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Common Core Algebra Rock the Regents 2016
Station 1:Linear Equations & Inequalities
Name: ____________________________________________
Teacher: _________________________________________
Date: _____________________________________________
Grade: 9 10 11 12 (circle one)
Period: __________________________________________
Topic: Modeling Expressions
Tips/Hints Look for keywords/hints within the word problem. Product = Multiply Total, all together, sum, more = Add Difference = Subtract
Example: To watch a varsity basketball game, spectators must buy a ticket at the door. The cost of an adult ticket is $3.00 and the cost of a student ticket is $1.50. If the number of adult tickets sold is represented by a and student tickets sold by s, which expression represents the amount of money collected at the door from the ticket sales?
a) 4.50as b) 4.50(a + s) c) (3.00a)(1.50s) d) 3.00a + 1.50s
A key word in this problem is TOTAL...so that means your are going to be adding something...so choices a and c are incorrect. Work Space
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Topic: Solving Linear Equations
Tips/Hints: Remember that you want to get rid of whatever is Adding or Subtracting first. Then you multiply or divide. Don’t forget that what you do to something on one side of the EQUAL sign, you must do EXACTLY THE SAME to the other side of the equal side.
Example: What is the value of x in the equation + = ?3
x − 261
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First Subtract from both sides of the equal sign. USE THE CALCULATOR!!!!!!! DO6
1 NOT ATTEMPT THE MATH IN YOUR HEAD!!!! Next you cross multiply. Last you solve for x. Work Space
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Topic: Graphing Linear Equations
Tips/Hints: Look for keywords/hints within the word problem. Product = Multiply Total, all together, sum, more = Add Difference = Subtract
Example: John has four more nickels than dimes in his pocket, for a total of $1.25. Which equation could be used to determine the number of dimes, x, in his pocket?
a) 0.10(x + 4) + 0.05(x) = $1.25 b) 0.05(x + 4) + 0.10(x) = $1.25 c) 0.10(4x) + 0.05(x) = $1.25 d) 0.05(4x) + 0.10(x) = $1.25
So if we read the question, we see the word MORE ...so that means we are going to be adding something. We also read the work nickels and dimes. We know that a nickle = .05 and a Dime = .10. The problem says that there are four more nickels than dimes….do we know how many dimes are there?? So we represent with an x. Amount of dimes = x There are four more nickels than dimes Amount of nickels = x + 4 SOOOOO… amount of dimes are x, each x is worth .10 so, 0.10(x) Amount of nickels are x + 4, each nickel is worth 0.05 so, 0.05(x+4) Work Space
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Topic: Modeling Linear Functions
Tips/Hints Look for keywords/hints within the word problem. Rate = this is your slope or your “m” Per day, month, pound, day, etc. = X Starting point, what you began with = your yintercept or your “b” Know that the equation should be in the form of, y = mx + b
Example: Each day Toni records the height of a plant for her science lab. Her data are shown in the table below.
The plant continues to grow at a constant daily rate. Write an equation to represent h(n), the height of the plant on the nth day. So we need to find m and b.
How do we find m? Then we pick a set from the table (x,y) we plug in the values for x, y and m, in order to find “b”. Work Space
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Topic: Graphing Linear Functions
Tips/Hints: Always start with the “b” in the equation. This is your starting point or you yintercept. Your slope tells you how to move from the yintercept. If the slope is a negative, you move either: Left and then up or down and to the right. If the slope is positive, you move up and to the right. REMEMBER, in order for you to graph a linear equation, in must be in the y = mx + b format.
Example: What is the value of the yintercept for the graph of 4x − 5y = 40 ? Remember, “y” has to be by itself on the left side of the equation, and it must look like y = mx + b First you get the x on the other side of the equal sign, so you must subtract 4x on BOTH sides. (Don’t forget, you CANNOT subtract 4x and 40….Both of them must have x!!!!) 5y = 4x + 40 Then you divide all terms by 5. Can you find out what the yintercept (“b”) is now? Work Space
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Topic: Transforming formulas
Tips/Hints: Transforming formulas is the same as solving a multistep equation.
