Practice · Place big digits high to maximize a decimal · Sensim Math

Variant 1 easy

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.

Your answer (check it yourself)

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 $\square.\square$ 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

\square.\square

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

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.

Your answer (check it yourself)

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 $\square.\square$ 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

\square.\square

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

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.

Your answer (check it yourself)

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 $\square.\square$ 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

\square.\square

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

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.

Your answer (check it yourself)

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 $\square.\square$ 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

\square.\square

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

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.

Your answer (check it yourself)

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 $\square.\square$ 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

\square.\square

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

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.

Your answer (check it yourself)

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 $\square.\square$ 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

\square.\square

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

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.

Your answer (check it yourself)

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 $\square.\square$ 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

\square.\square

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

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.

Your answer (check it yourself)

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 $\square.\square$ 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

\square.\square

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

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.

Your answer (check it yourself)

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 $\square.\square$ 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

\square.\square

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

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.

Your answer (check it yourself)

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 $\square.\square$ 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

\square.\square

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!

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

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