Mathematical Literacy in a Numerate Society



Mathematical Literacy in a Numerate Society

Introduction to the workbook:

Please read first to be successful in this course. To be successful in this course you must have buy-in from both teacher and student. The goal of the lesson plans is to make the transition from a traditional style to make the student quantitatively literate. This will take a different approach. Think of it as going from chalk and talk to transitioning to teaching through conversation. You’ll still use the chalk, just not so much as with the old form of the traditional style of lecture.

Constructionist learning style. Have what the student’s knowledge to base future lesson plans.

One: Number Sense

Lesson One: Numeracy – Enormity of the Geologic Time Scale

Earth’s age of 5 billion years, the formation of life 700 million years ago, the first appearance of dinosaurs 2000 million years ago, and the Birth of Christ 2000 years ago is impossible to visualize. But if we reduce the earth’s age to a week, we can visualize the relative sizes between the numbers.

Lesson Two: Numeracy – Vastness of Space

Issac Newton

The table illustrating the exact size of the planets and the true distances between them is the same problem for the astronomer as the geologic time scale is to the geologist. Thousand, millions, billions and trillions of miles is impossible to visualize. Can a student close their eyes and see the sun, a sphere whose diameter is nearly 865,000 miles in diameter. Of course not, but if we reduce the scale to 100,000 miles for every inch, we can provide a visualization as clear as day.

The visualization takes place in two steps.

Step 1: Create a new Scaled back Table, with small objects for the plants and paces you would take from the sun to place the planets.

Step 2: Create a new table, with the planets size and distance from the sun compared in relative terms to the size and distance from the sun for earth. In other words, to cement the size of the solar system in the student’s mind, have them calculate that the Sun has a volume over a million times that of earth and Pluto is 40 times farther from the Sun than Earth.

Closure: Have the student re-examine the Table with the real data, and have the students notice how much better they can comprehend the magnitude of these number. s compared to

Lesson Three: Set Theory – Introduction to Set Operations

Cantor and Dodgson. The infinite set and set across to the complex number plane

Set operations union, intersection, minus and complement

Cardinality of a set

Goal:

Motivation/ice breakers

Expectations:

Assessments:

Knowledge gained by the student and the teacher success:

Include over heads

a) [pic] b) [pic] c) [pic] d) [pic]

e) [pic] f) [pic] g) [pic] h) [pic]

and N subset Z, N subset Q, Z subset Q, Q subset R

➢ Numeracy

➢ Correlations vs causation

➢ Coincidence means to coincide, not to cause

➢ How many ways can something be done? And this has an implication in our every day lives

➢ Any two particular events have an extremely high number of associated facts, events and meanings associated with them. The fact that some of these occurrences appear to overlap or coincide should not surprise us. It is just a reflection of the multiplication principle that we wil soon see.

➢ “you and I were talking about so and so last night

➢ Two people meet on a plane, they find they have a common acquaintance. So what.

➢ Even the bizarre notion of the Kennedy and Lincoln coincidences that seem to captivate the chat rooms and fill the web pages.

➢ ? Take any two seemingly unrelated events, ‘google’ the topics on the Internet, and I assure you, you will get just as many “coincidences” or “what’s the odds of that’s” as the Kennedy/Lincoln coincidences. You try it.

➢ spin your own conspiracy theories about two presidents. Write a book, if you want. I often thought a good book could be written if we employed the same notion with the vice-presidents. Common thread or little known conspiracies about those really in charge. From John Adams to Dick Cheney

➢ December 5th. Boats. Sinking. The result: I found out that on December 5th, 1664, a ship off north Wales with 81 passengers aboard, sank. There was one survivor. A man named Hugh Williams. On the same date, December 5th, 1785, a ship sank with 60 passengers aboard. There was one survivor. A man named Hugh Williams. On December 5th, 1860, a ship with 25 passengers abroad sank. There was one survivor. A man named Hugh William

➢ We are compelled to find significance to a string of facts

➢ Nearly 7000 aumsement park deaths - stands out in a frightening way. But, let’s examine this number more closely. Roughly 317,000,000 or over 300 million people visit fixed site amusement parks in a given year. Now, let’s compare this number to those injured each year in other recreational activities: 82,722 people are injured on trampolines, 62,812 are injured in swimming pools, 544,561 are injured on bicycles, 20,000 are injured at music concerts, and 200,000 are injured in preschool and elementary school playgrounds.

