Practice · Exactly one prime sum makes a function · Sensim Math

Variant 1 easy

For two variables $x$ and $y$ , the values of $x$ are $1$ , $2$ , $a$ and the values of $y$ are $4$ , $5$ . Each $x$ is matched to every $y$ that makes $x+y$ a prime number. Find the sum of all values of $a$ for which $y$ is a function of $x$ . ( $a$ is a natural number with $3 \le a \le 9$ .)

Your answer

Show solution

1 · Understandwhat's really being asked

Each $x$ goes to whichever of $y=4$ and $y=5$ makes $x+y$ prime. For $y$ to be a function of $x$ , every $x$ must get exactly one $y$ , so for each $x$ exactly one of $x+4$ and $x+5$ must be prime. Find the allowed $a$ from $3$ to $9$ and add them.

Givens

$x$ takes the values $1$ , $2$ , $a$ .
$y$ takes the values $4$ , $5$ .
$x$ is matched to $y$ when $x+y$ is prime.

Unknowns

The sum of all $a$ that make $y$ a function of $x$ .

Constraints

$a$ is a natural number with $3 \le a \le 9$ .

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #7 Identify Subproblems#3 Eliminate Possibilities

Split the job: first check the fixed inputs $1$ and $2$ , then test each candidate $a$ from $3$ to $9$ in a list, crossing out any $a$ that gets no $y$ or two $y$ values.

3 · Execute4 carry out the plan

1What a function needs here

#7 Identify Subproblems 8.F.A.1

An

x

with no prime sum has no partner, and an

x

with two prime sums has two partners. Either way

y

is not a function, so each

x

needs exactly one of

x+4

,

x+5

to be prime.

A function gives every input exactly one output, not zero and not two.

2Check x = 1 and x = 2

#2 Make a Systematic List 4.OA.B.4

For

x=1

:

1+4=5

is prime and

1+5=6

is not, so

1

goes to

4

. For

x=2

:

2+4=6

is not prime and

2+5=7

is, so

2

goes to

5

.

Both fixed inputs already behave, so only

a

matters.

3Test each a

#3 Eliminate Possibilities 4.OA.B.4

List

a+4

and

a+5

for

a=3

to

9

:

(7,8)

,

(8,9)

,

(9,10)

,

(10,11)

,

(11,12)

,

(12,13)

,

(13,14)

. For

a=4

and

a=5

neither sum is prime, so they are crossed out. Every other

a

has exactly one prime sum.

a+4

and

a+5

are consecutive, so one of them is even and bigger than

2

; they can never both be prime. The only way to fail is to have no prime at all.

4Add the allowed values

#2 Make a Systematic List 4.OA.B.4

The allowed values are

3

,

6

,

7

,

8

and

9

.

3+6+7+8+9=33

Seven candidates, two crossed out, five left.

Answer:

33

4 · Reviewdoes it hold up?

Spot-check $a=3$ : $3+5=8$ is not prime and $3+4=7$ is, so $3$ goes only to $4$ . Spot-check $a=4$ : $8$ and $9$ are both composite, so $4$ has no partner. The sum $33$ is less than $3+4+\dots+9=42$ , as it should be after removing $4$ and $5$ .

Another way: Count the complement: the sum of all candidates is $3+4+5+6+7+8+9=42$ ; remove the two failures $4+5=9$ to get $42-9=33$ .

Standardsmin grade 8

8.F.A.1 Understand that a function is a rule that assigns exactly one output to each input — Requiring each input to have exactly one output.
4.OA.B.4 Find all factor pairs and recognize multiples; determine prime or composite — Deciding whether each sum is prime or composite.

💡Takeaway. A function needs exactly one output per input, so check for 'none' as carefully as for 'two'.

Variant 2 easy

For two variables $x$ and $y$ , the values of $x$ are $1$ , $2$ , $a$ and the values of $y$ are $3$ , $4$ . Each $x$ is matched to every $y$ that makes $x+y$ a prime number. Find the sum of all values of $a$ for which $y$ is a function of $x$ . ( $a$ is a natural number with $3 \le a \le 10$ .)

Your answer

Show solution

1 · Understandwhat's really being asked

Each $x$ goes to whichever of $y=3$ and $y=4$ makes $x+y$ prime. For $y$ to be a function of $x$ , every $x$ must get exactly one $y$ , so for each $x$ exactly one of $x+3$ and $x+4$ must be prime. Find the allowed $a$ from $3$ to $10$ and add them.

Givens

$x$ takes the values $1$ , $2$ , $a$ .
$y$ takes the values $3$ , $4$ .
$x$ is matched to $y$ when $x+y$ is prime.

Unknowns

The sum of all $a$ that make $y$ a function of $x$ .

Constraints

$a$ is a natural number with $3 \le a \le 10$ .

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #7 Identify Subproblems#3 Eliminate Possibilities

Split the job: first check the fixed inputs $1$ and $2$ , then test each candidate $a$ from $3$ to $10$ in a list, crossing out any $a$ that gets no $y$ or two $y$ values.

3 · Execute4 carry out the plan

1What a function needs here

#7 Identify Subproblems 8.F.A.1

An

x

with no prime sum has no partner, and an

x

with two prime sums has two partners. Either way

y

is not a function, so each

x

needs exactly one of

x+3

,

x+4

to be prime.

A function gives every input exactly one output, not zero and not two.

2Check x = 1 and x = 2

#2 Make a Systematic List 4.OA.B.4

For

x=1

:

1+3=4

is not prime and

1+4=5

is, so

1

goes to

4

. For

x=2

:

2+3=5

is prime and

2+4=6

is not, so

2

goes to

3

.

