K-8 Mathematics

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K-8 Mathematics

Week Fifteen

EXAM 3

Week Thirteen

Practice Problems 2

a) 11% of 45 is what number?

0.11 * 45 = 4.95

Multiply the decimal and the whole number together and then add.

b) 9% of what number is 17

0.09 * n = 17

–––– ––––

0.09 0.09

n = 188.8 <–– Repeating decimal / 189

Divide both sides by the decimal.

c) 17% is what % of 25

17/25 = 0.68

|

V

68%

Divide the percentage by the whole number.

Week Eleven

EXAM 2

Week Nine

Multiplying and Dividing Fractions

Multiplication (Part of a Part):

1/2 of 1/2

1/2 x 1/2 = 1/4

1/2 x 1/4 = 1/8

1/3 x 1/8 = 1/24

Division:

2/3 ÷ 4/5 =

(Keep –> Change –> Flip)

2/3 x 5/4 = 10/12 = 5/6

2/3 ÷ 4/5 =

2/3 / 4/5 = 2/3 x 5/4 / 4/5 x 5/4

V V

10/12 20/20 = 1

10/12 / 1 = 10/12 = 5/6

Adding and Subtracting Fractions

Addition:

1/4 + 2/4 = 3/4

3/12 + 2/12 = 5/12

Subtraction:

5/8 - 4/8 = 1/8

8/9 - 2/9 = 6/9 = 2/3

Improper Fractions:

5/6 + 2/3 =

V V

5/6 + 4/6 = 9/6 = 1 3/6 –> 1 1/2 (Mixed Number)

6/10 - 2/5 =

V V

6/10 - 4/10 = 2/10 = 1/5

Week Seven

Prime Factor Trees

Prime Factor Trees

24

/ \

6 4

/ \ / \

3 2 2 2

2 * 2 * 2 * 3 = 24

48

/ \

12 4

/ \ / \

6 2 2 2

/ \

2 3

2 * 2 * 2 * 2 * 3 = 48

GCF and LCM

GCF: Greatest Common Factor

GCF (24, 36):

1. List Method:

24: 1, 2, 3, 4, 6, 8, 12, 24

36: 1, 2, 3, 4, 6, 9, 12, 18, 36

GCF (24, 36) = 12

2. Prime Factorization Method:

*USE TREE*

24: 2 * 2 * 2 * 3

36: 2 * 2 * 3 * 3

GCF (24, 36) = 2 * 2 * 3 = 12

24

/ \

6 4

/ \ / \

3 2 2 2

2 * 2 * 2 * 3 = 24

36

/ \

6 6

/ \ / \

3 2 3 2

2 * 2 * 3 * 3 = 36

LCM: Least Common Multiple

LCM (24, 36):

1.List Method

24: 24, 48, 72, 96

36: 36, 72, 108

LCM (24, 36) = 72

2. Prime Factorization Method

LCM (24, 36) =

GCF * 2 * 3 =

^

Unused Factors from the GCF

12 * 2 * 3 = 72

Divisibility Rules

a is divisible by b if there is a number c that meets the requirement: b * c = a

Ex. 10 is divisible by 5 because 2 * 5 = 10

2 * 5 = 10 5 and 2 are factors of 10

5 * 2 = 10 5 and 2 are divisors of 10

10 ÷ 2 = 5 10 is divisible by 2 and 5

10 ÷ 5 = 2 10 is a multiple of 2 and 5

Divisibility Rules:

Ending:

• By 2: 0, 2, 4, 6, 8
• By 5: 0, 5
• By 10: 0

Sum of Digits:

• By 3: Sum of digits is divisible by 3
• By 9: Sum of digits is divisible by 9

Last Digits:

• By 4: Last two digits are divisible by 4
• By 8: Last three digits are divisible by 8

Extras:

• By 6: If it is divisible by both 2 and 3
• By 7: Double the last digit then subtract from the new number to see if it is divisible by 7.
• By 11: "Chop off", add, check if divisible by 11.

