Practice · Recover each letter's sign from a quadrant · Sensim Math

Variant 1 easy

The point $(ab, a-b)$ lies in Quadrant II. Which quadrant contains the point $(b^2, 2b-a)$ ?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A point built from $a$ and $b$ sits in Quadrant II. Use that to pin down the signs of $a$ and $b$ , then decide where a second point built from them lands.

Givens

The point $(ab, a-b)$ is in Quadrant II.

Unknowns

The quadrant containing $(b^2, 2b-a)$ .

Constraints

In Quadrant II the $x$ -coordinate is negative and the $y$ -coordinate is positive.
Points on an axis belong to no quadrant, so every coordinate here is strictly positive or strictly negative.

2 · Planchoose the strategy

#15 Organize Information in More Ways · also uses: #2 Make a Systematic List

Turn the quadrant into two sign facts, then list the only two sign patterns a negative product allows and keep the one that fits the other fact. Once the signs of $a$ and $b$ are known, every coordinate of the new point has a definite sign.

3 · Execute5 carry out the plan

1Translate the quadrant into signs

#15 Organize Information in More Ways 6.NS.C.6

Quadrant II means the

x

-coordinate is negative and the

y

-coordinate is positive.

ab<0, \quad a-b>0

A quadrant is just a pair of signs.

2List the sign patterns for a negative product

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

A product is negative exactly when its two factors have opposite signs, so either

a

is positive and

b

negative, or the other way round.

ab<0 \Rightarrow (a>0,\ b<0) \text{ or } (a<0,\ b>0)

Only two cases are possible, so we can test both.

3Keep the case that fits $a-b>0$

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

a-b>0

means

b<a

. If

a

were negative and

b

positive,

b

would be the larger one, which breaks

b<a

. So

a>0

and

b<0

.

b<a \Rightarrow a>0,\ b<0

A positive number is always bigger than a negative one.

4Find the signs of the new coordinates

#15 Organize Information in More Ways 7.NS.A.2

b^2

is positive because

b\ne 0

. Since

b<0

,

2b

is negative, and since

a>0

,

-a

is negative, and

2b-a

is a negative number plus a negative number, so it is negative.

b^2>0, \quad 2b-a = 2b+(-a)<0

Squaring removes the sign; then add two numbers of the same sign.

5Read off the quadrant

#15 Organize Information in More Ways 6.NS.C.6

A positive

x

-coordinate with a negative

y

-coordinate is Quadrant IV.

(+,\ -) \Rightarrow \text{Quadrant IV}

Right of the

y

-axis and below the

x

-axis.

Answer: Quadrant IV

4 · Reviewdoes it hold up?

Try $a=1$ , $b=-1$ : the first point is $(-1, 2)$ , which is in Quadrant II, and the second is $(1, -3)$ , which is in Quadrant IV.

Another way: Organize the facts in a sign table: from $ab<0$ and $b<a$ fill in $a:+$ , $b:-$ , then fill the rows $b^2:+$ and $2b-a:-$ to read the quadrant directly.

Standardsmin grade 7

7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Deciding signs of products and sums of signed numbers.
6.NS.C.6 Rational numbers on the number line and coordinate plane — Reading a quadrant from the signs of a point's coordinates.

💡Takeaway. A quadrant is a pair of signs -- read the signs and the letters give themselves away.

Variant 2 easy

The point $(a-b, ab)$ lies in Quadrant III. Which quadrant contains the point $(2a-b, b^2)$ ?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A point built from $a$ and $b$ sits in Quadrant III. Use that to pin down the signs of $a$ and $b$ , then decide where a second point built from them lands.

Givens

The point $(a-b, ab)$ is in Quadrant III.

Unknowns

The quadrant containing $(2a-b, b^2)$ .

Constraints

In Quadrant III the $x$ -coordinate and the $y$ -coordinate are both negative.
Points on an axis belong to no quadrant, so every coordinate here is strictly positive or strictly negative.

2 · Planchoose the strategy

#15 Organize Information in More Ways · also uses: #2 Make a Systematic List

Turn the quadrant into two sign facts, then list the only two sign patterns a negative product allows and keep the one that fits the other fact. Once the signs of $a$ and $b$ are known, every coordinate of the new point has a definite sign.

3 · Execute5 carry out the plan

1Translate the quadrant into signs

#15 Organize Information in More Ways 6.NS.C.6

Quadrant III means the

x

-coordinate and the

y

-coordinate are both negative.

a-b<0, \quad ab<0

A quadrant is just a pair of signs.

2List the sign patterns for a negative product

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

A product is negative exactly when its two factors have opposite signs, so either

a

is positive and

b

negative, or the other way round.

ab<0 \Rightarrow (a>0,\ b<0) \text{ or } (a<0,\ b>0)

Only two cases are possible, so we can test both.

