Welcome to Algebra I The following slides will highlight some of the skills you are expected to possess upon beginning this class

Multiplication Facts Fill out the multiplication table by finding the product of each number across the top row with each number down the left column.

So how do you memorize all of these? You Don’t. Just memorize some benchmark answers

So how do you memorize all of these? You Don’t. Just memorize some benchmark answers

So how do you memorize all of these? You Don’t. Just memorize some benchmark answers

So how do you memorize all of these? You Don’t. Just memorize some benchmark answers

So how do you memorize all of these? You Don’t. Just memorize some benchmark answers

What is 8 x 6? Since 8 x 5 = 40, 8 x 6 = 40 + 8 = 48

What is 6 x 7? Since 7 x 7 = 49 6 x 7 = 49 – 7 = 42

Factors and Multiples Let a, b and c represent any counting numbers

What are all of the factors of 32?

What are all of the factors of 12?

What are all of the factors of 12?

And these are some of the multiples of 8

Exponents and Order of Operations Simplify 32 + 62 – 14 • 3. 32 + 62 – 14 • 3 = 32 + 36 – 14 • 3 Simplify the power: 62 = 6 • 6 = 36. = 32 + 36 – 42 Multiply 14 and 3. = 68 – 42 Add and subtract in order from left to right. = 26 Subtract.

Exponents and Order of Operations Evaluate 5x = 32 ÷ p for x = 2 and p = 3. = 5 • 2 + 32 ÷ 3 Substitute 2 for x and 3 for p. = 5 • 2 + 9 ÷ 3 Simplify the power. = 10 + 3 Multiply and divide from left to right. = 13 Add.

Exponents and Order of Operations Find the total cost of a pair of jeans that cost $32 and have an 8% sales tax. total cost      original price      sales tax C = p + r • p sales tax rate C = p + r • p = 32 + 0.08 • 32 Substitute 32 for p. Change 8% to 0.08 and substitute 0.08 for r. = 32 + 2.56 Multiply first. = 34.56 Then add. The total cost of the jeans is $34.56.

Exponents and Order of Operations Simplify 3(8 + 6) ÷ (42 – 10). = 3(8 + 6) ÷ (16 – 10) Simplify the power. = 3(14) ÷ 6 Simplify within parentheses. = 42 ÷ 6 Multiply and divide from left to right. = 7 Divide.

Exponents and Order of Operations Evaluate each expression for x = 11 and z = 16. a. (xz)2 b. xz2 (xz)2 = (11 • 16)2 Substitute 11 for x and 16 for z. xz2 = 11 • 162 = (176)2 Simplify within parentheses. Multiply. = 11 • 256 = 30,976 Simplify. = 2816

Exponents and Order of Operations Simplify 4[(2 • 9) + (15 ÷ 3)2]. = 4[18 + (5)2] First simplify (2 • 9) and (15 ÷ 3). = 4[18 + 25] Simplify the power. = 4[43] Add within brackets. = 172 Multiply.

Exponents and Order of Operations A carpenter wants to build three decks in the shape of regular hexagons. The perimeter p of each deck will be 60 ft. The perpendicular distance a from the center of each deck to one of the sides will be 8.7 ft. Use the formula A = 3 ( ) to find the total area of all three decks. pa 2 A = 3 ( ) pa 2 = 3 ( ) 60 • 8.7 Substitute 60 for p and 8.7 for a. = 3 ( ) 522 2 Simplify the numerator. = 3(261) Simplify the fraction. = 783 Multiply. The total area of all three decks is 783 ft2.

Exploring Real Numbers Name the set(s) of numbers to which each number belongs. a. –13 b. 3.28 integers rational numbers rational numbers

Exploring Real Numbers Which set of numbers is most reasonable for displaying outdoor temperatures? integers

Exploring Real Numbers Determine whether the statement is true or false. If it is false, give a counterexample. All negative numbers are integers. A negative number can be a fraction, such as – . This is not an integer. 2 3 The statement is false.

Exploring Real Numbers Write – , – , and – , in order from least to greatest. 3 4 7 12 5 8 – = –0.75 Write each fraction as a decimal. – = –0.583 – = –0.625 3 4 7 12 5 8 –0.75 < –0.625 < –0.583 Order the decimals from least to greatest. From least to greatest, the fractions are – , – , and – . 3 4 7 12 5 8

Exploring Real Numbers Find each absolute value. a. |–2.5| b. |7| –2.5 is 2.5 units from 0 on a number line. 7 is 7 units from 0 on a number line. |–2.5| = 2.5 |7| = 7

Adding Real Numbers Simplify each expression. a. 12 + (–23) = The difference of the absolute values is 11. Since both addends are negative, add their absolute values. The negative addend has the greater absolute value, so the sum is negative. The sum is negative. a. 12 + (–23) = –11 b. –6.4 + (–8.6) = –15

Adding Real Numbers Simplify each expression.

