← Each new chord adds as many regions as it crosses, plus one · Generalize a Growing Pattern into a Rule

Each new chord adds as many regions as it crosses, plus one · 12 practice problems

4.OA.C.55.OA.B.3

Generated variants — 12

Freshly produced from the archetype’s parameters — problem, figure, and solution derived together.

Variant 1 easy answer: 16 regions

Line segments are drawn so that the number of regions the circle is divided into is as large as possible. When 55 line segments are drawn, into how many regions is the circle divided?

The figures show segments drawn one after another inside a circle. In the 1st figure, 11 segment divides the circle into 22 regions. In the 2nd figure, 22 segments are drawn so they cross each other, dividing it into 44 regions. In the 3rd figure, 33 segments are drawn so they all cross one another, making the number of regions as large as possible. In this way, each new segment is drawn to cross all the previous segments so that the number of regions is maximized.

Show solution
1 · Understandwhat's really being asked

Chords are drawn in a circle so each new one crosses every earlier one, making as many regions as possible. We need the number of regions after 5 of them.

Givens
  • 1 segment gives 2 regions.
  • 2 crossing segments give 4 regions.
  • 3 mutually crossing segments give 7 regions.
  • Every new segment is drawn to cross all the earlier ones.
Unknowns
  • The number of regions after 5 segments.
Constraints
  • No three segments meet at one point -- that would waste a region.
2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#1 Draw a Diagram

Drawing 5 crossing chords by hand is hopeless. Do the first three, look at how much each new chord adds, and turn that into a rule that can be run straight up to 5.

3 · Execute4 carry out the plan

1Count the small cases by drawing

#9 Solve an Easier Related Problem 4.OA.C.5
Use the pictures. With 1 segment the circle has 2 regions. The 2nd segment crosses the 1st, giving 4. The 3rd crosses both earlier ones, giving 7.
1 seg→2,2 seg→4,3 seg→71\text{ seg} \to 2,\quad 2\text{ seg} \to 4,\quad 3\text{ seg} \to 7
2, 4, 7 -- not doubling.

2See how many regions each new segment adds

#5 Look for a Pattern 5.OA.B.3
Look at the jumps: 2, 4, 7. From 1 to 2 segments we add 2 regions; from 2 to 3 we add 3. The reason is that the k-th segment crosses the k-1 already there, so it is chopped into k pieces, and each piece splits a region in two.
2→4 adds 2,4→7 adds 32 \to 4\text{ adds }2,\quad 4 \to 7\text{ adds }3
The k-th segment adds exactly k regions.

3State the rule

#5 Look for a Pattern 5.OA.B.3
Start from the whole circle, which is 1 region before any segment is drawn, then add 1 for the first segment, 2 for the second, and so on.
R(n)=1+(1+2+3+⋯+n)R(n) = 1 + (1 + 2 + 3 + \cdots + n)
One rule instead of a drawing.

4Add up to 5 segments

#5 Look for a Pattern 4.OA.C.5
Add the whole circle plus the running total from 1 up to 5. That sum is 15, so the total is 1 + 15.
R(5)=1+(1+2+3+4+⋯+5)=1+15=16R(5) = 1 + (1 + 2 + 3 + 4 + \cdots + 5) = 1 + 15 = 16
16 regions.
Answer: 16 regions
4 · Reviewdoes it hold up?

Check the rule on a case we drew: R(3) = 1 + (1 + 2 + 3) = 1 + 6 = 7, which is what the third picture shows.

Another way: The sum 1 + 2 + ... + 5 can be done by pairing the ends: 5 x 6 / 2 = 15. Adding the circle itself gives 16.

Standardsmin grade 5
  • 4.OA.C.5 Generate a number or shape pattern that follows a given rule — Generating the sequence of region counts from the drawings.
  • 5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships — Relating 'segments drawn' to 'regions added' and using it.
💡Takeaway. When a pattern is too big to draw, look at how much each step adds -- the step size is usually the simple part.
Variant 2 easy answer: 22 regions

Line segments are drawn so that the number of regions the circle is divided into is as large as possible. When 66 line segments are drawn, into how many regions is the circle divided?

The figures show segments drawn one after another inside a circle. In the 1st figure, 11 segment divides the circle into 22 regions. In the 2nd figure, 22 segments are drawn so they cross each other, dividing it into 44 regions. In the 3rd figure, 33 segments are drawn so they all cross one another, making the number of regions as large as possible. In this way, each new segment is drawn to cross all the previous segments so that the number of regions is maximized.

Show solution
1 · Understandwhat's really being asked

Chords are drawn in a circle so each new one crosses every earlier one, making as many regions as possible. We need the number of regions after 6 of them.

Givens
  • 1 segment gives 2 regions.
  • 2 crossing segments give 4 regions.
  • 3 mutually crossing segments give 7 regions.
  • Every new segment is drawn to cross all the earlier ones.
Unknowns
  • The number of regions after 6 segments.
Constraints
  • No three segments meet at one point -- that would waste a region.
2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#1 Draw a Diagram

Drawing 6 crossing chords by hand is hopeless. Do the first three, look at how much each new chord adds, and turn that into a rule that can be run straight up to 6.

3 · Execute4 carry out the plan

1Count the small cases by drawing

#9 Solve an Easier Related Problem 4.OA.C.5
Use the pictures. With 1 segment the circle has 2 regions. The 2nd segment crosses the 1st, giving 4. The 3rd crosses both earlier ones, giving 7.
1 seg→2,2 seg→4,3 seg→71\text{ seg} \to 2,\quad 2\text{ seg} \to 4,\quad 3\text{ seg} \to 7
2, 4, 7 -- not doubling.