3Keep the case that fits $a-b<0$

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

a-b<0

means

a<b

. If

b

were negative and

a

positive,

a

would be the larger one, which breaks

a<b

. So

b>0

and

a<0

.

a<b \Rightarrow b>0,\ a<0

A positive number is always bigger than a negative one.

4Find the signs of the new coordinates

#15 Organize Information in More Ways 7.NS.A.2

Since

a<0

,

2a

is negative, and since

b>0

,

-b

is negative, and

2a-b

is a negative number plus a negative number, so it is negative.

b^2

is positive because

b\ne 0

.

2a-b = 2a+(-b)<0, \quad b^2>0

Squaring removes the sign; then add two numbers of the same sign.

5Read off the quadrant

#15 Organize Information in More Ways 6.NS.C.6

A negative

x

-coordinate with a positive

y

-coordinate is Quadrant II.

(-,\ +) \Rightarrow \text{Quadrant II}

Left of the

y

-axis and above the

x

-axis.

Answer: Quadrant II

4 · Reviewdoes it hold up?

Try $a=-1$ , $b=1$ : the first point is $(-2, -1)$ , which is in Quadrant III, and the second is $(-3, 1)$ , which is in Quadrant II.

Another way: Organize the facts in a sign table: from $ab<0$ and $a<b$ fill in $a:-$ , $b:+$ , then fill the rows $2a-b:-$ and $b^2:+$ to read the quadrant directly.

Standardsmin grade 7

7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Deciding signs of products and sums of signed numbers.
6.NS.C.6 Rational numbers on the number line and coordinate plane — Reading a quadrant from the signs of a point's coordinates.

💡Takeaway. A quadrant is a pair of signs -- read the signs and the letters give themselves away.

Variant 3 easy

The point $(a-b, ab)$ lies in Quadrant IV. Which quadrant contains the point $(a^2, b-2a)$ ?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A point built from $a$ and $b$ sits in Quadrant IV. Use that to pin down the signs of $a$ and $b$ , then decide where a second point built from them lands.

Givens

The point $(a-b, ab)$ is in Quadrant IV.

Unknowns

The quadrant containing $(a^2, b-2a)$ .

Constraints

In Quadrant IV the $x$ -coordinate is positive and the $y$ -coordinate is negative.
Points on an axis belong to no quadrant, so every coordinate here is strictly positive or strictly negative.

2 · Planchoose the strategy

#15 Organize Information in More Ways · also uses: #2 Make a Systematic List

Turn the quadrant into two sign facts, then list the only two sign patterns a negative product allows and keep the one that fits the other fact. Once the signs of $a$ and $b$ are known, every coordinate of the new point has a definite sign.

3 · Execute5 carry out the plan

1Translate the quadrant into signs

#15 Organize Information in More Ways 6.NS.C.6

Quadrant IV means the

x

-coordinate is positive and the

y

-coordinate is negative.

a-b>0, \quad ab<0

A quadrant is just a pair of signs.

2List the sign patterns for a negative product

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

A product is negative exactly when its two factors have opposite signs, so either

a

is positive and

b

negative, or the other way round.

ab<0 \Rightarrow (a>0,\ b<0) \text{ or } (a<0,\ b>0)

Only two cases are possible, so we can test both.

3Keep the case that fits $a-b>0$

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

a-b>0

means

b<a

. If

a

were negative and

b

positive,

b

would be the larger one, which breaks

b<a

. So

a>0

and

b<0

.

b<a \Rightarrow a>0,\ b<0

A positive number is always bigger than a negative one.

4Find the signs of the new coordinates

#15 Organize Information in More Ways 7.NS.A.2

a^2

is positive because

a\ne 0

. Since

a>0

,

-2a

is negative, and

b-2a

is a negative number plus a negative number, so it is negative.

a^2>0, \quad b-2a = b+(-2a)<0

Squaring removes the sign; then add two numbers of the same sign.

5Read off the quadrant

#15 Organize Information in More Ways 6.NS.C.6

A positive

x

-coordinate with a negative

y

-coordinate is Quadrant IV.

(+,\ -) \Rightarrow \text{Quadrant IV}

Right of the

y

-axis and below the

x

-axis.

Answer: Quadrant IV

4 · Reviewdoes it hold up?

Try $a=1$ , $b=-1$ : the first point is $(2, -1)$ , which is in Quadrant IV, and the second is $(1, -3)$ , which is in Quadrant IV.

Another way: Organize the facts in a sign table: from $ab<0$ and $b<a$ fill in $a:+$ , $b:-$ , then fill the rows $a^2:+$ and $b-2a:-$ to read the quadrant directly.

Standardsmin grade 7

7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Deciding signs of products and sums of signed numbers.
6.NS.C.6 Rational numbers on the number line and coordinate plane — Reading a quadrant from the signs of a point's coordinates.

💡Takeaway. A quadrant is a pair of signs -- read the signs and the letters give themselves away.

