
626 F.3d 756 (2010)
UNITED STATES of America, Plaintiff-Appellee,
v.
Kevin Anthony HICKMAN, Defendant-Appellant.
No. 08-4764.
United States Court of Appeals, Fourth Circuit.
Argued: September 22, 2010.
Decided: November 29, 2010.
*759 ARGUED: Francis Albert Pommett, III, Baltimore, Maryland, for Appellant. Benjamin M. Block, Office of the United States Attorney, Baltimore, Maryland, for Appellee. ON BRIEF: Rod J. Rosenstein, United States Attorney, Bryan M. Giblin, Assistant United States Attorney, Office of the United States Attorney, Baltimore, Maryland, for Appellee.
Before TRAXLER, Chief Judge, and DAVIS and KEENAN, Circuit Judges.
Affirmed in part, vacated in part, and remanded with instructions by published opinion. Judge DAVIS wrote the opinion, in which Chief Judge TRAXLER and Judge KEENAN joined.


*760 OPINION
DAVIS, Circuit Judge:
Appellant Kevin Hickman was charged in two of the 11 counts in a superseding indictment, namely (1) conspiracy to distribute and to possess with intent to distribute one kilogram or more of heroin (Count I) and (2) possession of heroin with intent to distribute (Count VI), in violation of 21 U.S.C. §§ 846 and 841(a)(1). Alone among the eight defendants, Hickman proceeded to trial. The jury convicted Hickman on both counts and, pursuant to the Government's notice of enhanced punishment, the district court imposed a mandatory life sentence on the conspiracy count and a concurrent sentence of 360 months of imprisonment on the possession with intent to distribute count. In this timely appeal, Hickman principally contends that the Government failed to offer sufficient evidence to support his conspiracy conviction and therefore the district court erred in denying his motion for judgment of acquittal on that count. He also assigns error with respect to (1) the district court's jury instructions; (2) its handling of jury questions; and (3) its use of a prior conviction in calculating Hickman's sentence.
We readily conclude that the evidence of Hickman's knowing participation in a conspiracy to distribute large amounts of heroin was overwhelming. Nevertheless, for the reasons stated within, we further conclude that the district court erred, in part, when it denied Hickman's motion for judgment of acquittal as to the one-kilogram conspiracy charged in the indictment. The Government's strained attempt, through extrapolations testified to by a drug enforcement agent, to prove beyond a reasonable doubt that the charged conspiracy involved at least one kilogram of heroin relies on impermissible speculation and cannot be sustained. Nevertheless, we hold that the evidence was sufficient to prove the lesser included offense of conspiracy to distribute 100 grams or more of heroin. Accordingly, we vacate Hickman's conspiracy conviction and remand with directions that the district court impose judgment on the lesser included offense. In all other respects, we affirm the judgment of the district court.

I.

A.
The evidence offered by the Government in this prosecution resulted from the confluence of two distinct investigations by federal and local law enforcement agencies into heroin distribution activities in Baltimore. In the course of the federal component of the investigation, agents of the Drug Enforcement Administration obtained wiretaps on the phones of Hickman's co-defendants James Jones (also known as "Fat Cat") and James Henderson, among others, and instituted surveillance of an inner-city store known as Fat Cat's Variety Store, run by Jones.
The wiretaps intercepted calls between Tony Caldwell (who was also a co-defendant) and Jones on April 24, 2007, in which Caldwell informed Jones he had found a buyer for him, whom he called "Hookie" (Hickman). Agents then set up surveillance at Fat Cat's Variety Store. Special Agent Bennet Strickland observed Hickman's car arrive at Fat Cat's Variety Store, and the wiretap confirmed that Hickman and Henderson spoke by phone only minutes before.
A few minutes after Hickman's arrival, Jones called Caldwell to ask if the person named "Kevin" at his store was, in fact, "Hookie," and was told that he was. Jones also told Caldwell that he had five more grams of heroin than Hickman could pay for, and asked whether he should give Hickman the extra five; Caldwell instructed *761 him to deliver only "what he [was] supposed to get," J.A. 405. Shortly afterward, Caldwell asked Jones, "You all know each other?" and Jones confirmed, "Yeah, yeah, yeah, I know him, I know him. Definitely yeah." J.A. 407. Caldwell then told Jones that Hickman was "my co-defendant," J.A. 407 (apparently alluding to an earlier drug prosecution in state court). Just after leaving the store, Hickman spoke with Henderson by phone; Henderson asked if he "ever ma[d]e it to the store" and Hickman confirmed that he had. J.A. 409.
The investigators directed city police to stop Hickman, and after allowing Hickman to drive a few blocks away from the store so as not to raise suspicion, Baltimore Police Officer Keith Sokolowski stopped Hickman's vehicle for a traffic violation. Hickman was driving and his girlfriend, Claudia Lake, was in the front passenger seat. Officer Sokolowski testified that he discovered and seized 32.14 grams of heroin in the passenger area of the car, which Lake had tried to hide. Subsequent analysis revealed that the heroin was 38% pure. Sokolowski also found 17 gold-topped vials of heroin hidden under the gas cap of the vehicle, though the heroin in the vials was never weighed.
Hickman was arrested and then released on bail. The day of his release, he called Henderson about arranging an additional purchase from Jones. Over the next few days, the two spoke several times about it; though Hickman told Henderson that he had spoken with Jones and was simply waiting on him, the Government produced no evidence that this plan was ever consummated.
Meanwhile, in a search of Caldwell's house on May 8, 2007 arising from a separate investigation, local law enforcement officers seized 139 grams of heroin, later found to be 29% pure. In a subsequent search of Fat Cat's Variety Store on June 7, 2007, federal agents seized more than 25,000 vials and a variety of colored tops, packaged by the hundred. The evidence showed that the vials were of the sort customarily employed to package street-level quantities of heroin (one-tenth of a gram).
Additional facts are set forth in the following analyses as needed.