2See how many regions each new segment adds

#5 Look for a Pattern 5.OA.B.3
Look at the jumps: 2, 4, 7. From 1 to 2 segments we add 2 regions; from 2 to 3 we add 3. The reason is that the k-th segment crosses the k-1 already there, so it is chopped into k pieces, and each piece splits a region in two.
2→4 adds 2,4→7 adds 32 \to 4\text{ adds }2,\quad 4 \to 7\text{ adds }3
The k-th segment adds exactly k regions.

3State the rule

#5 Look for a Pattern 5.OA.B.3
Start from the whole circle, which is 1 region before any segment is drawn, then add 1 for the first segment, 2 for the second, and so on.
R(n)=1+(1+2+3+⋯+n)R(n) = 1 + (1 + 2 + 3 + \cdots + n)
One rule instead of a drawing.

4Add up to 6 segments

#5 Look for a Pattern 4.OA.C.5
Add the whole circle plus the running total from 1 up to 6. That sum is 21, so the total is 1 + 21.
R(6)=1+(1+2+3+4+⋯+6)=1+21=22R(6) = 1 + (1 + 2 + 3 + 4 + \cdots + 6) = 1 + 21 = 22
22 regions.
Answer: 22 regions
4 · Reviewdoes it hold up?

Check the rule on a case we drew: R(3) = 1 + (1 + 2 + 3) = 1 + 6 = 7, which is what the third picture shows.

Another way: The sum 1 + 2 + ... + 6 can be done by pairing the ends: 6 x 7 / 2 = 21. Adding the circle itself gives 22.

Standardsmin grade 5
  • 4.OA.C.5 Generate a number or shape pattern that follows a given rule — Generating the sequence of region counts from the drawings.
  • 5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships — Relating 'segments drawn' to 'regions added' and using it.
💡Takeaway. When a pattern is too big to draw, look at how much each step adds -- the step size is usually the simple part.
Variant 3 easy answer: 29 regions

Line segments are drawn so that the number of regions the circle is divided into is as large as possible. When 77 line segments are drawn, into how many regions is the circle divided?

The figures show segments drawn one after another inside a circle. In the 1st figure, 11 segment divides the circle into 22 regions. In the 2nd figure, 22 segments are drawn so they cross each other, dividing it into 44 regions. In the 3rd figure, 33 segments are drawn so they all cross one another, making the number of regions as large as possible. In this way, each new segment is drawn to cross all the previous segments so that the number of regions is maximized.

Show solution
1 · Understandwhat's really being asked

Chords are drawn in a circle so each new one crosses every earlier one, making as many regions as possible. We need the number of regions after 7 of them.

Givens
  • 1 segment gives 2 regions.
  • 2 crossing segments give 4 regions.
  • 3 mutually crossing segments give 7 regions.
  • Every new segment is drawn to cross all the earlier ones.
Unknowns
  • The number of regions after 7 segments.
Constraints
  • No three segments meet at one point -- that would waste a region.
2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#1 Draw a Diagram

Drawing 7 crossing chords by hand is hopeless. Do the first three, look at how much each new chord adds, and turn that into a rule that can be run straight up to 7.

3 · Execute4 carry out the plan

1Count the small cases by drawing

#9 Solve an Easier Related Problem 4.OA.C.5
Use the pictures. With 1 segment the circle has 2 regions. The 2nd segment crosses the 1st, giving 4. The 3rd crosses both earlier ones, giving 7.
1 seg→2,2 seg→4,3 seg→71\text{ seg} \to 2,\quad 2\text{ seg} \to 4,\quad 3\text{ seg} \to 7
2, 4, 7 -- not doubling.

2See how many regions each new segment adds

#5 Look for a Pattern 5.OA.B.3
Look at the jumps: 2, 4, 7. From 1 to 2 segments we add 2 regions; from 2 to 3 we add 3. The reason is that the k-th segment crosses the k-1 already there, so it is chopped into k pieces, and each piece splits a region in two.
2→4 adds 2,4→7 adds 32 \to 4\text{ adds }2,\quad 4 \to 7\text{ adds }3
The k-th segment adds exactly k regions.

3State the rule

#5 Look for a Pattern 5.OA.B.3
Start from the whole circle, which is 1 region before any segment is drawn, then add 1 for the first segment, 2 for the second, and so on.
R(n)=1+(1+2+3+⋯+n)R(n) = 1 + (1 + 2 + 3 + \cdots + n)
One rule instead of a drawing.

4Add up to 7 segments

#5 Look for a Pattern 4.OA.C.5
Add the whole circle plus the running total from 1 up to 7. That sum is 28, so the total is 1 + 28.
R(7)=1+(1+2+3+4+⋯+7)=1+28=29R(7) = 1 + (1 + 2 + 3 + 4 + \cdots + 7) = 1 + 28 = 29
29 regions.
Answer: 29 regions
4 · Reviewdoes it hold up?

Check the rule on a case we drew: R(3) = 1 + (1 + 2 + 3) = 1 + 6 = 7, which is what the third picture shows.

Another way: The sum 1 + 2 + ... + 7 can be done by pairing the ends: 7 x 8 / 2 = 28. Adding the circle itself gives 29.