Variant 4 easy

The point $(b-a, ab)$ lies in Quadrant IV. Which quadrant contains the point $(-a^2, 2b-a)$ ?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A point built from $a$ and $b$ sits in Quadrant IV. Use that to pin down the signs of $a$ and $b$ , then decide where a second point built from them lands.

Givens

The point $(b-a, ab)$ is in Quadrant IV.

Unknowns

The quadrant containing $(-a^2, 2b-a)$ .

Constraints

In Quadrant IV the $x$ -coordinate is positive and the $y$ -coordinate is negative.
Points on an axis belong to no quadrant, so every coordinate here is strictly positive or strictly negative.

2 · Planchoose the strategy

#15 Organize Information in More Ways · also uses: #2 Make a Systematic List

Turn the quadrant into two sign facts, then list the only two sign patterns a negative product allows and keep the one that fits the other fact. Once the signs of $a$ and $b$ are known, every coordinate of the new point has a definite sign.

3 · Execute5 carry out the plan

1Translate the quadrant into signs

#15 Organize Information in More Ways 6.NS.C.6

Quadrant IV means the

x

-coordinate is positive and the

y

-coordinate is negative.

b-a>0, \quad ab<0

A quadrant is just a pair of signs.

2List the sign patterns for a negative product

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

A product is negative exactly when its two factors have opposite signs, so either

a

is positive and

b

negative, or the other way round.

ab<0 \Rightarrow (a>0,\ b<0) \text{ or } (a<0,\ b>0)

Only two cases are possible, so we can test both.

3Keep the case that fits $b-a>0$

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

b-a>0

means

a<b

. If

b

were negative and

a

positive,

a

would be the larger one, which breaks

a<b

. So

b>0

and

a<0

.

a<b \Rightarrow b>0,\ a<0

A positive number is always bigger than a negative one.

4Find the signs of the new coordinates

#15 Organize Information in More Ways 7.NS.A.2

a^2

is positive because

a\ne 0

, so

-a^2

is negative. Since

b>0

,

2b

is positive, and since

a<0

,

-a

is positive, and

2b-a

is a positive number plus a positive number, so it is positive.

-a^2<0, \quad 2b-a = 2b+(-a)>0

Squaring removes the sign, and the minus in front makes it negative; then add two numbers of the same sign.

5Read off the quadrant

#15 Organize Information in More Ways 6.NS.C.6

A negative

x

-coordinate with a positive

y

-coordinate is Quadrant II.

(-,\ +) \Rightarrow \text{Quadrant II}

Left of the

y

-axis and above the

x

-axis.

Answer: Quadrant II

4 · Reviewdoes it hold up?

Try $a=-1$ , $b=1$ : the first point is $(2, -1)$ , which is in Quadrant IV, and the second is $(-1, 3)$ , which is in Quadrant II.

Another way: Organize the facts in a sign table: from $ab<0$ and $a<b$ fill in $a:-$ , $b:+$ , then fill the rows $-a^2:-$ and $2b-a:+$ to read the quadrant directly.

Standardsmin grade 7

7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Deciding signs of products and sums of signed numbers.
6.NS.C.6 Rational numbers on the number line and coordinate plane — Reading a quadrant from the signs of a point's coordinates.

💡Takeaway. A quadrant is a pair of signs -- read the signs and the letters give themselves away.

Variant 5 medium

The point $(ab, b-a)$ lies in Quadrant II. Which quadrant contains the point $(a-2b, a^2)$ ?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A point built from $a$ and $b$ sits in Quadrant II. Use that to pin down the signs of $a$ and $b$ , then decide where a second point built from them lands.

Givens

The point $(ab, b-a)$ is in Quadrant II.

Unknowns

The quadrant containing $(a-2b, a^2)$ .

Constraints

In Quadrant II the $x$ -coordinate is negative and the $y$ -coordinate is positive.
Points on an axis belong to no quadrant, so every coordinate here is strictly positive or strictly negative.

2 · Planchoose the strategy

#15 Organize Information in More Ways · also uses: #2 Make a Systematic List

Turn the quadrant into two sign facts, then list the only two sign patterns a negative product allows and keep the one that fits the other fact. Once the signs of $a$ and $b$ are known, every coordinate of the new point has a definite sign.

3 · Execute5 carry out the plan

1Translate the quadrant into signs

#15 Organize Information in More Ways 6.NS.C.6

Quadrant II means the

x

-coordinate is negative and the

y

-coordinate is positive.

ab<0, \quad b-a>0

A quadrant is just a pair of signs.

2List the sign patterns for a negative product

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

A product is negative exactly when its two factors have opposite signs, so either

a

is positive and

b

negative, or the other way round.

ab<0 \Rightarrow (a>0,\ b<0) \text{ or } (a<0,\ b>0)

Only two cases are possible, so we can test both.

3Keep the case that fits $b-a>0$

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

b-a>0

means

a<b

. If

b

were negative and

a

positive,

a

would be the larger one, which breaks

a<b

. So

b>0

and

a<0

.

a<b \Rightarrow b>0,\ a<0

A positive number is always bigger than a negative one.

