You might believe that you don’t understand some of the math concepts on the SAT, but if I had a dollar for every student who wrongly thought they didn’t understand linear relationships...
Let’s say it’s been a thousand students; then, you’d probably be able to figure out pretty quickly that I would have 1,000 dollars. If it had been twice as many students, I’d have twice as much money. If this makes sense, then you already get the essence of linear relationships. We often encounter real-life situations in which two quantities are related in such a way that a change in one of them results in a corresponding change in the other. If you get paid hourly, then working twice as many hours will double your pay. If your new school is only half as far away as your old one, it will take you half as long to get there if you travel at the same speed.
The phrase "linear relationships" doesn’t necessarily make sense as a result of this knowledge because the phrase comes from the shapes of the graphs of these relationships, and we will get to those eventually. To get started though, we just need to be able to model these relationships by writing mathematical expressions that allow us to calculate values without simply listing a long series of numbers. You don’t have to understand lines in an abstract way in order to express these relationships. Before we get into an algebraic understanding of lines, we can use common sense to work our way through some questions based on this type of relationship.
For example, Mindy lives within walking distance of three buffet restaurants that have different price structures.
The first buffet charges 20 dollars to enter and dine, no matter how much food she eats. Whether Mindy eats 4 plates of food or 20 plates of food, she will pay a fixed amount of 20 dollars to eat there.
The second buffet only charges by the plate (there is no entrance fee). Each plate of food Mindy eats costs 4 dollars. Here, Mindy will pay 4 dollars times the number of plates of food she eats. If she eats one plate of food, her bill will be 4 dollars; if she eats 4 plates, her bill will be . If she eats plates of food, her bill will be dollars.
The third buffet combines the approaches of both of the other buffets. This third buffet imposes a smaller entrance charge than does the first buffet: entrance to the third buffet only costs 5 dollars. The third buffet also charges less per plate than does the second buffet: each plate of food costs 2 dollars. Here, Mindy will pay 5 dollars plus 2 dollars per plate of food. In order to enter the restaurant, she pays 5 dollars; if she has a heart attack while waiting in line and gets 0 plates of food, the whole trip will still cost her only 5 dollars (aside from medical bills). If she eats one plate of food, the trip will cost 2 dollars for the one plate of food (2 dollars times 1 plate) plus the 5 dollars entrance fee (a total of 7 dollars). If she eats two plates of food, the cost will be 4 dollars for the two plates of food (2 dollars times 2 plates) plus the 5 dollars entrance charge for a total of 9 dollars. If she eats plates of food, her bill will be (2 dollars times plates) plus the 5 dollars entrance charge for a total of dollars.
The different styles of buffet demonstrate different facets of linear relationships. In the simplest case, there can just be a constant amount that never changes (the entrance charge). More commonly, the value produced can be changed by some amount (here, the price per plate) every time a variable (the number of plates) is changed. And finally, in the most general situation, there can be a combination of both of these attributes.
Now suppose that George starts a pen collection with 20 pens that were gifted to him by his grandfather. He plans on buying 3 new pens every month to add to his collection. How could we write an expression to show the total number of pens he would have after months?
The number of new pens George would have after months follows the same pattern we have already talked about. If he adds 3 pens a month for months, he would have new pens. All we have to do is add the original 20 pens to find out the total number of pens he has after months: .
Let’s look at this situation as represented by a table to see how this expression makes sense as a way of representing the pattern of how his collection grows.
Let’s say it’s been a thousand students; then, you’d probably be able to figure out pretty quickly that I would have 1,000 dollars. If it had been twice as many students, I’d have twice as much money. If this makes sense, then you already get the essence of linear relationships. We often encounter real-life situations in which two quantities are related in such a way that a change in one of them results in a corresponding change in the other. If you get paid hourly, then working twice as many hours will double your pay. If your new school is only half as far away as your old one, it will take you half as long to get there if you travel at the same speed.
The phrase "linear relationships" doesn’t necessarily make sense as a result of this knowledge because the phrase comes from the shapes of the graphs of these relationships, and we will get to those eventually. To get started though, we just need to be able to model these relationships by writing mathematical expressions that allow us to calculate values without simply listing a long series of numbers. You don’t have to understand lines in an abstract way in order to express these relationships. Before we get into an algebraic understanding of lines, we can use common sense to work our way through some questions based on this type of relationship.
For example, Mindy lives within walking distance of three buffet restaurants that have different price structures.
The first buffet charges 20 dollars to enter and dine, no matter how much food she eats. Whether Mindy eats 4 plates of food or 20 plates of food, she will pay a fixed amount of 20 dollars to eat there.
The second buffet only charges by the plate (there is no entrance fee). Each plate of food Mindy eats costs 4 dollars. Here, Mindy will pay 4 dollars times the number of plates of food she eats. If she eats one plate of food, her bill will be 4 dollars; if she eats 4 plates, her bill will be . If she eats plates of food, her bill will be dollars.
The third buffet combines the approaches of both of the other buffets. This third buffet imposes a smaller entrance charge than does the first buffet: entrance to the third buffet only costs 5 dollars. The third buffet also charges less per plate than does the second buffet: each plate of food costs 2 dollars. Here, Mindy will pay 5 dollars plus 2 dollars per plate of food. In order to enter the restaurant, she pays 5 dollars; if she has a heart attack while waiting in line and gets 0 plates of food, the whole trip will still cost her only 5 dollars (aside from medical bills). If she eats one plate of food, the trip will cost 2 dollars for the one plate of food (2 dollars times 1 plate) plus the 5 dollars entrance fee (a total of 7 dollars). If she eats two plates of food, the cost will be 4 dollars for the two plates of food (2 dollars times 2 plates) plus the 5 dollars entrance charge for a total of 9 dollars. If she eats plates of food, her bill will be (2 dollars times plates) plus the 5 dollars entrance charge for a total of dollars.
The different styles of buffet demonstrate different facets of linear relationships. In the simplest case, there can just be a constant amount that never changes (the entrance charge). More commonly, the value produced can be changed by some amount (here, the price per plate) every time a variable (the number of plates) is changed. And finally, in the most general situation, there can be a combination of both of these attributes.
Now suppose that George starts a pen collection with 20 pens that were gifted to him by his grandfather. He plans on buying 3 new pens every month to add to his collection. How could we write an expression to show the total number of pens he would have after months?
The number of new pens George would have after months follows the same pattern we have already talked about. If he adds 3 pens a month for months, he would have new pens. All we have to do is add the original 20 pens to find out the total number of pens he has after months: .
Let’s look at this situation as represented by a table to see how this expression makes sense as a way of representing the pattern of how his collection grows.
| Number of Months | Total Number of Pens | Pattern Representation |
|---|---|---|
| 0 | 20 | |
| 1 | 23 | |
| 2 | 26 | |