Practice · Sum pyramid: symmetric terms cancel at the top · Sensim Math

Variant 1 easy

The rectangular bricks at right follow this rule: the sum of the expressions in two neighboring bricks is written in the brick directly above them. For example, if a bottom row is $1$ , $2$ , $3$ , the row above it is $1+2=3$ and $2+3=5$ .

The bottom row of the four-row pyramid holds, from left to right, $x+y$ , $x+2y$ , $x-2y$ , $x-y$ . What expression goes in the top brick (A)?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Each brick is the sum of the two bricks under it. Starting from four expressions on the bottom row, build the pyramid up and find the expression at the top.

Givens

Bottom row: $x+y$ , $x+2y$ , $x-2y$ , $x-y$ .
Each brick equals the sum of the two bricks directly below it.
The pyramid has $4$ rows, so the top is $1$ brick.

Unknowns

The expression in the top brick (A).

Constraints

Each row has one fewer brick than the row below it.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #13 Convert to Algebra#5 Look for a Pattern

Fill the pyramid one row at a time, combining like terms at every step so the expressions stay short. Then look at why the $y$ parts disappear: the bottom row is symmetric.

3 · Execute4 carry out the plan

1Second row

#1 Draw a Diagram 7.EE.A.1

Add each pair of neighbors in the bottom row.

2x+3y,\quad 2x,\quad 2x-3y

For example

(x+y)+(x+2y)=2x+3y

: add the

x

's and the

y

's separately.

2Third row

#13 Convert to Algebra 6.EE.A.3

Add the two pairs.

(2x+3y)+2x=4x+3y,\quad 2x+(2x-3y)=4x-3y

Like terms combine into one term.

3Top brick

#13 Convert to Algebra 7.EE.A.1

The last sum: the

3y

and

-3y

cancel.

(4x+3y)+(4x-3y)=8x

Only

x

is left.

4Why the y's vanish

#5 Look for a Pattern 6.EE.A.3

Each bottom brick is counted

1

,

3

,

3

,

1

times on its way to the top. That gives

1+3+3+1=8

, so

8

copies of

x

while the

y

parts are mirror images and cancel.

1\times1+2\times3+(-2)\times3+(-1)\times1=0

Symmetric left and right halves cancel.

Answer:

8x

4 · Reviewdoes it hold up?

Try $x=1$ , $y=1$ : the bottom row is $2,3,-1,0$ , then $5,2,-1$ , then $7,1$ , then $8$ . And $8x=8$ . It matches.

Another way: Use the counting weights $1$ , $3$ , $3$ , $1$ directly: $(x+y)+3(x+2y)+3(x-2y)+(x-y)=8x$ .

Standardsmin grade 7

7.EE.A.1 Apply properties of operations to add, subtract, factor, and expand linear expressions — Adding linear expressions and combining like terms.
6.EE.A.3 Apply properties to generate equivalent expressions — Collecting like terms to write equivalent expressions.

💡Takeaway. Add the

x

's with the

x

's and the

y

's with the

y

's, and watch symmetry do the cancelling.

Variant 2 easy

The rectangular bricks at right follow this rule: the sum of the expressions in two neighboring bricks is written in the brick directly above them. For example, if a bottom row is $1$ , $2$ , $3$ , the row above it is $1+2=3$ and $2+3=5$ .

The bottom row of the four-row pyramid holds, from left to right, $x+3y$ , $x+y$ , $x-y$ , $x-3y$ . What expression goes in the top brick (A)?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Each brick is the sum of the two bricks under it. Starting from four expressions on the bottom row, build the pyramid up and find the expression at the top.

Givens

Bottom row: $x+3y$ , $x+y$ , $x-y$ , $x-3y$ .
Each brick equals the sum of the two bricks directly below it.
The pyramid has $4$ rows, so the top is $1$ brick.

Unknowns

The expression in the top brick (A).

Constraints

Each row has one fewer brick than the row below it.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #13 Convert to Algebra#5 Look for a Pattern

Fill the pyramid one row at a time, combining like terms at every step so the expressions stay short. Then look at why the $y$ parts disappear: the bottom row is symmetric.

3 · Execute4 carry out the plan

1Second row

#1 Draw a Diagram 7.EE.A.1

Add each pair of neighbors in the bottom row.

2x+4y,\quad 2x,\quad 2x-4y

For example

(x+3y)+(x+y)=2x+4y

: add the

x

's and the

y

's separately.

2Third row

#13 Convert to Algebra 6.EE.A.3

Add the two pairs.

(2x+4y)+2x=4x+4y,\quad 2x+(2x-4y)=4x-4y

Like terms combine into one term.

3Top brick

#13 Convert to Algebra 7.EE.A.1

The last sum: the

4y

and

-4y

cancel.

(4x+4y)+(4x-4y)=8x

Only

x

is left.

4Why the y's vanish

#5 Look for a Pattern 6.EE.A.3

Each bottom brick is counted

1

,

3

,

3

,

1

times on its way to the top. That gives

1+3+3+1=8

, so

8

copies of

x

while the

y

parts are mirror images and cancel.

