Interest. Calculating percentages of a number and a number based on a known percentage, expressing a ratio as a percentage
Methodical comment
At the center of the study of the material of this paragraph is the task: to determine how many percent one value is from another. An approach has been adopted according to which we first find what part one value is from the other, and then we express this part as a percentage. Therefore, it is important to focus on two points: to repeat the solution of the problems considered at the beginning of the year (paragraph 1.4 of the textbook, problems of the type 65 -67 ), and work out the ability to move from decimal and ordinary fractions to percentages (exercises 533 -536 ).
Solving problems 537 -543 it is advisable to carry out in two stages: express a part (share) of the value as a fraction and express the fraction as a percentage.
When solving problems 544 and 545 as well as tasks 550 and 551 it is recommended to check the answer by composing and solving the inverse problem. For example, having solved the problem 551 "A", we get the answer: the share price fell by 20%. Now you can create and solve the following problem: “In September the share cost 250 rubles, and in October its price fell by 20%. What was the share price in October? "
Considerable attention is paid to estimation tasks aimed at developing a "feeling" of a percentage as a certain fraction of a value (exercises 546 -549 ).
Exercise commentary
536. In this example, it is expedient to switch from an ordinary fraction to a decimal using the basic property of a fraction.
537. To answer the question of the problem, you must first answer the question: "Which part ...?"
544, 545. The first question is: "To which part ...?"; the second: "By what percentage ...?".
548.
One can argue as follows: a) the shaded part is slightly more than a quarter of the circle and much less than half of it, that is, the answer can be B - 27%; d) a third of the figure is shaded, that is, approximately 33% - answer B;
f) less than 50% of the circle is shaded, that is, you must choose the answer B - 45%.
551. The choice of the value, in relation to which the percentage of the increase or decrease in price is calculated, requires attention.
554. You can organize your work into groups and then combine the results.
Chapter 7. Symmetry (8 Lessons)
| Tutorial item | Number of lessons | Workbook | |
| 7.1. Axial symmetry | 47-50 (p. 74-76) | Recognize flat shapes that are symmetrical about a straight line. Cut two shapes out of paper that are symmetrical about a straight line. Use the tools to build a figure (segment, polyline, triangle, rectangle, circle) symmetrical to a given one with respect to a straight line, draw by hand. Draw a straight line with respect to which two figures are symmetrical. Construct ornaments and parquets using the property of symmetry. Formulate the properties of two shapes that are symmetrical about a straight line. Explore the properties of figures that are symmetrical about a plane using experiment, observation, modeling. Describe their properties | |
| 7.2. The axis of symmetry of the figure | 51-56 (p. 77-78), 79, 80 (p. 87), 94 (p. 96) | Find flat and spatial symmetrical figures in the surrounding world. Recognize shapes that have an axis of symmetry. Cut them out of paper, draw them by hand and using tools. The symmetry of the figure was carried out. Formulate the properties of isosceles and equilateral triangles, rectangles, squares, circles associated with axial symmetry. Formulate the properties of a parallelepiped, cube, cone, cylinder, ball, associated with symmetry about the plane. Construct shapes using the property of symmetry, including using computer programs | |
| 7.3. Central symmetry | 57-65 (p. 79-81) | Recognize flat shapes that are symmetrical about a point. Build a figure symmetrical to a given point with the help of tools, complete it, draw by hand. Find the center of symmetry of a figure, configuration. Design ornaments and parquets using the property of symmetry, including using computer programs. Formulate the properties of figures that are symmetrical about a point. Investigate the properties of figures with an axis and center of symmetry, using experiment, observation, measurement, modeling. Put forward hypotheses, formulate, substantiate, refute with the help of counterexamples statements about the axial and central symmetry of figures | |
| Review and control |
Basic goals: give an idea of the symmetry in the world around; to acquaint with the basic types of symmetry on the plane and in space; gain experience in constructing symmetrical figures; to expand the understanding of known shapes, introducing the properties associated with symmetry; show the possibilities of using symmetry in solving various problems and constructions.
Chapter Overview. The chapter deals with axial and central symmetry, as well as examples of symmetry in space.