Both fixed inputs already behave, so only

a

matters.

3Test each a

#3 Eliminate Possibilities 4.OA.B.4

List

a+3

and

a+4

for

a=3

to

10

:

(6,7)

,

(7,8)

,

(8,9)

,

(9,10)

,

(10,11)

,

(11,12)

,

(12,13)

,

(13,14)

. For

a=5

and

a=6

neither sum is prime, so they are crossed out. Every other

a

has exactly one prime sum.

a+3

and

a+4

are consecutive, so one of them is even and bigger than

2

; they can never both be prime. The only way to fail is to have no prime at all.

4Add the allowed values

#2 Make a Systematic List 4.OA.B.4

The allowed values are

3

,

4

,

7

,

8

,

9

and

10

.

3+4+7+8+9+10=41

Eight candidates, two crossed out, six left.

Answer:

41

4 · Reviewdoes it hold up?

Spot-check $a=3$ : $3+3=6$ is not prime and $3+4=7$ is, so $3$ goes only to $4$ . Spot-check $a=5$ : $8$ and $9$ are both composite, so $5$ has no partner. The sum $41$ is less than $3+4+\dots+10=52$ , as it should be after removing $5$ and $6$ .

Another way: Count the complement: the sum of all candidates is $3+4+5+6+7+8+9+10=52$ ; remove the two failures $5+6=11$ to get $52-11=41$ .

Standardsmin grade 8

8.F.A.1 Understand that a function is a rule that assigns exactly one output to each input — Requiring each input to have exactly one output.
4.OA.B.4 Find all factor pairs and recognize multiples; determine prime or composite — Deciding whether each sum is prime or composite.

💡Takeaway. A function needs exactly one output per input, so check for 'none' as carefully as for 'two'.

Variant 3 easy

For two variables $x$ and $y$ , the values of $x$ are $1$ , $2$ , $a$ and the values of $y$ are $10$ , $11$ . Each $x$ is matched to every $y$ that makes $x+y$ a prime number. Find the sum of all values of $a$ for which $y$ is a function of $x$ . ( $a$ is a natural number with $3 \le a \le 7$ .)

Your answer

Show solution

1 · Understandwhat's really being asked

Each $x$ goes to whichever of $y=10$ and $y=11$ makes $x+y$ prime. For $y$ to be a function of $x$ , every $x$ must get exactly one $y$ , so for each $x$ exactly one of $x+10$ and $x+11$ must be prime. Find the allowed $a$ from $3$ to $7$ and add them.

Givens

$x$ takes the values $1$ , $2$ , $a$ .
$y$ takes the values $10$ , $11$ .
$x$ is matched to $y$ when $x+y$ is prime.

Unknowns

The sum of all $a$ that make $y$ a function of $x$ .

Constraints

$a$ is a natural number with $3 \le a \le 7$ .

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #7 Identify Subproblems#3 Eliminate Possibilities

Split the job: first check the fixed inputs $1$ and $2$ , then test each candidate $a$ from $3$ to $7$ in a list, crossing out any $a$ that gets no $y$ or two $y$ values.

3 · Execute4 carry out the plan

1What a function needs here

#7 Identify Subproblems 8.F.A.1

An

x

with no prime sum has no partner, and an

x

with two prime sums has two partners. Either way

y

is not a function, so each

x

needs exactly one of

x+10

,

x+11

to be prime.

A function gives every input exactly one output, not zero and not two.

2Check x = 1 and x = 2

#2 Make a Systematic List 4.OA.B.4

For

x=1

:

1+10=11

is prime and

1+11=12

is not, so

1

goes to

10

. For

x=2

:

2+10=12

is not prime and

2+11=13

is, so

2

goes to

11

.

Both fixed inputs already behave, so only

a

matters.

3Test each a

#3 Eliminate Possibilities 4.OA.B.4

List

a+10

and

a+11

for

a=3

to

7

:

(13,14)

,

(14,15)

,

(15,16)

,

(16,17)

,

(17,18)

. For

a=4

and

a=5

neither sum is prime, so they are crossed out. Every other

a

has exactly one prime sum.

a+10

and

a+11

are consecutive, so one of them is even and bigger than

2

; they can never both be prime. The only way to fail is to have no prime at all.

4Add the allowed values

#2 Make a Systematic List 4.OA.B.4

The allowed values are

3

,

6

and

7

.

3+6+7=16

five candidates, two crossed out, three left.

Answer:

16

4 · Reviewdoes it hold up?

Spot-check $a=3$ : $3+11=14$ is not prime and $3+10=13$ is, so $3$ goes only to $10$ . Spot-check $a=4$ : $14$ and $15$ are both composite, so $4$ has no partner. The sum $16$ is less than $3+4+\dots+7=25$ , as it should be after removing $4$ and $5$ .

Another way: Count the complement: the sum of all candidates is $3+4+5+6+7=25$ ; remove the two failures $4+5=9$ to get $25-9=16$ .

Standardsmin grade 8

8.F.A.1 Understand that a function is a rule that assigns exactly one output to each input — Requiring each input to have exactly one output.
4.OA.B.4 Find all factor pairs and recognize multiples; determine prime or composite — Deciding whether each sum is prime or composite.

💡Takeaway. A function needs exactly one output per input, so check for 'none' as carefully as for 'two'.

