Download Integer = מִסְפָּר שָׁלֵם כל המספרים השלמים, חיוביים ושליליים

Integer =‫ִמ ְספָּ ר ָּשלֵ ם‬
‫ כולל הסיפרה אפס‬,‫ חיוביים ושליליים‬,‫כל המספרים השלמים‬
Even integers = )2 ‫מספרים זוגיים (כפולות של‬
Odd integers = ‫זוגיים‬-‫מספרים אי‬
Consecutive integers = ‫מספרים עוקבים‬: 57, 58, 59…..
Multiples of 7 = 7, 14, 21, 28 …
Factors of 18 = 1,2,3,6,9,18
Divisors of 18: 1,2,3,6,9,18
Vocabulary:
Sum: result of addition
Difference: result of subtraction
Product: result of multiplication
Quotient: result of division
)....,‫איך יודעים שמספר מתחלק ב? (חמש – ספרה אחרונה חמש או אפס‬
.‫יעזור ללמוד את זה – יש הרבה שאלות שמצריכות את הידע הבסיסי הזה‬
Prime number =‫ִמ ְספָּ ר ִראשֹונִ י‬
.)1( ‫ ובמספר‬,‫מספרים שלמים שניתנים לחלוקה במספר עצמו‬
2 (the only even prime),,,,,1,11,1,,11,13,2,,23,
1 is not a prime!
Average/Mean = ‫ממֻ צָּ ע‬,
ְ Mode = ‫ שכיח‬Median = ‫חציון‬
Average = the average of a list of n numbers is equal to the sum of the numbers divided by n.
Example: the mean of 2, 3, 5,7,13 =
=6
If the average is given, you can find the sum:
If the average of X+y=20
= 20  x+y = 40
If the average of 6 numbers is 12, the sum of these 6 numbers is 12*6 = 72.
Rules for performing basic arithmetic operations:
‫הכפלה באפס תמיד תיתן אפס‬
‫מוגדרת‬-‫חילוק באפס יוצר תוצאה לא‬
‫) תמיד תיתן תוצאה שלילית‬+ ‫ או‬-( ‫הכפלה או חלוקה של מספרים בעלי סימן שונה‬
‫) תמיד תיתן תוצאה חיובית‬-( ‫הכפלה או חלוקה של מספרים שליליים‬
Divisible = ‫שניתן לחלק אותו במספר מסוים ללא שארית‬
Fractions = ‫שבר פשוט‬
Numerator = )‫מונה (המספר העליון בשבר‬
Denominator = )‫מונה (המספר התחתון בשבר‬
To add or subtract 2 fractions with the same denominator,
You add the numerators and keep the denominator the same:
Mixed fraction = ‫שבר מעורב‬
=
+ =
If the denominators are not the same, need to find a common denominator:
To multiply fractions  multiply the numerators and the denominators
(They don’t have to be the same):
*- =
*3=
=
To divide fractionsinvert the 2nd fraction, and then proceed as in multiplication.
Decimals = all numbers can be express in decimal form using base 10
To determine the position of the decimal point in the product, you add the number of digits to the right
of the decimal points in the decimals being multiplied.
Example:
15.381 (3 decimal places) * 0.14 (2 decimal places) = 61524+15381 = 2.15332 (5 decimal places).
Convert a decimal to an equivalent fraction:
use the power of 10. Example:
84.1 =
9.17 =
0.612 =
Percent (= per hundred)
43% =
= 0.43
300% =
=3
0.5% =
= 0.005‫ש‬
Example: what 30% of 350? 
350*0.3 = 105. Or: (350)*(
) = (350)*(
)=
= 105
What percent 0f 80 is 5? 
=
x=
= 6.25 5 is 6.25% of 80.
Example Question:
Mixture of 12 ounces of vinegar and oil is 40% vinegar (by weight).
How many ounces of oil must be added to the mixture to produce a new mixture that is only 25%
vinegar?
X the number of ounces of oil to be added
4.8 = 3+0.25x x = 7.2
= 0.25 (0.40) (12) = (12+x) (0.25)
List of fractions and their equivalent per cents
= 50%
= 66 2/3%
= 75%
= 16 2/3%
= 12 ½%
= 62 ½%
= 33 1/3%
= 25%
= 20%
= 83 1/3%
= 37 ½%
= 87 ½%
:‫נוסחה לחישוב השינוי באחוזים‬
.‫ לחלק בגודל המקורי‬,‫השינוי במספרים‬
Example: if the quantity increases from 600 to 750 (
)=
Ratio:
The ratio of the number 9 to the number 21 can be expressed:
9 to 21= 3 to 7, or 3:7, or
=
= 22%
Exponent = ‫חזקה‬
Rules of exponents:
X-a =
 Example: 4-3 =
=
 Example:
 Example:
 Example:
 Example:
 Example:
=
=
=
 Example:
If x≠0, then
= 1  Example:
=
Perfect squares: 1,4,9,16,25,36,49,64,81,100,121,144,169,…..
