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Topic 1: Solving one step equations Name: ________________________ Per: ________________________ Date: ________________________ Class: Algebra 1A________________ Notes: Integer Review: Adding and subtracting integers A ________ number a ________ number = a ________ number A ________ number Example: 11 + 8 = _________ a _________ number = Example: 6 8 = ________ a _________ number So, two negatives make a _________ ____________. And, two positives make a ________ _____________. When adding, a positive and a negative _____________ and take the sign of the __________ ___________. Examples: A. Simplify each expression C. –3 – 9 B. –4 + 8 6 – 13 D. 7+15 A double negative _____________ and becomes a __________ Examples: Simplify each expression D. – 5 + ( – 16) E. – 6 + (– 14) F. – 6.2 – (– 6.9) G. 8.8 – (– 7.9) Multiplying and dividing When multiplying or dividing, integers A positive # divided by a positive # = ______________________. A positive # divided by a negative # = _____________________. A negative # divided by a positive # = _____________________. Note: The same is true if you multiply. A negative # divided by a negative # = ____________________. When several terms are multiplying or dividing, an ________ number of negatives become __________________. And an ______ number of negatives stays _________. 1 Examples: I. – 6 (5)(– 7) J. (–24)/6 K. 6(–6) L. 12 3 M. 4 9 N. 15 3 O. (2)(7) P. 18 2 Simplify each expression More Examples: The goal of Algebra The goal of solving equations is to get ________________ (you’re trying to get __ by __________). Opposite Operation We may have to move terms to get the variable by itself. We do this by using the opposite operation. So, if something is adding to the variable, we ____________. If something is subtracting from the variable, we _______. If something is multiplying to the variable, we _________. If something is dividing from the variable, we _________. Examples Solve each equation for x. Be sure to show your work. Q. x + 4 = 11 R. x + 2 = –5 S. x – 12 = 10 T. x – 6 = –1 V. 5x 15 U. 3x 18 2 More Examples Solve each equation for x. Show your work. W. x 6 12 X. x 2 6 Y. x 4 4 Z. x 4 2 DAY 2: STAR skills: Key words or phrases that will be used throughout your math classes! Deductive Reasoning Reasoning _____________ from given statements to form a _____________________. If a student is on the cross country team, then the student can run a Example long way. What can we conclude? STAR released Question #18 Example 2 If a student wears plaid, then the student must be Scottish. Conclude: ___________________________________________ Example 3 If a number is a prime number, then the number can only be divided by itself and one. Conclude: ___________________________________________ Inductive vs. Deductive ___________ reasoning goes from _____________ examples to a Reasoning ______________ conclusion. Many times inductive reasoning will have examples that use ______________ (which are specific). ___________ reasoning goes from _____________ examples to a ______________ conclusion. 3 Which statement is an A) Bob concluded that 5 8 a positive number, because when a example of deductive positive number and a negative number are added together the reasoning and which is answer is always the sign of the larger number. an example of inductive reasoning? B) Betsy concluded that two negative numbers always add to make a larger negative by calculating the sums of several pairs of negative numbers. Q #1: Which of the following best describes deductive reasoning? A using logic to draw conclusions based on accepted statements B accepting the meaning of a term without definition C defining mathematical terms to correspond with physical objects D inferring a general truth by examining a number of specific examples Q #3: Consider the arguments below. I. Every multiple of 4 is even. 376 is a multiple of 4. Therefore, 376 is even. II. A number can be written as a repeating decimal if it is rational. Pi cannot be written as a repeating decimal. Therefore, pi is not rational. Which one(s), if any, use deductive reasoning? Conditional Statements If-then statements (like the _____________ _________) These are often used in logical reasoning. Hypothesis The hypothesis of a conditional follows the _______ Circle the hypothesis: If x 2 4 , then x 2 or x 2 . Conclusion The conclusion of a conditional follows the ________. Circle the conclusion: If you want to find the perimeter of a shape, then you add up all the sides lengths. 