“On his mother’s birthday, Juan has cooked dinner for his mother and some guests. He made a huge pot of rice with 4 cups of rice and 8 cups of water. He made two
(pork shoulders), and baked 3 cakes for dessert . . .”
What follows the word problem above (and all word problems) is that one
sentence that causes a great deal of agitation in my students. That final sentence causes a major amount of fear, dread, and even loathing, but it is only until students resolve within themselves the sheer inevitability of it that they can then begin to attempt to understand what the word problem has in store. The final sentence is the sentence by which students are informed of their objective. It communicates what students will be required to find, the eventual prize, that ultimate solution to the jumble of words, symbols, letters, and numbers. Despite the fact that the dynamics of word problems are simple, it is within the language of the words that complexity is found.
The students know that they are reading a problem, and that this problem will require a solution. However, there is a real barrier that forms between the words, the numbers, and the requisite symbols in a word problem. The reality of the situation that I as a bilingual teacher try to convey is that it matters not whether one understands completely the words, but the substance of what is required. The words are mere tools, apparatus, vehicles, if you will. These vehicles of meaning are meant to deliver ideas, and sometimes the vehicles can be sleek, awesome crafts that convey information succinctly, while other times the vehicles can be ancient, rundown jalopies that convey very little. What is the ultimate objective of these problems is to help students acquire the ability to parse meaning from the problem in such a way as to configure correctly the comparisons being made. That is what a ratio does.
Ratios are comparisons between numbers by way of division. They do what math in a more general way aspires to accomplish. Ratios have as their ultimate objective the discovery of relationships between numbers, the asking of “How does this number relate to that number?” or “How does this procedure relate to that procedure?” or “If I follow a certain list of strictures in one type of problem will I need to follow consistently the same list of strictures for these other types of problems.” What makes math fascinating is that sometimes the connections between numbers are concrete and overt, such as the comparison of cups of rice with cups of water in the problem above, and sometimes the connections are more tenuous, like the cooking of two
compared to the cooking of one pot of rice, and three cakes. Whatever the case, these comparisons whether subtle or obvious can be tremendously fruitful in discussions about establishing relationships between numbers.
Again using the example above, students are informed that there are various parts of a meal prepared by someone for his mother and her guests. These various parts are all related because they are all prepared by one person for several people as one meal. If we were to illustrate the possible comparisons through mathematical means, a few examples would include, but not be limited by:
1:1 - to signify 1 pot of rice for 1 meal,,
1:2 - to signify 1 pot of rice or 1 meal compared to 2
1:3 - to signify 1 pot of rice compared to 3 cakes,,
1:2:3 - to signify 1 pot of rice and 2
compared to 3 cakes,
3:2:1:1 to compare every facet of the meal.
Although there are numerous ratios that can be explored by far the most significant ratio is 1:2 for cups of rice to cups of water, because this is the way you almost always cook rice. The major point of comparison is to illustrate that if there are more or less people attending the dinner, then in order to feed the people attending there will have to be more or less food cooked in similar proportions. Which brings us to another word that is exceedingly important when dealing with ratio word problems.
A proportion is an equality of two ratios. Proportions as ratios hinge on an adequate understanding of division and equivalent fractions. They depend on students understanding that 1/2 = 2/4 = 3/6.
This is a major key. It is a major key because once students acknowledge an understanding of this concept; they
will know it and can refer back to it, even if sometimes they need reminding. As a consequence, whenever dealing with ratio word problems that compare one thing to another (proportions), students know that they have to approach the problem as an equation where one ratio equals another ratio. This equals that.
For example, if students were to further consider the word problem above, they can clearly note the ratio that has been purposely set up by the wording of how the one pot of rice is made by four cups of rice and eight cups of water (4/8).Now, if Juan’s mother decides to double the amount of people that were invited to the dinner, then Juan knows that he has to cook twice as much food. This sets up a proportion in which Juan needs to realize that in order to cook enough rice in that one bowl he will need to create an equation 4/8 = 8/16. If to cook for x number of people, Juan needs 4 cups of rice and 8 cups of water then, for 2x number of people Juan needs to multiply both the numerator and the denominator by 2. If the students can realize the veracity of the equation then they will have no difficulty when it comes to algebraic equations that propose relationships between specific sets of data.