FOXBORO -- On Tuesday, Bill Belichick broke out a new throwback hoodie with Pat Patriot on the sleeve. A special bye-week hoodie, perhaps?

“Might be,” he said with a smile. “You don’t want to make it too popular and take over the gray one.”

Needless to say, it was a laid back atmosphere at Gillette Stadium on Tuesday, an indicator the Patriots’ bye week had arrived. Belichick said Tuesday the franchise will use the time to get a jump on preparations for Pittsburgh -- New England’s first opponent following the bye, a game set for Oct. 30 -- as well as engage in some self-scouting.

“We have quite a few things going on,” Belichick said Tuesday. “We’ll look back on last week’s game, the first six games -- actually all 10 games -- [and] try to evaluate how we’re doing some things, where we can improve, Maybe what things we want to do more of, things like that.”

“I think it’s really a combination of things,” said Patriots director of player personnel Nick Caserio when asked about New England’s course of action during the bye wee. “We’ll practice this week; we’ll work on a few things. The most important thing is making sure that we’re prepared to play a few weeks from now when we go down to Pittsburgh, so we’ll try to balance as much as we can and start our preparations here for Pittsburgh along the way.”

Self-scouting is really the chance to turn the focus to your own organization. Belichick said that involves a combination of things -- there’s individual techniques, combinations of personnel, as well as situational football.

“We cover all those things, absolutely,” Belichick said. “You have a list of things, you take the priorities -- we have to do this, we have to do that, this is less important than something else. You work on things you feel like are more important.”

One NFC personnel man said that the most important part of the self-scouting process involves looking “at the efficiency of your schemes vs. what the opponent has done versus you.” In that same vein, an NFC scout who has watched the Patriots this season has three points of emphasis New England will likely examine during the bye week.

“The Patriots do a good job of attacking you were you are weakest personnel or scheme wise. During the bye week, I think they’ll look at the efficiency of players in that offensive scheme,” he said. “Two, how they get better at third-down conversions on defense. And three, try to find ways to improve the run game in personnel groups and formations in which they have had success throwing the ball, in an effort to become more unpredictable and give Brady more options to adjust based on opponents adjustments.”

•Admittedly, the Patriots set a high bar for themselves when it comes to offensive efficiency -- remember, this is a team that leads the league in total offense and passing offense -- but over the bye week, New England will likely look to focus on some of the things that dragged them down Sunday against the Cowboys. New England had three series where they had three plays or less, and had a season-high seven negative plays from scrimmage (three sacks of Brady to go along with negative runs from Stevan Ridley, Danny Woodhead and two from BenJarvus Green-Ellis). In all, through six games, the Patriots have had 22 negative plays (excluding kneeldowns). And New England is 10th in the league in penalties with 39, and seventh in the NFL in penalty yardage with 353.

•The Patriots are 21st in the league in third-down defense with a 42.5 percent conversion rate. Not as bad as the 47.1 percent they allowed over the course of the 2010 season (easily the worst in the league), but still the sort of thing that will cause the coaching staff to take a serious look at personnel and scheme going forward. For what it’s worth, it looks like the numbers are headed in the right direction -- whether it has been matchup or scheme-based changes, the last two weeks against the Jets and the Cowboys, the Patriots have held opposing offenses to 7-for-23 (30 percent) on third down.

•Running the football out of “personnel groups and formations in which they have had success throwing the ball” boils down to trying to find a way to run when you initially show pass in an attempt to keep defenses guessing. The Patriots do have some flexibility when it comes to these sorts of scenarios, as the presence  of young tight ends Rob Gronkowski and Aaron Hernandez on the field simultaneously allows the New England offense to present one look and ultimately go with another.

One way they were able to run out of what might be considered a passing set happened last Sunday was when they got some motion with players in and out of the backfield. On the Patriots’ first play from scrimmage in the third quarter, Green-Ellis was initially split wide to the right, but as Brady called out the signals, he came back to the backfield to line up behind the quarterback. New England was able to make the most of the change, as Green-Ellis went for seven yards.