Variant 4 easy

For two variables $x$ and $y$ , the values of $x$ are $1$ , $2$ , $a$ and the values of $y$ are $11$ , $12$ . Each $x$ is matched to every $y$ that makes $x+y$ a prime number. Find the sum of all values of $a$ for which $y$ is a function of $x$ . ( $a$ is a natural number with $4 \le a \le 12$ .)

Your answer

Show solution

1 · Understandwhat's really being asked

Each $x$ goes to whichever of $y=11$ and $y=12$ makes $x+y$ prime. For $y$ to be a function of $x$ , every $x$ must get exactly one $y$ , so for each $x$ exactly one of $x+11$ and $x+12$ must be prime. Find the allowed $a$ from $4$ to $12$ and add them.

Givens

$x$ takes the values $1$ , $2$ , $a$ .
$y$ takes the values $11$ , $12$ .
$x$ is matched to $y$ when $x+y$ is prime.

Unknowns

The sum of all $a$ that make $y$ a function of $x$ .

Constraints

$a$ is a natural number with $4 \le a \le 12$ .

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #7 Identify Subproblems#3 Eliminate Possibilities

Split the job: first check the fixed inputs $1$ and $2$ , then test each candidate $a$ from $4$ to $12$ in a list, crossing out any $a$ that gets no $y$ or two $y$ values.

3 · Execute4 carry out the plan

1What a function needs here

#7 Identify Subproblems 8.F.A.1

An

x

with no prime sum has no partner, and an

x

with two prime sums has two partners. Either way

y

is not a function, so each

x

needs exactly one of

x+11

,

x+12

to be prime.

A function gives every input exactly one output, not zero and not two.

2Check x = 1 and x = 2

#2 Make a Systematic List 4.OA.B.4

For

x=1

:

1+11=12

is not prime and

1+12=13

is, so

1

goes to

12

. For

x=2

:

2+11=13

is prime and

2+12=14

is not, so

2

goes to

11

.

Both fixed inputs already behave, so only

a

matters.

3Test each a

#3 Eliminate Possibilities 4.OA.B.4

List

a+11

and

a+12

for

a=4

to

12

:

(15,16)

,

(16,17)

,

(17,18)

,

(18,19)

,

(19,20)

,

(20,21)

,

(21,22)

,

(22,23)

,

(23,24)

. For

a=4

,

a=9

and

a=10

neither sum is prime, so they are crossed out. Every other

a

has exactly one prime sum.

a+11

and

a+12

are consecutive, so one of them is even and bigger than

2

; they can never both be prime. The only way to fail is to have no prime at all.

4Add the allowed values

#2 Make a Systematic List 4.OA.B.4

The allowed values are

5

,

6

,

7

,

8

,

11

and

12

.

5+6+7+8+11+12=49

Nine candidates, three crossed out, six left.

Answer:

49

4 · Reviewdoes it hold up?

Spot-check $a=5$ : $5+11=16$ is not prime and $5+12=17$ is, so $5$ goes only to $12$ . Spot-check $a=4$ : $15$ and $16$ are both composite, so $4$ has no partner. The sum $49$ is less than $4+5+\dots+12=72$ , as it should be after removing $4$ , $9$ and $10$ .

Another way: Count the complement: the sum of all candidates is $4+5+6+7+8+9+10+11+12=72$ ; remove the three failures $4+9+10=23$ to get $72-23=49$ .

Standardsmin grade 8

8.F.A.1 Understand that a function is a rule that assigns exactly one output to each input — Requiring each input to have exactly one output.
4.OA.B.4 Find all factor pairs and recognize multiples; determine prime or composite — Deciding whether each sum is prime or composite.

💡Takeaway. A function needs exactly one output per input, so check for 'none' as carefully as for 'two'.

Variant 5 medium

For two variables $x$ and $y$ , the values of $x$ are $2$ , $3$ , $a$ and the values of $y$ are $8$ , $9$ . Each $x$ is matched to every $y$ that makes $x+y$ a prime number. Find the sum of all values of $a$ for which $y$ is a function of $x$ . ( $a$ is a natural number with $6 \le a \le 12$ .)

Your answer

Show solution

1 · Understandwhat's really being asked

Each $x$ goes to whichever of $y=8$ and $y=9$ makes $x+y$ prime. For $y$ to be a function of $x$ , every $x$ must get exactly one $y$ , so for each $x$ exactly one of $x+8$ and $x+9$ must be prime. Find the allowed $a$ from $6$ to $12$ and add them.

Givens

$x$ takes the values $2$ , $3$ , $a$ .
$y$ takes the values $8$ , $9$ .
$x$ is matched to $y$ when $x+y$ is prime.

Unknowns

The sum of all $a$ that make $y$ a function of $x$ .

Constraints

$a$ is a natural number with $6 \le a \le 12$ .

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #7 Identify Subproblems#3 Eliminate Possibilities

Split the job: first check the fixed inputs $2$ and $3$ , then test each candidate $a$ from $6$ to $12$ in a list, crossing out any $a$ that gets no $y$ or two $y$ values.

3 · Execute4 carry out the plan

1What a function needs here

#7 Identify Subproblems 8.F.A.1

An

x

with no prime sum has no partner, and an

x

with two prime sums has two partners. Either way

y

is not a function, so each

x

needs exactly one of

x+8

,

x+9

to be prime.

A function gives every input exactly one output, not zero and not two.

2Check x = 2 and x = 3

#2 Make a Systematic List 4.OA.B.4

For

x=2

:

2+8=10

is not prime and

2+9=11

is, so

2

goes to

9

. For

x=3

:

3+8=11

is prime and

3+9=12

is not, so

3

goes to

8

.

