Practice · Small angles combine into larger angles · Sensim Math

Variant 1 answer: 18 acute angles

Find how many acute angles, large and small, can be found in the figure.

From a single point on a straight line (the base), several rays extend upward, forming $7$ small angles labeled $1, 2, 3, 4, 5, 6, 7$ between neighboring rays. Each of the $7$ small angles is acute, and neighboring small angles can be combined to make larger angles. Count every acute angle (single or combined) in the figure.

Show solution

Understand

Several rays come out of one point on a straight line, making 7 small acute angles in a row labeled 1-7. By joining neighboring small angles we get bigger angles too. We must count every acute angle (small and large) that appears in the figure.

Givens

7 small angles in a row sit between neighboring rays above the base line.
Each of the small angles is acute (less than 90 degrees).
The small angles together fill the half-turn above the line, so they add up to 180 degrees.
Neighboring small angles can be combined into a single larger angle.

Unknowns

How many of the angles formed (single or combined) are acute.

Constraints

Only acute angles (strictly less than 90 degrees) are counted.
A combined angle must use neighboring small angles with no gaps.

Plan

#2 Make a Systematic List · also uses: #7 Identify Subproblems#1 Draw a Diagram

List every angle you can build from consecutive small angles in an orderly way (length 1, then 2, then 3...). Group them by how many small angles they contain, then keep only the ones small enough to stay acute.

Execute

#2 Make a Systematic List 4.MD.C.5

The 7 angles 1-7 each stand alone. All 7 are acute, so all 7 single angles count.

7 \text{ single acute angles}

An angle is the opening between two rays; each gap between neighboring rays is one such angle.

#2 Make a Systematic List 4.MD.C.7

Joining two neighboring small angles gives (1+2), (2+3), (3+4), (4+5), (5+6), (6+7): 6 larger angles, of which 6 still open less than 90 degrees and so are acute.

(1+2), (2+3), (3+4), (4+5), (5+6), (6+7) \rightarrow 6 \text{ acute angles}

Angle measure adds: two angles laid side by side make one angle whose size is their sum.

#7 Identify Subproblems 4.MD.C.7

Runs of three or more small angles add up to larger openings. Of all 28 possible runs, 5 of the longer ones reach 90 degrees or more, so they are not acute and are not counted.

28 - 7 - 6 = 15 \text{ non-acute or longer runs}

Keep adding pieces of the 180-degree half-turn and the angle eventually reaches a right angle and beyond, so it stops being acute.

#2 Make a Systematic List 4.MD.C.7

Acute angles are the 7 singles plus the 6 pairs (plus any acute longer runs).

7 + 6 + 5 = 18

Just total up the groups we kept.

Answer: 18 acute angles

Review

There are 28 angles in all (7 + 6 + 5 + 4 + 3 + 2 + 1). The longer combinations open toward and past a right angle, so it makes sense that only the 18 shorter ones stay acute. 18 is less than 28, as it must be.

Guess and check (tool 6): sketch the rays roughly equally spaced and estimate each small angle, then add neighbors until the running total first reaches 90 degrees to see where acute stops.

Standards · min grade 4

4.MD.C.5 Recognize angles as geometric shapes formed when two rays share an endpoint — Identifying each opening between neighboring rays as an angle.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Combining neighboring small angles and judging whether the total stays acute.

💡 List the angles by how many pieces they use, then keep only the ones still smaller than a right angle - that is Grade 4 angle sense you already have!

Variant 2 answer: 4 acute angles

Find how many acute angles, large and small, can be found in the figure.

From a single point on a straight line (the base), several rays extend upward, forming $4$ small angles labeled $1, 2, 3, 4$ between neighboring rays. Each of the $4$ small angles is acute, and neighboring small angles can be combined to make larger angles. Count every acute angle (single or combined) in the figure.

Show solution

Understand

Several rays come out of one point on a straight line, making 4 small acute angles in a row labeled 1-4. By joining neighboring small angles we get bigger angles too. We must count every acute angle (small and large) that appears in the figure.