Standardsmin grade 5
  • 4.OA.C.5 Generate a number or shape pattern that follows a given rule — Generating the sequence of region counts from the drawings.
  • 5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships — Relating 'segments drawn' to 'regions added' and using it.
💡Takeaway. When a pattern is too big to draw, look at how much each step adds -- the step size is usually the simple part.
Variant 4 easy answer: 37 regions

Line segments are drawn so that the number of regions the circle is divided into is as large as possible. When 88 line segments are drawn, into how many regions is the circle divided?

The figures show segments drawn one after another inside a circle. In the 1st figure, 11 segment divides the circle into 22 regions. In the 2nd figure, 22 segments are drawn so they cross each other, dividing it into 44 regions. In the 3rd figure, 33 segments are drawn so they all cross one another, making the number of regions as large as possible. In this way, each new segment is drawn to cross all the previous segments so that the number of regions is maximized.

Show solution
1 · Understandwhat's really being asked

Chords are drawn in a circle so each new one crosses every earlier one, making as many regions as possible. We need the number of regions after 8 of them.

Givens
  • 1 segment gives 2 regions.
  • 2 crossing segments give 4 regions.
  • 3 mutually crossing segments give 7 regions.
  • Every new segment is drawn to cross all the earlier ones.
Unknowns
  • The number of regions after 8 segments.
Constraints
  • No three segments meet at one point -- that would waste a region.
2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#1 Draw a Diagram

Drawing 8 crossing chords by hand is hopeless. Do the first three, look at how much each new chord adds, and turn that into a rule that can be run straight up to 8.

3 · Execute4 carry out the plan

1Count the small cases by drawing

#9 Solve an Easier Related Problem 4.OA.C.5
Use the pictures. With 1 segment the circle has 2 regions. The 2nd segment crosses the 1st, giving 4. The 3rd crosses both earlier ones, giving 7.
1 seg→2,2 seg→4,3 seg→71\text{ seg} \to 2,\quad 2\text{ seg} \to 4,\quad 3\text{ seg} \to 7
2, 4, 7 -- not doubling.

2See how many regions each new segment adds

#5 Look for a Pattern 5.OA.B.3
Look at the jumps: 2, 4, 7. From 1 to 2 segments we add 2 regions; from 2 to 3 we add 3. The reason is that the k-th segment crosses the k-1 already there, so it is chopped into k pieces, and each piece splits a region in two.
2→4 adds 2,4→7 adds 32 \to 4\text{ adds }2,\quad 4 \to 7\text{ adds }3
The k-th segment adds exactly k regions.

3State the rule

#5 Look for a Pattern 5.OA.B.3
Start from the whole circle, which is 1 region before any segment is drawn, then add 1 for the first segment, 2 for the second, and so on.
R(n)=1+(1+2+3+⋯+n)R(n) = 1 + (1 + 2 + 3 + \cdots + n)
One rule instead of a drawing.

4Add up to 8 segments

#5 Look for a Pattern 4.OA.C.5
Add the whole circle plus the running total from 1 up to 8. That sum is 36, so the total is 1 + 36.
R(8)=1+(1+2+3+4+⋯+8)=1+36=37R(8) = 1 + (1 + 2 + 3 + 4 + \cdots + 8) = 1 + 36 = 37
37 regions.
Answer: 37 regions
4 · Reviewdoes it hold up?

Check the rule on a case we drew: R(3) = 1 + (1 + 2 + 3) = 1 + 6 = 7, which is what the third picture shows.

Another way: The sum 1 + 2 + ... + 8 can be done by pairing the ends: 8 x 9 / 2 = 36. Adding the circle itself gives 37.

Standardsmin grade 5
  • 4.OA.C.5 Generate a number or shape pattern that follows a given rule — Generating the sequence of region counts from the drawings.
  • 5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships — Relating 'segments drawn' to 'regions added' and using it.
💡Takeaway. When a pattern is too big to draw, look at how much each step adds -- the step size is usually the simple part.
Variant 5 medium answer: 46 regions

Line segments are drawn so that the number of regions the circle is divided into is as large as possible. When 99 line segments are drawn, into how many regions is the circle divided?

The figures show segments drawn one after another inside a circle. In the 1st figure, 11 segment divides the circle into 22 regions. In the 2nd figure, 22 segments are drawn so they cross each other, dividing it into 44 regions. In the 3rd figure, 33 segments are drawn so they all cross one another, making the number of regions as large as possible. In this way, each new segment is drawn to cross all the previous segments so that the number of regions is maximized.

Show solution
1 · Understandwhat's really being asked

Chords are drawn in a circle so each new one crosses every earlier one, making as many regions as possible. We need the number of regions after 9 of them.

Givens
  • 1 segment gives 2 regions.
  • 2 crossing segments give 4 regions.
  • 3 mutually crossing segments give 7 regions.
  • Every new segment is drawn to cross all the earlier ones.
Unknowns
  • The number of regions after 9 segments.
Constraints
  • No three segments meet at one point -- that would waste a region.
2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#1 Draw a Diagram

Drawing 9 crossing chords by hand is hopeless. Do the first three, look at how much each new chord adds, and turn that into a rule that can be run straight up to 9.

3 · Execute4 carry out the plan

1Count the small cases by drawing

#9 Solve an Easier Related Problem 4.OA.C.5
Use the pictures. With 1 segment the circle has 2 regions. The 2nd segment crosses the 1st, giving 4. The 3rd crosses both earlier ones, giving 7.
1 seg→2,2 seg→4,3 seg→71\text{ seg} \to 2,\quad 2\text{ seg} \to 4,\quad 3\text{ seg} \to 7
2, 4, 7 -- not doubling.

