Practice · Place numbers by sign to maximize and minimize · Sensim Math

Variant 1 easy

Put the three rational numbers $+\frac{1}{2}$ , $+\frac{3}{4}$ , and $-3$ into the boxes of the expression below, one number per box with each used exactly once, and compute. Of all the values that can result, find the sum of the greatest and the least.

$\square \div \square - \square$

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Three numbers, $\frac{1}{2}$ , $\frac{3}{4}$ , $-3$ , fill the pattern (first) $\div$ (second) $-$ (third). Different placements give different values; we want the largest plus the smallest.

Givens

The numbers $+\frac{1}{2}$ , $+\frac{3}{4}$ , $-3$ , each used once.
The expression $\square \div \square - \square$ .

Unknowns

The greatest value plus the least value.

Constraints

Division comes before subtraction.
There are only $3 \times 2 \times 1 = 6$ placements.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #3 Eliminate Possibilities

With only six placements we can list them all, but sign reasoning tells us which ones can win. To make the result large, subtract the negative number; to make it small, put $-3$ into the quotient so it is negative, then subtract a positive. Rule out the rest and compare the survivors.

3 · Execute4 carry out the plan

1Greatest: subtract the negative number

#3 Eliminate Possibilities 7.NS.A.2

Subtracting

-3

adds

3

, and the quotient of the two positive numbers is positive. Dividing the larger by the smaller gives the bigger quotient.

\frac{3}{4} \div \frac{1}{2} - (-3) = \frac{3}{4} \times 2 + 3 = \frac{3}{2} + 3 = \frac{9}{2}

The other order,

\frac{1}{2} \div \frac{3}{4} + 3 = \frac{2}{3} + 3 = \frac{11}{3}

, is smaller, since

\frac{11}{3} < \frac{9}{2}

.

2Least: make the quotient negative and subtract a positive

#2 Make a Systematic List 7.NS.A.2

Put

-3

in the quotient. A large-sized negative quotient comes from

-3

on top.

(-3) \div \frac{1}{2} - \frac{3}{4} = -6 - \frac{3}{4} = -\frac{27}{4},\quad (-3) \div \frac{3}{4} - \frac{1}{2} = -4 - \frac{1}{2} = -\frac{9}{2}

Since

\frac{27}{4} > \frac{9}{2}

, the number

-\frac{27}{4}

is the smaller one.

3Check the remaining placements

#2 Make a Systematic List 7.NS.A.2

That leaves the two placements with

-3

in the second box: the quotient is negative, then a positive is subtracted.

\frac{1}{2} \div (-3) - \frac{3}{4} = -\frac{11}{12},\quad \frac{3}{4} \div (-3) - \frac{1}{2} = -\frac{3}{4}

Both are negative but greater than

-\frac{27}{4}

, so neither beats

-\frac{27}{4}

or

\frac{9}{2}

.

4Add the extremes

#2 Make a Systematic List 7.NS.A.1

Greatest

\frac{9}{2}

, least

-\frac{27}{4}

.

\frac{9}{2} + \left(-\frac{27}{4}\right) = -\frac{9}{4}

The sum is

-2\frac{1}{4}

.

Answer:

-2\frac{1}{4}

4 · Reviewdoes it hold up?

All six values are $\frac{9}{2}$ , $\frac{11}{3}$ , $-\frac{27}{4}$ , $-\frac{9}{2}$ , $-\frac{11}{12}$ , $-\frac{3}{4}$ . The largest is $\frac{9}{2}$ and the smallest is $-\frac{27}{4}$ ; $\frac{9}{2} + \left(-\frac{27}{4}\right) = -\frac{9}{4}$ .

Another way: Skip the reasoning and just compute all six placements in a table, then pick the largest and smallest values.

Standardsmin grade 7

7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Dividing signed fractions and predicting the sign of each quotient.
7.NS.A.1 Apply and extend understanding of addition and subtraction to rational numbers — Subtracting signed numbers and adding the two extremes.

💡Takeaway. Decide the signs first; then only a couple of placements are worth computing.

Variant 2 easy

Put the three rational numbers $+\frac{2}{3}$ , $+\frac{3}{2}$ , and $-4$ into the boxes of the expression below, one number per box with each used exactly once, and compute. Of all the values that can result, find the sum of the greatest and the least.

$\square \div \square - \square$

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Three numbers, $\frac{2}{3}$ , $\frac{3}{2}$ , $-4$ , fill the pattern (first) $\div$ (second) $-$ (third). Different placements give different values; we want the largest plus the smallest.

Givens

The numbers $+\frac{2}{3}$ , $+\frac{3}{2}$ , $-4$ , each used once.
The expression $\square \div \square - \square$ .

Unknowns

The greatest value plus the least value.

Constraints

Division comes before subtraction.
There are only $3 \times 2 \times 1 = 6$ placements.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #3 Eliminate Possibilities

With only six placements we can list them all, but sign reasoning tells us which ones can win. To make the result large, subtract the negative number; to make it small, put $-4$ into the quotient so it is negative, then subtract a positive. Rule out the rest and compare the survivors.

3 · Execute4 carry out the plan

1Greatest: subtract the negative number

#3 Eliminate Possibilities 7.NS.A.2

Subtracting

-4

adds

4

, and the quotient of the two positive numbers is positive. Dividing the larger by the smaller gives the bigger quotient.

\frac{3}{2} \div \frac{2}{3} - (-4) = \frac{3}{2} \times \frac{3}{2} + 4 = \frac{9}{4} + 4 = \frac{25}{4}

The other order,

\frac{2}{3} \div \frac{3}{2} + 4 = \frac{4}{9} + 4 = \frac{40}{9}

, is smaller, since

\frac{40}{9} < \frac{25}{4}

.

2Least: make the quotient negative and subtract a positive

#2 Make a Systematic List 7.NS.A.2

Put

-4

in the quotient. A large-sized negative quotient comes from

-4

on top.

(-4) \div \frac{2}{3} - \frac{3}{2} = -6 - \frac{3}{2} = -\frac{15}{2},\quad (-4) \div \frac{3}{2} - \frac{2}{3} = -\frac{8}{3} - \frac{2}{3} = -\frac{10}{3}

Since

\frac{15}{2} > \frac{10}{3}

, the number

-\frac{15}{2}

is the smaller one.