B.
Hickman was charged with seven others in an 11-count superseding indictment. He was named in two counts; count one alleged a "conspir[acy] ... to distribute, and possess with intent to distribute, one kilogram or more of ... heroin," and count six alleged "possess[ion] with the intent to distribute a quantity ... of heroin." J.A. 10-11, 16. Although all of Hickman's co-defendants pled guilty, none of them testified at trial, nor did the Government call as a witness any other participant in the overall conspiracy. Rather, the Government adduced the testimony of the following witnesses: Special Agent Strickland, who was conducting surveillance outside of Fat Cat's Variety Store at the time of Hickman's purchase; Officers Sokolowski and Michael Woodlon, who took part in the traffic stop of Hickman; Detective Constantine Passamichalis, who assisted in the raid on Caldwell's residence; and criminologists Anthony Rumber and Theodis Warnick, Jr., who tested the narcotics seized from Hickman and Caldwell.
Perhaps most significant for purposes of this appeal was extensive testimony from Special Agent Brendan O'Meara, who monitored the Jones wiretap and who was accepted by the court as an expert in narcotics investigations. The Government relied heavily on the content of the wiretap recordings, and it was Agent O'Meara who *762 interpreted them for the jury, explaining the vague and coded terminology used by drug dealers. In its effort to prove that the conspiracy (which the indictment alleged subsisted for only four months, from February 2007 through May 2007) involved more than one kilogram of heroin, the Government asked Agent O'Meara to explain how heroin is typically cut down from its raw, high-purity state to user-strength level of approximately 8% via mixture with mannite and quinine. Critical to the Government's theory of the case was O'Meara's opinion that the 25,000 vials seized from Fat Cat's Variety Store would be enough to hold one kilogram of user-strength heroin.
After the Government rested, the defense moved for a judgment of acquittal, which the district court denied. The defense at no time specifically argued that the Government's proof established only a conspiracy involving a lesser amount than one kilogram, and the defense never requested a lesser included offense instruction. Nor did the defense call any witnesses.
The jury found Hickman guilty on both the conspiracy and possession with intent to distribute counts. The jury was asked on the verdict form to determine whether the amount of heroin involved in the conspiracy and reasonably foreseeable to Hickman was (1) one kilogram or more, (2) less than one kilogram but greater than or equal to one hundred grams, or (3) less than one hundred grams. It found the conspiracy involved one kilogram or more and that such amount was foreseeable to Hickman. After the verdict, Hickman moved for a new trial pursuant to Rule 33, which was denied. Because Hickman had two predicate felony drug convictions which the Government had noticed pursuant to 21 U.S.C. § 851, he was sentenced to life imprisonment on the conspiracy count and to a concurrent sentence of 360 months for possession with intent to distribute.

II.
We first address Hickman's contention that the evidence was insufficient to prove the one-kilogram conspiracy. We easily conclude that the evidence adduced by the Government proved the existence of a heroin conspiracy, but we are unable to find in the record adequate evidence to prove beyond a reasonable doubt that the conspiracy involved one kilogram or more. After examining the evidence surrounding each of the transactions proven at trial, we consider the two inferences the Government asked the jury to draw in order to reach its finding that the conspiracy involved at least one kilogrammost notably, that the conspirators' seeming familiarity with the drug trade indicated that this conspiracy concerned some greater amount of heroin than was otherwise proved at trial. As we explain, the direct and circumstantial evidence in the record simply will not do the work the Government assigned it, and we hold the evidence is insufficient to prove beyond a reasonable doubt the charged one-kilogram conspiracy. Accordingly, we vacate the conviction and sentence on Count I and remand with directions to the district court to enter judgment against Hickman for conspiracy to distribute one hundred grams or more of heroin, a lesser included offense encompassed by Count I of the superseding indictment amply supported by sufficient evidence.

A.
We review the denial of Hickman's motion for judgment of acquittal de novo. See United States v. Green, 599 F.3d 360, 367 (4th Cir.2010), cert. denied, ___ U.S. ___, 131 S.Ct. 271, 340, ___ L.Ed.2d ___ (2010). "[V]iewing the evidence *763 in the light most favorable to the Government," United States v. Bynum, 604 F.3d 161, 166 (4th Cir.) (internal quotation marks omitted), cert. denied, ___ U.S. ___, 130 S.Ct. 3442, 177 L.Ed.2d 347 (2010), we are to determine whether the conviction is supported by "substantial evidence," where "substantial evidence is evidence that a reasonable finder of fact could accept as adequate and sufficient to support a conclusion of a defendant's guilt beyond a reasonable doubt," United States v. Young, 609 F.3d 348, 355 (4th Cir.2010).
Though this is a "heavy burden," id., it is by no means insuperable. The ultimate question is whether "any rational trier of facts could have found the defendant guilty beyond a reasonable doubt." Bynum, 604 F.3d at 166 (internal quotation marks omitted). Where, as here, the trier of fact would have had to rely on attenuated inferences of drug quantity, the finding of quantity cannot stand.