4Find the signs of the new coordinates

#15 Organize Information in More Ways 7.NS.A.2

Since

b>0

,

-2b

is negative, and

a-2b

is a negative number plus a negative number, so it is negative.

a^2

is positive because

a\ne 0

.

a-2b = a+(-2b)<0, \quad a^2>0

Squaring removes the sign; then add two numbers of the same sign.

5Read off the quadrant

#15 Organize Information in More Ways 6.NS.C.6

A negative

x

-coordinate with a positive

y

-coordinate is Quadrant II.

(-,\ +) \Rightarrow \text{Quadrant II}

Left of the

y

-axis and above the

x

-axis.

Answer: Quadrant II

4 · Reviewdoes it hold up?

Try $a=-1$ , $b=1$ : the first point is $(-1, 2)$ , which is in Quadrant II, and the second is $(-3, 1)$ , which is in Quadrant II.

Another way: Organize the facts in a sign table: from $ab<0$ and $a<b$ fill in $a:-$ , $b:+$ , then fill the rows $a-2b:-$ and $a^2:+$ to read the quadrant directly.

Standardsmin grade 7

7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Deciding signs of products and sums of signed numbers.
6.NS.C.6 Rational numbers on the number line and coordinate plane — Reading a quadrant from the signs of a point's coordinates.

💡Takeaway. A quadrant is a pair of signs -- read the signs and the letters give themselves away.

Variant 6 medium

The point $(ab, b-a)$ lies in Quadrant III. Which quadrant contains the point $(3a-b, -a^2)$ ?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A point built from $a$ and $b$ sits in Quadrant III. Use that to pin down the signs of $a$ and $b$ , then decide where a second point built from them lands.

Givens

The point $(ab, b-a)$ is in Quadrant III.

Unknowns

The quadrant containing $(3a-b, -a^2)$ .

Constraints

In Quadrant III the $x$ -coordinate and the $y$ -coordinate are both negative.
Points on an axis belong to no quadrant, so every coordinate here is strictly positive or strictly negative.

2 · Planchoose the strategy

#15 Organize Information in More Ways · also uses: #2 Make a Systematic List

Turn the quadrant into two sign facts, then list the only two sign patterns a negative product allows and keep the one that fits the other fact. Once the signs of $a$ and $b$ are known, every coordinate of the new point has a definite sign.

3 · Execute5 carry out the plan

1Translate the quadrant into signs

#15 Organize Information in More Ways 6.NS.C.6

Quadrant III means the

x

-coordinate and the

y

-coordinate are both negative.

ab<0, \quad b-a<0

A quadrant is just a pair of signs.

2List the sign patterns for a negative product

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

A product is negative exactly when its two factors have opposite signs, so either

a

is positive and

b

negative, or the other way round.

ab<0 \Rightarrow (a>0,\ b<0) \text{ or } (a<0,\ b>0)

Only two cases are possible, so we can test both.

3Keep the case that fits $b-a<0$

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

b-a<0

means

b<a

. If

a

were negative and

b

positive,

b

would be the larger one, which breaks

b<a

. So

a>0

and

b<0

.

b<a \Rightarrow a>0,\ b<0

A positive number is always bigger than a negative one.

4Find the signs of the new coordinates

#15 Organize Information in More Ways 7.NS.A.2

Since

a>0

,

3a

is positive, and since

b<0

,

-b

is positive, and

3a-b

is a positive number plus a positive number, so it is positive.

a^2

is positive because

a\ne 0

, so

-a^2

is negative.

3a-b = 3a+(-b)>0, \quad -a^2<0

Squaring removes the sign, and the minus in front makes it negative; then add two numbers of the same sign.

5Read off the quadrant

#15 Organize Information in More Ways 6.NS.C.6

A positive

x

-coordinate with a negative

y

-coordinate is Quadrant IV.

(+,\ -) \Rightarrow \text{Quadrant IV}

Right of the

y

-axis and below the

x

-axis.

Answer: Quadrant IV

4 · Reviewdoes it hold up?

Try $a=1$ , $b=-1$ : the first point is $(-1, -2)$ , which is in Quadrant III, and the second is $(4, -1)$ , which is in Quadrant IV.

Another way: Organize the facts in a sign table: from $ab<0$ and $b<a$ fill in $a:+$ , $b:-$ , then fill the rows $3a-b:+$ and $-a^2:-$ to read the quadrant directly.

Standardsmin grade 7

7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Deciding signs of products and sums of signed numbers.
6.NS.C.6 Rational numbers on the number line and coordinate plane — Reading a quadrant from the signs of a point's coordinates.

💡Takeaway. A quadrant is a pair of signs -- read the signs and the letters give themselves away.

