Place big digits high to maximize a decimal

4.NF.C.74.NF.C.6 · take · grade 4

Archetype: Build the Largest or Smallest Value from Digit Cards · step in a 7-type progression

Choose 2 of the given number cards and use each at most once to build a decimal of the form $\square.\square$ . Find every decimal you can build that is greater than $3$ .

$\boxed{2} \quad \boxed{3} \quad \boxed{7} \;\;\longrightarrow\;\; \square.\square$

The number cards are $2$ , $3$ , and $7$ , and each decimal you make has one digit in the ones place and one digit in the tenths place.

Show solution

Understand

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.

Givens

Digit cards available: 2, 3, 7
Each decimal uses two cards: one in the ones place, one in the tenths place
Each card is used at most once in a given decimal

Unknowns

All decimals of the form box.box greater than 3

Constraints

The decimal must be strictly greater than 3
The two chosen cards must be different (each used at most once)

Plan

#2 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.

Execute

#2 Make a Systematic List 4.NF.C.7

A decimal box.box is greater than 3 mostly because of its ones digit. If the ones digit is 2 (giving 2.x), the value is below 3. So the ones digit must be 3 or 7.

2.7 < 3 \quad \text{but} \quad 3.x \ge 3,\; 7.x > 3

Comparing decimals starts at the highest place, so the ones digit decides first.

#2 Make a Systematic List 4.NF.C.6

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 > 3,\quad 3.7 > 3

Any positive tenths added to a ones digit of 3 makes the number exceed 3.

#2 Make a Systematic List 4.NF.C.6

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 > 3,\quad 7.3 > 3

A ones digit of 7 already exceeds 3 no matter the tenths digit.

#2 Make a Systematic List 4.NF.C.7

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.

Answer: 3.2, 3.7, 7.2, 7.3

Review

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.

Eliminate possibilities (tool 3): of the 6 arrangements, cross out the two beginning with 2 (2.3, 2.7) because they are less than 3, leaving the same four.

Standards · min grade 4

4.NF.C.7 Compare two decimals to hundredths by reasoning about their size — Deciding which decimals are greater than 3 by their ones digit
4.NF.C.6 Use decimal notation for fractions with denominators 10 or 100 — Writing each two-card arrangement in decimal form

💡 Read the ones place first, list each case in order, and you'll catch every decimal that beats 3!