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# What is the term”exclamation mark” in mathematics? The answer is easy. Below are some approaches to inform as soon as an equation can be really actually a multiplication equation and also how to add and subtract using”exclamation marks”. For instance,”2x + 3″ imply”multiply two integers from about a few, making a value add up to 2 and 3″. By way of example,”2x + 3″ are multiplication by write a paper online three. In addition, we can add the values of 2 and 3 collectively. To add the worth, we’ll use”e”that I” (or”E”). With”I” means”include the worth of one to the worth of 2″. To add the worth , we can certainly do this similar to that:”x – y” means”multiply x by y, making a worth equal to zero”. For”x”y”, we’ll use”t” (or”TE”) for the subtraction and we’ll use”x y y” to solve the equation. You might feel that you are not supposed to utilize”e” in addition as”I” implies”subtract” however it’s perhaps not so easy. https://www.masterpapers.com For instance, to express”2 – 3″ approaches subtract from three. So, to add the values that we utilize”t”x” (or”TE”), which might be the numbers of the worth to be included. We will use”x” to subtract the price of a person in the price of both 2 plus that will provide us precisely exactly the outcome. To multiply the worth , we certainly can certainly do this like this:”2x + y” necessarily imply”multiply two integers by y, creating a price add up to two plus one”. You may know this is really just a multiplication equation when we use”x ray” to subtract one from 2. Or, it can be”x – y” to subtract from 2. Be aware that you can publish the equation using a decimal position and parentheses. Now, let us take an example. Let’s say that we would like to multiply the value of”nine” by”5″ and we all have”x nine”y https://www.eurachem.org/images/stories/Guides/pdf/recovery.pdf twenty-five”. Then we’ll use”x – y” to subtract the price of one in the worth of two.

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# What is the term”exclamation mark” in mathematics? The answer is easy. Below are some approaches to inform as soon as an equation can be really actually a multiplication equation and also how to add and subtract using”exclamation marks”. For instance,”2x + 3″ imply”multiply two integers from about a few, making a value add up to 2 and 3″. By way of example,”2x + 3″ are multiplication by write a paper online three. In addition, we can add the values of 2 and 3 collectively. To add the worth, we’ll use”e”that I” (or”E”). With”I” means”include the worth of one to the worth of 2″. To add the worth , we can certainly do this similar to that:”x – y” means”multiply x by y, making a worth equal to zero”. For”x”y”, we’ll use”t” (or”TE”) for the subtraction and we’ll use”x y y” to solve the equation. You might feel that you are not supposed to utilize”e” in addition as”I” implies”subtract” however it’s perhaps not so easy. https://www.masterpapers.com For instance, to express”2 – 3″ approaches subtract from three. So, to add the values that we utilize”t”x” (or”TE”), which might be the numbers of the worth to be included. We will use”x” to subtract the price of a person in the price of both 2 plus that will provide us precisely exactly the outcome. To multiply the worth , we certainly can certainly do this like this:”2x + y” necessarily imply”multiply two integers by y, creating a price add up to two plus one”. You may know this is really just a multiplication equation when we use”x ray” to subtract one from 2. Or, it can be”x – y” to subtract from 2. Be aware that you can publish the equation using a decimal position and parentheses. Now, let us take an example. Let’s

say that we would like to multiply the value of”nine” by”5″ and we all have”x nine”y https://www.eurachem.org/images/stories/Guides/pdf/recovery.pdf twenty-five”. Then we’ll use”x – y” to subtract the price of one in the worth of two.

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Our curriculum is spiral Please note that our virtual Singapore Math Grade 3 curriculum is spiral and it provides for the review of the important concepts that students learned in Grade 2. Our online K-5 math curriculum is aligned with all standard Singapore Math textbook series and it includes all content that these series cover, from Kindergarten grade through 5th grade. Our Singapore Math for 3rd Grade may introduce some topics one grade level earlier or postpone coverage of some topics until grade 4. In the few instances where 3rd grade level units don’t exactly align between our curriculum and textbooks, you will still be able to easily locate the corresponding unit in our program by referring to the table of contents one grade below or above. Correspondence to 3A and 3B For your reference, the following topics in our curriculum correspond to Singapore Math practice Grade 3 in levels 3A and 3B: Singapore Math 3a Multiplication and division, multiplication tables of 2, 5, and 10, multiplication tables of 3 and 4, multiplication tables of 6, 7, 8, and 9, solving problems involving multiplication and division, and mental math computation and estimation. Singapore Math 3b Understanding fractions, time, volume, mass, representing and interpreting data, area and perimeter, and attributes of two-dimensional shapes. Student prior knowledge Prior to starting third grade Singapore Math, students should already know how to relate three-digit numbers to place value, use place-value charts to form a number and compare three-digit numbers. The initial lessons in the Singapore Math 3rd Grade are both a review and an extension of content covered in the prior grade that include mental addition of 1-digit number to a 2-digit number and counting by 2s, 5s, and 10s. 3rd Grade Singapore Math Scope and Sequence • Multiplication And Division This unit covers understanding of multiplication and division. In this unit students will extend their knowledge of making equal groups to formalize their understanding of multiplication and division. The focus of this unit is on understanding multiplication and division using equal groups, not on memorizing facts. Students will learn how the multiplication symbol to represent addition of quantities in groups. • Multiplication Tables Of 2, 5, And 10 This unit covers multiplying by 2 using skip-counting, multiplying by 2 using dot paper, multiplying by 5 using skip-counting, multiplying by 5 using dot paper, multiplying by 10 using skip-counting, dividing using related multiplication facts of 2, 5, or 10. Students will learn building multiplication tables of 2, 5, and 10 to formalize their understanding of multiplication and division for facts 2, 5, and 10. Students will learn to find their division facts by thinking of corresponding multiplication facts. • Multiplication Tables Of 3 And 4 This unit covers multiplying by 3 using skip-counting, multiplying by 3 using dot paper, multiplying by 4 using skip-counting, multiplying by 4 using dot paper, dividing using related multiplication facts of 3 or 4. Students will learn building multiplication tables of 3 and 4 to formalize their understanding of multiplication and division for facts 3 and 14. Students will learn to find their division facts by thinking of corresponding multiplication facts. • Multiplication Tables Of 6, 7, 8, And 9 This unit covers multiplication properties, multiplying by 6, multiplying by 7, multiplying by 8, multiplying by 9, dividing using related multiplication facts of 6, 7, 8, or 9. Students will learn building multiplication tables of 6, 7, 8, and 9 to formalize their understanding of multiplication and division for facts 6, 7, 8, and 9. Students will learn to find their division facts by thinking of corresponding multiplication facts. • Solving Problems Involving Multiplication And Division This unit covers solving one- and two-step word problems involving multiplication and division. Students will use a part-whole and comparison models to solve word problems involving multiplication and division. • Mental Computation And Estimation This unit covers learning mental math strategies to solve multiplication and division problems. Students will use place value to round whole numbers. • Understanding Fractions This unit covers parts and wholes, fractions and number lines, comparing unit fractions, equivalence of fractions, and comparing like fractions. Students will learn fractional notation that include the terms “numerator” and “denominator.” Students will understand that a common fraction is composed of unit fractions and they will learn to compare unit fractions. • Time This unit covers telling time, adding time, subtracting time, and time intervals. Students will review and practice to tell time to the minute, learn telling intervals of time in hours, convert units of time between hours, minutes, seconds, days and weeks. • Volume This unit covers understanding of volume, comparing volume, measuring and estimating volume, and word problems involving volume. • Mass This unit covers measuring mass in kilograms, comparing mass in kilograms, measuring mass in grams, comparing mass in grams, and word problems involving mass. • Representing And Interpreting Data This unit covers scaled picture graphs and scaled bar graphs, reading and interpreting bar graphs, and line plots. Students will learn to sort data into groups and categories and use numerical data to interpret bar graphs and line plots. • Area And Perimeter This unit covers understanding of area, measuring area using square centimeters and square inches, measuring area using square meters and square feet, area and perimeter, solving problems involving area and perimeter. Students will learn to find and measure area of figures in square units that include square centimeters, square inches, square meters and square feet. • Attributes Of Two-Dimensional Shapes This unit covers categories and attributes of shapes and partitioning shapes into equal areas. Students will understand that figures with different shapes can have the same area.

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Our curriculum is spiral Please note that our virtual Singapore Math Grade 3 curriculum is spiral and it provides for the review of the important concepts that students learned in Grade 2. Our online K-5 math curriculum is aligned with all standard Singapore Math textbook series and it includes all content that these series cover, from Kindergarten grade through 5th grade. Our Singapore Math for 3rd Grade may introduce some topics one grade level earlier or postpone coverage of some topics until grade 4. In the few instances where 3rd grade level units don’t exactly align between our curriculum and textbooks, you will still be able to easily locate the corresponding unit in our program by referring to the table of contents one grade below or above. Correspondence to 3A and 3B For your reference, the following topics in our curriculum correspond to Singapore Math practice Grade 3 in levels 3A and 3B: Singapore Math 3a Multiplication and division, multiplication tables of 2, 5, and 10, multiplication tables of 3 and 4, multiplication tables of 6, 7, 8, and 9, solving problems involving multiplication and division, and mental math computation and estimation. Singapore Math 3b Understanding fractions, time, volume, mass, representing and interpreting data, area and perimeter, and attributes of two-dimensional shapes. Student prior knowledge Prior to starting third grade Singapore Math, students should already know how to relate three-digit numbers to place value, use place-value charts to form a number and compare three-digit numbers. The initial lessons in the Singapore Math 3rd Grade are both a review and an extension of content covered in the prior grade that include mental addition of 1-digit number to a 2-digit number and counting by 2s, 5s, and 10s. 3rd Grade Singapore Math Scope and Sequence • Multiplication And Division This unit covers understanding of multiplication and division. In this unit students will extend their knowledge of making equal groups to formalize their understanding of multiplication and division. The focus of this unit is on understanding multiplication and division using equal groups, not on memorizing facts. Students will learn how the multiplication symbol to represent addition of quantities in groups. • Multiplication Tables Of 2, 5, And 10 This unit covers multiplying by 2 using skip-counting, multiplying by 2 using dot paper, multiplying by 5 using skip-counting, multiplying by 5 using dot paper, multiplying by 10 using skip-counting, dividing using related multiplication facts of 2, 5, or 10. Students will learn building multiplication tables of 2, 5, and 10 to formalize their understanding of multiplication and division for facts 2, 5, and 10. Students will learn to find their division facts by thinking of corresponding multiplication facts. • Multiplication Tables Of 3 And 4 This unit covers multiplying by 3 using skip-counting, multiplying by 3 using dot paper, multiplying by 4 using skip-counting, multiplying by 4 using dot paper, dividing using related multiplication facts of 3 or 4. Students will learn building multiplication tables of 3 and 4 to formalize their understanding of multiplication and division for facts 3 and 14. Students will learn to find their division facts by thinking of corresponding multiplication facts. • Multiplication Tables Of 6, 7, 8, And 9 This unit covers multiplication properties, multiplying by 6, multiplying by 7, multiplying by 8, multiplying by 9, dividing using related multiplication facts of 6, 7, 8, or 9. Students will learn building multiplication tables of 6, 7, 8, and 9 to formalize their understanding of multiplication and division for facts 6, 7, 8, and 9. Students will learn to find their division facts by thinking of corresponding multiplication facts. • Solving Problems Involving Multiplication And Division This unit covers solving one- and two-step word problems involving multiplication and division. Students will use a part-whole and comparison models to solve word problems involving multiplication and division. • Mental Computation And Estimation This unit covers learning mental math strategies to solve multiplication and division problems. Students will use place value to round whole numbers. • Understanding Fractions This unit covers parts and wholes, fractions and number lines, comparing unit fractions, equivalence of fractions, and comparing like fractions. Students will learn fractional notation that include the terms “numerator” and “denominator.” Students will understand that a common fraction is composed of unit fractions and they will learn to compare unit fractions. • Time This unit covers telling time, adding time, subtracting time, and time intervals. Students will review and practice to tell time to the minute, learn telling intervals of time in hours, convert units of time between hours, minutes, seconds, days and weeks. • Volume This unit covers understanding of volume, comparing volume, measuring and estimating volume, and word problems involving volume. • Mass This unit covers measuring mass in kilograms, comparing mass in kilograms, measuring mass in grams, comparing mass in grams, and word problems involving mass. • Representing And Interpreting Data This unit covers scaled picture graphs and scaled bar graphs, reading and interpreting bar graphs, and line plots. Students will learn to sort data into groups and categories and use numerical data to interpret bar

graphs and line plots. • Area And Perimeter This unit covers understanding of area, measuring area using square centimeters and square inches, measuring area using square meters and square feet, area and perimeter, solving problems involving area and perimeter. Students will learn to find and measure area of figures in square units that include square centimeters, square inches, square meters and square feet. • Attributes Of Two-Dimensional Shapes This unit covers categories and attributes of shapes and partitioning shapes into equal areas. Students will understand that figures with different shapes can have the same area.

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# Fractions Bundle "Twist" 12 Worksheets Subject Resource Type Product Rating File Type Word Document File 181 KB|20 pages Share Product Description FRACTIONS BUNDLE "TWIST" 12 WORKSHEETS You receive 12 FULL worksheets all on the 4 OPERATIONS of FRACTIONS including WORD PROBLEMS for EACH operation! The "Twist"? - ALL these examples are written out IN WORDS to make your students THINK a little more. They must first READ the fractions correctly, then WRITE them correctly and then SOLVE them correctly (hopefully!) to lowest terms; Worksheet 1 - ADDITION OF FRACTIONS; all types of addition of fraction examples with like and unlike denominators; 15 examples plus a BONUS at the end; Worksheet 2 - SUBTRACTION OF FRACTIONS; all types of subtraction of fractions examples with and without borrowing; 18 examples plus 4 EXTRA bonus questions; Worksheet 3 - MULTIPLICATION OF FRACTIONS; these 16 examples involve cancelling when possible plus 2 EXTRA fraction questions; Worksheet 4 - DIVISION OF FRACTIONS; these 15 examples also involve cancelling when possible PLUS a GEOMETRY BONUS question; Worksheets 5 and 6 - MULTIPLICATION AND DIVISION OF FRACTIONS COMBINED; students must multiply or divide these examples using cancelling; 21 examples plus many BONUS fraction questions included; Worksheets 7 and 8 - ALL OPERATIONS PLUS!; these TWO worksheets combine ALL 4 Operations; 16 examples, 4 for each operation included; PLUS 8 BONUS questions on fractions; Worksheets 9 and 10 - WORD PROBLEMS for ADDITION & SUBTRACTION OF FRACTIONS; NO FLUFF! Just 25 word problems all on addition and/or subtraction of fractions; students must solve not only for the correct operation, but to lowest terms also; Worksheets 11 and 12 - WORD PROBLEMS for MULTIPLICATION & DIVISION of FRACTIONS; NO FLUFF AGAIN! 25 word problems on multiplication and/or division of fractions; read and solve; A HUGE packet of worksheets involving ALL OPERATIONS of fractions! The thumbnails only give you 4 items to look at. The download preview gives you more. Take a look! * ALSO AVAILABLE FOR YOU OR A COLLEAGUE! - CLICK ANY LINK YOU WANT: - FRACTIONS REVIEW AND REINFORCEMENT - 10 FULL worksheets on various aspects of fractions to be used as great review or part of your unit. A LOT of good work here for your students. This link will describe all 10 pages in detail. - FRACTIONS POWERPOINT FUN QUIZ - 60 slides in all! This Powerpoint program starts over again EACH time students get even one answer wrong! Challenging, fun and SELF-CORRECTING! Great activity with great graphics will keep students engaged. NOT for the beginner! Did I say it was self-correcting? - NUMBER LINE POWERPOINT LESSON - 50 slide Powerpoint LESSON dealing with the number line and positive and negative numbers. Designed as a whole-class lesson getting students actively involved in the answers. 5 different sections including less than and greater than. Great graphics and animation will hold their attention. Different and effective lesson! The Number Line! * * * GREAT TEACHING TO YOU! Total Pages 20 pages Included Teaching Duration N/A Report this Resource \$5.00

