Practice · Build largest and smallest mixed numbers from cards · Sensim Math

Variant 1 easy

From $4$ number cards, choose $2$ of them and use each one exactly once to make a mixed number whose denominator is $7$ . Find the sum of the largest mixed number and the smallest mixed number you can make this way.

Number cards: $5$ , $8$ , $4$ , $6$

Your answer

Show solution

1 · Understandwhat's really being asked

From the four cards 5, 8, 4, 6 I pick two cards. One becomes the whole part and the other becomes the numerator over a denominator of 7, making a mixed number like W N/7. I want the biggest and smallest such mixed numbers, then their sum.

Givens

Number cards available: 5, 8, 4, 6.
Use exactly two cards, each once, to form a mixed number.
The denominator is fixed at 7.
A mixed number has the form (whole) (numerator/7).

Unknowns

The largest mixed number that can be made.
The smallest mixed number that can be made.
The sum of the largest and smallest.

Constraints

Numerator must be less than 7 for a proper fraction part.
Each card is used at most once within one mixed number.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #2 Make a Systematic List

To get the biggest number, make the whole part as big as possible, then the numerator as big as possible; to get the smallest, make the whole part as small as possible. Test the card choices and check the constraint that a numerator must be under 7.

3 · Execute3 carry out the plan

1Build the largest mixed number

#6 Guess and Check 4.NF.A.2

The whole part matters most, so use the biggest card, 8, as the whole number. For the fraction part the numerator should be as large as possible but under 7; the largest valid numerator is 6. So the largest mixed number is 8

\frac{6}{7}

.

8\tfrac{6}{7}

A bigger whole part beats any fraction part, so maximize the whole number first.

2Build the smallest mixed number

#2 Make a Systematic List 4.NF.A.2

Now make the whole part as small as possible: the smallest card is 4. For the smallest value the numerator should be as small as possible but still under 7; the smallest valid numerator is 5. The smallest mixed number is 4

\frac{5}{7}

.

4\tfrac{5}{7}

A smaller whole part makes a smaller number; the numerator must still be a legal fraction under 7.

3Add the largest and smallest

#6 Guess and Check 4.NF.B.3

Add 8

\frac{6}{7}

+ 4

\frac{5}{7}

. Whole parts: 8 + 4 = 12. Fraction parts:

\frac{6}{7}

+

\frac{5}{7}

=

\frac{11}{7}

. Combine to get 13

\frac{4}{7}

= 13

\frac{4}{7}

.

8\tfrac{6}{7}+4\tfrac{5}{7}=13\tfrac{4}{7}=13\tfrac{4}{7}

Add wholes and like-denominator fractions separately, then regroup any improper fraction.

Answer:

13\frac{4}{7}

4 · Reviewdoes it hold up?

Largest is about 8.86 and smallest about 4.71, summing to about 13.57, which matches 13 $\frac{4}{7}$ . The rule that a numerator must be under 7 was respected in both extremes.

Another way: Systematically list valid mixed numbers (tool 2): whole from the cards, numerator from the remaining cards under 7. The max is 8 $\frac{6}{7}$ and the min is 4 $\frac{5}{7}$ , giving the same sum 13 $\frac{4}{7}$ .

Standardsmin grade 4

4.NF.A.2 Compare two fractions with different numerators and different denominators — Reasoning about which card choices make the mixed number largest or smallest.
4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Adding the two mixed numbers and regrouping an improper fraction into a whole plus a proper fraction.

💡Takeaway. This only needs Grade 4 comparing and adding fractions — pick the biggest whole part first, but keep the numerator under 7!

Variant 2 medium

From $4$ number cards, choose $2$ of them and use each one exactly once to make a mixed number whose denominator is $7$ . Find the sum of the largest mixed number and the smallest mixed number you can make this way.

Number cards: $2$ , $8$ , $6$ , $4$

Your answer

Show solution

1 · Understandwhat's really being asked

From the four cards 2, 8, 6, 4 I pick two cards. One becomes the whole part and the other becomes the numerator over a denominator of 7, making a mixed number like W N/7. I want the biggest and smallest such mixed numbers, then their sum.

Givens

Number cards available: 2, 8, 6, 4.
Use exactly two cards, each once, to form a mixed number.
The denominator is fixed at 7.
A mixed number has the form (whole) (numerator/7).

Unknowns

The largest mixed number that can be made.
The smallest mixed number that can be made.
The sum of the largest and smallest.

