Brian Urlacher leads the Bears against the Packers in one of several big inter-division Week 3 matchups. (Charles Rex Arbogast/AP)

Kind of makes you wonder why we haven’t been doing this all along.

The NFL should open each season with division games in Weeks 1, 2 and 3, and close it with rematches in Weeks 15, 16 and 17. Squeeze the byes and all the random 49ers-Bengals and Ravens-Rams games in between.

With yardage and scoring through the roof, there are few things the NFL could do to make the on-field product better. Starting and ending each season with division games is one.

Imagine if each Cowboys’ season started with the Redskins in Week 1, a trip to New York in Week 2, and Philly in Week 3? Or what about the Browns coming out of the gate against the Bengals, Steelers and Ravens? Wouldn’t Packers fans love to get an edge on the Vikings, Bears and Lions right away?

NFC South games sure would beat Carolina’s opening schedule this season: at Arizona, home against Green Bay and Jacksonville. Competitive games, but not exactly natural rivals.

The NFL has already taken steps in this direction, by trying its darndest to have teams play division rivals in Weeks 16 and 17. But why not just go whole hog?

First, it would eliminate the scheduling quirk of having to play a team twice in a three- or four-week span. Seeing a team in Week 1 and Week 15 is hardly like seeing the same team at all. Think of the Cowboys last season. In Week 1 they were led by Wade Phillips and Tony Romo and had dreams of playing in the Super Bowl on their home field. By Week 15, Jason Garrett was the coach, Jon Kitna was the QB and the franchise was trying to find its direction.

Second, it would guarantee great matchups to open the season, and even better ones down the stretch. The teams in the postseason hunt would need division and conference wins to bolster their playoff hopes, and the format would reward teams playing their best at the end. Fans of the teams out of the hunt would at least have reason to go to games. Raiders-Broncos is Raiders-Broncos whether it’s all on the line or nothing at all is.

Josh Freeman and the Bucs host the Falcons in a critical early-season NFC South matchup. (Andy King/AP)

In 2002, the NFL did a wise thing in expanding to 32 and switching to eight four-team divisions — the perfect number to award a home playoff game against a wild-card team to each champion. At the time, the NFL’s rearranging wasn’t considered a no-brainer. In hindsight, it preserved rivalries and even enhanced new ones. The Bucs are a much better fit alongside the Falcons, Saints and Panthers than they were in the old NFC North. With the tight, four-team divisions, we know exactly what to expect, and it works beautifully.

Look no further than this week’s schedule — which, in addition to the NFC East and NFC North clashes, features Falcons at Bucs, Patriots at Bills, Chiefs at Chargers and Cardinals at Seahawks — to remind yourself what we love about division games in the NFL.

Then imagine each season starting and ending with a three-week division round robin. The NFL could actually get more perfect.