The study of axial and central symmetry is based on the same scheme: in the course of a physical action, the concept of points symmetric with respect to a straight line (center) is introduced; the features of their location relative to the axis (center) of symmetry are analyzed and on the basis of this a method for constructing symmetric points is formulated; figures are considered that are symmetrical about a straight line (point), and the fact of their equality is fixed; the concept of the axis (center) of symmetry of the figure is introduced; the presence of the axes (center) of symmetry in the known figures is established.
The study of the types of symmetry and its properties is based on actual actions and physical experiment. For axial symmetry, this is bending along the axis of symmetry; for central symmetry, this is a 180 ° rotation.
As the main vehicle for the formation of ideas about symmetry, these actions should be a constant part of all lessons.
So, the introduction of the concept of points symmetric with respect to a straight line (point) should be accompanied by the practical steps described in the textbook (pp. 145, 149). Likewise, students should make sure that the symmetrical shapes are equal with the actual superimposition. (For this, it is convenient to transfer the drawing to tracing paper and bend or rotate 180 °.) It is also advisable to resort to experimental verification in order to confirm or refute the conclusion to which the student came as a result of mental actions. So, for example, to make sure that the triangles in the problem 560 asymmetrical, you can transfer the drawing to tracing paper and bend along a given straight line.
One of the main skills that students must master is the construction of a figure (point, segment, triangle, etc.) symmetrical to a given one. Note that along with teaching the construction of symmetrical figures by points with the help of tools, one should strive to ensure that students can present the entire symmetrical image, draw it by hand. We would like to emphasize that when constructing symmetrical points, students have the right to use any tool. As for the constructions with a compass and a ruler, they should be considered as additional material with which it is advisable to familiarize strong students.
We draw the teacher's attention to the fact that of the two types of symmetry - axial and central - the most difficult to learn is central symmetry. In this regard, the ability to construct a figure symmetrical to a given one relative to the center is not included in the mandatory learning outcomes. The main goal of studying this material is to form an idea of central symmetry as a 180 ° rotation. Therefore, it is important to ensure that students understand the 180 ° turn of speech and can perform this turn. When rotated by 180 °, the point occupies a position opposite to the center, that is, it turns out to be on the same straight line (passing through it and through the center), but on the other side of the center.
It is helpful for students to experiment with different centrally symmetrical shapes. For example, you can draw a rectangle in a notebook, draw its diagonals and make sure that the points of their intersection are the center of symmetry of the rectangle. To do this, you need to transfer the drawing to tracing paper, fix it at the point of intersection of the diagonals and rotate the rectangle on the tracing paper around this point by 180 °. Both rectangles will be aligned again. Next, you should discuss which vertices are aligned during this turn, which sides, angles, etc.
Among the shapes that students experiment with, there should be an equilateral triangle. By bending over, students can make sure that it has three axes of symmetry. If the inflexions are done carefully, the students will get the point of intersection of the axes of symmetry. Here you can make sure that this point is not its center of symmetry.
Control materials.
Manual "Test work". Verification works: 5. Axial symmetry; 6. Center and axis of symmetry of the figure.
Axial symmetry
Exercise commentary
560. You can transfer the drawing to tracing paper and perform folding.
562. We remind you that on checkered paper, constructions are made using its properties.
567. When completing the task, you can use the mirror.
569. Ask the students to first explain how the axis of symmetry should go relative to two symmetrical points.
570. The fastest will be the staining in which after the first bend, 2 colored squares will be obtained, after the second - 4, after the third - 8, and the fourth will be the last - all 16 squares will be colored. One of the possible color options is shown in Figure 8. (The number inside the square shows how the square turned out to be colored.)
If desired, the answer can be obtained using an experiment. To do this, on a separate sheet of paper, you need to reproduce the drawing and paint over the black square with a very soft pencil.
The axis of symmetry of the figure
Exercise commentary
581. It is advisable to illustrate the answer by folding an equilateral triangle cut out of paper.
584.
A triangle has 3, a quadrangle has 4, a pentagon has 5,
the hexagon has 6, etc.
586, 587. Students can use a mirror when completing assignments.