Examples:

*770: 2, 5, 7, 10, 11

*136: 2, 4, 8

Factors:

28: 1, 2, 4, 7, 14, 28

60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

7: 1, 7

91: 1, 7, 13, 91

1 –> Identity Multiplication Element

0 –> Identity Addition Element

Prime Numbers (0-60)

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59

Week Five

Subtraction Algorithms

Subtraction Algorithms

American Standard:

4 13

V V

4 5 3

-2 3 6

––––––

217

European/ Mexican:

1V

4 5 3

4V

-2 3 6

––––––

217

Reverse Indian:

1V

4 5 3

-2 3 6

––––––

2

2

1 7

217

Left - to - right:

13

V

4 5 3

-2 3 6

––––––

2 0 0

2 0

10

7

–––––

217

Expanded Notation:

40 13

453 = 400 + 50 + 3

236 = 200 + 30 + 6

––––––––––––

200 + 10 + 7

217

Integer Subtraction:

4 5 3

-2 3 6

–––––

-3

20

200

––––

217

Addition Algorithms

Addition Algorithms

American Standard:

1

V

384

+235

––––––

619

Partial Sums:

3| 8| 4

+2| 3| 5

––|––|–––

| |9

1 | 1 |

5 | |

––––––––––

619

Partial Sums With Place Value:

3| 8| 4

+2| 3| 5

––|––|–––

| |9

1 | 1 |0

5 | 0 |0

––––––––––

619

Left - to - Right:

384

+235

––––––

500

110

+ 9

––––––

619

Expanded Notation:

100

V

384 = 300 + 80 + 4

+235 = 200 + 30 +5

–––––––––––––––––––

600 + 10 + 9 = 619

Week Three

Multiplication Properties

`Multiplication ––> Repeated addition

3 x 4 = 12 ––> Product

^ ^

Factors

3 x 2 = 3 groups of 2

1. Identity property of multiplication
2. Any number multiplied by 1, equals one).
3. a x 1= a
4. 7 x 1 = 7
5. -3 x 1 = -3
6. 3/5 x 1 = 3/5
7. 7 x 0 = 0 (any number multiplied by 0, equals 0).
8. Commutative property of multiplication
9. The order of numbers being multiplied does not matter.
10. a x b = b x a
11. 7 x 3 = 3 x 7
12. Associative property of multiplication
13. When multiplying numbers together in an equation, the grouping does not matter.
14. (a x b) x c = a x (b x c)
15. (3 x 7) x 2 = 3 x (7 x 2)
16. Distributive property of multiplication
17. When a number is multiplied by a sum, it is the same as multiplying that number by adding the sum and partial products together.
18. a x (b + c) = (a x b) + (a x c)
19. 3 x 7 = 3 x (5 + 2) = (3 x 5) + (3 x 2)

Addition and Subtraction Properties

Meaning:

Addition (Put together/ Join)

1. Identity Property
2. When adding the number 0 to any value, the identity remains the same.
3. a + 0 = a
4. 4 + 0 = 4
5. Commutative Property
6. The order of numbers in an equation will equal the same value.
7. a + b = b + a
8. 3 + 4 = 4 + 3
9. Associative Property
10. Grouping of numbers does not matter when adding values.
11. (a + b) + c = a + (b +c)

Addends

3 + 4 = 7 <–––– Sum

^ ^

Addends

Meanings:

Subtraction:

1. Take away (5 - 2 = 3)
2. Comparison**
3. Missing addend** (3 + __ = 7)

7 - 3 = 4 <–––– Difference

^. ^--- Subtrahend

Minuend

Week One

Problem Solving

I have three 5-cent stamps and two 9-cent stamps. Using one of more of these stamps, how many different amounts of postage can I make?