4 Examples What is the conclusion of the statement below? (circle it) If x 5 9 , then x 5 5 9 5. What is the conclusion of the statement below? (underline it) If 2 r c , then r Perimeter c 2 To find the perimeter of any shape, ________ all the side lengths. Example The lengths of the sides of a triangle are 6, 12, and 14 centimeters. Find the perimeter of the triangle. The lengths of the sides of a triangle are y, y+2, and 8 centimeters. STAR released Q#15 If the perimeter is 48 centimeters, what is the value of y? Reminder: To find the perimeter of any shape, you ____ _____ ________ A tougher perimeter If the sides lengths of one triangle are x, 2x + 3, and 3x, and the problem sides of a second triangle are x, 2x – 1, and x + 4, what is the Step 1: _________ difference between the perimeters of the two triangles? your ______________. Step 2: Difference means ____________. Step 3: Set up an _______________. Step 4: __________ the equation. 5 Sometimes/always/ Statements made may be true sometimes, always, or never. never For example, if x 2 5 , then x 7 is _________________ true. Or, two negatives make a positive is __________________ true (explain here) Counterexample A counterexample is an example that _________________ an argument. For example, if I say that every odd number is prime, a counterexample to that would be the odd number ___________. Example The chart below shows an expression evaluated for one value of x. Finish filling in the table for the three missing values. x x2 3 1 4 3 7 9 Reading this chart, it appears that for every positive value of x, x 2 3 produces a number divisible by 4. Can you find a value of x that would serve as a counterexample? Now turn the page and fill out the summary. 6 Summary: Below is a model used to show adding negative and positive integers. Write the expression you think is being modeled and then simplify it. _________________ _________________ What did represent in your expression? What did represent in your expression? 7 Topic 2: Solving two step equations (Day 3) Name: ________________________ Per: ________________________ Date: ________________________ Class: Algebra 1A________________ Notes: Variable Usually ____, but any letter that represents the ______________ quantity. Constants The terms in an equation that don’t have a _______________. Example Identify the variables and the constants in the following equation: 2x 3 x 7 Like terms If two or more terms are the same variable, they are ______ _____. If two or more terms are constants, they are _______ _________. If like terms are on the same side of the ________ sign, they ______ their signs and ____________ together. If terms are on the _______________ side of the ________ sign, you must use _____________ _____________ to move the like terms to the same side of the equal sign. Example Identify the like terms in each equation(circle the variables and underline the constants). A) 2x 4 12 B) 3x 4x 13 C) 14 2x 4 D) 12 4x 2x E) 4x 12 16 F) 6 2x 6 8 A helpful tool: the To determine whether you need to use _____________ equal sign ______________ before combining like terms, it can help to draw a _____________ _________ through the equal sign. Do this on examples A through F on the last page. When do like terms need to move? Solving two step equations Goal: Get all of the _________________ on one side, and all of the _____________on the other side. Then (usually) divide to get ___________ or ____________ Try this 1) 3x 2 14 2) 4x 4 8 Solve for x. 3) 2x 3x 11 More Practice 5. 5 7 x 2 4) 3 5x 18 6. 7 3x 2 8. 12 2 7x What value of x is the solution to the equation? 7. 8 5 3x 9 More Practice 9. 4x 20 4 10. 7 13 2x What value of x is the solution to the equation? 11. 6 x 10 12. 