Constraints

Numerator must be less than 7 for a proper fraction part.
Each card is used at most once within one mixed number.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #2 Make a Systematic List

To get the biggest number, make the whole part as big as possible, then the numerator as big as possible; to get the smallest, make the whole part as small as possible. Test the card choices and check the constraint that a numerator must be under 7.

3 · Execute3 carry out the plan

1Build the largest mixed number

#6 Guess and Check 4.NF.A.2

The whole part matters most, so use the biggest card, 8, as the whole number. For the fraction part the numerator should be as large as possible but under 7; the largest valid numerator is 6. So the largest mixed number is 8

\frac{6}{7}

.

8\tfrac{6}{7}

A bigger whole part beats any fraction part, so maximize the whole number first.

2Build the smallest mixed number

#2 Make a Systematic List 4.NF.A.2

Now make the whole part as small as possible: the smallest card is 2. For the smallest value the numerator should be as small as possible but still under 7; the smallest valid numerator is 4. The smallest mixed number is 2

\frac{4}{7}

.

2\tfrac{4}{7}

A smaller whole part makes a smaller number; the numerator must still be a legal fraction under 7.

3Add the largest and smallest

#6 Guess and Check 4.NF.B.3

Add 8

\frac{6}{7}

+ 2

\frac{4}{7}

. Whole parts: 8 + 2 = 10. Fraction parts:

\frac{6}{7}

+

\frac{4}{7}

=

\frac{10}{7}

. Combine to get 11

\frac{3}{7}

= 11

\frac{3}{7}

.

8\tfrac{6}{7}+2\tfrac{4}{7}=11\tfrac{3}{7}=11\tfrac{3}{7}

Add wholes and like-denominator fractions separately, then regroup any improper fraction.

Answer:

11\frac{3}{7}

4 · Reviewdoes it hold up?

Largest is about 8.86 and smallest about 2.57, summing to about 11.43, which matches 11 $\frac{3}{7}$ . The rule that a numerator must be under 7 was respected in both extremes.

Another way: Systematically list valid mixed numbers (tool 2): whole from the cards, numerator from the remaining cards under 7. The max is 8 $\frac{6}{7}$ and the min is 2 $\frac{4}{7}$ , giving the same sum 11 $\frac{3}{7}$ .

Standardsmin grade 4

4.NF.A.2 Compare two fractions with different numerators and different denominators — Reasoning about which card choices make the mixed number largest or smallest.
4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Adding the two mixed numbers and regrouping an improper fraction into a whole plus a proper fraction.

💡Takeaway. This only needs Grade 4 comparing and adding fractions — pick the biggest whole part first, but keep the numerator under 7!

Variant 3 medium

From $4$ number cards, choose $2$ of them and use each one exactly once to make a mixed number whose denominator is $5$ . Find the sum of the largest mixed number and the smallest mixed number you can make this way.

Number cards: $4$ , $7$ , $2$ , $8$

Your answer

Show solution

1 · Understandwhat's really being asked

From the four cards 4, 7, 2, 8 I pick two cards. One becomes the whole part and the other becomes the numerator over a denominator of 5, making a mixed number like W N/5. I want the biggest and smallest such mixed numbers, then their sum.

Givens

Number cards available: 4, 7, 2, 8.
Use exactly two cards, each once, to form a mixed number.
The denominator is fixed at 5.
A mixed number has the form (whole) (numerator/5).

Unknowns

The largest mixed number that can be made.
The smallest mixed number that can be made.
The sum of the largest and smallest.

Constraints

Numerator must be less than 5 for a proper fraction part.
Each card is used at most once within one mixed number.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #2 Make a Systematic List

To get the biggest number, make the whole part as big as possible, then the numerator as big as possible; to get the smallest, make the whole part as small as possible. Test the card choices and check the constraint that a numerator must be under 5.

3 · Execute3 carry out the plan

1Build the largest mixed number

#6 Guess and Check 4.NF.A.2

The whole part matters most, so use the biggest card, 8, as the whole number. For the fraction part the numerator should be as large as possible but under 5; the largest valid numerator is 4. So the largest mixed number is 8

\frac{4}{5}

.

8\tfrac{4}{5}

A bigger whole part beats any fraction part, so maximize the whole number first.

2Build the smallest mixed number

#2 Make a Systematic List 4.NF.A.2

Now make the whole part as small as possible: the smallest card is 2. For the smallest value the numerator should be as small as possible but still under 5; the smallest valid numerator is 4. The smallest mixed number is 2

\frac{4}{5}

.