3Check the remaining placements

#2 Make a Systematic List 7.NS.A.2

That leaves the two placements with

-4

in the second box: the quotient is negative, then a positive is subtracted.

\frac{2}{3} \div (-4) - \frac{3}{2} = -\frac{5}{3},\quad \frac{3}{2} \div (-4) - \frac{2}{3} = -\frac{25}{24}

Both are negative but greater than

-\frac{15}{2}

, so neither beats

-\frac{15}{2}

or

\frac{25}{4}

.

4Add the extremes

#2 Make a Systematic List 7.NS.A.1

Greatest

\frac{25}{4}

, least

-\frac{15}{2}

.

\frac{25}{4} + \left(-\frac{15}{2}\right) = -\frac{5}{4}

The sum is

-1\frac{1}{4}

.

Answer:

-1\frac{1}{4}

4 · Reviewdoes it hold up?

All six values are $\frac{25}{4}$ , $\frac{40}{9}$ , $-\frac{15}{2}$ , $-\frac{10}{3}$ , $-\frac{5}{3}$ , $-\frac{25}{24}$ . The largest is $\frac{25}{4}$ and the smallest is $-\frac{15}{2}$ ; $\frac{25}{4} + \left(-\frac{15}{2}\right) = -\frac{5}{4}$ .

Another way: Skip the reasoning and just compute all six placements in a table, then pick the largest and smallest values.

Standardsmin grade 7

7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Dividing signed fractions and predicting the sign of each quotient.
7.NS.A.1 Apply and extend understanding of addition and subtraction to rational numbers — Subtracting signed numbers and adding the two extremes.

💡Takeaway. Decide the signs first; then only a couple of placements are worth computing.

Variant 3 easy

Put the three rational numbers $+\frac{1}{3}$ , $+\frac{3}{4}$ , and $-2$ into the boxes of the expression below, one number per box with each used exactly once, and compute. Of all the values that can result, find the sum of the greatest and the least.

$\square \div \square - \square$

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Three numbers, $\frac{1}{3}$ , $\frac{3}{4}$ , $-2$ , fill the pattern (first) $\div$ (second) $-$ (third). Different placements give different values; we want the largest plus the smallest.

Givens

The numbers $+\frac{1}{3}$ , $+\frac{3}{4}$ , $-2$ , each used once.
The expression $\square \div \square - \square$ .

Unknowns

The greatest value plus the least value.

Constraints

Division comes before subtraction.
There are only $3 \times 2 \times 1 = 6$ placements.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #3 Eliminate Possibilities

With only six placements we can list them all, but sign reasoning tells us which ones can win. To make the result large, subtract the negative number; to make it small, put $-2$ into the quotient so it is negative, then subtract a positive. Rule out the rest and compare the survivors.

3 · Execute4 carry out the plan

1Greatest: subtract the negative number

#3 Eliminate Possibilities 7.NS.A.2

Subtracting

-2

adds

2

, and the quotient of the two positive numbers is positive. Dividing the larger by the smaller gives the bigger quotient.

\frac{3}{4} \div \frac{1}{3} - (-2) = \frac{3}{4} \times 3 + 2 = \frac{9}{4} + 2 = \frac{17}{4}

The other order,

\frac{1}{3} \div \frac{3}{4} + 2 = \frac{4}{9} + 2 = \frac{22}{9}

, is smaller, since

\frac{22}{9} < \frac{17}{4}

.

2Least: make the quotient negative and subtract a positive

#2 Make a Systematic List 7.NS.A.2

Put

-2

in the quotient. A large-sized negative quotient comes from

-2

on top.

(-2) \div \frac{1}{3} - \frac{3}{4} = -6 - \frac{3}{4} = -\frac{27}{4},\quad (-2) \div \frac{3}{4} - \frac{1}{3} = -\frac{8}{3} - \frac{1}{3} = -3

Since

\frac{27}{4} > 3

, the number

-\frac{27}{4}

is the smaller one.

3Check the remaining placements

#2 Make a Systematic List 7.NS.A.2

That leaves the two placements with

-2

in the second box: the quotient is negative, then a positive is subtracted.

\frac{1}{3} \div (-2) - \frac{3}{4} = -\frac{11}{12},\quad \frac{3}{4} \div (-2) - \frac{1}{3} = -\frac{17}{24}

Both are negative but greater than

-\frac{27}{4}

, so neither beats

-\frac{27}{4}

or

\frac{17}{4}

.

4Add the extremes

#2 Make a Systematic List 7.NS.A.1

Greatest

\frac{17}{4}

, least

-\frac{27}{4}

.

\frac{17}{4} + \left(-\frac{27}{4}\right) = -\frac{5}{2}

The sum is

-2\frac{1}{2}

.

Answer:

-2\frac{1}{2}

4 · Reviewdoes it hold up?

All six values are $\frac{17}{4}$ , $\frac{22}{9}$ , $-\frac{27}{4}$ , $-3$ , $-\frac{11}{12}$ , $-\frac{17}{24}$ . The largest is $\frac{17}{4}$ and the smallest is $-\frac{27}{4}$ ; $\frac{17}{4} + \left(-\frac{27}{4}\right) = -\frac{5}{2}$ .

Another way: Skip the reasoning and just compute all six placements in a table, then pick the largest and smallest values.

Standardsmin grade 7

7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Dividing signed fractions and predicting the sign of each quotient.
7.NS.A.1 Apply and extend understanding of addition and subtraction to rational numbers — Subtracting signed numbers and adding the two extremes.

💡Takeaway. Decide the signs first; then only a couple of placements are worth computing.

Variant 4 medium

Put the three rational numbers $+\frac{3}{5}$ , $+\frac{5}{4}$ , and $-2$ into the boxes of the expression below, one number per box with each used exactly once, and compute. Of all the values that can result, find the sum of the greatest and the least.

$\square \div \square - \square$

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Three numbers, $\frac{3}{5}$ , $\frac{5}{4}$ , $-2$ , fill the pattern (first) $\div$ (second) $-$ (third). Different placements give different values; we want the largest plus the smallest.

