ISEE Middle Level

Entrance · ERBGeneral Reasoning (Verbal, Quantitative, Reading)노트 30개, 폴더 9개, 122 KB

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Notes for the ERB's ISEE Middle Level, following the published content areas: folders for Numbers and operations, Algebraic concepts, Geometry, Measurement, and Data analysis and probability, with definitions, methods and worked examples in US units and notation. Further folders give method notes for quantitative comparisons and word problems, Verbal Reasoning, Reading Comprehension and the essay, with examples written for these notes.

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Numbers and operations
- Integers and order of operations4 KB
- Factors, multiples and primes3 KB
- Fractions and decimals5 KB
- Percents4 KB
- Ratios, proportions and rates3 KB
- Exponents and roots4 KB
Algebraic concepts
- Patterns and functions4 KB
- Expressions3 KB
- Equations and inequalities4 KB
- Linear relationships4 KB
Geometry
- Angles and triangles4 KB
- Polygons and circles3 KB
- Congruence and similarity3 KB
- Coordinate geometry and transformations4 KB
Measurement
- Area and circumference3 KB
- Solids, surface area and volume4 KB
- Units and conversions4 KB
Data analysis and probability
- Displaying data5 KB
- Mean, median, mode and range3 KB
- Probability and counting5 KB
Comparing quantities and solving word problems
- Quantitative comparisons4 KB
- Solving word problems4 KB
Verbal reasoning
- Working out word meanings from context4 KB
- Roots, prefixes and suffixes4 KB
- Sentence completion5 KB
Reading comprehension
- Main idea, details and organization5 KB
- Inference and vocabulary in context4 KB
- Tone, style and figurative language4 KB
Writing an essay
- Planning and structuring a short essay5 KB
- Sentences, punctuation and revising5 KB

첫 번째 노트

Numbers and operations / Integers and order of operations

Integers and the number line

The integers are the whole numbers and their opposites: $\dots, -3, -2, -1, 0, 1, 2, 3, \dots$ . On a number line the numbers increase to the right, so any number is greater than every number to its left. That means $-2 > -7$ , because $-2$ sits further right, even though 7 is the larger digit.

The absolute value of a number is its distance from 0, written $\lvert n \rvert$ , so it is never negative. For example $\lvert -9 \rvert = 9$ and $\lvert 4 \rvert = 4$ . Two numbers with the same absolute value and different signs are opposites, and they add to 0.

To order integers, put the negatives first and ordered backwards by size: $-8 < -5 < -1 < 0 < 3$ . The negative number with the larger absolute value is the smaller number.

Adding and subtracting integers

When the signs match, add the absolute values and keep the sign: $-6 + (-4) = -10$ .

When the signs differ, subtract the smaller absolute value from the larger and keep the sign of the number with the larger absolute value: $-7 + 12 = 5$ and $3 + (-9) = -6$ .

Subtracting a number is the same as adding its opposite, which turns every subtraction into an addition:

\begin{aligned} -3 - (-8) &= -3 + 8 = 5 \\ 5 - 11 &= 5 + (-11) = -6 \end{aligned}

Multiplying and dividing integers

Two numbers with the same sign give a positive result, and two with different signs give a negative result. This holds for both multiplication and division.

(-4)(-6) = 24 \qquad (-5)(3) = -15 \qquad \frac{-36}{9} = -4 \qquad \frac{-28}{-7} = 4

A product of several factors is negative when an odd number of them are negative and positive when an even number are. So $(-2)(-3)(-5) = -30$ , but $(-2)(-3)(5) = 30$ . Zero times anything is 0, and division by 0 is undefined.

Properties of operations

These properties let you rearrange a calculation so that it is easier to do in your head.

Property	Statement	Example
Commutative	$a + b = b + a$ and $ab = ba$	$8 \times 25 = 25 \times 8$
Associative	$(a + b) + c = a + (b + c)$ , and the same for multiplication	$(17 + 38) + 62 = 17 + (38 + 62) = 117$
Distributive	$a(b + c) = ab + ac$	$7 \times 98 = 7(100 - 2) = 700 - 14 = 686$
Identity	$a + 0 = a$ and $a \times 1 = a$	$-15 + 0 = -15$
Inverse	$a + (-a) = 0$ and $a \times \frac{1}{a} = 1$ for $a \neq 0$	$9 \times \frac{1}{9} = 1$
Zero	$a \times 0 = 0$	$-412 \times 0 = 0$

Subtraction and division are not commutative or associative: $10 - 4 \neq 4 - 10$ , and $24 \div 6 \neq 6 \div 24$ . The distributive property works in reverse too, which is called factoring out a common factor: $6 \times 13 + 6 \times 7 = 6(13 + 7) = 120$ .

Order of operations

A calculation with several operations has one agreed value, because the operations are done in a fixed order:

Work inside parentheses (and other grouping symbols, such as brackets and the top or bottom of a fraction bar), innermost first.
Evaluate exponents and roots.
Multiply and divide, moving from left to right.
Add and subtract, moving from left to right.

Multiplication and division share a level, so neither comes first: $48 \div 6 \times 2 = 8 \times 2 = 16$ , not $48 \div 12$ . Addition and subtraction work the same way.

\begin{aligned} 18 - 2(4 + 1)^2 \div 5 &= 18 - 2(5)^2 \div 5 \\ &= 18 - 2(25) \div 5 \\ &= 18 - 50 \div 5 \\ &= 18 - 10 \\ &= 8 \end{aligned}

An exponent applies only to the number directly beside it, so the sign matters. $-3^2$ means the opposite of $3^2$ , which is $-9$ , while $(-3)^2 = (-3)(-3) = 9$ .

The absolute value bars act as grouping symbols: $12 - \lvert 5 - 9 \rvert = 12 - \lvert -4 \rvert = 12 - 4 = 8$ .

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