Mathematics
Scale Factors
Scale factors in mathematics refer to the ratio of the length, area, or volume of an original object to the length, area, or volume of a similar object. They are used to describe how much larger or smaller the new object is compared to the original. Scale factors are commonly used in geometry and are essential for understanding proportional relationships between objects.
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8 Key excerpts on "Scale Factors"
eBook - PDF- Doug French(Author)
- 2004(Publication Date)
- Continuum(Publisher)
Proportionality is the key property of similarity and it is an idea that is a considerable source of difficulty. That is a major reason for linking similarity to enlargement and for introducing Scale Factors because they offer a more accessible approach than equal ratios. Emphasis should be given initially to situations which involve doubling and other simple whole-number Scale Factors, which make better immediate sense to many students. Successful use of fractional Scale Factors is obviously very dependent on having acquired a real understanding of fractions and fluency with their use. Situations involving halving are considerably simpler than those involving any other fraction. A variety of examples which relate to familiar situations should be used alongside more traditional abstract configurations to develop understanding and fluency in determining Scale Factors and using them to calculate lengths, and at a later stage relating them to areas and volumes. The midpoint theorem and its generalization, the intercept theorem, provide powerful tools for solving problems and proving results. It is important to offer motivating examples which stimulate curiosity and have an aesthetic appeal, and which challenge students to think independently.
eBook - PDFMaking Sense of Number
Improving Personal Numeracy
- Annette Hilton, Geoff Hilton(Authors)
- 2021(Publication Date)
- Cambridge University Press(Publisher)
It is common to see Scale Factors on maps, house plans and scale models in museums. Scale Factors are usually represented as ratios. Some may include the units for ease of interpretation: for example, a regional tourist map in a brochure might have a scale of 4 cm : 20 km. Other Scale Factors do not include units: for example, an architect’s house plan might simply say 1:1 000 and we know that every 1 mm on the plan represents 1 m (i.e. 1 000 mm) in reality. Map reading is not as common as it once was now that we have access to digital maps on our phones, computers and satellite navigation in our cars. The maps on these devices (and on some hard-copy maps too) tend not to use ratio representations of scale. Rather, they have a linear scale with increments (e.g. centimetres) that indicate real distance, as shown in Figure 8.10. How would this be used? Learning activity 8.13 Look again at the graph in Figure 8.9 and answer the following questions: 1. How many students got a B score after the intervention? 2. How many fewer students got a C score after the intervention than before? (Answers in Appendix.) 0 125 250 375 m Figure 8.10 Example of scale factor on maps 154 Making Sense of Number It is useful to use scale factor when planning things around the house because drawings to scale allow you to make practical decisions. For example, you might draw a room plan for relocating or purchasing new furniture to make sure it will fit; you might be renovating the bathroom and want to estimate the amount of tiles or paint you will need; or you might be designing a garden or landscape layout. Learning activity 8.14 1. It is interesting to see what happens to the scale representation on digital maps. On your phone, open a maps app. Using your fingers, zoom in and zoom out on the map. As you do this, a scale factor diagram similar to that in Figure 8.10 should appear. Watch what happens to it as you zoom in or out on the map.
eBook - PDFGeometry Transformed
Euclidean Plane Geometry Based on Rigid Motions
- James R. King(Author)
- 2021(Publication Date)
- American Mathematical Society(Publisher)
C' D' A' B' C B A D Figure 7. Similar Figures with Scale Factor and Internal Ratios An example is shown in Figure 7, where the scale factor is 3/5 , so ‖? ′ ? ′ ‖ ‖??‖ = ‖? ′ ? ′ ‖ ‖??‖ = ‖? ′ ? ′ ‖ ‖??‖ = ‖? ′ ? ′ ‖ ‖??‖ = .6. But there are also internal ratios in these geometrical figures. A ratio of distances within ???? is equal to the ratio of corresponding distances within ? ′ ? ′ ? ′ ? ′ , but there is no reason for such internal ratios within a single figure such as ???? to be equal to each other. Here are two examples from Figure 7: ‖??‖ ‖??‖ = ‖? ′ ? ′ ‖ ‖? ′ ? ′ ‖ = 2; ‖??‖ ‖??‖ = ‖? ′ ? ′ ‖ ‖? ′ ? ′ ‖ = .75. The reason that the corresponding internal ratios are equal follows from the scal-ing, as in this example: ‖? ′ ? ′ ‖ ‖? ′ ? ′ ‖ = ?‖??‖ ?‖??‖ = ‖??‖ ‖??‖ . Theorem 8.14. Similitudes preserve internal ratios. More specifically, if ? is a simili-tude for any ?, ?, ?, ? with ? -images ? ′ , ? ′ ? ′ , ? ′ , these ratios are equal: ‖??‖/‖??‖ = ‖? ′ ? ′ ‖/‖? ′ ? ′ ‖ . Proof. The proof is essentially contained in the equation displayed before the theorem. □ The choice of two kinds of equal ratios can sometimes lead to confusion. The important point is that the scale factor is a ratio that is different from the others in that it controls the change in size for all distances. The other internal ratios represent relationships within the figures that are preserved, unrelated to the scale factor ? . To emphasize this distinction, we will try to use the term scale “factor” rather than scale “ratio”. 124 8. Dilations and Similarity Signed Ratios. A third ratio that plays an important role with similitudes is the signed ratio of points on a line. This ratio is only defined when the points are collinear. Definition 8.15. If ?, ?, ?, ? are points on a line 𝑚 with ? ≠ ? , the signed ratio ??/?? is a real number with |??/??| = ‖??‖/‖??‖ .