Variant 7 medium

The point $(b-a, ab)$ lies in Quadrant III. Which quadrant contains the point $(-b^2, a-3b)$ ?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A point built from $a$ and $b$ sits in Quadrant III. Use that to pin down the signs of $a$ and $b$ , then decide where a second point built from them lands.

Givens

The point $(b-a, ab)$ is in Quadrant III.

Unknowns

The quadrant containing $(-b^2, a-3b)$ .

Constraints

In Quadrant III the $x$ -coordinate and the $y$ -coordinate are both negative.
Points on an axis belong to no quadrant, so every coordinate here is strictly positive or strictly negative.

2 · Planchoose the strategy

#15 Organize Information in More Ways · also uses: #2 Make a Systematic List

Turn the quadrant into two sign facts, then list the only two sign patterns a negative product allows and keep the one that fits the other fact. Once the signs of $a$ and $b$ are known, every coordinate of the new point has a definite sign.

3 · Execute5 carry out the plan

1Translate the quadrant into signs

#15 Organize Information in More Ways 6.NS.C.6

Quadrant III means the

x

-coordinate and the

y

-coordinate are both negative.

b-a<0, \quad ab<0

A quadrant is just a pair of signs.

2List the sign patterns for a negative product

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

A product is negative exactly when its two factors have opposite signs, so either

a

is positive and

b

negative, or the other way round.

ab<0 \Rightarrow (a>0,\ b<0) \text{ or } (a<0,\ b>0)

Only two cases are possible, so we can test both.

3Keep the case that fits $b-a<0$

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

b-a<0

means

b<a

. If

a

were negative and

b

positive,

b

would be the larger one, which breaks

b<a

. So

a>0

and

b<0

.

b<a \Rightarrow a>0,\ b<0

A positive number is always bigger than a negative one.

4Find the signs of the new coordinates

#15 Organize Information in More Ways 7.NS.A.2

b^2

is positive because

b\ne 0

, so

-b^2

is negative. Since

b<0

,

-3b

is positive, and

a-3b

is a positive number plus a positive number, so it is positive.

-b^2<0, \quad a-3b = a+(-3b)>0

Squaring removes the sign; multiplying a negative by

-3

makes it positive.

5Read off the quadrant

#15 Organize Information in More Ways 6.NS.C.6

A negative

x

-coordinate with a positive

y

-coordinate is Quadrant II.

(-,\ +) \Rightarrow \text{Quadrant II}

Left of the

y

-axis and above the

x

-axis.

Answer: Quadrant II

4 · Reviewdoes it hold up?

Try $a=1$ , $b=-1$ : the first point is $(-2, -1)$ , which is in Quadrant III, and the second is $(-1, 4)$ , which is in Quadrant II.

Another way: Organize the facts in a sign table: from $ab<0$ and $b<a$ fill in $a:+$ , $b:-$ , then fill the rows $-b^2:-$ and $a-3b:+$ to read the quadrant directly.

Standardsmin grade 7

7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Deciding signs of products and sums of signed numbers.
6.NS.C.6 Rational numbers on the number line and coordinate plane — Reading a quadrant from the signs of a point's coordinates.

💡Takeaway. A quadrant is a pair of signs -- read the signs and the letters give themselves away.

Variant 8 hard

The point $(ab, b-a)$ lies in Quadrant II. Which quadrant contains the point $(b^2, 4a-b)$ ?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A point built from $a$ and $b$ sits in Quadrant II. Use that to pin down the signs of $a$ and $b$ , then decide where a second point built from them lands.

Givens

The point $(ab, b-a)$ is in Quadrant II.

Unknowns

The quadrant containing $(b^2, 4a-b)$ .

Constraints

In Quadrant II the $x$ -coordinate is negative and the $y$ -coordinate is positive.
Points on an axis belong to no quadrant, so every coordinate here is strictly positive or strictly negative.

2 · Planchoose the strategy

#15 Organize Information in More Ways · also uses: #2 Make a Systematic List

Turn the quadrant into two sign facts, then list the only two sign patterns a negative product allows and keep the one that fits the other fact. Once the signs of $a$ and $b$ are known, every coordinate of the new point has a definite sign.

3 · Execute5 carry out the plan

1Translate the quadrant into signs

#15 Organize Information in More Ways 6.NS.C.6

Quadrant II means the

x

-coordinate is negative and the

y

-coordinate is positive.

ab<0, \quad b-a>0

A quadrant is just a pair of signs.

2List the sign patterns for a negative product

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

A product is negative exactly when its two factors have opposite signs, so either

a

is positive and

b

negative, or the other way round.

ab<0 \Rightarrow (a>0,\ b<0) \text{ or } (a<0,\ b>0)

Only two cases are possible, so we can test both.

3Keep the case that fits $b-a>0$

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

b-a>0

means

a<b

. If

b

were negative and

a

positive,

a

would be the larger one, which breaks

a<b

. So

b>0

and

a<0

.

a<b \Rightarrow b>0,\ a<0

A positive number is always bigger than a negative one.

