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spcs:winter2014:math:day1:test

# Pre - Test

Directions for Students: Answer all the questions that are easy FIRST. Then go back and try to figure out the more challenging questions. Please show all your work and make your best guess, if you are unsure.

This is my rough-draft/answer sheet and notes of a pre-test for Math and Technology course in Santiago, Chile. Stanford University sends instructors to countries all over the world to teach exploratory courses. For this one, I have twenty-two 11-13 year old students, grades 6-8. I was going to teach more programming, but there weren't computers, so I decided to bring a bunch of Arduino's with me (Thanks Stanford!). It's a 2 week course.

I basically don't have a clear idea of how much these students know, so I thought I'd pretest them :)

## Part 1: Permutations and Combinations

Permutations: How many rainbows can you draw with these colors (red, orange, yellow, green, blue)? Each rainbow can only have 3 colors, no repeated colors.

```Where n is the number of things to choose from, and you choose r of them (No repetition, order matters)

n!/(n-r)!
```

Permutations: In how many ways can the letters of the word 'LEADER' be arranged?

```The word 'LEADER' contains 6 letters, namely 1L, 2E, 1A, 1D and 1R.

Where n is the number of things to choose from, and you choose r of them (No repetition, order matters)

(6!)/(1!2!1!1!1!) = 360
```

Permutations with Repetition: Each passcode is 5 numbers long. How many passcodes are possible?

`n^r`

Combinations: How many ways can a group of three students be selected from a class of 21 students?

```(21!)/(18!)(3!) = 1330

"n choose r"

Where n is the number of things to choose from, and you choose r of them (No repetition, order doesn't matter)
```

Combinations with Repetition: You get 4 scoops of ice cream, how many different combinations can you make from Chocolate, Vanilla, and Strawberry?

`"(n + r - 1) choose r"`

## Part 2: Game Theory

What is the Prisoners Dilemma according to this image? Try to explain what you thinks this represents.

`http://en.wikipedia.org/wiki/Prisoner's_dilemma`

What game does this matrix represent? Explain what the numbers mean.

R P S (0,0) (-1,1) (1,-1) (1,-1) (0,0) (-1,1) (-1,1) (1,-1) (0,0)
`rock paper scissors`

## Part 3: Percentages

I have 128 pieces of candy

• I give you 50% of my candies. How many do I have left?
• I give you 75% of my candies. How many do I have left?
• I give you 75% of my candies. How many do YOU have?
`Cumulative responses are acceptable as are non-cumulative`

You give me a quiz, and I get 19 out of 22 problems correct. What percentage did I get correct?

`19/22`

## Part 4: Probabilities

(Independent) My coin has two sides (heads and tails).

• What is the probability that the coin is heads when I flip it once?
• What is the probability that I flip 3 heads in a row?
`p(heads_heads_heads) = p(heads)*p(heads)*p(heads)`

(Independent) I roll two dice (each has 6 sides).

• What is the probability that the sum of the two sides is 12?
• What is the probability that the sum of the two sides is 7?
```p(roll6) = 1/6
p(roll12_2dice) = 1/36

1 + 6 = 7
2 + 5 = 7
3 + 4 = 7
4 + 3 = 7
5 + 2 = 7
6 + 1 = 7

p(roll7_2dice) = 6/36```

(Dependent) I have a bag of 5 balls: 2 red and 3 blue. Once I remove a ball, I do not put it back in the bag.

• What is the probability that I pick a blue ball?
• What is the probability that I pick a blue ball and then another blue ball?
```p(A and B) = p(A) * p(B|A)
2/5 * 1/4 = 1/10```

## Part 5: Geometry

Give me the area:

• a square where one side measures 4 cm
• a circle where the radius is 10 cm
• a triangle where the base is 8 and the height is 9
```s*s
pi*r^2
(1/2)bh```
`Using the 45°-45°-90° special triangle ratio, the length of the side of the square is 2.`
```Sum of angles in a triangle = 180°.
So, y° + 40° + 80° = 180°
y° + 120° = 180°
y° = 60°```

Two circles both of radii 6 have exactly one point in common. If A is a point on one circle and B is a point on the other circle, what is the maximum possible length for the line segment AB?

`If the radius is 6 then the diameter is 2 × 6 = 12 and the distance from A to B would be 2 × 12 = 24`

3-Dimensional Shapes

• What is the volume?
• What is the surface area?
```A = 2πrh
V = π(r^2)h```

Coordinates: You have 2 points on a graph, (2,3) and (4,9). How do you find the distance between the 2 points?

`sqrt( (x1-x2)^2 + (y1-y2)^2 )`

## Part 6: Trees

Prefix Notation: (+ (- 7 8) (/ 10 5) (+ 2 2 (x 9 10))

• Draw the tree representation for this equation
• Write this in infix notation
• What is does this return?
```(7-8) + (10 / 5) + (2 + 2 + (9 x 10))
```

Game Search: This is a tic-tac-toe game tree

Draw the game tree that starts with:

```|o|x| |
|x| |o|
|o|x| |```

## Part 7: Graph Theory

• How many vertices?
• How many edges?
• Which vertex has the highest degree?

Draw a graph that is fully connected with 5 vertices. How many edges does it have?

