Quiz 1-3 Quiz Solve for x: 3. Simplify: 4. What property is illustrated below?

1. 2. Wood shop: A piece of wood is 72” long. You cut the wood into 3 pieces. The 2 nd piece is 6” longer than the 1 st piece. The 3 rd piece is 6” longer than the than the 2 nd piece. a. Draw a diagram showing relative lengths of the pieces. b. Write an equation showing the length of each piece (use only one variable: x = length of 1 st piece. c. How long is the 1 st piece? The able shows the altitude of an airplane. What is its altitude at the 8 minute point? Time (min)01234 Height (ft)36,00032,80029,60026,40023,200 Quiz 1-5

1. 2. A car is traveling at 88 feet per second. How long will it take to travel 120 miles? The table shows the altitude of an airplane. What is its altitude at the 8 minute point? Time (min)01234 Height (ft)36,00032,80029,60026,40023,200 Quiz 1-5

1. 2. You leave Roy and travel south on the freeway at 65 mph. At the same time your friend travels north from Roy at a speed of 50 mph. How long would it take for you to be 200 miles apart? A salesperson has a base salary of $20,000 per year. She earns a commission based upon her total sales. Her commission is 10% of her total sales. If her total annual income was $55,000, what was her total sales? Quiz 1-5

No quiz today What problem from the homework do you want me to work?

Finish Section 1-5 Time-Distance word problems Finish Section 1-5 Time-Distance word problems

Speed/Distance Model: (involves time) Distance = (speed) (time) Speed is a “rate” Speed is a “rate” (distance per unit time) This is a ‘gotcha.’ All units (hours, minutes, feet, miles, etc., MUST be CONSISTENT throughout the problem!!!! d = r*t

Speed/Distance Model: (involves time) Distance = (speed) (time) Example: d = r*t It takes you 5 hours to drive to St. George. St. George is 300 miles away. How fast were you going? 1. Write the formula 2.Identify the quantities from the formula that are given in the problem: d = ?, r = ?, t = ? the problem: d = ?, r = ?, t = ? d = 300 miles, r = ?, t = 5 hours d = r*t

Speed/Distance Model: (involves time) Distance = (speed) (time) Example: d = r*t It takes you 5 hours to drive to St. George. St. George is 300 miles away. How fast were you going? 3. Replace the values given into the formula. 300 miles = r * 5 hours 4. Solve for the unknown variable. 300 miles = r * 5 hours ÷ 5 hours ÷ 5 hours

Speed/Distance Model: (involves time) Distance = (speed) (time) Example: d = r*t A plane flew at a speed of 300 miles/hr for 7 hours. How far did it fly? 1. Write the formula 2.Identify the quantities from the formula that are given in the problem: d = ?, r = ?, t = ? the problem: d = ?, r = ?, t = ? d = ?, r = 300 miles/hr, t = 7 hours d = r*t

Speed/Distance Model: (involves time) Distance = (speed) (time) Example: d = r*t A plane flew at a speed of 300 miles/hr for 7 hours. How far did it fly? 1. Write the formula 2.Identify the quantities from the formula that are given in the problem: d = ?, r = ?, t = ? the problem: d = ?, r = ?, t = ? d = ?, r = 300 miles/hr, t = 7 hours d = r*t 3. Replace the values given into the formula. d= 300 miles/hr * 7 hours 4. Solve for the unknown variable.

Speed/Distance Model: (involves time) Distance = (speed) (time) Example: d = r*t A plane flew 4000 miles in 7 hours. What was its speed? 3. Replace the values given into the formula = x miles/hr * 7 hours 4. Solve for the unknown variable.

Your turn ½ point of the equation, ½ point for the solution. Distance = (speed) (time) 1. What would the speed have to be to travel 1000 miles in 1. What would the speed have to be to travel 1000 miles in 6 hours? 6 hours? d = r*t 2. How long would it take to travel 1500 miles if your speed was 200 miles per hour? speed was 200 miles per hour?