Example: Michael borrows money from his uncle, who is charging him simple interest using the formula I = Prt. To figure out what the interest rate, r, is, Michael rearranges the formula to find r. His new formula is r equals to what? The question is asking you to find “r”, or, to isolate “r”...Get “r” by itself. Prt are all multiplying so we must divide….what do we want to stay and what do we want to get rid of? We want to get rid of P and t, so we divide Pt from both sides. Can you find out what “r” equals?? Work Space:
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Topic: Inequalities
Tips/Hints: You solve inequalities the same way you would go about solving a linear equation. Remember: If your last step is to divide by a NEGATIVE, then you must flip the arrow to the opposite direction.
Example:
You would solve this inequality the same way you would go about solving a linear equation. First you want to get the two “x” together on one side of the inequality sign. I would bring the x in x8, on the left side by subtracting x from both sides. (Do not be afraid of the fraction...USE the CALCULATOR) Next, I would bring the 7 over from the left to the right by subtracting 7 from both sides of the inequality sign. Then, since there is a fraction x, I would multiply both sides by the denominator (the3
−5 number 3. Last, I would divide both sides by negative 5….BUT...I will not forget to flip the arrow towards the opposite direction. Now you try to work it out. Work Space
Topic: Tips/Hints:
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Interpreting Solutions
You solve inequalities the same way you would go about solving a linear equation. Remember: If your last step is to divide by a NEGATIVE, then you must flip the arrow to the opposite direction. Other things to remember: If there is a line under the arrow, > or <, this means “equal to or”.... Meaning, you will include the answer you found. Example: x < 4, this means that x could be 4 or any number smaller than 4. If there is no line under the arrow, < or > , this means you must go either down or up one number. Example: x < 4, this means that x are numbers smaller than 4.
Example: Determine the smallest integer that makes −3x + 7 − 5x < 15 true. First...since 3x and 5x are on the SAME SIDE of the inequality sign, I would combine (add) them. This will give me 8x + 7 < 15 Then I would subtract 7 from both sides of the inequality sign. Next I would divide 8 from both sides, REMEMBERING to flip the arrow in the opposite direction because I divided by a NEGATIVE sign. Work Space
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Topic: Modeling Linear Inequalities
Tips/Hints: If there is a line under the arrow, > or <, this means “equal to or”.... Meaning, you will include the answer you found. Example: x < 4, this means that x could be 4 or any number smaller than 4. If there is no line under the arrow, < or > , this means you must go either down or up one number. Example: x < 4, this means that x are numbers smaller than 4.
Example: The cost of a pack of chewing gum in a vending machine is $0.75. The cost of a bottle of juice in the same machine is $1.25. Julia has $22.00 to spend on chewing gum and bottles of juice for her team and she must buy seven packs of chewing gum. If b represents the number of bottles of juice, which inequality represents the maximum number of bottles she can buy?
a) 0.75b + 1.25(7) ≥ 22 b) 0.75b + 1.25(7) ≤ 22 c) 0.75(7) + 1.25b ≥ 22 d) 0.75(7) + 1.25b ≤ 22
Ok..so the key word in this word problem is Maximum. This means that Julia cannot go over the amount of $22 when she adds up the cost for the gum and the bottles of juice. First, we need to find set up the inequality. Gum = 0.75 and a bottle of juice cost = 1.25 We know that she must buy 7 packs of gum → 7(0.75) But, we don’t how many bottles of water she could buy → 1.25b Can you now try to find the correct equation for this word problem Work Space
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Topic: Graphing Linear Inequalities
Tips/Hints: Things to remember < or > means a dotted line ( >) < or > means a solid line ( → ) Hint: if there is a solid line under the arrow, there is a solid line on the graph. y < 3 this means, you go to the number 3 on the y axis and shade down because, y < 3 means (numbers on) y is less than 3…. y > 7 this means, you go to the number 7 on the y axis and shade up because, y > 7 means (numbers on) y is more that 7
Example:
Always start at the end with the yintercept and then work with the slope (rise over run) This has a yintercept of 4 so choices 3 and 4 are eliminated. The line is solid so there should be a solid line under the arrow. The graph is being shaded up so up also means greater than, so we need to look for a greater than sign. Can you figure out which is the correct answer?
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Practice Problems
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Teacher Version Station #1:
Topic: Modeling Expressions
Tips/Hints Look for keywords/hints within the word problem. Product = Multiply Total, all together, sum, more = Add Difference = Subtract
Example: To watch a varsity basketball game, spectators must buy a ticket at the door. The cost of an adult ticket is $3.00 and the cost of a student ticket is $1.50. If the number of adult tickets sold is represented by a and student tickets sold by s, which expression represents the amount of money collected at the door from the ticket sales?
e) 4.50as f) 4.50(a + s) g) (3.00a)(1.50s) h) 3.00a + 1.50s
Topic: Solving Linear Equations
Tips/Hints: Remember that you want to get rid of whatever is Adding or Subtracting first. Then you multiply or divide. Don’t forget that what you do to something on one side of the EQUAL sign, you must do EXACTLY THE SAME to the other side of the equal side.
