Problem
Place big digits high to maximize a decimal
Using the digit cards 2, 3, and 7 (each at most once), build decimals of the form (ones).(tenths). List every such decimal that is greater than 3.
Fractions
Your answer
Watch out Don't list all six possible decimals - only the ones with ones digit 3 or 7 pass 3; 2.3 and 2.7 are both less than 3.
How to solve
Strategy Make a Systematic List — There are only a few two-card decimals, so listing them by the ones digit guarantees we find all that exceed 3 without missing any.
1STEP 1
The ones digit controls the size
The ones digit decides most of the size: a ones digit of 2 gives 2.x (below 3), so the ones digit must be 3 or 7.
2.7 less than 3, but 3.x and 7.x greater than 3
Comparing decimals starts at the highest place, so the ones digit decides first.
4.NF.C.7Make A Systematic ListA two-card decimal of the form (ones).(tenths) is decided by its ones digit: a ones digit of 2 keeps the value below 3, while a ones digit of 3 or 7 makes the value 3 or more.
Why?
The number equals its ones digit plus a tenths part, and that tenths part is always less than one whole, so the ones digit alone fixes which two whole numbers the value sits between.
Why?
A single tenths digit gives at most nine tenths, but it takes ten tenths bundled together to make one whole, so the tenths can never reach a full one.
🧱Place-value groupingTen ones make one ten and ten tens make one hundred — each place bundles ten of the last.
Why?
The decimal splits into a ones part and a tenths part with no gap and no overlap, so adding those two parts back gives its exact value.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
2STEP 2
Ones digit 3
With 3 in the ones place, the tenths digit can be 2 or 7 (the remaining cards): 3.2 and 3.7. Both are greater than 3.
3.2 greater than 3, 3.7 greater than 3
Any positive tenths added to a ones digit of 3 makes the number exceed 3.
4.NF.C.6Make A Systematic List3STEP 3
Ones digit 7
With 7 in the ones place, the tenths digit can be 2 or 3: 7.2 and 7.3. Both are greater than 3.
7.2 greater than 3, 7.3 greater than 3
A ones digit of 7 already exceeds 3 no matter the tenths digit.
4.NF.C.6Make A Systematic List4STEP 4
Collect all valid decimals
Listing every case with ones digit 3 or 7 gives four decimals greater than 3.
{3.2, 3.7, 7.2, 7.3}
A complete by-ones-digit list ensures none are missed and none repeat.
4.NF.C.7Make A Systematic ListAnswer
3.2, 3.7, 7.2, 7.3
Decimals with ones digit 2 (2.3, 2.7) are below 3 and correctly excluded; the four listed all start with 3 or 7, so each exceeds 3. There are 6 possible two-card decimals total and exactly 4 are greater than 3, which matches.
Takeaway
Read the ones place first, list each case in order, and you'll catch every decimal that beats 3!
- The ones digit controls the size
- Ones digit 3
- Ones digit 7
- Collect all valid decimals
Where next?
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