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# Fractions Bundle "Twist" 12 Worksheets Subject Resource Type Product Rating File Type Word Document File 181 KB|20 pages Share Product Description FRACTIONS BUNDLE "TWIST" 12 WORKSHEETS You receive 12 FULL worksheets all on the 4 OPERATIONS of FRACTIONS including WORD PROBLEMS for EACH operation! The "Twist"? - ALL these examples are written out IN WORDS to make your students THINK a little more. They must first READ the fractions correctly, then WRITE them correctly and then SOLVE them correctly (hopefully!) to lowest terms; Worksheet 1 - ADDITION OF FRACTIONS; all types of addition of fraction examples with like and unlike denominators; 15 examples plus a BONUS at the end; Worksheet 2 - SUBTRACTION OF FRACTIONS; all types of subtraction of fractions examples with and without borrowing; 18 examples plus 4 EXTRA bonus questions; Worksheet 3 - MULTIPLICATION OF FRACTIONS; these 16 examples involve cancelling when possible plus 2 EXTRA fraction questions; Worksheet 4 - DIVISION OF FRACTIONS; these 15 examples also involve cancelling when possible PLUS a GEOMETRY BONUS question; Worksheets 5 and 6 - MULTIPLICATION AND DIVISION OF FRACTIONS COMBINED; students must multiply or divide these examples using cancelling; 21 examples plus many BONUS fraction questions included; Worksheets 7 and 8 - ALL OPERATIONS PLUS!; these TWO worksheets combine ALL 4 Operations; 16 examples, 4 for each operation included; PLUS 8 BONUS questions on fractions; Worksheets 9 and 10 - WORD PROBLEMS for ADDITION & SUBTRACTION OF FRACTIONS; NO FLUFF! Just 25 word problems all on addition and/or subtraction of fractions; students must solve not only for the correct operation, but to lowest terms also; Worksheets 11 and 12 - WORD PROBLEMS for MULTIPLICATION & DIVISION of FRACTIONS; NO FLUFF AGAIN! 25 word problems on multiplication and/or division of fractions; read and solve; A HUGE packet of worksheets involving ALL OPERATIONS of fractions! The thumbnails only give you 4 items to look at. The download preview gives you more. Take a look! * ALSO AVAILABLE FOR YOU OR A COLLEAGUE! - CLICK ANY LINK YOU WANT: - FRACTIONS REVIEW AND REINFORCEMENT - 10 FULL worksheets on various aspects of fractions to be used as great review or part of your unit. A LOT of good work here for your students. This link will describe all 10 pages in detail. - FRACTIONS POWERPOINT FUN QUIZ - 60 slides in all! This Powerpoint program starts over again EACH time students get even one answer wrong! Challenging, fun and SELF-CORRECTING! Great activity with great graphics will keep students engaged. NOT for the beginner! Did I say it was self-correcting? - NUMBER LINE POWERPOINT LESSON - 50 slide Powerpoint LESSON dealing with the number line and positive and negative numbers. Designed as a whole-class

lesson getting students actively involved in the answers. 5 different sections including less than and greater than. Great graphics and animation will hold their attention. Different and effective lesson! The Number Line! * * * GREAT TEACHING TO YOU! Total Pages 20 pages Included Teaching Duration N/A Report this Resource \$5.00

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# Height of the room Given the floor area of a room as 24 feet by 48 feet and space diagonal of a room as 56 feet. Can you find the height of the room? Correct result: c =  16 ft #### Solution: We would be pleased if you find an error in the word problem, spelling mistakes, or inaccuracies and send it to us. Thank you! Tips to related online calculators Pythagorean theorem is the base for the right triangle calculator. #### You need to know the following knowledge to solve this word math problem: We encourage you to watch this tutorial video on this math problem: ## Next similar math problems: • Diagonal Determine the dimensions of the cuboid, if diagonal long 53 dm has an angle with one edge 42° and with another edge 64°. • Cuboidal room Length of cuboidal room is 2m breadth of cuboidal room is 3m and height is 6m find the length of the longest rod that can be fitted in the room • Ratio of edges The dimensions of the cuboid are in a ratio 3: 1: 2. The body diagonal has a length of 28 cm. Find the volume of a cuboid. • Four sided prism Calculate the volume and surface area of a regular quadrangular prism whose height is 28.6cm and the body diagonal forms a 50-degree angle with the base plane. • Cuboid diagonals The cuboid has dimensions of 15, 20 and 40 cm. Calculate its volume and surface, the length of the body diagonal and the lengths of all three wall diagonals. • Space diagonal angles Calculate the angle between the body diagonal and the side edge c of the block with dimensions: a = 28cm, b = 45cm and c = 73cm. Then, find the angle between the body diagonal and the plane of the base ABCD. • The room The room has a cuboid shape with dimensions: length 50m and width 60dm and height 300cm. Calculate how much this room will cost paint (floor is not painted) if the window and door area is 15% of the total area and 1m2 cost 15 euro. • Jared's room painting Jared wants to paint his room. The dimensions of the room are 12 feet by 15 feet, and the walls are 9 feet high. There are two windows that measure 6 feet by 5 feet each. There are two doors, whose dimensions are 30 inches by 6 feet each. If a gallon of p • Solid cuboid A solid cuboid has a volume of 40 cm3. The cuboid has a total surface area of 100 cm squared. One edge of the cuboid has a length of 2 cm. Find the length of a diagonal of the cuboid. Give your answer correct to 3 sig. Fig. • Find diagonal Find the length of the diagonal of a cuboid with length=20m width=25m height=150m

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# Height of the room Given the floor area of a room as 24 feet by 48 feet and space diagonal of a room as 56 feet. Can you find the height of the room? Correct result: c = 16 ft #### Solution: We would be pleased if you find an error in the word problem, spelling mistakes, or inaccuracies and send it to us. Thank you! Tips to related online calculators Pythagorean theorem is the base for the right triangle calculator. #### You need to know the following knowledge to solve this word math problem: We encourage you to watch this tutorial video on this math problem: ## Next similar math problems: • Diagonal Determine the dimensions of the cuboid, if diagonal long 53 dm has an angle with one edge 42° and with another edge 64°. • Cuboidal room Length of cuboidal room is 2m breadth of cuboidal room is 3m and height is 6m find the length of the longest rod that can be fitted in the room • Ratio of edges The dimensions of the cuboid are in a ratio 3: 1: 2. The body diagonal has a length of 28 cm. Find the volume of a cuboid. • Four sided prism Calculate the volume and surface area of a regular quadrangular prism whose height is 28.6cm and the body diagonal forms a 50-degree angle with the base plane. • Cuboid diagonals The cuboid has dimensions of 15, 20 and 40 cm. Calculate its volume and surface, the length of the body diagonal and the lengths of all three wall diagonals. • Space diagonal angles Calculate the angle between the body diagonal and the side edge c of the block with dimensions: a = 28cm, b = 45cm and c = 73cm. Then, find the angle between the body diagonal and the plane of the base ABCD. • The room The room has a cuboid shape with dimensions: length 50m and width 60dm and height 300cm. Calculate how much this room will cost paint (floor is not painted) if the window and door area is 15% of the total area and 1m2 cost 15 euro. • Jared's room painting Jared wants to paint his room. The dimensions of the room are 12 feet by 15 feet, and the walls are 9 feet high. There are two windows that measure 6 feet by 5 feet each. There are two doors, whose dimensions are 30 inches by 6 feet each. If a gallon of p • Solid cuboid A solid cuboid has a volume of 40 cm3. The cuboid has a total surface

area of 100 cm squared. One edge of the cuboid has a length of 2 cm. Find the length of a diagonal of the cuboid. Give your answer correct to 3 sig. Fig. • Find diagonal Find the length of the diagonal of a cuboid with length=20m width=25m height=150m

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math solve by using subsitution: 4x+y=2 3y+2x=-1 1. 0 2. 0 Similar Questions 1. algebra solve by subsitution 2x+y=9 8x+4y=36 2. alg. 2 solve by subsitution 2x+3y=10 x+6y=32 3. math solve by using subsitution: 4x+y=2 3y+2x=-1 4. math solve by using subsitution: 4x+y=2 3y+2x=-1 5. math solve by using subsitution: 4x+y=2 3y+2x=-1 6. math solve by using subsitution: 4x+y=2 3y+2x=-1 7. math use subsitution system to solve 4x+5y=21 Y=3x-11 8. agebra 2 honors subsitution with 3 equations (solve for x,y,z) x+y-2z=5 -x-y+z=2 -x+y+32=4 9. algebra solve by subsitution 6x + 5y = 27 x = 17 -8y my answer was (3,-7) 10. math -w-z=-2 and 4w+5z=16 im trying to solve this equation by using subsitution More Similar Questions

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math solve by using subsitution: 4x+y=2 3y+2x=-1 1. 0 2. 0 Similar Questions 1. algebra solve by subsitution 2x+y=9 8x+4y=36 2. alg. 2 solve by subsitution 2x+3y=10 x+6y=32 3. math solve by using subsitution: 4x+y=2 3y+2x=-1 4. math solve by using subsitution: 4x+y=2 3y+2x=-1 5. math solve by using subsitution: 4x+y=2 3y+2x=-1 6. math solve by using subsitution: 4x+y=2 3y+2x=-1 7. math use subsitution system to solve 4x+5y=21 Y=3x-11 8. agebra 2 honors subsitution with 3 equations (solve for x,y,z) x+y-2z=5 -x-y+z=2 -x+y+32=4 9. algebra solve by subsitution 6x + 5y = 27 x = 17 -8y my answer was (3,-7) 10. math -w-z=-2 and 4w+5z=16

im trying to solve this equation by using subsitution More Similar Questions

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[OK] Privacy Policy  -  Terms & Conditions  -  See DetailsWe use cookies to personalise content and ads, to provide social media features and to analyse our traffic. We also share information about your use of our site with our social media, advertising and analytics partners who may combine it with other information you've provided to them or they've collected from your use of their services. Puzzle Details In farmer Brown's hay loft there are a number of animals, in particular crows, mice and cockroaches. Being bored one day, I decided to count the animals and found there were exactly 150 feet and 50 heads in total, and there were twice as many cockroaches as mice. How many of each animal were there? [Ref: ZVYU] © Kevin Stone Answer: 10 cockroaches, 5 mice and 35 birds. Cockroaches have 6 feet, mice have 4 and birds have 2. For every mouse there are two cockroaches, so every mouse is worth 3 heads and 16 feet (4 + 6 + 6). We can now write down an expression for the heads and the feet (calling birds B and mice M): B +  3M = 50       (1) Feet gives us: 2B + 16M = 150      (2) If we double (1) we get: 2B +  6M = 100      (3) We can now do (2) - (3) to give: 10M = 50 M = 5 So we have 5 mice (and 10 cockroaches). We can use M = 5 in (1) to give: B + 3 x 5 = 50 B + 15 = 50 B = 35 So, C = 10, M = 5 and B = 35. Checking that 10 + 5 + 35 = 50, and 10 x 6 + 5 x 4 + 35 x 2 = 150. QED. Our Favourite Illusions Shadow Illusion Are the squares A and B the same colour? Spinning Dancer Which way is the dancer spinning? Impossible Waterfall? Is the water flowing uphill in this impossible Escher type waterfall? The Butterfly A colourful butterfly? Duck Or Rabbit? Is this a duck or a rabbit? Hidden Faces Can you find his three daughters as well? Blind Spot An amazing demonstration of your blind spot. Impossible Prongs? Impossible prongs? What Am I? Can you tell what this is a picture of? Who Turned To? Who is missing? Same Eyes? Are her eyes the same colour? Parallel Cafe Wall Lines? Are the horizontal lines parallel?

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[OK] Privacy Policy - Terms & Conditions - See DetailsWe use cookies to personalise content and ads, to provide social media features and to analyse our traffic. We also share information about your use of our site with our social media, advertising and analytics partners who may combine it with other information you've provided to them or they've collected from your use of their services. Puzzle Details In farmer Brown's hay loft there are a number of animals, in particular crows, mice and cockroaches. Being bored one day, I decided to count the animals and found there were exactly 150 feet and 50 heads in total, and there were twice as many cockroaches as mice. How many of each animal were there? [Ref: ZVYU] © Kevin Stone Answer: 10 cockroaches, 5 mice and 35 birds. Cockroaches have 6 feet, mice have 4 and birds have 2. For every mouse there are two cockroaches, so every mouse is worth 3 heads and 16 feet (4 + 6 + 6). We can now write down an expression for the heads and the feet (calling birds B and mice M): B + 3M = 50 (1) Feet gives us: 2B + 16M = 150 (2) If we double (1) we get: 2B + 6M = 100 (3) We can now do (2) - (3) to give: 10M = 50 M = 5 So we have 5 mice (and 10 cockroaches). We can use M = 5 in (1) to give: B + 3 x 5 = 50 B + 15 = 50 B = 35 So, C = 10, M = 5 and B = 35. Checking that 10 + 5 + 35 = 50, and 10 x 6 + 5 x 4 + 35 x 2 = 150. QED. Our Favourite Illusions Shadow Illusion Are the squares A and B the same colour? Spinning Dancer Which way is the dancer spinning? Impossible Waterfall? Is the water flowing uphill in this impossible Escher type waterfall? The Butterfly A colourful butterfly? Duck Or Rabbit? Is this a duck or a rabbit? Hidden Faces Can you find his three daughters as well? Blind Spot An amazing demonstration of your

blind spot. Impossible Prongs? Impossible prongs? What Am I? Can you tell what this is a picture of? Who Turned To? Who is missing? Same Eyes? Are her eyes the same colour? Parallel Cafe Wall Lines? Are the horizontal lines parallel?

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# You are out shopping with \$N, and you find an item whose price has a random value between \$0 and \$N. 27 views You are out shopping one day with \$N, and you find an item whose price has a random value between \$0 and \$N. You buy as many of these items as you can with your \$N. What is the expected value of the money you have left over? (Assume that \$N is large compared to a penny, so that the distribution of prices is essentially continuous.) posted Jun 9 ## 1 Solution expected value of the item=average of all the random numbers between 0 and N which is N/2. Hence he can buy 2 of these items and he will be left with nothing. solution Jun 19 Similar Puzzles A shipment of butterflies was mixed up by the dock workers, and they could not find who bought which species, where it was from, and what was the price. All the workers know is that Alejandro, Faye, Yvette, Sophie, and Zachary could have each bought butterflies that cost \$60, \$75, \$90, \$105, or \$120. Each could have bought the Clearwing, the Emperor, the Grayling, the Swallowtail, or the Torturix butterflies. Each butterfly could have lived in Australia, Jordan, Luxembourg, Panama, or Qatar. It is up to you to find out who bought which butterfly, what was the price, and where did it come from with the provided clues: 1. Neither the butterfly from Luxembourg nor the one from Australia sold for \$90. 2. The Emperor butterfly cost \$30 more than the Torturix butterfly. 3. Zachary's purchase was \$75. 4. The butterfly from Australia cost less than the one from Luxembourg. 5. Alejandro's purchase was from Luxembourg. 6. Of Yvette's purchase and the purchase for \$60, one was from Qatar and the other was the Torturix. 7. The butterfly that sold for \$120 was not from Panama. 8. The insect from Australia was not the Torturix. 9. Faye bought the Torturix. 10. Sophie did not buy the Grayling. 11. Of the Emperor and the insect worth \$105, one was won by Yvette and the other was from Luxembourg. 12. The insect that sold for \$105 was the Swallowtail. A family, made up of 2 parents with children, has an average age of 20. If you exclude one parent, who is 40, the average age drops to 15. How many kids are in the family? There is an equilateral triangle and three bugs are sitting on the three corners of the triangle. Each of the bugs picks up a random direction and starts walking along the edge of the equilateral triangle. What is the probability that none of the bugs crash into each other? +1 vote If A = 1 B = 2 C = 3 ... ... Z = 26. Based on above rule, you need to find an eleven letter word whose letter sum is equal to 52.

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# You are out shopping with \$N, and you find an item whose price has a random value between \$0 and \$N. 27 views You are out shopping one day with \$N, and you find an item whose price has a random value between \$0 and \$N. You buy as many of these items as you can with your \$N. What is the expected value of the money you have left over? (Assume that \$N is large compared to a penny, so that the distribution of prices is essentially continuous.) posted Jun 9 ## 1 Solution expected value of the item=average of all the random numbers between 0 and N which is N/2. Hence he can buy 2 of these items and he will be left with nothing. solution Jun 19 Similar Puzzles A shipment of butterflies was mixed up by the dock workers, and they could not find who bought which species, where it was from, and what was the price. All the workers know is that Alejandro, Faye, Yvette, Sophie, and Zachary could have each bought butterflies that cost \$60, \$75, \$90, \$105, or \$120. Each could have bought the Clearwing, the Emperor, the Grayling, the Swallowtail, or the Torturix butterflies. Each butterfly could have lived in Australia, Jordan, Luxembourg, Panama, or Qatar. It is up to you to find out who bought which butterfly, what was the price, and where did it come from with the provided clues: 1. Neither the butterfly from Luxembourg nor the one from Australia sold for \$90. 2. The Emperor butterfly cost \$30 more than the Torturix butterfly. 3. Zachary's purchase was \$75. 4. The butterfly from Australia cost less than the one from Luxembourg. 5. Alejandro's purchase was from Luxembourg. 6. Of Yvette's purchase and the purchase for \$60, one was from Qatar and the other was the Torturix. 7. The butterfly that sold for \$120 was not from Panama. 8. The insect from Australia was not the Torturix. 9. Faye bought the Torturix. 10. Sophie did not buy the Grayling. 11. Of the Emperor and the insect worth \$105, one was won by Yvette and the other was from Luxembourg. 12. The insect that sold for \$105 was the Swallowtail. A family, made up of 2 parents with children, has an average age of 20. If you exclude one parent, who is 40, the average age drops to 15. How many kids are in the family? There is an equilateral triangle and three bugs are sitting on the three corners of the triangle. Each of the bugs picks up a random direction and starts walking along the edge of the equilateral

triangle. What is the probability that none of the bugs crash into each other? +1 vote If A = 1 B = 2 C = 3 ... ... Z = 26. Based on above rule, you need to find an eleven letter word whose letter sum is equal to 52.