4Find the signs of the new coordinates

#15 Organize Information in More Ways 7.NS.A.2

b^2

is positive because

b\ne 0

. Since

a<0

,

4a

is negative, and since

b>0

,

-b

is negative, and

4a-b

is a negative number plus a negative number, so it is negative.

b^2>0, \quad 4a-b = 4a+(-b)<0

Squaring removes the sign; then add two numbers of the same sign.

5Read off the quadrant

#15 Organize Information in More Ways 6.NS.C.6

A positive

x

-coordinate with a negative

y

-coordinate is Quadrant IV.

(+,\ -) \Rightarrow \text{Quadrant IV}

Right of the

y

-axis and below the

x

-axis.

Answer: Quadrant IV

4 · Reviewdoes it hold up?

Try $a=-1$ , $b=1$ : the first point is $(-1, 2)$ , which is in Quadrant II, and the second is $(1, -5)$ , which is in Quadrant IV.

Another way: Organize the facts in a sign table: from $ab<0$ and $a<b$ fill in $a:-$ , $b:+$ , then fill the rows $b^2:+$ and $4a-b:-$ to read the quadrant directly.

Standardsmin grade 7

7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Deciding signs of products and sums of signed numbers.
6.NS.C.6 Rational numbers on the number line and coordinate plane — Reading a quadrant from the signs of a point's coordinates.

💡Takeaway. A quadrant is a pair of signs -- read the signs and the letters give themselves away.

Variant 9 medium

The point $(ab, a-b)$ lies in Quadrant III. Which quadrant contains the point $(-b^2, b-4a)$ ?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A point built from $a$ and $b$ sits in Quadrant III. Use that to pin down the signs of $a$ and $b$ , then decide where a second point built from them lands.

Givens

The point $(ab, a-b)$ is in Quadrant III.

Unknowns

The quadrant containing $(-b^2, b-4a)$ .

Constraints

In Quadrant III the $x$ -coordinate and the $y$ -coordinate are both negative.
Points on an axis belong to no quadrant, so every coordinate here is strictly positive or strictly negative.

2 · Planchoose the strategy

#15 Organize Information in More Ways · also uses: #2 Make a Systematic List

Turn the quadrant into two sign facts, then list the only two sign patterns a negative product allows and keep the one that fits the other fact. Once the signs of $a$ and $b$ are known, every coordinate of the new point has a definite sign.

3 · Execute5 carry out the plan

1Translate the quadrant into signs

#15 Organize Information in More Ways 6.NS.C.6

Quadrant III means the

x

-coordinate and the

y

-coordinate are both negative.

ab<0, \quad a-b<0

A quadrant is just a pair of signs.

2List the sign patterns for a negative product

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

A product is negative exactly when its two factors have opposite signs, so either

a

is positive and

b

negative, or the other way round.

ab<0 \Rightarrow (a>0,\ b<0) \text{ or } (a<0,\ b>0)

Only two cases are possible, so we can test both.

3Keep the case that fits $a-b<0$

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

a-b<0

means

a<b

. If

b

were negative and

a

positive,

a

would be the larger one, which breaks

a<b

. So

b>0

and

a<0

.

a<b \Rightarrow b>0,\ a<0

A positive number is always bigger than a negative one.

4Find the signs of the new coordinates

#15 Organize Information in More Ways 7.NS.A.2

b^2

is positive because

b\ne 0

, so

-b^2

is negative. Since

a<0

,

-4a

is positive, and

b-4a

is a positive number plus a positive number, so it is positive.

-b^2<0, \quad b-4a = b+(-4a)>0

Squaring removes the sign, and the minus in front makes it negative; then add two numbers of the same sign.

5Read off the quadrant

#15 Organize Information in More Ways 6.NS.C.6

A negative

x

-coordinate with a positive

y

-coordinate is Quadrant II.

(-,\ +) \Rightarrow \text{Quadrant II}

Left of the

y

-axis and above the

x

-axis.

Answer: Quadrant II

4 · Reviewdoes it hold up?

Try $a=-1$ , $b=1$ : the first point is $(-1, -2)$ , which is in Quadrant III, and the second is $(-1, 5)$ , which is in Quadrant II.

Another way: Organize the facts in a sign table: from $ab<0$ and $a<b$ fill in $a:-$ , $b:+$ , then fill the rows $-b^2:-$ and $b-4a:+$ to read the quadrant directly.

Standardsmin grade 7

7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Deciding signs of products and sums of signed numbers.
6.NS.C.6 Rational numbers on the number line and coordinate plane — Reading a quadrant from the signs of a point's coordinates.

💡Takeaway. A quadrant is a pair of signs -- read the signs and the letters give themselves away.

Variant 10 hard

The point $(ab, a-b)$ lies in Quadrant II. Which quadrant contains the point $(b-5a, -b^2)$ ?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A point built from $a$ and $b$ sits in Quadrant II. Use that to pin down the signs of $a$ and $b$ , then decide where a second point built from them lands.

