Kimi and Jordan are each working during the summer to earn money in addition to their weekly allowance. Kimi earns $9 per hour at her job, and her allowance is $8 per week. Jordan earns $7.50 per hour, and his allowance is $16 per week.
The weekly total of Kimi's allowance and job earnings, \($ K\), is the sum of her $8 allowance and \($9 \cdot h\), where \(h\) is the number of hours she works. In other words, \(K = 8+9h\). Similarly, the weekly total of Jordan's allowance and job earnings, \(J\), is given by \(J = 16+7.5h\). The graphs of these two equations are shown below.
The graphs show that if they work less than \(5\frac13\) hours, Jordan has a greater total. But if they work more than \(5\frac13\) hours, Kimi has a greater total.
Their totals are the same at the point of intersection of the graphs. We find the \(h\)-coordinate of this point of intersection by finding the number of hours for which Kimi and Jordan will have the same total, assuming they both work with same number of hours in a week. We do this by setting the expressions for Kimi's total and Jordan's total equal and solving for \(h\):
\begin{align} 9h+8&=7.5h+16\ 1.5h &= 8\ h&=8/1.5=5\frac13. \end{align} The graphs show that Kimi and Jordan's totals are the same if they both work for \(5\frac13\) hours.
Hours worked, \(h\) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
Kimi's weekly total, \(K\) | 8 | 17 | 26 | 35 | 44 | 53 | 62 | 71 |
Jordan's weekly total, \(J\) | 16 | 23.5 | 31 | 38.5 | 46 | 53.5 | 61 | 68.5 |
From the table above, it looks like their incomes will be the same somewhere between 5 and 6 hours. An income table for quarter hours shows that the their incomes might be equal between \(5\frac14\) and \(5\frac12\) hours. Similarly, a table for tenths of hours shows that the incomes might be equal between 5.3 and 5.4 hours.
A more efficient strategy is to notice from the table that the difference between their totals decreases by $1.50 each hour, which makes sense as this is the difference between their hourly rates. At 5 hours, the difference is $0.50, which is \(\frac13\) of $1.50. So it makes sense to try \(\frac13\) hour more. During this 20 minutes, Kimi will make $3.00 and Jordan will make $2.50. So in \(5\frac13\) hours they both will make $56.
Commentary
In the middle grades, students have lots of experience analyzing and comparing linear functions using graphs, tables, symbolic expressions, and verbal descriptions. In this task, students may choose a representation that suits them and then reason from within that representation.
To find the point of intersection in part (b), tabular approaches must extend beyond integer domain values. Halves, quarters, and tenths of hours can help students gradually hone in on the intersection, but more efficient methods use general equation solving techniques or proportional reasoning from values in the table.
Parts (c) and (d) require that students imagine changes in one of the two linear graphs (foreshadowing F-BF.3) and then predict what will happen to the point of intersection.
When used in instruction, this task provides opportunities to compare representations and to make connections among them.
While this problem involves linear functions, other tasks in this set illustrate F.BF.1a in the context of quadratic (Skeleton Tower), exponential (Rumors), and rational (Summer Intern) functions.