OPINION
Present and former executive protection agents (“Agents”) appeal from a grant of summary judgment in favor of their employer Vance Executive Protection, Inc., (“Vance”) on the Agents’ Fair Labor Standards Act (“FLSA”) claim seeking overtime compensation. We affirm in part, reverse in part, and remand with instructions.
I.
Agents employed by Vance serve as bodyguards for corporate officers and visiting dignitaries who contract with Vance for security services. The parties agree that the Agents typically work twelve-hour shifts, but on occasion are required to work more than twelve hours depending on the needs of the client. According to Vance’s Operational Guidelines Manual, an Agent is paid a “Daily Rate” for working a shift. J.A. 734. The most junior Agents earn $145 per shift and the most senior Agents earn in excess of $200 per shift. If circumstances require that an Agent work, for example, thirteen or fourteen hours, the Agent will not receive more than the established shift rate.
The dispute between the parties centers on whether Vance’s compensation plan violates the FLSA’s requirement that an employee receive “one and one-half times the regular rate at which he is employed” for hours worked in excess of a forty-hour workweek. 29 U.S.C.A. § 207(a)(1) (West 1998). Vance contends that the FLSA’s overtime requirement is already built into its compensation system. According to Vance, Agents are paid straight time for the first eight hours of a shift and double-time for the remaining four. The Agents, on the other hand, argue that the eight-four split is but an accounting mechanism used to avoid the FLSA’s overtime requirements. The Agents assert that they are paid a day rate and that their regular hourly rate should be determined by totaling all sums received in a workweek and dividing by the total hours worked. See 29 C.F.R. § 778.112 (2000). Such a calculation yields a much higher regular hourly rate than does Vance’s.
The following example illustrates the dispute between the parties. If an Agent earns $165 per shift and works six twelve-hour shifts in one workweek, the gross amount he receives from Vance is $990'for the seventy-two hours worked. Assuming this is a day rate, the Agent would calculate his regular hourly rate by dividing $990 by seventy-two hours, which yields an hourly rate of $13.75. Under the FLSA the first forty hours should be paid at $13.75 per hour for a total of $550, and the remaining thirty-two hours of overtime at time and a half ($20.625 per hour) for an overtime amount of $660. The overtime and non-overtime payments yield a total of $1,210, and hence the Agent argues he is due $220 from Vance ($1,210 — $990).
Under Vance’s more complicated calculation, which requires use of a multiplier of %, the Agent’s hourly wage for the first eight hours is $10.3125 (1/16 of $165). *163Thus, the Agent earns $412.50 for the first forty hours worked and $577.50 for the remaining thirty-two. Dividing $577.50 by thirty-two hours yields an overtime rate in excess of $18, well above the time and a half required by law. Hence, Vance denies the Agent is owed any compensation.
Hearing cross motions for summary judgment, the district court rejected the Agents’ assertion that they are paid a day rate and concluded that Vance’s division of the workday is not prohibited by the FLSA. However, the court held that Vance’s compensation plan passes muster only if the Agents actually work twelve hours every shift, or a limited number of additional hours which are covered by Vance’s double-time payments. Though the court recognized that it is possible for the Agents to have worked hours for which they were not compensated, the court found that the Agents had failed to offer sufficient proof of the number of uncompensated hours worked. The Agents appeal the district court’s grant of summary judgment as well as issues relating to the statute of limitations.
II.
A summary judgment motion should be granted only if there is no genuine dispute as to an issue of material fact and the moving party is entitled to judgment as a matter of law. See Anderson v. Liberty Lobby, Inc., 477 U.S. 242, 247-48, 106 S.Ct. 2505, 91 L.Ed.2d 202 (1986). The court must consider the evidence in the light most favorable to the non-moving party and draw all reasonable inferences from the facts in the non-movant’s favor. See United States v. Diebold, Inc., 369 U.S. 654, 655, 82 S.Ct. 993, 8 L.Ed.2d 176 (1962). “When reviewing cross-motions for summary judgment, we may, if appropriate, direct entry of judgment in favor of the party whose motion was denied by the district court.” Bakery & Confectionery Union & Indus. Int’l Pension Fund v. Ralph’s Grocery Co., 118 F.3d 1018, 1020 (4th Cir.1997); see also Monahan v. County of Chesterfield, 95 F.3d 1263, 1265 (4th Cir.1996).