Givens

4 small angles in a row sit between neighboring rays above the base line.
Each of the small angles is acute (less than 90 degrees).
The small angles together fill the half-turn above the line, so they add up to 180 degrees.
Neighboring small angles can be combined into a single larger angle.

Unknowns

How many of the angles formed (single or combined) are acute.

Constraints

Only acute angles (strictly less than 90 degrees) are counted.
A combined angle must use neighboring small angles with no gaps.

Plan

#2 Make a Systematic List · also uses: #7 Identify Subproblems#1 Draw a Diagram

List every angle you can build from consecutive small angles in an orderly way (length 1, then 2, then 3...). Group them by how many small angles they contain, then keep only the ones small enough to stay acute.

Execute

#2 Make a Systematic List 4.MD.C.5

The 4 angles 1-4 each stand alone. All 4 are acute, so all 4 single angles count.

4 \text{ single acute angles}

An angle is the opening between two rays; each gap between neighboring rays is one such angle.

#2 Make a Systematic List 4.MD.C.7

Joining two neighboring small angles gives (1+2), (2+3), (3+4): 3 larger angles, of which 0 still open less than 90 degrees and so are acute.

(1+2), (2+3), (3+4) \rightarrow 0 \text{ acute angles}

Angle measure adds: two angles laid side by side make one angle whose size is their sum.

#7 Identify Subproblems 4.MD.C.7

Runs of three or more small angles add up to larger openings. Of all 10 possible runs, 0 of the longer ones reach 90 degrees or more, so they are not acute and are not counted.

10 - 4 - 0 = 6 \text{ non-acute or longer runs}

Keep adding pieces of the 180-degree half-turn and the angle eventually reaches a right angle and beyond, so it stops being acute.

#2 Make a Systematic List 4.MD.C.7

Acute angles are the 4 singles plus the 0 pairs (plus any acute longer runs).

4 + 0 = 4

Just total up the groups we kept.

Answer: 4 acute angles

Review

There are 10 angles in all (4 + 3 + 2 + 1). The longer combinations open toward and past a right angle, so it makes sense that only the 4 shorter ones stay acute. 4 is less than 10, as it must be.

Guess and check (tool 6): sketch the rays roughly equally spaced and estimate each small angle, then add neighbors until the running total first reaches 90 degrees to see where acute stops.

Standards · min grade 4

4.MD.C.5 Recognize angles as geometric shapes formed when two rays share an endpoint — Identifying each opening between neighboring rays as an angle.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Combining neighboring small angles and judging whether the total stays acute.

💡 List the angles by how many pieces they use, then keep only the ones still smaller than a right angle - that is Grade 4 angle sense you already have!

Variant 3 answer: 4 acute angles

Find how many acute angles, large and small, can be found in the figure.

From a single point on a straight line (the base), several rays extend upward, forming $4$ small angles labeled $1, 2, 3, 4$ between neighboring rays. Each of the $4$ small angles is acute, and neighboring small angles can be combined to make larger angles. Count every acute angle (single or combined) in the figure.

Show solution

Understand

Several rays come out of one point on a straight line, making 4 small acute angles in a row labeled 1-4. By joining neighboring small angles we get bigger angles too. We must count every acute angle (small and large) that appears in the figure.

Givens

4 small angles in a row sit between neighboring rays above the base line.
Each of the small angles is acute (less than 90 degrees).
The small angles together fill the half-turn above the line, so they add up to 180 degrees.
Neighboring small angles can be combined into a single larger angle.

Unknowns

How many of the angles formed (single or combined) are acute.

Constraints

Only acute angles (strictly less than 90 degrees) are counted.
A combined angle must use neighboring small angles with no gaps.

Plan

#2 Make a Systematic List · also uses: #7 Identify Subproblems#1 Draw a Diagram

List every angle you can build from consecutive small angles in an orderly way (length 1, then 2, then 3...). Group them by how many small angles they contain, then keep only the ones small enough to stay acute.

Execute

#2 Make a Systematic List 4.MD.C.5

The 4 angles 1-4 each stand alone. All 4 are acute, so all 4 single angles count.