3\times1+1\times3+(-1)\times3+(-3)\times1=0

Symmetric left and right halves cancel.

Answer:

8x

4 · Reviewdoes it hold up?

Try $x=1$ , $y=1$ : the bottom row is $4,2,0,-2$ , then $6,2,-2$ , then $8,0$ , then $8$ . And $8x=8$ . It matches.

Another way: Use the counting weights $1$ , $3$ , $3$ , $1$ directly: $(x+3y)+3(x+y)+3(x-y)+(x-3y)=8x$ .

Standardsmin grade 7

7.EE.A.1 Apply properties of operations to add, subtract, factor, and expand linear expressions — Adding linear expressions and combining like terms.
6.EE.A.3 Apply properties to generate equivalent expressions — Collecting like terms to write equivalent expressions.

💡Takeaway. Add the

x

's with the

x

's and the

y

's with the

y

's, and watch symmetry do the cancelling.

Variant 3 easy

The rectangular bricks at right follow this rule: the sum of the expressions in two neighboring bricks is written in the brick directly above them. For example, if a bottom row is $1$ , $2$ , $3$ , the row above it is $1+2=3$ and $2+3=5$ .

The bottom row of the five-row pyramid holds, from left to right, $x+3y$ , $x+y$ , $x$ , $x-y$ , $x-3y$ . What expression goes in the top brick (A)?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Each brick is the sum of the two bricks under it. Starting from five expressions on the bottom row, build the pyramid up and find the expression at the top.

Givens

Bottom row: $x+3y$ , $x+y$ , $x$ , $x-y$ , $x-3y$ .
Each brick equals the sum of the two bricks directly below it.
The pyramid has $5$ rows, so the top is $1$ brick.

Unknowns

The expression in the top brick (A).

Constraints

Each row has one fewer brick than the row below it.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #13 Convert to Algebra#5 Look for a Pattern

Fill the pyramid one row at a time, combining like terms at every step so the expressions stay short. Then look at why the $y$ parts disappear: the bottom row is symmetric.

3 · Execute5 carry out the plan

1Second row

#1 Draw a Diagram 7.EE.A.1

Add each pair of neighbors in the bottom row.

2x+4y,\quad 2x+y,\quad 2x-y,\quad 2x-4y

For example

(x+3y)+(x+y)=2x+4y

: add the

x

's and the

y

's separately.

2Third row

#13 Convert to Algebra 7.EE.A.1

Add neighbors again.

4x+5y,\quad 4x,\quad 4x-5y

(2x+y)+(2x-y)=4x

: the

y

's cancel.

3Fourth row

#13 Convert to Algebra 6.EE.A.3

Add the two pairs.

(4x+5y)+4x=8x+5y,\quad 4x+(4x-5y)=8x-5y

Like terms combine into one term.

4Top brick

#13 Convert to Algebra 7.EE.A.1

The last sum: the

5y

and

-5y

cancel.

(8x+5y)+(8x-5y)=16x

Only

x

is left.

5Why the y's vanish

#5 Look for a Pattern 6.EE.A.3

Each bottom brick is counted

1

,

4

,

6

,

4

,

1

times on its way to the top. That gives

1+4+6+4+1=16

, so

16

copies of

x

while the

y

parts are mirror images and cancel.

3\times1+1\times4+0\times6+(-1)\times4+(-3)\times1=0

Symmetric left and right halves cancel.

Answer:

16x

4 · Reviewdoes it hold up?

Try $x=1$ , $y=1$ : the bottom row is $4,2,1,0,-2$ , then $6,3,1,-2$ , then $9,4,-1$ , then $13,3$ , then $16$ . And $16x=16$ . It matches.

Another way: Use the counting weights $1$ , $4$ , $6$ , $4$ , $1$ directly: $(x+3y)+4(x+y)+6x+4(x-y)+(x-3y)=16x$ .

Standardsmin grade 7

7.EE.A.1 Apply properties of operations to add, subtract, factor, and expand linear expressions — Adding linear expressions and combining like terms.
6.EE.A.3 Apply properties to generate equivalent expressions — Collecting like terms to write equivalent expressions.

💡Takeaway. Add the

x

's with the

x

's and the

y

's with the

y

's, and watch symmetry do the cancelling.

Variant 4 easy

The rectangular bricks at right follow this rule: the sum of the expressions in two neighboring bricks is written in the brick directly above them. For example, if a bottom row is $1$ , $2$ , $3$ , the row above it is $1+2=3$ and $2+3=5$ .

The bottom row of the five-row pyramid holds, from left to right, $x+2y$ , $x+y$ , $x$ , $x-y$ , $x-2y$ . What expression goes in the top brick (A)?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Each brick is the sum of the two bricks under it. Starting from five expressions on the bottom row, build the pyramid up and find the expression at the top.

Givens

Bottom row: $x+2y$ , $x+y$ , $x$ , $x-y$ , $x-2y$ .
Each brick equals the sum of the two bricks directly below it.
The pyramid has $5$ rows, so the top is $1$ brick.

Unknowns

The expression in the top brick (A).