Two people traveling (1) Same direction You leave Roy and travel south at 65 mph. Your You leave Roy and travel south at 65 mph. Your friend leaves 2 hours later. How long would she have to friend leaves 2 hours later. How long would she have to travel to catch up to you if she is going 75 mph? travel to catch up to you if she is going 75 mph? d = r*t Who traveled further, you or your friend?

Two people traveling (1) Same direction You leave Roy and travel south at 65 mph. Your You leave Roy and travel south at 65 mph. Your friend leaves 2 hours later. How long would she have to friend leaves 2 hours later. How long would she have to travel to catch up to you if she is going 75 mph? travel to catch up to you if she is going 75 mph? d = r*t How do we relate her travel time to your travel time?

Summary (1) Same direction d = r*t Replace the variables with numbers from the problem. If there are 2 unknown variables, you need to find a way to relate the two. For example: to relate the two. For example: When you have only one variable, you can solve for it.

Your turn (1) Same direction 3. Your friend travels north at 20 mph for an hour, then you 3. Your friend travels north at 20 mph for an hour, then you follow at 30 mph. How long will it take to catch up? follow at 30 mph. How long will it take to catch up? d = r*t 4. Your friend travels east at 50 mph for 3 hours, then you 4. Your friend travels east at 50 mph for 3 hours, then you follow. It takes you 5 hours to catch up. How fast were you follow. It takes you 5 hours to catch up. How fast were you going? going?

Two people traveling (2) Opposite direction You leave Roy and travel south at 65 mph. Your You leave Roy and travel south at 65 mph. Your friend travels north at 50 mph. How long will it be until you are friend travels north at 50 mph. How long will it be until you are 350 miles apart? 350 miles apart? d = r*t 350 miles Total distance problem. How do we relate her travel time to your travel time?

Summary (2) Opposite direction d = r*t Replace the variables with numbers from the problem. If there are 2 unknown variables, you need to find a way to relate the two. For example: to relate the two. For example: When you have only one variable, you can solve for it.

Two people traveling (2) Opposite direction 5. You and your friend both leave Roy at the same time in 5. You and your friend both leave Roy at the same time in opposite directions. Your speed is 30 mph and his speed opposite directions. Your speed is 30 mph and his speed is 55 mph. How long will it be until you are 280 miles apart? is 55 mph. How long will it be until you are 280 miles apart? d = r*t 6. You and your friend both leave Roy at the same time in 6. You and your friend both leave Roy at the same time in opposite directions. Your speed is 50 mph. After 8 hours opposite directions. Your speed is 50 mph. After 8 hours you are 600 miles apart. What was his speed? you are 600 miles apart. What was his speed?

Homework: Section 1 – 5 Finish the time-distance problems from the assignment work sheet.

Section 1-4 Rewrite Formulas and Equations. Section 1-4 Rewrite Formulas and Equations.

Homework: Section 1 – 4 Problems (evens): 2-16 (for problems 8-14 these are two part problems; 1 st solve for the indicated variable then plug in a value) 22-28: book says solve and plug in like 2-16 above but I just want you to solve for the indicated variable (don’t plug in) (19 problems)

Vocabulary Solve for a variable (more then one variable in the equation): Use properties of equality to rewrite the equation as an equivalent equation with the specified variable on one side of the equal sign and all other terms on the other side. Solve the single variable equation: Use properties of equality to rewrite the equation as an equivalent equation with the variable on one side of the equal sign and a number on the other side.

x + 1 = 5 = 4 = 4 x = - 1 Solve for “x” Solve for the variable: Use properties of equality to rewrite the equation as an equivalent equation with the variable on one side of the equal sign and a number on the other side.

Solve for ‘x’ 4 + 2x + y = 6 ÷ 2 ÷ x + y = 2 2x + y = 2 - y - y 2x = 2 – y 2x = 2 – y Solve for the variable: Use properties of equality to rewrite the equation as an equivalent equation with the specified variable on one side of the equal sign and all other terms on the other side.