2\tfrac{4}{5}

A smaller whole part makes a smaller number; the numerator must still be a legal fraction under 5.

3Add the largest and smallest

#6 Guess and Check 4.NF.B.3

Add 8

\frac{4}{5}

+ 2

\frac{4}{5}

. Whole parts: 8 + 2 = 10. Fraction parts:

\frac{4}{5}

+

\frac{4}{5}

=

\frac{8}{5}

. Combine to get 11

\frac{3}{5}

= 11

\frac{3}{5}

.

8\tfrac{4}{5}+2\tfrac{4}{5}=11\tfrac{3}{5}=11\tfrac{3}{5}

Add wholes and like-denominator fractions separately, then regroup any improper fraction.

Answer:

11\frac{3}{5}

4 · Reviewdoes it hold up?

Largest is about 8.80 and smallest about 2.80, summing to about 11.60, which matches 11 $\frac{3}{5}$ . The rule that a numerator must be under 5 was respected in both extremes.

Another way: Systematically list valid mixed numbers (tool 2): whole from the cards, numerator from the remaining cards under 5. The max is 8 $\frac{4}{5}$ and the min is 2 $\frac{4}{5}$ , giving the same sum 11 $\frac{3}{5}$ .

Standardsmin grade 4

4.NF.A.2 Compare two fractions with different numerators and different denominators — Reasoning about which card choices make the mixed number largest or smallest.
4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Adding the two mixed numbers and regrouping an improper fraction into a whole plus a proper fraction.

💡Takeaway. This only needs Grade 4 comparing and adding fractions — pick the biggest whole part first, but keep the numerator under 5!

Variant 4 easy

From $4$ number cards, choose $2$ of them and use each one exactly once to make a mixed number whose denominator is $7$ . Find the sum of the largest mixed number and the smallest mixed number you can make this way.

Number cards: $6$ , $1$ , $8$ , $4$

Your answer

Show solution

1 · Understandwhat's really being asked

From the four cards 6, 1, 8, 4 I pick two cards. One becomes the whole part and the other becomes the numerator over a denominator of 7, making a mixed number like W N/7. I want the biggest and smallest such mixed numbers, then their sum.

Givens

Number cards available: 6, 1, 8, 4.
Use exactly two cards, each once, to form a mixed number.
The denominator is fixed at 7.
A mixed number has the form (whole) (numerator/7).

Unknowns

The largest mixed number that can be made.
The smallest mixed number that can be made.
The sum of the largest and smallest.

Constraints

Numerator must be less than 7 for a proper fraction part.
Each card is used at most once within one mixed number.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #2 Make a Systematic List

To get the biggest number, make the whole part as big as possible, then the numerator as big as possible; to get the smallest, make the whole part as small as possible. Test the card choices and check the constraint that a numerator must be under 7.

3 · Execute3 carry out the plan

1Build the largest mixed number

#6 Guess and Check 4.NF.A.2

The whole part matters most, so use the biggest card, 8, as the whole number. For the fraction part the numerator should be as large as possible but under 7; the largest valid numerator is 6. So the largest mixed number is 8

\frac{6}{7}

.

8\tfrac{6}{7}

A bigger whole part beats any fraction part, so maximize the whole number first.

2Build the smallest mixed number

#2 Make a Systematic List 4.NF.A.2

Now make the whole part as small as possible: the smallest card is 1. For the smallest value the numerator should be as small as possible but still under 7; the smallest valid numerator is 4. The smallest mixed number is 1

\frac{4}{7}

.

1\tfrac{4}{7}

A smaller whole part makes a smaller number; the numerator must still be a legal fraction under 7.

3Add the largest and smallest

#6 Guess and Check 4.NF.B.3

Add 8

\frac{6}{7}

+ 1

\frac{4}{7}

. Whole parts: 8 + 1 = 9. Fraction parts:

\frac{6}{7}

+

\frac{4}{7}

=

\frac{10}{7}

. Combine to get 10

\frac{3}{7}

= 10

\frac{3}{7}

.

8\tfrac{6}{7}+1\tfrac{4}{7}=10\tfrac{3}{7}=10\tfrac{3}{7}

Add wholes and like-denominator fractions separately, then regroup any improper fraction.

Answer:

10\frac{3}{7}

4 · Reviewdoes it hold up?