Perfect cubes: 1, 8, 27, 64,125…..
Perfect fourth powers: 1, 16, 81,256……
Exponents:
List of common exponents
(My suggestion: make a list of common exponents, and memorize them)
Example:
34 = 81
25 = 32
106 = 1,000,000
(-4)3 = -64
( )4 =
If x = 0, then these expressions are not defined.
Square root = ‫שורש ריבועי‬
All positive No. have 2 square roots that differ only in sign
Negative No. does not have square roots because the square of a real number can’t be negative.
Rules: (a>0, b>0)
(
)(
=
)=
(
.
=
= √(16)(3) = (√16)( √3) = 4√3
Examples:
5
+
=5
+
=5
+
=5
+
=8
56
‫) דברים שונים אחד מהשני‬+( ‫אי אפשר לחבר‬

)(


)=
):(
(
)(
)-(
)=
)= (
=
=
)-(
)=
=5
-
=
-
= -2
Ordering and real Number line
All integers and all numbers with values between them have a natural ordering. Examples:
-
Absolute value = ‫ערך מוחלט‬
N –> if N is positive or zero
-N –> if N is negative. Example:
| |=
| |=
|
|=
Algebraic expressions
Coefficient – ‫מ ַקדֵּ ם‬,
ְ constant = )‫קבוע (מספר‬
In 2x²+7x-5  2 is the coefficient of the x² term, 7 is the coefficient of the x term, -5 is the constant
term.
Terms with the same variable part can be combined. Examples:
2x+5x = 7x
X²-3x+6x²= (1-3+6) x² = 4 x²
Factored out –‫ הוצאת גורם משותף‬.
Type of factoring:
X²+2x = x(x+2)
X²-1 = (x+1) (x-1)
X²+2x+1 = (x+1) (x-1) = (x+1)²
2x²+5x-3= (2x-1) (x+3)
Examples:
4x+12 = 4(x+3)
15y²-9y = 3y (5y-3)
Rule:
A²-b² = (a+b)(a-b)
To multiply 2 expressions:
Each term of the 1st expression is multiplied by each term of the 2nd, and the results added. Example:
(x+2)(3x-7) = x (3x) +x (-7) +2(3x) +2(-7) = 3x²-7x+6x-14 = 3x²-x-14
Probability = ‫הסתברות‬
The chance that a specific outcome can occur =
Example: a jar contains 13 red marbles and 7 green marbles, the probability that marble
selected randomly will be green is:
= or 0.35
If the outcome can never occur – the probability is 0, if it’s certain to occur, its probability is 1.
Linear equations = ‫משוואות עם נעלמים‬
1 variable: - find the value of the variable that makes the equation true.
Rules for solving linear equation:
When you add or subtracted the same constant to both sides of the equation – the equality is
preserved. The new equation is equivalent to the original.
When both sides of an equation are multiplied or divided by the same nonzero constant – the
equality is preserved. The new equation is equivalent to the original.
Example: 3x-4 = 8
3x-4+4 = 8+4 (4 added to both sides)
3x=12
3x/3=12/3 (both sides divided by 3)
X=4
2 variables:
X+y = 16
5x-y = 2
Better to solve by add/subtract:
6x+ y-y = 18 6x = 18 x = 18/6 = 3
Back to the 1st one x+y = 16, so: y = 13
Second degree – 1 variable: (always 2 solutions)
:‫משוואה ריבועית‬
X²+6x-9 = x+15
-x, -15 :‫ מבצעים את אותן פעולות בשני הצדדים‬.‫יש להשוות את אחד מצדי המשוואה לאפס‬
X²+5x-24 = 0
Factoring ask “what 2 numbers that if you multiply them you get 5, and add them get -24”?
8, -3.  (x+8)(X-3) = 0
X+8=0 or: x-3=0.  x=-8, x=3.
Another way to solve: ax²+bx+c=0
Formula
=
=
= -8/3.
Inequalities = ‫משוואות עם נעלמים‬
-m, -10 -21 ≥ 6m /:6 
≥

≥
Or:
-7m, +11-6m≥21 divide by -6 m≤When divide or multiply by a negative – you have to switch the sign (≥ to ≤, etc.)