2See how many regions each new segment adds

#5 Look for a Pattern 5.OA.B.3
Look at the jumps: 2, 4, 7. From 1 to 2 segments we add 2 regions; from 2 to 3 we add 3. The reason is that the k-th segment crosses the k-1 already there, so it is chopped into k pieces, and each piece splits a region in two.
2→4 adds 2,4→7 adds 32 \to 4\text{ adds }2,\quad 4 \to 7\text{ adds }3
The k-th segment adds exactly k regions.

3State the rule

#5 Look for a Pattern 5.OA.B.3
Start from the whole circle, which is 1 region before any segment is drawn, then add 1 for the first segment, 2 for the second, and so on.
R(n)=1+(1+2+3+⋯+n)R(n) = 1 + (1 + 2 + 3 + \cdots + n)
One rule instead of a drawing.

4Add up to 9 segments

#5 Look for a Pattern 4.OA.C.5
Add the whole circle plus the running total from 1 up to 9. That sum is 45, so the total is 1 + 45.
R(9)=1+(1+2+3+4+⋯+9)=1+45=46R(9) = 1 + (1 + 2 + 3 + 4 + \cdots + 9) = 1 + 45 = 46
46 regions.
Answer: 46 regions
4 · Reviewdoes it hold up?

Check the rule on a case we drew: R(3) = 1 + (1 + 2 + 3) = 1 + 6 = 7, which is what the third picture shows.

Another way: The sum 1 + 2 + ... + 9 can be done by pairing the ends: 9 x 10 / 2 = 45. Adding the circle itself gives 46.

Standardsmin grade 5
  • 4.OA.C.5 Generate a number or shape pattern that follows a given rule — Generating the sequence of region counts from the drawings.
  • 5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships — Relating 'segments drawn' to 'regions added' and using it.
💡Takeaway. When a pattern is too big to draw, look at how much each step adds -- the step size is usually the simple part.
Variant 6 medium answer: 56 regions

Line segments are drawn so that the number of regions the circle is divided into is as large as possible. When 1010 line segments are drawn, into how many regions is the circle divided?

The figures show segments drawn one after another inside a circle. In the 1st figure, 11 segment divides the circle into 22 regions. In the 2nd figure, 22 segments are drawn so they cross each other, dividing it into 44 regions. In the 3rd figure, 33 segments are drawn so they all cross one another, making the number of regions as large as possible. In this way, each new segment is drawn to cross all the previous segments so that the number of regions is maximized.

Show solution
1 · Understandwhat's really being asked

Chords are drawn in a circle so each new one crosses every earlier one, making as many regions as possible. We need the number of regions after 10 of them.

Givens
  • 1 segment gives 2 regions.
  • 2 crossing segments give 4 regions.
  • 3 mutually crossing segments give 7 regions.
  • Every new segment is drawn to cross all the earlier ones.
Unknowns
  • The number of regions after 10 segments.
Constraints
  • No three segments meet at one point -- that would waste a region.
2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#1 Draw a Diagram

Drawing 10 crossing chords by hand is hopeless. Do the first three, look at how much each new chord adds, and turn that into a rule that can be run straight up to 10.

3 · Execute4 carry out the plan

1Count the small cases by drawing

#9 Solve an Easier Related Problem 4.OA.C.5
Use the pictures. With 1 segment the circle has 2 regions. The 2nd segment crosses the 1st, giving 4. The 3rd crosses both earlier ones, giving 7.
1 seg→2,2 seg→4,3 seg→71\text{ seg} \to 2,\quad 2\text{ seg} \to 4,\quad 3\text{ seg} \to 7
2, 4, 7 -- not doubling.

2See how many regions each new segment adds

#5 Look for a Pattern 5.OA.B.3
Look at the jumps: 2, 4, 7. From 1 to 2 segments we add 2 regions; from 2 to 3 we add 3. The reason is that the k-th segment crosses the k-1 already there, so it is chopped into k pieces, and each piece splits a region in two.
2→4 adds 2,4→7 adds 32 \to 4\text{ adds }2,\quad 4 \to 7\text{ adds }3
The k-th segment adds exactly k regions.

3State the rule

#5 Look for a Pattern 5.OA.B.3
Start from the whole circle, which is 1 region before any segment is drawn, then add 1 for the first segment, 2 for the second, and so on.
R(n)=1+(1+2+3+⋯+n)R(n) = 1 + (1 + 2 + 3 + \cdots + n)
One rule instead of a drawing.

4Add up to 10 segments

#5 Look for a Pattern 4.OA.C.5
Add the whole circle plus the running total from 1 up to 10. That sum is 55, so the total is 1 + 55.
R(10)=1+(1+2+3+4+⋯+10)=1+55=56R(10) = 1 + (1 + 2 + 3 + 4 + \cdots + 10) = 1 + 55 = 56
56 regions.
Answer: 56 regions
4 · Reviewdoes it hold up?

Check the rule on a case we drew: R(3) = 1 + (1 + 2 + 3) = 1 + 6 = 7, which is what the third picture shows.

Another way: The sum 1 + 2 + ... + 10 can be done by pairing the ends: 10 x 11 / 2 = 55. Adding the circle itself gives 56.

Standardsmin grade 5
  • 4.OA.C.5 Generate a number or shape pattern that follows a given rule — Generating the sequence of region counts from the drawings.
  • 5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships — Relating 'segments drawn' to 'regions added' and using it.
💡Takeaway. When a pattern is too big to draw, look at how much each step adds -- the step size is usually the simple part.
Variant 7 medium answer: 67 regions

Line segments are drawn so that the number of regions the circle is divided into is as large as possible. When 1111 line segments are drawn, into how many regions is the circle divided?

