← Place big digits high to maximize a decimal · Build the Largest or Smallest Value from Digit Cards

Place big digits high to maximize a decimal · 10 practice problems

4.NF.C.74.NF.C.6

Generated variants — 10

Freshly produced from the archetype’s parameters — problem, figure, and solution derived together.

Variant 1 easy answer: 3.2, 3.5, 5.2, 5.3

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 $3$ .

$\boxed{2} \quad \boxed{3} \quad \boxed{5} \;\;\longrightarrow\;\; \square.\square$

The number cards are 2, 3, and 5, and each decimal you make has one digit in the ones place and one digit in the tenths place.

Show solution

1 · Understandwhat's really being asked

Using the digit cards 2, 3, 5 (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, 5
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)

2 · Planchoose the strategy

#2 Make a Systematic List

There are only 6 two-card decimals, so listing them by the ones digit guarantees we find all that exceed 3 without missing any.

3 · Execute4 carry out the plan

1The ones digit controls the size

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

A decimal box.box is greater than 3 mostly because of its ones digit. A ones digit below 3 gives a value under 3, so the ones digit must be at least 3.

\text{ones} \ge 3 \Rightarrow \text{decimal} > 3

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

2Ones digit 3

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

With 3 in the ones place, the tenths digit can be 2 or 5 (the remaining cards): 3.2, 3.5. All are greater than 3.

3.2 > 3,\quad 3.5 > 3

A ones digit of 3 keeps the number above 3 for any tenths digit.

3Ones digit 5

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

With 5 in the ones place, the tenths digit can be 2 or 3 (the remaining cards): 5.2, 5.3. All are greater than 3.

5.2 > 3,\quad 5.3 > 3

A ones digit of 5 keeps the number above 3 for any tenths digit.

4Collect all valid decimals

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

Listing every case with a qualifying ones digit gives 4 decimals greater than 3.

\{3.2,\; 3.5,\; 5.2,\; 5.3\}

A complete by-ones-digit list ensures none are missed and none repeat.

Answer: 3.2, 3.5, 5.2, 5.3

4 · Reviewdoes it hold up?

Decimals with too-small ones digit (2) are at or below 3 and correctly excluded (2.3, 2.5); the 4 listed all exceed 3. There are 6 possible two-card decimals total and exactly 4 are greater than 3, which matches.

Another way: Eliminate possibilities (tool 3): of the 6 arrangements, cross out those at or below 3 (2.3, 2.5), leaving the same 4.

Standardsmin 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

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

Variant 2 easy answer: 5.2, 5.6, 6.2, 6.5

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 $5$ .

$\boxed{2} \quad \boxed{5} \quad \boxed{6} \;\;\longrightarrow\;\; \square.\square$

The number cards are 2, 5, and 6, and each decimal you make has one digit in the ones place and one digit in the tenths place.

Show solution

1 · Understandwhat's really being asked

Using the digit cards 2, 5, 6 (each at most once), build decimals of the form (ones).(tenths). List every such decimal that is greater than 5.

Givens

Digit cards available: 2, 5, 6
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 5

Constraints

The decimal must be strictly greater than 5
The two chosen cards must be different (each used at most once)

2 · Planchoose the strategy

#2 Make a Systematic List

There are only 6 two-card decimals, so listing them by the ones digit guarantees we find all that exceed 5 without missing any.

3 · Execute4 carry out the plan

1The ones digit controls the size

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

A decimal box.box is greater than 5 mostly because of its ones digit. A ones digit below 5 gives a value under 5, so the ones digit must be at least 5.

\text{ones} \ge 5 \Rightarrow \text{decimal} > 5

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

2Ones digit 5

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

With 5 in the ones place, the tenths digit can be 2 or 6 (the remaining cards): 5.2, 5.6. All are greater than 5.

5.2 > 5,\quad 5.6 > 5

A ones digit of 5 keeps the number above 5 for any tenths digit.

3Ones digit 6

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

With 6 in the ones place, the tenths digit can be 2 or 5 (the remaining cards): 6.2, 6.5. All are greater than 5.