14 2x 18 Next step: Moving two Identify which example you need to use opposite operations before terms combining like terms. A) x + 13 = – x + 21 Practice C) 4x + 14 = 6x – 38 B) 3y – 5y = 27 + 19 D) 5x + 14 = 3x + 6 What value of x is the solution to the equation? E) x – 6 = 2x + 12 F) -2x + 8 = x - 7 10 More practice G) 3m – 21 = 28 – 3m H) 3x – 6 = x + 11 What value of x is the solution to the equation? Moving the variable to x = 5 is equivalent (means “the ______ ___” ) to ____ = ______ the right side and Sometimes you may want or need to move your variable to the right solving. side of the equation to solve (usually either to keep your variables ___________ or because your constants are only on the _______ side of the equation). For example, Case 1) –x + 12 = 2x – 6 Case 2) 3x + 52 = 7x Try this Solve each of the two cases above by moving the variables to the right side of the equation. In which case was it better to use the right side for variables? Why? When like terms are Sometimes your like terms may be on the same side of the equation. on the same side of For example, 4n + 14 – 8n = 38. Explain in sentences the two ways the equation you would combine like terms in this equation. Explain here: 11 Examples: 1) 4x 3x 5 2x 25 2) 10x 4x 2x 32 What value of x is the solution to the equation? 3) 2x 5x 15 x 21 Try this 5) 12x 6x 7 x 10 16 4) 22 12x 7 7 x 2x 6) 6x 4x 8x 13 5 What value of x is the solution to the equation? 12 What value of x is the solution to the 7) 2x x 6 18 9x 8) 14 2x 7 5x 2x equation? Summary: In the model below the number 1. So 3x 2 looks like represents a variable and and 2x 4 looks like represents . Explain how to solve = 13 Topic 3: Problem solving strategies: Drawing a picture and using rate. (Day 4) Draw a _____________ Name: ________________________ Per: ________________________ Date: ________________________ Class: Algebra 1A________________ Notes: To help ____________ a word problem, sometimes it is helpful to draw a _________________. Depending on the problem, you could _________ a _____________ of a ________, or a ________________, or a _________ or ___________________. Then, you can ____________ the _____________ or __________ and try to set up an ______________ which you can solve. A 60-foot-long rope is cut into 3 pieces. The first piece of rope is STAR released Q#12 twice as long as the second piece of rope. The third piece of rope is three times as long as the second piece of rope. What is the length of the longest piece of rope? Hint: 1) Draw a ___________ 2) Label each _________ 3) Set up an __________ Try again A 24-inch-long piece of Fruit Rollups is divided among three people. The first piece of Rollups is three times as long as the second piece of Rollups. The third piece of Rollups is twice as long as the second piece of rollups. What is the length of the longest piece of Fruit Rollups? A. 16 inches B. 12 inches C. 8 inches D. 4 inches 14 Another example Three friends earned $120 by creating a wedding picture montage. Bob spent twice as long as Amy working on the video, while George spent three times as many hours working as Amy. How much money should each person make? Rate problems Distance is the _______________ of something Rate is the _____________ of something Time is the _____________ of time (usually in hours or minutes) _______________ = __________ x __________ or __ = __ x __ Conceptual If Ed drives 40 miles per hour, he travels ____ miles in 1 hour. understanding If Kay drives 50 miles per hour, she travels ____ miles in 2 hours. If Bob drives 65 mph, he travels _____ miles in 2 hours. If Vern drives 45 mph for 4 hours, he travels _____ miles total. If Lily drives 35 mph for 3 hours, she travels _____ miles total. Andy’s average driving speed for a 4-hour trip was 50 miles per STAR released Q#73 hour. During the first 3 hours he drove 40 miles per hour. What was his average speed for the last hour of his trip? Hint: How many total miles did Andy drive in four hours? How far did Andy drive in the first three hours? 15 Similar question Miranda’s average driving speed for a 4-hour trip was 60 miles per hour. During the first three hours she drove 65 miles per hour. What was her average driving speed for the last hour of her trip? A harder skill Suppose Pat drove for five hours at an average speed of 50 miles per hour. If Pat drove the first three hours at 40 miles per hour, what was Pat’s average speed for the last two hours? 