eBook - PDF- Paul A. Calter, Michael A. Calter(Authors)
- 2011(Publication Date)
- Wiley(Publisher)
The ratio of distances on the drawing to corresponding distances on the actual object is called the scale of the drawing. Use Statement 100 to convert between the areas on the drawing and areas on the actual object. ◆◆◆ Example 16: A certain map has a scale of . How many acres on the land are represented by on the map? Solution: If the area on the land, then, by Statement 100, Converting to acres, we have ◆◆◆ Volumes of Similar Solids Just as we thought of an irregular area as made up of many small squares, we can think of any solid as being made up of many tiny cubes, each of which has a vol- ume equal to the cube of its side. Thus if the dimensions of the solid are multiplied by a factor of k, the volume of each cube (and hence the entire solid) will increase by a factor of Volumes of similar solid figures are proportional to the cubes of any two corresponding dimensions. 101 Volumes of Similar Figures k 3 . A 4.20 10 9 in. 2 a 1 ft 2 144 in. 2 b a 1 acre 43,560 ft 2 b 670 acres A 168(25,000,000) 4.20 10 9 in. 2 A 168 a 5000 1 b 2 25,000,000 A 168 in. 2 1 : 5000 A 4.84 a 5.26 3.15 b 2 13.5 in. 2 A 4.84 a 5.26 3.15 b 2 4.84 in. 2 19 in. 2 19 in. 2 . 4.84 in. 2 . Areas of Similar Figures 500 Chapter 17 ◆ Ratio, Proportion, and Variation ◆◆◆ Example 17: If the volume of the smaller solid in Fig. 17–13 is find the volume V of the larger solid. Solution: By Statement 101, the volume V is to 15.6 as the cube of the ratio of 5.26 to 3.15. So we have the proportion Note that we know very little about the size and shape of these solids, yet we are able to compute the volume of the larger solid. The methods of this chapter give us another powerful tool for making estimates. ◆◆◆ Students often forget to square corresponding dimensions when finding areas and to cube corresponding dimensions when finding volumes. A scale model of something, such as a building, and the object itself, are similar solids.
eBook - PDFEngineering Drawing
Principles and Applications
- Lakhwinder Pal Singh, Harwinder Singh(Authors)
- 2021(Publication Date)
- Cambridge University Press(Publisher)
No. Category Recommended Scales 1. Reducing Scales 1:2 1:5 1:10 1:20 1:50 1:100 1:200 1:500 1:1000 1:2000 1:5000 1:10000 2. Enlarging Scales 50:1 20:1 10:1 5:1 2:1 -- 3. Full Size Scales 1:1 6.2 Representative Fraction or Scale Factor The ratio of the drawing to the object is called the representative fraction, abbreviated as RF. In detailed words, representative fraction (RF) or scale factor (SF) is the ratio of length of a line in the drawing to actual length of the object represented. Representative Fraction = Length of a line on the drawing Actu al length of a line on the object The dimensions in both numerator and denominator of the fraction must be in the same units. For example, if we wish to represent a dimension 2 m by a line 4 cm long, it will be evident that when the line is divided into 4 equal parts, each part will be 1 cm long and that this 1 cm length of the line represents 0.5 m on the actual object. So we might say that we are using a scale of 1 cm to 0.5 m. This is quite a common scale and is recommended by the BIS. Its R F = × = 1 0 5 100 1 50 cm cm . We call this scale to be 1:50 without any mention of cm, m or other linear units. 6.2.1 Construction of scales When the required scale is not available out of a set of recommended scales, it has to be constructed on the drawing sheet. For constructing a scale, the following information is needed: • The RF of the scale • The units it has to represent • The maximum length, required to be measured Length of scale = RF × maximum length to be measured by a scale; if it is not given, then we can assume the length of the scale 15 cm to 30 cm or in other words, we shall assume the maximum distance to be measured such that the length of the scale on calculation comes out to be 15 cm or so. 69 SCALES Scales used in engineering practice are mentioned by one of the ways given below: 1. 1:2 or ½ (RF) 2. half full size, etc. 3. 1 cm = 2 cm, etc.
