• Chinese math examples: counting boards, number representations and calculation methods, systems
  of equations, magic squares, Pythagorean theorem, Chinese remainder theorem
• Mayan math: calendars, pure base-20 system vs. positional almost-base-20 solar calendar system,
  zero
• Inca math: what were quipus for, how were numbers represented, yupanas (weird abacuses) and
  calculation

Discussion: Inequality in Our World 

Inequalities are present in our everyday lives. You may budget to spend less than $100 on groceries or plan to spend more than 2 hours a week exercising. Both of these statements involve budgeting resources, such as time and money. However, unlike last week where only one value could solve our T=15h equation, these statements allow for a variety of situations that meet the requirements for a solution. For example, in looking at the grocery situation, a bill of $90, $85, or $98 would all fit the criteria of spending less than $100. 

To break this down further, you might assume that a grocery bill of $90 would include items purchased from different departments of the grocery store. Meats, vegetables, dairy, and paper products might be subgroups represented by the overall group—groceries. The value of each subgroup would not be equally represented in your grocery bill. 

What other situations can you imagine where the variables, or subgroups, do not equally represent the whole group?

In this Discussion, you examine the groups and subgroups that may exist in your field of study or current workplace. You also explore how to diagram these groups using abstract math. 

To prepare for this Discussion:

• Review the TED Talk on abstract math, paying particular attention to how Eugenia Cheng (2018) explains how pure mathematics models social inequality.
• Think about an overall group that may exist in your environment.
• Identify three subgroups within the overall group, and diagram these groups as Cheng (2018) did in the presentation using the following format where a/b/c are your individual subgroups:
  • {a,b,c}
  • {a,b}, {a,c}, {b,c}
  • {a}, {b}, {c}
  • { }
• Think about two inequality statements that can be inferred from the diagram referring to the specific groups that you have just created. For example, if a represents dogs and c represents cats then and inequality could be: dogs>cats.
• Using the problem-solving techniques from Week 1, decide if these inequalities are true based on the overall group you selected.
• Consider one potential bias or inequality that may exist in either Level 2 or Level 3 of your diagram and think about how it would create an unequal ranking between the elements on this level.
• Think about what the inequality would be in the context of your situation and think about how it would be expressed as a mathematical inequality.
• Consider who might be interested in these results, and why.

Post at least 2 paragraphs responding to the following prompts: 

• Provide diagram created based on your example of social inequality.
• Write one inequality statements that can be inferred from your diagram, referring to your specific sub-groups (not the variables a/b/c).
• Explain whether you feel these inequalities are true.
• Express your conclusion as a mathematical inequality.
• Explain who might be interested in these results, and why.

Assignment 5a: Methods for Dividing and Interpreting Remainders

Submission Instructions: Write your answers to the problems on paper and then scan or take photos of those pages.  If there are multiple pages to your completed assignment, you must submit them as one multi-page document (pdf, docx, jpg, or png).

Directions: Complete each of the following. Reminder: The “quotient” of two numbers is the result of dividing two numbers.

Find the quotient of 128 and 4.

Use drawings of base-ten blocks to find the quotient. First represent the problem with the base-ten blocks then show any trading and grouping needed to find the quotient.

What is the quotient of 128 and 4?

Find the quotient of 741 and 3.

Use the partial quotients algorithm to find the quotient.

1. What is the quotient of 741 and 3?

Find the quotient of 924 and 7.

1. Use the standard algorithm for division to find the product.
2. What is the quotient of 924 and 7?
3. Write a division word problem involving a remainder such that the remainder indicates that the whole number part of the quotient needs to be rounded up in order to correctly answer the question asked in the word problem.

Write a division word problem involving a remainder such that the remainder itself is the answer to the question asked in the word problem.

1. section 2 
2. Submission Instructions:Write your answers to the problems on paper and then scan or take photos of those pages.  If there are multiple pages to your completed assignment, you must submit them as one multi-page document (pdf, docx, jpg, or png).
3. Directions: Use order of operations to evaluate each of the following expressions. Show each step. An example is shown below.

Example:   32 ÷ 8 x 22 + 3(4 – 1)

1. 32 ÷ 8 x 22 + 3(3)
2.                      32 ÷ 8 x 4 + 3(3)
3.                       4 x 4 + 3(3)
4.                        16 + 3(3)

             16 + 9

                             25

14 – 4 + 15 ÷ 3

12 ÷ 3 x 2 + 5

5 x 4 – 2 x 3 + 16 ÷ 4

32 – 542  8  2 – 2

(24 ÷ 3 x 4) – (16 – 6 + 2)