B.
Overwhelming evidence supports the jury's finding that Hickman was a knowing member of the basic conspiracy alleged in Count I of the superseding indictment.
Conviction for conspiracy to distribute narcotics under 21 U.S.C. § 846 requires proof beyond a reasonable doubt of three elements: (1) "an agreement between two or more persons to engage in conduct that violates a federal drug law" here, to distribute or possess narcotics with intent to distribute; "(2) the defendant's knowledge of the conspiracy; and (3) the defendant's knowing and voluntary participation in the conspiracy." United States v. Kellam, 568 F.3d 125, 139 (4th Cir.), cert. denied, ___ U.S. ___, 130 S.Ct. 657, 175 L.Ed.2d 501 (2009). Proof of a conspiratorial agreement need not be by direct evidence, and rather may "be proven inferentially and by circumstantial evidence." United States v. Godwin, 272 F.3d 659 (4th Cir.2001).
Given the plethora of direct evidence, including but not limited to the content of the telephone communications and the circumstances surrounding the traffic stop of Hickman, only very modest inferences, if any, were required here to show the existence of a conspiratorial agreement and Hickman's knowing membership in a heroin conspiracy. The damning wiretap recordings reveal Hickman and Henderson's coordination before[1] and after[2] Hickman purchased 32 grams of heroin from Jones; Henderson discussing his stake in the transaction;[3] and later exchanges between Hickman and Henderson concerning the possibility of a subsequent transaction after the police seized the heroin initially purchased.[4] Moreover, agents observed *764 Hickman entering and exiting Jones's store[5]; in the interim, wiretaps record Jones discussing the Hickman sale with Caldwell;[6] and a police stop some minutes later found Hickman in possession of those 32 grams.[7] In the face of this mountain of evidence of Hickman's knowing participation in a heroin distribution conspiracy, Hickman's broader challenge to the sufficiency of the evidence is plainly unavailing.[8]

C.
Although there is overwhelming evidence of Hickman's knowing membership in a heroin distribution conspiracy, his challenge to the jury's finding that the conspiracy of which he became a member involved at least one kilogram of heroin has merit. Sufficient evidence supported a finding that Hickman knowingly became a member of a large-scale heroin distribution conspiracy which involved, at the least Jones, Henderson, and Caldwell, and that many of the acts of the coconspirators were reasonably foreseeable to Hickman. Yet, no matter how generously we indulge the available reasonable inferences in favor of the Government, adding the post-dilution weight of heroin from all known and reasonably inferable transactionswhether completed, attempted, or merely agreed upon by any of Hickman's co-conspiratorsto reach a sum of one kilogram, if not a mathematical impossibility, would require reasoning so attenuated as to provide insufficient support for the jury's verdict on the one-kilogram verdict.
The jury heard evidence of the amounts and purity of heroin seized during the traffic stop of Hickman and the raid on Caldwell's home; evidence of discussion between Hickman and Henderson of an attempt in the days after Hickman's initial arrest to "do the same thing [transaction] again," J.A. 450, presumably with Jones; and a vague statement the defense itself elicited on recross-examination of a Government witness concerning the "recover[y] *765 [of] heroin ... the week prior" from someone exiting Jones's store, J.A. 335.
The indictment alleged a four-month conspiracy. The only definite amounts of heroin established by the Government were the 32.14 grams of 38% pure heroin recovered during the stop of Hickman's vehicle; 17 gold-topped vials also recovered during the traffic stop, the contents of which were never weighed; the 5 grams of heroin Jones kept from Hickman; and 139 grams of 29% pure heroin seized from the raid on Caldwell's apartment. Expert testimony from Agent O'Meara provided a basis for inferring that the co-conspirators intended that the heroin be substantially diluted before reaching end-users. To "step on" or "cut down" the heroin, a dealer would mix one part raw heroin with one, two, or three parts of "cut"the solvent used to dilute the heroin, often mannite or quinineto reach a concentration fit for most users, which Agent O'Meara estimated to be 8%. J.A. 286-88.
Agent O'Meara's discussion of the dilution process was the jury's only basis for inflating the weight of the recovered heroin, and as such the method of dilution he described must constrain the trier of fact's reasonable inferences. As Agent O'Meara described, raw heroin is said, in the "slang vernacular," to "take[] a one, two, or a three" depending on how many parts of solvent are mixed with each part of heroin, J.A. 288; because "the drug dealers on the street don't have a laboratory," J.A. 289, they are limited to this simple mixing process. Agent O'Meara's lengthy discussion, and the vernacular terms he cites, indicate that this mixing of heroin with "cut" is always in the ratio of 1: n, where n is a natural number. This, of course, would bound the maximum dilution of heroin of a given purity: without "a laboratory" capable of more precise measurements and mixtures, dilution to precisely 8% will ordinarily be impossible. Thus the 38% pure heroin recovered from Hickman would only "take a four"be mixed 1:4 with solventsince a 1:5 mixture would decrease the purity below 8%; similarly, the heroin seized from Caldwell, 29% pure, would "take a three."[9] These dilutions would result in 128.56 grams of 9.5% purity and 417 grams of 9 2/3% puritytogether, approximately 546 grams of heroin. But we will assume, arguendo, that the Hickman conspiracy would have diluted this heroin precisely to 8% purity, generating 153 grams and 504 grams for Hickman and Caldwell, respectively. Together with the 5 grams Jones kept from Hickman which, assuming it was of the same purity as the 32 grams Jones sold him, would add another 23.75 gramsthe Government would have established a total of 681 grams.
As for the 17 gold-top vials recovered during the traffic stop of Hickman, their contents were never weighed. J.A. 177-78. *766 Rumber, one of the Government's two criminologists at trial, testified that the gross weight of the vials was 24.30 grams, and confirmed that the net weightthe weight of the heroin itselfwas never measured. The only testimony on point is from Agent O'Meara, who explained that a vial of heroin typically contains 0.1 grams. Thus the 17 vials support an additional 1.7 grams of heroin. Added to the 681 grams assumed above, this would support a finding of no more than 683 grams.
Evidence concerning the remaining two potential transactions is scant, to say the least. During the week after Hickman's initial arrest and release, he and Henderson spoke about attempting to "do the same thing again," J.A. 450, and Hickman claimed he had made several telephone calls to a potential narcotics supplier, apparently Jones. Though Hickman remained free until his arrest in August 2007, no evidence of this transaction was offered. Aside from Henderson's reference to "do[ing] the same thing again," J.A. 450 (emphasis added), there is no evidence concerning the amount of heroin Hickman and Henderson sought. Attributing any more than another 32.14 grams to this potential transactionlike the earlier purchase from Joneswould be purely speculative; thus, at a maximum, the jury could find that the conspiracy involved another 153 grams of "cut" heroin, giving them a hypothetical sum of 836 grams.
Evidence of the final alleged transactiona brief statement on recross-examination by Agent O'Meara about the seizure of some amount of heroin from some individual exiting Jones's store a week before Hickman's purchaseis extremely vague:
Q. How many people did you stop coming out of Fat Cat's and recover heroin from?
A. We recovered heroin out ofthe week prior. I think it was April 17th.
Q. And you recovered it in terms of the stop of Hickman; is that right?
A. Yes, sir.
J.A. 335. There is no evidence that this seizure had anything to do with Hickman, Henderson, or Caldwell. Moreover, given that Jones was in the business of selling drug paraphernalia and so would likely be visited by those who had recently bought (or intended to buy) narcotics elsewhere, there is little reason to believe that the seized heroin had just been sold through Jones's store. That the month-long surveillance of Jones's store bore no more fruit than this single drug seizure further undermines the circumstantial evidence of drug quantity implicating Jones.[10] Even if, squinting our eyes, it were to appear nonspeculative to attach an amount to this earlier seizure and link it to the Count I conspiracy, the evidence could by no means bear any more than the "stepped on" amount of Hickman's purchase from Jones, 153 grams. And even with this final amount, the sum would fall short of one kilogram.

