RATIO LESSON PLAN PART A

LESSON PLAN PROPORTIONALAMOUNTS PART A΄ Stella Seremetaki Pure Mathematician

Defining the cognitive object.a Similar amounts - Properties of similar amountsb. Define the goal.• Identify whether there is a proportion of the two-dimensional change• Fill in tables of similar amounts when giving their reason• Calculate the ratio of two similar amounts when giving their tables• Use rate as a special case rate factorc. Organizing matter.(1) Development of proposed activities (20 min):1st Students discover (Bruner) that when two amounts change together the changeis not always proportional2nd The use of the coefficient and the proportional tables.(2) Suggested applications presentation (20 min)

d. Method selection.Method: Building knowledge (according to Knowledge Theory) through the active participationof students (Piaget) through the development of activities.1st activity: Students compare 56: 1.60, 81: 1.80, 63: 1.75, 68: 1.70and find the ratio ratios: 35, 45, 36 and 40, which are different from each other. They areexpected to conclude on their own that the original claim does not apply.2nd activity: If the students divide the proceeds with the value of kilo 0.4e they will find thepounds they sold each time and in total, ie 15 + 7 + 13 + 8 + 9 + 12 + 6 + 4 + 11 + 5 =90 So it would seem that he forgot to score 10 pounds for 4e.An attempt is made to draw students conclusions about:(a) the wording of the definition of the relevant amounts or sizes; (b) the relationship thatlinks them; and (c) the role of the proportion as a ratio factor.

e. Assignment of work.(3) Exercises are offered for exercises - problems (5 min)f. Recap.(4) The suggested exercises - problems (20 min)(5) Self-assessment test (20 min)(6) The definitions and rules (5 min)d. Materials and Supervisory Tools: Student Book, Table.

Chapter firstSimilar Amounts - Conversely Amounts Amounts1.1. Showing points at level1.2. Two Number Ratio - Ratio1.3. Similar amounts - Properties of similar amounts1.4. Analog proportional graph1.5. Proportional problems1.6. Inversely proportional amountsRepetitive Self-Assessment Questions

Showing points at level1.1 I design a wheelchair system I find the coordinates of a point I find a point when given its coordinates Two Number Ratio - Ratio1.2 I understand the meaning of speech and the meaning of analogy I solve equations of the form αx = β, by searching for the fourth analogof the relationship. I know that it is generally

Similar amounts - Properties of similar amounts1.3 I recognize if there is a proportion of the two-dimensional change I fill out tables of similar amounts when given their reason I calculate the ratio of the two proportional sums when their tables aregiven. I use the rate as a special case rate factor1.4. Analog proportional graph I graphically represent a ratio I find that the points with coordinates of the pairs of the correspondingvalues of two similar sums are in a semi-alignment beginning at thebeginning of the axes

Proportional problems1.5 I organize the data of a table-size problem and buildit on the basis of this table, where necessary, and graph1.6 I solve problems by applying, where necessary, theproperties of such sums in two frames: numerical andgraphic

In order to determine the location of a point in the plane: We designtwo perpendicular half-axes Ox and Oy. On each of them we define thesame unit of measurement. These semi-axles are a semi-axle system.Picture.The half-axle Οx is called half-axle of the abscissa or x-axis of x.The semi-axle Οy is called semi-axle of the ordinate or semi-axle of y.Point C is called the beginning of the axles3 is the abscissa of point A(3,1) 1 is the ordinate of point A(3,1)

The abscissa and ordinate of point A are named coordinates of A and usuallywhen we want to refer to point A, we write A (3.1).The pair (3,1) whose firstnumber 3 is the abscissa of point A and the second number 1 is the ordinate ofpoint A is called ordered pair, because the order, ie the order in which thenumbers that make up it.With this system, we assign to each point A a pair ofnumbers (3,1), that is, an ordered pair whose numbers are called coordinatesof the point.Conversely, any ordered pair of positive numbers e.g. (2,4)corresponds to a point M of the plane.The semi-axle system we use is called ortho-normal, because the semi-axles areintersected vertically (ortho-) and we have the same measuring unit on them(normal).