6.2 > 5,\quad 6.5 > 5

A ones digit of 6 keeps the number above 5 for any tenths digit.

4Collect all valid decimals

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

Listing every case with a qualifying ones digit gives 4 decimals greater than 5.

\{5.2,\; 5.6,\; 6.2,\; 6.5\}

A complete by-ones-digit list ensures none are missed and none repeat.

Answer: 5.2, 5.6, 6.2, 6.5

4 · Reviewdoes it hold up?

Decimals with too-small ones digit (2) are at or below 5 and correctly excluded (2.5, 2.6); the 4 listed all exceed 5. There are 6 possible two-card decimals total and exactly 4 are greater than 5, which matches.

Another way: Eliminate possibilities (tool 3): of the 6 arrangements, cross out those at or below 5 (2.5, 2.6), leaving the same 4.

Standardsmin grade 4

4.NF.C.7 Compare two decimals to hundredths by reasoning about their size — Deciding which decimals are greater than 5 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

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

Variant 3 easy answer: 2.1, 2.6, 6.1, 6.2

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 $2$ .

$\boxed{1} \quad \boxed{2} \quad \boxed{6} \;\;\longrightarrow\;\; \square.\square$

The number cards are 1, 2, and 6, and each decimal you make has one digit in the ones place and one digit in the tenths place.

Show solution

1 · Understandwhat's really being asked

Using the digit cards 1, 2, 6 (each at most once), build decimals of the form (ones).(tenths). List every such decimal that is greater than 2.

Givens

Digit cards available: 1, 2, 6
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 2

Constraints

The decimal must be strictly greater than 2
The two chosen cards must be different (each used at most once)

2 · Planchoose the strategy

#2 Make a Systematic List

There are only 6 two-card decimals, so listing them by the ones digit guarantees we find all that exceed 2 without missing any.

3 · Execute4 carry out the plan

1The ones digit controls the size

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

A decimal box.box is greater than 2 mostly because of its ones digit. A ones digit below 2 gives a value under 2, so the ones digit must be at least 2.

\text{ones} \ge 2 \Rightarrow \text{decimal} > 2

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

2Ones digit 2

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

With 2 in the ones place, the tenths digit can be 1 or 6 (the remaining cards): 2.1, 2.6. All are greater than 2.

2.1 > 2,\quad 2.6 > 2

A ones digit of 2 keeps the number above 2 for any tenths digit.

3Ones digit 6

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

With 6 in the ones place, the tenths digit can be 1 or 2 (the remaining cards): 6.1, 6.2. All are greater than 2.

6.1 > 2,\quad 6.2 > 2

A ones digit of 6 keeps the number above 2 for any tenths digit.

4Collect all valid decimals

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

Listing every case with a qualifying ones digit gives 4 decimals greater than 2.

\{2.1,\; 2.6,\; 6.1,\; 6.2\}

A complete by-ones-digit list ensures none are missed and none repeat.

Answer: 2.1, 2.6, 6.1, 6.2

4 · Reviewdoes it hold up?

Decimals with too-small ones digit (1) are at or below 2 and correctly excluded (1.2, 1.6); the 4 listed all exceed 2. There are 6 possible two-card decimals total and exactly 4 are greater than 2, which matches.

Another way: Eliminate possibilities (tool 3): of the 6 arrangements, cross out those at or below 2 (1.2, 1.6), leaving the same 4.

Standardsmin grade 4

4.NF.C.7 Compare two decimals to hundredths by reasoning about their size — Deciding which decimals are greater than 2 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

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

Variant 4 medium answer: 3.2, 3.7, 7.2, 7.3

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

1 · Understandwhat's really being asked

Using the digit cards 2, 3, 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)

2 · Planchoose the strategy

#2 Make a Systematic List

There are only 6 two-card decimals, so listing them by the ones digit guarantees we find all that exceed 3 without missing any.

3 · Execute4 carry out the plan

1The ones digit controls the size

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

A decimal box.box is greater than 3 mostly because of its ones digit. A ones digit below 3 gives a value under 3, so the ones digit must be at least 3.