Discuss poverty in this country

➢ Set Theory

➢ Counting Theory – Multiplication Principle, Permutations, Combinations

PROJECT

Below is a grid of a city showing three streets that run east-west and three streets that run north-south. The numbers you see are the number of minutes it usually takes to travel the segment of the street shown during peek time, which is weekdays from 6:30 to 8:00 am. The obvious question, you want to get from the “start” to the “end”, which route will enable you to travel the quickest?

[pic]

To solve this question, most of us would try to calculate the travel time for each possible route one at a time and then choose to maneuver through traffic by using the route that has the smallest number of minutes associated with it. So, let’s examine the travel times for all possible routes. The natural question arises is ‘how many different routes are there?’ We don’t want to use a method that would bog us down with too many calculations.

For notation purposes, so we are not bogged down, let’s use the notation E for east and N for north. Note, that at any intersection, we would either go east north. So, for the route ENEN, it would take 3 + 5 + 6 + 4 minutes or 18 minutes. And for the route EENN, it would take 3 + 7 + 4 + 5 minutes or 19 minutes.

Alright, let’s try to exact the fastest path by hand calculations:

1. ENEN: 18 minutes

2. EENN: 19 minutes

3. ENNE: 13 minutes

4. NEEN: 18 minutes

5. NNEE: 17 minutes

6. NENE: 14 minutes

So, the fastest path would be going east for 2 blocks and then north for two blocks.

We had to try 6 possible paths. Where did the number 6 come from. Well, for each single block we travel, at each intersection, we have a choice of two directions, either E or N. And how many blocks do we need to travel along? Four. So, we have 4 in a group, with 2 choices for each item. Does order matter? Well, we are looking for any list of two E’s and two N’s. So, we have [pic]paths. Note, this is the same calculation as [pic]

Now, let’s zoom our grid out and thus show a larger sector of this city.

[pic]

Again, we ask, which is the fastest path? Well, first we ask ourselves, would it be just as easy to try to calculate the time for each possible path again? We need to find the number of paths possible to know whether or not it would to cumbersome to list each individual path’s travel time.

Again, there are 2 choices at any intersection, E or N. And we travel a total of 6 blocks. So, we are looking for lists of 3 E’s and 3 N’s. So, we have [pic] paths. Note, this is the same number as [pic] paths to test.

How many different paths would we need to test for each of the following?

[pic]

or

[pic]

For the 4 by 4 grid, we would have to try [pic] and for the 5 by 5 grid, we would have to try [pic]routes to try.

Two: Probability and Statistics

ACTIVITY: Introduction to Permutations and Combinations

Have the class gather in a circle. Count the number of students in the circle. Have the students shake hands with everyone in the circle.

➢ How many hands did each person shake?

For ex: if there were 19 people, each person shook 18 hands.

➢ How many handshakes occurred?

For ex: if there were 19 people in the class, each person shook 18 hands. 19 people shaking 18 hands each is 19(18) = 342 handshakes total. But, each of the 342 handshakes were counted twice, so there were actually ½ of the 342 handshakes, or 171 handshakes.

➢ 342 and we divided out the order. [pic].

➢ How many hands were shook? 342. Or [pic]

Now, line the students up along the wall. Ask them in how many ways can we elect a president, vice president and a treasurer to represent the class.

1 person is president, 1 person is vice-president and we have 17 other people standing that could be treasurer. 19(18)(17) = 5,814 ways the elections could turn out.

Now, say you want to form committees of three. How many committees are possible?

Take groups of three people. Give each person in the first group a card that reads A, B and C. Have a president line, vice president line and a treasurer line. {ABC, ACB, BAC, BCA, CBA, CAB} or 6 ways to order 6 people. [pic]committees are possible.

Try with a smaller group. A group of five. Give each a name card ALAN, BILLY, CINDY, DEANN and EVELYN. Want a P, VP and T. Show all ways. Want a committee of three. Show all 60/6 = 10 ways.