Both fixed inputs already behave, so only

a

matters.

3Test each a

#3 Eliminate Possibilities 4.OA.B.4

List

a+8

and

a+9

for

a=6

to

12

:

(14,15)

,

(15,16)

,

(16,17)

,

(17,18)

,

(18,19)

,

(19,20)

,

(20,21)

. For

a=6

,

a=7

and

a=12

neither sum is prime, so they are crossed out. Every other

a

has exactly one prime sum.

a+8

and

a+9

are consecutive, so one of them is even and bigger than

2

; they can never both be prime. The only way to fail is to have no prime at all.

4Add the allowed values

#2 Make a Systematic List 4.OA.B.4

The allowed values are

8

,

9

,

10

and

11

.

8+9+10+11=38

Seven candidates, three crossed out, four left.

Answer:

38

4 · Reviewdoes it hold up?

Spot-check $a=8$ : $8+8=16$ is not prime and $8+9=17$ is, so $8$ goes only to $9$ . Spot-check $a=6$ : $14$ and $15$ are both composite, so $6$ has no partner. The sum $38$ is less than $6+7+\dots+12=63$ , as it should be after removing $6$ , $7$ and $12$ .

Another way: Count the complement: the sum of all candidates is $6+7+8+9+10+11+12=63$ ; remove the three failures $6+7+12=25$ to get $63-25=38$ .

Standardsmin grade 8

8.F.A.1 Understand that a function is a rule that assigns exactly one output to each input — Requiring each input to have exactly one output.
4.OA.B.4 Find all factor pairs and recognize multiples; determine prime or composite — Deciding whether each sum is prime or composite.

💡Takeaway. A function needs exactly one output per input, so check for 'none' as carefully as for 'two'.

Variant 6 medium

For two variables $x$ and $y$ , the values of $x$ are $1$ , $3$ , $a$ and the values of $y$ are $10$ , $11$ . Each $x$ is matched to every $y$ that makes $x+y$ a prime number. Find the sum of all values of $a$ for which $y$ is a function of $x$ . ( $a$ is a natural number with $4 \le a \le 13$ .)

Your answer

Show solution

1 · Understandwhat's really being asked

Each $x$ goes to whichever of $y=10$ and $y=11$ makes $x+y$ prime. For $y$ to be a function of $x$ , every $x$ must get exactly one $y$ , so for each $x$ exactly one of $x+10$ and $x+11$ must be prime. Find the allowed $a$ from $4$ to $13$ and add them.

Givens

$x$ takes the values $1$ , $3$ , $a$ .
$y$ takes the values $10$ , $11$ .
$x$ is matched to $y$ when $x+y$ is prime.

Unknowns

The sum of all $a$ that make $y$ a function of $x$ .

Constraints

$a$ is a natural number with $4 \le a \le 13$ .

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #7 Identify Subproblems#3 Eliminate Possibilities

Split the job: first check the fixed inputs $1$ and $3$ , then test each candidate $a$ from $4$ to $13$ in a list, crossing out any $a$ that gets no $y$ or two $y$ values.

3 · Execute4 carry out the plan

1What a function needs here

#7 Identify Subproblems 8.F.A.1

An

x

with no prime sum has no partner, and an

x

with two prime sums has two partners. Either way

y

is not a function, so each

x

needs exactly one of

x+10

,

x+11

to be prime.

A function gives every input exactly one output, not zero and not two.

2Check x = 1 and x = 3

#2 Make a Systematic List 4.OA.B.4

For

x=1

:

1+10=11

is prime and

1+11=12

is not, so

1

goes to

10

. For

x=3

:

3+10=13

is prime and

3+11=14

is not, so

3

goes to

10

.

Both fixed inputs already behave, so only

a

matters.

3Test each a

#3 Eliminate Possibilities 4.OA.B.4

List

a+10

and

a+11

for

a=4

to

13

:

(14,15)

,

(15,16)

,

(16,17)

,

(17,18)

,

(18,19)

,

(19,20)

,

(20,21)

,

(21,22)

,

(22,23)

,

(23,24)

. For

a=4

,

a=5

,

a=10

and

a=11

neither sum is prime, so they are crossed out. Every other

a

has exactly one prime sum.

a+10

and

a+11

are consecutive, so one of them is even and bigger than

2

; they can never both be prime. The only way to fail is to have no prime at all.

4Add the allowed values

#2 Make a Systematic List 4.OA.B.4

The allowed values are

6

,

7

,

8

,

9

,

12

and

13

.

6+7+8+9+12+13=55

Ten candidates, four crossed out, six left.

Answer:

55

4 · Reviewdoes it hold up?

Spot-check $a=6$ : $6+10=16$ is not prime and $6+11=17$ is, so $6$ goes only to $11$ . Spot-check $a=4$ : $14$ and $15$ are both composite, so $4$ has no partner. The sum $55$ is less than $4+5+\dots+13=85$ , as it should be after removing $4$ , $5$ , $10$ and $11$ .

Another way: Count the complement: the sum of all candidates is $4+5+6+7+8+9+10+11+12+13=85$ ; remove the four failures $4+5+10+11=30$ to get $85-30=55$ .

Standardsmin grade 8

8.F.A.1 Understand that a function is a rule that assigns exactly one output to each input — Requiring each input to have exactly one output.
4.OA.B.4 Find all factor pairs and recognize multiples; determine prime or composite — Deciding whether each sum is prime or composite.

💡Takeaway. A function needs exactly one output per input, so check for 'none' as carefully as for 'two'.

