THINGS TO REMEMBER

ALGEBRA 1

THINGS TO REMEMBER

ALGEBRA 1

PREMIER CURRICULUM SERIESBased on the Sunshine State Standards for Secondary Education,

established by the State of Florida, Department of Education

Copyright 2009Revision Date:12/2009

Copyright 2009Revision Date:12/2009

Pre Algebra Things to Remember

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Lesson 1 Remember the simple definition of absolute value is the distance between any 2 numbers on a number line. -2 -1 0 1 2 What would be the absolute value of -2 to 2? Find -2 and count the number of units between it and 2. That should be 4. Evaluate the segment length 3- ׀ to 2 ׀ = ________________________________________________________________________ -5 -4 -3 -2 -1 0 1 2 3 4 5 .Find -3 and count the number of units between it and 2. That should be 5, which is the distance between -3 and 2 on the number line. ______________________________________________________________________ Review multiplication tables. You should know up to 12 x 12. ______________________________________________________________________ Square roots are the reverse of squares. √ is a square root symbol also called a radical sign.√4 reads square root of 4. Finding the square root means finding the number that was multiplied by itself to get the square. The square root of 36 equals 6. When the square is a small number we can do it mentally. If the square is a big number we can use a prime factor tree to find the root. Example 1: Use prime factors to find the square root of 64. 64 = 16 x 4 (none of these is a prime number) 2 x 8 2 x 2 2 x 4 2 x 2 Write only the prime factors as a product = 2 x 2 x 2 x 2 x 2 x 2 Select one 2 for each pair of twos; instead of writing 2 six times we write it 3 times. 2 x 2 x 2 = 8 √64 = 8 Example 2: Use prime factors to find the square root of 100. 100 = 5 x 20 (Find the prime factors of all the composite numbers) 20 = 4 x 5 4 = 2 x 2 Write only the prime factors as a product = 2 x 2 x 5 x 5 (write only one 2 and one 5) √100 = 2 x 5 = 10

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Recall that the √100 = 10 x 10; √64 = 2 x 2 x 2. We could have written that in a shorter way by using powers and exponents. Example: √100 = 102 or 10 to the second power. We also say 10 squared. 10 is the base, 2 is the exponent. √64 = 23 or 2 to the third power, also 2 cubed. 2 is the base, 3 is the exponent. ______________________________________________________________________ The base is the number we multiply. The exponent is how many times we multiply the base. 23 is 2 raised to the power of 3; 103 is 10 raised to the 3rd power. We can expand numbers raised to a power by multiplying the base by itself the number of times indicated by the power. Examples: 42 = 4 x 4 63 = 6 x 6 x 6 124 = 12 x 12 x 12 x 12 One more example: Write 200,000 in scientific notation. 100,000 is 10 x 10 x 10 x 10 x 10 = 105. Then 2x105 ______________________________________________________________________ Numbers can have interesting patterns. An Arithmetic Sequence is made by adding some value each time.

1,4,7,10,13, 16, … The above sequence has a difference of 3 between each number. The pattern continues by adding 3 to the last number each time.

3, 8, 13, 18, 23, 28 …. The above sequence has a difference of 5 between each number. The pattern continues by adding 5 to the last number each time.

A Geometric Sequence is made by multiplying by some value each time.

2, 4, 8,1 6, 32, ….. The above sequence has a factor of 2 between each number. The pattern continues by multiplying the last number by 2 each time. 3, 9, 27, 81, 243, ….. The above sequence has a factor of 3 between each number. The pattern continues by multiplying the last number by 3 each time

Triangular Numbers

1, 3, 6, 10, 15, 21, 28, ….. Do you see the pattern? Add 2 then add 3 that add 4 then add 5 etc. Square Numbers 1, 4, 9, 16, 25, 36, …

The next number is made by squaring where it is in the pattern (its position in the sequence.) The second number is 2 squared, then 3 squared etc.

Cube Numbers 1, 8, 27, 64, 125, 216, …

Pre Algebra Things to Remember

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The next number is made by cubing where it is in the pattern. The second number is 2 cubed then 3 cubed then 4 cubed etc.

Fibonacci Numbers 0, 1, 1, 2, 3, 5, 8, 13, 21 …

The next number is found by adding the two numbers before it together. The 2 is found by adding the two numbers in front of it (1+1) The 21 is found by adding the two numbers in front of it (8+13

Can you figure out the next number in the following sequence? 50, 49,7,44, 24, 10, 18, 30, 36, 13, 28….. Probably not! They came from a random number generator. ________________________________________________________________________Scientific notation is used to abbreviate or shorten large numbers. Numbers expressed this way are always written as the product of a factor and a power of 10. The diameter or distance through the center of the earth is just about 8,000 miles. In scientific notation that would be written as 8 x 103 (because 10x10x10 is 1,000). ________________________________________________________________________ Recall that √100 = 10 x 10; √64 = 2 x 2 x 2. It could have been written in a shorter way by using powers and exponents. Example: √100 = 102 or 10 to the second power. We also say 10 squared. 10 is the base, 2 is the exponent. √64 = 23 or 2 to the third power, also 2 cubed. 2 is the base, 3 is the exponent. 23 is 2 raised to the power of 3; 103 is 10 raised to the 3rd power. We can expand numbers raised to a power by multiplying the base by itself the number of times indicated by the power. Example: 42 = 4 x 4 63 = 6 x 6 x 6 124 = 12 x 12 x 12 x 12 ________________________________________________________________________ Go back to the beginning of this part of the review and use the number line that goes from -5 to 5. ________________________________________________________________________ Review arithmetic sequence above. ________________________________________________________________________

Lesson 2 A number is divisible by 5 if the its last digit is a 0 or 5 ________________________________________________________________________ Order of operations: In algebra there is a specific order to follow. Many students already know the mnemonic PEMDAS (Please Excuse My Dear Aunt Sally). This is a tool to help us remember this important order: Parentheses (work these first), Exponents, Multiplication, Division, Addition, and Subtraction. If there is only multiplication in a problem, work from left to right. If there is only addition and subtraction in a problem, work from left to right. Example: If a = 4, b = 6, and c = 2, find the value of: ab + bc 4 x 6 + 6 x 2 = 24 + 12 = 36 Hint: apply the order of operations to this problem; multiplication first.