D.
To reach its finding that the Count I conspiracy involved at least one kilogram, the jury would have had to rely on either or both of two grounds: (1) that Hickman was criminally liable for the distribution of heroin by buyers of paraphernalia from *767 Jonesa theory that fails as a matter of law; or that, (2) on account of the conspirators' apparent familiarity with the drug trade, they must have undertaken to distribute some amount beyond the amounts involved in the evidenced transactions. Neither theory can sustain Hickman's conviction on the one-kilogram conspiracy.

1.
The Government told jurors during closing argument that Jones "basically runs a one-stop shop[] for heroin," with "thousands and thousands of vials that heroin goes in." Tr. in United States v. Hickman, No. 1:07-cr-00261, vol. 3, at 33-34 (D. Md. April 30, 2008). Indeed, the evidence well supports the inference that Jones was in the business of selling drug paraphernalia. As Agent O'Meara testified, a raid on Jones's store produced bins containing more than 25,000 glass vials of the sort used to distribute heroin, packaged in groups of one hundred, in various sizes and with variously colored tops, also packaged in groups of one hundred; customers were apprised of the merchandise by a three-ring binder on the counter. J.A. 276-77, 284-85, 292.
Yet there is absolutely no evidence to support a finding that Hickman is liable for heroin distributions by those who had purchased empty vials from Jones. Indeed, it strains credulity even to think that, on the evidence in the record, Jones himself could be convicted of a widespread conspiracy to distribute heroin on the mere fact that he sells drug paraphernalia and had such packaged merchandise at his store. Finding such conspiracy liability for Jones would have required the trier of fact to find beyond a reasonable doubt that Jones's purchase of paraphernalia for resale evidenced a conspiracy to conspire with future paraphernalia buyers concerning subsequent distribution. While paraphernalia vendors can certainly become parties to the distribution conspiracies of their buyers, such cases present substantial evidence of the vendor's involvement in and/or knowing facilitation of, a distributor's operations. See, e.g., United States v. Meredith, 824 F.2d 1418, 1428 (4th Cir. 1987).
Tellingly, Agent O'Meara testified that law enforcement "did not stop a lot of people coming out of [Jones's store]" because they knew that Jones was "also selling paraphernalia, like vials, and ... we're not going to go out and arrest a particular person just for having the actual vials ... empty vials." J.A. 331-32. "Unless there is a specific reason, [and] we know that a drug deal is taking place," he testified, "we're not just stopping everybody that's coming out of there and ... trying to pull a bunch of empty vials off the street." J.A. 332. In fact, the jury was told that in more than a month of surveillance of Jones's store, only a single patron other than Hickman was found with narcotics on his person. Where the Government failed to present evidence about even a single buyer of the paraphernalia or any plans on the part of Jones to involve himself in the distribution activities of his paraphernalia customers, it would be mere speculation to find that Jones's own distribution activities were coextensive with his paraphernalia business.[11] On this paltry evidence, no *768 reasonable inference as to the scope of Jones's conspiracy to distribute narcotics could be drawn from the amount of paraphernalia merchandise he hoped to sell.
Furthermore, even if the evidence had supported widespread conspiracy liability for Jones on the basis of his trafficking in drug paraphernalia, this liability could not be transferred to Hickman on this record. As an initial matter, while we have assumed that Hickman and Jones conspired together with respect to the distribution chain that included Henderson and Caldwell, no evidence was presented to suggest that the selling of paraphernalia entered into this agreement. Moreover, as there was no evidence that Hickman had or planned to have any involvement with (or even any knowledge of) Jones's paraphernalia buyers, such a conspiracy (having Jones at the center) would have been a classic "rimless," "hub-and-spokes" conspiracy, which has long been held to make out multiple, distinct conspiracies and not a single, large one, see Kotteakos v. United States, 328 U.S. 750, 754-55, 66 S.Ct. 1239, 90 L.Ed. 1557 (1946); Dickson v. Microsoft Corp., 309 F.3d 193, 203-04 (4th Cir. 2002). Thus evidence of the number of packaged, empty vials in Jones's store is not a sufficient basis for further inflating the amount of heroin ascribable to the Count I conspiracy.