Solve for “x” yx – 2 = Solve for the variable: Use properties of equality to rewrite the equation as an equivalent equation with the specified variable on one side of the equal sign and all other terms on the other side. ÷ y ÷ y yx = 6 yx = 6

Your turn: 7. Solve for ‘k’ 8. Solve for ‘k’ 9. Solve for ‘k’

Vocabulary Formula: An equation that relates two or more quantities, usually represented by variables. Quantity: An measure of a real world physical property (length, width, temperature, pressure, weight, mass, etc.).

Formulas are used extensively in science. Science and math come together when mathematical equations are used to describe the physical world. Once a formula is known then scientists can use the equation to predict the value of unknown variables in the formula.

Circumference C = πd C = π*2r What real world quantity does “d” represent? does “d” represent? d = 2r What real world quantity does “r” represent? does “r” represent? Since d = 2r, we can replace ‘d’ in the circumference formula with ‘2r’. C = 2πr What property allows us to re-write the formula like this ? the formula like this ?

Solve for radius We will now solve for “r” In this form, we say that ‘c’ is a function of ‘r’. ‘c’ is a function of ‘r’. ÷ 2π ÷ 2π In this form, we say that ‘r’ is a function of ‘c’. ‘r’ is a function of ‘c’.

Circumference Your turn: for the area of a triangle formula: (This is ‘A’ is a function of ‘b’ and ‘h’.) 10. Solve for “b” 11. Solve for “h”. We call this new version of the formula “b” is a function of “h” and “A” “b” is a function of “h” and “A” 12. What do you call this new version of the formula?

Your Turn: 13. Solve for ‘h’. 14. Solve for (Area of a trapezoid: where the length of the parallel bases are of the parallel bases are and the distance between them is ‘h’.) and the distance between them is ‘h’.)

What if two terms have the variable you’re trying to solve for? Solve for ‘x’. ‘x’ is common to both terms  factor it out (reverse distributive property). factor it out (reverse distributive property). How do you turn (3y – 2) into a “one” so that it disappears a “one” so that it disappears on the left side of the equation? on the left side of the equation? ÷(3y – 2) ÷(3y – 2)

Example Solve the equation for “y”. Use “reverse distributive property 9y + 6xy = 30 ÷(9 + 6x) ÷ (9 + 6x) What is “common” to both of the left side terms? the left side terms? “Factor out” the common term “same thing left/right”

15. Solve for ‘x’. 16. In problem #15, if y = 5, x = ? Your turn: Your turn: 17. Solve for ‘y’.

Solving formula Problems The perimeter of a rectangular back yard is 41 feet. Its length is 12 feet. What is its width? Draw the picture Write the formula Replace known variables in the formula with constants ÷2 ÷2 Solve for the variable

Solving formula Problems 1. Draw the picture (it helps to see it) 2. Write the formula 3. Replace known variables in the formula with constants 4. Solve for the variable

18. If the base of a triangle is 4 inches and its area is 15 square inches, what is its height? 20. The perimeter of a rectangle is 100 miles. It is 22 miles long. How wide is the rectangle? Your turn: Your turn: 19. The area of a trapezoid is 40 square feet. The length of one base is 8 feet and its height is 3 feet, what is the length of the other base?

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Using formulas so solve real world problems. What is the profit model (words describing the relationship between profit, $ from sales, and $ going for expenses.) ? (From a previous example) the profit for selling ‘c’ candles at $3 each when the cost to rent the booth and buy supplies is $120. How many candles must be sold to have a profit of $500 ? have a profit of $500 ? P = 3c - 120

Using formulas so solve real world problems. How many candles must be sold to have a profit of $525 ? 1. Solve for the variable P = 3c Method ÷3 ÷3 2. Plug numbers into the formula.

Using formulas so solve real world problems. How many candles must be sold to have a profit of $525 ? 2. Solve for the variable P = 3c Method ÷3 ÷3 1. Plug numbers into the formula. Same answer