Constraints

Each row has one fewer brick than the row below it.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #13 Convert to Algebra#5 Look for a Pattern

Fill the pyramid one row at a time, combining like terms at every step so the expressions stay short. Then look at why the $y$ parts disappear: the bottom row is symmetric.

3 · Execute5 carry out the plan

1Second row

#1 Draw a Diagram 7.EE.A.1

Add each pair of neighbors in the bottom row.

2x+3y,\quad 2x+y,\quad 2x-y,\quad 2x-3y

For example

(x+2y)+(x+y)=2x+3y

: add the

x

's and the

y

's separately.

2Third row

#13 Convert to Algebra 7.EE.A.1

Add neighbors again.

4x+4y,\quad 4x,\quad 4x-4y

(2x+y)+(2x-y)=4x

: the

y

's cancel.

3Fourth row

#13 Convert to Algebra 6.EE.A.3

Add the two pairs.

(4x+4y)+4x=8x+4y,\quad 4x+(4x-4y)=8x-4y

Like terms combine into one term.

4Top brick

#13 Convert to Algebra 7.EE.A.1

The last sum: the

4y

and

-4y

cancel.

(8x+4y)+(8x-4y)=16x

Only

x

is left.

5Why the y's vanish

#5 Look for a Pattern 6.EE.A.3

Each bottom brick is counted

1

,

4

,

6

,

4

,

1

times on its way to the top. That gives

1+4+6+4+1=16

, so

16

copies of

x

while the

y

parts are mirror images and cancel.

2\times1+1\times4+0\times6+(-1)\times4+(-2)\times1=0

Symmetric left and right halves cancel.

Answer:

16x

4 · Reviewdoes it hold up?

Try $x=1$ , $y=1$ : the bottom row is $3,2,1,0,-1$ , then $5,3,1,-1$ , then $8,4,0$ , then $12,4$ , then $16$ . And $16x=16$ . It matches.

Another way: Use the counting weights $1$ , $4$ , $6$ , $4$ , $1$ directly: $(x+2y)+4(x+y)+6x+4(x-y)+(x-2y)=16x$ .

Standardsmin grade 7

7.EE.A.1 Apply properties of operations to add, subtract, factor, and expand linear expressions — Adding linear expressions and combining like terms.
6.EE.A.3 Apply properties to generate equivalent expressions — Collecting like terms to write equivalent expressions.

💡Takeaway. Add the

x

's with the

x

's and the

y

's with the

y

's, and watch symmetry do the cancelling.

Variant 5 medium

The rectangular bricks at right follow this rule: the sum of the expressions in two neighboring bricks is written in the brick directly above them. For example, if a bottom row is $1$ , $2$ , $3$ , the row above it is $1+2=3$ and $2+3=5$ .

The bottom row of the five-row pyramid holds, from left to right, $x-2y$ , $x-y$ , $x$ , $x+y$ , $x+2y$ . What expression goes in the top brick (A)?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Each brick is the sum of the two bricks under it. Starting from five expressions on the bottom row, build the pyramid up and find the expression at the top.

Givens

Bottom row: $x-2y$ , $x-y$ , $x$ , $x+y$ , $x+2y$ .
Each brick equals the sum of the two bricks directly below it.
The pyramid has $5$ rows, so the top is $1$ brick.

Unknowns

The expression in the top brick (A).

Constraints

Each row has one fewer brick than the row below it.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #13 Convert to Algebra#5 Look for a Pattern

Fill the pyramid one row at a time, combining like terms at every step so the expressions stay short. Then look at why the $y$ parts disappear: the bottom row is symmetric.

3 · Execute5 carry out the plan

1Second row

#1 Draw a Diagram 7.EE.A.1

Add each pair of neighbors in the bottom row.

2x-3y,\quad 2x-y,\quad 2x+y,\quad 2x+3y

For example

(x-2y)+(x-y)=2x-3y

: add the

x

's and the

y

's separately.

2Third row

#13 Convert to Algebra 7.EE.A.1

Add neighbors again.

4x-4y,\quad 4x,\quad 4x+4y

(2x-y)+(2x+y)=4x

: the

y

's cancel.

3Fourth row

#13 Convert to Algebra 6.EE.A.3

Add the two pairs.

(4x-4y)+4x=8x-4y,\quad 4x+(4x+4y)=8x+4y

Like terms combine into one term.

4Top brick

#13 Convert to Algebra 7.EE.A.1

The last sum: the

-4y

and

4y

cancel.

(8x-4y)+(8x+4y)=16x

Only

x

is left.

5Why the y's vanish

#5 Look for a Pattern 6.EE.A.3

Each bottom brick is counted

1

,

4

,

6

,

4

,

1

times on its way to the top. That gives

1+4+6+4+1=16

, so

16

copies of

x

while the

y

parts are mirror images and cancel.

(-2)\times1+(-1)\times4+0\times6+1\times4+2\times1=0

Symmetric left and right halves cancel.

Answer:

16x

4 · Reviewdoes it hold up?

Try $x=1$ , $y=1$ : the bottom row is $-1,0,1,2,3$ , then $-1,1,3,5$ , then $0,4,8$ , then $4,12$ , then $16$ . And $16x=16$ . It matches.