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# Problem: The rate constant of a reaction at 32°C is 0.055/s. If the frequency factor is 1.2 x 1013/s, what is the activation barrier? ###### FREE Expert Solution We’re being asked to determine the activation barrier (activation energy, Ea) of a reaction given the rate constant and frequency factor. We can use the two-point form of the Arrhenius Equation to calculate activation energy: where: k = rate constant Ea = activation energy (in J/mol) R = gas constant (8.314 J/mol • K) T = temperature (in K) A = Arrhenius constant or frequency factor 94% (270 ratings) ###### Problem Details The rate constant of a reaction at 32°C is 0.055/s. If the frequency factor is 1.2 x 1013/s, what is the activation barrier?

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# Problem: The rate constant of a reaction at 32°C is 0.055/s. If the frequency factor is 1.2 x 1013/s, what is the activation barrier? ###### FREE Expert Solution We’re being asked to determine the activation barrier (activation energy, Ea) of a reaction given the rate constant and frequency factor. We can use the two-point form of the Arrhenius Equation to calculate activation energy: where: k = rate constant Ea = activation energy (in J/mol) R = gas constant (8.314 J/mol • K) T = temperature (in K) A = Arrhenius constant or frequency factor 94% (270 ratings) ###### Problem Details The rate constant of a reaction at 32°C is 0.055/s.

If the frequency factor is 1.2 x 1013/s, what is the activation barrier?