588. You need to start the solution by looking at Figure 7.14 of the textbook. It can be seen from the figure that the vertex that does not belong to the base lies on the axis of symmetry of the triangle.
The sequence of constructions will be as follows: a segment equal to
6 cm; a straight line is drawn through its middle, perpendicular to this segment; on this straight line any point is chosen and connected to the ends of the segment. Construction can be performed using any tools, as well as on checkered paper using its properties.
589. First, using two inflections, we get two perpendicular lines. With the third bend, you need to bend the formed right angle. Expanding a sheet of paper, we will see four isosceles triangles, one of which must be outlined with a pencil. It is useful to mark its equal sides and equal angles.
591. The first body has two planes of symmetry, the second has one, the third has none, the fourth has one.
Central symmetry
Exercise commentary
598. If in some cases it is easier for students to construct a point symmetrical about a given point, not by cells, but with the help of a ruler, they can do it.
601. Students may find it easier to draw if they represent the vertices of the shape with letters.
607. You can use the pictures in this chapter of the tutorial.
Chapter 8. Expressions, formulas, equations (15 lessons)
Approximate lesson planning of teaching material
| Tutorial item | Number of lessons | Didactic materials | Characteristics of the main activities of students |
| 8.1. About mathematical language | O-44, P-34 | Discuss the features of the mathematical language. Write mathematical expressions taking into account the rules of the syntax of the mathematical language, compose expressions according to the conditions of problems with literal data. Use letters to write mathematical sentences, general statements; translate from mathematical language into natural language and vice versa. Illustrate general statements written in letter form with numerical examples | |
| 8.2. Literal Expressions and Numeric Substitutions | - | Build speech constructions using new terminology (literal expression, numeric substitution, meaning of an alphabetic expression, acceptable values of letters). Calculate the numeric values of literal expressions given the values of the letters. Find valid letter values in an expression. Answer questions on letter data problems by constructing appropriate expressions | |
| 8.3. Formulas. Formula calculations | O-45, P-35, P-36 | Draw up formulas expressing dependencies between quantities, including according to the conditions specified in the figure. Calculate by formulas, express one quantity from a formula through others | |
| 8.4. Formulas for circumference, area of a circle and volume of a sphere | Find experimentally the ratio of the circumference to the diameter. Discuss the singularities of the number π; find more information about this number. Get acquainted with the formulas for the circumference, area of a circle, volume of a ball; calculate by these formulas. Calculate the dimensions of shapes bounded by circles and their arcs. Round off calculation results using formulas | ||
| 8.5. What is an equation | O-46, "Check yourself", P-37 | Build speech constructions using the words "equation", "equation root". Check if the specified number is the root of the equation in question. Solve equations based on dependencies between action components. Create mathematical models (equations) according to the conditions of word problems | |
| Review and control |
Basic goals: to develop students' ideas about the use of alphabetic symbols, to form elementary skills in composing alphabetic expressions and calculating their values, as well as working with formulas, to give an initial idea of an equation with one variable.
Chapter Overview. The chapter includes material related to the algebraic block of the content of the mathematics course for grades 5-6. It is grouped around three fundamental algebraic concepts: expression, formula, equation. The presentation of the material is based on acquaintance with the mathematical language, translation from natural language into mathematical, the use of mathematical language to describe reality.
First, the issue of using letters to denote numbers is discussed, the concept of an alphabetic expression and such related concepts as "numeric substitution", "the meaning of an alphabetic expression", "acceptable values of letters" are introduced. At the elementary level, relevant practical skills are practiced.
Experience with literal expressions is the basis for the next segment, which explores the issue of formulas. A formula for students is literal equality, which describes a rule in symbolic language. Students write down in the form of formulas the rules for calculating certain quantities known to them (perimeter and area of a rectangle and square, volume of a rectangular parallelepiped, etc.) and get acquainted with new geometric concepts and corresponding formulas (circumference, area of a circle, volume of a ball).
The chapter ends with a discussion of the question of equations. The equation appears as a result of translating the condition of a word problem into mathematical language. Equations are solved at this stage of studying the course by a technique known from elementary school - based on the dependence between the components of actions. We emphasize that this fragment, in its didactic role, serves as an introductory stage to the topic "Equations", the study of which will begin in the course of 7th grade algebra.