4 \text{ single acute angles}

An angle is the opening between two rays; each gap between neighboring rays is one such angle.

#2 Make a Systematic List 4.MD.C.7

Joining two neighboring small angles gives (1+2), (2+3), (3+4): 3 larger angles, of which 0 still open less than 90 degrees and so are acute.

(1+2), (2+3), (3+4) \rightarrow 0 \text{ acute angles}

Angle measure adds: two angles laid side by side make one angle whose size is their sum.

#7 Identify Subproblems 4.MD.C.7

Runs of three or more small angles add up to larger openings. Of all 10 possible runs, 0 of the longer ones reach 90 degrees or more, so they are not acute and are not counted.

10 - 4 - 0 = 6 \text{ non-acute or longer runs}

Keep adding pieces of the 180-degree half-turn and the angle eventually reaches a right angle and beyond, so it stops being acute.

#2 Make a Systematic List 4.MD.C.7

Acute angles are the 4 singles plus the 0 pairs (plus any acute longer runs).

4 + 0 = 4

Just total up the groups we kept.

Answer: 4 acute angles

Review

There are 10 angles in all (4 + 3 + 2 + 1). The longer combinations open toward and past a right angle, so it makes sense that only the 4 shorter ones stay acute. 4 is less than 10, as it must be.

Guess and check (tool 6): sketch the rays roughly equally spaced and estimate each small angle, then add neighbors until the running total first reaches 90 degrees to see where acute stops.

Standards · min grade 4

4.MD.C.5 Recognize angles as geometric shapes formed when two rays share an endpoint — Identifying each opening between neighboring rays as an angle.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Combining neighboring small angles and judging whether the total stays acute.

💡 List the angles by how many pieces they use, then keep only the ones still smaller than a right angle - that is Grade 4 angle sense you already have!

Variant 4 answer: 4 acute angles

Find how many acute angles, large and small, can be found in the figure.

From a single point on a straight line (the base), several rays extend upward, forming $4$ small angles labeled $1, 2, 3, 4$ between neighboring rays. Each of the $4$ small angles is acute, and neighboring small angles can be combined to make larger angles. Count every acute angle (single or combined) in the figure.

Show solution

Understand

Several rays come out of one point on a straight line, making 4 small acute angles in a row labeled 1-4. By joining neighboring small angles we get bigger angles too. We must count every acute angle (small and large) that appears in the figure.

Givens

4 small angles in a row sit between neighboring rays above the base line.
Each of the small angles is acute (less than 90 degrees).
The small angles together fill the half-turn above the line, so they add up to 180 degrees.
Neighboring small angles can be combined into a single larger angle.

Unknowns

How many of the angles formed (single or combined) are acute.

Constraints

Only acute angles (strictly less than 90 degrees) are counted.
A combined angle must use neighboring small angles with no gaps.

Plan

#2 Make a Systematic List · also uses: #7 Identify Subproblems#1 Draw a Diagram

List every angle you can build from consecutive small angles in an orderly way (length 1, then 2, then 3...). Group them by how many small angles they contain, then keep only the ones small enough to stay acute.

Execute

#2 Make a Systematic List 4.MD.C.5

The 4 angles 1-4 each stand alone. All 4 are acute, so all 4 single angles count.

4 \text{ single acute angles}

An angle is the opening between two rays; each gap between neighboring rays is one such angle.

#2 Make a Systematic List 4.MD.C.7

Joining two neighboring small angles gives (1+2), (2+3), (3+4): 3 larger angles, of which 0 still open less than 90 degrees and so are acute.

(1+2), (2+3), (3+4) \rightarrow 0 \text{ acute angles}

Angle measure adds: two angles laid side by side make one angle whose size is their sum.

#7 Identify Subproblems 4.MD.C.7

Runs of three or more small angles add up to larger openings. Of all 10 possible runs, 0 of the longer ones reach 90 degrees or more, so they are not acute and are not counted.