Largest is about 8.86 and smallest about 1.57, summing to about 10.43, which matches 10 $\frac{3}{7}$ . The rule that a numerator must be under 7 was respected in both extremes.

Another way: Systematically list valid mixed numbers (tool 2): whole from the cards, numerator from the remaining cards under 7. The max is 8 $\frac{6}{7}$ and the min is 1 $\frac{4}{7}$ , giving the same sum 10 $\frac{3}{7}$ .

Standardsmin grade 4

4.NF.A.2 Compare two fractions with different numerators and different denominators — Reasoning about which card choices make the mixed number largest or smallest.
4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Adding the two mixed numbers and regrouping an improper fraction into a whole plus a proper fraction.

💡Takeaway. This only needs Grade 4 comparing and adding fractions — pick the biggest whole part first, but keep the numerator under 7!

Variant 5 easy

From $4$ number cards, choose $2$ of them and use each one exactly once to make a mixed number whose denominator is $5$ . Find the sum of the largest mixed number and the smallest mixed number you can make this way.

Number cards: $1$ , $6$ , $4$ , $8$

Your answer

Show solution

1 · Understandwhat's really being asked

From the four cards 1, 6, 4, 8 I pick two cards. One becomes the whole part and the other becomes the numerator over a denominator of 5, making a mixed number like W N/5. I want the biggest and smallest such mixed numbers, then their sum.

Givens

Number cards available: 1, 6, 4, 8.
Use exactly two cards, each once, to form a mixed number.
The denominator is fixed at 5.
A mixed number has the form (whole) (numerator/5).

Unknowns

The largest mixed number that can be made.
The smallest mixed number that can be made.
The sum of the largest and smallest.

Constraints

Numerator must be less than 5 for a proper fraction part.
Each card is used at most once within one mixed number.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #2 Make a Systematic List

To get the biggest number, make the whole part as big as possible, then the numerator as big as possible; to get the smallest, make the whole part as small as possible. Test the card choices and check the constraint that a numerator must be under 5.

3 · Execute3 carry out the plan

1Build the largest mixed number

#6 Guess and Check 4.NF.A.2

The whole part matters most, so use the biggest card, 8, as the whole number. For the fraction part the numerator should be as large as possible but under 5; the largest valid numerator is 4. So the largest mixed number is 8

\frac{4}{5}

.

8\tfrac{4}{5}

A bigger whole part beats any fraction part, so maximize the whole number first.

2Build the smallest mixed number

#2 Make a Systematic List 4.NF.A.2

Now make the whole part as small as possible: the smallest card is 1. For the smallest value the numerator should be as small as possible but still under 5; the smallest valid numerator is 4. The smallest mixed number is 1

\frac{4}{5}

.

1\tfrac{4}{5}

A smaller whole part makes a smaller number; the numerator must still be a legal fraction under 5.

3Add the largest and smallest

#6 Guess and Check 4.NF.B.3

Add 8

\frac{4}{5}

+ 1

\frac{4}{5}

. Whole parts: 8 + 1 = 9. Fraction parts:

\frac{4}{5}

+

\frac{4}{5}

=

\frac{8}{5}

. Combine to get 10

\frac{3}{5}

= 10

\frac{3}{5}

.

8\tfrac{4}{5}+1\tfrac{4}{5}=10\tfrac{3}{5}=10\tfrac{3}{5}

Add wholes and like-denominator fractions separately, then regroup any improper fraction.

Answer:

10\frac{3}{5}

4 · Reviewdoes it hold up?

Largest is about 8.80 and smallest about 1.80, summing to about 10.60, which matches 10 $\frac{3}{5}$ . The rule that a numerator must be under 5 was respected in both extremes.

Another way: Systematically list valid mixed numbers (tool 2): whole from the cards, numerator from the remaining cards under 5. The max is 8 $\frac{4}{5}$ and the min is 1 $\frac{4}{5}$ , giving the same sum 10 $\frac{3}{5}$ .

Standardsmin grade 4

4.NF.A.2 Compare two fractions with different numerators and different denominators — Reasoning about which card choices make the mixed number largest or smallest.
4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Adding the two mixed numbers and regrouping an improper fraction into a whole plus a proper fraction.

💡Takeaway. This only needs Grade 4 comparing and adding fractions — pick the biggest whole part first, but keep the numerator under 5!