2.
Nor could the jury properly convict Hickman of the one-kilogram conspiracy in reliance on the Government's repeated assertions during closing argument that the four-month Count I conspiracy encompassed far more drug distribution activity (and that Hickman could reasonably foresee such quantity) than that of which the Government could produce competent evidence. The Government strenuously urged jurors to draw such an inference, claiming that the evidence presented was only "a window into the conspiracy," merely "part of something larger"; that the evidenced transactions were simply "two brief episodes ... in the life of this conspiracy" or "a few days in the life of a heroin conspiracy"; that "this [was] an ongoing course of business," an "ongoing thing." Tr. in United States v. Hickman, No. 1:07-cr-00261, vol. 3, at 33, 38, 41-42 (D. Md. April 30, 2008). "[I]t's clear," said the prosecutor, "that this is a regular course of business for these gentlemen. It's clear that this isn't the first time they've done this before [sic]. It's clear that this wasn't going to be the last time.... [Hickman] would have continued to [seek out drug transactions]."
This line of argument is troubling, not just because it seems to urge jurors to convict the defendant for what he "would have continued to do" which, to the extent these hypothesized future bad acts were not captured by an agreement within the charged period, is clearly improper, see United States v. Cunningham, 54 F.3d 295 (7th Cir.1995) ("The Government may not attempt to obtain a conviction by appealing to jurors to prevent futures crimes...."); United States v. Monaghan, 741 F.2d 1434 (D.C.Cir.1984) ("A prosecutor may not urge jurors to convict a criminal defendant in order to ... deter future lawbreaking."), but also because it invites the jury to speculate as to the amount of heroin involved in the conspiracy.
Where no evidence exists to guide the trier of fact in determining the outer scope of a conspiracy, the trier may not *769 simply guess at the magnitude or frequency of unknown criminal activity. Unbridled speculation is an impermissible basis for conviction beyond a reasonable doubt.
Under more than two decades of federal law, it is impermissible to enhance drug amounts without particularized evidence of narcotics transactions. Federal district courts have long struggled with extrapolating drug amounts under the U.S. Sentencing Guidelines, which instruct that, "[w]here ... the amount [of narcotics] seized does not reflect the scale of the offense, the court shall approximate the quantity of the controlled substance." U.S.S.G. § 2D1.1 n. 12. In reviewing district courts' findings, the courts of appeals have developed a sizable body of case law narrowing the range of permissible inferences. Its teachings are instructive here.
As the First Circuit has explained, "[w]here drug-quantity extrapolations have been upheld, the Government managed to demonstrate an adequate basis in fact and that the drug quantities were determined in a manner consistent with the accepted standards of reasonable reliability." United States v. Rivera-Maldonado, 194 F.3d 224, 232 (1st Cir.1999) (citing to Second and Third Circuit cases). Concerned about the methodologies district courts employ to calculate their extrapolations, courts have cautioned against using small sample sizes. See, e.g., Rivera-Maldonado, 194 F.3d at 231-33 (warning that "the smaller the sampling, the less reliable the resulting probability estimate," the court reversed where the sample size was "minuscule"); United States v. Howard, 80 F.3d 1194 (7th Cir. 1996) ("[A]bsent a reliable evidentiary basis supporting the ... assumption as to the amounts [buyers] were able to purchase from [defendant]... we are unwilling to rest on speculation [from the details of a known purchase].... On remand, it will be the Government's responsibility to proffer some evidentiary basis from which a reasonable and reliable estimate may be made of the amounts [buyers] purchased from [defendant] on the other occasions.").
To be sure, where courts have evidence of a number of transactions, they have been permitted to multiply that number by an average weight-per-transaction to reach an estimate, see, e.g., United States v. Correa-Alicea, 585 F.3d 484 (1st Cir.2009) (upholding such a calculation using "highly conservative" estimates), cert. denied, ___ U.S. ___, 130 S.Ct. 1909, 176 L.Ed.2d 384 (2010); United States v. Rodriguez, 525 F.3d 85, 107-08 (1st Cir.2008) (approving of a calculation using a "conservative estimate" of weight-per-transaction); United States v. Durham, 211 F.3d 437, 444 (7th Cir.2000) ("[I]t is also permissible for a court to take witness' estimates of the amount of drugs they purchased and multiply that by the minimum quantity sold on each occasion...."), though the circuit courts have urged district courts to "make conservative approximations," Henderson, 58 F.3d at 1151; see also United States v. Sklar, 920 F.2d 107 (1st Cir.1990) ("[W]hen choosing between a number of plausible estimates of drug quantity ... a court must err on the side of caution.") (quoting United States v. Walton, 908 F.2d 1289, 1302 (6th Cir.1990)).
Such conservative extrapolation as to the amount sold in evidenced transactions is perfectly proper; but the courts of appeals have refused to allow a trier of fact to extend this extrapolation so far as to fabricate transactions of which there is no evidence. Thus the First Circuit has held that evidence that a drug conspiracy "did a substantial amount of narcotics business" was insufficient, "[i]n the absence of particularized findings," to support a determination by a preponderance of the evidence of an amount of narcotics that *770 "seem[ed] attainable given the appellant's role in the conspiracy," explaining that a court "cannot uphold a drug quantity calculation on the basis of hunch or intuition." United States v. Marrero-Ortiz, 160 F.3d 768, 779-80 (1st Cir.1998) (emphasis added). Likewise the Third Circuit has reversed and remanded for more particularized findings despite proof that defendant's "involvement with cocaine was extensive and continued over a long period of time," holding that such evidence "does not translate readily into a specific drug quantity finding." United States v. Miele, 989 F.2d 659, 668 (3d Cir.1993).
Here the Government contends, unpersuasively, that Hickman's use of "vague and coded language" establishes "familiar[ity] with drug trafficking" and that "only someone heavily involved in repeated drug trafficking would be so brazen and unrepentant to return almost immediately to trying to find more heroin to sell." Br. of Appellee at 27-28. In a case like Hickman's, where a lack of evidence would make particularized findings impossible, courts would have reversed a sentencing court's findingmade by a mere preponderance of the evidencethat Hickman's conspiracy involved one kilogram or more of heroin. And if such an inference would have been too speculative to satisfy the Sentencing Guidelines, which merely require an "approximat[ion]," U.S.S.G. § 2D1.1 n. 12, based on evidence with "sufficient indicia of reliability to support a conclusion that they are probably accurate," U.S.S.G. § 6A1.3, then a fortiori it would fail to satisfy a rational trier of fact tasked with making findings beyond a reasonable doubt.

