What multiplication or subtraction problem

is being modeled?

Take-away

Missing addend

Repeated addition

Repeated subtraction

-5 -4 -3 -2 -1 0 1 2 3

-4 -3 -2 -1 0 1 2 3 4

What multiplication or subtraction problem

is being modeled?

Rectangular array

Missing factor

-3

-5 -4 -3 -2 -1 0 1 2 3 4 5 6

a2 a3 a4 a5

0 0 0 0 0

1 1 1 1 1

2 4 8 16 32

3 9 27 81 243

4 16 64 256 1024

5 25 125 625 3125

6 36 216 1296 7776

7 49 343 2401 16807

8 64 512 4096 32768

9 81 729 6561 59049

10 100 1000 10000 100000

5 52 2 16 6 216 243

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

216

63

36

62

6

61

0

60

2 0 4

Divide 346six

23 = 25b

What base number will make the equation true?

2b + 5 = 232b +5 (-5) = 23 – 5(÷ 2) 2b = 18 (÷ 2)

b = 9

331four = 75b

First, change 331four into base 1048 +12 + 1 = 61

So, 7b + 5 = 617b + 5 (-5) = 61 – 57b = 567b ÷ 7 = 56 ÷ 7b = 8

75eight = 61

Check:7 x 8 = 56 56 + 5 = 61

16

42

4

41

0

40

3 3 1

64

82

8

81

0

80

7 5

Divisibility TestsDivisor When Proof

2 0,2,4,6,8

3 Sum of digits is divisible by 3

4 Last 2 digits are divisible by 4

5 0, 5

6 Is divisible by 2 and 3

7

8 Divide 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.

a

b

a

b

a

b

35,

610,

1220

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 2 - 4

Combine

Red

2 – 4Red

Chip Model

(-2) – (-7) RedStart with 2

How many factors does 500 have?

500 = 22 ∙ 53

Possibilities for 2 are: 20, 21, 22

Possibilities for 3 are: 30, 31, 32, 33

3 × 4 = 12 Possibilities

2 8 are there in

9 9

8 2 ÷

9 9How many groups of ?

The whole IS ….

2 84 groups of in

9 9

The whole is ____.The answer is ____.

8?in thereare 4

3 of groupsmany How

1 2 3 4 5 6 7 8

4

3

1

8 is what s,other wordIn

1 2 3 4 5 6 8 9 107

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

.454545 .615384 .6

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.

3 4Find a number between and

5 5

3 6 4 8 = and =

5 10 5 10

so,

7 3 4 is between and

10 5 5

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

-5 -4 -3 -2 -1 0 1 2 3 4 5 6

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.

a c c a + = + Communitive

b b b b

a c d a d + + = + Associative

b b b b b

c

b

Additive identity for fractions is 0. There is a unique number such that

0 a a 0 a + = + =

b b b bb

IC

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

a

b

C W F Z Q

?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 additionY Y Y Y Y

Closed under subtractionN N N Y Y

Closed under multiplication Y Y Y Y Y

Closed under divisionN N N N Y

C W F Z Q

?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 additionY Y Y Y Y

Closed under subtractionN N N Y Y

Closed under multiplicationY Y Y Y Y

Closed under divisionN N N N Y

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

Base 5 Addition

+ 1 2 3 4

1 2 3 4 10

2 3 4 10 11

3 4 10 11 12

4 10 11 12 13

434five

+ 213five

If the 4th term in an arithmetic sequence is 43 and the 11th term is 127, what is the 15th term?

Describe your strategy in words.

Find the difference between terms. 127(11th term) – 43 (4th term) = 84

84 ÷ 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 1 2 3 4

1 1 2 3 4

2 2 4 11 13

3 3 11 14 22

4 4 13 22 31

442 X12

1434 4420

11404five

1

Base 5 Addition

+ 1 2 3 4

1 2 3 4 10

2 3 4 10 11

3 4 10 11 12

4 10 11 12 13

Now try344X32

Sieve of Erathanous (spelling)

There are an infinite number of primes.If there were a finite number of primes, then the product of all ofthe 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 5th 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 1,2,3,4,6,9,12,182(18k)=m, 3(12k)=m, 4(9k)=m, 6(6k)=m, ….

Quadratric functions are functions that can be written in theForm f(x) = ax2 + 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 ofthe 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 5th number, EVEN IF IT HAS ALREADY BEEN CROSSED OUT.Keep doing this for every number that is left.

Check your answer

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

Theorem 1 a,m,n,k whole numbers with a ≠ 0if a | m and a | n then a | m + nif a | m and a | n then a | (m - n) for m ≥ nif a | m then a | km

Proof for divisibility of 2

100a Prove that a | b and a | c, then a2 | bcAssume a | b and a | cSince a | b, then there is a whole number x such that ax=bSince a | c, then there is a whole number x such that ay=cNow bc = (ax) (ay) = a2 xySince whole numbers are closed under multiplication (xy is a whole number)Then a2 | 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.

1

21

3

21

3

2

21

3

3Reflexive

SymmetricTransitive

An exponential function has the variable in the exponent. f(x) = 2x g(x)=3x

You can’t put any number in x that will equal 0 for output 30 = 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.

x f(x)

-1 .5

0 1

1 2

2 4

x g(x)

-1

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 + 6f(x) = 4x2 -1

Parabola Range – if a > 0 smallest vertex yif 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 ?

3

212

What do you do next?

4

316

100

x

4

67

3

38

100

x

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.

9

_A ∩ B ∩ C

(A∩B)UC

(A U B) ∩ C

_ (A ∩ B) U C

AB

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

ABA

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.

1

21

3

21

3

2

21

3

3Reflexive

SymmetricTransitive

An exponential function has the variable in the exponent. f(x) = 2x g(x)=3x

You can’t put any number in x that will equal 0 for output 30 = 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.

x f(x)

-1 .5

0 1

1 2

2 4

x g(x)

-1

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 + 6f(x) = 4x2 -1

Parabola Range – if a > 0 smallest vertex yif 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 ?

3

212

What do you do next?

4

316

100

x

4

67

3

38

100

x

Finish the problem.

Multiplication as repeated addition

4 x ⅓ = ⅓ + ⅓ + ⅓ + ⅓