Givens

The numbers $+\frac{3}{5}$ , $+\frac{5}{4}$ , $-2$ , each used once.
The expression $\square \div \square - \square$ .

Unknowns

The greatest value plus the least value.

Constraints

Division comes before subtraction.
There are only $3 \times 2 \times 1 = 6$ placements.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #3 Eliminate Possibilities

With only six placements we can list them all, but sign reasoning tells us which ones can win. To make the result large, subtract the negative number; to make it small, put $-2$ into the quotient so it is negative, then subtract a positive. Rule out the rest and compare the survivors.

3 · Execute4 carry out the plan

1Greatest: subtract the negative number

#3 Eliminate Possibilities 7.NS.A.2

Subtracting

-2

adds

2

, and the quotient of the two positive numbers is positive. Dividing the larger by the smaller gives the bigger quotient.

\frac{5}{4} \div \frac{3}{5} - (-2) = \frac{5}{4} \times \frac{5}{3} + 2 = \frac{25}{12} + 2 = \frac{49}{12}

The other order,

\frac{3}{5} \div \frac{5}{4} + 2 = \frac{12}{25} + 2 = \frac{62}{25}

, is smaller, since

\frac{62}{25} < \frac{49}{12}

.

2Least: make the quotient negative and subtract a positive

#2 Make a Systematic List 7.NS.A.2

Put

-2

in the quotient. A large-sized negative quotient comes from

-2

on top.

(-2) \div \frac{3}{5} - \frac{5}{4} = -\frac{10}{3} - \frac{5}{4} = -\frac{55}{12},\quad (-2) \div \frac{5}{4} - \frac{3}{5} = -\frac{8}{5} - \frac{3}{5} = -\frac{11}{5}

Since

\frac{55}{12} > \frac{11}{5}

, the number

-\frac{55}{12}

is the smaller one.

3Check the remaining placements

#2 Make a Systematic List 7.NS.A.2

That leaves the two placements with

-2

in the second box: the quotient is negative, then a positive is subtracted.

\frac{3}{5} \div (-2) - \frac{5}{4} = -\frac{31}{20},\quad \frac{5}{4} \div (-2) - \frac{3}{5} = -\frac{49}{40}

Both are negative but greater than

-\frac{55}{12}

, so neither beats

-\frac{55}{12}

or

\frac{49}{12}

.

4Add the extremes

#2 Make a Systematic List 7.NS.A.1

Greatest

\frac{49}{12}

, least

-\frac{55}{12}

.

\frac{49}{12} + \left(-\frac{55}{12}\right) = -\frac{1}{2}

The sum is

-\frac{1}{2}

.

Answer:

-\frac{1}{2}

4 · Reviewdoes it hold up?

All six values are $\frac{49}{12}$ , $\frac{62}{25}$ , $-\frac{55}{12}$ , $-\frac{11}{5}$ , $-\frac{31}{20}$ , $-\frac{49}{40}$ . The largest is $\frac{49}{12}$ and the smallest is $-\frac{55}{12}$ ; $\frac{49}{12} + \left(-\frac{55}{12}\right) = -\frac{1}{2}$ .

Another way: Skip the reasoning and just compute all six placements in a table, then pick the largest and smallest values.

Standardsmin grade 7

7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Dividing signed fractions and predicting the sign of each quotient.
7.NS.A.1 Apply and extend understanding of addition and subtraction to rational numbers — Subtracting signed numbers and adding the two extremes.

💡Takeaway. Decide the signs first; then only a couple of placements are worth computing.

Variant 5 medium

Put the three rational numbers $+\frac{2}{5}$ , $+\frac{3}{4}$ , and $-3$ into the boxes of the expression below, one number per box with each used exactly once, and compute. Of all the values that can result, find the sum of the greatest and the least.

$\square \div \square - \square$

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Three numbers, $\frac{2}{5}$ , $\frac{3}{4}$ , $-3$ , fill the pattern (first) $\div$ (second) $-$ (third). Different placements give different values; we want the largest plus the smallest.

Givens

The numbers $+\frac{2}{5}$ , $+\frac{3}{4}$ , $-3$ , each used once.
The expression $\square \div \square - \square$ .

Unknowns

The greatest value plus the least value.

Constraints

Division comes before subtraction.
There are only $3 \times 2 \times 1 = 6$ placements.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #3 Eliminate Possibilities

With only six placements we can list them all, but sign reasoning tells us which ones can win. To make the result large, subtract the negative number; to make it small, put $-3$ into the quotient so it is negative, then subtract a positive. Rule out the rest and compare the survivors.

3 · Execute4 carry out the plan

1Greatest: subtract the negative number

#3 Eliminate Possibilities 7.NS.A.2

Subtracting

-3

adds

3

, and the quotient of the two positive numbers is positive. Dividing the larger by the smaller gives the bigger quotient.

\frac{3}{4} \div \frac{2}{5} - (-3) = \frac{3}{4} \times \frac{5}{2} + 3 = \frac{15}{8} + 3 = \frac{39}{8}

The other order,

\frac{2}{5} \div \frac{3}{4} + 3 = \frac{8}{15} + 3 = \frac{53}{15}

, is smaller, since

\frac{53}{15} < \frac{39}{8}

.

2Least: make the quotient negative and subtract a positive

#2 Make a Systematic List 7.NS.A.2

Put

-3

in the quotient. A large-sized negative quotient comes from

-3

on top.

(-3) \div \frac{2}{5} - \frac{3}{4} = -\frac{15}{2} - \frac{3}{4} = -\frac{33}{4},\quad (-3) \div \frac{3}{4} - \frac{2}{5} = -4 - \frac{2}{5} = -\frac{22}{5}

Since

\frac{33}{4} > \frac{22}{5}

, the number

-\frac{33}{4}

is the smaller one.

3Check the remaining placements

#2 Make a Systematic List 7.NS.A.2

That leaves the two placements with

-3

in the second box: the quotient is negative, then a positive is subtracted.