Variant 7 medium

For two variables $x$ and $y$ , the values of $x$ are $2$ , $4$ , $a$ and the values of $y$ are $8$ , $9$ . Each $x$ is matched to every $y$ that makes $x+y$ a prime number. Find the sum of all values of $a$ for which $y$ is a function of $x$ . ( $a$ is a natural number with $6 \le a \le 14$ .)

Your answer

Show solution

1 · Understandwhat's really being asked

Each $x$ goes to whichever of $y=8$ and $y=9$ makes $x+y$ prime. For $y$ to be a function of $x$ , every $x$ must get exactly one $y$ , so for each $x$ exactly one of $x+8$ and $x+9$ must be prime. Find the allowed $a$ from $6$ to $14$ and add them.

Givens

$x$ takes the values $2$ , $4$ , $a$ .
$y$ takes the values $8$ , $9$ .
$x$ is matched to $y$ when $x+y$ is prime.

Unknowns

The sum of all $a$ that make $y$ a function of $x$ .

Constraints

$a$ is a natural number with $6 \le a \le 14$ .

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #7 Identify Subproblems#3 Eliminate Possibilities

Split the job: first check the fixed inputs $2$ and $4$ , then test each candidate $a$ from $6$ to $14$ in a list, crossing out any $a$ that gets no $y$ or two $y$ values.

3 · Execute4 carry out the plan

1What a function needs here

#7 Identify Subproblems 8.F.A.1

An

x

with no prime sum has no partner, and an

x

with two prime sums has two partners. Either way

y

is not a function, so each

x

needs exactly one of

x+8

,

x+9

to be prime.

A function gives every input exactly one output, not zero and not two.

2Check x = 2 and x = 4

#2 Make a Systematic List 4.OA.B.4

For

x=2

:

2+8=10

is not prime and

2+9=11

is, so

2

goes to

9

. For

x=4

:

4+8=12

is not prime and

4+9=13

is, so

4

goes to

9

.

Both fixed inputs already behave, so only

a

matters.

3Test each a

#3 Eliminate Possibilities 4.OA.B.4

List

a+8

and

a+9

for

a=6

to

14

:

(14,15)

,

(15,16)

,

(16,17)

,

(17,18)

,

(18,19)

,

(19,20)

,

(20,21)

,

(21,22)

,

(22,23)

. For

a=6

,

a=7

,

a=12

and

a=13

neither sum is prime, so they are crossed out. Every other

a

has exactly one prime sum.

a+8

and

a+9

are consecutive, so one of them is even and bigger than

2

; they can never both be prime. The only way to fail is to have no prime at all.

4Add the allowed values

#2 Make a Systematic List 4.OA.B.4

The allowed values are

8

,

9

,

10

,

11

and

14

.

8+9+10+11+14=52

Nine candidates, four crossed out, five left.

Answer:

52

4 · Reviewdoes it hold up?

Spot-check $a=8$ : $8+8=16$ is not prime and $8+9=17$ is, so $8$ goes only to $9$ . Spot-check $a=6$ : $14$ and $15$ are both composite, so $6$ has no partner. The sum $52$ is less than $6+7+\dots+14=90$ , as it should be after removing $6$ , $7$ , $12$ and $13$ .

Another way: Count the complement: the sum of all candidates is $6+7+8+9+10+11+12+13+14=90$ ; remove the four failures $6+7+12+13=38$ to get $90-38=52$ .

Standardsmin grade 8

8.F.A.1 Understand that a function is a rule that assigns exactly one output to each input — Requiring each input to have exactly one output.
4.OA.B.4 Find all factor pairs and recognize multiples; determine prime or composite — Deciding whether each sum is prime or composite.

💡Takeaway. A function needs exactly one output per input, so check for 'none' as carefully as for 'two'.

Variant 8 medium

For two variables $x$ and $y$ , the values of $x$ are $1$ , $3$ , $a$ and the values of $y$ are $3$ , $4$ . Each $x$ is matched to every $y$ that makes $x+y$ a prime number. Find the sum of all values of $a$ for which $y$ is a function of $x$ . ( $a$ is a natural number with $6 \le a \le 15$ .)

Your answer

Show solution

1 · Understandwhat's really being asked

Each $x$ goes to whichever of $y=3$ and $y=4$ makes $x+y$ prime. For $y$ to be a function of $x$ , every $x$ must get exactly one $y$ , so for each $x$ exactly one of $x+3$ and $x+4$ must be prime. Find the allowed $a$ from $6$ to $15$ and add them.

Givens

$x$ takes the values $1$ , $3$ , $a$ .
$y$ takes the values $3$ , $4$ .
$x$ is matched to $y$ when $x+y$ is prime.

Unknowns

The sum of all $a$ that make $y$ a function of $x$ .

Constraints

$a$ is a natural number with $6 \le a \le 15$ .

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #7 Identify Subproblems#3 Eliminate Possibilities

Split the job: first check the fixed inputs $1$ and $3$ , then test each candidate $a$ from $6$ to $15$ in a list, crossing out any $a$ that gets no $y$ or two $y$ values.

3 · Execute4 carry out the plan

1What a function needs here

#7 Identify Subproblems 8.F.A.1

An

x

with no prime sum has no partner, and an

x

with two prime sums has two partners. Either way

y

is not a function, so each

x

needs exactly one of

x+3

,

x+4

to be prime.

A function gives every input exactly one output, not zero and not two.

2Check x = 1 and x = 3

#2 Make a Systematic List 4.OA.B.4

For

x=1

:

1+3=4

is not prime and

1+4=5

is, so

1

goes to

4

. For

x=3

:

3+3=6

is not prime and

3+4=7

is, so

3

goes to

4

.