The figures show segments drawn one after another inside a circle. In the 1st figure, 11 segment divides the circle into 22 regions. In the 2nd figure, 22 segments are drawn so they cross each other, dividing it into 44 regions. In the 3rd figure, 33 segments are drawn so they all cross one another, making the number of regions as large as possible. In this way, each new segment is drawn to cross all the previous segments so that the number of regions is maximized.

Show solution
1 · Understandwhat's really being asked

Chords are drawn in a circle so each new one crosses every earlier one, making as many regions as possible. We need the number of regions after 11 of them.

Givens
  • 1 segment gives 2 regions.
  • 2 crossing segments give 4 regions.
  • 3 mutually crossing segments give 7 regions.
  • Every new segment is drawn to cross all the earlier ones.
Unknowns
  • The number of regions after 11 segments.
Constraints
  • No three segments meet at one point -- that would waste a region.
2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#1 Draw a Diagram

Drawing 11 crossing chords by hand is hopeless. Do the first three, look at how much each new chord adds, and turn that into a rule that can be run straight up to 11.

3 · Execute4 carry out the plan

1Count the small cases by drawing

#9 Solve an Easier Related Problem 4.OA.C.5
Use the pictures. With 1 segment the circle has 2 regions. The 2nd segment crosses the 1st, giving 4. The 3rd crosses both earlier ones, giving 7.
1 seg→2,2 seg→4,3 seg→71\text{ seg} \to 2,\quad 2\text{ seg} \to 4,\quad 3\text{ seg} \to 7
2, 4, 7 -- not doubling.

2See how many regions each new segment adds

#5 Look for a Pattern 5.OA.B.3
Look at the jumps: 2, 4, 7. From 1 to 2 segments we add 2 regions; from 2 to 3 we add 3. The reason is that the k-th segment crosses the k-1 already there, so it is chopped into k pieces, and each piece splits a region in two.
2→4 adds 2,4→7 adds 32 \to 4\text{ adds }2,\quad 4 \to 7\text{ adds }3
The k-th segment adds exactly k regions.

3State the rule

#5 Look for a Pattern 5.OA.B.3
Start from the whole circle, which is 1 region before any segment is drawn, then add 1 for the first segment, 2 for the second, and so on.
R(n)=1+(1+2+3+⋯+n)R(n) = 1 + (1 + 2 + 3 + \cdots + n)
One rule instead of a drawing.

4Add up to 11 segments

#5 Look for a Pattern 4.OA.C.5
Add the whole circle plus the running total from 1 up to 11. That sum is 66, so the total is 1 + 66.
R(11)=1+(1+2+3+4+⋯+11)=1+66=67R(11) = 1 + (1 + 2 + 3 + 4 + \cdots + 11) = 1 + 66 = 67
67 regions.
Answer: 67 regions
4 · Reviewdoes it hold up?

Check the rule on a case we drew: R(3) = 1 + (1 + 2 + 3) = 1 + 6 = 7, which is what the third picture shows.

Another way: The sum 1 + 2 + ... + 11 can be done by pairing the ends: 11 x 12 / 2 = 66. Adding the circle itself gives 67.

Standardsmin grade 5
  • 4.OA.C.5 Generate a number or shape pattern that follows a given rule — Generating the sequence of region counts from the drawings.
  • 5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships — Relating 'segments drawn' to 'regions added' and using it.
💡Takeaway. When a pattern is too big to draw, look at how much each step adds -- the step size is usually the simple part.
Variant 8 medium answer: 79 regions

Line segments are drawn so that the number of regions the circle is divided into is as large as possible. When 1212 line segments are drawn, into how many regions is the circle divided?

The figures show segments drawn one after another inside a circle. In the 1st figure, 11 segment divides the circle into 22 regions. In the 2nd figure, 22 segments are drawn so they cross each other, dividing it into 44 regions. In the 3rd figure, 33 segments are drawn so they all cross one another, making the number of regions as large as possible. In this way, each new segment is drawn to cross all the previous segments so that the number of regions is maximized.

Show solution
1 · Understandwhat's really being asked

Chords are drawn in a circle so each new one crosses every earlier one, making as many regions as possible. We need the number of regions after 12 of them.

Givens
  • 1 segment gives 2 regions.
  • 2 crossing segments give 4 regions.
  • 3 mutually crossing segments give 7 regions.
  • Every new segment is drawn to cross all the earlier ones.
Unknowns
  • The number of regions after 12 segments.
Constraints
  • No three segments meet at one point -- that would waste a region.
2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#1 Draw a Diagram

Drawing 12 crossing chords by hand is hopeless. Do the first three, look at how much each new chord adds, and turn that into a rule that can be run straight up to 12.

3 · Execute4 carry out the plan

1Count the small cases by drawing

#9 Solve an Easier Related Problem 4.OA.C.5
Use the pictures. With 1 segment the circle has 2 regions. The 2nd segment crosses the 1st, giving 4. The 3rd crosses both earlier ones, giving 7.
1 seg→2,2 seg→4,3 seg→71\text{ seg} \to 2,\quad 2\text{ seg} \to 4,\quad 3\text{ seg} \to 7
2, 4, 7 -- not doubling.