The reason for two similar sizes, expressed in the same unit of measurement,is the quotient of their measures. Equality of words is called analogy. Two shapes are said to be the same when one is smaller or larger than theother. The ratio of the two-point distance of an image of an object to the actualdistance of the two corresponding points of the object is called a scale. If the ratios of the respective sides of two parallels are equal, then theywill be equal to the ratio of their perimeters. Each image ratio ratio is equivalent to the Image relationship

We measure a distance on a map with a scale of 1: 10,000,000and we find it equal to 2.4 cm. What is the true distance of thetwo points?After the scale of 1: 10,000,000 is given, 1 cm of the map corresponds to10,000,000 cm in reality.Therefore, if the 2.4 cm of the map corresponds to x cm in realitywe will have: Picture. It therefore follows that:1 · x = 2.4 · 10.000.000 or x = 24.000.000 cm = 240.000 m = 240 Km.

Two sums are called accordingly, if they are changed in sucha way that when the values of one are multiplied by a numberthen the respective values of the other are multiplied by thesame number.Two sums x and y are proportional when their respectivevalues always give the same quotient: Figure. Quotient a iscalled a ratio factor.

The proportional sums x and y are associated with the relation:Image where α is the ratio coefficient. When the amount y is a percentage of the x, the two sums arelinked to the Image relationship and are proportional, with aratio factor of the Image or%.The relation y = α · x expresses an interaction of the x and ysums.In particular, doubling, tripling, and so on. of one amountresults in doubling, tripling, and so on. of the other amount.

Which of the following amounts are appropriate:(Place an \"x\" in the corresponding position)TRUE OR FALSE(a)The number of beverages and the money they cost(b) The floor area and the number of slabs that are paved(c) The number of workers and the time required to complete a project.(d) The length and width of a rectangle of a given area(e) The speed and time required to cover a distance.(f) The side of a square and its area.(g) The age of a person and his property.(h) The amount that someone spends to buy lotteries and the amount theyearn.

Ratio / Definition: Equality of ratios is calledanalogy, ie the relations α / β = γ / δ, α / β = 3/5express ratios

Exercise 1. page 92 FigFind the reasons for the aligned parts of the drawing on page 92of the school book,at the link belowebooks.edu.grSolution: I see from the exercise scheme of Fig. 6 on page 92 that AB= 4 pm, DG = 1 pm, EZ = 5 μm, RH = 2 μm, K = 3 μmTherefore the proportions are as follows(a) AB / DT = 4/1, EZ / RH = 5/2, CL / AB = 3/4, AB / CL =4/3, RH / EZ = 2/5, DG / AB = 1/4b) DG / RU = 1/5, RU / RL = 2/3, AB / AB = 4/4 = 1, RU / 1/1=1

Exercise 2 p.92 figRectangle ABDC is given. Design another rectangle with sides similar tothe sides of the ABGB so that the ratio of their respective sides is 2: 1Solution: AB = 4.5m and DB = 2.5mLet the desired rectangle with x and y sides. From the exercise case, Imake the ratios of 4.5 / χ = 2/1 (1) and 2.5 / β = 2/1 (2)From (1) multiplying the buckle I have 2x = 4.5 division and bothmembers with 2 and i have x = 2.25From (2) multiplying buckle I have 2y = 2.5 divided and both memberswith 2 and I have y = 1.25

Exercise 3. page 92 FigAt the link http://ebooks.edu.gr/modules/document/Actual height 1.76 m = 176 cm, Fantastic 4cm. HowMuch Are Items Diminished?Solution: The ratio is P / Φ = 176/4 = 44. Now I haveshrunk by 44 times

Lost exercises similar to the book for practice:1. Give the ABGB rectangle with side AB = 4 and the AB relationship is valid= 6.GD. (1) Check whether the AB and DD sides are analogous.Solution: From relationship (1) I have: AB / DG = 6/1. So I have equality ofreasons, so it's analogy.Generally: A relation of the form a = w, where k is a physical number expressesa ratio between α and β2. Give a rectangle with sides x and x + 1 .1) Find the perimeter of P. 2) If x =2, examine whether the x and p sums are analogousSolution: 1) P = x + 1 + x + 1 + x + x = 4x + 2. 2) For x = 2 I have Π = 10,therefore Π / χ = 10/2 = 5. The sums Π and x are analogous

2. Give rectangle with sides x and x + 1 .1) Find the perimeter of P. 2)aIf x = 2, examine whether the x and p sums are analogousSolution: 1) P = x + 1 + x + 1 + x + x = 4x + 2. 2) For x = 2 I haveΠ = 10, therefore Π / χ = 10/2 = 5. The sums Π and x are analogous3. If the dimensions of a room in a 1: 100 scale is 2 x 5. Find theactual dimensions of the room.Solution: Let x and y be the actual dimensions of the room. I will have2 / x = 1/100 equivalents x = 200 and 5 / y = 1/100 equivalent y =500. Then the actual dimensions are 200x500

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