Both fixed inputs already behave, so only

a

matters.

3Test each a

#3 Eliminate Possibilities 4.OA.B.4

List

a+3

and

a+4

for

a=6

to

15

:

(9,10)

,

(10,11)

,

(11,12)

,

(12,13)

,

(13,14)

,

(14,15)

,

(15,16)

,

(16,17)

,

(17,18)

,

(18,19)

. For

a=6

,

a=11

and

a=12

neither sum is prime, so they are crossed out. Every other

a

has exactly one prime sum.

a+3

and

a+4

are consecutive, so one of them is even and bigger than

2

; they can never both be prime. The only way to fail is to have no prime at all.

4Add the allowed values

#2 Make a Systematic List 4.OA.B.4

The allowed values are

7

,

8

,

9

,

10

,

13

,

14

and

15

.

7+8+9+10+13+14+15=76

Ten candidates, three crossed out, seven left.

Answer:

76

4 · Reviewdoes it hold up?

Spot-check $a=7$ : $7+3=10$ is not prime and $7+4=11$ is, so $7$ goes only to $4$ . Spot-check $a=6$ : $9$ and $10$ are both composite, so $6$ has no partner. The sum $76$ is less than $6+7+\dots+15=105$ , as it should be after removing $6$ , $11$ and $12$ .

Another way: Count the complement: the sum of all candidates is $6+7+8+9+10+11+12+13+14+15=105$ ; remove the three failures $6+11+12=29$ to get $105-29=76$ .

Standardsmin grade 8

8.F.A.1 Understand that a function is a rule that assigns exactly one output to each input — Requiring each input to have exactly one output.
4.OA.B.4 Find all factor pairs and recognize multiples; determine prime or composite — Deciding whether each sum is prime or composite.

💡Takeaway. A function needs exactly one output per input, so check for 'none' as carefully as for 'two'.

Variant 9 hard

For two variables $x$ and $y$ , the values of $x$ are $2$ , $3$ , $a$ and the values of $y$ are $9$ , $10$ . Each $x$ is matched to every $y$ that makes $x+y$ a prime number. Find the sum of all values of $a$ for which $y$ is a function of $x$ . ( $a$ is a natural number with $6 \le a \le 15$ .)

Your answer

Show solution

1 · Understandwhat's really being asked

Each $x$ goes to whichever of $y=9$ and $y=10$ makes $x+y$ prime. For $y$ to be a function of $x$ , every $x$ must get exactly one $y$ , so for each $x$ exactly one of $x+9$ and $x+10$ must be prime. Find the allowed $a$ from $6$ to $15$ and add them.

Givens

$x$ takes the values $2$ , $3$ , $a$ .
$y$ takes the values $9$ , $10$ .
$x$ is matched to $y$ when $x+y$ is prime.

Unknowns

The sum of all $a$ that make $y$ a function of $x$ .

Constraints

$a$ is a natural number with $6 \le a \le 15$ .

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #7 Identify Subproblems#3 Eliminate Possibilities

Split the job: first check the fixed inputs $2$ and $3$ , then test each candidate $a$ from $6$ to $15$ in a list, crossing out any $a$ that gets no $y$ or two $y$ values.

3 · Execute4 carry out the plan

1What a function needs here

#7 Identify Subproblems 8.F.A.1

An

x

with no prime sum has no partner, and an

x

with two prime sums has two partners. Either way

y

is not a function, so each

x

needs exactly one of

x+9

,

x+10

to be prime.

A function gives every input exactly one output, not zero and not two.

2Check x = 2 and x = 3

#2 Make a Systematic List 4.OA.B.4

For

x=2

:

2+9=11

is prime and

2+10=12

is not, so

2

goes to

9

. For

x=3

:

3+9=12

is not prime and

3+10=13

is, so

3

goes to

10

.

Both fixed inputs already behave, so only

a

matters.

3Test each a

#3 Eliminate Possibilities 4.OA.B.4

List

a+9

and

a+10

for

a=6

to

15

:

(15,16)

,

(16,17)

,

(17,18)

,

(18,19)

,

(19,20)

,

(20,21)

,

(21,22)

,

(22,23)

,

(23,24)

,

(24,25)

. For

a=6

,

a=11

,

a=12

and

a=15

neither sum is prime, so they are crossed out. Every other

a

has exactly one prime sum.

a+9

and

a+10

are consecutive, so one of them is even and bigger than

2

; they can never both be prime. The only way to fail is to have no prime at all.

4Add the allowed values

#2 Make a Systematic List 4.OA.B.4

The allowed values are

7

,

8

,

9

,

10

,

13

and

14

.

7+8+9+10+13+14=61

Ten candidates, four crossed out, six left.

Answer:

61

4 · Reviewdoes it hold up?

Spot-check $a=7$ : $7+9=16$ is not prime and $7+10=17$ is, so $7$ goes only to $10$ . Spot-check $a=6$ : $15$ and $16$ are both composite, so $6$ has no partner. The sum $61$ is less than $6+7+\dots+15=105$ , as it should be after removing $6$ , $11$ , $12$ and $15$ .

Another way: Count the complement: the sum of all candidates is $6+7+8+9+10+11+12+13+14+15=105$ ; remove the four failures $6+11+12+15=44$ to get $105-44=61$ .

Standardsmin grade 8

8.F.A.1 Understand that a function is a rule that assigns exactly one output to each input — Requiring each input to have exactly one output.
4.OA.B.4 Find all factor pairs and recognize multiples; determine prime or composite — Deciding whether each sum is prime or composite.

