Similar to a square, a rectangle is also referred to as an equiangular quadrilateral. Since both a square and a rectangle have an equal number of sides, i.
We can observe the similarities in the properties of both a square and a rectangle in the following section. Both a square and a rectangle have certain special properties that distinguish them from a general quadrilateral. We can draw the comparison of the requisite properties of both the shapes from the following table and conclude whether a square has all the properties that define a rectangle.
From the comparison drawn above for the common properties shared between a square and a rectangle, we observe that a square has all the properties that define a rectangle, which makes them alike in a certain manner. This means that a square can also be referred to as a type of rectangle. Yes, a square is a special type of rectangle because it possesses all the properties of a rectangle. Similar to a rectangle, a square has:. A square is called a special kind of rectangle because it possesses some additional properties which do not apply to rectangles.
They are:. Example 1: Write a few common properties between squares and rectangles. We know that a square is a special rectangle. Thus, all the properties of a rectangle will also be the properties of a square. Some of the properties include:. Example 2: There is a park near Sam's house. The length of the sides of the park is meters, meters, meters, and meters. Determine whether the park is a square or a rectangle. The length of each side of the park is given as meters, meters, meters, and meters.
Now, we know that all four sides of a square are equal. Hence, this is not a square. Given that a rectangle has opposite sides equal, we can see that the given dimensions are of a rectangle, and not a square.
The answer is Yes. Popular pages mathwarehouse. Surface area of a Cylinder. Unit Circle Game. Pascal's Triangle demonstration. Create, save share charts. Interactive simulation the most controversial math riddle ever!
Calculus Gifs. How to make an ellipse. Volume of a cone. Similarly, the problem of counting all the chairs in a house requires that we make decisions about objects that may be as disparate as easy chairs, stools, chaise lounges and probably even step ladders. For each object considered, color, form, size and materials need to be taken into account and a single dichotomous decision arrived at, i.
It goes without saying that the only sort of number that can result from the act of counting is an integer. On the other hand, in the case of mass nouns, we can introduce adjectival quantity only after we have singled out the attribute of the referent object that we wish to quantify. We say, for example, three hundred grams of clay. In so doing we ignore the shape of the clay and its color and focus solely on its weight more properly its mass, in this case.
Because of these considerations, it is clear that the result of assigning size to some attribute of a mass noun, an act we normally call measuring , necessarily results in a non-negative rational number. Perceptually available simple measures. These include:. Composite measures - Qualitative relations among simple measures.
We can form combinations of perceptually available measures in order to generate more complex measures that may be more informative in more complex situations. Clearly the simplest form of combination is one that combines two simple measures. We consider first a form of combination that leads to Direct Variation. We refer to such combinations as Product Measures. Assume we have a collection of objects say, pickup truck, fast pitched baseball, car, motorcycle, bicycle, etc.
And if so, how much closer? First thoughts would suggest that the relevant properties of the moving objects that we want to pay attention to are how heavy they are and how fast they are moving. Each of the moving objects has both a mass and a speed.
Assuming we have the proper instruments for measuring speed and mass, we could measure both the speed and the mass for each of the moving objects. A baseball, which is clearly less massive than a car, can be much harder to stop that a slowly rolling car. By the same token, even a very slowly rolling locomotive can be much harder to stop than a fast moving car [a fact that can be attested to by many drivers injured in attempting to beat a train across a grade crossing].
Let us put on hold for the moment making such a measure quantitative and turn to its logical complement - a form of combination that leads to Inverse Variation. We refer to such combinations as Quotient Measures.
A Quotient Measure is a composite measure C that depends on two component measures A and B such that. Clearly each piece of aluminum, pine and polystyrene has both a volume and a mass. Composite Measures - Quantitative Relations among simple measures. In many situations we go beyond the qualitative variation considered in the previous section.
Consider, for example a product measure such as momentum. Returning to the qualitative attempt at formulating a measure that captures a direct variation — the problem of stopping a moving object. This sort of measure avoids the difficulties that the previous proposed measure had.
Here, for example is a task that requires the formation of a product measure — one that is well-known to be difficult for middle school, and even upper elementary students.
Two Sandwich Shops. Shop A offers 3 kinds of cheese, 6 kinds of meat, and 3 kinds of bread. Shop B offers 4 kinds of cheese, 4 kinds of meat, and 4 kinds of bread.
Which shop has the larger offering? How much larger? Design a menu offering that is. Lest one think that formulating product measures is a problem only for young students consider the following example of a measure of efficiency for different aircraft types.
Efficiency of Airplanes. Data of the following sorts might be of interest. Number of passengers that can be carried. Average range [mi].
Typically, product measures are Cartesian Products — here are some examples. Such Product Measures are almost always extensive quantities and are rarely if ever dimensionless. Returning to the qualitative attempt at formulating a measure — one that captures an inverse variation — the problem of characterizing a material rather than an object made of that material.
Assuming we have the proper instruments for measuring volume and mass, we could measure both the volume and the mass for each of the pieces of aluminum, pine and polystyrene. We can find some pieces of pine with more mass than pieces of aluminum and some pieces of pine with less mass than pieces of aluminum. Here are data from the measurement of masses and volumes and a scatter plot of those data. A scatter plot of these data looks like this:. Normally, we present the concept of density somewhere in middle school as the ratio of mass to volume.
If however, we proceed from the data, we see that doing that constitutes a leap! We have a finite number of data points — the assertion that.
For all non-negative values of mass and volume is an assertion of a model. It is a predictive model allowing us to predict the magnitude of the mass of a piece of material whose volume we have measured but whose mass we have not measured. Similarly, the model allows us to predict the value of the volume of a piece of material whose mass we have measured but whose volume we have not measured. Quotient Measures such as density are almost always intensive quantities and are sometimes dimensionless.
There is a caveat that must be expressed here about this model of density. This is fundamentally a formulation of a macroscopic model of density.
It depends on the assumption that seems to work perfectly well in our macroscopic world. The reader is invited to consider whether these assumptions are true on a microscopic scale or on a galactic scale. Some perceptually based Quotient measures. As a way of ramping up to the introduction of measure formulating tasks into the curriculum we describe here a series of such tasks that are largely based on visual perception and that require little or no external or prior knowledge to begin the task.
The full tasks are to be found on the challenge section of this website. Nonetheless it presents a nice opportunity to have students devise measures — several perfectly reasonable ones are likely to arise in discussion — and to discuss their relative merits. Students are shown a collection of rectangles of varying sizes and aspect ratios and presented with the following questions -.
Given a coin, a tuna-fish can and a piece of uncooked spaghetti:. By how much? Students are shown pictures of a tennis ball, an orange, a basketball and the Earth and asked -. A maximum smooth-ness? Densities can be linear densities as in the case of traffic flow. Densities can be area densities as in the illustrative example. Densities can be volume densities as in the classic case of mass density.
Other volume densities that we have found students have much difficulties with include electric charge density and energy density in energy storage media such as batteries or capacitors as well as energy density in propagating electromagnetic waves. Additive measures. Are there measures that are more complex than product measures and quotient measures that play a role in K education? The cost of renting a car can depend on both the number of days the car is rented as well as the number of miles driven.
Renting a Car. Time Cost is a product measure. Distance Cost is a product measure. Additive Measures of the form. In general, measures in the social sciences have this sort of structure.
Probably the best known example is Gross National Product GNP which is a linear combination of a large number of indicators with suitable coefficients so that the combined measure is in dollars. Choosing a measure to characterize the state of a system is an exercise in deciding what features of the system one wants to include in ones characterization and what features are sufficiently unimportant for the purpose at hand that they may be neglected, at least at first.
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