As a general rule, the FLSA provides that an employer must pay an employee one and one-half times the employee’s “regular rate” for all hours worked in excess of forty per week. See 29 U.S.C.A. § 207(a)(1). The “regular rate” is the hourly rate that the employer pays the employee “for the normal, non-overtime workweek.” Walling v. Youngerman-Reynolds Hardwood Co., 325 U.S. 419, 424, 65 S.Ct. 1242, 89 L.Ed. 1705 (1945). Normally, the regular rate is arrived at by “dividing [the] total remuneration for employment (except statutory exclusions) in any workweek by the total number of hours actually worked.” 29 C.F.R. § 778.109 (2000).
The exclusion central to the present case is found in 29 U.S.C.A § 207(e)(5). This statutory provision permits employers to exclude “extra compensation provided by a premium rate paid for certain hours worked by the employee in any day or workweek because such hours are hours worked ... in excess of the employee’s normal working hours or regular working hours.” 29 U.S.C.A. § 207(e)(5). For example,
[i]f an employee whose maximum hours standard is 40 hours is hired at the rate of $5.75 an hour and receives, as overtime compensation under his contract, $6.25 per hour for each hour actually worked in excess of 8 per day (or in excess of his normal or regular daily working hours), his employer may exclude the premium portion of the overtime rate from the employee’s regular rate and credit the total of the extra 50 cent payments thus made for daily overtime hours against the overtime compen*164sation which is due under the statute for hours in excess of 40 in that workweek.
29 C.F.R. § 778.202(c) (2000). Exemptions must be narrowly construed and the burden is on Vance “to show that it is entitled to the benefits of th[e] exceptions.” Johnson v. City of Columbia, 949 F.2d 127, 130 (4th Cir.1991) (en banc).
Though specifically permitting a premium for hours worked in excess of regular working hours, the regulations prohibit split-day compensation schemes in which “the normal workday is artificially divided into a ‘straight time’ period to which one rate is assigned, followed by a so-called ‘overtime’ period for which a higher ‘rate’ is specified.” 29 C.F.R. § 778.202(c); see also Walling v. Helmerich & Payne, Inc., 323 U.S. 37, 38-39, 65 S.Ct. 11, 89 L.Ed. 29 (1944); 29 C.F.R. § 778.501 (2000). To permit such divisions would undermine the FLSA’s objectives of spreading work and compensating employees for excessive hours worked. See Helmerich & Payne, Inc., 323 U.S. at 40, 65 S.Ct. 11.
Of course, the Department of Labor through its interpretation of the FLSA recognizes that not all employees entitled to overtime compensation are paid by the hour. Some workers, for example, contract to be paid on a piece rate or a day rate basis and the regulations explain how to calculate the regular rate for these employees. See 29 C.F.R. §§ 778.111-.112 (2000). The Agents contend that they are paid a flat sum for a day’s work and consequently their overtime rate should be calculated pursuant to 29 C.F.R. § 778.112, which provides:
If the employee is paid a flat sum for a day’s work or for doing a particular job, without regard to the number of hours worked in the day or at the job, and if he receives no other form of compensation for services, his regular rate is determined by totaling all the sums received at such day rates or job rates in the workweek and dividing by the total hours actually worked. He is then entitled to extra half-time pay at this rate for all hours worked in excess of 40 in the workweek.
29 C.F.R. § 778.112.
The primary issue in this case is whether the Agents are paid a day rate. Resolution of this issue then dictates the proper result under the FLSA. Because this question is before us on cross motions for summary judgment and because there are no genuine issues of material fact, our decision becomes one of law. We believe, however, the law dictates a conclusion different from that of the district court.
Vance’s Operational Guidelines Manual describes the Agents’ pay structure as a “Daily Rate” which increases as an Agent gains seniority. J.A. 734. The Manual further provides that Agents normally work a minimum of twelve hours per day “with additional hours often arising from relief response, protectee’s movements, advances, luggage details or other unforseen circumstances.” J.A. 677. Vance officials testified that if an Agent is required to work, for instance, fourteen or fifteen hours in a day because of some exigency, the Agent receives no additional compensation. In the rare case that an Agent receives an assignment requiring him to work only eight hours, the Agent still earns the same amount he normally receives for a twelve- or fourteen-hour day. Documents provided to detail leaders attending a Vance seminar listed the “average daily rate” paid to Agents as $153.93, and made no mention of hourly rates. When filling out employment verification forms for entities dealing with the Agents, Vance officials noted that the Agents were paid “per day” even though there was a “per hour” option on the forms. J.A. 650-51. In addition, the Agents testified that *165when they were hired Vance officials never mentioned anything about an hourly rate; the Agents’ compensation was always explained to them in terms of a daily rate.