Adding Real Numbers The water level in a lake rose 6 inches and then fell 11 inches. Write an addition statement to find the total change in water level. 6 + (–11) = –5   The water level fell 5 inches.

Adding Real Numbers Evaluate 3.6 + (–t) for t = –1.7. 3.6 + (–t) = 3.6 + [–(–1.7)] Substitute –1.7 for t. = 3.6 + [1.7] –(–1.7) means the opposite of –1.7, which is 1.7. = 5.3 Simplify.

Adding Real Numbers A scuba diver who is 88 ft below sea level begins to ascend to the surface. a.  Write an expression to represent the diver’s depth below sea level after rising any number of feet. Relate: 88 ft below sea level plus  feet diver rises   Define: Let r = the number of feet the diver rises. Write: –88 + r –88 + r b.  Find the new depth of the scuba diver after rising 37 ft. = –88 + 37 Substitute 37 for r. = – 51 Simplify. The scuba diver is 51 ft below sea level.

Subtracting Real Numbers Simplify –11.6 – (–14). = –11.6 + 14 The opposite of –14 is 14. = 2.4 Add.

Subtracting Real Numbers Simplify |–13 – (–21)|. = | –13 + 21|   The opposite of –21 is 21. = | 8 | Add within absolute value symbols. = 8 Find the absolute value.

Subtracting Real Numbers Evaluate x – (–y) for x = –3 and y = –6. = –3 – [–(–6)] Substitute –3 for x and –6 for y. = –3 – 6 The opposite of –6 is 6. = –9 Subtract.

Subtracting Real Numbers The temperature in Montreal, Canada, at 6:00 P.M. was –8°C. Find the temperature at 10:00 P.M. if it fell 7°C. Find the temperature at 10:00 P.M. by subtracting 7°C from the temperature at 6:00 P.M. = –8 – 7 Is like combining – 8 and – 7 . = –15 Simplify. The temperature at 10:00 P.M. was –15°C.

Subtracting Real Numbers Simplify each expression.

Adding Matrices 7 –5.4 –2 11.1 3 –1 –6 8.6 11 2.3 5 –3 Add + 7 –5.4 –2 7 –5.4 –2 11.1 3 –1 –6 8.6 11 2.3 5 –3 Add + 7 –5.4 –2 11.1 3 –1 –6 8.6 11 2.3 5 –3 + –6 + 7 8.6 + (–5.4) 11 + (–2) 2.3 + 11.1 5 + 3 –3 + (–1) Add corresponding elements. = 1 3.2 9 13.4 8 –4 Simplify. =

Subtracting Matrices 7 –5.4 –2 11.1 3 –1 –6 8.6 11 2.3 5 –3 Subtract – 7 –5.4 –2 11.1 3 –1 –6 8.6 11 2.3 5 –3 Subtract – 7 –5.4 –2 11.1 3 –1 –6 8.6 11 2.3 5 –3 – –6 – 7 8.6 – (–5.4) 11 – (–2) 2.3 – 11.1 5 – 3 –3 – (–1) Subtract corresponding elements. = – 13 14 13 – 8.8 2 –2 Simplify. =

Multiplying and Dividing Real Numbers Simplify each expression. b. –6( ) 3 4 a. –3(–11) The product of a positive number and a negative number is negative. –6( ) = – 3 4 18 –3(–11) = 33 The product of two negative numbers is positive. = –4 1 2 Write – as a mixed number. 18 4

Multiplying and Dividing Real Numbers Evaluate 5rs for r = –18 and s = –5. 5rs = 5(–18)(–5) Substitute –18 for r and –5 for s. = –90(–5) 5(–18) results in a negative number, –90. = 450 –90(–5) results in a positive number, 450.

Multiplying and Dividing Real Numbers Use the expression –5.5( ) to calculate the change in temperature for an increase in altitude a of 7200 ft. a 1000 –5.5( ) = –5.5 ( ) 7200 1000 a Substitute 7200 for a. = –5.5(7.2) Divide within parentheses. = –39.6°F Multiply. The change in temperature is –39.6°F.