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# HOW MUCH WILL I SPEND ON GAS? Save this PDF as: Size: px Start display at page: ## Transcription 1 HOW MUCH WILL I SPEND ON GAS? Outcome (lesson objective) The students will use the current and future price of gasoline to construct T-charts, write algebraic equations, and plot the equations on a graph. Student/Class Goal Students will determine their gasoline cost for a month s time. They will use this information to calculate yearly gas cost. Time Frame Two 1 ½ hour classes One 3 hour class Standard Use Math to Solve Problems and Communicate NRS EFL 3-6 COPS Understand, interpret, and work with pictures, numbers, and symbolic information. Apply knowledge of mathematical concepts and procedures to figure out how to answer a question, solve a problem, make a prediction, or carry out a task that has a mathematical dimension. Define and select data to be used in solving the problem. Determine the degree of precision required by the situation. Solve problem using appropriate quantitative procedures and verify that the results are reasonable. Communicate results using a variety of mathematical representations, including graphs, charts, tables, and algebraic models. Materials Graph paper 40x30 paper is included at the end of the lesson (Worksheet 4) Colored pencils (if available) Calculators How Do I Find My Gas Mileage? Handout T-Chart Illustration How to Graph a Linear Equation in 5 Quick Steps Student Resource Activity Addresses Components of Performance Students will work with basic operations and patterns to complete a T-chart. Students will construct a graph and plot points on it. Students will use problem solving to determine how to figure out their monthly gas cost. Students will determine appropriate intervals on the x and y axes of the graph. Students will round the number when necessary to plot the points on the graph. Students will recognize if a data set is not reasonable by observing the other data sets and observing the points on the graph (is it on the line?) Students will communicate the results of the data by constructing a graph, T-chart, algebraic formula and in writing. Learner Prior Knowledge Basic understanding of gas prices and what is meant by miles per gallon (mpg), found at lesson, Pumped Up Gas Prices. Instructional Activities Step 1 - Discuss with students the current gasoline price and what they expect to happen to gas prices in the future. Ask students about how much they are spending on gas each month and how many miles the car they are driving gets per gallon? Students may need to research this information by actually calculating their mileage on the handout How Do I Find My Gas Mileage? or by visiting Cars and mpg This web site lists the in-city and highway mileage for 1985 or newer vehicles. If using this site, students will need to decide what type of driving they do during a month (city or highway) and estimate their car s mileage. Step 2 - Using \$4 per gallon as the price of gas, construct, with the class, a T-chart showing the relationship between the number of gallons purchased(x) and the total cost of the gas(y). See T-Chart Illustration for an example. As a class, look at the T-chart that was constructed. Give the students time to discover the relationship between the number of gallons purchased(x) and the total price of the gas(y). Express this relationship verbally (The number of gallons purchased times \$4 will equal the total cost.). Discuss how this relationship can be written as an equation as a function of x? (4x=y) Step 3 Next, with the students, graph the relationship/equation the class discovered during Step 2. The How to Graph a Linear Equation in 5 Quick Steps, a student resource from their math journals, provides basic guidelines for graphing equations. Help the students determine appropriate intervals and labels for the x and y axes and a title for the graph. Plot several points together and then let the students plot the remainder of the points. Draw a line passing through the points and label the line with the equation written in Step 2. Step 4 - Discuss with the students how many miles they drive each month. Share with the class that many people drive about 1,000 4 Gallons purchased (x) T-CHART ILLUSTRATION Total gas cost (y) 1 \$ \$ x30 Grid 5 How to Graph a Linear Equation in 5 Quick Steps Step 1 Construct a T-chart of Values Using your equation, construct a T-chart of values if one has not been done already. Substitute some simple numbers into the equation for x or y. If x=1, what is y? If x=10, what is y? If y=0, what is x? Each pair of values in your T-chart will become a point on the graph. (See illustration 1 for an example of a T-chart) Step 2 Decide on the interval for each axis Before starting the graph, look at the T-chart to determine the highest value for y found on the chart. Look at the values needed for x. Using graph paper, count the number of lines on the x and y axes. Use these numbers to determine the intervals on each axis. (If you use the graph paper at the end of this lesson there are 30 spaces on the x axis and 40 spaces on the y axis.) If the largest total cost/y value that needs to be graphed is \$80 and there are 40 lines on the y axis, let each line on the y axis represent \$2. The number of gallons of gas/x value that goes with \$80 is 20. There are 30 lines, so to make it simple one line will equal one gallon. Be sure the students realize they do not need to put a number next to every line. For example, the x might be labeled on every 5 th line (five gallons) and the y axis might also be labeled on every 5 th line (or \$10). This is a good step to do in pencil. That way if the interval you selected did not work out, the numbers can be erased any you can start over. Step 3 Label each Axis Decide what labels need to be added to the x and y axis. What do the numbers on the x-axis represent? What do the numbers on the y-axis represent? Usually the labels will match the descriptions/labels of x and y on the T-chart. (Note: When graphing equations involving elapsed time, time is traditionally represented by x) Step 4 Plot the points Using each pair of points from the T-chart, plot the points on the graph. Every point does not need to be plotted. Just be sure you have at least 3. Using a ruler, draw a line through the points you have plotted. Write your equation next to the line. Step 5 Give the graph a title Decide on a title for the graph. Make sure it accurately represents what is being shown on the graph. Does it explain the relationship between x and y? How to Graph a Linear Equation in 5 Quick Steps Student Resource ### Solving Systems of Linear Equations Substitutions Solving Systems of Linear Equations Substitutions Outcome (lesson objective) Students will accurately solve a system of equations algebraically using substitution. Student/Class Goal Students thinking ### Solving Systems of Linear Equations Elimination (Addition) Solving Systems of Linear Equations Elimination (Addition) Outcome (lesson objective) Students will accurately solve systems of equations using elimination/addition method. Student/Class Goal Students ### Solving Systems of Linear Equations Substitutions Solving Systems of Linear Equations Substitutions Outcome (learning objective) Students will accurately solve a system of equations algebraically using substitution. Student/Class Goal Students thinking ### Converting Units of Measure Measurement Converting Units of Measure Measurement Outcome (lesson objective) Given a unit of measurement, students will be able to convert it to other units of measurement and will be able to use it to solve contextual ### Solving Systems of Linear Equations Graphing Solving Systems of Linear Equations Graphing Outcome (learning objective) Students will accurately solve a system of equations by graphing. Student/Class Goal Students thinking about continuing their academic ### Solving Systems of Linear Equations Putting it All Together Solving Systems of Linear Equations Putting it All Together Outcome (lesson objective) Students will determine the best method to use when solving systems of equation as they solve problems using graphing, ### TEMPERATURE BAR GRAPH TEMPERATURE BAR GRAPH Outcome (lesson objective) Students will figure mean, median and mode using weather, temperature data, create a bar graph charting one city s high and low temperatures, and formulate ### Lesson 4: Solving and Graphing Linear Equations Lesson 4: Solving and Graphing Linear Equations Selected Content Standards Benchmarks Addressed: A-2-M Modeling and developing methods for solving equations and inequalities (e.g., using charts, graphs, ### Solving Systems of Linear Equations Putting it All Together Solving Systems of Linear Equations Putting it All Together Student/Class Goal Students thinking about continuing their academic studies in a post-secondary institution will need to know and be able to ### Lesson 2: Constructing Line Graphs and Bar Graphs Lesson 2: Constructing Line Graphs and Bar Graphs Selected Content Standards Benchmarks Assessed: D.1 Designing and conducting statistical experiments that involve the collection, representation, and analysis ### Tennessee Department of Education. Task: Sally s Car Loan Tennessee Department of Education Task: Sally s Car Loan Sally bought a new car. Her total cost including all fees and taxes was \$15,. She made a down payment of \$43. She financed the remaining amount ### Basic Understandings Activity: TEKS: Exploring Transformations Basic understandings. (5) Tools for geometric thinking. Techniques for working with spatial figures and their properties are essential to understanding underlying ### Acquisition Lesson Plan for the Concept, Topic or Skill---Not for the Day Acquisition Lesson Plan Concept: Linear Systems Author Name(s): High-School Delaware Math Cadre Committee Grade: Ninth Grade Time Frame: Two 45 minute periods Pre-requisite(s): Write algebraic expressions ### Linear Equations. 5- Day Lesson Plan Unit: Linear Equations Grade Level: Grade 9 Time Span: 50 minute class periods By: Richard Weber Linear Equations 5- Day Lesson Plan Unit: Linear Equations Grade Level: Grade 9 Time Span: 50 minute class periods By: Richard Weber Tools: Geometer s Sketchpad Software Overhead projector with TI- 83 ### Lines, Lines, Lines!!! Slope-Intercept Form ~ Lesson Plan Lines, Lines, Lines!!! Slope-Intercept Form ~ Lesson Plan I. Topic: Slope-Intercept Form II. III. Goals and Objectives: A. The student will write an equation of a line given information about its graph. ### Teacher: Maple So School: Herron High School. Comparing the Usage Cost of Electric Vehicles Versus Internal Combustion Vehicles Teacher: Maple So School: Herron High School Name of Lesson: Comparing the Usage Cost of Electric Vehicles Versus Internal Combustion Vehicles Subject/ Course: Mathematics, Algebra I Grade Level: 9 th ### Have pairs share with the class. Students can work in pairs to categorize their food according to the food groups in the food pyramid. FOOD PYRAMID MENU Outcome (lesson objective) Students will read charts about nutrition, exercise, and caloric intake, practice vocabulary strategies and integrate that information with their own experience ### Solving Systems of Equations Introduction Solving Systems of Equations Introduction Outcome (learning objective) Students will write simple systems of equations and become familiar with systems of equations vocabulary terms. Student/Class Goal ### Unit 1 Equations, Inequalities, Functions Unit 1 Equations, Inequalities, Functions Algebra 2, Pages 1-100 Overview: This unit models real-world situations by using one- and two-variable linear equations. This unit will further expand upon pervious ### Activity- The Energy Choices Game Activity- The Energy Choices Game Purpose Energy is a critical resource that is used in all aspects of our daily lives. The world s supply of nonrenewable resources is limited and our continued use of ### What does the number m in y = mx + b measure? To find out, suppose (x 1, y 1 ) and (x 2, y 2 ) are two points on the graph of y = mx + b. PRIMARY CONTENT MODULE Algebra - Linear Equations & Inequalities T-37/H-37 What does the number m in y = mx + b measure? To find out, suppose (x 1, y 1 ) and (x 2, y 2 ) are two points on the graph of PAY CHECK ADVANCE LOANS Outcome (lesson objective) Learners will use their knowledge of interest to compare data from various pay check advance companies and evaluate the positives and negatives of pay ### Overview. Observations. Activities. Chapter 3: Linear Functions Linear Functions: Slope-Intercept Form Name Date Linear Functions: Slope-Intercept Form Student Worksheet Overview The Overview introduces the topics covered in Observations and Activities. Scroll through the Overview using " (! to review, ### Open-Ended Problem-Solving Projections MATHEMATICS Open-Ended Problem-Solving Projections Organized by TEKS Categories TEKSING TOWARD STAAR 2014 GRADE 7 PROJECTION MASTERS for PROBLEM-SOLVING OVERVIEW The Projection Masters for Problem-Solving ### THE LEAST SQUARES LINE (other names Best-Fit Line or Regression Line ) Sales THE LEAST SQUARES LINE (other names Best-Fit Line or Regression Line ) 1 Problem: A sales manager noticed that the annual sales of his employees increase with years of experience. To estimate the ### Drive It Green. Time Frame: 3 class periods Drive It Green Lesson Overview: Millions of people around the globe are moving every day, creating a myriad of energy challenges and opportunities for new innovation. In this three-part lesson, students ### Graphing Linear Equations in Two Variables Math 123 Section 3.2 - Graphing Linear Equations Using Intercepts - Page 1 Graphing Linear Equations in Two Variables I. Graphing Lines A. The graph of a line is just the set of solution points of the ### SPIRIT 2.0 Lesson: A Point Of Intersection SPIRIT 2.0 Lesson: A Point Of Intersection ================================Lesson Header============================= Lesson Title: A Point of Intersection Draft Date: 6/17/08 1st Author (Writer): Jenn ### Title: Patterns and Puzzles X Stackable Cert. Documentation Technology Study / Life skills EL-Civics Career Pathways Police Paramedic Fire Rescue Medical Asst. EKG / Cardio Phlebotomy Practical Nursing Healthcare Admin Pharmacy Tech ### Objectives. Day Day 1 Day 2 Day 3 Day 4 Day 5 Day 6 Day 7. Inventory 2,782 2,525 2,303 2,109 1,955 1,788 1,570 Activity 4 Objectives Use the CellSheet App to predict future trends based on average daily use Use a linear regression model to predict future trends Introduction Have you ever gone to buy a new CD or ### AP Physics 1 Summer Assignment AP Physics 1 Summer Assignment AP Physics 1 Summer Assignment Welcome to AP Physics 1. This course and the AP exam will be challenging. AP classes are taught as college courses not just college-level courses, ### Graphs of Proportional Relationships Graphs of Proportional Relationships Student Probe Susan runs three laps at the track in 12 minutes. A graph of this proportional relationship is shown below. Explain the meaning of points A (0,0), B (1,), ### Solving Systems of Linear Equations Putting it All Together Solving Systems of Linear Equations Putting it All Together Outcome (learning objective) Students will determine the best method to use when solving systems of equation as they solve problems using graphing, ### Algebra EOC Practice Test #2 Class: Date: Algebra EOC Practice Test #2 Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Which of the following lines is perpendicular to the line y = ### #1 Automobile Problems (answer key at end) #1 Automobile Problems (answer key at end) 1. A car is traveling at 60 mph and is tailgating another car at distance of only 30 ft. If the reaction time of the tailgater is 0.5 seconds (time between seeing ### Tennessee Department of Education Tennessee Department of Education Task: Pool Patio Problem Algebra I A hotel is remodeling their grounds and plans to improve the area around a 20 foot by 40 foot rectangular pool. The owner wants to use ### In A Heartbeat (Algebra) The Middle School Math Project In A Heartbeat (Algebra) Objective Students will apply their knowledge of scatter plots to discover the correlation between heartbeats per minute before and after aerobic ### Acquisition Lesson Planning Form Key Standards addressed in this Lesson: MM2A2c Time allotted for this Lesson: 5 Hours Acquisition Lesson Planning Form Key Standards addressed in this Lesson: MM2A2c Time allotted for this Lesson: 5 Hours Essential Question: LESSON 2 Absolute Value Equations and Inequalities How do you ### Linear Equations. Find the domain and the range of the following set. {(4,5), (7,8), (-1,3), (3,3), (2,-3)} Linear Equations Domain and Range Domain refers to the set of possible values of the x-component of a point in the form (x,y). Range refers to the set of possible values of the y-component of a point in ### Lesson 1: Review of Decimals: Addition, Subtraction, Multiplication LESSON 1: REVIEW OF DECIMALS: ADDITION AND SUBTRACTION Weekly Focus: whole numbers and decimals Weekly Skill: place value, add, subtract, multiply Lesson Summary: In the warm up, students will solve a ### Scope and Sequence KA KB 1A 1B 2A 2B 3A 3B 4A 4B 5A 5B 6A 6B Scope and Sequence Earlybird Kindergarten, Standards Edition Primary Mathematics, Standards Edition Copyright 2008 [SingaporeMath.com Inc.] The check mark indicates where the topic is first introduced ### EQUATIONS and INEQUALITIES EQUATIONS and INEQUALITIES Linear Equations and Slope 1. Slope a. Calculate the slope of a line given two points b. 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Excel ### Lesson 4: Convert Fractions, Review Order of Operations Lesson 4: Convert Fractions, Review Order of Operations LESSON 4: Convert Fractions, Do Order of Operations Weekly Focus: fractions, decimals, percent, order of operations Weekly Skill: convert, compute ### 1) Write the following as an algebraic expression using x as the variable: Triple a number subtracted from the number 1) Write the following as an algebraic expression using x as the variable: Triple a number subtracted from the number A. 3(x - x) B. x 3 x C. 3x - x D. x - 3x 2) Write the following as an algebraic expression ### Application of Function Composition Math Objectives Given functions f and g, the student will be able to determine the domain and range of each as well as the composite functions defined by f ( g( x )) and g( f ( x )). Students will interpret ### Accommodated Lesson Plan on Solving Systems of Equations by Elimination for Diego Accommodated Lesson Plan on Solving Systems of Equations by Elimination for Diego Courtney O Donovan Class: Algebra 1 Day #: 6-7 Grade: 8th Number of Students: 25 Date: May 12-13, 2011 Goal: Students will ### Solutions of Equations in Two Variables 6.1 Solutions of Equations in Two Variables 6.1 OBJECTIVES 1. Find solutions for an equation in two variables 2. Use ordered pair notation to write solutions for equations in two variables We discussed ### High School Algebra Reasoning with Equations and Inequalities Solve equations and inequalities in one variable. Performance Assessment Task Quadratic (2009) Grade 9 The task challenges a student to demonstrate an understanding of quadratic functions in various forms. A student must make sense of the meaning of relations ### Years after 2000. US Student to Teacher Ratio 0 16.048 1 15.893 2 15.900 3 15.900 4 15.800 5 15.657 6 15.540 To complete this technology assignment, you should already have created a scatter plot for your data on your calculator and/or in Excel. 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Indicator 4 Solve open sentences by representing an expression in more than one way using ### Technology: CBR2, Graphing Calculator & Cords, Overhead Projector, & Overhead Unit for Calculator Analyzing Graphs Lisa Manhard Grade Level: 7 Technology: CBR2, Graphing Calculator & Cords, Overhead Projector, & Overhead Unit for Calculator Materials: Student Worksheets (3) Objectives Evaluate what Ohio Standards Connection Geometry and Spatial Sense Benchmark C Specify locations and plot ordered pairs on a coordinate plane. Indicator 6 Extend understanding of coordinate system to include points ### GETTING TO THE CORE: THE LINK BETWEEN TEMPERATURE AND CARBON DIOXIDE DESCRIPTION This lesson plan gives students first-hand experience in analyzing the link between atmospheric temperatures and carbon dioxide ( ) s by looking at ice core data spanning hundreds of thousands ### Algebra Cheat Sheets Sheets Algebra Cheat Sheets provide you with a tool for teaching your students note-taking, problem-solving, and organizational skills in the context of algebra lessons. These sheets teach the concepts ### Summer Math Exercises. For students who are entering. Pre-Calculus Summer Math Eercises For students who are entering Pre-Calculus It has been discovered that idle students lose learning over the summer months. 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# HOW MUCH WILL I SPEND ON GAS? Save this PDF as: Size: px Start display at page: ## Transcription 1 HOW MUCH WILL I SPEND ON GAS? Outcome (lesson objective) The students will use the current and future price of gasoline to construct T-charts, write algebraic equations, and plot the equations on a graph. Student/Class Goal Students will determine their gasoline cost for a month s time. They will use this information to calculate yearly gas cost. Time Frame Two 1 ½ hour classes One 3 hour class Standard Use Math to Solve Problems and Communicate NRS EFL 3-6 COPS Understand, interpret, and work with pictures, numbers, and symbolic information. Apply knowledge of mathematical concepts and procedures to figure out how to answer a question, solve a problem, make a prediction, or carry out a task that has a mathematical dimension. Define and select data to be used in solving the problem. Determine the degree of precision required by the situation. Solve problem using appropriate quantitative procedures and verify that the results are reasonable. Communicate results using a variety of mathematical representations, including graphs, charts, tables, and algebraic models. Materials Graph paper 40x30 paper is included at the end of the lesson (Worksheet 4) Colored pencils (if available) Calculators How Do I Find My Gas Mileage? Handout T-Chart Illustration How to Graph a Linear Equation in 5 Quick Steps Student Resource Activity Addresses Components of Performance Students will work with basic operations and patterns to complete a T-chart. Students will construct a graph and plot points on it. Students will use problem solving to determine how to figure out their monthly gas cost. Students will determine appropriate intervals on the x and y axes of the graph. Students will round the number when necessary to plot the points on the graph. Students will recognize if a data set is not reasonable by observing the other data sets and observing the points on the graph (is it on the line?) Students will communicate the results of the data by constructing a graph, T-chart, algebraic formula and in writing. Learner Prior Knowledge Basic understanding of gas prices and what is meant by miles per gallon (mpg), found at lesson, Pumped Up Gas Prices. Instructional Activities Step 1 - Discuss with students the current gasoline price and what they expect to happen to gas prices in the future. Ask students about how much they are spending on gas each month and how many miles the car they are driving gets per gallon? Students may need to research this information by actually calculating their mileage on the handout How Do I Find My Gas Mileage? or by visiting Cars and mpg This web site lists the in-city and highway mileage for 1985 or newer vehicles. If using this site, students will need to decide what type of driving they do during a month (city or highway) and estimate their car s mileage. Step 2 - Using \$4 per gallon as the price of gas, construct, with the class, a T-chart showing the relationship between the number of gallons purchased(x) and the total cost of the gas(y). See T-Chart Illustration for an example. As a class, look at the T-chart that was constructed. Give the students time to discover the relationship between the number of gallons purchased(x) and the total price of the gas(y). Express this relationship verbally (The number of gallons purchased times \$4 will equal the total cost.). Discuss how this relationship can be written as an equation as a function of x? (4x=y) Step 3 Next, with the students, graph the relationship/equation the class discovered during Step 2. The How to Graph a Linear Equation in 5 Quick Steps, a student resource from their math journals, provides basic guidelines for graphing equations. Help the students determine appropriate intervals and labels for the x and y axes and a title for the graph. Plot several points together and then let the students plot the remainder of the points. Draw a line passing through the points and label the line with the equation written in Step 2. Step 4 - Discuss with the students how many miles they drive each month. Share with the class that many people drive about 1,000 4 Gallons purchased (x) T-CHART ILLUSTRATION Total gas cost (y) 1 \$ \$ x30 Grid 5 How to Graph a Linear Equation in 5 Quick Steps Step 1 Construct a T-chart of Values Using your equation, construct a T-chart of values if one has not been done already. Substitute some simple numbers into the equation for x or y. If x=1, what is y? If x=10, what is y? If y=0, what is x? Each pair of values in your T-chart will become a point on the graph. (See illustration 1 for an example of a T-chart) Step 2 Decide on the interval for each axis Before starting the graph, look at the T-chart to determine the highest value for y found on the chart. Look at the values needed for x. Using graph paper, count the number of lines on the x and y axes. Use these numbers to determine the intervals on each axis. (If you use the graph paper at the end of this lesson there are 30 spaces on the x axis and 40 spaces on the y axis.) If the largest total cost/y value that needs to be graphed is \$80 and there are 40 lines on the y axis, let each line on the y axis represent \$2. The number of gallons of gas/x value that goes with \$80 is 20. There are 30 lines, so to make it simple one line will equal one gallon. Be sure the students realize they do not need to put a number next to every line. For example, the x might be labeled on every 5 th line (five gallons) and the y axis might also be labeled on every 5 th line (or \$10). This is a good step to do in pencil. That way if the interval you selected did not work out, the numbers can be erased any you can start over. Step 3 Label each Axis Decide what labels need to be added to the x and y axis. What do the numbers on the x-axis represent? What do the numbers on the y-axis represent? Usually the labels will match the descriptions/labels of x and y on the T-chart. (Note: When graphing equations involving elapsed time, time is traditionally represented by x) Step 4 Plot the points Using each pair of points from the T-chart, plot the points on the graph. Every point does not need to be plotted. Just be sure you have at least 3. Using a ruler, draw a line through the points you have plotted. Write your equation next to the line. Step 5 Give the graph a title Decide on a title for the graph. Make sure it accurately represents what is being shown on the graph. Does it explain the relationship between x and y? How to Graph a Linear Equation in 5 Quick Steps Student Resource ### Solving Systems of Linear Equations Substitutions Solving Systems of Linear Equations Substitutions Outcome (lesson objective) Students will accurately solve a system of equations algebraically using substitution. Student/Class Goal Students thinking ### Solving Systems of Linear Equations Elimination (Addition) Solving Systems of Linear Equations Elimination (Addition) Outcome (lesson objective) Students will accurately solve systems of equations using elimination/addition method. Student/Class Goal Students ### Solving Systems of Linear Equations Substitutions Solving Systems of Linear Equations Substitutions Outcome (learning objective) Students will accurately solve a system of equations algebraically using substitution. 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Find the domain and the range of the following set. {(4,5), (7,8), (-1,3), (3,3), (2,-3)} Linear Equations Domain and Range Domain refers to the set of possible values of the x-component of a point in the form (x,y). Range refers to the set of possible values of the y-component of a point in ### Lesson 1: Review of Decimals: Addition, Subtraction, Multiplication LESSON 1: REVIEW OF DECIMALS: ADDITION AND SUBTRACTION Weekly Focus: whole numbers and decimals Weekly Skill: place value, add, subtract, multiply Lesson Summary: In the warm up, students will solve a ### Scope and Sequence KA KB 1A 1B 2A 2B 3A 3B 4A 4B 5A 5B 6A 6B Scope and Sequence Earlybird Kindergarten, Standards Edition Primary Mathematics, Standards Edition Copyright 2008 [SingaporeMath.com Inc.] The check mark indicates where the topic is first introduced ### EQUATIONS and INEQUALITIES EQUATIONS and INEQUALITIES Linear Equations and Slope 1. Slope a. Calculate the slope of a line given two points b. Calculate the slope of a line parallel to a given line. c. Calculate the slope of a line ### Charlesworth School Year Group Maths Targets Charlesworth School Year Group Maths Targets Year One Maths Target Sheet Key Statement KS1 Maths Targets (Expected) These skills must be secure to move beyond expected. I can compare, describe and solve ### F.IF.7b: Graph Root, Piecewise, Step, & Absolute Value Functions F.IF.7b: Graph Root, Piecewise, Step, & Absolute Value Functions F.IF.7b: Graph Root, Piecewise, Step, & Absolute Value Functions Analyze functions using different representations. 7. Graph functions expressed ### Subject: Math Grade Level: 5 Topic: The Metric System Time Allotment: 45 minutes Teaching Date: Day 1 Subject: Math Grade Level: 5 Topic: The Metric System Time Allotment: 45 minutes Teaching Date: Day 1 I. (A) Goal(s): For student to gain conceptual understanding of the metric system and how to convert ### Math-in-CTE Lesson Plan: Marketing Math-in-CTE Lesson Plan: Marketing Lesson Title: Break-Even Point Lesson 01 Occupational Area: Marketing Ed./Accounting CTE Concept(s): Math Concepts: Lesson Objective: Fixed Costs, Variable Costs, Total ### ALGEBRA I (Common Core) Thursday, January 28, 2016 1:15 to 4:15 p.m., only ALGEBRA I (COMMON CORE) The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION ALGEBRA I (Common Core) Thursday, January 28, 2016 1:15 to 4:15 p.m., only Student Name: School Name: The ### Prentice Hall Mathematics: Course 1 2008 Correlated to: Arizona Academic Standards for Mathematics (Grades 6) PO 1. Express fractions as ratios, comparing two whole numbers (e.g., ¾ is equivalent to 3:4 and 3 to 4). Strand 1: Number Sense and Operations Every student should understand and use all concepts and ### Part 1 Expressions, Equations, and Inequalities: Simplifying and Solving Section 7 Algebraic Manipulations and Solving Part 1 Expressions, Equations, and Inequalities: Simplifying and Solving Before launching into the mathematics, let s take a moment to talk about the words ### A synonym is a word that has the same or almost the same definition of Slope-Intercept Form Determining the Rate of Change and y-intercept Learning Goals In this lesson, you will: Graph lines using the slope and y-intercept. Calculate the y-intercept of a line when given ### Temperature Scales. The metric system that we are now using includes a unit that is specific for the representation of measured temperatures. Temperature Scales INTRODUCTION The metric system that we are now using includes a unit that is specific for the representation of measured temperatures. The unit of temperature in the metric system is ### Activity 6 Graphing Linear Equations Activity 6 Graphing Linear Equations TEACHER NOTES Topic Area: Algebra NCTM Standard: Represent and analyze mathematical situations and structures using algebraic symbols Objective: The student will be ### Current California Math Standards Balanced Equations Balanced Equations Current California Math Standards Balanced Equations Grade Three Number Sense 1.0 Students understand the place value of whole numbers: 1.1 Count, read, and write whole numbers to 10,000. ### Title ID Number Sequence and Duration Age Level Essential Question Learning Objectives. Lead In Title ID Number Sequence and Duration Age Level Essential Question Learning Objectives Lesson Activity Barbie Bungee (75-80 minutes) MS-M-A1 Lead In (15-20 minutes) Activity (45-50 minutes) Closure (10 ### PREPARING A PERSONAL LETTER PREPARING A PERSONAL LETTER Outcome (lesson objective) Students will identify the parts and format of a personal (friendly) letter then write a letter using the appropriate format with proper spelling, ### Convert between units of area and determine the scale factor of two similar figures. CHAPTER 5 Units of Area c GOAL Convert between units of area and determine the scale factor of two. You will need a ruler centimetre grid paper a protractor a calculator Learn about the Math The area of ### The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION INTEGRATED ALGEBRA. Tuesday, January 22, 2013 9:15 a.m. to 12:15 p.m. INTEGRATED ALGEBRA The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION INTEGRATED ALGEBRA Tuesday, January 22, 2013 9:15 a.m. to 12:15 p.m., only Student Name: School Name: The possession ### Algebra I Sample Questions. 1 Which ordered pair is not in the solution set of (1) (5,3) (2) (4,3) (3) (3,4) (4) (4,4) 1 Which ordered pair is not in the solution set of (1) (5,3) (2) (4,3) (3) (3,4) (4) (4,4) y 1 > x + 5 and y 3x 2? 2 5 2 If the quadratic formula is used to find the roots of the equation x 2 6x 19 = 0, ### EXCEL Tutorial: How to use EXCEL for Graphs and Calculations. EXCEL Tutorial: How to use EXCEL for Graphs and Calculations. Excel is powerful tool and can make your life easier if you are proficient in using it. You will need to use Excel to complete most of your ### The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION INTEGRATED ALGEBRA. Wednesday, June 12, 2013 1:15 to 4:15 p.m. INTEGRATED ALGEBRA The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION INTEGRATED ALGEBRA Wednesday, June 12, 2013 1:15 to 4:15 p.m., only Student Name: School Name: The possession ### Scientific Graphing in Excel 2010 Scientific Graphing in Excel 2010 When you start Excel, you will see the screen below. Various parts of the display are labelled in red, with arrows, to define the terms used in the remainder of this overview. ### Chapter 4 -- Decimals Chapter 4 -- Decimals \$34.99 decimal notation ex. The cost of an object. ex. The balance of your bank account ex The amount owed ex. The tax on a purchase. Just like Whole Numbers Place Value - 1.23456789 ### Systems of Linear Equations: Two Variables OpenStax-CNX module: m49420 1 Systems of Linear Equations: Two Variables OpenStax College This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 4.0 In this section, Factoring Quadratic Trinomials Student Probe Factor Answer: Lesson Description This lesson uses the area model of multiplication to factor quadratic trinomials Part 1 of the lesson consists of circle puzzles ### Unit 7 Quadratic Relations of the Form y = ax 2 + bx + c Unit 7 Quadratic Relations of the Form y = ax 2 + bx + c Lesson Outline BIG PICTURE Students will: manipulate algebraic expressions, as needed to understand quadratic relations; identify characteristics ### Part 1: Background - Graphing Department of Physics and Geology Graphing Astronomy 1401 Equipment Needed Qty Computer with Data Studio Software 1 1.1 Graphing Part 1: Background - Graphing In science it is very important to find and ### Title: Basic Metric Measurements Conversion (police) X X Stackable Certificate Documentation Technology Study / Life skills EL-Civics Career Pathways Police Paramedic Fire Rescue Medical Asst. EKG / Cardio Phlebotomy Practical Nursing Healthcare Admin Pharmacy ### Graphs of Proportional Relationships Graphs of Proportional Relationships Student Probe Susan runs three laps at the track in 12 minutes. A graph of this proportional relationship is shown below. Explain the meaning of points A (0,0), B (1,4), ### N Q.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. Performance Assessment Task Swimming Pool Grade 9 The task challenges a student to demonstrate understanding of the concept of quantities. A student must understand the attributes of trapezoids, how to A Correlation of to the Minnesota Academic Standards Grades K-6 G/M-204 Introduction This document demonstrates the high degree of success students will achieve when using Scott Foresman Addison Wesley READING THE NEWSPAPER Outcome (lesson objective) Students will comprehend and critically evaluate text as they read to find the main idea. They will construct meaning as they analyze news articles and ### Lesson 18: Introduction to Algebra: Expressions and Variables LESSON 18: Algebra Expressions and Variables Weekly Focus: expressions Weekly Skill: write and evaluate Lesson Summary: For the Warm Up, students will solve a problem about movie tickets sold. In Activity ### Answers Teacher Copy. Systems of Linear Equations Monetary Systems Overload. Activity 3. Solving Systems of Two Equations in Two Variables of 26 8/20/2014 2:00 PM Answers Teacher Copy Activity 3 Lesson 3-1 Systems of Linear Equations Monetary Systems Overload Solving Systems of Two Equations in Two Variables Plan Pacing: 1 class period Chunking ### ALGEBRA 1 ~ Cell Phone Task Group: Kimberly Allen, Matt Blundin, Nancy Bowen, Anna Green, Lee Hale, Katie Owens ALGEBRA 1 ~ Cell Phone Task Group: Kimberly Allen, Matt Blundin, Nancy Bowen, Anna Green, Lee Hale, Katie Owens Math Essential Standards Approximate and interpret rates of change from graphical and numerical ### Relationships Between Two Variables: Scatterplots and Correlation Relationships Between Two Variables: Scatterplots and Correlation Example: Consider the population of cars manufactured in the U.S. What is the relationship (1) between engine size and horsepower? (2) ### High School Algebra Reasoning with Equations and Inequalities Solve systems of equations. Performance Assessment Task Graphs (2006) Grade 9 This task challenges a student to use knowledge of graphs and their significant features to identify the linear equations for various lines. A student ### HiMAP Pull-Out Section: Spring 1988 References The Average of Rates and the Average Rate Peter A. Lindstrom In our everyday lives, we often hear phrases involving averages: the student has a grade-point average of 3.12, the baseball player has a batting ### Teens and Budgeting. { http://youth.macu.com } Teens and Budgeting { http://youth.macu.com } Mountain America Credit Union knows that it s never too early to start learning critical money management skills. That s why we ve put this information together ### Comparing Simple and Compound Interest Comparing Simple and Compound Interest GRADE 11 In this lesson, students compare various savings and investment vehicles by calculating simple and compound interest. Prerequisite knowledge: Students should ### Excel -- Creating Charts Excel -- Creating Charts The saying goes, A picture is worth a thousand words, and so true. Professional looking charts give visual enhancement to your statistics, fiscal reports or presentation. Excel ### Lesson 4: Convert Fractions, Review Order of Operations Lesson 4: Convert Fractions, Review Order of Operations LESSON 4: Convert Fractions, Do Order of Operations Weekly Focus: fractions, decimals, percent, order of operations Weekly Skill: convert, compute ### 1) Write the following as an algebraic expression using x as the variable: Triple a number subtracted from the number 1) Write the following as an algebraic expression using x as the variable: Triple a number subtracted from the number A. 3(x - x) B. x 3 x C. 3x - x D. x - 3x 2) Write the following as an algebraic expression ### Application of Function Composition Math Objectives Given functions f and g, the student will be able to determine the domain and range of each as well as the composite functions defined by f ( g( x )) and g( f ( x )). Students will interpret ### Accommodated Lesson Plan on Solving Systems of Equations by Elimination for Diego Accommodated Lesson Plan on Solving Systems of Equations by Elimination for Diego Courtney O Donovan Class: Algebra 1 Day #: 6-7 Grade: 8th Number of Students: 25 Date: May 12-13, 2011 Goal: Students will ### Solutions of Equations in Two Variables 6.1 Solutions of Equations in Two Variables 6.1 OBJECTIVES 1. Find solutions for an equation in two variables 2. Use ordered pair notation to write solutions for equations in two variables We discussed ### High School Algebra Reasoning with Equations and Inequalities Solve equations and inequalities in one variable. Performance Assessment

