Illustrative Mathematics (2024)

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Task

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.

1. Jordan wonders who will have more income in a week if they both work the same number of hours. Kimi says, "It depends." Explain what she means.
2. Is there a number of hours worked for which they will have the same income? If so, find that number of hours. If not, why not?
3. What would happen to your answer to part (b) if Kimi were to get a raise in her hourly rate? Explain.
4. What would happen to your answer to part (b) if Jordan were no longer to get an allowance? Explain.

IM 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.

Solutions

Solution:Kimi and Jordan

1. 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.

2. 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.

3. If Kimi's hourly rate were to increase, the dotted line above would have an increased slope and intersect the vertical axis at the same point, which would move the point of intersection to the left. So Kimi and Jordan's total would be the same if they both worked a number of hours less than $5\frac13$.
4. If Jordan were to have no allowance, the solid line would intersect the vertical axis at $0$ and have the same slope. Because the slope of the solid line is less than that of the dotted line, the lines would not intersect in the first quadrant. So Jordan and Kimi's totals would never be the same.

Solution:Reasoning from the context

1. Kimi's allowance is less than Jordan's, but her hourly rate is higher. Sofor only a few hours worked, she will have less job earnings per week than Jordan, but ifshe works a lot of hours, her higher hourly rate will more than make up for the lower allowance. We can find the number of hours at which this change occurs by calculating how many hours,at $\$1.50$ more per hour, it will take to make up the $\$8$ difference in their allowances: $\$8/(\$1.50/\mbox{hour}) = 5\frac13$ hours, or $5$ hours and $20$ minutes.
2. See part (a).
3. If Kimi's hourly rate were higher she would be able to make up the allowance difference more quickly. In the above calculation, the denominator would increase and the resulting number of hours would decrease.
4. If Jordan were without an allowance, it would be impossible for him to make up for Kimi's allowance because his hourly rate is lower.

Solution:Tabular solution

1. Based on their allowances and hourly rates, the following table compares their weekly income from 0 to 7 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 data in the table, Jordan has a greater total if they both work 0 to 5 hours, and Kimi has a greater total if they both work 6 or 7 hours. Since Kimi's hourly rate is greater, she will have a greater total for any number of hours worked beyond this as well.



2. 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.

3. If Kimi's hourly rate were to increase, then the numbers in the middle row of the table (after the 8) would increase, exceeding the bottom row of the table sooner in the row. So Kimi and Jordan's totals be the same at less than $5\frac13$ hours.
4. If Jordan were to have no allowance, all of the entries in the bottom row of the table would decrease by \$16. Because Jordan's total increases more slowly than Kimi's, it would fall further and further behind Kimi's, and their totals would never be the same.

Solution:Purely algebraic solution

1. The weekly total of Kimi's allowance and job earnings, $K$, is the sum of her \$8 allowance and (\$9 * $h$), 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 $J = 16+7.5h$. Both $K$ and $J$ depend on $h$, so there is no obvious answer to Jordan's question without further investigation.
2. First we find 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}Because Jordan's allowance is greater, this also tells us that Jordan will have a greater total than Kimi if they work less than $5\frac13$ hours. If however, they work more than $5\frac13$ hours, then Kimi's total will be greater.
3. If Kimi's hourly rate were greater, the 1.5 in the above solution would be a greater number, and the result of the division would be less. So they would have the same income at less than $5\frac13$ hours.
4. Without an allowance Jordan's total would be given by $J=7.5h$. And the solution process above would become as follows:\begin{align}9h+8&=7.5h\\1.5h &= -8\\h&=-8/1.5=-5\frac13\end{align}This means that their total would be the same if they both worked $-5\frac13$ hours. This solution to the algebraic equation does not make sense in terms of the context, since they both work a positive number of hours. So their totals would never be equal.