Math Basics 1|Page

Grade 4 Semester 1

  1. How to read a number in exponential form

an - a to the nth power, OR, a to the power of n

Special cases for n = 2 and 3 :

n = 2 read “a square”

n = 3 read “ a cube”

  1. Exponent Rules

n0 = 1 (except 0)

n1 = n

  1. Rounding rules

Look at the first digit to the right of the place to which you are rounding. If the digit is less than 5, round down. If the digit is 5 or more, round up.

  1. Inverse Operations: Operations that “Undo” each other.

Addition and Subtraction are inverse operations. Multiplication and Division are inverse operations.

  1. Related Sentences ( Or Fact Family)

Addition: a + b = s Related sentences: s – a = b, s – b = a

Subtraction: s – a = b Related sentences: s – b = a, s = a + b (or , a + b = s )

Multiplication: a × b = p Related sentences: p ÷ a = b, p ÷ b = a

Division: p ÷ a = b Related sentences: p ÷ b = a, p = a × b (or, a × b = p)

  1. Number properties

1)Commutative Property (only for addition and multiplication):

a + b = b + a a × b = b × a

Application: Can be used to check Addition/Multiplication calculation result

2)Associative Property of addition and multiplication

a + (b + c) = (a + b) + c a × (b × c) = (a × b) × c

3)Distributive Property of multiplication over addition and subtraction

a × (b + c) = a × b + a × c a x (b – c) = a x b – a x c

  1. Order of operations

Parentheses -> Exponent -> Multiplication -> Addition

Division Subtraction

Example: (2+2)3+ 5 x 8 – 4 ÷ 2 = (4)3 + 40 – 2 = 64 + 40 – 2 = 102

  1. Calculation Technique : 5 x 2 = 10 25 x 4 = 100 125 x 8 = 1000
  1. Calculation Technique : Multiply by 11 technique
  1. Prime number: A number that only has two factors, itself and 1.

Composite number: A number that has more than two factors

1 is neither a prime nor a composite number.

Prime factorization: A way to write a composite number as the product of prime factors.

  1. Divisibility Test

2- Even (last digit is 0, 2, 4, 6, or 8).

3- Sum of digits is divisible by 3.

4- Last two digits divisible by 4 (even0, even4, even8, odd2, odd6).

5- Last digit is 0 or 5.

6- Divisible by 2 and 3 (even, and sum of digits is divisible by 3).

7- Double the last digit and subtract it from the rest of the number. If the answer is 0 or divisible by 7

8- Last three digits divisible by 8.

9- Sum of digits is divisible by 9.

10- Last digit is 0.

11- The difference between the sum of the odd numbered digits (1st, 3rd, 5th...) and the sum of the even numbered digits (2nd, 4th...) is divisible by 11.

Terminologies:

Addition: Addend + Addend = Sum

Subtraction: Minuend – Subtraend = Difference

Multiplication: Factor x Factor = Product

Division: Divident ÷ Divisor = Quotient R Remainder

Grade 4 Semester 2

1. In a division problem, if there is a remainder, the remainder MUST be smaller than the

divisor. The divisor MUST be bigger than the remainder. Example,

A ÷ 28 = 14 …. B The largest value of B is 27.

A ÷ B = 14 … 29 The smallest value of B is 30 .

2. In a division problem, if there is a remainder, the (dividend – remainder) is divisible by the

divisor. Example,

A ÷ 28 = 14 … 24 Then ( A – 24 ) ÷ 28 = 0

(Check: A = 28 x 14 + 24 => (A – 24) ÷ 28 = 14 )

3. Multiply by 10, 100, 1000 ===> Move decimal points to the right. Add zeros to right if needed

Divide by 10, 100, 1000 ===> Move decimal points to the left. Add zeros to the left if needed

4. Word Problems

  1. 归一问题
  • Total Work = Unit Work x Days x Number Workers
  • Days = Total Work ÷ Unit Work ÷ Number Workers
  • Number Workers = Total Work ÷ Unit Work ÷ Days
  • Unit Work = Total Work ÷ Days ÷ Number Workers
  1. 植树问题
  • Open ending: Tree count = Segment Count + 1
  • Close ending: Tree count = Segment Count

c. 和差问题

  • Bigger Value = (Sum + Difference) ÷ 2
  • Smaller Value = (Sum – Difference) ÷ 2

d. 和倍问题

  • Smaller Value = Sum ÷ (Multiples + 1)
  • Bigger Value = Smaller Value x Multiples

e. 差倍问题

  • Smaller Value = Difference ÷(Multiples -1)
  • Bigger Value = Smaller Value x Multiples OR
  • Bigger Value = Smaller Value + Difference