Multiplying and Dividing Real Numbers Use the order of operations to simplify each expression. a. –0.24 Write as repeated multiplication. –(0.2 • 0.2 • 0.2 • 0.2) = = –0.0016 Simplify. b. (–0.2)4 Write as repeated multiplication. (–0.2)(–0.2)(–0.2)(–0.2) = = 0.0016 Simplify.

Multiplying and Dividing Real Numbers Simplify each expression. The quotient of a positive number and a negative number is negative. = –14 a. 70 ÷ (–5) b. –54 ÷ (–9) The quotient of a negative number and a negative number is positive. = 6

Multiplying and Dividing Real Numbers x y Evaluate – – 4z2 for x = 4, y = –2, and z = –4. – – 4z2 = – 4(–4)2 Substitute 4 for x, –2 for y, and –4 for z. x y –4 –2 = – 4(16) Simplify the power. –4 –2 = 2 – 64 Divide and multiply. = –62 Subtract.

Multiplying and Dividing Real Numbers 3 2 3 4 Evaluate for p = and r = – . = p ÷ r Rewrite the equation. p r = ÷ Substitute for p and – for r. 3 2 4 (– ) = Multiply by – , the reciprocal of – . 3 2 4 (– ) = –2 Simplify.

The Distributive Property Simplify 3(4m – 7). 3(4m – 7) = 3(4m) – 3(7) Use the Distributive Property. = 12m – 21 Simplify.

The Distributive Property Simplify –(5q – 6). –(5q – 6) = –1(5q – 6) Rewrite the expression using –1. = –1(5q) – 1(–6) Use the Distributive Property. = –5q + 6 Simplify.

The Distributive Property Simplify –2w2 + w2. = (–2 + 1)w2 Use the Distributive Property. = –w2 Simplify.

The Distributive Property Write an expression for the product of –6 and the quantity 7 minus m. Relate:  –6  times the quantity 7 minus m   Write: –6 • (7 – m) –6(7 – m)

Properties of Real Numbers Name the property each equation illustrates. a. 3 • a = a • 3 Commutative Property of Multiplication, because the order of the factors changes b. p • 0 = 0 Multiplication Property of Zero, because a factor multiplied by zero is zero c. 6 + (–6) = 0 Inverse Property of Addition, because the sum of a number and its inverse is zero

Properties of Real Numbers Suppose you buy a shirt for $14.85, a pair of pants for $21.95, and a pair of shoes for $25.15. Find the total amount you spent. 14.85 + 21.95 + 25.15 = 14.85 + 25.15 + 21.95 Commutative Property of Addition = (14.85 + 25.15) + 21.95 Associative Property of Addition = 40.00 + 21.95 Add within parentheses first. = 61.95 Simplify. The total amount spent was $61.95.

Properties of Real Numbers Simplify 3x – 4(x – 8). Justify each step. 3x – 4(x – 8) = 3x – 4x + 32 Distributive Property = (3 – 4)x + 32 Distributive Property = –1x + 32 Subtraction = –x + 32 Identity Property of Multiplication

Graphing Data on the Coordinate Plane Name the coordinates of point A in the graph. Move 2 units to the left of the origin. Then move 3 units up. The coordinates of A are (–2, 3).

Graphing Data on the Coordinate Plane Graph the point B(–4, –2) on the coordinate plane. Move 4 units to the left of the origin. Then move 2 units down.

Graphing Data on the Coordinate Plane In which quadrant or on which axis would you find each point? a. (2, –5) b. (6, 0) Since the x-coordinate is positive and the y-coordinate is negative, the point is in Quadrant IV. Since the y-coordinate is 0, the point is on the x-axis.

Graphing Data on the Coordinate Plane The table shows the number of hours worked and the amount of money each person earned. Make a scatter plot of the data. Name Hours Amount worked earned Janel 6 $25.50 Roscoe 12 $51.00 Victoria 11 $46.75 Alex 9 $38.25 Jordan 15 $63.75 Jennifer 10 $42.50 For 6 hours worked and earnings of $25.50, plot (6, 25.50). The highest amount earned is $63.75. So a reasonable scale on the vertical axis is from 0 to 70 with every 10 points labeled.

Graphing Data on the Coordinate Plane Use the scatter plot in your previous answer to answer the following question: Is there a positive correlation, negative correlation, or no correlation between the number of hours worked and the amount earned? Explain. As the number of hours worked increases, the earnings increase. There is a positive correlation between hours worked and earnings.