2See how many regions each new segment adds

#5 Look for a Pattern 5.OA.B.3
Look at the jumps: 2, 4, 7. From 1 to 2 segments we add 2 regions; from 2 to 3 we add 3. The reason is that the k-th segment crosses the k-1 already there, so it is chopped into k pieces, and each piece splits a region in two.
2→4 adds 2,4→7 adds 32 \to 4\text{ adds }2,\quad 4 \to 7\text{ adds }3
The k-th segment adds exactly k regions.

3State the rule

#5 Look for a Pattern 5.OA.B.3
Start from the whole circle, which is 1 region before any segment is drawn, then add 1 for the first segment, 2 for the second, and so on.
R(n)=1+(1+2+3+⋯+n)R(n) = 1 + (1 + 2 + 3 + \cdots + n)
One rule instead of a drawing.

4Add up to 12 segments

#5 Look for a Pattern 4.OA.C.5
Add the whole circle plus the running total from 1 up to 12. That sum is 78, so the total is 1 + 78.
R(12)=1+(1+2+3+4+⋯+12)=1+78=79R(12) = 1 + (1 + 2 + 3 + 4 + \cdots + 12) = 1 + 78 = 79
79 regions.
Answer: 79 regions
4 · Reviewdoes it hold up?

Check the rule on a case we drew: R(3) = 1 + (1 + 2 + 3) = 1 + 6 = 7, which is what the third picture shows.

Another way: The sum 1 + 2 + ... + 12 can be done by pairing the ends: 12 x 13 / 2 = 78. Adding the circle itself gives 79.

Standardsmin grade 5
  • 4.OA.C.5 Generate a number or shape pattern that follows a given rule — Generating the sequence of region counts from the drawings.
  • 5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships — Relating 'segments drawn' to 'regions added' and using it.
💡Takeaway. When a pattern is too big to draw, look at how much each step adds -- the step size is usually the simple part.
Variant 9 hard answer: 106 regions

Line segments are drawn so that the number of regions the circle is divided into is as large as possible. When 1414 line segments are drawn, into how many regions is the circle divided?

The figures show segments drawn one after another inside a circle. In the 1st figure, 11 segment divides the circle into 22 regions. In the 2nd figure, 22 segments are drawn so they cross each other, dividing it into 44 regions. In the 3rd figure, 33 segments are drawn so they all cross one another, making the number of regions as large as possible. In this way, each new segment is drawn to cross all the previous segments so that the number of regions is maximized.

Show solution
1 · Understandwhat's really being asked

Chords are drawn in a circle so each new one crosses every earlier one, making as many regions as possible. We need the number of regions after 14 of them.

Givens
  • 1 segment gives 2 regions.
  • 2 crossing segments give 4 regions.
  • 3 mutually crossing segments give 7 regions.
  • Every new segment is drawn to cross all the earlier ones.
Unknowns
  • The number of regions after 14 segments.
Constraints
  • No three segments meet at one point -- that would waste a region.
2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#1 Draw a Diagram

Drawing 14 crossing chords by hand is hopeless. Do the first three, look at how much each new chord adds, and turn that into a rule that can be run straight up to 14.

3 · Execute4 carry out the plan

1Count the small cases by drawing

#9 Solve an Easier Related Problem 4.OA.C.5
Use the pictures. With 1 segment the circle has 2 regions. The 2nd segment crosses the 1st, giving 4. The 3rd crosses both earlier ones, giving 7.
1 seg→2,2 seg→4,3 seg→71\text{ seg} \to 2,\quad 2\text{ seg} \to 4,\quad 3\text{ seg} \to 7
2, 4, 7 -- not doubling.

2See how many regions each new segment adds

#5 Look for a Pattern 5.OA.B.3
Look at the jumps: 2, 4, 7. From 1 to 2 segments we add 2 regions; from 2 to 3 we add 3. The reason is that the k-th segment crosses the k-1 already there, so it is chopped into k pieces, and each piece splits a region in two.
2→4 adds 2,4→7 adds 32 \to 4\text{ adds }2,\quad 4 \to 7\text{ adds }3
The k-th segment adds exactly k regions.

3State the rule

#5 Look for a Pattern 5.OA.B.3
Start from the whole circle, which is 1 region before any segment is drawn, then add 1 for the first segment, 2 for the second, and so on.
R(n)=1+(1+2+3+⋯+n)R(n) = 1 + (1 + 2 + 3 + \cdots + n)
One rule instead of a drawing.

4Add up to 14 segments

#5 Look for a Pattern 4.OA.C.5
Add the whole circle plus the running total from 1 up to 14. That sum is 105, so the total is 1 + 105.
R(14)=1+(1+2+3+4+⋯+14)=1+105=106R(14) = 1 + (1 + 2 + 3 + 4 + \cdots + 14) = 1 + 105 = 106
106 regions.
Answer: 106 regions
4 · Reviewdoes it hold up?

Check the rule on a case we drew: R(3) = 1 + (1 + 2 + 3) = 1 + 6 = 7, which is what the third picture shows.

Another way: The sum 1 + 2 + ... + 14 can be done by pairing the ends: 14 x 15 / 2 = 105. Adding the circle itself gives 106.

Standardsmin grade 5
  • 4.OA.C.5 Generate a number or shape pattern that follows a given rule — Generating the sequence of region counts from the drawings.
  • 5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships — Relating 'segments drawn' to 'regions added' and using it.
💡Takeaway. When a pattern is too big to draw, look at how much each step adds -- the step size is usually the simple part.
Variant 10 hard answer: 121 regions

Line segments are drawn so that the number of regions the circle is divided into is as large as possible. When 1515 line segments are drawn, into how many regions is the circle divided?

