Dilations
Dilations in mathematics refer to transformations that change the size of an object without altering its shape. They are performed by multiplying the coordinates of each point by a constant factor. Dilations can either enlarge or reduce the size of the original figure, and the center of dilation determines the fixed point around which the transformation occurs.
3 Key excerpts on "Dilations"
- Caryl Lorandini(Author)
- 2021(Publication Date)
- Barrons Educational Services(Publisher)
...Pupils become smaller when the light is bright, and larger in the dark. Eye doctors dilate eyes to check for signs of disease. This does not change the shape of your pupils, just the size. The same holds true for a dilation in mathematics. DEFINITION Dilation A transformation that stretches or shrinks a function or graph both horizontally and vertically by the same scale factor. There is a scale factor that indicates the size of the dilation and a center of dilation. EXAMPLE: Dilate Δ ABC by a scale factor of 3 about the origin. A (1, 2) → A ′ (3, 6) B (2, 4) → B ′ (6, 12) C (3, 1) → C ′ (9, 3) When you dilate, you multiply both the x -value and the y -value by the scale factor. When the scale factor is greater than one, the figure is enlarged. EXAMPLE: Dilate Δ ABC by a scale factor of about the origin. A (–3, 3) → A ′ (–1, 1) B (0, 6) → B ′ (0, 2) C (3, 3) → C ′ (1, 1) When you dilate, you multiply both the x -value and the y -value by the scale factor. When the scale factor is greater than one, the figure is enlarged. When you create a dilation, the idea is that the pre-image is made bigger or smaller, depending on the scale factor. The image and the pre-image are not congruent, but the shapes are similar or in proportion. EXAMPLE 13.5 1) Graph Δ ABC and then draw the image of ABC after a dilation of a scale factor of. Label the image A ′ B ′ C ′. A (–8, 0) B (0, 8) C (4, 4) 2) Graph Δ ABC and then draw the image of ABC after a dilation of a scale factor of 2. Label the image A ′ B ′ C ′. A (–1, –3) B (0, 2) C (3, 1) SOLUTIONS 1) 2) 13.6 What Is Symmetry? When you look at a Valentine’s Day heart, it appears to have two halves. The shape is said to have symmetry. Also, if you look at a pinwheel, it appears to have the same shape repeated; it has symmetry as well. DEFINITION Symmetry The property of having the same size and shape across a dividing line or around a point. There are two types of symmetry...
eBook - ePubUnderstanding Lesson Study for Mathematics
A Practical Guide for Improving Teaching and Learning
- Rosa Archer, Siân Morgan, David Swanson(Authors)
- 2020(Publication Date)
- Routledge(Publisher)
...In terms of van Hiele levels, when learning about transformations an important task in order to move from level 1, analysis, to level 2, abstraction, is to understand which variables remain unchanged after the transformation. This can pose significant difficulties since children are used to classifying transformed shapes as equivalent. Another difficulty that arises is linking the idea of enlargement with that of the ratio between sides. While enlargement can be intuitively understood, the idea of ratio might pose more difficulties and therefore get in the way of solving problems with enlargement (Johnston-Wilder & Mason, 2005). The concept of enlargement is also associated with multiplication and division; children are able to apply multiplicative reasoning to abstract concepts related to number but might struggle when applying these ideas to continuous quantities and geometric relationships (Nunes et al., 2009). For this lesson, we were primarily concerned with guiding learners to form a mental image of enlargement, recognising what changes and what stays the same when enlarging shapes using a COE. The étude also encourages a degree of freedom and creativity as well as making sure learners become procedurally fluent in enlargement. We will see in the analysis how the activity helps to develop relational understanding, as well as becoming fluent as described in the national curriculum in England (Department for Education, 2014). We will also return to the van Hieles levels both in the analysis and the ‘what have we learned’ sections. Suggested reading Foster, C. (2014). Mathematical fluency without drill and practice. Mathematics Teaching, 240, 5–7. Foster, C. (2017). Mathematical etudes. NRICH article available at: https://nrich.maths.org/13206 Jones, K. (1998)...
eBook - ePubTeaching Fractions and Ratios for Understanding
Essential Content Knowledge and Instructional Strategies for Teachers
- Susan J. Lamon(Author)
- 2020(Publication Date)
- Routledge(Publisher)
...So squaring the scale factor describes the change in area. The area of B is 4 times the area of A. Similarly, the volume for scaled figures (3-dimensional cubes) is (scale factor) 3 = 2 3 = 8. That is, the volume of a 4 by 4 by 4 cube is 8 times greater than the volume of a 2 by 2 by 2 cube. Similarity Here are two pictures of commonly recognized objects. What do you think they are? Why do you have to concentrate on the images and perhaps even make guesses as to what you are seeing? The images are distorted. They have been resized in a way that does not preserve the proportions of the original objects. When an object is shrunk or enlarged in such a way that all of its dimensions preserve the proportions within the original object, then we say that the objects are similar. Similar objects have to have the same shape; that is, one is just a smaller or a larger version of the other. One of the objects can be shrunk or enlarged until it is an exact copy of the other, that is, the original and the resized copy are congruent (same shape, same size). Unfortunately, the mathematical use of the term similar does not agree with the colloquial use of the term. These sailboats are similar. These dragons are similar. These rabbits are not similar. Human beings seem to have a built-in capability to recognize objects whether they are large or small in size, whether they are the real thing, or whether they are photographs or models with only some of the characteristics of real things. For example, even very small children recognize that the above sketches are models of a bunny. They will call each one a bunny, implicitly understanding that the picture does not have all of the characteristics of a real bunny—life, movement, furriness, size, and so on—and even though each is portrayed to a different degree of abstraction. In our everyday use of the word similar we mean showing some resemblance, but mathematically speaking, similarity is a more precise notion...