Task Quadratic (2009) Grade 9 The task challenges a student to demonstrate an understanding of quadratic functions in various forms. A student must make sense of the meaning of relations ### Years after 2000. US Student to Teacher Ratio 0 16.048 1 15.893 2 15.900 3 15.900 4 15.800 5 15.657 6 15.540 To complete this technology assignment, you should already have created a scatter plot for your data on your calculator and/or in Excel. You could do this with any two columns of data, but for demonstration ### 1 BPS Math Year at a Glance (Adapted from A Story of Units Curriculum Maps in Mathematics P-5) Grade 5 Key Areas of Focus for Grades 3-5: Multiplication and division of whole numbers and fractions-concepts, skills and problem solving Expected Fluency: Multi-digit multiplication Module M1: Whole ### Unit #3: Investigating Quadratics (9 days + 1 jazz day + 1 summative evaluation day) BIG Ideas: Unit #3: Investigating Quadratics (9 days + 1 jazz day + 1 summative evaluation day) BIG Ideas: Developing strategies for determining the zeroes of quadratic functions Making connections between the meaning Ohio Standards Connection Patterns, Functions and Algebra Benchmark E Solve open sentences and explain strategies. Indicator 4 Solve open sentences by representing an expression in more than one way using ### Technology: CBR2, Graphing Calculator & Cords, Overhead Projector, & Overhead Unit for Calculator Analyzing Graphs Lisa Manhard Grade Level: 7 Technology: CBR2, Graphing Calculator & Cords, Overhead Projector, & Overhead Unit for Calculator Materials: Student Worksheets (3) Objectives Evaluate what Ohio Standards Connection Geometry and Spatial Sense Benchmark C Specify locations and plot ordered pairs on a coordinate plane. Indicator 6 Extend understanding of coordinate system to include points ### GETTING TO THE CORE: THE LINK BETWEEN TEMPERATURE AND CARBON DIOXIDE DESCRIPTION This lesson plan gives students first-hand experience in analyzing the link between atmospheric temperatures and carbon dioxide ( ) s by looking at ice core data spanning hundreds of thousands ### Algebra Cheat Sheets Sheets Algebra Cheat Sheets provide you with a tool for teaching your students note-taking, problem-solving, and organizational skills in the context of algebra lessons. These sheets teach the concepts ### Summer Math Exercises. For students who are entering. Pre-Calculus Summer Math Eercises For students who are entering Pre-Calculus It has been discovered that idle students lose learning over the summer months. To help you succeed net fall and perhaps to help you learn ### Grade Level Year Total Points Core Points % At Standard 9 2003 10 5 7 % Performance Assessment Task Number Towers Grade 9 The task challenges a student to demonstrate understanding of the concepts of algebraic properties and representations. A student must make sense of the ### Charts, Tables, and Graphs Charts, Tables, and Graphs The Mathematics sections of the SAT also include some questions about charts, tables, and graphs. You should know how to (1) read and understand information that is given; (2) ### Indicator 2: Use a variety of algebraic concepts and methods to solve equations and inequalities. 3 rd Grade Math Learning Targets Algebra: Indicator 1: Use procedures to transform algebraic expressions. 3.A.1.1. Students are able to explain the relationship between repeated addition and multiplication.

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### Theory: Let us draw the graph of the equation using the coordinates of $$x$$ and $$y$$-intercepts. The graph can be obtained by plotting the $$x$$ and $$y$$-intercepts and then drawing a line joining these points. Example: Draw the graph of the equation $$4y-3x = 6$$ using the $$x$$ and $$y$$-intercepts. Solution: To find the $$x$$-intercept, put $$y = 0$$ in the given equation. $$4(0)-3x = 6$$ $$0-3x = 6$$ $$-3x = 6$$ $$x = \frac{6}{-3}$$ $$x = -2$$ Thus, the $$x$$-intercept is $$x = -2$$. Similarly, to find the $$y$$-intercept, put $$x = 0$$ in the given equation. $$4y-3(0) = 6$$ $$4y-0 = 6$$ $$4y = 6$$ $$y = \frac{6}{4}$$ $$y = \frac{3}{2}$$ Thus, the $$y$$-intercept is $$\frac{3}{2}$$. We shall plot the graph using these two coordinates $$(-2,0)$$, and $$(0,\frac{3}{2})$$ and then, draw a line through the two points.

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### Theory: Let us draw the graph of the equation using the coordinates of $$x$$ and $$y$$-intercepts. The graph can be obtained by plotting the $$x$$ and $$y$$-intercepts and then drawing a line joining these points. Example: Draw the graph of the equation $$4y-3x = 6$$ using the $$x$$ and $$y$$-intercepts. Solution: To find the $$x$$-intercept, put $$y = 0$$ in the given equation. $$4(0)-3x = 6$$ $$0-3x = 6$$ $$-3x = 6$$ $$x = \frac{6}{-3}$$ $$x = -2$$ Thus, the $$x$$-intercept is $$x = -2$$. Similarly, to find the $$y$$-intercept, put $$x = 0$$ in the given equation. $$4y-3(0) = 6$$ $$4y-0 = 6$$ $$4y = 6$$ $$y = \frac{6}{4}$$ $$y = \frac{3}{2}$$ Thus, the $$y$$-intercept is $$\frac{3}{2}$$. We shall plot the graph using these

two coordinates $$(-2,0)$$, and $$(0,\frac{3}{2})$$ and then, draw a line through the two points.

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# How do you factor 4y² -13y -12? May 30, 2015 Use a version of the AC Method. $A = 4$, $B = 13$, $C = 12$ Look for a pair of factors of $A C = 48$ whose difference is $B = 13$. $16$ and $3$ work. Use this pair to split the middle term then factor by grouping... $4 {y}^{2} - 13 y - 12$ $= 4 {y}^{2} + 3 y - 16 y - 12$ $= \left(4 {y}^{2} + 3 y\right) - \left(16 y + 12\right)$ $= y \left(4 y + 3\right) - 4 \left(4 y + 3\right)$ $= \left(y - 4\right) \left(4 y + 3\right)$

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# How do you factor 4y² -13y -12? May 30, 2015 Use a version of the AC Method. $A = 4$, $B = 13$, $C = 12$ Look for a pair of factors of $A C = 48$ whose difference is $B = 13$. $16$ and $3$ work. Use this pair to split the middle term then factor by grouping... $4 {y}^{2} - 13 y - 12$ $= 4 {y}^{2} + 3 y - 16 y - 12$ $= \left(4 {y}^{2} + 3 y\right) - \left(16 y + 12\right)$ $= y \left(4 y + 3\right) - 4 \left(4

y + 3\right)$ $= \left(y - 4\right) \left(4 y + 3\right)$

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# SBI Clerk Pre Quantitative Aptitude Quiz- 07 ## SBI Clerk Pre Quantitative Aptitude Quiz Quantitative aptitude measures a candidate’s numerical proficiency and problem-solving abilities. It is the most important section of almost all competitive exams. Candidates are often stymied by the complexity of Quantitative Aptitude Questions but if they practice more and more questions, it will become quite easy. So, here we are providing you with the SBI Clerk Pre-Quantitative Aptitude Quiz to enhance your preparation for your upcoming examination. Questions given in this SBI Clerk Pre-Quantitative Aptitude Quiz are based on the most recent and the latest exam pattern. A detailed explanation for each question will be given in this SBI Clerk Pre-Quantitative Aptitude Quiz. This SBI Clerk Pre-Quantitative Aptitude Quiz is entirely free of charge. This SBI Clerk Pre-Quantitative Aptitude Quiz will assist aspirants in achieving a good score in their upcoming examinations. 1. A dishonest cloth merchant sales cloth at the cost price but uses false scale which measures 80 cm in lieu of 1 m. Find his gain percentage? (a) 20% (b) 25% (c) 15% (d) 12% (e) 22% 2. The area of two squares is in the ratio 225 : 256. Find ratio of their diagonals? 3. ‘P’ sells his watch at 20% profit to Q while Q sales it to R at a loss of 10%. If R pays Rs. 2160. Find at what price P sold watch to Q? (a) Rs. 2000 (b) Rs. 2200 (c) Rs. 2400 (d) Rs. 1800 (e) Rs. 2500 4. In how many ways can letter of word ‘PROMISE’ be arranged such that all vowels always come together? (a) 720 (b) 120 (c) 960 (d) 880 (e) 480 5. Find ratio between S.I. and C.I. on a sum of money invested for 3 years at 5% rate of interest per annum? (a) 15 : 1261 (b) 151 : 156 (c) 121 : 441 (d) 1200 : 1261 (e) 121 : 484 Directions (6-10): In each of these questions, two equations (I) and (II) are given. You have to solve both the equations and give answer. Click to Buy Bank MahaCombo Package Recommended PDF’s for: #### Most important PDF’s for Bank, SSC, Railway and Other Government Exam : Download PDF Now AATMA-NIRBHAR Series- Static GK/Awareness Practice Ebook PDF Get PDF here The Banking Awareness 500 MCQs E-book| Bilingual (Hindi + English) Get PDF here AATMA-NIRBHAR Series- Banking Awareness Practice Ebook PDF Get PDF here Computer Awareness Capsule 2.O Get PDF here AATMA-NIRBHAR Series Quantitative Aptitude Topic-Wise PDF Get PDF here AATMA-NIRBHAR Series Reasoning Topic-Wise PDF Get PDF Here Memory Based Puzzle E-book | 2016-19 Exams Covered Get PDF here Caselet Data Interpretation 200 Questions Get PDF here Puzzle & Seating Arrangement E-Book for BANK PO MAINS (Vol-1) Get PDF here ARITHMETIC DATA INTERPRETATION 2.O E-book Get PDF here 3

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# SBI Clerk Pre Quantitative Aptitude Quiz- 07 ## SBI Clerk Pre Quantitative Aptitude Quiz Quantitative aptitude measures a candidate’s numerical proficiency and problem-solving abilities. It is the most important section of almost all competitive exams. Candidates are often stymied by the complexity of Quantitative Aptitude Questions but if they practice more and more questions, it will become quite easy. So, here we are providing you with the SBI Clerk Pre-Quantitative Aptitude Quiz to enhance your preparation for your upcoming examination. Questions given in this SBI Clerk Pre-Quantitative Aptitude Quiz are based on the most recent and the latest exam pattern. A detailed explanation for each question will be given in this SBI Clerk Pre-Quantitative Aptitude Quiz. This SBI Clerk Pre-Quantitative Aptitude Quiz is entirely free of charge. This SBI Clerk Pre-Quantitative Aptitude Quiz will assist aspirants in achieving a good score in their upcoming examinations. 1. A dishonest cloth merchant sales cloth at the cost price but uses false scale which measures 80 cm in lieu of 1 m. Find his gain percentage? (a) 20% (b) 25% (c) 15% (d) 12% (e) 22% 2. The area of two squares is in the ratio 225 : 256. Find ratio of their diagonals? 3. ‘P’ sells his watch at 20% profit to Q while Q sales it to R at a loss of 10%. If R pays Rs. 2160. Find at what price P sold watch to Q? (a) Rs. 2000 (b) Rs. 2200 (c) Rs. 2400 (d) Rs. 1800 (e) Rs. 2500 4. In how many ways can letter of word ‘PROMISE’ be arranged such that all vowels always come together? (a) 720 (b) 120 (c) 960 (d) 880 (e) 480 5. Find ratio between S.I. and C.I. on a sum of money invested for 3 years at 5% rate of interest per annum? (a) 15 : 1261 (b) 151 : 156 (c) 121 : 441 (d) 1200 : 1261 (e) 121 : 484 Directions (6-10): In each of these questions, two equations (I) and (II) are given. You have to solve both the equations and give answer. Click to Buy Bank MahaCombo Package Recommended PDF’s for: #### Most important PDF’s for Bank, SSC, Railway and Other Government Exam : Download PDF Now AATMA-NIRBHAR Series- Static GK/Awareness Practice Ebook PDF Get PDF here The Banking Awareness 500 MCQs E-book| Bilingual (Hindi + English) Get PDF here AATMA-NIRBHAR Series- Banking Awareness Practice Ebook PDF Get PDF here Computer Awareness Capsule 2.O Get PDF here AATMA-NIRBHAR Series Quantitative Aptitude Topic-Wise PDF Get PDF here AATMA-NIRBHAR