Another way: Use the counting weights $1$ , $4$ , $6$ , $4$ , $1$ directly: $(x-2y)+4(x-y)+6x+4(x+y)+(x+2y)=16x$ .

Standardsmin grade 7

7.EE.A.1 Apply properties of operations to add, subtract, factor, and expand linear expressions — Adding linear expressions and combining like terms.
6.EE.A.3 Apply properties to generate equivalent expressions — Collecting like terms to write equivalent expressions.

💡Takeaway. Add the

x

's with the

x

's and the

y

's with the

y

's, and watch symmetry do the cancelling.

Variant 6 medium

The rectangular bricks at right follow this rule: the sum of the expressions in two neighboring bricks is written in the brick directly above them. For example, if a bottom row is $1$ , $2$ , $3$ , the row above it is $1+2=3$ and $2+3=5$ .

The bottom row of the four-row pyramid holds, from left to right, $2x+2y$ , $2x+y$ , $2x-y$ , $2x-2y$ . What expression goes in the top brick (A)?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Each brick is the sum of the two bricks under it. Starting from four expressions on the bottom row, build the pyramid up and find the expression at the top.

Givens

Bottom row: $2x+2y$ , $2x+y$ , $2x-y$ , $2x-2y$ .
Each brick equals the sum of the two bricks directly below it.
The pyramid has $4$ rows, so the top is $1$ brick.

Unknowns

The expression in the top brick (A).

Constraints

Each row has one fewer brick than the row below it.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #13 Convert to Algebra#5 Look for a Pattern

Fill the pyramid one row at a time, combining like terms at every step so the expressions stay short. Then look at why the $y$ parts disappear: the bottom row is symmetric.

3 · Execute4 carry out the plan

1Second row

#1 Draw a Diagram 7.EE.A.1

Add each pair of neighbors in the bottom row.

4x+3y,\quad 4x,\quad 4x-3y

For example

(2x+2y)+(2x+y)=4x+3y

: add the

x

's and the

y

's separately.

2Third row

#13 Convert to Algebra 6.EE.A.3

Add the two pairs.

(4x+3y)+4x=8x+3y,\quad 4x+(4x-3y)=8x-3y

Like terms combine into one term.

3Top brick

#13 Convert to Algebra 7.EE.A.1

The last sum: the

3y

and

-3y

cancel.

(8x+3y)+(8x-3y)=16x

Only

x

is left.

4Why the y's vanish

#5 Look for a Pattern 6.EE.A.3

Each bottom brick is counted

1

,

3

,

3

,

1

times on its way to the top. That gives

1+3+3+1=8

, so

8

copies of

2x

, which is

16x

, while the

y

parts are mirror images and cancel.

2\times1+1\times3+(-1)\times3+(-2)\times1=0

Symmetric left and right halves cancel.

Answer:

16x

4 · Reviewdoes it hold up?

Try $x=1$ , $y=1$ : the bottom row is $4,3,1,0$ , then $7,4,1$ , then $11,5$ , then $16$ . And $16x=16$ . It matches.

Another way: Use the counting weights $1$ , $3$ , $3$ , $1$ directly: $(2x+2y)+3(2x+y)+3(2x-y)+(2x-2y)=16x$ .

Standardsmin grade 7

7.EE.A.1 Apply properties of operations to add, subtract, factor, and expand linear expressions — Adding linear expressions and combining like terms.
6.EE.A.3 Apply properties to generate equivalent expressions — Collecting like terms to write equivalent expressions.

💡Takeaway. Add the

x

's with the

x

's and the

y

's with the

y

's, and watch symmetry do the cancelling.

Variant 7 medium

The rectangular bricks at right follow this rule: the sum of the expressions in two neighboring bricks is written in the brick directly above them. For example, if a bottom row is $1$ , $2$ , $3$ , the row above it is $1+2=3$ and $2+3=5$ .

The bottom row of the five-row pyramid holds, from left to right, $2x+y$ , $2x+2y$ , $2x$ , $2x-2y$ , $2x-y$ . What expression goes in the top brick (A)?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Each brick is the sum of the two bricks under it. Starting from five expressions on the bottom row, build the pyramid up and find the expression at the top.

Givens

Bottom row: $2x+y$ , $2x+2y$ , $2x$ , $2x-2y$ , $2x-y$ .
Each brick equals the sum of the two bricks directly below it.
The pyramid has $5$ rows, so the top is $1$ brick.

Unknowns

The expression in the top brick (A).

Constraints

Each row has one fewer brick than the row below it.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #13 Convert to Algebra#5 Look for a Pattern

Fill the pyramid one row at a time, combining like terms at every step so the expressions stay short. Then look at why the $y$ parts disappear: the bottom row is symmetric.

3 · Execute5 carry out the plan

1Second row

#1 Draw a Diagram 7.EE.A.1

Add each pair of neighbors in the bottom row.

4x+3y,\quad 4x+2y,\quad 4x-2y,\quad 4x-3y

For example

(2x+y)+(2x+2y)=4x+3y

: add the

x

's and the

y

's separately.