💡Takeaway. A function needs exactly one output per input, so check for 'none' as carefully as for 'two'.

Variant 10 hard

For two variables $x$ and $y$ , the values of $x$ are $2$ , $4$ , $a$ and the values of $y$ are $14$ , $15$ . Each $x$ is matched to every $y$ that makes $x+y$ a prime number. Find the sum of all values of $a$ for which $y$ is a function of $x$ . ( $a$ is a natural number with $5 \le a \le 9$ .)

Your answer

Show solution

1 · Understandwhat's really being asked

Each $x$ goes to whichever of $y=14$ and $y=15$ makes $x+y$ prime. For $y$ to be a function of $x$ , every $x$ must get exactly one $y$ , so for each $x$ exactly one of $x+14$ and $x+15$ must be prime. Find the allowed $a$ from $5$ to $9$ and add them.

Givens

$x$ takes the values $2$ , $4$ , $a$ .
$y$ takes the values $14$ , $15$ .
$x$ is matched to $y$ when $x+y$ is prime.

Unknowns

The sum of all $a$ that make $y$ a function of $x$ .

Constraints

$a$ is a natural number with $5 \le a \le 9$ .

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #7 Identify Subproblems#3 Eliminate Possibilities

Split the job: first check the fixed inputs $2$ and $4$ , then test each candidate $a$ from $5$ to $9$ in a list, crossing out any $a$ that gets no $y$ or two $y$ values.

3 · Execute4 carry out the plan

1What a function needs here

#7 Identify Subproblems 8.F.A.1

An

x

with no prime sum has no partner, and an

x

with two prime sums has two partners. Either way

y

is not a function, so each

x

needs exactly one of

x+14

,

x+15

to be prime.

A function gives every input exactly one output, not zero and not two.

2Check x = 2 and x = 4

#2 Make a Systematic List 4.OA.B.4

For

x=2

:

2+14=16

is not prime and

2+15=17

is, so

2

goes to

15

. For

x=4

:

4+14=18

is not prime and

4+15=19

is, so

4

goes to

15

.

Both fixed inputs already behave, so only

a

matters.

3Test each a

#3 Eliminate Possibilities 4.OA.B.4

List

a+14

and

a+15

for

a=5

to

9

:

(19,20)

,

(20,21)

,

(21,22)

,

(22,23)

,

(23,24)

. For

a=6

and

a=7

neither sum is prime, so they are crossed out. Every other

a

has exactly one prime sum.

a+14

and

a+15

are consecutive, so one of them is even and bigger than

2

; they can never both be prime. The only way to fail is to have no prime at all.

4Add the allowed values

#2 Make a Systematic List 4.OA.B.4

The allowed values are

5

,

8

and

9

.

5+8+9=22

five candidates, two crossed out, three left.

Answer:

22

4 · Reviewdoes it hold up?

Spot-check $a=5$ : $5+15=20$ is not prime and $5+14=19$ is, so $5$ goes only to $14$ . Spot-check $a=6$ : $20$ and $21$ are both composite, so $6$ has no partner. The sum $22$ is less than $5+6+\dots+9=35$ , as it should be after removing $6$ and $7$ .

Another way: Count the complement: the sum of all candidates is $5+6+7+8+9=35$ ; remove the two failures $6+7=13$ to get $35-13=22$ .

Standardsmin grade 8

8.F.A.1 Understand that a function is a rule that assigns exactly one output to each input — Requiring each input to have exactly one output.
4.OA.B.4 Find all factor pairs and recognize multiples; determine prime or composite — Deciding whether each sum is prime or composite.

💡Takeaway. A function needs exactly one output per input, so check for 'none' as carefully as for 'two'.

Variant 11 hard

For two variables $x$ and $y$ , the values of $x$ are $1$ , $2$ , $a$ and the values of $y$ are $15$ , $16$ . Each $x$ is matched to every $y$ that makes $x+y$ a prime number. Find the sum of all values of $a$ for which $y$ is a function of $x$ . ( $a$ is a natural number with $4 \le a \le 8$ .)

Your answer

Show solution

1 · Understandwhat's really being asked

Each $x$ goes to whichever of $y=15$ and $y=16$ makes $x+y$ prime. For $y$ to be a function of $x$ , every $x$ must get exactly one $y$ , so for each $x$ exactly one of $x+15$ and $x+16$ must be prime. Find the allowed $a$ from $4$ to $8$ and add them.

Givens

$x$ takes the values $1$ , $2$ , $a$ .
$y$ takes the values $15$ , $16$ .
$x$ is matched to $y$ when $x+y$ is prime.

Unknowns

The sum of all $a$ that make $y$ a function of $x$ .

Constraints

$a$ is a natural number with $4 \le a \le 8$ .

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #7 Identify Subproblems#3 Eliminate Possibilities

Split the job: first check the fixed inputs $1$ and $2$ , then test each candidate $a$ from $4$ to $8$ in a list, crossing out any $a$ that gets no $y$ or two $y$ values.

3 · Execute4 carry out the plan

1What a function needs here

#7 Identify Subproblems 8.F.A.1

An

x

with no prime sum has no partner, and an

x

with two prime sums has two partners. Either way

y

is not a function, so each

x

needs exactly one of

x+15

,

x+16

to be prime.

A function gives every input exactly one output, not zero and not two.