Series Reasoning Topic-Wise PDF Get PDF Here Memory Based Puzzle E-book | 2016-19 Exams Covered Get PDF here Caselet Data Interpretation 200 Questions Get PDF here Puzzle & Seating Arrangement E-Book for BANK PO MAINS (Vol-1) Get PDF here ARITHMETIC DATA INTERPRETATION 2.O E-book Get PDF here 3

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Search 75,714 tutors 0 0 ## find the composite function Given the  f(x)= 4/x2 and g(x) = 3-2x, find the composite function (f º g) and simplify. Given f(x) and g(x),   (f º g)(x) = f(g(x)) with f(x) = 4/x2   and g(x) = 3-2x,  First apply g, then apply f to that result as follows: g(x) = 3x-2 f(g(x)) =  4/ (3 - 2x)2  = 4/ ((3-2x)(3-2x)) ## f(g(x)) = 4/ (4x2-12x+9) f(x)= 4/x2 and g(x) = 3-2x f(g(x)) = 4/ (3 - 2x)2 f(g(x)) = 4/(4x2 - 12x + 9)

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Search 75,714 tutors 0 0 ## find the composite function Given the f(x)= 4/x2 and g(x) = 3-2x, find the composite function (f º g) and simplify. Given f(x) and g(x), (f º g)(x) = f(g(x)) with f(x) = 4/x2 and g(x) = 3-2x, First apply g, then apply f to that result as follows: g(x) = 3x-2 f(g(x)) = 4/ (3 - 2x)2 = 4/ ((3-2x)(3-2x)) ## f(g(x)) = 4/ (4x2-12x+9) f(x)= 4/x2 and g(x) = 3-2x f(g(x)) = 4/

(3 - 2x)2 f(g(x)) = 4/(4x2 - 12x + 9)

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### Present Remotely Send the link below via email or IM • Invited audience members will follow you as you navigate and present • People invited to a presentation do not need a Prezi account • This link expires 10 minutes after you close the presentation Do you really want to delete this prezi? Neither you, nor the coeditors you shared it with will be able to recover it again. # Maths Revision: Ratios and Rates A revision about ratios and rates. by ## Tiffany Chin on 1 October 2012 Report abuse #### Transcript of Maths Revision: Ratios and Rates Ratios Ratios and Rates A comparison between 2/more quantities of the same kind Ratios can be shown as... 1:1 Ratio Form 1 to 1 Word Form 1/1 Fraction Form 1:n/m:1 Unit Ratio Please do not copy! Equivalent Ratios Ratios with the same meaning Simplify e.g. 2:4 1:2 1/2 2/4 Prime Factorization Unit Ratio Ratios with a denominator to 1 (Simplify) A:B/A 1:B/A 1:n A:B/B 1:B/A m:1 Conversion of A and B Comparing Ratios Change order to unit ratio e.g. Which class has the highest BOYS:GIRLS ratio? Class 1 Class 2 Class 3 2:3 3:4 1:5 1. Compare your question to ratio given If same order, m:1 e.g. Highest B:G If not, 1:n e.g. Lowest G:B 2. Convert 3. Order Use m:1 Class 1 Class 2 Class 3 2/3:1= 0.6: 1 3/4:1=0.75:1 1/5:1=0.2:1 Class 2 Class 1 Class 3 4. Choose your answer Highest B:G Ratio Multiply Method 1: Find the Unknown e.g. Solve for Unknown 3:4=a:8 =6:8 a=6 Method 2: Fractions e.g. 3:4=c:10 10x3/4=c/28.5x10 30/4=c 7.5=c Find A:B:C from A:B/B:C Cross Product e.g. A:B=4:3 B:C=7:5 4 : 3 7 : 5 28:21:15 Multiply as follow Line up common value Rearrange and simplify as needed Modeling Using Ratios e.g. 250 ml lemon juice, how

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### Present Remotely Send the link below via email or IM • Invited audience members will follow you as you navigate and present • People invited to a presentation do not need a Prezi account • This link expires 10 minutes after you close the presentation Do you really want to delete this prezi? Neither you, nor the coeditors you shared it with will be able to recover it again. # Maths Revision: Ratios and Rates A revision about ratios and rates. by ## Tiffany Chin on 1 October 2012 Report abuse #### Transcript of Maths Revision: Ratios and Rates Ratios Ratios and Rates A comparison between 2/more quantities of the same kind Ratios can be shown as... 1:1 Ratio Form 1 to 1 Word Form 1/1 Fraction Form 1:n/m:1 Unit Ratio Please do not copy! Equivalent Ratios Ratios with the same meaning Simplify e.g. 2:4 1:2 1/2 2/4 Prime Factorization Unit Ratio Ratios with a denominator to 1 (Simplify) A:B/A 1:B/A 1:n A:B/B 1:B/A m:1 Conversion of A and B Comparing Ratios Change order to unit ratio e.g. Which class has the highest BOYS:GIRLS ratio? Class 1 Class 2 Class 3 2:3 3:4 1:5 1. Compare your question to ratio given If same order, m:1 e.g. Highest B:G If not, 1:n e.g. Lowest G:B 2. Convert 3. Order Use m:1 Class 1 Class 2 Class 3 2/3:1= 0.6: 1 3/4:1=0.75:1 1/5:1=0.2:1 Class 2 Class 1 Class 3 4. Choose your answer Highest B:G Ratio Multiply Method 1: Find the Unknown e.g. Solve for Unknown 3:4=a:8 =6:8 a=6 Method 2: Fractions e.g. 3:4=c:10 10x3/4=c/28.5x10 30/4=c 7.5=c Find A:B:C from A:B/B:C Cross Product

e.g. A:B=4:3 B:C=7:5 4 : 3 7 : 5 28:21:15 Multiply as follow Line up common value Rearrange and simplify as needed Modeling Using Ratios e.g. 250 ml lemon juice, how

https://ambitiousbaba.com/quant-quiz-for-sbi-clerk-pre-2021-16-march-2021/

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# Quant Quiz for SBI clerk pre 2021| 16 March 2021 ## Quant Quiz for SBI clerk pre 2021 Quant Quiz to improve your Quantitative Aptitude for SBI Po & SBI clerk exam IBPS PO Reasoning , IBPS Clerk Reasoning , IBPS RRB Reasoning, LIC AAO ,LIC Assistant  and other competitive exam. Direction (1 -5): The Pie-chart given below shows the percentage distribution of the monthly income of Rakesh. Note: (i) Total monthly income of Rakesh = expenditure on (Rent + Education + Electricity) + Monthly saving of Rakesh (ii) Monthly saving of Rakesh = Rs. 4000 Q1. Rakesh’s expenditure on education is how much more or less than his expenditure on rent? 1. 500 Rs. 2. 400 Rs. 3. 600 Rs. 4. 300 Rs. 5. 700 Rs. Rakesh’s saving is what percentage of his expenditure on electricity. 1. 250% 2. 266 2/3% 3. 225% 4. 233 1/3% 5. 275% Q3. Find Rakesh’s total expenditure on electricity and education together. 1. 3000 Rs. 2. 4500 Rs. 3. 4000 Rs. 4. 5000 Rs. 5. 3500 Rs. Q4. Find out Rakesh’s annual income? 1. 120000 Rs. 2. 144000 Rs. 3. 110000 Rs. 4. 125000 Rs. 5. 136000 Rs. Q5. Rakesh’s expenditure on rent and education together is what percent more or less than his saving? 1. 8% 2. 11% 3. 9% 4. 10% 5. 12.5% Direction (6 -10): What will come in the place of question (?) mark in the following number series. Q6. 90, 117, 145, 174, 204, ? 1. 225 2. 220 3. 230 4. 235 5. 240 Q7. ?, 10, 100, 1500, 30000, 750000 1. 2 2. 1 3. 5 4. 4 5. 10 Q8. 145, 170, 197, 226, 257, ? 1. 332 2. 325 3. 401 4. 290 5. 360 Q9. 473, 460, 434, 382, 278, ? 1. 50 2. 90 3. 60 4. 80 5. 70 Q10. 4200, 5600, 7200, 9000, 11000, ? 1. 12800 2. 13200 3. 13100 4. 14120 5. 15000 Solutions Q1. Ans(1) Q2. Ans(2) Q3. Ans(3) Q4. Ans(1) Q5. Ans(5) Q6. Ans(4) Q7. Ans(1) Q8. Ans(4) Q9. Ans(5) Q10. Ans(2) Recommended PDF’s for 2021: ### 2021 Preparation Kit PDF #### Most important PDF’s for Bank, SSC, Railway and Other Government Exam : Download PDF Now AATMA-NIRBHAR Series- Static GK/Awareness Practice Ebook PDF Get PDF here The Banking Awareness 500 MCQs E-book| Bilingual (Hindi + English) Get PDF here AATMA-NIRBHAR Series- Banking Awareness Practice Ebook PDF Get PDF here Computer Awareness Capsule 2.O Get PDF here AATMA-NIRBHAR Series Quantitative Aptitude Topic-Wise PDF 2020 Get PDF here Memory Based Puzzle E-book | 2016-19 Exams Covered Get PDF here Caselet Data Interpretation 200 Questions Get PDF here Puzzle & Seating Arrangement E-Book for BANK PO MAINS (Vol-1) Get PDF here ARITHMETIC DATA INTERPRETATION 2.O E-book Get PDF here 3

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# Quant Quiz for SBI clerk pre 2021| 16 March 2021 ## Quant Quiz for SBI clerk pre 2021 Quant Quiz to improve your Quantitative Aptitude for SBI Po & SBI clerk exam IBPS PO Reasoning , IBPS Clerk Reasoning , IBPS RRB Reasoning, LIC AAO ,LIC Assistant and other competitive exam. Direction (1 -5): The Pie-chart given below shows the percentage distribution of the monthly income of Rakesh. Note: (i) Total monthly income of Rakesh = expenditure on (Rent + Education + Electricity) + Monthly saving of Rakesh (ii) Monthly saving of Rakesh = Rs. 4000 Q1. Rakesh’s expenditure on education is how much more or less than his expenditure on rent? 1. 500 Rs. 2. 400 Rs. 3. 600 Rs. 4. 300 Rs. 5. 700 Rs. Rakesh’s saving is what percentage of his expenditure on electricity. 1. 250% 2. 266 2/3% 3. 225% 4. 233 1/3% 5. 275% Q3. Find Rakesh’s total expenditure on electricity and education together. 1. 3000 Rs. 2. 4500 Rs. 3. 4000 Rs. 4. 5000 Rs. 5. 3500 Rs. Q4. Find out Rakesh’s annual income? 1. 120000 Rs. 2. 144000 Rs. 3. 110000 Rs. 4. 125000 Rs. 5. 136000 Rs. Q5. Rakesh’s expenditure on rent and education together is what percent more or less than his saving? 1. 8% 2. 11% 3. 9% 4. 10% 5. 12.5% Direction (6 -10): What will come in the place of question (?) mark in the following number series. Q6. 90, 117, 145, 174, 204, ? 1. 225 2. 220 3. 230 4. 235 5. 240 Q7. ?, 10, 100, 1500, 30000, 750000 1. 2 2. 1 3. 5 4. 4 5. 10 Q8. 145, 170, 197, 226, 257, ? 1. 332 2. 325 3. 401 4. 290 5. 360 Q9. 473, 460, 434, 382, 278, ? 1. 50 2. 90 3. 60 4. 80 5. 70 Q10. 4200, 5600, 7200, 9000, 11000, ? 1. 12800 2. 13200 3. 13100 4. 14120 5. 15000 Solutions Q1. Ans(1) Q2. Ans(2) Q3. Ans(3) Q4. Ans(1) Q5. Ans(5) Q6. Ans(4) Q7. Ans(1) Q8. Ans(4) Q9. Ans(5) Q10. Ans(2) Recommended PDF’s for 2021: ### 2021 Preparation Kit PDF #### Most important PDF’s for Bank, SSC, Railway and Other Government Exam : Download PDF Now AATMA-NIRBHAR Series- Static GK/Awareness Practice Ebook PDF Get PDF here The Banking Awareness 500 MCQs E-book| Bilingual (Hindi + English) Get PDF here AATMA-NIRBHAR Series- Banking Awareness Practice Ebook PDF Get PDF here Computer Awareness Capsule 2.O Get PDF here AATMA-NIRBHAR Series Quantitative Aptitude

Topic-Wise PDF 2020 Get PDF here Memory Based Puzzle E-book | 2016-19 Exams Covered Get PDF here Caselet Data Interpretation 200 Questions Get PDF here Puzzle & Seating Arrangement E-Book for BANK PO MAINS (Vol-1) Get PDF here ARITHMETIC DATA INTERPRETATION 2.O E-book Get PDF here 3

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# Is 0/4 undefined? $\frac{0}{4} = 0$ is defined. $\frac{4}{0}$ is not. If $\frac{4}{0} = k$ for some $k \in \mathbb{R}$, then we would have $0 \cdot k = 4$, but $0 \cdot$anything$= 0 \ne 4$, so there is no such $k$.

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# Is 0/4 undefined? $\frac{0}{4} = 0$ is defined. $\frac{4}{0}$ is not. If $\frac{4}{0} = k$ for some $k \in \mathbb{R}$, then we would have $0 \cdot k = 4$, but $0 \cdot$anything$= 0 \ne 4$, so

there is no such $k$.

