We can use these proportions to compare two ratios with different denominators by finding a denominator that their proportions have in common. Thus the common denominator of 2/3 and 3/4 is 12. We then can use the common denominator to add/subtract and divide common ratios (fractions). This approach to division is quite different from the traditional approach and does not rely on the mechanical process of inverting the divisor and multiplying which most students find difficult to understand. Using common denominators means that to divide two fractions we simply divide the numerators of their common denominators, because when we divide common denominators they =1 since both have the same value.
Category: Labs
Ratio and Proportion
We think about ratio tables in terms of motion. Move up 2 and over 1, or move up 1 and over 2. In this way we build proportional patterns. By coloring the cells we land on like knights in a chess table, we can see the proportions of different ratios. These proportions build linear patterns on the ratio table and introduce us to the very important concept of slope
Division and Ratio
We can make a division table just like we made a multiplication table. Division is surprisingly our most important operation in terms of most of the problems we solve in our daily lives. Division produces numbers we call fractions or rationals and functions we call ratios. With spreadsheets we concentrate on ratios and on the patterns of ratios.
Square Numbers
The square numbers form an interesting pattern on the times (multiplication) table. They run along a diagonal from 1 to the top right of the table separating the table into two halves. This is the first step in looking at patterns in the multiplication table. Students build a new square number table by using a rule and then graph that the square numbers. The square numbers form a parabola on a graph.
Magic Rectangle
Multiplication tables have some wonderful and quite surprising patterns. This is one of them. Draw any rectangle in a multiplication table and you will find that the products of opposite corners are equal. For example a rectangle around a full 12 by 12 table will be 1144 and 1212. Try it, is it always true? Why?