\text{ones} \ge 3 \Rightarrow \text{decimal} > 3

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

2Ones digit 3

#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, 3.7. All are greater than 3.

3.2 > 3,\quad 3.7 > 3

A ones digit of 3 keeps the number above 3 for any tenths digit.

3Ones digit 7

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

With 7 in the ones place, the tenths digit can be 2 or 3 (the remaining cards): 7.2, 7.3. All are greater than 3.

7.2 > 3,\quad 7.3 > 3

A ones digit of 7 keeps the number above 3 for any tenths digit.

4Collect all valid decimals

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

Listing every case with a qualifying ones digit gives 4 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

4 · Reviewdoes it hold up?

Decimals with too-small ones digit (2) are at or below 3 and correctly excluded (2.3, 2.7); the 4 listed all exceed 3. There are 6 possible two-card decimals total and exactly 4 are greater than 3, which matches.

Another way: Eliminate possibilities (tool 3): of the 6 arrangements, cross out those at or below 3 (2.3, 2.7), leaving the same 4.

Standardsmin 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

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

Variant 5 medium answer: 6.1, 6.7, 7.1, 7.6

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 $6$ .

$\boxed{1} \quad \boxed{6} \quad \boxed{7} \;\;\longrightarrow\;\; \square.\square$

The number cards are 1, 6, and 7, and each decimal you make has one digit in the ones place and one digit in the tenths place.

Show solution

1 · Understandwhat's really being asked

Using the digit cards 1, 6, 7 (each at most once), build decimals of the form (ones).(tenths). List every such decimal that is greater than 6.

Givens

Digit cards available: 1, 6, 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 6

Constraints

The decimal must be strictly greater than 6
The two chosen cards must be different (each used at most once)

2 · Planchoose the strategy

#2 Make a Systematic List

There are only 6 two-card decimals, so listing them by the ones digit guarantees we find all that exceed 6 without missing any.

3 · Execute4 carry out the plan

1The ones digit controls the size

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

A decimal box.box is greater than 6 mostly because of its ones digit. A ones digit below 6 gives a value under 6, so the ones digit must be at least 6.

\text{ones} \ge 6 \Rightarrow \text{decimal} > 6

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

2Ones digit 6

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

With 6 in the ones place, the tenths digit can be 1 or 7 (the remaining cards): 6.1, 6.7. All are greater than 6.

6.1 > 6,\quad 6.7 > 6

A ones digit of 6 keeps the number above 6 for any tenths digit.

3Ones digit 7

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

With 7 in the ones place, the tenths digit can be 1 or 6 (the remaining cards): 7.1, 7.6. All are greater than 6.

7.1 > 6,\quad 7.6 > 6

A ones digit of 7 keeps the number above 6 for any tenths digit.

4Collect all valid decimals

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

Listing every case with a qualifying ones digit gives 4 decimals greater than 6.

\{6.1,\; 6.7,\; 7.1,\; 7.6\}

A complete by-ones-digit list ensures none are missed and none repeat.

Answer: 6.1, 6.7, 7.1, 7.6

4 · Reviewdoes it hold up?

Decimals with too-small ones digit (1) are at or below 6 and correctly excluded (1.6, 1.7); the 4 listed all exceed 6. There are 6 possible two-card decimals total and exactly 4 are greater than 6, which matches.

Another way: Eliminate possibilities (tool 3): of the 6 arrangements, cross out those at or below 6 (1.6, 1.7), leaving the same 4.

Standardsmin grade 4

4.NF.C.7 Compare two decimals to hundredths by reasoning about their size — Deciding which decimals are greater than 6 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

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

Variant 6 medium answer: 4.1, 4.8, 8.1, 8.4

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 $4$ .

$\boxed{1} \quad \boxed{4} \quad \boxed{8} \;\;\longrightarrow\;\; \square.\square$

The number cards are 1, 4, and 8, and each decimal you make has one digit in the ones place and one digit in the tenths place.

Show solution

1 · Understandwhat's really being asked

Using the digit cards 1, 4, 8 (each at most once), build decimals of the form (ones).(tenths). List every such decimal that is greater than 4.