2Third row

#13 Convert to Algebra 7.EE.A.1

Add neighbors again.

8x+5y,\quad 8x,\quad 8x-5y

(4x+2y)+(4x-2y)=8x

: the

y

's cancel.

3Fourth row

#13 Convert to Algebra 6.EE.A.3

Add the two pairs.

(8x+5y)+8x=16x+5y,\quad 8x+(8x-5y)=16x-5y

Like terms combine into one term.

4Top brick

#13 Convert to Algebra 7.EE.A.1

The last sum: the

5y

and

-5y

cancel.

(16x+5y)+(16x-5y)=32x

Only

x

is left.

5Why the y's vanish

#5 Look for a Pattern 6.EE.A.3

Each bottom brick is counted

1

,

4

,

6

,

4

,

1

times on its way to the top. That gives

1+4+6+4+1=16

, so

16

copies of

2x

, which is

32x

, while the

y

parts are mirror images and cancel.

1\times1+2\times4+0\times6+(-2)\times4+(-1)\times1=0

Symmetric left and right halves cancel.

Answer:

32x

4 · Reviewdoes it hold up?

Try $x=1$ , $y=1$ : the bottom row is $3,4,2,0,1$ , then $7,6,2,1$ , then $13,8,3$ , then $21,11$ , then $32$ . And $32x=32$ . It matches.

Another way: Use the counting weights $1$ , $4$ , $6$ , $4$ , $1$ directly: $(2x+y)+4(2x+2y)+6\times2x+4(2x-2y)+(2x-y)=32x$ .

Standardsmin grade 7

7.EE.A.1 Apply properties of operations to add, subtract, factor, and expand linear expressions — Adding linear expressions and combining like terms.
6.EE.A.3 Apply properties to generate equivalent expressions — Collecting like terms to write equivalent expressions.

💡Takeaway. Add the

x

's with the

x

's and the

y

's with the

y

's, and watch symmetry do the cancelling.

Variant 8 hard

The rectangular bricks at right follow this rule: the sum of the expressions in two neighboring bricks is written in the brick directly above them. For example, if a bottom row is $1$ , $2$ , $3$ , the row above it is $1+2=3$ and $2+3=5$ .

The bottom row of the six-row pyramid holds, from left to right, $x+5y$ , $x+3y$ , $x+y$ , $x-y$ , $x-3y$ , $x-5y$ . What expression goes in the top brick (A)?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Each brick is the sum of the two bricks under it. Starting from six expressions on the bottom row, build the pyramid up and find the expression at the top.

Givens

Bottom row: $x+5y$ , $x+3y$ , $x+y$ , $x-y$ , $x-3y$ , $x-5y$ .
Each brick equals the sum of the two bricks directly below it.
The pyramid has $6$ rows, so the top is $1$ brick.

Unknowns

The expression in the top brick (A).

Constraints

Each row has one fewer brick than the row below it.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #13 Convert to Algebra#5 Look for a Pattern

Fill the pyramid one row at a time, combining like terms at every step so the expressions stay short. Then look at why the $y$ parts disappear: the bottom row is symmetric.

3 · Execute6 carry out the plan

1Second row

#1 Draw a Diagram 7.EE.A.1

Add each pair of neighbors in the bottom row.

2x+8y,\quad 2x+4y,\quad 2x,\quad 2x-4y,\quad 2x-8y

For example

(x+5y)+(x+3y)=2x+8y

: add the

x

's and the

y

's separately.

2Third row

#13 Convert to Algebra 7.EE.A.1

Add neighbors again.

4x+12y,\quad 4x+4y,\quad 4x-4y,\quad 4x-12y

(2x+4y)+2x=4x+4y

: like terms combine.

3Fourth row

#13 Convert to Algebra 7.EE.A.1

Add neighbors again.

8x+16y,\quad 8x,\quad 8x-16y

(4x+4y)+(4x-4y)=8x

: the

y

's cancel.

4Fifth row

#13 Convert to Algebra 6.EE.A.3

Add the two pairs.

(8x+16y)+8x=16x+16y,\quad 8x+(8x-16y)=16x-16y

Like terms combine into one term.

5Top brick

#13 Convert to Algebra 7.EE.A.1

The last sum: the

16y

and

-16y

cancel.

(16x+16y)+(16x-16y)=32x

Only

x

is left.

6Why the y's vanish

#5 Look for a Pattern 6.EE.A.3

Each bottom brick is counted

1

,

5

,

10

,

10

,

5

,

1

times on its way to the top. That gives

1+5+10+10+5+1=32

, so

32

copies of

x

while the

y

parts are mirror images and cancel.

5\times1+3\times5+1\times10+(-1)\times10+(-3)\times5+(-5)\times1=0

Symmetric left and right halves cancel.

Answer:

32x

4 · Reviewdoes it hold up?

Try $x=1$ , $y=1$ : the bottom row is $6,4,2,0,-2,-4$ , then $10,6,2,-2,-6$ , then $16,8,0,-8$ , then $24,8,-8$ , then $32,0$ , then $32$ . And $32x=32$ . It matches.