Example: What is the value of x in the equation + = ?3
x − 261
65
16
Topic: Modeling Linear Equations
Tips/Hints: Look for keywords/hints within the word problem. Product = Multiply Total, all together, sum, more = Add Difference = Subtract
Example: John has four more nickels than dimes in his pocket, for a total of $1.25. Which equation could be used to determine the number of dimes, x, in his pocket?
e) 0.10(x + 4) + 0.05(x) = $1.25 f) 0.05(x + 4) + 0.10(x) = $1.25 g) 0.10(4x) + 0.05(x) = $1.25 h) 0.05(4x) + 0.10(x) = $1.25
Topic: Modeling Linear Functions
Tips/Hints Look for keywords/hints within the word problem. Rate = this is your slope or your “m” Per day, month, pound, day, etc. = X Starting point, what you began with = your yintercept or your “b” Know that the equation should be in the form of, y = mx + b
Example: Each day Toni records the height of a plant for her science lab. Her data are shown in the table below.
The plant continues to grow at a constant daily rate. Write an equation to represent h(n), the height of the plant on the nth day.
Topic: Graphing Linear Functions
Tips/Hints: Always start with the “b” in the equation. This is your starting point or you yintercept.
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Your slope tells you how to move from the yintercept. If the slope is a negative, you move either: Left and then up or down and to the right. If the slope is positive, you move up and to the right. REMEMBER, in order for you to graph a linear equation, in must be in the y = mx + b format.
Example: What is the value of the yintercept for the graph of 4x − 5y = 40 ?
Topic: Transforming formulas
Tips/Hints: Transforming formulas is the same as solving a multistep equation.
Example: Michael borrows money from his uncle, who is charging him simple interest using the formula I = Prt. To figure out what the interest rate, r, is, Michael rearranges the formula to find r. His new formula is r equals to what?
Topic: Inequalities
Tips/Hints: You solve inequalities the same way you would go about solving a linear equation. Remember: If your last step is to divide by a
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NEGATIVE, then you must flip the arrow to the opposite direction.
Example:
Topic: Interpreting Solutions
Example: Determine the smallest integer that makes −3x + 7 − 5x < 15 true.
Tips/Hints: You solve inequalities the same way you would go about solving a linear equation. Remember: If your last step is to divide by a NEGATIVE, then you must flip the arrow to the opposite direction. Other things to remember: If there is a line under the arrow, > or <, this means “equal to or”.... Meaning, you will include the answer you found. Example: x < 4, this means that x could be 4 or any number smaller than 4. If there is no line under the arrow, < or > , this means you must go either down or up one number. Example: x < 4, this means that x are numbers smaller than 4.
Topic: Modeling Linear Inequalities
Tips/Hints: If there is a line under the arrow, > or <, this
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means “equal to or”.... Meaning, you will include the answer you found. Example: x < 4, this means that x could be 4 or any number smaller than 4. If there is no line under the arrow, < or > , this means you must go either down or up one number. Example: x < 4, this means that x are numbers smaller than 4.
Example: The cost of a pack of chewing gum in a vending machine is $0.75. The cost of a bottle of juice in the same machine is $1.25. Julia has $22.00 to spend on chewing gum and bottles of juice for her team and she must buy seven packs of chewing gum. If b represents the number of bottles of juice, which inequality represents the maximum number of bottles she can buy?
e) 0.75b + 1.25(7) ≥ 22 f) 0.75b + 1.25(7) ≤ 22 g) 0.75(7) + 1.25b ≥ 22 h) 0.75(7) + 1.25b ≤ 22
Topic: Graphing Linear Inequalities
Tips/Hints: Things to remember < or > means a dotted line ( >) < or > means a solid line ( → )
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Hint: if there is a solid line under the arrow, there is a solid line on the graph. y < 3 this means, you go to the number 3 on the y axis and shade down because, y < 3 means (numbers on) y is less than 3…. y > 7 this means, you go to the number 7 on the y axis and shade up because, y > 7 means (numbers on) y is more that 7
Example:
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