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 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 1
Topic: Solving Linear Equations 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. x 2 1 5 What is the value of x in the equation + =? 1 6 First Subtract NOT ATTEMPT THE MATH IN YOUR HEAD!!!! Next you cross multiply. Last you solve for x. Work Space 3 6 6 from both sides of the equal sign. USE THE CALCULATOR!!!!!!! DO 2
Topic: Graphing Linear Equations Look for keywords/hints within the word problem. Product = Multiply Total, all together, sum, more = Add Difference = Subtract 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 3
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 y intercept or your b Know that the equation should be in the form of, y = mx + b 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 4
Topic: Graphing Linear Functions Always start with the b in the equation. This is your starting point or you y intercept. Your slope tells you how to move from the y intercept. 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. What is the value of the y intercept 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 y intercept ( b ) is now? Work Space 5
Topic: Transforming formulas Transforming formulas is the same as solving a multi step equation. 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: 6
Topic: Inequalities 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. 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 x 8, 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 number 3. 5 3 x, I would multiply both sides by the denominator (the 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: 7
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. 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. x < 4, this means that x are numbers smaller than 4. 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 8
Topic: Modeling Linear Inequalities If there is a line under the arrow, > or <, this means equal to or... Meaning, you will include the answer you found. 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. x < 4, this means that x are numbers smaller than 4. 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 9
Topic: Graphing Linear Inequalities 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 Always start at the end with the y intercept and then work with the slope (rise over run) This has a y intercept 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? 10
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Teacher Version Topic: Modeling Expressions Station #1: Tips/Hints Look for keywords/hints within the word problem. Product = Multiply Total, all together, sum, more = Add Difference = Subtract 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 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. x 2 1 5 What is the value of x in the equation + =? 3 6 6 16
Topic: Modeling Linear Equations Look for keywords/hints within the word problem. Product = Multiply Total, all together, sum, more = Add Difference = Subtract 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 y intercept or your b Know that the equation should be in the form of, y = mx + b 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 Always start with the b in the equation. This is your starting point or you y intercept. 17
Your slope tells you how to move from the y intercept. 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. What is the value of the y intercept for the graph of 4x 5y = 40? Topic: Transforming formulas Transforming formulas is the same as solving a multi step equation. 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 You solve inequalities the same way you would go about solving a linear equation. Remember: If your last step is to divide by a 18
NEGATIVE, then you must flip the arrow to the opposite direction. Topic: Interpreting Solutions Determine the smallest integer that makes 3x + 7 5x < 15 true. 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. 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. x < 4, this means that x are numbers smaller than 4. Topic: Modeling Linear Inequalities If there is a line under the arrow, > or <, this 19
means equal to or... Meaning, you will include the answer you found. 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. x < 4, this means that x are numbers smaller than 4. 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 Things to remember < or > means a dotted line ( >) < or > means a solid line ( ) 20
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 21