16 Two airplanes left the same airport traveling in opposite directions. similar to: STAR released Q#89 If one airplane averages 300 miles per hour and the other airplane averages 230 miles per hour, in how many hours will the distance between the two planes be 1325 miles? Do you add or subtract the speeds? ___________ What word determines that in the above problem? ____________ Set up the equation to solve this: Now solve (expect a decimal/fraction answer) Similar skill What if two cars leave the same location at the same time and travel in the same direction on a road. The first car travels at a rate of 50 miles per hour, and the second car travels at a rate of 30 miles per hour. How much time will pass before the cars are exactly 80 miles apart? Do you add or subtract the speeds? ___________ What word determines that in the above problem? ____________ Set up the equation to solve this: Now solve Now turn the page and fill out the summary. 17 Summary: A grocer is stacking soup cans in a pyramid. The bottom row will have 7 cans. Each row will have 1 can fewer than the row below it. How many rows of cans will be stacked in the pyramid? How many cans will be stacked in all? Draw a picture, and then explain how you drew your picture and how it helped you solve the problem. 18 Topic 4: Problem solving strategies: Error Analysis. Name: ________________________ Per: ________________________ Date: ________________________ Class: Algebra 1A________________ (Day 5) Notes: Correct vs. incorrect When presented with an equation that is solved step by step, you solutions may be asked to identify whether there is an error in the solution. To do this, it might be helpful to: also known as 1) __________________________________________________ Error Analysis _____________________________________________________ 2) __________________________________________________ _____________________________________________________ Usually, ____________ the problem on ______ _____ next to the solution _________ to identify _________ the _________ is located. STAR released Q#11 Solve: 3 x 5 2 x 20 Step 1: 3x 15 2x 20 Step 2: 5x 15 20 Step 3: 5x 5 Step 4: x 1 Which is the first incorrect step in the solution shown above? 19 Stan’s solution to an equation is shown below. STAR released Q#21 Are all of the steps correct? Or is there an error in one of the steps? Given: b 6 b 10 80 Step 1: b 6b 10 80 Step 2: 7b 10 80 Step 3: 7b 80 10 Step 4: 7b 70 Step 5: 7b 70 7 7 Step 6: b 10 What is true about Stan’s solution? Similar STAR question Four steps used to simplify the equation are shown below. skills 4(2 4 x) 6 2(1 3x) similar to: STAR released Q#54 I 8 16x 4 6x II 22x 4 III 8 16x 6 2 6x IV 22x 4 Hint: Try solving this out for yourself, showing your work. What is the correct order for these steps? 20 Jeremy, Michael, Shanan, and Brenda each worked the same math problem at the chalkboard. Each student’s work is shown below. Their teacher said that while two of them had the correct answer, only one of them had arrived at the correct conclusion using correct steps. Jeremy’s work Shanan’s work 6 4 2x 8 6 4 2x 8 6 4 2x 8 6 4 2x 8 2 2x 8 2 2x 8 2x 6 2 x 10 x3 x 5 Michael’s work Brenda’s work 6 4 2x 8 6 4 2x 8 6 4 2x 8 6 4 2x 8 2 2x 8 2 2x 8 2x 6 2 x 10 x3 x5 Which is a completely correct solution? Now turn the page and fill out the summary. 21 Summary: Tanya solved the equation 3 6x 15 as follows. 3 6x 15 6x 3 15 6 x 18 x3 Myra discovered that Tanya’s solution was wrong. Explain how she knew that Tanya’s solution was wrong. 22 Topic 5: The Distributive Property (Day 7) Name: ________________________ Per: ________________________ Date: ________________________ Class: Algebra 1A________________ Notes: The Distributive Property When parentheses are present, this indicates that you need to distribute the term immediately in front of the parentheses (so you are _____________). This looks like 2(x + 3) and becomes ___ + ___ . In an equation, you should distribute BEFORE combining like terms or performing opposite operations. Why? Why do we distribute before adding? PEMDAS = we __________ before we _____ or ___________. And, the parentheses show that we are multiplying the term outside the parentheses to ______ term inside. Examples as a class 1) 3 x 4 6 3) 15 5 x 1 2) 4 x 2 12 4) 4 x 4 16 23 Distributing a negative If the term in front of the parentheses is negative, then the signs term of the terms inside the parentheses must ____________. Example: 2 x 4 6 Why? 3 x 2 9 A negative times a negative is a ______________________. A negative times a positive is a ______________________. Ex) -4(-2) = Ex) -9(4) = Ex) 3(-12) = If there is not a number (shown) before of the parentheses, but only a sign, then distribute the sign. A positive sign or no sign means all signs _______ the _______. A negative sign means all signs _____________. 