E.
In this case, where evidence of unknown transactions was meager and offered virtually no guide as to the amounts that may have been involved, we hold that the jury verdict finding the heroin conspiracy involved one kilogram or more was not supported by sufficient evidence.
Yet it is within our power to direct entry of judgment on a lesser included offense when vacating a greater offense for insufficient evidence. See Rutledge v. United States, 517 U.S. 292, 306, 116 S.Ct. 1241, 134 L.Ed.2d 419 (1996) (noting that "federal appellate courts appear to have uniformly concluded that they may direct the entry of judgment for a lesser included offense when a conviction for a greater offense is reversed on grounds that affect only the greater offense"); accord United States v. Silvers, 90 F.3d 95 (4th Cir.1996); United States v. Plenty Arrows, 946 F.2d 62, 66-67 (8th Cir.1991) (doing so); Government of Virgin Islands v. Brown, 685 F.2d 834, 841-42 (3d Cir.1982) (same). Some circuits have introduced limitations on this remedy, chief among them that the jury have been instructed on the lesser included offense. See, e.g., United States v. Dhinsa, 243 F.3d 635, 674-76 (2d Cir. 2001); United States v. Vasquez-Chan, 978 F.2d 546, 553-54 (9th Cir.1992), overruled on other grounds by United States v. Nevils, 598 F.3d 1158 (9th Cir.2010). We decline to consider that question here, where even the most stringent limitations would be functionally satisfied.
Because we conclude that the record contains sufficient evidence to persuade a rational fact finder beyond a reasonable doubt of Hickman's guilt on the lesser included offense of conspiracy to distribute one hundred grams or more of heroin, we direct entry of judgment against Hickman under Count I of the indictment for conspiracy to distribute and to possess with intent to distribute heroin in the amount of *771 one hundred grams or more.[12]

III.
Hickman's second contention, that the trial judge erred in rejecting his proposed instructions regarding the amount of heroin attributable to him, is without merit.
Reviewing the refusal to give a jury instruction for abuse of discretion, we will reverse only when the requested instruction "(1) was correct; (2) was not substantially covered by the court's charge to the jury; and (3) dealt with some point in the trial so important, that failure to give the requested instruction seriously impaired the defendant's ability to conduct his defense." United States v. Lighty, 616 F.3d 321, 366 (4th Cir.2010) (internal quotation marks omitted). This review is holistic: "we do not view a single instruction in isolation; rather we consider whether[,] taken as a whole and in the context of the entire charge, the instructions accurately and fairly state the controlling law." Id. (internal quotation marks omitted).
At trial, defense counsel requested that the judge instruct the jury that Hickman could only be held responsible for the "amount of drugs that were foreseeable to the Defendant and within the scope of his agreement." J.A. 347. And counsel objected to the verdict form's second question"If you find the Defendant ... guilty as to Count 1, how do you find as to the amount of heroin involved?"asking that the following be appended: "and that is foreseeable to him and within the scope of his agreement." J.A. 348.
The trial court repeatedly instructed the jury that coconspirators' actual or intended distribution of narcotics could only be charged to Hickman "so long as it was reasonably foreseeable to [him] that such a type and quantity of drugs would be involved in the conspiracy which he joined." J.A. 358 (twice); see also J.A. 359 ("acts or plans" of coconspirators only attributable to Hickman if "reasonably foreseeable" to him). With respect to the special verdict form, the judge specifically instructed the jury that the second question asked "the type and amount of drugs reasonably foreseeable to the Defendant that would be involved in the conspiracy, or that he possessed or distributed," and explained that "[t]he burden is on the Government to establish the type and amount of drugs beyond a reasonable doubt." J.A. 359a.
These instructions accord with the principles of Pinkerton v. United States, 328 U.S. 640, 66 S.Ct. 1180, 90 L.Ed. 1489 (1946), and our case law, see, e.g., United States v. Brooks, 524 F.3d 549, 558 (4th Cir.2008) ("[A] trial court is obliged to instruct a jury to use Pinkerton principles to determine the quantity of drugs attributable to each individual defendant involved in a drug conspiracy."); United States v. Williams, 986 F.2d 86, 90 (4th Cir.1993) ("A defendant convicted of conspiracy should be sentenced not only on the basis of his conduct, but also on the basis of conduct of coconspirators in furtherance of the conspiracy that was known to the defendant or reasonably foreseeable to him.").
To the extent that defendant's proposed instruction differs from the instructions given, it misstates the law. It is true that *772 Hickman is responsible only for the amount of drugs "within the scope of his agreement," if this "scope" is properly understood as encompassing any co-conspirators' conduct in furtherance of the conspiracy and reasonably foreseeable to him; but the trial court instructions adequately covered this. If the requested instruction is taken to mean that Hickman is responsible only for an amount of drugs he personally knew of and explicitly agreed to distribute or have distributed, then it mistakes a basic tenant of our conspiracy law, see Pinkerton, 328 U.S. at 647-48, 66 S.Ct. 1180; Williams, 986 F.2d at 90. There was no abuse of discretion in the district court's rejection of Hickman's proposed instruction.