10 - 4 - 0 = 6 \text{ non-acute or longer runs}

Keep adding pieces of the 180-degree half-turn and the angle eventually reaches a right angle and beyond, so it stops being acute.

#2 Make a Systematic List 4.MD.C.7

Acute angles are the 4 singles plus the 0 pairs (plus any acute longer runs).

4 + 0 = 4

Just total up the groups we kept.

Answer: 4 acute angles

Review

There are 10 angles in all (4 + 3 + 2 + 1). The longer combinations open toward and past a right angle, so it makes sense that only the 4 shorter ones stay acute. 4 is less than 10, as it must be.

Guess and check (tool 6): sketch the rays roughly equally spaced and estimate each small angle, then add neighbors until the running total first reaches 90 degrees to see where acute stops.

Standards · min grade 4

4.MD.C.5 Recognize angles as geometric shapes formed when two rays share an endpoint — Identifying each opening between neighboring rays as an angle.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Combining neighboring small angles and judging whether the total stays acute.

💡 List the angles by how many pieces they use, then keep only the ones still smaller than a right angle - that is Grade 4 angle sense you already have!

Variant 5 answer: 9 acute angles

Find how many acute angles, large and small, can be found in the figure.

From a single point on a straight line (the base), several rays extend upward, forming $5$ small angles labeled $1, 2, 3, 4, 5$ between neighboring rays. Each of the $5$ small angles is acute, and neighboring small angles can be combined to make larger angles. Count every acute angle (single or combined) in the figure.

Show solution

Understand

Several rays come out of one point on a straight line, making 5 small acute angles in a row labeled 1-5. By joining neighboring small angles we get bigger angles too. We must count every acute angle (small and large) that appears in the figure.

Givens

5 small angles in a row sit between neighboring rays above the base line.
Each of the small angles is acute (less than 90 degrees).
The small angles together fill the half-turn above the line, so they add up to 180 degrees.
Neighboring small angles can be combined into a single larger angle.

Unknowns

How many of the angles formed (single or combined) are acute.

Constraints

Only acute angles (strictly less than 90 degrees) are counted.
A combined angle must use neighboring small angles with no gaps.

Plan

#2 Make a Systematic List · also uses: #7 Identify Subproblems#1 Draw a Diagram

List every angle you can build from consecutive small angles in an orderly way (length 1, then 2, then 3...). Group them by how many small angles they contain, then keep only the ones small enough to stay acute.

Execute

#2 Make a Systematic List 4.MD.C.5

The 5 angles 1-5 each stand alone. All 5 are acute, so all 5 single angles count.

5 \text{ single acute angles}

An angle is the opening between two rays; each gap between neighboring rays is one such angle.

#2 Make a Systematic List 4.MD.C.7

Joining two neighboring small angles gives (1+2), (2+3), (3+4), (4+5): 4 larger angles, of which 4 still open less than 90 degrees and so are acute.

(1+2), (2+3), (3+4), (4+5) \rightarrow 4 \text{ acute angles}

Angle measure adds: two angles laid side by side make one angle whose size is their sum.

#7 Identify Subproblems 4.MD.C.7

Runs of three or more small angles add up to larger openings. Of all 15 possible runs, 0 of the longer ones reach 90 degrees or more, so they are not acute and are not counted.

15 - 5 - 4 = 6 \text{ non-acute or longer runs}

Keep adding pieces of the 180-degree half-turn and the angle eventually reaches a right angle and beyond, so it stops being acute.

#2 Make a Systematic List 4.MD.C.7

Acute angles are the 5 singles plus the 4 pairs (plus any acute longer runs).

5 + 4 = 9

Just total up the groups we kept.

Answer: 9 acute angles

Review

There are 15 angles in all (5 + 4 + 3 + 2 + 1). The longer combinations open toward and past a right angle, so it makes sense that only the 9 shorter ones stay acute. 9 is less than 15, as it must be.

Guess and check (tool 6): sketch the rays roughly equally spaced and estimate each small angle, then add neighbors until the running total first reaches 90 degrees to see where acute stops.