Another way: Use the counting weights $1$ , $5$ , $10$ , $10$ , $5$ , $1$ directly: $(x+5y)+5(x+3y)+10(x+y)+10(x-y)+5(x-3y)+(x-5y)=32x$ .

Standardsmin grade 7

7.EE.A.1 Apply properties of operations to add, subtract, factor, and expand linear expressions — Adding linear expressions and combining like terms.
6.EE.A.3 Apply properties to generate equivalent expressions — Collecting like terms to write equivalent expressions.

💡Takeaway. Add the

x

's with the

x

's and the

y

's with the

y

's, and watch symmetry do the cancelling.

Variant 9 hard

The rectangular bricks at right follow this rule: the sum of the expressions in two neighboring bricks is written in the brick directly above them. For example, if a bottom row is $1$ , $2$ , $3$ , the row above it is $1+2=3$ and $2+3=5$ .

The bottom row of the six-row pyramid holds, from left to right, $x+3y$ , $x+2y$ , $x+y$ , $x-y$ , $x-2y$ , $x-3y$ . What expression goes in the top brick (A)?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Each brick is the sum of the two bricks under it. Starting from six expressions on the bottom row, build the pyramid up and find the expression at the top.

Givens

Bottom row: $x+3y$ , $x+2y$ , $x+y$ , $x-y$ , $x-2y$ , $x-3y$ .
Each brick equals the sum of the two bricks directly below it.
The pyramid has $6$ rows, so the top is $1$ brick.

Unknowns

The expression in the top brick (A).

Constraints

Each row has one fewer brick than the row below it.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #13 Convert to Algebra#5 Look for a Pattern

Fill the pyramid one row at a time, combining like terms at every step so the expressions stay short. Then look at why the $y$ parts disappear: the bottom row is symmetric.

3 · Execute6 carry out the plan

1Second row

#1 Draw a Diagram 7.EE.A.1

Add each pair of neighbors in the bottom row.

2x+5y,\quad 2x+3y,\quad 2x,\quad 2x-3y,\quad 2x-5y

For example

(x+3y)+(x+2y)=2x+5y

: add the

x

's and the

y

's separately.

2Third row

#13 Convert to Algebra 7.EE.A.1

Add neighbors again.

4x+8y,\quad 4x+3y,\quad 4x-3y,\quad 4x-8y

(2x+3y)+2x=4x+3y

: like terms combine.

3Fourth row

#13 Convert to Algebra 7.EE.A.1

Add neighbors again.

8x+11y,\quad 8x,\quad 8x-11y

(4x+3y)+(4x-3y)=8x

: the

y

's cancel.

4Fifth row

#13 Convert to Algebra 6.EE.A.3

Add the two pairs.

(8x+11y)+8x=16x+11y,\quad 8x+(8x-11y)=16x-11y

Like terms combine into one term.

5Top brick

#13 Convert to Algebra 7.EE.A.1

The last sum: the

11y

and

-11y

cancel.

(16x+11y)+(16x-11y)=32x

Only

x

is left.

6Why the y's vanish

#5 Look for a Pattern 6.EE.A.3

Each bottom brick is counted

1

,

5

,

10

,

10

,

5

,

1

times on its way to the top. That gives

1+5+10+10+5+1=32

, so

32

copies of

x

while the

y

parts are mirror images and cancel.

3\times1+2\times5+1\times10+(-1)\times10+(-2)\times5+(-3)\times1=0

Symmetric left and right halves cancel.

Answer:

32x

4 · Reviewdoes it hold up?

Try $x=1$ , $y=1$ : the bottom row is $4,3,2,0,-1,-2$ , then $7,5,2,-1,-3$ , then $12,7,1,-4$ , then $19,8,-3$ , then $27,5$ , then $32$ . And $32x=32$ . It matches.

Another way: Use the counting weights $1$ , $5$ , $10$ , $10$ , $5$ , $1$ directly: $(x+3y)+5(x+2y)+10(x+y)+10(x-y)+5(x-2y)+(x-3y)=32x$ .

Standardsmin grade 7

7.EE.A.1 Apply properties of operations to add, subtract, factor, and expand linear expressions — Adding linear expressions and combining like terms.
6.EE.A.3 Apply properties to generate equivalent expressions — Collecting like terms to write equivalent expressions.

💡Takeaway. Add the

x

's with the

x

's and the

y

's with the

y

's, and watch symmetry do the cancelling.

Variant 10 medium

The rectangular bricks at right follow this rule: the sum of the expressions in two neighboring bricks is written in the brick directly above them. For example, if a bottom row is $1$ , $2$ , $3$ , the row above it is $1+2=3$ and $2+3=5$ .

The bottom row of the five-row pyramid holds, from left to right, $3x+2y$ , $3x+y$ , $3x$ , $3x-y$ , $3x-2y$ . What expression goes in the top brick (A)?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Each brick is the sum of the two bricks under it. Starting from five expressions on the bottom row, build the pyramid up and find the expression at the top.

Givens

Bottom row: $3x+2y$ , $3x+y$ , $3x$ , $3x-y$ , $3x-2y$ .
Each brick equals the sum of the two bricks directly below it.
The pyramid has $5$ rows, so the top is $1$ brick.