The figures show segments drawn one after another inside a circle. In the 1st figure, 11 segment divides the circle into 22 regions. In the 2nd figure, 22 segments are drawn so they cross each other, dividing it into 44 regions. In the 3rd figure, 33 segments are drawn so they all cross one another, making the number of regions as large as possible. In this way, each new segment is drawn to cross all the previous segments so that the number of regions is maximized.

Show solution
1 · Understandwhat's really being asked

Chords are drawn in a circle so each new one crosses every earlier one, making as many regions as possible. We need the number of regions after 15 of them.

Givens
  • 1 segment gives 2 regions.
  • 2 crossing segments give 4 regions.
  • 3 mutually crossing segments give 7 regions.
  • Every new segment is drawn to cross all the earlier ones.
Unknowns
  • The number of regions after 15 segments.
Constraints
  • No three segments meet at one point -- that would waste a region.
2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#1 Draw a Diagram

Drawing 15 crossing chords by hand is hopeless. Do the first three, look at how much each new chord adds, and turn that into a rule that can be run straight up to 15.

3 · Execute4 carry out the plan

1Count the small cases by drawing

#9 Solve an Easier Related Problem 4.OA.C.5
Use the pictures. With 1 segment the circle has 2 regions. The 2nd segment crosses the 1st, giving 4. The 3rd crosses both earlier ones, giving 7.
1 seg→2,2 seg→4,3 seg→71\text{ seg} \to 2,\quad 2\text{ seg} \to 4,\quad 3\text{ seg} \to 7
2, 4, 7 -- not doubling.

2See how many regions each new segment adds

#5 Look for a Pattern 5.OA.B.3
Look at the jumps: 2, 4, 7. From 1 to 2 segments we add 2 regions; from 2 to 3 we add 3. The reason is that the k-th segment crosses the k-1 already there, so it is chopped into k pieces, and each piece splits a region in two.
2→4 adds 2,4→7 adds 32 \to 4\text{ adds }2,\quad 4 \to 7\text{ adds }3
The k-th segment adds exactly k regions.

3State the rule

#5 Look for a Pattern 5.OA.B.3
Start from the whole circle, which is 1 region before any segment is drawn, then add 1 for the first segment, 2 for the second, and so on.
R(n)=1+(1+2+3+⋯+n)R(n) = 1 + (1 + 2 + 3 + \cdots + n)
One rule instead of a drawing.

4Add up to 15 segments

#5 Look for a Pattern 4.OA.C.5
Add the whole circle plus the running total from 1 up to 15. That sum is 120, so the total is 1 + 120.
R(15)=1+(1+2+3+4+⋯+15)=1+120=121R(15) = 1 + (1 + 2 + 3 + 4 + \cdots + 15) = 1 + 120 = 121
121 regions.
Answer: 121 regions
4 · Reviewdoes it hold up?

Check the rule on a case we drew: R(3) = 1 + (1 + 2 + 3) = 1 + 6 = 7, which is what the third picture shows.

Another way: The sum 1 + 2 + ... + 15 can be done by pairing the ends: 15 x 16 / 2 = 120. Adding the circle itself gives 121.

Standardsmin grade 5
  • 4.OA.C.5 Generate a number or shape pattern that follows a given rule — Generating the sequence of region counts from the drawings.
  • 5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships — Relating 'segments drawn' to 'regions added' and using it.
💡Takeaway. When a pattern is too big to draw, look at how much each step adds -- the step size is usually the simple part.
Variant 11 hard answer: 172 regions

Line segments are drawn so that the number of regions the circle is divided into is as large as possible. When 1818 line segments are drawn, into how many regions is the circle divided?

The figures show segments drawn one after another inside a circle. In the 1st figure, 11 segment divides the circle into 22 regions. In the 2nd figure, 22 segments are drawn so they cross each other, dividing it into 44 regions. In the 3rd figure, 33 segments are drawn so they all cross one another, making the number of regions as large as possible. In this way, each new segment is drawn to cross all the previous segments so that the number of regions is maximized.

Show solution
1 · Understandwhat's really being asked

Chords are drawn in a circle so each new one crosses every earlier one, making as many regions as possible. We need the number of regions after 18 of them.

Givens
  • 1 segment gives 2 regions.
  • 2 crossing segments give 4 regions.
  • 3 mutually crossing segments give 7 regions.
  • Every new segment is drawn to cross all the earlier ones.
Unknowns
  • The number of regions after 18 segments.
Constraints
  • No three segments meet at one point -- that would waste a region.
2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#1 Draw a Diagram

Drawing 18 crossing chords by hand is hopeless. Do the first three, look at how much each new chord adds, and turn that into a rule that can be run straight up to 18.

3 · Execute4 carry out the plan

1Count the small cases by drawing

#9 Solve an Easier Related Problem 4.OA.C.5
Use the pictures. With 1 segment the circle has 2 regions. The 2nd segment crosses the 1st, giving 4. The 3rd crosses both earlier ones, giving 7.
1 seg→2,2 seg→4,3 seg→71\text{ seg} \to 2,\quad 2\text{ seg} \to 4,\quad 3\text{ seg} \to 7
2, 4, 7 -- not doubling.