Givens

The point $(ab, a-b)$ is in Quadrant II.

Unknowns

The quadrant containing $(b-5a, -b^2)$ .

Constraints

In Quadrant II the $x$ -coordinate is negative and the $y$ -coordinate is positive.
Points on an axis belong to no quadrant, so every coordinate here is strictly positive or strictly negative.

2 · Planchoose the strategy

#15 Organize Information in More Ways · also uses: #2 Make a Systematic List

Turn the quadrant into two sign facts, then list the only two sign patterns a negative product allows and keep the one that fits the other fact. Once the signs of $a$ and $b$ are known, every coordinate of the new point has a definite sign.

3 · Execute5 carry out the plan

1Translate the quadrant into signs

#15 Organize Information in More Ways 6.NS.C.6

Quadrant II means the

x

-coordinate is negative and the

y

-coordinate is positive.

ab<0, \quad a-b>0

A quadrant is just a pair of signs.

2List the sign patterns for a negative product

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

A product is negative exactly when its two factors have opposite signs, so either

a

is positive and

b

negative, or the other way round.

ab<0 \Rightarrow (a>0,\ b<0) \text{ or } (a<0,\ b>0)

Only two cases are possible, so we can test both.

3Keep the case that fits $a-b>0$

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

a-b>0

means

b<a

. If

a

were negative and

b

positive,

b

would be the larger one, which breaks

b<a

. So

a>0

and

b<0

.

b<a \Rightarrow a>0,\ b<0

A positive number is always bigger than a negative one.

4Find the signs of the new coordinates

#15 Organize Information in More Ways 7.NS.A.2

Since

a>0

,

-5a

is negative, and

b-5a

is a negative number plus a negative number, so it is negative.

b^2

is positive because

b\ne 0

, so

-b^2

is negative.

b-5a = b+(-5a)<0, \quad -b^2<0

Squaring removes the sign, and the minus in front makes it negative; then add two numbers of the same sign.

5Read off the quadrant

#15 Organize Information in More Ways 6.NS.C.6

A negative

x

-coordinate with a negative

y

-coordinate is Quadrant III.

(-,\ -) \Rightarrow \text{Quadrant III}

Left of the

y

-axis and below the

x

-axis.

Answer: Quadrant III

4 · Reviewdoes it hold up?

Try $a=1$ , $b=-1$ : the first point is $(-1, 2)$ , which is in Quadrant II, and the second is $(-6, -1)$ , which is in Quadrant III.

Another way: Organize the facts in a sign table: from $ab<0$ and $b<a$ fill in $a:+$ , $b:-$ , then fill the rows $b-5a:-$ and $-b^2:-$ to read the quadrant directly.

Standardsmin grade 7

7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Deciding signs of products and sums of signed numbers.
6.NS.C.6 Rational numbers on the number line and coordinate plane — Reading a quadrant from the signs of a point's coordinates.

💡Takeaway. A quadrant is a pair of signs -- read the signs and the letters give themselves away.

Variant 11 hard

The point $(b-a, ab)$ lies in Quadrant III. Which quadrant contains the point $(3b-2a, a^2)$ ?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A point built from $a$ and $b$ sits in Quadrant III. Use that to pin down the signs of $a$ and $b$ , then decide where a second point built from them lands.

Givens

The point $(b-a, ab)$ is in Quadrant III.

Unknowns

The quadrant containing $(3b-2a, a^2)$ .

Constraints

In Quadrant III the $x$ -coordinate and the $y$ -coordinate are both negative.
Points on an axis belong to no quadrant, so every coordinate here is strictly positive or strictly negative.

2 · Planchoose the strategy

#15 Organize Information in More Ways · also uses: #2 Make a Systematic List

Turn the quadrant into two sign facts, then list the only two sign patterns a negative product allows and keep the one that fits the other fact. Once the signs of $a$ and $b$ are known, every coordinate of the new point has a definite sign.

3 · Execute5 carry out the plan

1Translate the quadrant into signs

#15 Organize Information in More Ways 6.NS.C.6

Quadrant III means the

x

-coordinate and the

y

-coordinate are both negative.

b-a<0, \quad ab<0

A quadrant is just a pair of signs.

2List the sign patterns for a negative product

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

A product is negative exactly when its two factors have opposite signs, so either

a

is positive and

b

negative, or the other way round.

ab<0 \Rightarrow (a>0,\ b<0) \text{ or } (a<0,\ b>0)

Only two cases are possible, so we can test both.

3Keep the case that fits $b-a<0$

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

b-a<0

means

b<a

. If

a

were negative and

b

positive,

b

would be the larger one, which breaks

b<a

. So

a>0

and

b<0

.

b<a \Rightarrow a>0,\ b<0

A positive number is always bigger than a negative one.