\frac{2}{5} \div (-3) - \frac{3}{4} = -\frac{53}{60},\quad \frac{3}{4} \div (-3) - \frac{2}{5} = -\frac{13}{20}

Both are negative but greater than

-\frac{33}{4}

, so neither beats

-\frac{33}{4}

or

\frac{39}{8}

.

4Add the extremes

#2 Make a Systematic List 7.NS.A.1

Greatest

\frac{39}{8}

, least

-\frac{33}{4}

.

\frac{39}{8} + \left(-\frac{33}{4}\right) = -\frac{27}{8}

The sum is

-3\frac{3}{8}

.

Answer:

-3\frac{3}{8}

4 · Reviewdoes it hold up?

All six values are $\frac{39}{8}$ , $\frac{53}{15}$ , $-\frac{33}{4}$ , $-\frac{22}{5}$ , $-\frac{53}{60}$ , $-\frac{13}{20}$ . The largest is $\frac{39}{8}$ and the smallest is $-\frac{33}{4}$ ; $\frac{39}{8} + \left(-\frac{33}{4}\right) = -\frac{27}{8}$ .

Another way: Skip the reasoning and just compute all six placements in a table, then pick the largest and smallest values.

Standardsmin grade 7

7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Dividing signed fractions and predicting the sign of each quotient.
7.NS.A.1 Apply and extend understanding of addition and subtraction to rational numbers — Subtracting signed numbers and adding the two extremes.

💡Takeaway. Decide the signs first; then only a couple of placements are worth computing.

Variant 6 easy

Put the three rational numbers $+\frac{2}{3}$ , $+\frac{4}{5}$ , and $-2$ into the boxes of the expression below, one number per box with each used exactly once, and compute. Of all the values that can result, find the sum of the greatest and the least.

$\square \div \square - \square$

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Three numbers, $\frac{2}{3}$ , $\frac{4}{5}$ , $-2$ , fill the pattern (first) $\div$ (second) $-$ (third). Different placements give different values; we want the largest plus the smallest.

Givens

The numbers $+\frac{2}{3}$ , $+\frac{4}{5}$ , $-2$ , each used once.
The expression $\square \div \square - \square$ .

Unknowns

The greatest value plus the least value.

Constraints

Division comes before subtraction.
There are only $3 \times 2 \times 1 = 6$ placements.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #3 Eliminate Possibilities

With only six placements we can list them all, but sign reasoning tells us which ones can win. To make the result large, subtract the negative number; to make it small, put $-2$ into the quotient so it is negative, then subtract a positive. Rule out the rest and compare the survivors.

3 · Execute4 carry out the plan

1Greatest: subtract the negative number

#3 Eliminate Possibilities 7.NS.A.2

Subtracting

-2

adds

2

, and the quotient of the two positive numbers is positive. Dividing the larger by the smaller gives the bigger quotient.

\frac{4}{5} \div \frac{2}{3} - (-2) = \frac{4}{5} \times \frac{3}{2} + 2 = \frac{6}{5} + 2 = \frac{16}{5}

The other order,

\frac{2}{3} \div \frac{4}{5} + 2 = \frac{5}{6} + 2 = \frac{17}{6}

, is smaller, since

\frac{17}{6} < \frac{16}{5}

.

2Least: make the quotient negative and subtract a positive

#2 Make a Systematic List 7.NS.A.2

Put

-2

in the quotient. A large-sized negative quotient comes from

-2

on top.

(-2) \div \frac{2}{3} - \frac{4}{5} = -3 - \frac{4}{5} = -\frac{19}{5},\quad (-2) \div \frac{4}{5} - \frac{2}{3} = -\frac{5}{2} - \frac{2}{3} = -\frac{19}{6}

Since

\frac{19}{5} > \frac{19}{6}

, the number

-\frac{19}{5}

is the smaller one.

3Check the remaining placements

#2 Make a Systematic List 7.NS.A.2

That leaves the two placements with

-2

in the second box: the quotient is negative, then a positive is subtracted.

\frac{2}{3} \div (-2) - \frac{4}{5} = -\frac{17}{15},\quad \frac{4}{5} \div (-2) - \frac{2}{3} = -\frac{16}{15}

Both are negative but greater than

-\frac{19}{5}

, so neither beats

-\frac{19}{5}

or

\frac{16}{5}

.

4Add the extremes

#2 Make a Systematic List 7.NS.A.1

Greatest

\frac{16}{5}

, least

-\frac{19}{5}

.

\frac{16}{5} + \left(-\frac{19}{5}\right) = -\frac{3}{5}

The sum is

-\frac{3}{5}

.

Answer:

-\frac{3}{5}

4 · Reviewdoes it hold up?

All six values are $\frac{16}{5}$ , $\frac{17}{6}$ , $-\frac{19}{5}$ , $-\frac{19}{6}$ , $-\frac{17}{15}$ , $-\frac{16}{15}$ . The largest is $\frac{16}{5}$ and the smallest is $-\frac{19}{5}$ ; $\frac{16}{5} + \left(-\frac{19}{5}\right) = -\frac{3}{5}$ .

Another way: Skip the reasoning and just compute all six placements in a table, then pick the largest and smallest values.

Standardsmin grade 7

7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Dividing signed fractions and predicting the sign of each quotient.
7.NS.A.1 Apply and extend understanding of addition and subtraction to rational numbers — Subtracting signed numbers and adding the two extremes.

💡Takeaway. Decide the signs first; then only a couple of placements are worth computing.

Variant 7 medium

Put the three rational numbers $+\frac{3}{4}$ , $+\frac{6}{5}$ , and $-3$ into the boxes of the expression below, one number per box with each used exactly once, and compute. Of all the values that can result, find the sum of the greatest and the least.

$\square \div \square - \square$

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Three numbers, $\frac{3}{4}$ , $\frac{6}{5}$ , $-3$ , fill the pattern (first) $\div$ (second) $-$ (third). Different placements give different values; we want the largest plus the smallest.

Givens

The numbers $+\frac{3}{4}$ , $+\frac{6}{5}$ , $-3$ , each used once.
The expression $\square \div \square - \square$ .

Unknowns

The greatest value plus the least value.