`A complete graph is a graph in which each pair of graph vertices is connected by an edge. `

Traveling Salesperson

• Create an adjacency matrix for this weighted graph
• The Traveling Salesperson starts at point A, what is the shortest path he can take, so that he returns to point A with the lowest cost. He must visit B, C, and D.
```For a given source vertex (node) in the graph,
the algorithm finds the path with lowest cost (i.e. the shortest path)
between that vertex and every other vertex. ```

## Part 8: Circuits

What is Ohm's law?

`V = IR`

Circuit Diagram:

• Describe what this diagram represents
• What is the current flowing through this diagram?
`I = V/R`

What do we use resistors for?

```Resistors are used to limit current in a circuit.
It follows the math and the practical implication is to not overload and burn out, for example, LEDs.  ```

What is an Arduino?

`Microcontroller`

What is the difference between a Digital and Analog Clock?

```"BTW, a digital clock gives the exact time to it's number of places like any digital device.
It's always 6:06 till it's 6:07 LOL.

Of course, it's easily possible to construct a digital clock that will allow more accuracy
than a 2 or even 3 hand analog clock. (given equally accurate mechanisms)"```
```"I'm not sure the difference between analog and digital is best described as approximate vs accurate,
but that of continuous vs discrete (as Steve points out).

Maybe a different choice of object such as what is the difference
between an analog signal and a digital signal.
One changes smoothly and the other abruptly.
Music is digital in an mp3 file, but it's analog when the speaker (and air) moves."```

Logic Gates:

• Which # pin do you plug the black wire?
• Which # pin do you plug the red wire?
• What does this chip do?
```red - 14
black - 7
or gate```

## Part 9: Boolean Algebra

Draw the truth table for: ~p v q (not p or q)

```00 1
01 1
10 0
11 1```

Use DeMorgan's Law to simplify this equation: ~(p ^ q)

`~p v ~q`

What is another way to represent p → q (p implies q)?

`~p v q`

## Part 10: Karnaugh Maps

Explain what is happening the in diagram above.

`Y = ~A v ~B`

Explain what is happening the in diagram above.

`Y = A(~C) v B`

Why do we use Karnaugh Maps?

`To detect redundancy`

## Part 11: Logic

For the following mark whether the argument is inductive or deductive:

• Jill and Bob are friends. Jill likes to dance, cook and write. Bob likes to dance and cook. Therefore it can be assumed he also likes to write.
• Jennifer leaves for school at 7:00 a.m. and is on time. Jennifer assumes, then, that she will always be on time if she leaves at 7:00 a.m.
• Robert is a teacher. All teachers are nice. Therefore, it can be assumed that Robert is nice.
• All cats that you have observed purr. Therefore, every cat must purr.
• All students that have been taught by Mrs. Smith are right handed. So, Mrs. Smith assumes that all students are right handed.
• All observed basketball players are tall, so all basketball players must be tall.
• Suzy is a doctor. Doctors are smart. Suzy is assumed to be smart.
• All observed brown dogs are small dogs. Therefore, all small dogs are brown.
• All observed children like to play with Legos. All children, therefore, enjoy playing with Legos.
• All observed police officers are under 50 years old. John is a police officer. John is under 50 years old.
• Mary and Sue are friends. Mary enjoys fishing, running and rock climbing. Sue likes fishing and rock climbing. Sue must also like running.
• Ray is a football player. All football players weigh more than 170 pounds. Ray weighs more than 170 pounds.
• All observed cats in the area are brown. Tiny is a cat. Tiny is brown.
• Bob is a sumo wrestler. Sumo wrestlers weigh a lot. Bob weighs a lot.
`It's debatable, but anything that starts with "All observed" is inductive.`

Match the logical statement with the English sentence:

 ∀x IsABunny(x) ⇒ IsCute(x) There is at least one student who doesn’t hate (any of) the Math homework ∀x IsAStudent(x) ∧ IsTakingMath(x) ⇒ IsCool(x) Everyone who eats ramen is either very hungry or a student ∀x IsABunny(x) ∧ IsAStudent(x) ∧ IsTakingMath(x) ⇒ IsCute(x) ∧ IsCool(x) Cats rule and dogs drool ∀x EatsRamen(x) ⇒ IsVeryHungry(x) ∨ IsStudent(x) Every bunny who is a student taking Math is cute and cool ∃s ∀h IsAStudent(s) ∧ IsTaking(s,Math) ∧ HomeworkFor(h,Math) ∧ ¬Hates(s,h) All bunnies are cute ∀a (IsACat(a) ⇒ Rules(a)) ∧ (IsADog(a) ⇒ Drools(a)) Everyone student who is taking Math is cool
`http://feature-space.com/en/document41.pdf`

Write the logical statement for “All men are mortal.”

`∀x IsMan(x) -> IsMortal(x)`

## Part 12: Number Systems

How do you convert 2 decimal to binary?

`10`

How do you convert 31 decimal to hexadecimal?

`1F`

How do you convert A3D2F in hexadecimal to binary?

`10100011110100101111`

What color is:

• #000000
• #FF0000
• #FFFF00
• #FFFFFF
• #FF1493
• #8A2BE2
```black
red
yellow
white
pink
purple``` 