Vance officials counter that they described the Agents’ pay as a “daily rate” because it was “the simplest term we have been able to use to tell our employees how much they can expect to be paid for a 12 hour shift.” J.A. 382. Vance officials also contended that the Agents did not “want to associate an hourly rate with the work that they do,” and therefore Vance used the term “daily rate” to assuage the Agents’ egos. J.A. 471. Other than payroll documents, Vance officials could point to no documents available to the Agents that specified an hourly rate of employment.
At appellate argument, Vance made much of the fact that the Agents’ pay stubs split the gross amount earned into “regular” and “doubletime.” See J.A. 779. However, the pay stubs never specified an hourly rate and testimony from Vance officials indicated that the actual calculation of the hourly rates was left to the human resources department and that such information was not readily available to the Agents.
In light of the foregoing, we can find no genuine issue of material fact regarding the Agents’ rate of pay. Unquestionably, Vance Agents are “paid a flat sum for a day’s work ... without regard to the number of hours worked in the day.” 29 C.F.R. § 788.112; see also Dufrene v. Browning-Ferris, Inc., 207 F.3d 264, 266 (5th Cir.) (describing employees who “were guaranteed a day’s pay, regardless of the hours worked that day” as day-rate employees), cert. denied, — U.S. —, 121 S.Ct. 72, 148 L.Ed.2d 36 (2000). Vance represented to its Agents and the world that the Agents were paid day rates. The pay stubs did reflect a regular and double-time component, but the stubs standing alone cannot compete with the plethora of evidence indicating that the Agents were paid a day rate. See Brennan v. Elmer’s Disposal Serv., Inc., 510 F.2d 84, 88 (9th Cir.1975) (holding that pay stubs alone were not sufficient to establish an employee’s regular rate); see also Anderson, 477 U.S. at 251, 106 S.Ct. 2505 (a scintilla of evidence is insufficient to bar summary judgment). The alleged hourly rate paid to the Agents is in reality but a form of mathematical legerdemain used by Vance to avoid the implications of the FLSA. Accordingly, the district court erred in awarding summary judgment to Vance on the grounds that the Agents were paid by the hour and that no overtime was owed in light of the premium paid to the Agents.1 We reverse the district court on this matter and remand for the entry of summary judgment in favor of the Agents. See Ralph’s Grocery Co., 118 F.3d at 1020.
III.
The Agents also seek compensation for hours worked in excess of forty per week that are not reflected in Vance’s records. As discussed in the previous section, the parties do not dispute that on occasion the Agents’ shifts lasted longer than twelve hours. The regulations implementing the FLSA mandate that the employer keep records of the “[hjours worked each workday” by all employees. 29 C.F.R. § 516.2(a)(7) (2000). Vance admits that it did not keep proper records, and the Supreme Court has held that in such a situation an employee should not be penalized “on the ground that he is unable to prove *166the precise extent of uncompensated work.” Anderson v. Mt. Clemens Pottery Co., 328 U.S. 680, 687, 66 S.Ct. 1187, 90 L.Ed. 1515 (1946).
When an employer’s records are inadequate or inaccurate, an employee carries his burden under the law by (1) “proving] that he has in fact performed work for which he was improperly compensated,” and (2) “produc[ing] sufficient evidence to show the amount and extent of that work as a matter of just and reasonable inference.” Id. If the employee carries his burden, “[t]he burden then shifts to the employer to come forward with evidence of the precise amount of work performed or with evidence to negative the reasonableness of the inference to be drawn from the employee’s evidence.” Id. at 687-88, 66 S.Ct. 1187. If the employer cannot “produce such evidence, the court may then award damages to the employee, even though the result be only approximate.” Id. at 688, 66 S.Ct. 1187.
Though the Agents have easily satisfied the first prong of their burden by proving that they performed work for which they were not compensated, we agree with the district court that the Agents have not come forward with “sufficient evidence to show the amount and extent of that work as a matter of just and reasonable inference.” Id. at 687, 66 S.Ct. 1187. The bulk of the Agents’ proof on the second prong concerns work done for demanding clients such as Princess Sarah of the Saudi royal family. Princess Sarah, for example, often asked the Agents to run errands for her after their shifts were scheduled to end. There is much general testimony about picking up gifts for Princess Sarah’s children and paying her bills, but the record is bereft of evidence showing the amount or extent of this extra work. While we acknowledge that the Agents need not “prove each hour of overtime work with unerring accuracy or certainty,” Pforr v. Food Lion, Inc., 851 F.2d 106, 108 (4th Cir.1988), enough evidence must be offered so that the court as “a matter of just and reasonable inference” may estimate the unrecorded hours, Mt. Clemens Pottery Co., 328 U.S. at 687, 66 S.Ct. 1187. Because there is insufficient evidence to estimate the amount and extent of hours worked in excess of forty per week, we affirm the grant of summary judgment as to the unrecorded hours.2
IV.