Givens

Digit cards available: 1, 4, 8
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 4

Constraints

The decimal must be strictly greater than 4
The two chosen cards must be different (each used at most once)

2 · Planchoose the strategy

#2 Make a Systematic List

There are only 6 two-card decimals, so listing them by the ones digit guarantees we find all that exceed 4 without missing any.

3 · Execute4 carry out the plan

1The ones digit controls the size

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

A decimal box.box is greater than 4 mostly because of its ones digit. A ones digit below 4 gives a value under 4, so the ones digit must be at least 4.

\text{ones} \ge 4 \Rightarrow \text{decimal} > 4

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

2Ones digit 4

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

With 4 in the ones place, the tenths digit can be 1 or 8 (the remaining cards): 4.1, 4.8. All are greater than 4.

4.1 > 4,\quad 4.8 > 4

A ones digit of 4 keeps the number above 4 for any tenths digit.

3Ones digit 8

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

With 8 in the ones place, the tenths digit can be 1 or 4 (the remaining cards): 8.1, 8.4. All are greater than 4.

8.1 > 4,\quad 8.4 > 4

A ones digit of 8 keeps the number above 4 for any tenths digit.

4Collect all valid decimals

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

Listing every case with a qualifying ones digit gives 4 decimals greater than 4.

\{4.1,\; 4.8,\; 8.1,\; 8.4\}

A complete by-ones-digit list ensures none are missed and none repeat.

Answer: 4.1, 4.8, 8.1, 8.4

4 · Reviewdoes it hold up?

Decimals with too-small ones digit (1) are at or below 4 and correctly excluded (1.4, 1.8); the 4 listed all exceed 4. There are 6 possible two-card decimals total and exactly 4 are greater than 4, which matches.

Another way: Eliminate possibilities (tool 3): of the 6 arrangements, cross out those at or below 4 (1.4, 1.8), leaving the same 4.

Standardsmin grade 4

4.NF.C.7 Compare two decimals to hundredths by reasoning about their size — Deciding which decimals are greater than 4 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

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

Variant 7 medium answer: 4.2, 4.8, 8.2, 8.4

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 $4$ .

$\boxed{2} \quad \boxed{4} \quad \boxed{8} \;\;\longrightarrow\;\; \square.\square$

The number cards are 2, 4, and 8, and each decimal you make has one digit in the ones place and one digit in the tenths place.

Show solution

1 · Understandwhat's really being asked

Using the digit cards 2, 4, 8 (each at most once), build decimals of the form (ones).(tenths). List every such decimal that is greater than 4.

Givens

Digit cards available: 2, 4, 8
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 4

Constraints

The decimal must be strictly greater than 4
The two chosen cards must be different (each used at most once)

2 · Planchoose the strategy

#2 Make a Systematic List

There are only 6 two-card decimals, so listing them by the ones digit guarantees we find all that exceed 4 without missing any.

3 · Execute4 carry out the plan

1The ones digit controls the size

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

A decimal box.box is greater than 4 mostly because of its ones digit. A ones digit below 4 gives a value under 4, so the ones digit must be at least 4.

\text{ones} \ge 4 \Rightarrow \text{decimal} > 4

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

2Ones digit 4

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

With 4 in the ones place, the tenths digit can be 2 or 8 (the remaining cards): 4.2, 4.8. All are greater than 4.

4.2 > 4,\quad 4.8 > 4

A ones digit of 4 keeps the number above 4 for any tenths digit.

3Ones digit 8

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

With 8 in the ones place, the tenths digit can be 2 or 4 (the remaining cards): 8.2, 8.4. All are greater than 4.

8.2 > 4,\quad 8.4 > 4

A ones digit of 8 keeps the number above 4 for any tenths digit.

4Collect all valid decimals

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

Listing every case with a qualifying ones digit gives 4 decimals greater than 4.

\{4.2,\; 4.8,\; 8.2,\; 8.4\}

A complete by-ones-digit list ensures none are missed and none repeat.

Answer: 4.2, 4.8, 8.2, 8.4

4 · Reviewdoes it hold up?