2See how many regions each new segment adds

#5 Look for a Pattern 5.OA.B.3
Look at the jumps: 2, 4, 7. From 1 to 2 segments we add 2 regions; from 2 to 3 we add 3. The reason is that the k-th segment crosses the k-1 already there, so it is chopped into k pieces, and each piece splits a region in two.
2→4 adds 2,4→7 adds 32 \to 4\text{ adds }2,\quad 4 \to 7\text{ adds }3
The k-th segment adds exactly k regions.

3State the rule

#5 Look for a Pattern 5.OA.B.3
Start from the whole circle, which is 1 region before any segment is drawn, then add 1 for the first segment, 2 for the second, and so on.
R(n)=1+(1+2+3+⋯+n)R(n) = 1 + (1 + 2 + 3 + \cdots + n)
One rule instead of a drawing.

4Add up to 18 segments

#5 Look for a Pattern 4.OA.C.5
Add the whole circle plus the running total from 1 up to 18. That sum is 171, so the total is 1 + 171.
R(18)=1+(1+2+3+4+⋯+18)=1+171=172R(18) = 1 + (1 + 2 + 3 + 4 + \cdots + 18) = 1 + 171 = 172
172 regions.
Answer: 172 regions
4 · Reviewdoes it hold up?

Check the rule on a case we drew: R(3) = 1 + (1 + 2 + 3) = 1 + 6 = 7, which is what the third picture shows.

Another way: The sum 1 + 2 + ... + 18 can be done by pairing the ends: 18 x 19 / 2 = 171. Adding the circle itself gives 172.

Standardsmin grade 5
  • 4.OA.C.5 Generate a number or shape pattern that follows a given rule — Generating the sequence of region counts from the drawings.
  • 5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships — Relating 'segments drawn' to 'regions added' and using it.
💡Takeaway. When a pattern is too big to draw, look at how much each step adds -- the step size is usually the simple part.
Variant 12 hard answer: 211 regions

Line segments are drawn so that the number of regions the circle is divided into is as large as possible. When 2020 line segments are drawn, into how many regions is the circle divided?

The figures show segments drawn one after another inside a circle. In the 1st figure, 11 segment divides the circle into 22 regions. In the 2nd figure, 22 segments are drawn so they cross each other, dividing it into 44 regions. In the 3rd figure, 33 segments are drawn so they all cross one another, making the number of regions as large as possible. In this way, each new segment is drawn to cross all the previous segments so that the number of regions is maximized.

Show solution
1 · Understandwhat's really being asked

Chords are drawn in a circle so each new one crosses every earlier one, making as many regions as possible. We need the number of regions after 20 of them.

Givens
  • 1 segment gives 2 regions.
  • 2 crossing segments give 4 regions.
  • 3 mutually crossing segments give 7 regions.
  • Every new segment is drawn to cross all the earlier ones.
Unknowns
  • The number of regions after 20 segments.
Constraints
  • No three segments meet at one point -- that would waste a region.
2 · Planchoose the strategy

#5 Look for a Pattern · also uses: #9 Solve an Easier Related Problem#1 Draw a Diagram

Drawing 20 crossing chords by hand is hopeless. Do the first three, look at how much each new chord adds, and turn that into a rule that can be run straight up to 20.

3 · Execute4 carry out the plan

1Count the small cases by drawing

#9 Solve an Easier Related Problem 4.OA.C.5
Use the pictures. With 1 segment the circle has 2 regions. The 2nd segment crosses the 1st, giving 4. The 3rd crosses both earlier ones, giving 7.
1 seg→2,2 seg→4,3 seg→71\text{ seg} \to 2,\quad 2\text{ seg} \to 4,\quad 3\text{ seg} \to 7
2, 4, 7 -- not doubling.

2See how many regions each new segment adds

#5 Look for a Pattern 5.OA.B.3
Look at the jumps: 2, 4, 7. From 1 to 2 segments we add 2 regions; from 2 to 3 we add 3. The reason is that the k-th segment crosses the k-1 already there, so it is chopped into k pieces, and each piece splits a region in two.
2→4 adds 2,4→7 adds 32 \to 4\text{ adds }2,\quad 4 \to 7\text{ adds }3
The k-th segment adds exactly k regions.

3State the rule

#5 Look for a Pattern 5.OA.B.3
Start from the whole circle, which is 1 region before any segment is drawn, then add 1 for the first segment, 2 for the second, and so on.
R(n)=1+(1+2+3+⋯+n)R(n) = 1 + (1 + 2 + 3 + \cdots + n)
One rule instead of a drawing.

4Add up to 20 segments

#5 Look for a Pattern 4.OA.C.5
Add the whole circle plus the running total from 1 up to 20. That sum is 210, so the total is 1 + 210.
R(20)=1+(1+2+3+4+⋯+20)=1+210=211R(20) = 1 + (1 + 2 + 3 + 4 + \cdots + 20) = 1 + 210 = 211
211 regions.
Answer: 211 regions
4 · Reviewdoes it hold up?

Check the rule on a case we drew: R(3) = 1 + (1 + 2 + 3) = 1 + 6 = 7, which is what the third picture shows.

Another way: The sum 1 + 2 + ... + 20 can be done by pairing the ends: 20 x 21 / 2 = 210. Adding the circle itself gives 211.

Standardsmin grade 5
  • 4.OA.C.5 Generate a number or shape pattern that follows a given rule — Generating the sequence of region counts from the drawings.
  • 5.OA.B.3 Generate two numerical patterns using two given rules and identify relationships — Relating 'segments drawn' to 'regions added' and using it.
💡Takeaway. When a pattern is too big to draw, look at how much each step adds -- the step size is usually the simple part.