Constraints

Division comes before subtraction.
There are only $3 \times 2 \times 1 = 6$ placements.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #3 Eliminate Possibilities

With only six placements we can list them all, but sign reasoning tells us which ones can win. To make the result large, subtract the negative number; to make it small, put $-3$ into the quotient so it is negative, then subtract a positive. Rule out the rest and compare the survivors.

3 · Execute4 carry out the plan

1Greatest: subtract the negative number

#3 Eliminate Possibilities 7.NS.A.2

Subtracting

-3

adds

3

, and the quotient of the two positive numbers is positive. Dividing the larger by the smaller gives the bigger quotient.

\frac{6}{5} \div \frac{3}{4} - (-3) = \frac{6}{5} \times \frac{4}{3} + 3 = \frac{8}{5} + 3 = \frac{23}{5}

The other order,

\frac{3}{4} \div \frac{6}{5} + 3 = \frac{5}{8} + 3 = \frac{29}{8}

, is smaller, since

\frac{29}{8} < \frac{23}{5}

.

2Least: make the quotient negative and subtract a positive

#2 Make a Systematic List 7.NS.A.2

Put

-3

in the quotient. A large-sized negative quotient comes from

-3

on top.

(-3) \div \frac{3}{4} - \frac{6}{5} = -4 - \frac{6}{5} = -\frac{26}{5},\quad (-3) \div \frac{6}{5} - \frac{3}{4} = -\frac{5}{2} - \frac{3}{4} = -\frac{13}{4}

Since

\frac{26}{5} > \frac{13}{4}

, the number

-\frac{26}{5}

is the smaller one.

3Check the remaining placements

#2 Make a Systematic List 7.NS.A.2

That leaves the two placements with

-3

in the second box: the quotient is negative, then a positive is subtracted.

\frac{3}{4} \div (-3) - \frac{6}{5} = -\frac{29}{20},\quad \frac{6}{5} \div (-3) - \frac{3}{4} = -\frac{23}{20}

Both are negative but greater than

-\frac{26}{5}

, so neither beats

-\frac{26}{5}

or

\frac{23}{5}

.

4Add the extremes

#2 Make a Systematic List 7.NS.A.1

Greatest

\frac{23}{5}

, least

-\frac{26}{5}

.

\frac{23}{5} + \left(-\frac{26}{5}\right) = -\frac{3}{5}

The sum is

-\frac{3}{5}

.

Answer:

-\frac{3}{5}

4 · Reviewdoes it hold up?

All six values are $\frac{23}{5}$ , $\frac{29}{8}$ , $-\frac{26}{5}$ , $-\frac{13}{4}$ , $-\frac{29}{20}$ , $-\frac{23}{20}$ . The largest is $\frac{23}{5}$ and the smallest is $-\frac{26}{5}$ ; $\frac{23}{5} + \left(-\frac{26}{5}\right) = -\frac{3}{5}$ .

Another way: Skip the reasoning and just compute all six placements in a table, then pick the largest and smallest values.

Standardsmin grade 7

7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Dividing signed fractions and predicting the sign of each quotient.
7.NS.A.1 Apply and extend understanding of addition and subtraction to rational numbers — Subtracting signed numbers and adding the two extremes.

💡Takeaway. Decide the signs first; then only a couple of placements are worth computing.

Variant 8 medium

Put the three rational numbers $+\frac{3}{4}$ , $+\frac{5}{6}$ , and $-2$ into the boxes of the expression below, one number per box with each used exactly once, and compute. Of all the values that can result, find the sum of the greatest and the least.

$\square \div \square - \square$

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Three numbers, $\frac{3}{4}$ , $\frac{5}{6}$ , $-2$ , fill the pattern (first) $\div$ (second) $-$ (third). Different placements give different values; we want the largest plus the smallest.

Givens

The numbers $+\frac{3}{4}$ , $+\frac{5}{6}$ , $-2$ , each used once.
The expression $\square \div \square - \square$ .

Unknowns

The greatest value plus the least value.

Constraints

Division comes before subtraction.
There are only $3 \times 2 \times 1 = 6$ placements.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #3 Eliminate Possibilities

With only six placements we can list them all, but sign reasoning tells us which ones can win. To make the result large, subtract the negative number; to make it small, put $-2$ into the quotient so it is negative, then subtract a positive. Rule out the rest and compare the survivors.

3 · Execute4 carry out the plan

1Greatest: subtract the negative number

#3 Eliminate Possibilities 7.NS.A.2

Subtracting

-2

adds

2

, and the quotient of the two positive numbers is positive. Dividing the larger by the smaller gives the bigger quotient.

\frac{5}{6} \div \frac{3}{4} - (-2) = \frac{5}{6} \times \frac{4}{3} + 2 = \frac{10}{9} + 2 = \frac{28}{9}

The other order,

\frac{3}{4} \div \frac{5}{6} + 2 = \frac{9}{10} + 2 = \frac{29}{10}

, is smaller, since

\frac{29}{10} < \frac{28}{9}

.

2Least: make the quotient negative and subtract a positive

#2 Make a Systematic List 7.NS.A.2

Put

-2

in the quotient. A large-sized negative quotient comes from

-2

on top.

(-2) \div \frac{3}{4} - \frac{5}{6} = -\frac{8}{3} - \frac{5}{6} = -\frac{7}{2},\quad (-2) \div \frac{5}{6} - \frac{3}{4} = -\frac{12}{5} - \frac{3}{4} = -\frac{63}{20}

Since

\frac{7}{2} > \frac{63}{20}

, the number

-\frac{7}{2}

is the smaller one.

3Check the remaining placements

#2 Make a Systematic List 7.NS.A.2

That leaves the two placements with

-2

in the second box: the quotient is negative, then a positive is subtracted.

\frac{3}{4} \div (-2) - \frac{5}{6} = -\frac{29}{24},\quad \frac{5}{6} \div (-2) - \frac{3}{4} = -\frac{7}{6}

Both are negative but greater than

-\frac{7}{2}

, so neither beats

-\frac{7}{2}

or

\frac{28}{9}

.

4Add the extremes

#2 Make a Systematic List 7.NS.A.1

Greatest

\frac{28}{9}

, least

-\frac{7}{2}

.

