What multiplication or subtraction problem is being modeled? Take-away Missing addend Repeated addition Repeated subtraction

What multiplication or subtraction problem is being modeled? Rectangular array Missing factor

a2a2 a3a3 a4a4 a5a

Draw a picture of how we could use Dienes blocks to represent The number 204 six

Divide 346 six

23 = 25 b What base number will make the equation true? 2b + 5 = 23 2b +5 (-5) = 23 – 5 (÷ 2) 2b = 18 (÷ 2) b = 9

331 four = 75 b First, change 331 four into base = 61 So, 7b + 5 = 61 7b + 5 (-5) = 61 – 5 7b = 56 7b ÷ 7 = 56 ÷ 7 b = 8 75 eight = 61 Check: 7 x 8 = =

Divisibility Tests DivisorWhenProof 20,2,4,6,8 3 Sum of digits is divisible by 3 4 Last 2 digits are divisible by 4 50, 5 6Is divisible by 2 and 3 7 8Divide by 4 and then that result by 2 9 Sum of digits is divisible by 3 or 9 10 Ones digit is 0 11

6.1 Fractions are used to indicate the number of parts of a whole to be considered. Part-to-whole, when b ≠ 0, and a & b are whole numbers Then, represents “a” “of b” equivalent parts You must know what the whole is. Quantity is a measurement of length, area, or volume. The numerator counts how many you have. The denominator tells us what’s being counted. A fraction is a number that can be represented by an ordered pair of whole numbers where b ≠ 0.

Show 4 + (-7) Look at the first number to see if you will take away or combine. Look at the second number to see what color of chips to use. Add one red and one black (opposites) chip ( + = zero) for each number to be subtracted. Take 7 away, then pair as many as possible to make zero. The answer is the number of chips that are not paired up after subtracting. 4 + (-7) = 3 Try Combine Red

2 – 4 Red Chip Model

(-2) – (-7) Red Start with 2

How many factors does 500 have? 500 = 2 2 ∙ 5 3 Possibilities for 2 are: 2 0, 2 1, 2 2 Possibilities for 3 are: 3 0, 3 1, 3 2, × 4 = 12 Possibilities

How many groups of ? The whole IS ….

The whole is ____. The answer is ____

Fractions whose denominators are powers of 2 and 5 (or both) with no other prime factors have terminating decimal representations. Count the number of places in a repeating decimal to find the number of periods. 2 periods 6 periods 1 period

Ratios are associated with a comparison involving “for every” How many ____ for every ____ ? - Miles / Gallon - Miles / Minute or Minutes / Mile -Dollars / Item A ratio is an ordered pair of numbers, written a:b with b ≠ 0. Used when we want to compare relative quantities. Some chickens lay white eggs, some lay brown eggs. The ratio of white to brown is 3:5. Ratio of brown to white is __:__. Ratio of white to eggs is __:__. Ratio of brown to eggs is __:__.

Between any 2 fractions there is another fraction. Therefore, the set of fractions is said to be dense.

When you see the number -4, you can refer to it as negative 4 or the opposite of 4. Call -(-4) the opposite of the opposite of 4. (Positive 4) Show (-2) – (-7) on a measurement model To subtract (-7) switch the direction of the arrow. (-2) – (-7) = 5

The set of fractions is closed. The sum of two fractions is a fraction. Additive identity for fractions is 0. There is a unique number such that

I C F W Relationship of Counting Numbers {1, 2, 3, 4 ….} Whole Numbers {0, 1, 2, 3, 4….} Fractions (all fractions were b 0, like ⅓, ⅜) Integers

CWFZQ ? Set of Counting Numbers {1,2,3,4,…} Set of Whole Numbers {0,1,2,3,…} Set of (Nonnegativ e) Fractions { | a, b are whole numbers with b ≠ 0} Set of integers {…-3,-2,- 1,0,1,2,..} Set of Rational Numbers { | a, b are integers with b ≠ 0} Closed under addition YYYYY Closed under subtraction NNNYY Closed under multiplication YYYYY Closed under division NNNNY

CWFZQ ? Set of Counting Numbers {1,2,3,4,…} Set of Whole Numbers {0,1,2,3,…} Set of (Nonnegative) Fractions { | a, b are whole numbers with b ≠ 0} Set of integers {…-3,-2,- 1,0,1,2,..} Set of Rational Numbers { | a, b are integers with b ≠ 0} Closed under addition YYYYY Closed under subtraction NNNYY Closed under multiplication YYYYY Closed under division NNNNY

Rational numbers any number that is a ratio of 2 integers such that the denominator is not zero. Includes C, W, F, Z

Base 5 Addition five five

If the 4 th term in an arithmetic sequence is 43 and the 11 th term is 127, what is the 15 th term? Describe your strategy in words. Find the difference between terms. 127(11 th term) – 43 (4 th term) = ÷ 7 = 12 Find the first term. 43– 36 = 7 Now use the formula a + (15) d, where a is the first term and d is the difference 7 + (15) 12 = 187

Base 5 Multiplication X X five 1 Base 5 Addition Now try 344 X32

Sieve of Erathanous (spelling) There are an infinite number of primes. If there were a finite number of primes, then the product of all of the primes plus 1 would also be a prime. None of the primes before it would divide evenly and have a remainder of 1. To find prime numbers, start with a 100’s chart. Circle the number 2. Now cross out every multiple of 2. Circle he number 3 and cross out every multiple of three, EVEN IF IT HAS ALREADY BEEN CROSSED OUT. Circle the number 5, then cross out every 5 th number, EVEN IF IT HAS ALREADY BEEN CROSSED OUT. Keep doing this for every number that is left. Check your answer

No 2 odd prime numbers will total an odd prime because 2 is the only even prime number. If 36/m what else divides 36 36|m 36x = m 1,2,3,4,6,9,12,18 2(18k)=m, 3(12k)=m, 4(9k)=m, 6(6k)=m, ….

