Tag: <span>Facts</span>

05 Apr

Basic Math Facts – Exponents
Posted in Education by anshbookstore on April 5, 2023 @ 7:26 am

Exponents comprise a juicy tidbit of basic-math-facts material. Exponents allow us to raise numbers, variables, and even expressions to powers, thus achieving repeated multiplication. The ever present exponent in all kinds of mathematical problems requires that the student be thoroughly conversant with its features and properties. Here we look at the laws, the knowledge of which, will allow any student to master this topic.

In the expression 3^2, which is read “3 squared,” or “3 to the second power,” 3 is the base and 2 is the power or exponent. The exponent tells us how many times to use the base as a factor. The same applies to variables and variable expressions. In x^3, this mean x*x*x. In (x + 1)^2, this means (x + 1)*(x + 1). Exponents are omnipresent in algebra and indeed all of mathematics, and understanding their properties and how to work with them is extremely important. Mastering exponents requires that the student be familiar with some basic laws and properties.

Product Law

When multiplying expressions involving the same base to different or equal powers, simply write the base to the sum of the powers. For example, (x^3)(x^2) is the same as x^(3 + 2) = x^5. To see why this is so, think of the exponential expression as pearls on a string. In x^3 = x*x*x, you have three x’s (pearls) on the string. In x^2, you have two pearls. Thus in the product you have five pearls, or x^5.

Quotient Law

When dividing expressions involving the same base, you simply subtract the powers. Thus in (x^4)/(x^2) = x^(4-2) = x^2. Why this is so depends on the cancellation property of the real numbers. This property says that when the same number or variable appears in both the numerator and denominator of a fraction, then this term can be canceled. Let us look at a numerical example to make this completely clear. Take (5*4)/4. Since 4 appears in both the top and bottom of this expression, we can kill it—well not kill, we don’t want to get violent, but you know what I mean—to get 5. Now let’s multiply and divide to see if this agrees with our answer: (5*4)/4 = 20/4 = 5. Check. Thus this cancellation property holds. In an expression such as (y^5)/(y^3), this is (y*y*y*y*y)/(y*y*y), if we expand. Since we have 3 y’s in the denominator, we can use those to cancel 3 y’s in the numerator to get y^2. This agrees with y^(5-3) = y^2.

Power of a Power Law

In an expression such as (x^4)^3, we have what is known as a power to a power. The power of a power law states that we simplify by multiplying the powers together. Thus (x^4)^3 = x^(4*3) = x^12. If you think about why this is so, notice that the base in this expression is x^4. The exponent 3 tells us to use this base 3 times. Thus we would obtain (x^4)*(x^4)*(x^4). Now we see this as a product of the same base to the same power and can thus use our first property to get x^(4 + 4+ 4) = x^12.

Distributive Property

This property tells us how to simplify an expression such as (x^3*y^2)^3. To simplify this, we distribute the power 3 outside parentheses inside, multiplying each power to get x^(3*3)*y^(2*3) = x^9*y^6. To understand why this is so, notice that the base in the original expression is x^3*y^2. The 3 outside parentheses tells us to multiply this base by itself 3 times. When you do that and then rearrange the expression using both the associative and commutative properties of multiplication, you can then apply the first property to get the answer.

Zero Exponent Property

Any number or variable—except 0—to the 0 power is always 1. Thus 2^0 = 1; x^0 = 1; (x + 1)^0 = 1. To see why this is so, let us consider the expression (x^3)/(x^3). This is clearly equal to 1, since any number (except 0) or expression over itself yields this result. Using our quotient property, we see this is equal to x^(3 – 3) = x^0. Since both expressions must yield the same result, we get that x^0 = 1.

Negative Exponent Property

When we raise a number or variable to a negative integer, we end up with the reciprocal. That is 3^(-2) = 1/(3^2). To see why this is so, let us consider the expression (3^2)/(3^4). If we expand this, we obtain (3*3)/(3*3*3*3). Using the cancellation property, we end up with 1/(3*3) = 1/(3^2). Using the quotient property we that (3^2)/(3^4) = 3^(2 – 4) = 3^(-2). Since both of these expressions must be equal, we have that 3^(-2) = 1/(3^2).

Understanding these six properties of exponents will give students the solid foundation they need to tackle all kinds of pre-algebra, algebra, and even calculus problems. Often times, a student’s stumbling blocks can be removed with the bulldozer of foundational concepts. Study these properties and learn them. You will then be on the road to mathematical mastery.



Source by Joe Pagano

15 Jun

Random Facts Versus Whole Science Approach to Homeschool Teaching
Posted in Education by anshbookstore on June 15, 2022 @ 12:17 am

When it comes to learning science, most of us were taught in the public school system, which is a big proponent of the random fact teaching methodology. In other words, science was a single subject taught in a vacuum separate from other subjects. When it comes to teaching difficult or complex subjects such as science, it makes more sense to take a holistic approach. Here’s why.

The Science Random Fact Junk Drawer

There has been much news lately about the American education crisis in regards to a lack of interest in STEM (science, technology, engineering, math) disciplines. The United States is falling behind other developed countries when it comes to new technologies and discoveries, mainly because it is producing fewer graduates with related degrees.

One of the reasons for this lack of interest in STEM disciplines is due to the way kids are taught. Students often learn a bit of science here and a bit of science there without being provided any logical way to connect the dots. This collection of random facts can be likened to your junk drawer at home – you know there’s a screwdriver in the midst of all those rubber bands and paper clips and batteries and gadgets somewhere, you just can’t find it amongst all the clutter.

The same holds true for kids learning science. For instance, if a child learns a little something about the earth and the moon and how the shadow of our planet can cause a lunar eclipse, that’s an interesting, but random, fact. You might also have taught your child some astronomy concepts and explained how the moon affects the ocean’s tides. Perhaps your child has also learned something about gravity and the moon’s gravitational pull. But if you are using many mainstream homeschool science curricula, those facts were never pulled together to show the student how the moon is at the core of all these facts and they are interrelated. That’s why it’s so difficult for many kids (and adults alike!) to make the leap between one science fact and how it impacts so many other areas of the world around us. This also makes it very hard to extract a random fact later because the child must rely on rote learning.

The Whole Science Teaching Approach

A better, more effective way to teach homeschool science is through an exponential approach. By helping kids make their own connection between subjects, they are much better equipped to draw broader conclusions. This is also a great way to encourage their natural curiosity and develop hands-on experimentation that offers exciting new discoveries in the child’s mind.

The whole science homeschool teaching approach is all about extrapolation. Once your student has assimilated some core concepts they are prepared to expand that knowledge and apply it to different, everyday situations.

For instance, let’s go back to that random fact about the moon’s gravitational pull on earth. That’s a physic concepts and that explains much about a lunar eclipse, which is a topic generally brought up in astronomy. Those same gravitational forces are at work when it comes to oceanic tide cycles, a topic that may be part of biology learning. By painting the bigger picture, a student can connect the dots between physics and astronomy and biology herself and become excited about learning more.

This approach also compartmentalizes and organizes bits of information so they can easily be retrieved at will and on demand. And it aids the homeschool science teacher, who often doesn’t understand the information herself, present complex concepts and help the student come to a conclusion that need not be foregone.

When it comes to teaching a difficult subject such as science, the homeschool teacher would be wise to use a whole science approach rather than relying on a random fact methodology.



Source by Dr Rebecca Keller