Unknowns

The expression in the top brick (A).

Constraints

Each row has one fewer brick than the row below it.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #13 Convert to Algebra#5 Look for a Pattern

Fill the pyramid one row at a time, combining like terms at every step so the expressions stay short. Then look at why the $y$ parts disappear: the bottom row is symmetric.

3 · Execute5 carry out the plan

1Second row

#1 Draw a Diagram 7.EE.A.1

Add each pair of neighbors in the bottom row.

6x+3y,\quad 6x+y,\quad 6x-y,\quad 6x-3y

For example

(3x+2y)+(3x+y)=6x+3y

: add the

x

's and the

y

's separately.

2Third row

#13 Convert to Algebra 7.EE.A.1

Add neighbors again.

12x+4y,\quad 12x,\quad 12x-4y

(6x+y)+(6x-y)=12x

: the

y

's cancel.

3Fourth row

#13 Convert to Algebra 6.EE.A.3

Add the two pairs.

(12x+4y)+12x=24x+4y,\quad 12x+(12x-4y)=24x-4y

Like terms combine into one term.

4Top brick

#13 Convert to Algebra 7.EE.A.1

The last sum: the

4y

and

-4y

cancel.

(24x+4y)+(24x-4y)=48x

Only

x

is left.

5Why the y's vanish

#5 Look for a Pattern 6.EE.A.3

Each bottom brick is counted

1

,

4

,

6

,

4

,

1

times on its way to the top. That gives

1+4+6+4+1=16

, so

16

copies of

3x

, which is

48x

, while the

y

parts are mirror images and cancel.

2\times1+1\times4+0\times6+(-1)\times4+(-2)\times1=0

Symmetric left and right halves cancel.

Answer:

48x

4 · Reviewdoes it hold up?

Try $x=1$ , $y=1$ : the bottom row is $5,4,3,2,1$ , then $9,7,5,3$ , then $16,12,8$ , then $28,20$ , then $48$ . And $48x=48$ . It matches.

Another way: Use the counting weights $1$ , $4$ , $6$ , $4$ , $1$ directly: $(3x+2y)+4(3x+y)+6\times3x+4(3x-y)+(3x-2y)=48x$ .

Standardsmin grade 7

7.EE.A.1 Apply properties of operations to add, subtract, factor, and expand linear expressions — Adding linear expressions and combining like terms.
6.EE.A.3 Apply properties to generate equivalent expressions — Collecting like terms to write equivalent expressions.

💡Takeaway. Add the

x

's with the

x

's and the

y

's with the

y

's, and watch symmetry do the cancelling.

Variant 11 hard

The rectangular bricks at right follow this rule: the sum of the expressions in two neighboring bricks is written in the brick directly above them. For example, if a bottom row is $1$ , $2$ , $3$ , the row above it is $1+2=3$ and $2+3=5$ .

The bottom row of the five-row pyramid holds, from left to right, $2x+4y$ , $2x+2y$ , $2x$ , $2x-2y$ , $2x-4y$ . What expression goes in the top brick (A)?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Each brick is the sum of the two bricks under it. Starting from five expressions on the bottom row, build the pyramid up and find the expression at the top.

Givens

Bottom row: $2x+4y$ , $2x+2y$ , $2x$ , $2x-2y$ , $2x-4y$ .
Each brick equals the sum of the two bricks directly below it.
The pyramid has $5$ rows, so the top is $1$ brick.

Unknowns

The expression in the top brick (A).

Constraints

Each row has one fewer brick than the row below it.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #13 Convert to Algebra#5 Look for a Pattern

Fill the pyramid one row at a time, combining like terms at every step so the expressions stay short. Then look at why the $y$ parts disappear: the bottom row is symmetric.

3 · Execute5 carry out the plan

1Second row

#1 Draw a Diagram 7.EE.A.1

Add each pair of neighbors in the bottom row.

4x+6y,\quad 4x+2y,\quad 4x-2y,\quad 4x-6y

For example

(2x+4y)+(2x+2y)=4x+6y

: add the

x

's and the

y

's separately.

2Third row

#13 Convert to Algebra 7.EE.A.1

Add neighbors again.

8x+8y,\quad 8x,\quad 8x-8y

(4x+2y)+(4x-2y)=8x

: the

y

's cancel.

3Fourth row

#13 Convert to Algebra 6.EE.A.3

Add the two pairs.

(8x+8y)+8x=16x+8y,\quad 8x+(8x-8y)=16x-8y

Like terms combine into one term.

4Top brick

#13 Convert to Algebra 7.EE.A.1

The last sum: the

8y

and

-8y

cancel.

(16x+8y)+(16x-8y)=32x

Only

x

is left.

5Why the y's vanish

#5 Look for a Pattern 6.EE.A.3

Each bottom brick is counted

1

,

4

,

6

,

4

,

1

times on its way to the top. That gives

1+4+6+4+1=16

, so

16

copies of

2x

, which is

32x

, while the

y

parts are mirror images and cancel.