2x 4 8 Solving two step 14 x 2 First step: ___________________. equations with distributing Then, solve normally. Examples Solve for x. A) 2 x 3 4 x 8 B) 3( x 3) 4( x 2) 24 More Examples Solve for x. C) 4 x 12 2 x 4 D) 2 x 3 4 x 2 Distribute first, then In equations with parentheses, and thus distributing, you should combine like terms _______________________ first, then ______________ _______ ____________. What is the value of x in E) 3 x 2 12 2 x 9 F) 4 x 5 24 2 x 6 the equations to the right? Equivalent Equivalent means _____ ___________ ______. Example: Is the equation 2 3x 6 18 equivalent to 6x 12 18 ? ________, by the __________________ Property. The Distributive Property When there are parentheses, and thus distributing, you should of Multiplication over always ___________________ before _________________ Addition ________ __________. Example: What is the next step in the equation 4 3 2 x 6 31 3x ? 25 Example What equation is equivalent to 4 x 2 6 x 1 12 x ? Simplify the left side (only) here Practice Solving 1) 3 2 x 4 1 12x 12x 2) 7 3 2 x 4 1 What is the value of x in each of these equations? 3) 6 x 3 10 4) 10 3x 6 2 5) 6 x 2 x 4 16 6) 4 2 x 3 4 2 3x 6 26 Summary: 4 x 3 means I have 4 groups of x 3 . We can show this with a model. 1st group 2nd group 3rd group 4th group Explain how this model shows that 4 x 3 4 x 12 . 27 Questions dropped from the notes Testable Questions The total cost of renting a truck for n days is given by the equation c 100 50n . If the total cost was $350, for how many days was STAR released Q#10 the truck rented? Hint: For which variable do you substitute 350? ______ The cost to rent a construction backhoe is $500 per day plus $200 STAR released Q#13 per hour of use. What is the maximum number of hours the backhoe can be used each day if the rental cost is not to exceed Hint: $1900 per day? Would drawing a picture be helpful here? _______ So we should _____ ___ ___ _______________. Solving equations with Some equations have decimals in them, especially certain word decimals problems. To solve, follow the steps of solving a normal equation. However, there are two ways to work with the decimals: 1) Keep the decimals in the equation until the end. Just remember, when you divide by a decimal, you have to move the decimal of the term on the __________ of the division sign. Example: .65 .26 2) Multiply by a power of 10 to clear the equation of decimals. 28 Multiplying by 10 moves the decimal _____ time, multiplying by 100 moves the decimal _____ times, etc. Remember, each term has to be multiplied (if there is already something multiplying into a parentheses, be careful to multiply that out first, and then multiply each term by the power of 10). Bob will make punch that is 20% fruit juice by adding pure fruit juice to a 2-liter mixture that is 10% fruit juice. How many liters of pure fruit juice does he need to add? similar to: STAR released Q#90 To solve: What percent is pure fruit juice? ___________ What variable can we use to represent how much fruit juice Bob is adding? ____ How much fruit juice does Bob start with in the mixture? _______ How much fruit juice will Bob have after he adds more? _____________ Set up an equation and solve .20 ( A harder problem similar to: STAR released Q#86 + ) = ____ ( ) + 2 (.10) A pharmacist mixed some 10%-saline solution with some 15%saline solution to obtain 100 mL of a 12%-saline solution. How much of the 10% saline solution did the pharmacist use in the mixture? To solve: What are we trying to solve for? What should we call x? ________________________ Still using x, how much of the other saline solution did the 29 pharmacist add if there is a total of 100 mL? _______________ Set up an equation and solve. .10 ( ) + .15 ( ) = .12 (100) Q #1: Which of the following best describes deductive reasoning? A using logic to draw conclusions based on accepted statements B accepting the meaning of a term without definition C defining mathematical terms to correspond with physical objects D inferring a general truth by examining a number of specific examples Q #3: Consider the arguments below. I. Every multiple of 4 is even. 376 is a multiple of 4. Therefore, 376 is even. II. A number can be written as a repeating decimal if it is rational. Pi cannot be written as a repeating decimal. Therefore, pi is not rational. Which one(s), if any, use deductive reasoning? 30