Quantitative comparison
How to solve?
.‫ תשובות‬2 ‫ ופוסלים בעזרתו‬-)‫מציבים דוגמה אחת (שעונה לתנאי שניתן בשאלה‬
Geometry
1 line going through 2 points.
Segment = AB ‫ למשל‬.‫מקטע‬
If 2 lines intersect:
c
d
b
a
the angels that vertical are equal
A=b, c=d, a+c=180°/ a full circle = 360°
Two perpendicular lines that intersect: right angle = 90°
Parallel: l3 ∏l4 a=a, b=b.
b
4 of the angels are equal to each other
a
b
a
Coordinate Geometry
a
b
a
b
X-axis – the horizontal number line
Y-axis – the vertical number line
To find a distance between 2 points: construct a right triangle, and then apply the Pythagoras Theorem:
Example: PQ =
= √56.25 = 7.5
Linear equation - y=mx+b
m the slope of the line
To find the slope of the line between 2 points:
. Example: the slope of the line passing through points (4, 1.5) and (-2,-3) is:
=
b the Y intercepts
To find the y intercept - b:
After you found the slope – choose a point on the line and substitute the x and y in the linear equation.
Polygons:
Triangle:
The basic polygon. The most important geometry fact: a+b+c = 180° (angles).
.‫הזוית הגדולה נמצאת מול הצלע הגדולה‬
Rule: Triangle can only be triangle if the sum of 2 sides is bigger than the 3rd side.
Example: which of the following could be the lengths of a triangle?
1) 14, 17,2
2) 11,15,27
3) 13,13,0.028
Only 1 and 3.
Isosceles –‫ ְמשֻׁ לָּש ְשוֵּה שוקיים‬. Two sides are equal, 2 angles are equal.
Equilateral = ‫ ְמשֻׁ לָּש ְשוֵּה צְ לָּעֹות‬. All the angles = 60°, all the sides are equal.
Right triangle = angle of 90°. It must be the biggest angle.
Hypotenuse = ‫י ֶֶתר‬. ‫הצלע שמול הזוית הגדולה במשולש ישר זוית‬
Rule for any right triangle: a²+b²=c²
:‫גדלים שחוזרים על עצמם במשולשים‬
3:4:5 and any multiple of them (9:16:25…)
5:12:13
8:15:17
7:24:25
* If you take a square and divide it to 2 - you get this special triangle:
1:1:√2 1:1:1.4≈
If you see a 45° angle it’s probably that.
* If you take an altitude in Equilateral triangle – you get this special triangle:
1: √3:2  1: 1.7≈:2
Angles- 30:60:90
30
2
2
2
2
2
60
2
1
2
2
1
2
2
Area of triangle: a = ½ *b*h.
(b=base, h = height).
The sum of the measures of the interior angles of an n-sided polygon is:
(n-2)*(180°) example: the sum for a hexagon (n=6) is 4*180° = 720°
Regular polygon = a polygon with all sides the same length and the measures of all interior angles equal.
Quadrilaterals = ‫ְמרֻ בָּ עים‬
Every quadrilateral has four sides and 4 interior angles whose measures sum to 360°.
Area of a quadrilaterals = base*height
Special quadrilaterals:
Rectangle = a quadrilateral with all interior angles of 90°. Opposite sides are parallel and have
equal length, two diagonals have equal length.
Square = a rectangle with all sides of equal length.
Parallelogram = a quadrilateral with both pairs of opposite sides parallel. Opposite sides have
equal length; opposite interior angles have equal measure.
Trapezoid = a quadrilateral with one pair of opposite sides parallel.
Area of a Trapezoid = ½ (b1+b2)*(h)
Circles
R = the radius
Circumference = 2
Area =
Arc = set of all points between 2 given points, can be measured in degrees. To find the length of
an arc it is important to know the ratio of arc length to circumference is equal to the ratio of arc
measure to 360°. Example – in a circle that circumference = 10
the arc interior angle = 50°,
so:
=
 ABC = (
)*(10
3 Dimensional Figures
Rectangular solid = has 6 rectangular surfaces called faces. The dimensions of rectangular
solid are length (l), width (w) and height (h).
Cube = a rectangular solid with l=w=h. Volume = l*w*h.
Surface area of a rectangular solid is the sum of the areas of the 6 facesA = 2(wl+lh+wh)
Cylinder = volume =
Surface area = sum of the 2 base areas and the area of the curved surface 2(