\frac{28}{9} + \left(-\frac{7}{2}\right) = -\frac{7}{18}

The sum is

-\frac{7}{18}

.

Answer:

-\frac{7}{18}

4 · Reviewdoes it hold up?

All six values are $\frac{28}{9}$ , $\frac{29}{10}$ , $-\frac{7}{2}$ , $-\frac{63}{20}$ , $-\frac{29}{24}$ , $-\frac{7}{6}$ . The largest is $\frac{28}{9}$ and the smallest is $-\frac{7}{2}$ ; $\frac{28}{9} + \left(-\frac{7}{2}\right) = -\frac{7}{18}$ .

Another way: Skip the reasoning and just compute all six placements in a table, then pick the largest and smallest values.

Standardsmin grade 7

7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Dividing signed fractions and predicting the sign of each quotient.
7.NS.A.1 Apply and extend understanding of addition and subtraction to rational numbers — Subtracting signed numbers and adding the two extremes.

💡Takeaway. Decide the signs first; then only a couple of placements are worth computing.

Variant 9 hard

Put the three rational numbers $+\frac{1}{2}$ , $+\frac{2}{3}$ , and $-\frac{3}{2}$ into the boxes of the expression below, one number per box with each used exactly once, and compute. Of all the values that can result, find the sum of the greatest and the least.

$\square \div \square - \square$

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Three numbers, $\frac{1}{2}$ , $\frac{2}{3}$ , $-\frac{3}{2}$ , fill the pattern (first) $\div$ (second) $-$ (third). Different placements give different values; we want the largest plus the smallest.

Givens

The numbers $+\frac{1}{2}$ , $+\frac{2}{3}$ , $-\frac{3}{2}$ , each used once.
The expression $\square \div \square - \square$ .

Unknowns

The greatest value plus the least value.

Constraints

Division comes before subtraction.
There are only $3 \times 2 \times 1 = 6$ placements.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #3 Eliminate Possibilities

With only six placements we can list them all, but sign reasoning tells us which ones can win. To make the result large, subtract the negative number; to make it small, put $-\frac{3}{2}$ into the quotient so it is negative, then subtract a positive. Rule out the rest and compare the survivors.

3 · Execute4 carry out the plan

1Greatest: subtract the negative number

#3 Eliminate Possibilities 7.NS.A.2

Subtracting

-\frac{3}{2}

adds

\frac{3}{2}

, and the quotient of the two positive numbers is positive. Dividing the larger by the smaller gives the bigger quotient.

\frac{2}{3} \div \frac{1}{2} - \left(-\frac{3}{2}\right) = \frac{2}{3} \times 2 + \frac{3}{2} = \frac{4}{3} + \frac{3}{2} = \frac{17}{6}

The other order,

\frac{1}{2} \div \frac{2}{3} + \frac{3}{2} = \frac{3}{4} + \frac{3}{2} = \frac{9}{4}

, is smaller, since

\frac{9}{4} < \frac{17}{6}

.

2Least: make the quotient negative and subtract a positive

#2 Make a Systematic List 7.NS.A.2

Put

-\frac{3}{2}

in the quotient. A large-sized negative quotient comes from

-\frac{3}{2}

on top.

\left(-\frac{3}{2}\right) \div \frac{1}{2} - \frac{2}{3} = -3 - \frac{2}{3} = -\frac{11}{3},\quad \left(-\frac{3}{2}\right) \div \frac{2}{3} - \frac{1}{2} = -\frac{9}{4} - \frac{1}{2} = -\frac{11}{4}

Since

\frac{11}{3} > \frac{11}{4}

, the number

-\frac{11}{3}

is the smaller one.

3Check the remaining placements

#2 Make a Systematic List 7.NS.A.2

That leaves the two placements with

-\frac{3}{2}

in the second box: the quotient is negative, then a positive is subtracted.

\frac{1}{2} \div \left(-\frac{3}{2}\right) - \frac{2}{3} = -1,\quad \frac{2}{3} \div \left(-\frac{3}{2}\right) - \frac{1}{2} = -\frac{17}{18}

Both are negative but greater than

-\frac{11}{3}

, so neither beats

-\frac{11}{3}

or

\frac{17}{6}

.

4Add the extremes

#2 Make a Systematic List 7.NS.A.1

Greatest

\frac{17}{6}

, least

-\frac{11}{3}

.

\frac{17}{6} + \left(-\frac{11}{3}\right) = -\frac{5}{6}

The sum is

-\frac{5}{6}

.

Answer:

-\frac{5}{6}

4 · Reviewdoes it hold up?

All six values are $\frac{17}{6}$ , $\frac{9}{4}$ , $-\frac{11}{3}$ , $-\frac{11}{4}$ , $-1$ , $-\frac{17}{18}$ . The largest is $\frac{17}{6}$ and the smallest is $-\frac{11}{3}$ ; $\frac{17}{6} + \left(-\frac{11}{3}\right) = -\frac{5}{6}$ .

Another way: Skip the reasoning and just compute all six placements in a table, then pick the largest and smallest values.

Standardsmin grade 7

7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Dividing signed fractions and predicting the sign of each quotient.
7.NS.A.1 Apply and extend understanding of addition and subtraction to rational numbers — Subtracting signed numbers and adding the two extremes.

💡Takeaway. Decide the signs first; then only a couple of placements are worth computing.

Variant 10 hard

Put the three rational numbers $+\frac{1}{4}$ , $+\frac{2}{3}$ , and $-\frac{1}{2}$ into the boxes of the expression below, one number per box with each used exactly once, and compute. Of all the values that can result, find the sum of the greatest and the least.

$\square \div \square - \square$

Your answer

Show solution

1 · Understandwhat's really being asked

Three numbers, $\frac{1}{4}$ , $\frac{2}{3}$ , $-\frac{1}{2}$ , fill the pattern (first) $\div$ (second) $-$ (third). Different placements give different values; we want the largest plus the smallest.

Givens

The numbers $+\frac{1}{4}$ , $+\frac{2}{3}$ , $-\frac{1}{2}$ , each used once.
The expression $\square \div \square - \square$ .

Unknowns

The greatest value plus the least value.