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# Equation solver with square root This Equation solver with square root supplies step-by-step instructions for solving all math troubles. Our website will give you answers to homework. ## The Best Equation solver with square root In this blog post, we will be discussing about Equation solver with square root. As any math student knows, calculus can be a difficult subject to grasp. The concepts are often complex and require a great deal of concentration to understand. Fortunately, there are now many calculus solvers available that can help to make the subject more manageable. These tools allow you to input an equation and see the steps involved in solving it. This can be a great way to learn how to solve problems on your own. In addition, calculus solvers with steps can also help you to check your work and ensure that you are getting the correct answer. With so many helpful features, it is no wonder that these tools are becoming increasingly popular among math students of all levels. Word phrase math is a mathematical technique that uses words instead of symbols to represent numbers and operations. This approach can be particularly helpful for students who struggle with traditional math notation. By using words, students can more easily visualize the relationships between numbers and operations. As a result, word phrase math can provide a valuable tool for understanding complex mathematical concepts. Additionally, this technique can also be used to teach basic math skills to young children. By representing numbers and operations with familiar words, children can develop a strong foundation for future mathematics learning. A rational function is any function which can be expressed as the quotient of two polynomials. In other words, it is a fraction whose numerator and denominator are both polynomials. The simplest example of a rational function is a linear function, which has the form f(x)=mx+b. More generally, a rational function can have any degree; that is, the highest power of x in the numerator and denominator can be any number. To solve a rational function, we must first determine its roots. A root is a value of x for which the numerator equals zero. Therefore, to solve a rational function, we set the numerator equal to zero and solve for x. Once we have determined the roots of the function, we can use them to find its asymptotes. An asymptote is a line which the graph of the function approaches but never crosses. A rational function can have horizontal, vertical, or slant asymptotes, depending on its roots. To find a horizontal asymptote, we take the limit of the function as x approaches infinity; that is, we let x get very large and see what happens to the value of the function. Similarly, to find a vertical asymptote, we take the limit of the function as x approaches zero. Finally, to find a slant asymptote, we take the limit of the function as x approaches one of its roots. Once we have determined all of these features of the graph, we can sketch it on a coordinate plane. A binomial solver is a math tool that helps solve equations with two terms. This type of equation is also known as a quadratic equation. The solver will usually ask for the coefficients of the equation, which are the numbers in front of the x terms. It will also ask for the constants, which are the numbers not attached to an x. With this information, the solver can find the roots, or solutions, to the equation. The roots tell where the line intersects the x-axis on a graph. There are two roots because there are two values of x that make the equation true. To find these roots, the solver will use one of several methods, such as factoring or completing the square. Each method has its own set of steps, but all require some algebraic manipulation. The binomial solver can help take care of these steps so that you can focus on understanding the concept behind solving quadratic equations. Basic mathematics is the study of mathematical operations and their properties. The focus of this branch of mathematics is on addition, subtraction, multiplication, and division. These operations are the foundation for all other types of math, including algebra, geometry, and trigonometry. In addition to studying how these operations work, students also learn how to solve equations and how to use basic concepts of geometry and trigonometry. Basic mathematics is an essential part of every student's education, and it provides a strong foundation for further study in math. ## Help with math It has helped me more than my math teacher, of course there are a few problems here and there like not scanning the exercises correctly and it never scans "±" this sign but apart from that I really love the app. Although it would be nice to not have to pay for explanations, maybe other features but explanation is an important part. Mila Flores Although the app doesn't solve everything it is pretty much my free math tutor. It's so easy to work with. Steps for solving are always easy to understand. I love it!!! The app is the best and it helped me with my studies at school and I now understand math better Bethany Young App that solves math problems with a picture Math solution calculator Free online math websites Math probability solver Geometry tutor online

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# Equation solver with square root This Equation solver with square root supplies step-by-step instructions for solving all math troubles. Our website will give you answers to homework. ## The Best Equation solver with square root In this blog post, we will be discussing about Equation solver with square root. As any math student knows, calculus can be a difficult subject to grasp. The concepts are often complex and require a great deal of concentration to understand. Fortunately, there are now many calculus solvers available that can help to make the subject more manageable. These tools allow you to input an equation and see the steps involved in solving it. This can be a great way to learn how to solve problems on your own. In addition, calculus solvers with steps can also help you to check your work and ensure that you are getting the correct answer. With so many helpful features, it is no wonder that these tools are becoming increasingly popular among math students of all levels. Word phrase math is a mathematical technique that uses words instead of symbols to represent numbers and operations. This approach can be particularly helpful for students who struggle with traditional math notation. By using words, students can more easily visualize the relationships between numbers and operations. As a result, word phrase math can provide a valuable tool for understanding complex mathematical concepts. Additionally, this technique can also be used to teach basic math skills to young children. By representing numbers and operations with familiar words, children can develop a strong foundation for future mathematics learning. A rational function is any function which can be expressed as the quotient of two polynomials. In other words, it is a fraction whose numerator and denominator are both polynomials. The simplest example of a rational function is a linear function, which has the form f(x)=mx+b. More generally, a rational function can have any degree; that is, the highest power of x in the numerator and denominator can be any number. To solve a rational function, we must first determine its roots. A root is a value of x for which the numerator equals zero. Therefore, to solve a rational function, we set the numerator equal to zero and solve for x. Once we have determined the roots of the function, we can use them to find its asymptotes. An asymptote is a line which the graph of the function approaches but never crosses. A rational function can have horizontal, vertical, or slant asymptotes, depending on its roots. To find a horizontal asymptote, we take the limit of the function as x approaches infinity; that is, we let x get very large and see what happens to the value of the function. Similarly, to find a vertical asymptote, we take the limit of the function as x approaches zero. Finally, to find a slant asymptote, we take the limit of the function as x approaches one of its roots. Once we have determined all of these features of the graph, we can sketch it on a coordinate plane. A binomial solver is a math tool that helps solve equations with two terms. This type of equation is also known as a quadratic equation. The solver will usually ask for the coefficients of the equation, which are the numbers in front of the x terms. It will also ask for the constants, which are the numbers not attached to an x. With this information, the solver can find the roots, or solutions, to the equation. The roots tell where the line intersects the x-axis on a graph. There are two roots because there are two values of x that make the equation true. To find these roots, the solver will use one of several methods, such as factoring or completing the square. Each method has its own set of steps, but all require some algebraic manipulation. The binomial solver can help take care of these steps so that you can focus on understanding the concept behind solving quadratic equations. Basic mathematics is the study of mathematical operations and their properties. The focus of this branch of mathematics is on addition, subtraction, multiplication, and division. These operations are the foundation for all other types of math, including algebra, geometry, and trigonometry. In addition to studying how these operations work, students also learn how to solve equations and how to use basic concepts of geometry and trigonometry. Basic mathematics is an essential part of every student's education, and it provides a strong foundation for further study in math. ## Help with math It has helped me more than my math teacher, of course there are a few problems here and there like not scanning the exercises correctly and it never scans "±" this sign but apart from that I really love the app. Although it would be nice to

not have to pay for explanations, maybe other features but explanation is an important part. Mila Flores Although the app doesn't solve everything it is pretty much my free math tutor. It's so easy to work with. Steps for solving are always easy to understand. I love it!!! The app is the best and it helped me with my studies at school and I now understand math better Bethany Young App that solves math problems with a picture Math solution calculator Free online math websites Math probability solver Geometry tutor online

https://members.simplenursing.com/quiz_bank_question/med-math-metric-conversions-question-7/

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## Med Math: Metric Conversions Question #7 330 mL = _____ L 1. 330, 000 • Rationale: 2. 3, 300 • Rationale: 3. 0.33 • Rationale: 4. 0. 033 • Rationale: ### Explanation DA solution = 1 L/1000mLx 330 mL/1 = 0.33 L Ratio solution = 1000 mL/1 L = 330 mL/x L

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## Med Math: Metric Conversions Question #7 330 mL = _____ L 1. 330, 000 • Rationale: 2. 3, 300 • Rationale: 3. 0.33 • Rationale: 4. 0. 033 • Rationale: ### Explanation DA solution = 1 L/1000mLx 330 mL/1 = 0.33 L Ratio solution = 1000

mL/1 L = 330 mL/x L

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### Select your language Suggested languages for you: Americas Europe Q8RP Expert-verified Found in: Page 415 ### Fundamentals Of Differential Equations And Boundary Value Problems Book edition 9th Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider Pages 616 pages ISBN 9780321977069 # In Problems 3-10, determine the Laplace transform of the given function.${\mathbf{\left(}\mathbf{t}\mathbf{+}\mathbf{3}\mathbf{\right)}}^{{\mathbf{2}}}{\mathbf{-}}{\mathbf{\left(}{\mathbf{e}}^{\mathbf{t}}\mathbf{+}\mathbf{3}\mathbf{\right)}}^{{\mathbf{2}}}$ Therefore, the solution is$\text{L}\left\{{\text{(t+3)}}^{\text{2}}\text{-}{\left({\text{e}}^{\text{t}}\text{+3}\right)}^{\text{2}}\right\}\text{=}\frac{\text{2}}{{\text{s}}^{\text{3}}}\text{+}\frac{\text{6}}{{\text{s}}^{\text{2}}}\text{-}\frac{\text{1}}{\text{s-2}}\text{-}\frac{\text{6}}{\text{s-1}}$. See the step by step solution ## Step 1: Given Information The given value is ${\left(\text{t+3}\right)}^{\text{2}}\text{-}{\left({\text{e}}^{\text{t}}\text{+3}\right)}^{\text{2}}$ ## Step 2: Determining the Laplace transform Using the following Laplace transform property, to find the Laplace of given integral: $\text{L}\left\{{\text{e}}^{\text{at}}\right\}\text{=}\frac{\text{1}}{\text{s-a}}$ $\text{L}\left\{{\text{t}}^{\text{n}}\right\}\text{=}\frac{\text{n!}}{{\text{s}}^{\text{n+1}}}$ Apply the Laplace transform property, we get: $\begin{array}{c}\text{L}\left\{{\text{(t+3)}}^{\text{2}}\text{-}{\left({\text{e}}^{\text{t}}\text{+3}\right)}^{\text{2}}\right\}\text{=L}\left\{{\text{(t+3)}}^{\text{2}}\right\}\text{-L}\left\{{\left({\text{e}}^{\text{t}}\text{+3}\right)}^{\text{2}}\right\}\\ \text{=L}\left\{{\text{t}}^{\text{2}}\text{+6t+3}\right\}\text{-L}\left\{{\text{e}}^{\text{2t}}{\text{+6e}}^{\text{t}}\text{+3}\right\}\\ \text{=L}\left\{{\text{t}}^{\text{2}}\right\}\text{+6L{t}+3L{1}-L}\left\{{\text{e}}^{\text{2t}}\right\}\text{+6L}\left\{{\text{e}}^{\text{t}}\right\}\text{-3L{1}}\\ \text{=L}\left\{{\text{t}}^{\text{2}}\right\}\text{+6L{t}-L}\left\{{\text{e}}^{\text{2t}}\right\}\text{+6L}\left\{{\text{e}}^{\text{t}}\right\}\end{array}$ Simplify further as follows $\begin{array}{c}\text{L}\left\{{\text{(t+3)}}^{\text{2}}\text{-}{\left({\text{e}}^{\text{t}}\text{+3}\right)}^{\text{2}}\right\}\text{=}\frac{\text{2!}}{{\text{s}}^{\text{3}}}\text{+6}\frac{\text{1!}}{{\text{s}}^{\text{2}}}\text{-}\frac{\text{1}}{\text{s-2}}\text{-6}\frac{\text{1}}{\text{s-1}}\\ \text{=}\frac{\text{2}}{{\text{s}}^{\text{3}}}\text{+}\frac{\text{6}}{{\text{s}}^{\text{2}}}\text{-}\frac{\text{1}}{\text{s-2}}\text{-}\frac{\text{6}}{\text{s-1}}\end{array}$ Therefore,$\text{L}\left\{{\text{(t+3)}}^{\text{2}}\text{-}{\left({\text{e}}^{\text{t}}\text{+3}\right)}^{\text{2}}\right\}\text{=}\frac{\text{2}}{{\text{s}}^{\text{3}}}\text{+}\frac{\text{6}}{{\text{s}}^{\text{2}}}\text{-}\frac{\text{1}}{\text{s-2}}\text{-}\frac{\text{6}}{\text{s-1}}$

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### Select your language Suggested languages for you: Americas Europe Q8RP Expert-verified Found in: Page 415 ### Fundamentals Of Differential Equations And Boundary Value Problems Book edition 9th Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider Pages 616 pages ISBN 9780321977069 # In Problems 3-10, determine the Laplace transform of the given function.${\mathbf{\left(}\mathbf{t}\mathbf{+}\mathbf{3}\mathbf{\right)}}^{{\mathbf{2}}}{\mathbf{-}}{\mathbf{\left(}{\mathbf{e}}^{\mathbf{t}}\mathbf{+}\mathbf{3}\mathbf{\right)}}^{{\mathbf{2}}}$ Therefore, the solution is$\text{L}\left\{{\text{(t+3)}}^{\text{2}}\text{-}{\left({\text{e}}^{\text{t}}\text{+3}\right)}^{\text{2}}\right\}\text{=}\frac{\text{2}}{{\text{s}}^{\text{3}}}\text{+}\frac{\text{6}}{{\text{s}}^{\text{2}}}\text{-}\frac{\text{1}}{\text{s-2}}\text{-}\frac{\text{6}}{\text{s-1}}$. See the step by step solution ## Step 1: Given Information The given value is ${\left(\text{t+3}\right)}^{\text{2}}\text{-}{\left({\text{e}}^{\text{t}}\text{+3}\right)}^{\text{2}}$ ## Step 2: Determining the Laplace transform Using the following Laplace transform property, to find the Laplace of given integral: $\text{L}\left\{{\text{e}}^{\text{at}}\right\}\text{=}\frac{\text{1}}{\text{s-a}}$ $\text{L}\left\{{\text{t}}^{\text{n}}\right\}\text{=}\frac{\text{n!}}{{\text{s}}^{\text{n+1}}}$ Apply the Laplace transform property, we

get: $\begin{array}{c}\text{L}\left\{{\text{(t+3)}}^{\text{2}}\text{-}{\left({\text{e}}^{\text{t}}\text{+3}\right)}^{\text{2}}\right\}\text{=L}\left\{{\text{(t+3)}}^{\text{2}}\right\}\text{-L}\left\{{\left({\text{e}}^{\text{t}}\text{+3}\right)}^{\text{2}}\right\}\\ \text{=L}\left\{{\text{t}}^{\text{2}}\text{+6t+3}\right\}\text{-L}\left\{{\text{e}}^{\text{2t}}{\text{+6e}}^{\text{t}}\text{+3}\right\}\\ \text{=L}\left\{{\text{t}}^{\text{2}}\right\}\text{+6L{t}+3L{1}-L}\left\{{\text{e}}^{\text{2t}}\right\}\text{+6L}\left\{{\text{e}}^{\text{t}}\right\}\text{-3L{1}}\\ \text{=L}\left\{{\text{t}}^{\text{2}}\right\}\text{+6L{t}-L}\left\{{\text{e}}^{\text{2t}}\right\}\text{+6L}\left\{{\text{e}}^{\text{t}}\right\}\end{array}$ Simplify further as follows $\begin{array}{c}\text{L}\left\{{\text{(t+3)}}^{\text{2}}\text{-}{\left({\text{e}}^{\text{t}}\text{+3}\right)}^{\text{2}}\right\}\text{=}\frac{\text{2!}}{{\text{s}}^{\text{3}}}\text{+6}\frac{\text{1!}}{{\text{s}}^{\text{2}}}\text{-}\frac{\text{1}}{\text{s-2}}\text{-6}\frac{\text{1}}{\text{s-1}}\\ \text{=}\frac{\text{2}}{{\text{s}}^{\text{3}}}\text{+}\frac{\text{6}}{{\text{s}}^{\text{2}}}\text{-}\frac{\text{1}}{\text{s-2}}\text{-}\frac{\text{6}}{\text{s-1}}\end{array}$ Therefore,$\text{L}\left\{{\text{(t+3)}}^{\text{2}}\text{-}{\left({\text{e}}^{\text{t}}\text{+3}\right)}^{\text{2}}\right\}\text{=}\frac{\text{2}}{{\text{s}}^{\text{3}}}\text{+}\frac{\text{6}}{{\text{s}}^{\text{2}}}\text{-}\frac{\text{1}}{\text{s-2}}\text{-}\frac{\text{6}}{\text{s-1}}$