4\times1+2\times4+0\times6+(-2)\times4+(-4)\times1=0

Symmetric left and right halves cancel.

Answer:

32x

4 · Reviewdoes it hold up?

Try $x=1$ , $y=1$ : the bottom row is $6,4,2,0,-2$ , then $10,6,2,-2$ , then $16,8,0$ , then $24,8$ , then $32$ . And $32x=32$ . It matches.

Another way: Use the counting weights $1$ , $4$ , $6$ , $4$ , $1$ directly: $(2x+4y)+4(2x+2y)+6\times2x+4(2x-2y)+(2x-4y)=32x$ .

Standardsmin grade 7

7.EE.A.1 Apply properties of operations to add, subtract, factor, and expand linear expressions — Adding linear expressions and combining like terms.
6.EE.A.3 Apply properties to generate equivalent expressions — Collecting like terms to write equivalent expressions.

💡Takeaway. Add the

x

's with the

x

's and the

y

's with the

y

's, and watch symmetry do the cancelling.

Variant 12 hard

The rectangular bricks at right follow this rule: the sum of the expressions in two neighboring bricks is written in the brick directly above them. For example, if a bottom row is $1$ , $2$ , $3$ , the row above it is $1+2=3$ and $2+3=5$ .

The bottom row of the six-row pyramid holds, from left to right, $2x+2y$ , $2x+y$ , $2x+y$ , $2x-y$ , $2x-y$ , $2x-2y$ . What expression goes in the top brick (A)?

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Each brick is the sum of the two bricks under it. Starting from six expressions on the bottom row, build the pyramid up and find the expression at the top.

Givens

Bottom row: $2x+2y$ , $2x+y$ , $2x+y$ , $2x-y$ , $2x-y$ , $2x-2y$ .
Each brick equals the sum of the two bricks directly below it.
The pyramid has $6$ rows, so the top is $1$ brick.

Unknowns

The expression in the top brick (A).

Constraints

Each row has one fewer brick than the row below it.

2 · Planchoose the strategy

#1 Draw a Diagram · also uses: #13 Convert to Algebra#5 Look for a Pattern

Fill the pyramid one row at a time, combining like terms at every step so the expressions stay short. Then look at why the $y$ parts disappear: the bottom row is symmetric.

3 · Execute6 carry out the plan

1Second row

#1 Draw a Diagram 7.EE.A.1

Add each pair of neighbors in the bottom row.

4x+3y,\quad 4x+2y,\quad 4x,\quad 4x-2y,\quad 4x-3y

For example

(2x+2y)+(2x+y)=4x+3y

: add the

x

's and the

y

's separately.

2Third row

#13 Convert to Algebra 7.EE.A.1

Add neighbors again.

8x+5y,\quad 8x+2y,\quad 8x-2y,\quad 8x-5y

(4x+2y)+4x=8x+2y

: like terms combine.

3Fourth row

#13 Convert to Algebra 7.EE.A.1

Add neighbors again.

16x+7y,\quad 16x,\quad 16x-7y

(8x+2y)+(8x-2y)=16x

: the

y

's cancel.

4Fifth row

#13 Convert to Algebra 6.EE.A.3

Add the two pairs.

(16x+7y)+16x=32x+7y,\quad 16x+(16x-7y)=32x-7y

Like terms combine into one term.

5Top brick

#13 Convert to Algebra 7.EE.A.1

The last sum: the

7y

and

-7y

cancel.

(32x+7y)+(32x-7y)=64x

Only

x

is left.

6Why the y's vanish

#5 Look for a Pattern 6.EE.A.3

Each bottom brick is counted

1

,

5

,

10

,

10

,

5

,

1

times on its way to the top. That gives

1+5+10+10+5+1=32

, so

32

copies of

2x

, which is

64x

, while the

y

parts are mirror images and cancel.

2\times1+1\times5+1\times10+(-1)\times10+(-1)\times5+(-2)\times1=0

Symmetric left and right halves cancel.

Answer:

64x

4 · Reviewdoes it hold up?

Try $x=1$ , $y=1$ : the bottom row is $4,3,3,1,1,0$ , then $7,6,4,2,1$ , then $13,10,6,3$ , then $23,16,9$ , then $39,25$ , then $64$ . And $64x=64$ . It matches.

Another way: Use the counting weights $1$ , $5$ , $10$ , $10$ , $5$ , $1$ directly: $(2x+2y)+5(2x+y)+10(2x+y)+10(2x-y)+5(2x-y)+(2x-2y)=64x$ .

Standardsmin grade 7

7.EE.A.1 Apply properties of operations to add, subtract, factor, and expand linear expressions — Adding linear expressions and combining like terms.
6.EE.A.3 Apply properties to generate equivalent expressions — Collecting like terms to write equivalent expressions.

💡Takeaway. Add the

x

's with the

x

's and the

y

's with the

y

's, and watch symmetry do the cancelling.

Sum pyramid: symmetric terms cancel at the top · 12 practice problems

Generated variants — 12

Givens

Unknowns

Constraints

Givens

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Givens

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Givens

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Givens

Unknowns

Constraints

Givens

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Givens

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Givens

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Givens

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Where next?