Constraints

Division comes before subtraction.
There are only $3 \times 2 \times 1 = 6$ placements.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #3 Eliminate Possibilities

With only six placements we can list them all, but sign reasoning tells us which ones can win. To make the result large, subtract the negative number; to make it small, put $-\frac{1}{2}$ into the quotient so it is negative, then subtract a positive. Rule out the rest and compare the survivors.

3 · Execute4 carry out the plan

1Greatest: subtract the negative number

#3 Eliminate Possibilities 7.NS.A.2

Subtracting

-\frac{1}{2}

adds

\frac{1}{2}

, and the quotient of the two positive numbers is positive. Dividing the larger by the smaller gives the bigger quotient.

\frac{2}{3} \div \frac{1}{4} - \left(-\frac{1}{2}\right) = \frac{2}{3} \times 4 + \frac{1}{2} = \frac{8}{3} + \frac{1}{2} = \frac{19}{6}

The other order,

\frac{1}{4} \div \frac{2}{3} + \frac{1}{2} = \frac{3}{8} + \frac{1}{2} = \frac{7}{8}

, is smaller, since

\frac{7}{8} < \frac{19}{6}

.

2Least: make the quotient negative and subtract a positive

#2 Make a Systematic List 7.NS.A.2

Put

-\frac{1}{2}

in the quotient. A large-sized negative quotient comes from

-\frac{1}{2}

on top.

\left(-\frac{1}{2}\right) \div \frac{1}{4} - \frac{2}{3} = -2 - \frac{2}{3} = -\frac{8}{3},\quad \left(-\frac{1}{2}\right) \div \frac{2}{3} - \frac{1}{4} = -\frac{3}{4} - \frac{1}{4} = -1

Since

\frac{8}{3} > 1

, the number

-\frac{8}{3}

is the smaller one.

3Check the remaining placements

#2 Make a Systematic List 7.NS.A.2

That leaves the two placements with

-\frac{1}{2}

in the second box: the quotient is negative, then a positive is subtracted.

\frac{1}{4} \div \left(-\frac{1}{2}\right) - \frac{2}{3} = -\frac{7}{6},\quad \frac{2}{3} \div \left(-\frac{1}{2}\right) - \frac{1}{4} = -\frac{19}{12}

Both are negative but greater than

-\frac{8}{3}

, so neither beats

-\frac{8}{3}

or

\frac{19}{6}

.

4Add the extremes

#2 Make a Systematic List 7.NS.A.1

Greatest

\frac{19}{6}

, least

-\frac{8}{3}

.

\frac{19}{6} + \left(-\frac{8}{3}\right) = \frac{1}{2}

The sum is

\frac{1}{2}

.

Answer:

\frac{1}{2}

4 · Reviewdoes it hold up?

All six values are $\frac{19}{6}$ , $\frac{7}{8}$ , $-\frac{8}{3}$ , $-1$ , $-\frac{7}{6}$ , $-\frac{19}{12}$ . The largest is $\frac{19}{6}$ and the smallest is $-\frac{8}{3}$ ; $\frac{19}{6} + \left(-\frac{8}{3}\right) = \frac{1}{2}$ .

Another way: Skip the reasoning and just compute all six placements in a table, then pick the largest and smallest values.

Standardsmin grade 7

7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Dividing signed fractions and predicting the sign of each quotient.
7.NS.A.1 Apply and extend understanding of addition and subtraction to rational numbers — Subtracting signed numbers and adding the two extremes.

💡Takeaway. Decide the signs first; then only a couple of placements are worth computing.

Variant 11 hard

Put the three rational numbers $+\frac{1}{2}$ , $+\frac{4}{3}$ , and $-\frac{2}{3}$ into the boxes of the expression below, one number per box with each used exactly once, and compute. Of all the values that can result, find the sum of the greatest and the least.

$\square \div \square - \square$

Your answer

Show solution

1 · Understandwhat's really being asked

Three numbers, $\frac{1}{2}$ , $\frac{4}{3}$ , $-\frac{2}{3}$ , fill the pattern (first) $\div$ (second) $-$ (third). Different placements give different values; we want the largest plus the smallest.

Givens

The numbers $+\frac{1}{2}$ , $+\frac{4}{3}$ , $-\frac{2}{3}$ , each used once.
The expression $\square \div \square - \square$ .

Unknowns

The greatest value plus the least value.

Constraints

Division comes before subtraction.
There are only $3 \times 2 \times 1 = 6$ placements.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #3 Eliminate Possibilities

With only six placements we can list them all, but sign reasoning tells us which ones can win. To make the result large, subtract the negative number; to make it small, put $-\frac{2}{3}$ into the quotient so it is negative, then subtract a positive. Rule out the rest and compare the survivors.

3 · Execute4 carry out the plan

1Greatest: subtract the negative number

#3 Eliminate Possibilities 7.NS.A.2

Subtracting

-\frac{2}{3}

adds

\frac{2}{3}

, and the quotient of the two positive numbers is positive. Dividing the larger by the smaller gives the bigger quotient.

\frac{4}{3} \div \frac{1}{2} - \left(-\frac{2}{3}\right) = \frac{4}{3} \times 2 + \frac{2}{3} = \frac{8}{3} + \frac{2}{3} = \frac{10}{3}

The other order,

\frac{1}{2} \div \frac{4}{3} + \frac{2}{3} = \frac{3}{8} + \frac{2}{3} = \frac{25}{24}

, is smaller, since

\frac{25}{24} < \frac{10}{3}

.

2Least: make the quotient negative and subtract a positive

#2 Make a Systematic List 7.NS.A.2

Put

-\frac{2}{3}

in the quotient. A large-sized negative quotient comes from

-\frac{2}{3}

on top.

\left(-\frac{2}{3}\right) \div \frac{1}{2} - \frac{4}{3} = -\frac{4}{3} - \frac{4}{3} = -\frac{8}{3},\quad \left(-\frac{2}{3}\right) \div \frac{4}{3} - \frac{1}{2} = -\frac{1}{2} - \frac{1}{2} = -1

Since

\frac{8}{3} > 1

, the number

-\frac{8}{3}

is the smaller one.