Control materials.
Manual "Test work". Test 7. Letters and formulas.
Manual "Thematic tests". Test 14. Letters and formulas.
About mathematical language
Methodical comment
Students already have experience of using letters to write the simplest expressions, properties of arithmetic operations, to designate an unknown number. They also know how to use such mathematical symbols as arithmetic signs, comparison signs, brackets. Now this knowledge and skills serve as the basis for a conversation about the mathematical language as a special language of science, which was created and improved along with the development of mathematics.
The exercises in the paragraph are aimed at developing the skills of reading and writing letter expressions and letter equalities. All work is carried out as an activity of translation from natural language to mathematical and vice versa. It is advisable to add tasks to the system of exercises of the textbook on meaningful interpretation of letter expressions, for example: “A kilogram of chocolates is worth a rubles, a kilogram of caramel costs b rubles. What could be bought if the purchase price (in rubles) is a+ b? 3b? 2a? 2a+ b? What is the meaning of the expression a– b?»
Lesson summary: Expression of attitude as a percentage.
6th grade. UMKDorofeeva G.V.
The purpose of the lesson: with formulate a rule for expressing the relationship as a percentage.
Regulatory goals: to teach to plan, control, evaluate their actions.
Communication goals: to teach to formulate their own opinions and positions, to teach to cooperate and to accept the opinions of their classmates.
Personal goals: to teach to use the information received to solve educational problems.
Metasubject goals: to teach to identify gaps in knowledge and be able to fill them.
Lesson Objectives:
Educational: teach techniques and methods of reasoning.Build skills solutionstasks, including tasks with practical content, with real data, to find the percentage of two values.
Developing: to develop the intellectual and creative abilities of students, logical thinking, mathematical speech (oral and written), attention, interest in mathematics, cognitive activity, outlook.
Educational: education of accuracy, accuracy, striving for continuous improvement of their knowledge, activity, sense of responsibility, self-confidence, education of elements of a culture of communication, respectful attitude to each other, mutual understanding.
Lesson type: combined.
Forms of work in the lesson : individual, frontal-collective.Teaching methods: verbal, visual, practical, problematic.
Equipment: interactive whiteboard (ID), drawing tools.
Lesson plan:
Lesson steps
Formable student learning activities
1. Organizational moment (1 min.)
Self-regulation
2. Actualization of knowledge (10 min.)
Compare and analyze, observe and refute wrong decisions. Assessment of available computational skills.
3. Goal setting and motivation (1 min.)
Forecasting, reflection
4. Learning new material (8 min.)
Understand the information presented. Building speech structures, rationalizing, applying an algorithm, proposing and testing hypotheses, the ability to analyze and respond to incoming answers
5. Physical exercise (2 minutes)
Aesthetic perception, health preservation, self-regulation
6. Securing the studied material
(18 minutes)
Formulate your thoughts verbally, be able to interact with a neighbor while completing an educational task; establish and compare different points of view before making a decision and making a choice. Compare your mode of action with the standard. Argue your point of view, argue and defend your position in a way that is not hostile to opponents
8. Summing up the results of the lesson, reflection
(5 minutes.)
Subject reflection, awareness of the relevance of the studied material. Comparison and comparison of personal successes with others.
During the classes
Stages
Teacher activity
Student activities
1. Organizing time
Greet and check the general readiness and individual students for the lesson.
Greet teachers, control their own readiness (on desks - notebooks, textbooks, pens, pencils, rulers, squares, diaries)
2. Knowledge update
Slide 1
Oral work:
№1. Questions: 1) What is percentage? 2) What is attitude? 3) What does the ratio show if the numerator is greater than the denominator? 4) What does the ratio show if the numerator is greater than the denominator? 5) How to express the ratio in the form of a decimal fraction?
№2.
Express as a decimal: 40%, 5%, 370%.
№3. Divide 480 by 5: 3.
№1. 1) One hundredth of the value.
