Then, suddenly there was a big flock of chickens and other unexplained creatures on the farm like turkeys, guineas, peacocks and occasionally a steer for beef. In our family, it all started with Bock-Bock, our first hen that didn’t mind being carried around for hours. It begins with owning six or so chickens which are cute and practical, providing eggs and entertainment daily.Ī little adding and subtracting along with multiplying results in a multi-generation flock with a few other animals thrown in just because it seemed like a good idea at the time. It’s more like a pesky word problem with a nonsensical answer. This means that for every additional pound of steak we want to buy, we have to decrease the amount of chicken by 2.71 pounds.Chicken math is not like ordinary math. We can buy 20.91 pounds of chicken and 20.91 pounds of steak for 100 dollars. If we want equal amounts of chicken and steak, we let $c=s$ and solve fore $s$. On the other extreme, the horizontal intercept shows that we can buy 28.65 pounds of steak if we don't buy any chicken. The vertical intercept shows that we can buy at most 77.52 pounds of chicken if we choose not to buy any steak at all. The graph shows all the options we have for chicken and steak combinations to buy for our barbeque. We can find the horizontal intercept by setting the function equal to zero and solving for $s$. The graph is a line with vertical intercept $(0, 77.52)$ and slope $-2.71.$ Since it does not make sense to talk about negative amounts of chicken or steak, we are only interested in the part of the line where both coordinates are non-negative, as shown below. We can graph this function with $s$, the amount of steak, on the horizontal axis and $c$, the amount of chicken, on the vertical axis. In other words, we can find an equation of the form $s=f(c)$: Let $c$ be the independent variable then we can find a equation that expresses $s$ in terms of $c$. Since the problem does not specify which of the two variables is the independent variable and which is the dependent variable, we can make a choice. To find a function that relates the two variables, we can solve for one variable in terms of the other. If we have only \$100 dollars to spend on chicken and steak together, we can write this relationship in the form of the equation Then $1.29c$ gives us the total amount we pay for chicken, and $3.49s$ gives us the total amount we pay for steak. So, let us assign variables to those two quantities. We don't know how many pounds of chicken and steak we are buying. Therefore, all solutions are approximate.Ĭhicken costs \$1.29 per pound and steak costs \$3.49. Note that all numbers are rounded to two digits after the decimal. It is especially useful for students to see that, for example, the "steak-intercept" is interpreted in the same way regardless of whether the vertical or horizontal axis is the "steak-axis." This task presents a good opportunity to have a discussion about choosing independent and dependent variables and interpreting slopes, intercepts, and generic points on the graph in the context. To fully explore part (b), students should interpret the horizontal and vertical intercepts of the graph, the slope of the graph, and the coordinates of points on the graph. Whenever we have a (non-constant) linear relationship between two quantities, we can always write either quantity as a function of the other, and students should understand that which variable is the dependent variable and which is the independent variable is a choice made by the modeler. Part (a) is relatively straightforward, although the wording is left intentionally ambiguous about whether the amount of steak should be written as a function of the amount of chicken or the amount of chicken as a function of the amount of steak (both approaches are presented in the solutions shown below). This task presents a real world situation that can be modeled with a linear function best suited for an instructional context.
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