3Check the remaining placements

#2 Make a Systematic List 7.NS.A.2

That leaves the two placements with

-\frac{2}{3}

in the second box: the quotient is negative, then a positive is subtracted.

\frac{1}{2} \div \left(-\frac{2}{3}\right) - \frac{4}{3} = -\frac{25}{12},\quad \frac{4}{3} \div \left(-\frac{2}{3}\right) - \frac{1}{2} = -\frac{5}{2}

Both are negative but greater than

-\frac{8}{3}

, so neither beats

-\frac{8}{3}

or

\frac{10}{3}

.

4Add the extremes

#2 Make a Systematic List 7.NS.A.1

Greatest

\frac{10}{3}

, least

-\frac{8}{3}

.

\frac{10}{3} + \left(-\frac{8}{3}\right) = \frac{2}{3}

The sum is

\frac{2}{3}

.

Answer:

\frac{2}{3}

4 · Reviewdoes it hold up?

All six values are $\frac{10}{3}$ , $\frac{25}{24}$ , $-\frac{8}{3}$ , $-1$ , $-\frac{25}{12}$ , $-\frac{5}{2}$ . The largest is $\frac{10}{3}$ and the smallest is $-\frac{8}{3}$ ; $\frac{10}{3} + \left(-\frac{8}{3}\right) = \frac{2}{3}$ .

Another way: Skip the reasoning and just compute all six placements in a table, then pick the largest and smallest values.

Standardsmin grade 7

7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Dividing signed fractions and predicting the sign of each quotient.
7.NS.A.1 Apply and extend understanding of addition and subtraction to rational numbers — Subtracting signed numbers and adding the two extremes.

💡Takeaway. Decide the signs first; then only a couple of placements are worth computing.

Variant 12 hard

Put the three rational numbers $+\frac{4}{5}$ , $+\frac{3}{2}$ , and $-\frac{5}{2}$ into the boxes of the expression below, one number per box with each used exactly once, and compute. Of all the values that can result, find the sum of the greatest and the least.

$\square \div \square - \square$

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Three numbers, $\frac{4}{5}$ , $\frac{3}{2}$ , $-\frac{5}{2}$ , fill the pattern (first) $\div$ (second) $-$ (third). Different placements give different values; we want the largest plus the smallest.

Givens

The numbers $+\frac{4}{5}$ , $+\frac{3}{2}$ , $-\frac{5}{2}$ , each used once.
The expression $\square \div \square - \square$ .

Unknowns

The greatest value plus the least value.

Constraints

Division comes before subtraction.
There are only $3 \times 2 \times 1 = 6$ placements.

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #3 Eliminate Possibilities

With only six placements we can list them all, but sign reasoning tells us which ones can win. To make the result large, subtract the negative number; to make it small, put $-\frac{5}{2}$ into the quotient so it is negative, then subtract a positive. Rule out the rest and compare the survivors.

3 · Execute4 carry out the plan

1Greatest: subtract the negative number

#3 Eliminate Possibilities 7.NS.A.2

Subtracting

-\frac{5}{2}

adds

\frac{5}{2}

, and the quotient of the two positive numbers is positive. Dividing the larger by the smaller gives the bigger quotient.

\frac{3}{2} \div \frac{4}{5} - \left(-\frac{5}{2}\right) = \frac{3}{2} \times \frac{5}{4} + \frac{5}{2} = \frac{15}{8} + \frac{5}{2} = \frac{35}{8}

The other order,

\frac{4}{5} \div \frac{3}{2} + \frac{5}{2} = \frac{8}{15} + \frac{5}{2} = \frac{91}{30}

, is smaller, since

\frac{91}{30} < \frac{35}{8}

.

2Least: make the quotient negative and subtract a positive

#2 Make a Systematic List 7.NS.A.2

Put

-\frac{5}{2}

in the quotient. A large-sized negative quotient comes from

-\frac{5}{2}

on top.

\left(-\frac{5}{2}\right) \div \frac{4}{5} - \frac{3}{2} = -\frac{25}{8} - \frac{3}{2} = -\frac{37}{8},\quad \left(-\frac{5}{2}\right) \div \frac{3}{2} - \frac{4}{5} = -\frac{5}{3} - \frac{4}{5} = -\frac{37}{15}

Since

\frac{37}{8} > \frac{37}{15}

, the number

-\frac{37}{8}

is the smaller one.

3Check the remaining placements

#2 Make a Systematic List 7.NS.A.2

That leaves the two placements with

-\frac{5}{2}

in the second box: the quotient is negative, then a positive is subtracted.

\frac{4}{5} \div \left(-\frac{5}{2}\right) - \frac{3}{2} = -\frac{91}{50},\quad \frac{3}{2} \div \left(-\frac{5}{2}\right) - \frac{4}{5} = -\frac{7}{5}

Both are negative but greater than

-\frac{37}{8}

, so neither beats

-\frac{37}{8}

or

\frac{35}{8}

.

4Add the extremes

#2 Make a Systematic List 7.NS.A.1

Greatest

\frac{35}{8}

, least

-\frac{37}{8}

.

\frac{35}{8} + \left(-\frac{37}{8}\right) = -\frac{1}{4}

The sum is

-\frac{1}{4}

.

Answer:

-\frac{1}{4}

4 · Reviewdoes it hold up?

All six values are $\frac{35}{8}$ , $\frac{91}{30}$ , $-\frac{37}{8}$ , $-\frac{37}{15}$ , $-\frac{91}{50}$ , $-\frac{7}{5}$ . The largest is $\frac{35}{8}$ and the smallest is $-\frac{37}{8}$ ; $\frac{35}{8} + \left(-\frac{37}{8}\right) = -\frac{1}{4}$ .

Another way: Skip the reasoning and just compute all six placements in a table, then pick the largest and smallest values.

Standardsmin grade 7

7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Dividing signed fractions and predicting the sign of each quotient.
7.NS.A.1 Apply and extend understanding of addition and subtraction to rational numbers — Subtracting signed numbers and adding the two extremes.

💡Takeaway. Decide the signs first; then only a couple of placements are worth computing.

Place numbers by sign to maximize and minimize · 12 practice problems

Generated variants — 12

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Where next?