There are three 5-cent stamps - ©©© 

There are also two 9-cent stamps - ©©

© 

©© 

©©© 

© 

©©

©©

©©©

©©©

©©©©

©©©©

©©©©©

11 different postage combinations

Explanation: First I started off with laying out the different combinations of the 5-cent and 9-cent stamps, there are three 5-cent (blue) stamps and two 9-cent (red) stamps. Each combination containing both types of stamps are all unique as they have different numbers of both the 5-cent and 9-cent stamps. I first started with different combinations of 5-cent stamps with the 9-cent stamps by adding one, two, or three 5-cent stamps to each combination of 9-cent stamps. For example, when I had one 5-cent stamp I needed to add one 9-cent stamp and then another 9-cent stamp to a different 5-cent stamp so it went up +1 each time until I ran out of 9-cent stamps. I did this with other combinations of 5-cent stamps such as having two or three 5-cent stamps with one or two 9-cent stamps. Adding up all the different combinations of stamps it is a total of 11 different postage combinations.

Polya's Four Steps

1. Understand the question/ problem:
2. What are you asked to find/ show?
3. Can you restate the problem?
4. Can you draw a picture or diagram to help you solve the problem?
5. Plan how to solve the problem:
6. Draw a picture/ diagram
7. Look for a pattern
8. Work backwards
9. Make the problem simpler
10. Implement the plan to solve the problem:
11. Try different strategies
12. Do not get discouraged
13. Carrying out the plan is usually easier than devising the plan
14. Look back (reflect):
15. Is it a reasonable answer?
16. Did all questions get answered?
17. Is there an easier way?
18. What did you learn?

Week Fourteen

Positive and Negative Numbers

-17 + (10) = -7

-10 - (8) = -18

3 * (-4) = -7

16 / -4 = -4

Week Twelve

Decimals and Place Value

Decimals:

a) 0.128 < 0.234 < 0.45 <0.9

b) 0.23 < 0.3 < 0.378 < 0.98

c) 0.003 < 0.03 < 0.033 < 0.303 < 0.33 < 3.003

Place Value:

Hundred Hundred

Thousands Thousands Tens Tenths Thousandths Thousandths

V V V V v V

100,000 10,000 1,000 100 10 1 . 1/10 1/100 1/1,000 1/10,000 1/100,000

^ ^ ^ ^ ^

Ten Hundreds Ones Hundredths Ten

Thousands Thousandths

**If decimals are repeating, only use the line to represent what is being repeated**

__

0.21212121 = 0.21

_

0.555555 = 0.5

__

0.2345454545 = 0.2345

Multiplying and Dividing Decimals

Multiplication:

0.34 x 2 = ?

0.34

x 2

-----------

0.68

Answer:

0.34 x 2 = 0.68

Division:

1 ÷ 8 = ?

0.1 2 5

8 ⟌ 1.0 0 0

-8 | |

---- V |

2 0 |

-1 6 |

----- V

4 0

-4 0

----------

0

Answer:

1 ÷ 8 = 0.125

0.125 = 12.5%

0.125 = 1/8

Adding and Subtracting Decimals

Adding:

1.24 + 1.35 = ?

1.24

+ 1.35

-------------

2.59

21.34 + 90.319 = ?

21.34

+ 90.319

---------------

111.659

Subtracting:

34.45 - 32.23 = ?

34.45

- 32.23

---------------

2.22

12.338 - 8.23 = ?

12.338

- 08.230

---------------

4.108

Week Ten

Practice Problems

There was ¾ of a pie in the refrigerator. John ate 2/3 of the left over pie. How much pie did he eat?

1. First start by drawing out 3/4 on paper.
2. Then shade 2/3 of whats left of the pizza.
3. It would be 3/6 still shaded.

Jim, Ken, Len, and Max have a bag of miniature candy bars from trick-or-treating together. Jim took ¼ of all the bars, and Ken and Len each took 1/3 of all the bars. Max got the remaining 4 bars. How many bars were in the bag originally? How many bars did Jim, Ken, and Len each get?