4Find the signs of the new coordinates

#15 Organize Information in More Ways 7.NS.A.2

Since

b<0

,

3b

is negative, and since

a>0

,

-2a

is negative, and

3b-2a

is a negative number plus a negative number, so it is negative.

a^2

is positive because

a\ne 0

.

3b-2a = 3b+(-2a)<0, \quad a^2>0

Squaring removes the sign; then add two numbers of the same sign.

5Read off the quadrant

#15 Organize Information in More Ways 6.NS.C.6

A negative

x

-coordinate with a positive

y

-coordinate is Quadrant II.

(-,\ +) \Rightarrow \text{Quadrant II}

Left of the

y

-axis and above the

x

-axis.

Answer: Quadrant II

4 · Reviewdoes it hold up?

Try $a=1$ , $b=-1$ : the first point is $(-2, -1)$ , which is in Quadrant III, and the second is $(-5, 1)$ , which is in Quadrant II.

Another way: Organize the facts in a sign table: from $ab<0$ and $b<a$ fill in $a:+$ , $b:-$ , then fill the rows $3b-2a:-$ and $a^2:+$ to read the quadrant directly.

Standardsmin grade 7

7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Deciding signs of products and sums of signed numbers.
6.NS.C.6 Rational numbers on the number line and coordinate plane — Reading a quadrant from the signs of a point's coordinates.

💡Takeaway. A quadrant is a pair of signs -- read the signs and the letters give themselves away.

Variant 12 hard

The point $(a-b, ab)$ lies in Quadrant III. Which quadrant contains the point $(2a-3b, -a^2)$ ?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

A point built from $a$ and $b$ sits in Quadrant III. Use that to pin down the signs of $a$ and $b$ , then decide where a second point built from them lands.

Givens

The point $(a-b, ab)$ is in Quadrant III.

Unknowns

The quadrant containing $(2a-3b, -a^2)$ .

Constraints

In Quadrant III the $x$ -coordinate and the $y$ -coordinate are both negative.
Points on an axis belong to no quadrant, so every coordinate here is strictly positive or strictly negative.

2 · Planchoose the strategy

#15 Organize Information in More Ways · also uses: #2 Make a Systematic List

Turn the quadrant into two sign facts, then list the only two sign patterns a negative product allows and keep the one that fits the other fact. Once the signs of $a$ and $b$ are known, every coordinate of the new point has a definite sign.

3 · Execute5 carry out the plan

1Translate the quadrant into signs

#15 Organize Information in More Ways 6.NS.C.6

Quadrant III means the

x

-coordinate and the

y

-coordinate are both negative.

a-b<0, \quad ab<0

A quadrant is just a pair of signs.

2List the sign patterns for a negative product

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

A product is negative exactly when its two factors have opposite signs, so either

a

is positive and

b

negative, or the other way round.

ab<0 \Rightarrow (a>0,\ b<0) \text{ or } (a<0,\ b>0)

Only two cases are possible, so we can test both.

3Keep the case that fits $a-b<0$

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

a-b<0

means

a<b

. If

b

were negative and

a

positive,

a

would be the larger one, which breaks

a<b

. So

b>0

and

a<0

.

a<b \Rightarrow b>0,\ a<0

A positive number is always bigger than a negative one.

4Find the signs of the new coordinates

#15 Organize Information in More Ways 7.NS.A.2

Since

a<0

,

2a

is negative, and since

b>0

,

-3b

is negative, and

2a-3b

is a negative number plus a negative number, so it is negative.

a^2

is positive because

a\ne 0

, so

-a^2

is negative.

2a-3b = 2a+(-3b)<0, \quad -a^2<0

Squaring removes the sign, and the minus in front makes it negative; then add two numbers of the same sign.

5Read off the quadrant

#15 Organize Information in More Ways 6.NS.C.6

A negative

x

-coordinate with a negative

y

-coordinate is Quadrant III.

(-,\ -) \Rightarrow \text{Quadrant III}

Left of the

y

-axis and below the

x

-axis.

Answer: Quadrant III

4 · Reviewdoes it hold up?

Try $a=-1$ , $b=1$ : the first point is $(-2, -1)$ , which is in Quadrant III, and the second is $(-5, -1)$ , which is in Quadrant III.

Another way: Organize the facts in a sign table: from $ab<0$ and $a<b$ fill in $a:-$ , $b:+$ , then fill the rows $2a-3b:-$ and $-a^2:-$ to read the quadrant directly.

Standardsmin grade 7

7.NS.A.2 Apply and extend understanding of multiplication and division of rational numbers — Deciding signs of products and sums of signed numbers.
6.NS.C.6 Rational numbers on the number line and coordinate plane — Reading a quadrant from the signs of a point's coordinates.

💡Takeaway. A quadrant is a pair of signs -- read the signs and the letters give themselves away.

Recover each letter's sign from a quadrant · 12 practice problems

Generated variants — 12

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Givens

Unknowns

Constraints

Where next?