The Agents also challenge the district court’s ruling that the statute of limitations for Agents Vance E. Morris and Todd K. Molter would only run back from December 21, 1999, the day these two Agents filed their consents. Under the FLSA, an employee may bring an action on behalf of similarly situated employees who must opt-in to the suit by filing written consents with the district court. See 29 U.S.C.A. § 216(b) (West 1998) (“An action to recover [unpaid overtime] ... may be maintained against any employer ... by any one or more employees for and in behalf of himself or themselves and other employees similarly situated. No employee shall be a party plaintiff to any such action unless he gives his consent in writing to become such a party and such consent is filed.”). An FLSA collective action is deemed commenced for an individual plaintiff “on the date when the complaint is filed, if he is specifically named as a party *167plaintiff in the complaint and his written consent to become a party plaintiff is filed on such date in the court.” 29 U.S.C.A. § 256(a) (West 1998). For plaintiffs not named in the original complaint, a collective action under the FLSA commences “on the subsequent date on which [the plaintiffs’] written consent is filed in the court.” 29 U.S.C.A § 256(b); see also Songu Mbriwa v. Davis Memorial Goodwill Indus., 144 F.R.D. 1, 2 (D.D.C.1992) (“Until a plaintiff, even a named plaintiff, has filed a written consent, [ ]he has not joined in the class action, at least for statute of limitations purposes.”). In other words, consents not filed with the complaint do not relate back.
Though not named in the original complaint, Morris and Molter were named in the amended complaint filed in September 1999. However, the consent forms for these two agents were not filed with the court until December 21,1999. Examining the statute of limitations as of the filing of the consents rather than the filing of the action, especially in the case of Morris, affects the amount of damages, if any, these two Agents can recover. See 29 U.S.C.A. § 255(a) (1998) (stating that the statute of limitations for an action seeking overtime compensation is two years and in cases of willful violation of the FLSA it is three years).3 The Agents argue that the suit against Vance was not a collective action and therefore no consent forms were required. The, fact that Morris and Molter did not file consent forms until December 1999, so the argument goes, should in no way affect the period of time for which they are entitled to recover damages.
The complaint and amended complaint, however, make clear that the suit against Vance is a collective action. The seventh paragraph in both documents states that “[pjlaintiffs bring this action on behalf of themselves and all other employees similarly situated.” J.A. 11, 68. Yet, the Agents argue that a collective action was never pursued and that the suit against Vance was simply fifteen individual actions joined under Federal Rule of Civil Procedure 20(a). We disagree.
The purpose of the consent forms is “to make ... uncertain plaintiffs certain, and actual participants, so that defendants could know the parties and the charges with which they were to be faced.” Deley v. Atlantic Box & Lumber Corp., 119 F.Supp. 727, 728 (D.N.J.1954). Otherwise, one employee could sue on behalf of similarly situated employees “without the specific rights of the others ever being actually considered.” Id. The consent form requirement clearly enures to the benefit of employers by making them aware of what allegations they face and from whom the allegations originate. We can find nothing in the record to indicate that the Agents ever considered their suit to be anything but a collective action or that the Agents represented to Vance or the district court that their action was not collective; therefore, we decline to inject uncertainty into the law by permitting plaintiffs to change the nature of their action in the middle of judicial proceedings without proper notice to the court and employer. If a group of plaintiffs decided to abandon a collective action and instead simply join their individual actions under Rule 20(a), they could, for example, move to amend the pleadings pursuant to Rule 15(a), *168which provides leave to freely amend pleadings “when justice so requires.” But nothing indicates to us that the Agents so changed their action, and therefore we affirm the district court’s decision regarding the necessity of consent forms for the present case.4
V.
For the foregoing reasons we reverse the district court’s grant of summary judgment for Vance on the Agents’ compensation scheme and remand with instructions for the entry of summary judgment in favor of the Agents. We affirm the district court’s decision that the Agents cannot recover overtime compensation for unrecorded hours worked in excess of forty per week because the Agents did not come forward with sufficient evidence to show the amount and extent of the work. Finally, we affirm the district court’s decision that the Agents pursued a collective action and therefore the statute of limitations for Morris and Molter runs back from the date their consent forms were filed with the district court.
AFFIRMED IN PART, REVERSED IN PART, AND REMANDED WITH INSTRUCTIONS.