2) The quotient of two numbers. 3) How many times the first number is greater than the second. 4) How much is the first number from the second. 5) Divide the first number by the second in a column.
№2. 40%=0,4
5%=0,05
300%=3,7
№3.
*5=480:8*5=60*5=300
*3=480:8*3=60*3=180
(or 480-300 = 180)
3. Goal setting and motivation
Today in the lesson we will continue to solve problems and learn how to express an attitude as a percentage. Who will try to formulate the purpose of the lesson?
Slide 2
Students write in a notebook: Cool work."Expression of attitude as a percentage."
Goal: Learn to express relationships as a percentage.
4. Learning new material
Task: 60 seeds were planted for growing cucumber seedlings. Sprouted 48 seeds. Determine how much of the seeds have sprouted?
What is known in the problem? How many seeds have you planted? How many seeds have sprouted?
What can you compose? What will this attitude show?
What ratio will show how much of the germinated seeds are from the planted seeds?
What fraction did you get?
Is it possible to convert this common fraction to a decimal? How?
Did you answer the problem question? How to formulate the answer correctly?
Can we answer the problem question using percentages?
What do I need to do?
How do I convert a decimal to a percentage?
Walk through the solution to this problem. Can we say that we expressed the ratio as a percentage? How did we do it? Create an algorithm for expressing the ratio as a percentage.
Students discuss solutions.
The number of planted and germinated seeds is given. 60 and 48. You can create a ratio that will show how much the first number is from the second.
Correct, reducible.
5. Fizminutka
Slides 3-5 ... + Write your name and surname with your eyes on the wall.
Students do eye exercises
6. Consolidation of the studied material
№ from the tutorial
№ 533 (a). 534, 535, 538 (a), 539 (a, b)
Slide 6
7. Summing up the lesson, reflection
Summarizes the lesson, assesses the work of students, reports homework.Slide 7 d.z. p.6.4 No. 533 (b), 538 (b), 539 (c, d)
What have you learned today? How do you express the ratio as a percentage?
Slide 8
Depict, depending on your self-esteem, in your notebooks one of the options for "smiley".
Slide 9
Thank you for the lesson.
Slide 10
How to express a relationship as a percentage. Calculate the ratio, give the answer as a decimal fraction. Multiply the resulting fraction by 100%.
Write down homework in diaries.
The percentage (or ratio) of two numbers is the ratio of one number to another multiplied by 100%.
The percentage of two numbers can be written using the following formula:
Percentage example
For example, there are two numbers: 750 and 1100.
The percentage of 750 to 1100 is
750 is 68.18% of 1100.
The percentage of 1100 to 750 is
The number 1100 is 146.67% of 750.
Example task 1
The plant's vehicle production rate is 250 vehicles per month. The plant assembled 315 vehicles in a month. Question: by what percentage did the plant exceed the plan?
Percentage of 315 to 250 = 315: 250 * 100 = 126%.
The plan was fulfilled by 126%. The plan was overfulfilled by 126% - 100% = 26%.
Example task 2
The company's profit for 2011 was $ 126 million, in 2012 the profit was $ 89 million. Question: by what percentage did profit fall in 2012?
The percentage of 89 million to 126 million = 89: 126 * 100 = 70.63%
Profit fell by 100% - 70.63% = 29.37%
The percentage expression of pot odds and the expression in the form of a ratio are two points that need to be seriously thought about and dealt with. This knowledge will be useful to you not only directly to improve your understanding of the pot odds themselves, but also give an idea of the chances of being able to complete your draw, and will also be useful during other mathematical calculations.
Below you will find two tables to help you learn how to convert ratios to percentages and vice versa.
- The first table shows the exact odds you will use based on the number of your improvement outs.
- The second table presents the rounded odds that you can use to quickly calculate pot odds. Those. if you need to call $ 5 to win a $ 20 pot, your odds are 4 to 1 (or 20% as a percentage).