Variant 6 hard

From $4$ number cards, choose $2$ of them and use each one exactly once to make a mixed number whose denominator is $8$ . Find the sum of the largest mixed number and the smallest mixed number you can make this way.

Number cards: $9$ , $3$ , $5$ , $7$

Your answer

Show solution

1 · Understandwhat's really being asked

From the four cards 9, 3, 5, 7 I pick two cards. One becomes the whole part and the other becomes the numerator over a denominator of 8, making a mixed number like W N/8. I want the biggest and smallest such mixed numbers, then their sum.

Givens

Number cards available: 9, 3, 5, 7.
Use exactly two cards, each once, to form a mixed number.
The denominator is fixed at 8.
A mixed number has the form (whole) (numerator/8).

Unknowns

The largest mixed number that can be made.
The smallest mixed number that can be made.
The sum of the largest and smallest.

Constraints

Numerator must be less than 8 for a proper fraction part.
Each card is used at most once within one mixed number.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #2 Make a Systematic List

To get the biggest number, make the whole part as big as possible, then the numerator as big as possible; to get the smallest, make the whole part as small as possible. Test the card choices and check the constraint that a numerator must be under 8.

3 · Execute3 carry out the plan

1Build the largest mixed number

#6 Guess and Check 4.NF.A.2

The whole part matters most, so use the biggest card, 9, as the whole number. For the fraction part the numerator should be as large as possible but under 8; the largest valid numerator is 7. So the largest mixed number is 9

\frac{7}{8}

.

9\tfrac{7}{8}

A bigger whole part beats any fraction part, so maximize the whole number first.

2Build the smallest mixed number

#2 Make a Systematic List 4.NF.A.2

Now make the whole part as small as possible: the smallest card is 3. For the smallest value the numerator should be as small as possible but still under 8; the smallest valid numerator is 5. The smallest mixed number is 3

\frac{5}{8}

.

3\tfrac{5}{8}

A smaller whole part makes a smaller number; the numerator must still be a legal fraction under 8.

3Add the largest and smallest

#6 Guess and Check 4.NF.B.3

Add 9

\frac{7}{8}

+ 3

\frac{5}{8}

. Whole parts: 9 + 3 = 12. Fraction parts:

\frac{7}{8}

+

\frac{5}{8}

=

\frac{12}{8}

. Combine to get 13

\frac{4}{8}

= 13

\frac{1}{2}

.

9\tfrac{7}{8}+3\tfrac{5}{8}=13\tfrac{4}{8}=13\tfrac{1}{2}

Add wholes and like-denominator fractions separately, then regroup any improper fraction.

Answer:

13\frac{1}{2}

4 · Reviewdoes it hold up?

Largest is about 9.88 and smallest about 3.62, summing to about 13.50, which matches 13 $\frac{1}{2}$ . The rule that a numerator must be under 8 was respected in both extremes.

Another way: Systematically list valid mixed numbers (tool 2): whole from the cards, numerator from the remaining cards under 8. The max is 9 $\frac{7}{8}$ and the min is 3 $\frac{5}{8}$ , giving the same sum 13 $\frac{1}{2}$ .

Standardsmin grade 4

4.NF.A.2 Compare two fractions with different numerators and different denominators — Reasoning about which card choices make the mixed number largest or smallest.
4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Adding the two mixed numbers and regrouping an improper fraction into a whole plus a proper fraction.

💡Takeaway. This only needs Grade 4 comparing and adding fractions — pick the biggest whole part first, but keep the numerator under 8!

Variant 7 medium

From $4$ number cards, choose $2$ of them and use each one exactly once to make a mixed number whose denominator is $6$ . Find the sum of the largest mixed number and the smallest mixed number you can make this way.

Number cards: $2$ , $5$ , $9$ , $3$

Your answer

Show solution

1 · Understandwhat's really being asked

From the four cards 2, 5, 9, 3 I pick two cards. One becomes the whole part and the other becomes the numerator over a denominator of 6, making a mixed number like W N/6. I want the biggest and smallest such mixed numbers, then their sum.

Givens

Number cards available: 2, 5, 9, 3.
Use exactly two cards, each once, to form a mixed number.
The denominator is fixed at 6.
A mixed number has the form (whole) (numerator/6).

Unknowns

The largest mixed number that can be made.
The smallest mixed number that can be made.
The sum of the largest and smallest.