Quadratric functions are functions that can be written in the Form f(x) = ax 2 + bx + c where a, b, c are real numbers with A ≠0

Sieve of Erathanous (spelling) There are an infinite number of primes. If there were a finite number of primes, then the product of all of the primes plus 1 would also be a prime. None of the primes before it would divide evenly and have a remainder of 1. To find prime numbers, start with a 100’s chart. Circle the number 2. Now cross out every multiple of 2. Circle he number 3 and cross out every multiple of three, EVEN IF IT HAS ALREADY BEEN CROSSED OUT. Circle the number 5, then cross out every 5 th number, EVEN IF IT HAS ALREADY BEEN CROSSED OUT. Keep doing this for every number that is left. Check your answer

Theorem 1 a,m,n,k whole numbers with a ≠ 0 if a | m and a | n then a | m + n if a | m and a | n then a | (m - n) for m ≥ n if a | m then a | km Proof for divisibility of 2 100a Prove that a | b and a | c, then a 2 | bc Assume a | b and a | c Since a | b, then there is a whole number x such that ax=b Since a | c, then there is a whole number x such that ay=c Now bc = (ax) (ay) = a 2 xy Since whole numbers are closed under multiplication (xy is a whole number) Then a 2 | bc

Linear functions Their graph is a straight line. A linear function can be written in the form y = mx + b. (or f(x) = mx + b) The “m” represents slope or steepness and is the constant rate of change of the line The “b” tells us what the y intercept is.

Reflexive Symmetric Transitive

An exponential function has the variable in the exponent. f(x) = 2 x g(x)=3 x You can’t put any number in x that will equal 0 for output 3 0 = 1 In a function with variable in the exponent, If base > 1, the function increases rapidly. Find g(x). The larger the base, the ___________ it increases. xf(x) xg(x) 0 1 2

Quadratic functions are functions that can be written in the form f(x) = ax2 + bx + c where a, b, c are real numbers with a ≠ 0. f(x) = x2 + 5x + 6 f(x) = 4x2 -1 Parabola Range – if a > 0 smallest vertex y if a < 0 – largest vertex y

Fractions whose denominators are powers of 2 and 5 (or both) with no other prime factors have terminating decimal representations.

What percent of 16 ¾ is 12 2/3 ? What do you do next?

Finish the problem.

Multiplication as repeated addition 4 x ⅓ = ⅓ + ⅓ + ⅓ + ⅓

Note to Self To be a function all inputs have to be mapped to one output. Domain –input, Range – output For –a, say the opposite of a Use the term “exchange” in place of “borrow” Don’t call a datapoint a coordinate, since there are two points in an ordered pair (2, 3). 1/3 to 3/1 or 3 ÷1 to 1 ÷ 3, is called a transformation? Finding the reciprocal is …

More Notes to Self A googooplex is 1 with 100 zeros (?). 1 mph = 1.61 kph (kilometers). The square root of negative 9 is not a real number. A proportion is a statement that two given ratios are equal.

_ A ∩ B ∩ C

(A∩B)UC

(A U B) ∩ C

_ (A ∩ B) U C

A B If A is a subset of B then then A union B is ___? If A is a subset of B, then A intersection B is ___? Discussion Question

A B A B If A is a subset of B then then A union B is ___? If A is a subset of B, then A intersection B is ___?

Improper subset less than or equal number of elements in A as in B Proper subset less elements in A as in B Set A is said to be a subset of B if and only if every element of A is also an element of B. We write A B.

Linear functions Their graph is a straight line. A linear function can be written in the form y = mx + b. (or f(x) = mx + b) The “m” represents slope or steepness and is the constant rate of change of the line The “b” tells us what the y intercept is.

Reflexive Symmetric Transitive

An exponential function has the variable in the exponent. f(x) = 2 x g(x)=3 x You can’t put any number in x that will equal 0 for output 3 0 = 1 In a function with variable in the exponent, If base > 1, the function increases rapidly. Find g(x). The larger the base, the ___________ it increases. xf(x) xg(x) 0 1 2

Quadratic functions are functions that can be written in the form f(x) = ax2 + bx + c where a, b, c are real numbers with a ≠ 0. f(x) = x2 + 5x + 6 f(x) = 4x2 -1 Parabola Range – if a > 0 smallest vertex y if a < 0 – largest vertex y

Fractions whose denominators are powers of 2 and 5 (or both) with no other prime factors have terminating decimal representations.

What percent of 16 ¾ is 12 2/3 ? What do you do next?

Finish the problem.

Multiplication as repeated addition 4 x ⅓ = ⅓ + ⅓ + ⅓ + ⅓