2Check x = 1 and x = 2

#2 Make a Systematic List 4.OA.B.4

For

x=1

:

1+15=16

is not prime and

1+16=17

is, so

1

goes to

16

. For

x=2

:

2+15=17

is prime and

2+16=18

is not, so

2

goes to

15

.

Both fixed inputs already behave, so only

a

matters.

3Test each a

#3 Eliminate Possibilities 4.OA.B.4

List

a+15

and

a+16

for

a=4

to

8

:

(19,20)

,

(20,21)

,

(21,22)

,

(22,23)

,

(23,24)

. For

a=5

and

a=6

neither sum is prime, so they are crossed out. Every other

a

has exactly one prime sum.

a+15

and

a+16

are consecutive, so one of them is even and bigger than

2

; they can never both be prime. The only way to fail is to have no prime at all.

4Add the allowed values

#2 Make a Systematic List 4.OA.B.4

The allowed values are

4

,

7

and

8

.

4+7+8=19

five candidates, two crossed out, three left.

Answer:

19

4 · Reviewdoes it hold up?

Spot-check $a=4$ : $4+16=20$ is not prime and $4+15=19$ is, so $4$ goes only to $15$ . Spot-check $a=5$ : $20$ and $21$ are both composite, so $5$ has no partner. The sum $19$ is less than $4+5+\dots+8=30$ , as it should be after removing $5$ and $6$ .

Another way: Count the complement: the sum of all candidates is $4+5+6+7+8=30$ ; remove the two failures $5+6=11$ to get $30-11=19$ .

Standardsmin grade 8

8.F.A.1 Understand that a function is a rule that assigns exactly one output to each input — Requiring each input to have exactly one output.
4.OA.B.4 Find all factor pairs and recognize multiples; determine prime or composite — Deciding whether each sum is prime or composite.

💡Takeaway. A function needs exactly one output per input, so check for 'none' as carefully as for 'two'.

Variant 12 hard

For two variables $x$ and $y$ , the values of $x$ are $1$ , $2$ , $a$ and the values of $y$ are $16$ , $17$ . Each $x$ is matched to every $y$ that makes $x+y$ a prime number. Find the sum of all values of $a$ for which $y$ is a function of $x$ . ( $a$ is a natural number with $6 \le a \le 12$ .)

Your answer

Show solution

1 · Understandwhat's really being asked

Each $x$ goes to whichever of $y=16$ and $y=17$ makes $x+y$ prime. For $y$ to be a function of $x$ , every $x$ must get exactly one $y$ , so for each $x$ exactly one of $x+16$ and $x+17$ must be prime. Find the allowed $a$ from $6$ to $12$ and add them.

Givens

$x$ takes the values $1$ , $2$ , $a$ .
$y$ takes the values $16$ , $17$ .
$x$ is matched to $y$ when $x+y$ is prime.

Unknowns

The sum of all $a$ that make $y$ a function of $x$ .

Constraints

$a$ is a natural number with $6 \le a \le 12$ .

2 · Planchoose the strategy

#2 Make a Systematic List · also uses: #7 Identify Subproblems#3 Eliminate Possibilities

Split the job: first check the fixed inputs $1$ and $2$ , then test each candidate $a$ from $6$ to $12$ in a list, crossing out any $a$ that gets no $y$ or two $y$ values.

3 · Execute4 carry out the plan

1What a function needs here

#7 Identify Subproblems 8.F.A.1

An

x

with no prime sum has no partner, and an

x

with two prime sums has two partners. Either way

y

is not a function, so each

x

needs exactly one of

x+16

,

x+17

to be prime.

A function gives every input exactly one output, not zero and not two.

2Check x = 1 and x = 2

#2 Make a Systematic List 4.OA.B.4

For

x=1

:

1+16=17

is prime and

1+17=18

is not, so

1

goes to

16

. For

x=2

:

2+16=18

is not prime and

2+17=19

is, so

2

goes to

17

.

Both fixed inputs already behave, so only

a

matters.

3Test each a

#3 Eliminate Possibilities 4.OA.B.4

List

a+16

and

a+17

for

a=6

to

12

:

(22,23)

,

(23,24)

,

(24,25)

,

(25,26)

,

(26,27)

,

(27,28)

,

(28,29)

. For

a=8

,

a=9

,

a=10

and

a=11

neither sum is prime, so they are crossed out. Every other

a

has exactly one prime sum.

a+16

and

a+17

are consecutive, so one of them is even and bigger than

2

; they can never both be prime. The only way to fail is to have no prime at all.

4Add the allowed values

#2 Make a Systematic List 4.OA.B.4

The allowed values are

6

,

7

and

12

.

6+7+12=25

Seven candidates, four crossed out, three left.

Answer:

25

4 · Reviewdoes it hold up?

Spot-check $a=6$ : $6+16=22$ is not prime and $6+17=23$ is, so $6$ goes only to $17$ . Spot-check $a=8$ : $24$ and $25$ are both composite, so $8$ has no partner. The sum $25$ is less than $6+7+\dots+12=63$ , as it should be after removing $8$ , $9$ , $10$ and $11$ .

Another way: Count the complement: the sum of all candidates is $6+7+8+9+10+11+12=63$ ; remove the four failures $8+9+10+11=38$ to get $63-38=25$ .

Standardsmin grade 8

8.F.A.1 Understand that a function is a rule that assigns exactly one output to each input — Requiring each input to have exactly one output.
4.OA.B.4 Find all factor pairs and recognize multiples; determine prime or composite — Deciding whether each sum is prime or composite.

💡Takeaway. A function needs exactly one output per input, so check for 'none' as carefully as for 'two'.

Exactly one prime sum makes a function · 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?