Chapter two

Following is a list of observances typically covered by the Census Bureau's Facts for Features series:

African-American History Month (February)                                   Back to School (August)

Valentine's Day (Feb. 14)                                                                      Labor Day (Sept. 1)

Women's History Month (March)                                                      Grandparents Day (Sept. 7)

St. Patrick's Day (March 17)                                                                 Hispanic Heritage Month (Sept. 15-Oct. 15)

Asian Pacific American Heritage Month (May)                               Halloween (Oct. 31)

Older Americans Month (May)                                                          American Indian/Alaska Native Heritage Month

Mother's Day (May 11)                                                                            (November)

Father's Day (June 15)                                                                          Veterans Day (Nov. 11)

The Fourth of July (July 4)                                                                  Thanksgiving Day (Nov. 27)

Anniversary of Americans with Disabilities Act (July 26)             The Holiday Season (December)

Editor's note: Some of the preceding data were collected in surveys and, therefore, are subject to sampling error. Questions or comments should be directed to the Census Bureau's Public Information Office: telephone:

(301) 763-3030; fax: (301) 457-3670; or e-mail: < pio@>.

PROJECT: Bayes Formula

Say nothing and just hand out the following poll to your class.

1. Should abortion be legal or illegal if "the woman's life is endangered"

Circle One: Legal Illegal

2. Should abortion be legal or illegal if "the woman's physical health is endangered"

Circle One: Legal Illegal

3. Should abortion be legal or illegal if "the pregnancy was caused by rape or incest"

Circle One: Legal Illegal

4. Should abortion be legal or illegal if "”there is evidence that the baby may be physically impaired"

Circle One: Legal Illegal

5. Should abortion be legal or illegal if "the woman or family cannot afford to raise the child"

Circle One: Legal Illegal

6. Which side of the political debate on abortion do you sympathize with more: the right-to-life movement that believes abortion is the taking of human life and should be outlawed, OR, the pro-choice movement that believes a woman has the right to choose what happens to her body, including the right to decide to have an abortion.

Circle One: Right-to-life Pro Choice

7. Which party most closely reflects your views on most political issues:

Circle One: Democrat Republican Independent

Collect the poll and ask the class these key Questions:

• Can we draw along party lines on the abortion issue?

• Does the abortion debate transcend party lines, polarizing us in directions we may not want to admit, as the media would like us to believe?”

Tell the class the first 5 questions were placed on the poll to get you to critically think about the abortion issue. This is a common practice on may polls. The key questions for deciphering the question “does abortion transcend party lines?” are the last two question. Count the results and then runs Baye’s Theorem. Show the class the tree diagram. Then discuss with the class the results. From your poll, is the abortion issue an issue that transcends political boundaries? How many arguments can we form from Baye’s Theorem.

Statistics:

Time: 5 minutes.

Introduce mean, median and mode.

Use the following transparency. Split your class into exactly three groups. This helps create enough data. Hand out one of the following questions to each of the three groups, and have the students answer the questions. Then have the students report to the class their findings.

Range, Mean, Median and Mode

Write the name of each partner in your group.

Find the range, mean, median and mode of for the partners in your group for each other’s:

Heights, in inches

Range, Mean, Median and Mode

Write the name of each partner in your group.

Find the range, mean, median and mode of for the partners in your group for each other’s

Distance, in miles, traveled each day to school

Range, Mean, Median and Mode

Write the name of each partner in your group.

Find the range, mean, median and mode of for the partners in your group for each other’s:

Number of siblings

(brothers and sisters)

Three: Finance

Four: Geometry

Five: Rates of Change and Calculus

Six: Logic, Argument and Proof

Activity: How observant are you? To be done after introducing logic and truth tables.

After introducing logic and truth tables, run a game of speed math. For the next six questions, use the following statements.

P: George Washington was the first president of the United States.

Q: Arizona is the only state that has no state capital.

R: 6 + 8 = 100

S: All swans are birds.

Tell your students to study these 4 statements carefully. Tell them to take time to observe the truth value of each. Then one at a time, write each of the six compound sentences down. Instruct the students to scream the answer out as quickly as possible. Then have the students explain what is the fastest way of deciding the truth value of the compound sentence, that is why they should have been able to answer the question so quickly.

Find the truth value:

1. [pic] 2. [pic] 3. [pic]

4. [pic] 5. [pic] 6. [pic]

Answers:

1. T It is quick because P and R have opposite truth values, so the conjunction is F, so the negation of the conjunction is T.

2. F It is quick because Q is F, then the conditional statement is T, so the negation of a true statement is F.

3. T It is quick because S is T, so the disjunction is T.

4. F It is quick because we are negating P and R, where one is true and one is false, so the result will have one T and one F statement. Since we have a conjunction, we need both to be T for the statement to be true.

5. T It is quick because the negation of P is F, so the conditional is true, so the disjunction is T.

6. T It is quick because P is T ands we have a disjunction.

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