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_______________________________________________________________________ Example: If a = 4, b = 6, and c = 2, find the value of b ÷ c. 4 ÷ 2 = 2 ________________________________________________________________________ The base is the number we multiply. The exponent is how many times we multiply the base. 23 is 2 raised to the power of 3; 103 is 10 raised to the 3rd power. We can expand numbers raised to a power by multiplying the base by itself the number of times indicated by the power. Examples: 42 = 4 x 4 63 = 6 x 6 x 6 124 = 12 x 12 x 12 x 12 ________________________________________________________________________ Rules for adding integers:

• A negative and a positive of the same value equal zero. • Add numbers with the same sign, whether they are negative or positive. • Subtract numbers with different signs. • The answer takes the sign of the larger number.

Example: A friend borrowed $5 one day, then need to borrow $3 more dollars. How much money was owed? -5 + -3 = -8 ─ (negatives) Since they are all negatives we simply add them together. ________________________________________________________________________ Example: What would 20 gallons of gas cost at $4.50 per gallon? Let g stand for gallon. g = $4.50 20g x $4.50 = $90 ________________________________________________________________________ What would be the total cost of 12 ink cartridges at $20 each and 4 reams of paper at $6.00 each? Taxes are 7 cents on the dollar (7%). 12 x $20 = $240 4 x $6 = $24 $240 + $24 = $264 $264 x .07 = $18.48 tax $264 + $18.48 = $282.48 ________________________________________________________________________ If 10 baseballs cost $89, what is the cost of 1? $89 ÷ 10 = $8.90 for one baseball. ________________________________________________________________________ The sum of 3 consecutive odd numbers 39. What are the number? First, what is the relationship between consecutive odd numbers? They are 2 numbers apart. Let n = the first odd number; then n + 2 = the next odd number and n + 4 = the next odd number after that. Then we have the following: n + n+2+n+4 = 39 3n + 6 = 39 3n = 33 n = 11 The three consecutive odd numbers are 11, 13, and 15. ________________________________________________________________________ A 30 pound wild boar takes 12 hours to cook. How much time will a 20 pound boar take? Use proportions to solve such a problem. 30 = 20 Use cross multiplication: 30t = 240 t = 240 ÷ 30 = 8 12 t It will take 8 hours to cook the 20 pound boar! ________________________________________________________________________

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Lesson 3 Bar graph: This type of graph is used to compare data. Histogram: A histogram is made from a stem and leaf plot but it looks a lot like a bar graph when complete. A rectangle is drawn above each value on the number line. However, they are not the same thing. All graphs use pictures or symbols. A bar graph demonstrates a relationship between variables. This type of graph is used to compare data. The median is the middle value in the data and is important. Renee Descartes was the famous mathematician who popularized math in his art. If the range of data is 5 5o 76, then 0 to 80 would be a good scale. When there are two values of y for each value of x then a graph is not a function. ________________________________________________________________________ What quadrant would the point P (3,5) be in? Quadrant I What quadrant would the point (-3,5) be in? Quadrant II

________________________________________________________________________ The mean, median, and the mode are called the measures of central tendency. The coordinate plane is a combination of two number lines that cross each other at the center or origin. The value of the origin is zero. The side to side, east-west, or horizontal number line is the X axis. The up and down, north-south, or vertical number line is the Y axis

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Lesson 4 Use the following geometric shapes to answer related questions. 3 cm 3 cm 4 cm 2 cm 3cm 5 cm 4 cm 2 cm Square Rhombus Parallelogram Rectangle Trapezoid Hexagon Cube Triangle Pentagon Oval Circle Rectangular Prism Cylinder An angle greater than 90 degrees is Obtuse. An angle less than 90 degrees is Acute.

Lesson 5 The interior angles of a triangle total 180degrees. The Pythagorean Theorem works only on right triangles. It says that the sum of squares of the two sides is equal to the square of the third side (the longest side) called the hypotenuse. That is a2 + b2 = c2

For a right triangle with sides of 6 and 8, find the length of side C. A2 + B2= C2 Substitute: 62 + 82 = C2 36 + 64 = 100 C2 = 100 C = √100 = 10 (C = 10) ________________________________________________________________________ The Area of a triangle: ½ bh or ½ lw. Find the area of a triangle with a base of 20 feet and a height of 40 feet. A = base X height divided by 2. 20 x 40 = 800 square feet divided by two = 400 square feet.

Pre Algebra Things to Remember

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___________________________________________________________________ Figures are similar when they have the same shape but not the same size. Dilated figures are similar because they are either smaller or larger than the original. The symbol for similarity is ≈ ______________________________________________________________________ Figures are congruent when they have the same measure in all aspects. We will work often with polygons that have the same measure of sides and angles. The symbol for congruency is ≡ Figures are similar when they have the same shape, but not the same size. Figures are congruent when they have the same measure in all aspects. A triangle where two sides are equal is called an isosceles triangle. For a right triangle, the sine of either of the acute angles is defined as opposite side divided by hypotenuse.

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Author: Bernice Stephens-AlleyneCopyright 2009

Revision Date:12/2009

Author: Bernice Stephens-AlleyneCopyright 2009

Revision Date:12/2009