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Scientific Notation, Exponents and Square Roots:

A conceptual approach

  • Exponents, Square Root and Scientific Notation: Have you figured out how is it that we talk of the cube of a number? The square of a number? The root of a number? We know that the calculator has a keyboard for each one of them? But what is the exact meaning? You will see that the lesson develops the concept behind the power of a number. It starts by presenting the square and cube geometric meanings and their roots. Students learn to figure out in between what whole numbers is a given root. The activity is extended to learn about powers of 10, and finalizes with scientific notation.








FRACTIONS: A conceptual approach

  • Adding and Subtracting: All of us have learned that a fraction is a number with a fraction bar and a numerator on the top and the denominator at the bottom. But do we understand what is the effect of changing those in a fraction? Will the fraction be a larger number? or a smaller number? Now, if I have several fractions, how do I add them? Do I add all the numbers of the top, and the ones on the bottom and put the answer in the corresponding place? In this lesson students like you learn to add and subtract fractions with fraction bars. They learn the concept of equivalent fraction and how to go from lowest terms to highest terms.
  • Multiplying: If you multiply two times three the answer is greater than two or three? What about if you multiply an integer by a fraction? or Two improper fractions? By working in this lesson you will see it presents how to get the fractional part of a number, and the concept of multiplying a given number by a fraction. Fraction bars are used and it is illustrated this in terms of the area of a rectangle.
  • Fractional Part of a Number and Improper Fractions to Mixed numbers: In a lesson before this you learned the effect of multiplying fractions. Nevertheless, the fractions had the denominator larger than the numerator. They were improper fractions. Now, you will do something similar when the numerator is equal or larger than the denominator. Those are proper fractions. In these exercises students are presented with the different formats that getting a part of a number may have. It extends the lesson before covered. It finalizes presenting the conceptual meaning of an improper fraction and how to change it to mixed number and vice versa.
  • Multiplying Mixed Numbers: You have in previous lessons already understood the concept of multipying by a fraction. You also learned that a proper fraction generates a mixed number or an integer. Working this presentation you will learn how to multiply a mixed number by a fraction, by a whole number and by another mixed number. They start with an interesting problem that involves mixed numbers and ribbon tape.
  • Dividing Fractions: For most of us is easy to understand the effects of diving integers. We know that six divided by two will give us three. Simple. But, what two fractions will give us also three when we divide one onto the other? In this activity students learn how many parts a number or a fraction fits into another whole number or fraction. This is done using fraction bars.








DECIMALS: A conceptual approach

  • Adding: Many of us learned to read a problem with a sentence like this: "0.001 of the water volume was spilled of the cup." {which we read: zero point zero zero one of the water...} (not all of us, but some of us) Now the question is what is the real meaning of 0.001? In this lesson students learn with the help of manipulatives (place value mat) the concept of a tenth, hundredth, thousandth and how to apply it to addition problems that involve decimals. The reading above might more properly be done: " A thousand of the water volume ..." Would you be able to answer how many mililiters were spilled if the volume was 77 millimeters?
  • Subtracting: All these lessons are part of the number sense. In a car shop you have millimetric tools that have measurements referred in millimetric units. others have units in the English system and use fractions where the millimetric tools use decimals. Sometimes you have the equivalent from one to the other, but not always. So if you dad tells you while fixing the car in the garage: "Could you pass me the three fourths wrench?" and you realize is missing... Would you be able to get the closest matching one of the millimetric set without looking at the size but just at the engraved numbers? The lesson of adding decimals is extended in this one to subtraction using the place value mat and the manipulatives. Students learn how to compare decimals by going from the factional representation to the decimal one.
  • Multiply by a decimal: You learned in the section for fractions the effect of multiplying fractions and whether the answer is a smaller or larger number that each one of the factors in the product. Now, you will do the same but with decimals. By covering the lesson you will learn in this lesson how to get the fractional part of a number and then its meaning as decimal part of that same number. They extend this understanding to multiplication of any number by a decimal. Once, you mastered the concept then the opportunity is given to solve several word problems.
  • Divide by a decimal: In previous lessons you practice fractions adding, subtracting, multiplying or dividing them. You learned that when you divide three fourths by a fourth you don't get a fraction, you get three. Because a fourth fits three times in three fourths. Now this is done with decimals. In this lesson you will use the place value mat and the manipulatives to master the conceptual meaning of a division, and it is extended to division by a decimal number.
  • Fractions to Decimals to Percent: You read in the news that only 5% of the starts in a constellation of 7000 trillion of starts are Nova starts. Is that an insignificant amount of stars? Then you hear somebody saying that he marked in the forest 500 trees. The forest has millions of them. Did he marked a large portion of them? In this lesson students learn how to extend the concept of a fraction represented as decimal and how it relates to the percentage representation. Students go from a geometric representation in a 10x10 grid to the formal notation. This lesson is the introduction for the unit dedicated to percents.









PERCENTS: A conceptual approach

  • Percent of a Number: If you had difficulties figuring out what a percent is in relation to its fractional representation. You will get help in studying this presentation that uses a graphic representation of the concept. This lesson presents the procedure to get the percent of a number using a grid of 10x10 and linking it to the fractional and decimal representations for the area that is shaded and this is extended to get the percent of a  number.
  • Comparing Numbers as a Percent: Have used the zoom feature of the camara in your cell phone? If you have your sister full figure in the eye of the camara. You have her "using 100%" of the usable viewable area for her figure. Now, if you zoom "in" now her face ocupies the 100% of the useful viewable area. If you zoom "out" depending of how much your camara may allow it. She might become a small spot somewhere in the screen. If you consider that the screen has pixels (small squares making up the figures), and the amount of pixels in the area of you screen is a fixed amount; then you may realize that when you zoom "in" or "out" you are playing with the number of pixels in that screen that are used for her full viewable figure or part of her figure. Sounds complex? In this lesson students are able to go from comparing two numbers to getting the percent of a given number of objects in a set.
  • Calculate the Percent of Increase or Decrease of a Quantity: Have you gone with mom or dad to the cloths section in a departmental store and see that the price changed by some amount of dollars from the day before? Can you determine the percent it went down? If you go to the store across street and you see the same shirt a few dollars above. Could you tell the percent it is higher than the one in the first store? This lesson shows students how to understand a percent of increase and decrease as a change in area, from one initial one to final one. This is extended to word problems that involve this same concept.
  • Finding the Amount of a Discount or a Commission: Now that you had the opportunity to go over fractions, decimals, and percents it is time to see some application problems. This lesson has been targeted to be expanded in the near future. Meanwhile, in the few problems you will apply everything learned about fractions, decimals and percents to figure out discounts and commissions in sale prices.
  • Calculating Simple Interest: As explained above this lesson presents three word problems that involve getting the simple interest of one amount which is an effort to apply the concepts learned in the previous lessons. The lesson is a short one, but will benefit to try to work the few presented problems. Hopefully, we will find the time to add more problems.