Jim: 1/4 = 3/12 = 12 bars

Ken: 1/3 = 4/12 = 16 bars

Len: 1/3 = 4/12 = 16 bars

Max: 4 bars = 1/12

Draw a 3 x 4 grid containing 12 squares inside of it.

Week Eight

Fractions

Fractions:

Meanings:

1. Part-Whole
2. Quotient
3. Ratio

Models:

1. Surface Area
2. Length
3. Sets (groups of things)

20 Students

13 girls

7 boys

Fractions:

Girls 13 Boys 7

------- = ----- --------- = ----

Whole 20 Whole 20

Not Fractions:

Boys 7 Girls 13

-------- = ---- ------- = ----

Girls 13 Boys 7

3/7 > 1/7

4/5 > 4/9

3/7 < 5/8

9/10 > 3/4

4/ 4 = 1

1.4./ 1.4 = 1

x/ x = 1

xy/ xy =1

Week Six

EXAM 1

Week Four

Multiplication and Division

Division

8 ÷ 4 -> ÷ Division Sign

8/2 -> / Division/ Fraction Bar

2 ⟌8 -> ⟌ The rinculum

Quotient

V

4

2⟌8 <- Dividend

^

Divisor

Standard Algorithm Place Value Explicit

1 4 6 1 4 6

4 ⟌5 8 5 4 ⟌5 8 5

- 4 | | - 4 0 0

------V | --------

1 8 | 1 8 5

- 16 | - 1 6 0

---------V --------

2 5 2 5

- 2 4 - 2 4

--------- ---------

1 <-- Remainder 1 <-- Remainder

Alternative Algorithm

167 Pokemon Cards

12 in each booster pack

How many Packs? 13 Packs

12 ⟌ 1 6 7

- 1 2 0 --> 10 packs

-----------

47

- 36 --> 3 packs

-------------

11

1. Standard Algorithm 2. Place Value 3. Expanded Notation

2

^ 19 19

19 x 13 x 13

x 13 ---------- ---------

---------- 3 x 9 = 27 -->20 10 + 9

5 7 3 x 10 = 30 10 + 3

+ 1 9 0 10 x 9 = 90 ----------

---------- + 10 x 10 = 100 110 + 7

2 4 7 --------------------- -----------

2 4 7 100 + 90 + 0

200 + 40 + 7

------------------

2 4 7

Week Two

Expanded Notation

Expanded Notation:

285 = 2 hundreds + 8 tens + 5 ones.

= 200 + 80 + 5

= (2 x 100) + (8 x 10) + (5 x 1)

= (2 x 10^2) + (8x10^1) + (5 x 10^0)

Note: When moving from left to right, the exponent must decrease by -1.

(Numbers to the power of 0 always equal 1)

For expanded notation, we are going to use the number 1342 with a base of 5.

13425

13425 = (1 x 5^3) + (3 x 5^2) + (4 x 5^1) + (2 x 5^0)

= (1 x 125) + (3 x 25) + (4 x 5) + (2 x 1)

= 125 + 75 + 20 + 2

= 222

2123 = (2 x 3^2) + (1 x 3^1) + (2 x 3^0)

= (2 x 9) + (1 x 3) + (2 x 1)

= 18 + 3 + 2

= 23

Numeration Systems

Base-10 (Decimal)

Using the number, 357.35, lets break down what each value represents.

Decimal

10's Point 1/100

V V V

3 5 7 . 3 5

^ ^ ^

100's 1's 1/10

<–––––––– –––––––––>

x10 ÷10

Base-10:

Ones - 10^0

Tens - 10^1

Hundreds - 10^2

Thousands - 10^3

When in base form, no value can exceed the number of the base. For example,

In base-10 there are 10 numeric digits:

0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.

Therefore, any value in base-form cannot be larger than 9.

So in base-5, the digits are 0, 1, 2, 3, and 4.

And again but in base-3, 0, 1, and 2.