Decimals with too-small ones digit (2) are at or below 4 and correctly excluded (2.4, 2.8); the 4 listed all exceed 4. There are 6 possible two-card decimals total and exactly 4 are greater than 4, which matches.

Another way: Eliminate possibilities (tool 3): of the 6 arrangements, cross out those at or below 4 (2.4, 2.8), leaving the same 4.

Standardsmin grade 4

4.NF.C.7 Compare two decimals to hundredths by reasoning about their size — Deciding which decimals are greater than 4 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

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

Variant 8 hard answer: 7.4, 7.8, 8.4, 8.7

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 $7$ .

$\boxed{4} \quad \boxed{7} \quad \boxed{8} \;\;\longrightarrow\;\; \square.\square$

The number cards are 4, 7, and 8, and each decimal you make has one digit in the ones place and one digit in the tenths place.

Show solution

1 · Understandwhat's really being asked

Using the digit cards 4, 7, 8 (each at most once), build decimals of the form (ones).(tenths). List every such decimal that is greater than 7.

Givens

Digit cards available: 4, 7, 8
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 7

Constraints

The decimal must be strictly greater than 7
The two chosen cards must be different (each used at most once)

2 · Planchoose the strategy

#2 Make a Systematic List

There are only 6 two-card decimals, so listing them by the ones digit guarantees we find all that exceed 7 without missing any.

3 · Execute4 carry out the plan

1The ones digit controls the size

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

A decimal box.box is greater than 7 mostly because of its ones digit. A ones digit below 7 gives a value under 7, so the ones digit must be at least 7.

\text{ones} \ge 7 \Rightarrow \text{decimal} > 7

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

2Ones digit 7

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

With 7 in the ones place, the tenths digit can be 4 or 8 (the remaining cards): 7.4, 7.8. All are greater than 7.

7.4 > 7,\quad 7.8 > 7

A ones digit of 7 keeps the number above 7 for any tenths digit.

3Ones digit 8

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

With 8 in the ones place, the tenths digit can be 4 or 7 (the remaining cards): 8.4, 8.7. All are greater than 7.

8.4 > 7,\quad 8.7 > 7

A ones digit of 8 keeps the number above 7 for any tenths digit.

4Collect all valid decimals

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

Listing every case with a qualifying ones digit gives 4 decimals greater than 7.

\{7.4,\; 7.8,\; 8.4,\; 8.7\}

A complete by-ones-digit list ensures none are missed and none repeat.

Answer: 7.4, 7.8, 8.4, 8.7

4 · Reviewdoes it hold up?

Decimals with too-small ones digit (4) are at or below 7 and correctly excluded (4.7, 4.8); the 4 listed all exceed 7. There are 6 possible two-card decimals total and exactly 4 are greater than 7, which matches.

Another way: Eliminate possibilities (tool 3): of the 6 arrangements, cross out those at or below 7 (4.7, 4.8), leaving the same 4.

Standardsmin grade 4

4.NF.C.7 Compare two decimals to hundredths by reasoning about their size — Deciding which decimals are greater than 7 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

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

Variant 9 hard answer: 5.3, 5.9, 9.3, 9.5

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 $5$ .

$\boxed{3} \quad \boxed{5} \quad \boxed{9} \;\;\longrightarrow\;\; \square.\square$

The number cards are 3, 5, and 9, and each decimal you make has one digit in the ones place and one digit in the tenths place.

Show solution

1 · Understandwhat's really being asked

Using the digit cards 3, 5, 9 (each at most once), build decimals of the form (ones).(tenths). List every such decimal that is greater than 5.

Givens

Digit cards available: 3, 5, 9
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 5

Constraints

The decimal must be strictly greater than 5
The two chosen cards must be different (each used at most once)

2 · Planchoose the strategy

#2 Make a Systematic List

There are only 6 two-card decimals, so listing them by the ones digit guarantees we find all that exceed 5 without missing any.

3 · Execute4 carry out the plan

1The ones digit controls the size

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

A decimal box.box is greater than 5 mostly because of its ones digit. A ones digit below 5 gives a value under 5, so the ones digit must be at least 5.

