Nope. High income may prevent having the IRA contribution reduce your taxes, thus the Roth (or backdoor Roth).
...but that's not the case for the OP, who isn't even close to the limits.
One suggestion for the OP: Don't go below the 10% bracket by putting money in an IRA. Do some Roth instead.
Standard deduction is $12,600. 4 Personal Exemptions (2 adults, 2 kids) @ $4000 = $16,000. So, the first $38,600 is ZERO tax. Taking your income down to $36k is pointless. Put the extra $2600 in a Roth.
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As Jack pointed out, it would be very easy to get down to the 50% saver's credit level. That can result in about a $1000 credit which should wipe out all taxes. Also if they have kids, they would increase their earned income credit by using traditional. The return on traditional for low income folks can exceed 40% because of these credits.