Constraints

Numerator must be less than 6 for a proper fraction part.
Each card is used at most once within one mixed number.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #2 Make a Systematic List

To get the biggest number, make the whole part as big as possible, then the numerator as big as possible; to get the smallest, make the whole part as small as possible. Test the card choices and check the constraint that a numerator must be under 6.

3 · Execute3 carry out the plan

1Build the largest mixed number

#6 Guess and Check 4.NF.A.2

The whole part matters most, so use the biggest card, 9, as the whole number. For the fraction part the numerator should be as large as possible but under 6; the largest valid numerator is 5. So the largest mixed number is 9

\frac{5}{6}

.

9\tfrac{5}{6}

A bigger whole part beats any fraction part, so maximize the whole number first.

2Build the smallest mixed number

#2 Make a Systematic List 4.NF.A.2

Now make the whole part as small as possible: the smallest card is 2. For the smallest value the numerator should be as small as possible but still under 6; the smallest valid numerator is 3. The smallest mixed number is 2

\frac{3}{6}

.

2\tfrac{3}{6}

A smaller whole part makes a smaller number; the numerator must still be a legal fraction under 6.

3Add the largest and smallest

#6 Guess and Check 4.NF.B.3

Add 9

\frac{5}{6}

+ 2

\frac{3}{6}

. Whole parts: 9 + 2 = 11. Fraction parts:

\frac{5}{6}

+

\frac{3}{6}

=

\frac{8}{6}

. Combine to get 12

\frac{2}{6}

= 12

\frac{1}{3}

.

9\tfrac{5}{6}+2\tfrac{3}{6}=12\tfrac{2}{6}=12\tfrac{1}{3}

Add wholes and like-denominator fractions separately, then regroup any improper fraction.

Answer:

12\frac{1}{3}

4 · Reviewdoes it hold up?

Largest is about 9.83 and smallest about 2.50, summing to about 12.33, which matches 12 $\frac{1}{3}$ . The rule that a numerator must be under 6 was respected in both extremes.

Another way: Systematically list valid mixed numbers (tool 2): whole from the cards, numerator from the remaining cards under 6. The max is 9 $\frac{5}{6}$ and the min is 2 $\frac{3}{6}$ , giving the same sum 12 $\frac{1}{3}$ .

Standardsmin grade 4

4.NF.A.2 Compare two fractions with different numerators and different denominators — Reasoning about which card choices make the mixed number largest or smallest.
4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Adding the two mixed numbers and regrouping an improper fraction into a whole plus a proper fraction.

💡Takeaway. This only needs Grade 4 comparing and adding fractions — pick the biggest whole part first, but keep the numerator under 6!

Variant 8 medium

From $4$ number cards, choose $2$ of them and use each one exactly once to make a mixed number whose denominator is $5$ . Find the sum of the largest mixed number and the smallest mixed number you can make this way.

Number cards: $3$ , $9$ , $2$ , $7$

Your answer

Show solution

1 · Understandwhat's really being asked

From the four cards 3, 9, 2, 7 I pick two cards. One becomes the whole part and the other becomes the numerator over a denominator of 5, making a mixed number like W N/5. I want the biggest and smallest such mixed numbers, then their sum.

Givens

Number cards available: 3, 9, 2, 7.
Use exactly two cards, each once, to form a mixed number.
The denominator is fixed at 5.
A mixed number has the form (whole) (numerator/5).

Unknowns

The largest mixed number that can be made.
The smallest mixed number that can be made.
The sum of the largest and smallest.

Constraints

Numerator must be less than 5 for a proper fraction part.
Each card is used at most once within one mixed number.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #2 Make a Systematic List

To get the biggest number, make the whole part as big as possible, then the numerator as big as possible; to get the smallest, make the whole part as small as possible. Test the card choices and check the constraint that a numerator must be under 5.

3 · Execute3 carry out the plan

1Build the largest mixed number

#6 Guess and Check 4.NF.A.2

The whole part matters most, so use the biggest card, 9, as the whole number. For the fraction part the numerator should be as large as possible but under 5; the largest valid numerator is 3. So the largest mixed number is 9

\frac{3}{5}

.

9\tfrac{3}{5}

A bigger whole part beats any fraction part, so maximize the whole number first.

2Build the smallest mixed number

#2 Make a Systematic List 4.NF.A.2

Now make the whole part as small as possible: the smallest card is 2. For the smallest value the numerator should be as small as possible but still under 5; the smallest valid numerator is 3. The smallest mixed number is 2

\frac{3}{5}

.