\text{ones} \ge 5 \Rightarrow \text{decimal} > 5

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

2Ones digit 5

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

With 5 in the ones place, the tenths digit can be 3 or 9 (the remaining cards): 5.3, 5.9. All are greater than 5.

5.3 > 5,\quad 5.9 > 5

A ones digit of 5 keeps the number above 5 for any tenths digit.

3Ones digit 9

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

With 9 in the ones place, the tenths digit can be 3 or 5 (the remaining cards): 9.3, 9.5. All are greater than 5.

9.3 > 5,\quad 9.5 > 5

A ones digit of 9 keeps the number above 5 for any tenths digit.

4Collect all valid decimals

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

Listing every case with a qualifying ones digit gives 4 decimals greater than 5.

\{5.3,\; 5.9,\; 9.3,\; 9.5\}

A complete by-ones-digit list ensures none are missed and none repeat.

Answer: 5.3, 5.9, 9.3, 9.5

4 · Reviewdoes it hold up?

Decimals with too-small ones digit (3) are at or below 5 and correctly excluded (3.5, 3.9); the 4 listed all exceed 5. There are 6 possible two-card decimals total and exactly 4 are greater than 5, which matches.

Another way: Eliminate possibilities (tool 3): of the 6 arrangements, cross out those at or below 5 (3.5, 3.9), leaving the same 4.

Standardsmin grade 4

4.NF.C.7 Compare two decimals to hundredths by reasoning about their size — Deciding which decimals are greater than 5 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

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

Variant 10 hard answer: 4.3, 4.9, 9.3, 9.4

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 $4$ .

$\boxed{3} \quad \boxed{4} \quad \boxed{9} \;\;\longrightarrow\;\; \square.\square$

The number cards are 3, 4, and 9, and each decimal you make has one digit in the ones place and one digit in the tenths place.

Show solution

1 · Understandwhat's really being asked

Using the digit cards 3, 4, 9 (each at most once), build decimals of the form (ones).(tenths). List every such decimal that is greater than 4.

Givens

Digit cards available: 3, 4, 9
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 4

Constraints

The decimal must be strictly greater than 4
The two chosen cards must be different (each used at most once)

2 · Planchoose the strategy

#2 Make a Systematic List

There are only 6 two-card decimals, so listing them by the ones digit guarantees we find all that exceed 4 without missing any.

3 · Execute4 carry out the plan

1The ones digit controls the size

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

A decimal box.box is greater than 4 mostly because of its ones digit. A ones digit below 4 gives a value under 4, so the ones digit must be at least 4.

\text{ones} \ge 4 \Rightarrow \text{decimal} > 4

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

2Ones digit 4

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

With 4 in the ones place, the tenths digit can be 3 or 9 (the remaining cards): 4.3, 4.9. All are greater than 4.

4.3 > 4,\quad 4.9 > 4

A ones digit of 4 keeps the number above 4 for any tenths digit.

3Ones digit 9

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

With 9 in the ones place, the tenths digit can be 3 or 4 (the remaining cards): 9.3, 9.4. All are greater than 4.

9.3 > 4,\quad 9.4 > 4

A ones digit of 9 keeps the number above 4 for any tenths digit.

4Collect all valid decimals

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

Listing every case with a qualifying ones digit gives 4 decimals greater than 4.

\{4.3,\; 4.9,\; 9.3,\; 9.4\}

A complete by-ones-digit list ensures none are missed and none repeat.

Answer: 4.3, 4.9, 9.3, 9.4

4 · Reviewdoes it hold up?

Decimals with too-small ones digit (3) are at or below 4 and correctly excluded (3.4, 3.9); the 4 listed all exceed 4. There are 6 possible two-card decimals total and exactly 4 are greater than 4, which matches.

Another way: Eliminate possibilities (tool 3): of the 6 arrangements, cross out those at or below 4 (3.4, 3.9), leaving the same 4.

Standardsmin grade 4

4.NF.C.7 Compare two decimals to hundredths by reasoning about their size — Deciding which decimals are greater than 4 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

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