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# Period Of Function by -2 views Period of sinusoidal functions from equation. You can figure this out without looking at a graph by dividing with the frequency which in this case is 2. Graphing Trigonometric Functions Graphing Trigonometric Functions Neon Signs ### Periodic functions are used throughout science to describe oscillations waves and other phenomena that exhibit periodicity. Period of function. This is why this function family is also called the periodic function family. Tap for more steps. Therefore in the case of the basic cosine function f x. When this occurs we call the horizontal shift the period of the function. If a function has a repeating pattern like sine or cosine it is called a periodic function. Amplitude a Let b be a real number. You might immediately guess that there is a connection here to finding points on a circle. F x k f x. Horizontal stretch is measured for sinusoidal functions as their periods. For basic sine and cosine functions the period is 2 π. Any function that is not periodic is called aperiodic. The period is defined as the length of one wave of the function. The period is the length on the x axis in one cycle. However the amplitude does not refer to the highest point on the graph or the distance from the highest point to the x axis. Midline of sinusoidal functions from equation. The x-value results in a unique output eg. The period of a sinusoid is the length of a complete cycle. By using this website you agree to our Cookie Policy. Periodic Functions A periodic function occurs when a specific horizontal shift P results in the original function. More formally we say that this type of function has a positive constant k where any input x. The period is defined as the length of a functions cycle. Period of a Function The time interval between two waves is known as a Period whereas a function that repeats its values at regular intervals or periods is known as a Periodic Function. According to periodic function definition the period of a function is represented like f x f x p p is equal to the real number and this is the period of the given function f x. In other words a periodic function is a function that repeats its values after every particular interval. Trig functions are cyclical and when you graph them youll see the ups and downs of the graph and youll see that these ups and downs. Where f x P f x for all values of x. Replace with in the formula for period. Amplitude and Period of Sine and Cosine Functions The amplitude of y a sin x and y a cos x represents half the distance between the maximum and minimum values of the function. This is the currently selected item. Any part of the graph that shows this pattern over one period is called a cycle. Notice that in the graph of the sine function shown that f x sin x has period. Period of sinusoidal functions from graph. The distance between and is. Midline amplitude and period review. The absolute value is the distance between a number and zero. The period of a periodic function is the interval of x -values on which the cycle of the graph thats repeated in both directions lies. In this case one full wave is 180 degrees or radians. So the period of or is. You are partially correct. Khan Academy is a 501c3 nonprofit organization. The period of a periodic function is the interval of x -values on which one copy of the repeated pattern occurs. This is the currently selected item. A periodic function repeats its values at set intervals called periods. Period can be defined as the time interval between the two occurrences of the wave. A periodic function is a function that repeats its values at regular intervals for example the trigonometric functions which repeat at intervals of 2π radians. The period of the function can be calculated using. Our mission is to provide a free world-class education to anyone anywhere. Grade 12 trigonometry problems and questions on how to find the period of trigonometric functions given its graph or formula are presented along with detailed solutions. So how to find the period of a function actually. Free function periodicity calculator – find periodicity of periodic functions step-by-step This website uses cookies to ensure you get the best experience. In the problems below we will use the formula for the period P of trigonometric functions of the form y a sin bx c d or y a cos bx c d and which is given by. The period is the length of the smallest interval that contains exactly one copy of the repeating pattern. A function is just a type of equation where every input eg. Amplitude Period Phase Shift Vertical Translation And Range Of Y 3 Math Videos Translation Period The Mathematics Of Sine Cosine Functions Was First Developed By The Ancient Indian Mathematician Aryabhata H Trigonometric Functions Mathematics Theorems Amplitude Phase Shift Vertical Shift And Period Change Of The Cosine Function Fun Learning Period Change Transformations Of Sin Function Love Math Math Graphing Step By Step Instructions Of How To Graph The Sine Function Graphing Trigfunction Trigonometry Sinusoidal Math Materials Math Graphic Organizers Calculus Graphing Trigonometric Functions Graphing Trigonometric Functions Trigonometry Geometry Trigonometry 9 Trig Graphs Amplitude And Period Trigonometry Graphing Classroom Posters Graphs Of Trigonometric Functions Poster Zazzle Com In 2021 Trigonometric Functions Functions Math Math Poster Applied Mathematics Period Of A Function Hard Sum Mathematics How To Apply Period Students Will Match 10 Graphs To 10 Sine Or Cosine Equations By Finding The Amplitude And Period Of Each Function Students Will Then Graphing Sines Equations Precal And Trig Function Posters Math Word Problems Mathematics Worksheets Word Problems Graphing Sin Cosine W Period Change 4 Terrific Examples Graphing Precalculus Trigonometry Translating Periodic Functions Learning Mathematics Math Methods Math Instruction Trigonometric Graphing Math Methods Learning Math Math Trigonometric Graphing Math Methods Learning Math Math Pin On Math Applied Mathematics Greatest Integer Function And Period Integers Mathematics How To Apply Applet Allows For Students To Drag 5 Key Points Of One Period Of A Sinusoidal Wave So That The Graph Displayed Match Trigonometric Functions Graphing Equations Functions Non Functions By Ryan Devoe Period 7 By Rmdevoe Teaching Mathematics Teaching Algebra Teaching Math READ:   Which One Of The Following Is An Example Of A Period Cost?

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# Period Of Function by -2 views Period of sinusoidal functions from equation. You can figure this out without looking at a graph by dividing with the frequency which in this case is 2. Graphing Trigonometric Functions Graphing Trigonometric Functions Neon Signs ### Periodic functions are used throughout science to describe oscillations waves and other phenomena that exhibit periodicity. Period of function. This is why this function family is also called the periodic function family. Tap for more steps. Therefore in the case of the basic cosine function f x. When this occurs we call the horizontal shift the period of the function. If a function has a repeating pattern like sine or cosine it is called a periodic function. Amplitude a Let b be a real number. You might immediately guess that there is a connection here to finding points on a circle. F x k f x. Horizontal stretch is measured for sinusoidal functions as their periods. For basic sine and cosine functions the period is 2 π. Any function that is not periodic is called aperiodic. The period is defined as the length of one wave of the function. The period is the length on the x axis in one cycle. However the amplitude does not refer to the highest point on the graph or the distance from the highest point to the x axis. Midline of sinusoidal functions from equation. The x-value results in a unique output eg. The period of a sinusoid is the length of a complete cycle. By using this website you agree to our Cookie Policy. Periodic Functions A periodic function occurs when a specific horizontal shift P results in the original function. More formally we say that this type of function has a positive constant k where any input x. The period is defined as the length of a functions cycle. Period of a Function The time interval between two waves is known as a Period whereas a function that repeats its values at regular intervals or periods is known as a Periodic Function. According to periodic function definition the period of a function is represented like f x f x p p is equal to the real number and this is the period of the given function f x. In other words a periodic function is a function that repeats its values after every particular interval. Trig functions are cyclical and when you graph them youll see the ups and downs of the graph and youll see that these ups and downs. Where f x P f x for all values of x. Replace with in the formula for period. Amplitude and Period of Sine and Cosine Functions The amplitude of y a sin x and y a cos x represents half the distance between the maximum and minimum values of the function. This is the currently selected item. Any part of the graph that shows this pattern over one period is called a cycle. Notice that in the graph of the sine function shown that f x sin x has period. Period of sinusoidal functions from graph. The distance between and is. Midline amplitude and period review. The absolute value is the distance between a number and zero. The period of a periodic function is the interval of x -values on which the cycle of the graph thats repeated in both directions lies. In this case one full wave is 180 degrees or radians. So the period of or is. You are partially correct. Khan Academy is a 501c3 nonprofit organization. The period of a periodic function is the interval of x -values on which one copy of the repeated pattern occurs. This is the currently selected item. A periodic function repeats its values at set intervals called periods. Period can be defined as the time interval between the two occurrences of the wave. A periodic function is a function that repeats its values at regular intervals for example the trigonometric functions which repeat at intervals of 2π radians. The period of the function can be calculated using. Our mission is to provide a free world-class education to anyone anywhere. Grade 12 trigonometry problems and questions on how to find the period of trigonometric functions given its graph or formula are presented along with detailed solutions. So how to find the period of a function actually. Free function periodicity calculator – find periodicity of periodic functions step-by-step This website uses cookies to ensure you get the best experience. In the problems below we will use the formula for the period P of trigonometric functions of the form y a sin bx c d or y a cos bx c d and which is given by. The period is the length of the smallest interval that contains exactly one copy of the repeating pattern. A function is just a type of equation where every input eg. Amplitude Period Phase Shift Vertical Translation And Range Of Y 3 Math Videos Translation Period The Mathematics Of Sine Cosine Functions Was First Developed By The Ancient Indian Mathematician Aryabhata H Trigonometric Functions Mathematics Theorems Amplitude Phase Shift Vertical Shift And Period Change Of The Cosine Function Fun Learning Period Change Transformations Of Sin Function Love Math Math Graphing Step By Step Instructions Of How To Graph The Sine Function Graphing Trigfunction Trigonometry Sinusoidal Math Materials Math Graphic Organizers Calculus Graphing Trigonometric Functions Graphing Trigonometric Functions Trigonometry Geometry Trigonometry 9 Trig Graphs Amplitude And Period Trigonometry Graphing Classroom Posters Graphs Of Trigonometric Functions Poster Zazzle Com In 2021 Trigonometric Functions Functions Math Math Poster Applied Mathematics Period Of A Function Hard Sum Mathematics How To Apply Period Students Will Match 10 Graphs To 10 Sine Or Cosine Equations By Finding The Amplitude And Period Of Each Function Students Will Then Graphing Sines Equations Precal And Trig Function Posters Math Word Problems

Mathematics Worksheets Word Problems Graphing Sin Cosine W Period Change 4 Terrific Examples Graphing Precalculus Trigonometry Translating Periodic Functions Learning Mathematics Math Methods Math Instruction Trigonometric Graphing Math Methods Learning Math Math Trigonometric Graphing Math Methods Learning Math Math Pin On Math Applied Mathematics Greatest Integer Function And Period Integers Mathematics How To Apply Applet Allows For Students To Drag 5 Key Points Of One Period Of A Sinusoidal Wave So That The Graph Displayed Match Trigonometric Functions Graphing Equations Functions Non Functions By Ryan Devoe Period 7 By Rmdevoe Teaching Mathematics Teaching Algebra Teaching Math READ: Which One Of The Following Is An Example Of A Period Cost?

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# Thread: Question involving a continuous function on a closed interval 1. ## Question involving a continuous function on a closed interval Intuitively this is obvious by graphing g(x) = x on [0,1] and seeing since f is continuous it has to intersect with g at some point. But I spent a long time and cannot figure out how to prove this. 2. It is the well-known fixed point theorem. Suppose $f$ is continuous on $[0,1]$ and $f(x)\epsilon[0,1]$ for every $x\epsilon[0,1]$. If $f(0)=0$ or $f(1)=1$, the theorem is proved. So we try to prove the theorem assuming $f(0)>0$ and $f(1)<1$. Let $g(x)=f(x)-x$ for all $x\epsilon[0,1]$. Hence $g(0)>0$ and $g(1)<0$ and $g$ is continuous on $[0,1]$, that is, $0$ is an intermediate value of $g$ on $[0,1]$. Hence by intermediate value theorem, there exists a point $c\epsilon(0,1)$ such that $g(c)=0$ --which means $f(c)=c.$ Hence the prrof. EDIT: $c$ is equivalent to $x_0$ 3. Originally Posted by paulrb Intuitively this is obvious by graphing g(x) = x on [0,1] and seeing since f is continuous it has to intersect with g at some point. But I spent a long time and cannot figure out how to prove this. Alternatively, suppose that $f(x)\ne x$ then the mapping $\displaystyle f:[0,1]\to\{-1,1}:\frac{|f(x)-x|}{f(x)-x}$ is a continuous surjection which is impossible.

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# Thread: Question involving a continuous function on a closed interval 1. ## Question involving a continuous function on a closed interval Intuitively this is obvious by graphing g(x) = x on [0,1] and seeing since f is continuous it has to intersect with g at some point. But I spent a long time and cannot figure out how to prove this. 2. It is the well-known fixed point theorem. Suppose $f$ is continuous on $[0,1]$ and $f(x)\epsilon[0,1]$ for every $x\epsilon[0,1]$. If $f(0)=0$ or $f(1)=1$, the theorem is proved. So we try to prove the theorem assuming $f(0)>0$ and $f(1)<1$. Let $g(x)=f(x)-x$ for all $x\epsilon[0,1]$. Hence $g(0)>0$ and $g(1)<0$ and $g$ is continuous on $[0,1]$, that is, $0$ is an intermediate value of $g$ on $[0,1]$. Hence by intermediate value theorem, there exists a point $c\epsilon(0,1)$ such that $g(c)=0$ --which means $f(c)=c.$ Hence the prrof. EDIT: $c$ is equivalent to $x_0$ 3. Originally Posted by paulrb Intuitively this is obvious by graphing g(x) = x on [0,1] and seeing since f is continuous it has to intersect with g at some point. But I spent a long time and cannot figure

out how to prove this. Alternatively, suppose that $f(x)\ne x$ then the mapping $\displaystyle f:[0,1]\to\{-1,1}:\frac{|f(x)-x|}{f(x)-x}$ is a continuous surjection which is impossible.

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## Vibrations of Continuous Systems: Lateral Vibrations of Beams ### Equation of Motion Consider a beam, shown in Figure 10.6(a) which has a length , density (mass per unit volume) and Young’s modulus which is acted upon by a distributed load (per unit length) acting laterally along the beam. Let measure the lateral deflection of the beam and assume that only small deformations occur. Figure 10.6(b) shows the FBD/MAD for an infinitesimal element of the beam with mass . By applying Newton’s Law’s and considering moments about the left end of the element we find As a first order approximation we can write that so that the above becomes or (10.23) which is a well known result from beam theory. In the vertical direction we then have Note: For small motions, the shear forces act vertically to a first order approximation. The actual vertical components would be for example where can be obtained from the slope of the beam. However, for small motions, , so is used here as the vertical component. A similar remark holds for the term. Now, using 10.23 we can write that so the equation of motion becomes (10.24) For small deformations the bending moment in the beam is related to the deflection by so that 10.24 becomes or (10.25) This is the general equation which governs the lateral vibrations of beams. If we limit ourselves to only consider free vibrations of uniform beams (, is constant), the equation of motion reduces to which can be written (10.26) where (10.27) Note that this is not the wave equation. ### Solution To Equation of Motion Once again we look for solutions which represent a mode shape undergoing \sshm of the form This results in so 10.26 becomes Separating the and terms we find Again, since the LHS depends only on and the RHS depends only on and they must be equal for all values of and , both sides must be equal to a constant, which we call , so that This results in equations or (10.28a) (10.28b) where (10.29) Note for future reference that so that (10.30) The solution to 10.28b is as we have seen To find the solution to 10.28a, which is a 4 order linear ODE with constant coefficients, we assume a solution of the form The equation of motion then becomes or The four roots to this equation are The total solution will be a linear combination of solutions, one for each of the above roots, (10.31) where each of the may be complex. However, by using Euler’s identity and introducing the hyperbolic and functions 10.31 can be written or (10.32) where The advantage of expressing the solution as in 10.32 is that all of the terms involved are real. We see that and are complex conjugates, while and are real. As can be seen there are four constants to be determined which requires that four boundary conditions be specified. These will often be determined with one pair of boundary conditions specified at each end, depending on the type of support. Figure 10.7 shows three of the most common support conditions and the pairs of boundary conditions that are associated with each type of support. #### EXAMPLE A uniform beam (density , cross–sectional area , flexural rigidity ) of length is fixed at one end and free at the other. Determine expressions for the natural frequencies and associated mode shapes for this beam. #### Complete Response of Lateral Motion of Beam We have again found an infinite number of solutions which satisfy the equation of motion 10.26 given by (10.33) where is the natural frequency and is the associated mode shape, both of which depend on the specific boundary conditions present. The general solution will then be a superposition of all of the solutions in 10.33, (10.34) The constants and are to be determined from the initial conditions of the beam. If the initial displacement and velocity of the beam are specified as then 10.34 gives and can then be found from (10.35) (10.36) where once again the orthogonality property of the mode shape has been used This is once again beyond the scope of this course.

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## Vibrations of Continuous Systems: Lateral Vibrations of Beams ### Equation of Motion Consider a beam, shown in Figure 10.6(a) which has a length , density (mass per unit volume) and Young’s modulus which is acted upon by a distributed load (per unit length) acting laterally along the beam. Let measure the lateral deflection of the beam and assume that only small deformations occur. Figure 10.6(b) shows the FBD/MAD for an infinitesimal element of the beam with mass . By applying Newton’s Law’s and considering moments about the left end of the element we find As a first order approximation we can write that so that the above becomes or (10.23) which is a well known result from beam theory. In the vertical direction we then have Note: For small motions, the shear forces act vertically to a first order approximation. The actual vertical components would be for example where can be obtained from the slope of the beam. However, for small motions, , so is used here as the vertical component. A similar remark holds for the term. Now, using 10.23 we can write that so the equation of motion becomes (10.24) For small deformations the bending moment in the beam is related to the deflection by so that 10.24 becomes or (10.25) This is the general equation which governs the lateral vibrations of beams. If we limit ourselves to only consider free vibrations of uniform beams (, is constant), the equation of motion reduces to which can be written (10.26) where (10.27) Note that this is not the wave equation. ### Solution To Equation of Motion Once again we look for solutions which represent a mode shape undergoing \sshm of the form This results in so 10.26 becomes Separating the and terms we find Again, since the LHS depends only on and the RHS depends only on and they must be equal for all values of and , both sides must be equal to a constant, which we call , so that This results in equations or (10.28a) (10.28b) where (10.29) Note for future reference that so that (10.30) The solution to 10.28b is as we have seen To find the solution to 10.28a, which is a 4 order linear ODE with constant coefficients, we assume a solution of the form The equation of motion then becomes or The four roots to this equation are The total solution will be a linear combination of solutions, one for each of the above roots, (10.31) where each of the may be complex. However, by using Euler’s identity and introducing the hyperbolic and functions 10.31 can be written or (10.32) where The advantage of expressing the solution as in 10.32 is that all of the terms involved are real. We see that and are complex conjugates, while and are real. As can be seen there are four constants to be determined which requires that four boundary conditions be specified. These will often be determined with one pair of boundary conditions specified at each end, depending on the type of support. Figure 10.7 shows three of the most common support conditions and the pairs of boundary conditions that are associated with each type of support. #### EXAMPLE A uniform beam (density , cross–sectional area , flexural rigidity ) of length is fixed at one end and free at the other. Determine expressions for the natural frequencies and associated mode shapes for this beam. #### Complete Response of Lateral Motion of Beam We have again found an infinite number of solutions which satisfy the equation of motion 10.26 given by (10.33) where is the natural frequency and is the associated mode shape, both of which depend on the specific boundary conditions present. The general solution will then be

a superposition of all of the solutions in 10.33, (10.34) The constants and are to be determined from the initial conditions of the beam. If the initial displacement and velocity of the beam are specified as then 10.34 gives and can then be found from (10.35) (10.36) where once again the orthogonality property of the mode shape has been used This is once again beyond the scope of this course.