2\tfrac{3}{5}

A smaller whole part makes a smaller number; the numerator must still be a legal fraction under 5.

3Add the largest and smallest

#6 Guess and Check 4.NF.B.3

Add 9

\frac{3}{5}

+ 2

\frac{3}{5}

. Whole parts: 9 + 2 = 11. Fraction parts:

\frac{3}{5}

+

\frac{3}{5}

=

\frac{6}{5}

. Combine to get 12

\frac{1}{5}

= 12

\frac{1}{5}

.

9\tfrac{3}{5}+2\tfrac{3}{5}=12\tfrac{1}{5}=12\tfrac{1}{5}

Add wholes and like-denominator fractions separately, then regroup any improper fraction.

Answer:

12\frac{1}{5}

4 · Reviewdoes it hold up?

Largest is about 9.60 and smallest about 2.60, summing to about 12.20, which matches 12 $\frac{1}{5}$ . The rule that a numerator must be under 5 was respected in both extremes.

Another way: Systematically list valid mixed numbers (tool 2): whole from the cards, numerator from the remaining cards under 5. The max is 9 $\frac{3}{5}$ and the min is 2 $\frac{3}{5}$ , giving the same sum 12 $\frac{1}{5}$ .

Standardsmin grade 4

4.NF.A.2 Compare two fractions with different numerators and different denominators — Reasoning about which card choices make the mixed number largest or smallest.
4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Adding the two mixed numbers and regrouping an improper fraction into a whole plus a proper fraction.

💡Takeaway. This only needs Grade 4 comparing and adding fractions — pick the biggest whole part first, but keep the numerator under 5!

Variant 9 hard

From $4$ number cards, choose $2$ of them and use each one exactly once to make a mixed number whose denominator is $8$ . Find the sum of the largest mixed number and the smallest mixed number you can make this way.

Number cards: $7$ , $4$ , $10$ , $6$

Your answer

Show solution

1 · Understandwhat's really being asked

From the four cards 7, 4, 10, 6 I pick two cards. One becomes the whole part and the other becomes the numerator over a denominator of 8, making a mixed number like W N/8. I want the biggest and smallest such mixed numbers, then their sum.

Givens

Number cards available: 7, 4, 10, 6.
Use exactly two cards, each once, to form a mixed number.
The denominator is fixed at 8.
A mixed number has the form (whole) (numerator/8).

Unknowns

The largest mixed number that can be made.
The smallest mixed number that can be made.
The sum of the largest and smallest.

Constraints

Numerator must be less than 8 for a proper fraction part.
Each card is used at most once within one mixed number.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #2 Make a Systematic List

To get the biggest number, make the whole part as big as possible, then the numerator as big as possible; to get the smallest, make the whole part as small as possible. Test the card choices and check the constraint that a numerator must be under 8.

3 · Execute3 carry out the plan

1Build the largest mixed number

#6 Guess and Check 4.NF.A.2

The whole part matters most, so use the biggest card, 10, as the whole number. For the fraction part the numerator should be as large as possible but under 8; the largest valid numerator is 7. So the largest mixed number is 10

\frac{7}{8}

.

10\tfrac{7}{8}

A bigger whole part beats any fraction part, so maximize the whole number first.

2Build the smallest mixed number

#2 Make a Systematic List 4.NF.A.2

Now make the whole part as small as possible: the smallest card is 4. For the smallest value the numerator should be as small as possible but still under 8; the smallest valid numerator is 6. The smallest mixed number is 4

\frac{6}{8}

.

4\tfrac{6}{8}

A smaller whole part makes a smaller number; the numerator must still be a legal fraction under 8.

3Add the largest and smallest

#6 Guess and Check 4.NF.B.3

Add 10

\frac{7}{8}

+ 4

\frac{6}{8}

. Whole parts: 10 + 4 = 14. Fraction parts:

\frac{7}{8}

+

\frac{6}{8}

=

\frac{13}{8}

. Combine to get 15

\frac{5}{8}

= 15

\frac{5}{8}

.

10\tfrac{7}{8}+4\tfrac{6}{8}=15\tfrac{5}{8}=15\tfrac{5}{8}

Add wholes and like-denominator fractions separately, then regroup any improper fraction.

Answer:

15\frac{5}{8}

4 · Reviewdoes it hold up?

Largest is about 10.88 and smallest about 4.75, summing to about 15.62, which matches 15 $\frac{5}{8}$ . The rule that a numerator must be under 8 was respected in both extremes.