IV.
Appellant next argues that the trial court abused its discretion in refusing to grant a new trial after allowing the jury to make limited use during deliberations of the wiretap transcripts, which were not in evidence. This contention lacks merit.
Under ordinary circumstances, this court will not consider alleged errors that were invited by the appellant. See United States v. Quinn, 359 F.3d 666, 674-75 (4th Cir.2004); United States v. Jackson, 124 F.3d 607, 617 (4th Cir.1997) ("The `invited error' doctrine recognizes that a court cannot be asked by counsel to take a step in a case and later be convicted of error, because it has complied with such request.") (internal quotation marks omitted). Here, the jury asked to rehear a wiretap recording "of Hickman after April 25 talking about getting something going." J.A. 363. The trial court and counsel agreed that this might refer to any of six different intercepted calls, and discussed several options. J.A. 363-65. After the court suggested playing all six calls for the jury, counsel for Hickmanlikely concerned about replaying so many recordings of his client planning a drug purchaseinterjected, "Well, I would like to ask them if they can take a look at the book and specify more carefully what they want to have played." J.A. 365. The court immediately confirmed that defense counsel "ha[d] no objection to them having the book, then," and counsel agreed, "I have no objection to them having the book, and then we can take the books back, to specify, as long as we get the books back after that." J.A. 365. When the prosecutor suggested allowing the deputy to take back the relevant book, defense again agreed: "Okay. Just send one book back, yeah." J.A. 365.
Hickman complains that when the jury concluded its deliberations less than thirty minutes later, it had yet to request a specific tape and return the transcript book, and that this was a misuse of the book that tainted the jury verdict. But defense counsel agreed to the method by which the jury was to use the transcript bookhaving the deputy leave the book with the jurors until they requested a particular tape. Defense did insist that "we get the books after [the jurors select a tape]," but this proviso was never violated; rather, the triggering conditionthe jury's selection of the tapenever occurred. Had the jury perused the book over thirty minutes, sent for a particular tape, and then arrived at a verdict before hearing it, the defense clearly would have invited any error that accrued. The mere fact that the jury never bothered to send for a particular tape is immaterial. Thus any error here was invited by Hickman.
In the absence of extraordinary circumstances like an apparent "miscarriage of justice" or doubt as to "the integrity of the judicial process," United States v. Herrera, 23 F.3d 74, 76 (4th Cir.1994), this court has never reviewed errors invited by the appellant. *773 We decline Hickman's invitation to do so here.

V.
Finally, Hickman challenges the district court's reliance on one of two predicate drug offenses to enhance his sentence under 21 U.S.C. § 841(b)(1)(A), claiming that the conviction was "illegal," Br. of Appellant at 28. Hickman was convicted in 2006 for attempted distribution of heroin in a Maryland court. Hickman contends that the underlying charging document wrongly charged this attempt under the common law, rather than under Maryland criminal statutes. This contention is unavailing.
Even if Hickman were able to evade the conclusive presumption that a prior conviction not set aside on direct or collateral review is valid by establishing a violation of the right to counsel, see Daniels v. United States, 532 U.S. 374, 380-82, 121 S.Ct. 1578, 149 L.Ed.2d 590 (2001), his claim would fail. In Maryland, attempt is a common-law crime, see Moore v. State, 388 Md. 623, 644-45, 882 A.2d 256 (2005) ("By Maryland common law, the attempt to commit a crime is, itself, a separate crime...."), and thus the attempted distribution was properly charged under the common law.