f. 平均数问题

  • Average = Total ÷ Count

g. 还原问题

  • Use line segment, table to illustrate

Grade 5 Semester 1

1. Decimal Number Multiplication:

(A.xx x B.y) = ( A.xx x 100) x (B.y x 10) ÷ 100 ÷ 10 = (Axx x By) ÷ 1000

2. Decimal Number Division:

(A.xx÷ B.y) = ( A.xx x 10) ÷ (B.y x 10) = Ax.x ÷ By Or

(A.xx÷ B.y) = ( A.xx x 100) ÷ (B.y x 100) = Axx ÷ By0

3. Calculation Technique : N ÷ 0.5 = 2 x N N ÷ 0.25 = 4 x N N ÷ 0.125 = 8 x N

N x 0.5 = N ÷ 2 N x 0.25 = N ÷ 4 N x 0.125 = N ÷ 8

4. Order of Operation and Number properties are all also applicable to decimal numbers

5. Word Problem:

a. 鸡兔同笼

兔count = [Total feet count – (Total head count x feet count per chicken)] ÷ (feet count per bunny – feet count per chicken)

鸡count = [(Total head count x feet count per bunny) – Total feet count] ÷ (feet count per bunny – feet count per chicken)

b.盈亏问题

Assume number of ITEMs are putting to groups or distributed to groups.

Group count = Total ITEM count difference in two cases ÷ ITEM count difference per group in two cases

ITEM count = Case1 ITEM count per group x Group count +/- (extra or missing ITEM counts)

OR

ITEM count = Case2 ITEM count per group x Group count +/- (extra of missing ITEM counts)

c. 相遇问题

  • Distance = (Speed1 + Speed2) x Time

d. 追及问题 (Chasing)

  • Distance Difference = (Fast Speed – Slow Speed) x Time

e. 流水行船问题(Traveling on the water)

  • Water and boat heading to the same direction: Combined Speed = Boat Speed + Water Speed
  • Water and boat heading to the opposite direction: Combined Speed = Boat Speed – Water Speed
  • Round Trip Speed = (BoatSpeed + WaterSpeed)+(BoatSpeed – WaterSpeed) = 2 x BoatSpeed
  • SpeedDifference between two one-way trip

= (BoatSpeed+WaterSpeed) – (BoatSpeed – WaterSpeed) = 2 x WaterSpeed

6.Introducing Algebra

Terminologies:

Variable - Letters that represent mystery numbers (i.e., numbers that are not yet known)

Example: x + 3 = 5 a – 2 = 6 y÷ 3 = 4 5z = 10 x + y = 7

Expression – A phrase in mathematical sentences

Example: monomial expression, such as, x, 3, 6y;

binomial expression, such as, x + 3, a – 4, x + y;

trinomial expression, such as, x + y – 3, 2x – 3y + 7;

polynomial expression, such as, x + y + z – 4, 2a + 3b – 4c + 2, etc. (two or more terms)

Mathematical sentence - Contains two mathematical phrases joined by an equal sign or an inequality sign.

Equation– A mathematical sentence in which the two phrases are joined by “=” sign

Example: 3 + 6 = 9, + 1 = 2, 7 = 14 There are TRUE equation and FALSE equation

Inequality– A mathematical sentence in which the two phrases are joined by “>” , “<”, “≥”, “≤”

Example: 6 > 5, 7 < 10

Coefficient – The number in front of the variable

Example: In expression 2, coefficient is ‘2’. In 3 , coefficient is ‘3’.

Operation symbols: +, −, •(4• or 4),÷, / (or )

Like Terms –Terms that consist only of numbers are like terms. Example, 5, 3, 0.4, , etc.

Terms that use the same variable to the same degree are like terms. Example, 3, , , etc.

Unlike Terms – a) A number and a variable are unlike terms. Example, 5, .

b) Terms that use different variables are unlike terms. Example, , , .

c) Terms that are to the different power of a variable are unlike terms. Example, , etc.

Addition/Subtraction Rule – To add/subtract like terms, add/subtract the coefficients of the terms. Example:

. You can NOT add/subtract unlike terms.

Multiplication Rule – a) Multiply a number and a variable. [Step1: Multipy the coefficients; Step2: Attach the variable at the end]

b) Multiply two like termsor unlike terms. [Step1: Multiply the coefficients; Step2: Multiply the variables; Step3: Multiply the answers from the first two steps]

Division Rule – Divide two like termsor unlike terms.[Step1: Divide the coefficients; Step2: Divide the variables; Step3: Multiply the answers from the first two steps]