Ratio and Percentage Representation of Outs
| Number of outs | Improvement on the next card - Attitude | Improvement on the next map -% |
| 1 | 46.0 to 1 | 2.1% |
| 2 | 22.5 to 1 | 4.3% |
| 3 | 14.7 to 1 | 6.4% |
| 4 (gutshot) | 10.8 to 1 | 8.5% |
| 5 | 8.4 to 1 | 10.6% |
| 6 | 6.8 to 1 | 12.8% |
| 7 | 5.7 to 1 | 14.9% |
| 8 (straight draw) | 4.9 to 1 | 17.0% |
| 9 (flush draw) | 4.2 to 1 | 19.1% |
| 10 | 3.7 to 1 | 21.3% |
| 11 | 3.3 to 1 | 23.4% |
| 12 | 2.9 to 1 | 25.5% |
| 13 | 2.6 to 1 | 27.7% |
| 14 | 2.4 to 1 | 29.8% |
| 15 (straight + flush draw) | 2.1 to 1 | 31.9% |
| 16 | 1.9 to 1 | 34.0% |
| 17 | 1.8 to 1 | 36.2% |
| 18 | 1.6 to 1 | 38.3% |
| 19 | 1.5 to 1 | 40.4% |
| 20 | 1.4 to 1 | 42.6% |
| 21 | 1.2 to 1 | 44.7% |
| 22 | 1.1 to 1 | 46.8% |
Simple conversion from relationship to interest and vice versa
| Attitude | Interest -% |
| 10 to 1 | 9% |
| 9 to 1 | 10% |
| 8 to 1 | 11% |
| 7 to 1 | 13% |
| 6 to 1 | 14% |
| 5 to 1 | 17% |
| 4 to 1 | 20% |
| 3 to 1 | 25% |
| 2.5 to 1 | 29% |
| 2 to 1 | 33% |
| 1.5 to 1 | 40% |
| 1 to 1 | 50% |
If you do not want to constantly refer to these tables, you can download yourself the hoRatio odds converter program, which will do all the dirty work for you.
Decoding a row of lines with outs
Gutshot Is a special kind of straight draw that requires only one card to complete. Here's a simple example: you have a board in your hands. You can only complete a straight combination if any comes on the turn or river.
Straight draw- a standard open street (OESD - open-ended straight draw) with a lot of outs for improvement. Example: on your board. You will be able to complete the combination of a straight if any or comes on the turn or river.
Flush draw- a situation when you are holding on the board and the emergence of another chirv card will complete your draw.
Straight + flush draw- a combination of OESD and flush draw at the same time. For example, when you have a board.
How to use conversion tables
The first table will be useful for comparing the ratio and percentage of probabilities versus the number of outs to improve your hand. Just by looking at the chart, you can see that the flush draw has 9 outs for improvement and the odds are 4.2: 1 as a ratio, or 19.1% as a percentage.
The second table will be useful for comparing and converting odds. Therefore, with this table at hand, you can calculate pot odds on the fly. For example, you need to call $ 10 to win a $ 50 pot. Pot odds are 5: 1. We look at the table and see that this corresponds to about 17%.
As we mentioned earlier, you can also use the hoRatio program to quickly convert any percentage expressions to ratios and vice versa. Perhaps it will turn out to be much more convenient and useful.
Converting odds in your mind
How to get a percentage from a fraction
To get a percentage from a fraction, you need to add two numbers from this fraction and divide the resulting number by 100.
For example, if you have a flush draw on the turn, the odds of completing your draw are 4.1: 1 (we will use an approximate value of 4: 1).
- The odds are 4 to 1, so we add two numbers from the ratio: 4 + 1 = 5.
- 100 / 5 = 20%.
Thus, if you have a 4: 1 chance of improving, then there is a 20% chance you will be able to complete your draw. It's simple.
How to get a fraction from a percentage
To get a fraction from a percentage, you need to divide 100 by the number of percent. Then subtract 1 (one) from the resulting number. As a result, you get the number "x", which can be substituted into the fraction "x: 1".
For example, if you have a flush draw on the turn and you know that the probability of completing your draw is 19.6% (assuming 20%), then you get the following:
- 100 / 20 = 5.
- 5 - 1 = 4.
Thus, the ratio will be 4 to 1.
Feel free to round up percentages to whole numbers to make it easier for you to divide in your head and make the calculations as easy as possible.