Unit 4: Number properties and more

Slope definition / formula?

C31) Slope = Rise/run

Or

(y2-y1)/(x2-x1)

The 4 types of slopes?

C31) Positive, negative, zero, undefined

Slope intercept equation?

31) Y=mx+b

Where m is the slope and b is the y-intercept

How to calculate the distance between two points (1,3) and (7,-5)?

C31) Find the rise (y2-y1) and run (x2-x1). Then, once you have the vertical and horizontal line use Pythagorean theorem.

Probability is (blank)/(blank)

C30) Number of desired or successful outcomes / total number of possible outcomes

Calculate the numbers and denominator separately.

in probability, for more than one event, what do AND & OR mean?

C30) AND means multiply , OR means add

P(A) + P(not A) = 1. So when probability problems set up an ‘at least’ or ‘at most’ scenario, what would you look for?

C30) The 1-x shortcut.

It’s easier to calculate the probability of what you do NOT WANT. And then subtract that from 1. Always look if this shortcut is applicable for ‘at least’ or ‘at most’ problems

In combinatorics, what do AND & OR refer to?

C29) AND means multiply and OR means add.

Example- If you can get a free side dish of soup or salad offered with the main, you can get 2 possible side dishes so add to find the number of side dishes to order.

If you want to find the possible combinations of 3 main dishes offers and side dish, then you need to decide on a main AND side dish. So list out all combinations. You make a decision for a main and a decision for a side as well.

3×2=6 possible combinations.

When a question asks how many possible ways something can happen, you have a combinatorics or a counting problem. Which method do you use?

C29) The slot method.

Each digit refers to the number of options (eg 1 OR 3 OR 5) that can be put in that blank.

______ x _____ x ______

(Digit 1) AND (Digit 2) AND (Digit 3)

How many arrangements are in the word EEL?

C29) use the anagram grid method.

it’s not 3! =6 arrangements because there are two identical letters.

E1E2L E1LE2. LE1E2

E2E1L. E2LE1. LE2E1

The two arrangements in each column are identical. So the answer is 3.

What is the anagram grid method used for?

C29) To arrange groups and find the number of ways things can be distributed.

So from the image, the Ss, Bs are indistinguishable so use the formula. The numerator is the largest number in the top row.

Top row/bottom row = 7!/(1!1!2!3!)

You can also have multiple groups so use the above method for all groups and then multiply the answers to get the final answer.

m=2k + 1, where k is an integer. Is m odd or even?

16/m =n , what does this convey?

C28) Even

16=mn ; so m and n are factors of 16

Even + even =?

Odd + odd =?

Odd + even = ?

Even x even = ?

Even x odd = ?

Odd x odd = ?

Even + even = Even

Odd + odd = Even

Odd + even = Odd

Even x even = Even

Even x odd = Even

Odd x odd = Odd

The sum of two primes is always ____ (Odd/even) unless one of prime numbers is ___.

C28) Even, 2

Rules of disability for 2,3,4,5,6,8,9

C27)

How would you find factors of 72?

C27) Using factor pairs and walking upwards from 1 and 72.

How would you find the prime factors of a number? What is the factor foundation rule?

C27) Prime factorization involves creating a prime factor tree.

Factor foundation rule : if a is a factor of b, and b is a factor of c, then a is a factor of c.

What is the prime box used for? What is a partial prime box? What is ‘quotient’?

C27) A box of all prime factors of a number (the lowest level building blocks). Always do repeat copies if the same prime number appears more than once.

You can use the box to test whether or not a specific number is a factor of another number.

A partial prime box has ‘…’ in the prime box indicating there are other prime factors not mentioned in it.

Eg- if n is divisible by 8 and 15, is it divisible by 12?

We only have a partial list of prime factors of n because n is an unknown number. There is no way of knowing what additional primes n has in its box. it is divisible by 12 in this case.

always strip out overlaps in the prime box if for example- an integer k is divisible by 8 and 10. Variables mostly have partial prime boxes.

Quotient is the number of times the divisor goes into the dividend COMPLETELY.

How are dividend, quotient, divisor and remainder related?

Dividend = quotient x divisor + remainder

Quotient x divisor = multiple of divisor

How would you count integers, inclusive? What about consecutive multiples? (Which formula)

C25)

Consecutive integers counting inclusive

= (Last - first + 1)

Consecutive mutliples counting inclusive

= (last multiple - first multiple / increment) + 1

The average and median are equal to each other in an evenly spaced set. True or false?

The mean and median of the evenly spaced set are equal to the average of the first and last terms. True or false?

C25) Both True

How would you calculate the sum of consecutive integers?

C25) Sum of evenly spaced set = average x number of terms

Which tool would you use to find overlapping sets for only two sets of data?

C23) The double-set matrix.

Note: the rows must correspond to the mutually exclusive options for one decision.

Many overlapping sets have % or fractions. You can pick smart numbers as the total- 100 for problems involving %

How are rate and time related to work and distance? (Formulae)

C22) Rate x time = distance

Rate x time = work

For rate/distance problems, you can also draw it out

If an elevator takes 4 seconds to move 1 floor, what is the rate?

C22) 1 Floor every 4 seconds

= ¼ floor /second

Always distance over time

How would you find the average rate?

C22) The average rate is never the ‘straight’ average of two rates given for different legs of the journey. The average rate will always be closer to the slower of the two rates.

Average speed = total distance / total time

If two workers are working together you ___ the rates.

C22) Add (If two workers are working together on the same task)

if one worker is undoing the work of another, subtract