Another way: Systematically list valid mixed numbers (tool 2): whole from the cards, numerator from the remaining cards under 8. The max is 10 $\frac{7}{8}$ and the min is 4 $\frac{6}{8}$ , giving the same sum 15 $\frac{5}{8}$ .

Standardsmin grade 4

4.NF.A.2 Compare two fractions with different numerators and different denominators — Reasoning about which card choices make the mixed number largest or smallest.
4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Adding the two mixed numbers and regrouping an improper fraction into a whole plus a proper fraction.

💡Takeaway. This only needs Grade 4 comparing and adding fractions — pick the biggest whole part first, but keep the numerator under 8!

Variant 10 hard

From $4$ number cards, choose $2$ of them and use each one exactly once to make a mixed number whose denominator is $6$ . Find the sum of the largest mixed number and the smallest mixed number you can make this way.

Number cards: $5$ , $11$ , $3$ , $9$

Your answer

Show solution

1 · Understandwhat's really being asked

From the four cards 5, 11, 3, 9 I pick two cards. One becomes the whole part and the other becomes the numerator over a denominator of 6, making a mixed number like W N/6. I want the biggest and smallest such mixed numbers, then their sum.

Givens

Number cards available: 5, 11, 3, 9.
Use exactly two cards, each once, to form a mixed number.
The denominator is fixed at 6.
A mixed number has the form (whole) (numerator/6).

Unknowns

The largest mixed number that can be made.
The smallest mixed number that can be made.
The sum of the largest and smallest.

Constraints

Numerator must be less than 6 for a proper fraction part.
Each card is used at most once within one mixed number.

2 · Planchoose the strategy

#6 Guess and Check · also uses: #2 Make a Systematic List

To get the biggest number, make the whole part as big as possible, then the numerator as big as possible; to get the smallest, make the whole part as small as possible. Test the card choices and check the constraint that a numerator must be under 6.

3 · Execute3 carry out the plan

1Build the largest mixed number

#6 Guess and Check 4.NF.A.2

The whole part matters most, so use the biggest card, 11, as the whole number. For the fraction part the numerator should be as large as possible but under 6; the largest valid numerator is 5. So the largest mixed number is 11

\frac{5}{6}

.

11\tfrac{5}{6}

A bigger whole part beats any fraction part, so maximize the whole number first.

2Build the smallest mixed number

#2 Make a Systematic List 4.NF.A.2

Now make the whole part as small as possible: the smallest card is 3. For the smallest value the numerator should be as small as possible but still under 6; the smallest valid numerator is 5. The smallest mixed number is 3

\frac{5}{6}

.

3\tfrac{5}{6}

A smaller whole part makes a smaller number; the numerator must still be a legal fraction under 6.

3Add the largest and smallest

#6 Guess and Check 4.NF.B.3

Add 11

\frac{5}{6}

+ 3

\frac{5}{6}

. Whole parts: 11 + 3 = 14. Fraction parts:

\frac{5}{6}

+

\frac{5}{6}

=

\frac{10}{6}

. Combine to get 15

\frac{4}{6}

= 15

\frac{2}{3}

.

11\tfrac{5}{6}+3\tfrac{5}{6}=15\tfrac{4}{6}=15\tfrac{2}{3}

Add wholes and like-denominator fractions separately, then regroup any improper fraction.

Answer:

15\frac{2}{3}

4 · Reviewdoes it hold up?

Largest is about 11.83 and smallest about 3.83, summing to about 15.67, which matches 15 $\frac{2}{3}$ . The rule that a numerator must be under 6 was respected in both extremes.

Another way: Systematically list valid mixed numbers (tool 2): whole from the cards, numerator from the remaining cards under 6. The max is 11 $\frac{5}{6}$ and the min is 3 $\frac{5}{6}$ , giving the same sum 15 $\frac{2}{3}$ .

Standardsmin grade 4

4.NF.A.2 Compare two fractions with different numerators and different denominators — Reasoning about which card choices make the mixed number largest or smallest.
4.NF.B.3 Understand a fraction with numerator greater than one as sum of unit fractions — Adding the two mixed numbers and regrouping an improper fraction into a whole plus a proper fraction.

💡Takeaway. This only needs Grade 4 comparing and adding fractions — pick the biggest whole part first, but keep the numerator under 6!

Build largest and smallest mixed numbers from cards · 10 practice problems

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