eBook - PDF- Robert Reys, Mary Lindquist, Diana V. Lambdin, Nancy L. Smith, Anna Rogers, Audrey Cooke, Bronwyn Ewing, Kylie Robson(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
They found that the majority of the children could give correct answers, and that those who struggled did not realise that the problems involved ratio and proportion, did not have sufficient understanding of geometry, or had difficulties working with proportion and geometry or measurement within the problems (p. 76). Avan and Isiksal-Bostan provided a specific example where a child tried to use additive thinking to solve a scaling problem involving length (p. 77), highlighting how some children’s difficulties to progress from additive thinking to multiplicative thinking can impact. Using a zoom in or out on a computer screen or when photocopying both illustrate proportions between the original and its new image. The concepts of ratio and proportion can be naturally connected to geometry problems, as similarity is based on proportions. Two figures are similar if their respective sides are in the same ratio (i.e. proportional). Thus, all squares are similar, but all rectangles are not. Figure 13.9 shows drawings of similar rectangles. Notice that in each of the five rectangles, the ratio between the vertical side and the horizontal size is constant. For example, in the smallest rectangle, the ratio is 2:6 and in the largest rectangle the ratio is 6:18. The rectangles in figure 13.9(b) show that a single line passes through the vertex of each of these rectangles, and this demonstrates that the slope of the diagonals (i.e. the ratio of the vertical height to the horizontal length) of each of these rectangles is the same. This is visible from figures 13.9(a) and 13.9(b). The ratio table shown in figure 13.9(c) shows these lengths and the equivalence of each of the ordered pairs shown. The pattern can be extended in the graphs as well as in the ratio table. An examination of these ratios, such as 2:6, 3:9 or 5:15, confirms that each of them is equivalent, and the resulting gradient is 1 3 .
eBook - PDFScale Models in Engineering
Fundamentals and Applications
- Richard I. Emori, Dieterich J. Schuring(Authors)
- 2013(Publication Date)
- Pergamon(Publisher)
A few examples will demonstrate the relations among the five primary and the nearly infinite number of secondary Scale Factors. We stated at the beginning that homologous behavior with respect to speed would require a constant speed scale factor for all corresponding speeds, that is, * i ' Speed can be expressed as the first derivative of length with respect to time, so that . Geometrical and temporal homology requires, however, that 2 1 = J*j; , I z = Γέ^ , e*c, ατκΐ t 1 = tt; , ^ = t , cue, sothat ^* ** uiUï ί*
eBook - PDFIntroductory Mathematics
Concepts with Applications
- Charles P. McKeague(Author)
- 2013(Publication Date)
- XYZ Textbooks(Publisher)
shape size vertices ratio similar width 1. Similar figures are two or more objects with the same but are of a different . 2. If two figures are , then their corresponding sides are proportional. 3. When labeling the of a triangle ABC, we label the corresponding sides abc. 4. For two similar rectangles, we write a proportion as the of the height of the larger rectangle to the height of the smaller rectangle is equal to the ratio of the of the larger rectangle to the width of the smaller rectangle. Problems A In problems 1–4, for each pair of similar triangles, set up a proportion in order to find the unknown. 1. 2. 3. 4. In problems 5–10, for each pair of similar figures, set up a proportion in order to find the unknown. 5. 6. 7. 6 h 4 6 18 15 10 h 8 y 12 21 y 15 10 4 x 16 12 9 a 48 54 36 a 15 3 5 5.6 Exercise Set 275 8. 9. 10. B For each problem, draw a figure on the grid on the right that is similar to the given figure. 11. AC is proportional to DF. 12. AB is proportional to DE. 13. BC is proportional to EF. 14. AC is proportional to DF. 15. DC is proportional to HG. 16. AD is proportional to EH. 17. AB is proportional to FG. 18. BC is proportional to FG. x 40 24 9 y 50 40 40 y 30 42 28 A C B D F A C B D E B A C E F A C B D F A D B C H G A B D C E H B A D C E F G A D B C G F 276 Chapter 5 Ratio and Proportion Applying the Concepts Recall that the perimeters of two chessboards are proportional to the length of each of their sides. 19. Size of a Chessboard The perimeter of a chessboard is 50 inches and the length of each side is 12.5 inches. If a life- sized chessboard has a perimeter of 1,000 inches, use proportions to find the length of each side. 20. Size of a Chessboard The perimeter of a chessboard is 1,280 mm and the length of each side is 320 mm. If a life-sized chessboard has a perimeter of 25,400 mm, use proportions to find the length of each side. 21. Video Resolution A new graphics card can increase the resolution of a computer’s monitor.
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