VI.
For the reasons explained, we conclude that the evidence was insufficient to establish beyond a reasonable doubt that Hickman knowingly became a member of a conspiracy to distribute or to possess with the intent to distribute more than one kilogram of heroin. On the other hand, there is overwhelming evidence in the record to show, beyond a reasonable doubt, that Hickman knowingly became a member of a lesser included conspiracy involving 100 grams or more of heroin, and that such amount was reasonably foreseeable to Hickman. Accordingly, we vacate the conviction for conspiracy to distribute heroin in the amount of one kilogram or more and the life sentence imposed thereon, and we remand the case to the district court with directions to enter a judgment of sentence as to the lesser offense and to resentence accordingly. In all other respects, the judgment is affirmed.
AFFIRMED IN PART, VACATED IN PART, AND REMANDED WITH INSTRUCTIONS
NOTES
[1]  See J.A. 400-01 (on the day of the buy, Henderson reporting to Hickman that Henderson is driving "[r]ight behind" him in "the gold car" at 3:04 p.m.).
[2]  See J.A. 408-09 (at 3:29 p.m., Henderson calling Hickman to ask if he "ever ma[d]e it to the store" and if he was "on [his] way back from seeing him [Jones]," and Henderson arranging that he will "just come on up" to Hickman's house to "get that shit out the way"that is, collect the purchased heroin).
[3]  See J.A. 426-27 (an hour after the buy, having just learned of Hickman's arrest, Henderson telling an unidentified man that Hickman "had my fucking bread yo" and complains "I ain't even get to get that [i.e., retrieve the narcotics] from him yo").
[4]  See J.A. 446 (two days after Hickman's purchase, consequent arrest and release on bail, Henderson asking Hickman to "check with him and see if we able to do something tomorrow man," to which Hickman replies, "I've been trying to call but yeah I'll do that"); J.A. 448 (the next day, Henderson checking in with Hickman, who reports that he's "waiting on him to call" but that "[everything]'s still good"); J.A. 450-51 (the day after that, Henderson asking Hickman if he can "see if that dude can do the same thing again"; Hickman confirming he "talked to him"); J.A. 457-58 (the following day, Henderson again asking after Hickman's progress and Hickman informing him that he is "trying now man" and that he is "going to go down there and see ... Fat Cat").
[5]  See J.A. 81 (Agent Strickland testifying to seeing a "gold sedan" pull up at Jones's store at 3:13 p.m. and a black male exiting the car and entering the store); J.A. 95-96, 123 (Detective Sokolowski testifying to stopping the sedan with the same license tag at 3:39 p.m.)
[6]  J.A. 402-05 (at 3:20 p.m., Jones calling Caldwell to confirm that Hickman is Caldwell's man and to ask Caldwell whether to spot Hickman Jones's last five grams of heroin, which Hickman's funds wouldn't cover).
[7]  J.A. 95-98, 123 (Detective Sokolowski testifying to stopping Hickman and finding the bag of heroin).
[8]  We pause to acknowledge the practical conundrum faced by counsel for Hickman. Rare is the lawyer who wants to make an argument to a jury that, "My client was not involved in a conspiracy, but if you disagree, it was at most a conspiracy involving less than one kilogram of heroin." Thus, in his arguments on the motion for judgment of acquittal, in his request for jury instructions (discussed infra), and in his closing argument to the jury, counsel largely sought to cast Hickman as involved merely in one or more buy-sell transactions rather than in a conspiracy, and that argument is pressed on us in this appeal. Nonetheless, we have no doubt that Hickman's argument on the motion for judgment of acquittal was adequate to alert the district court to the deficiency in the Government's proof of drug quantity and to preserve the sufficiency of evidence issue for purposes of this appeal.
[9]  We note that 8%, though Agent O'Meara gave it as a "rough" estimate of the "average" for lowest-purity "scramble" heroin in Baltimore City, J.A. 286, was the lowest purity discussed at trial. The Government's criminologist, who had worked for the Baltimore City Police Department for eleven years and tested "thousands" of samples for the presence of controlled dangerous substances, J.A. 168-69, put the weakest heroin submitted to the crime lab for testing at "around 9 to about 11" percent, J.A. 173. (In fact, Government counsel pegged "[w]hat's on the street" at "10%" during a colloquy with the district judge and defense counsel, in which he went on to explain to the judge that the Government's theory was that the seized heroin would have been "cut down further ... [d]own to 10%." J.A. 206.) As we view the facts in the light most favorable to the Government, we assume 8% purity was the conspiracy's intended target. But finding, beyond a reasonable doubt, even higher weights via speculation as to a lower purity target would be wholly unreasonable on this record.
[10]  During a pre-trial hearing on a suppression motion, Officer Sokolowski testified that he knew of only two other seizures related to Jones's store, the first involving "over 100 gel caps of heroin," and the other, "without having notes in front of [him], [he recalled to be] a large quantity of heroin and cocaine." J.A. 58-59. When the Government attempted to elicit this testimony from Sokolowski during trial, the court sustained a defense objection as the witness began to answer, and the Government was forced to move on. J.A. 94-95.
[11]  Even where co-conspirators have had no connection to paraphernalia sales, and thus any packaging materials would presumably be used for their own distribution, some circuit courts have been skeptical of relying on unused drug packaging materials to increase drug amounts charged to a conspiracy. See United States v. Henderson, 58 F.3d 1145 (7th Cir.1995) (explaining that the court "question[ed] whether multiple boxes of unused baggies present the same degree of reliability [as the used baggies relied on in an earlier case]" and declining to rely upon them to support lower court's finding as to amount of narcotics within the scope of the conspiracy, where lower court's methodology appeared to lack sufficient indicia of reliability).
[12]  Because Hickman had been convicted of two prior felony drug offenses, the Government filed a notice of enhanced punishment mandating a minimum mandatory sentence for the one-kilogram conspiracy of life imprisonment. See 21 U.S.C. §§ 846, 841(b)(1)(A). We of course express no view as to what might be an appropriate sentence after further proceedings.