Standards · min grade 4

4.MD.C.5 Recognize angles as geometric shapes formed when two rays share an endpoint — Identifying each opening between neighboring rays as an angle.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Combining neighboring small angles and judging whether the total stays acute.

💡 List the angles by how many pieces they use, then keep only the ones still smaller than a right angle - that is Grade 4 angle sense you already have!

Variant 6 answer: 3 acute angles

Find how many acute angles, large and small, can be found in the figure.

From a single point on a straight line (the base), several rays extend upward, forming $3$ small angles labeled $1, 2, 3$ between neighboring rays. Each of the $3$ small angles is acute, and neighboring small angles can be combined to make larger angles. Count every acute angle (single or combined) in the figure.

Show solution

Understand

Several rays come out of one point on a straight line, making 3 small acute angles in a row labeled 1-3. By joining neighboring small angles we get bigger angles too. We must count every acute angle (small and large) that appears in the figure.

Givens

3 small angles in a row sit between neighboring rays above the base line.
Each of the small angles is acute (less than 90 degrees).
The small angles together fill the half-turn above the line, so they add up to 180 degrees.
Neighboring small angles can be combined into a single larger angle.

Unknowns

How many of the angles formed (single or combined) are acute.

Constraints

Only acute angles (strictly less than 90 degrees) are counted.
A combined angle must use neighboring small angles with no gaps.

Plan

#2 Make a Systematic List · also uses: #7 Identify Subproblems#1 Draw a Diagram

List every angle you can build from consecutive small angles in an orderly way (length 1, then 2, then 3...). Group them by how many small angles they contain, then keep only the ones small enough to stay acute.

Execute

#2 Make a Systematic List 4.MD.C.5

The 3 angles 1-3 each stand alone. All 3 are acute, so all 3 single angles count.

3 \text{ single acute angles}

An angle is the opening between two rays; each gap between neighboring rays is one such angle.

#2 Make a Systematic List 4.MD.C.7

Joining two neighboring small angles gives (1+2), (2+3): 2 larger angles, of which 0 still open less than 90 degrees and so are acute.

(1+2), (2+3) \rightarrow 0 \text{ acute angles}

Angle measure adds: two angles laid side by side make one angle whose size is their sum.

#7 Identify Subproblems 4.MD.C.7

Runs of three or more small angles add up to larger openings. Of all 6 possible runs, 0 of the longer ones reach 90 degrees or more, so they are not acute and are not counted.

6 - 3 - 0 = 3 \text{ non-acute or longer runs}

Keep adding pieces of the 180-degree half-turn and the angle eventually reaches a right angle and beyond, so it stops being acute.

#2 Make a Systematic List 4.MD.C.7

Acute angles are the 3 singles plus the 0 pairs (plus any acute longer runs).

3 + 0 = 3

Just total up the groups we kept.

Answer: 3 acute angles

Review

There are 6 angles in all (3 + 2 + 1). The longer combinations open toward and past a right angle, so it makes sense that only the 3 shorter ones stay acute. 3 is less than 6, as it must be.

Guess and check (tool 6): sketch the rays roughly equally spaced and estimate each small angle, then add neighbors until the running total first reaches 90 degrees to see where acute stops.

Standards · min grade 4

4.MD.C.5 Recognize angles as geometric shapes formed when two rays share an endpoint — Identifying each opening between neighboring rays as an angle.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Combining neighboring small angles and judging whether the total stays acute.

💡 List the angles by how many pieces they use, then keep only the ones still smaller than a right angle - that is Grade 4 angle sense you already have!

Variant 7 answer: 11 acute angles

Find how many acute angles, large and small, can be found in the figure.

From a single point on a straight line (the base), several rays extend upward, forming $6$ small angles labeled $1, 2, 3, 4, 5, 6$ between neighboring rays. Each of the $6$ small angles is acute, and neighboring small angles can be combined to make larger angles. Count every acute angle (single or combined) in the figure.

Show solution

Understand

Several rays come out of one point on a straight line, making 6 small acute angles in a row labeled 1-6. By joining neighboring small angles we get bigger angles too. We must count every acute angle (small and large) that appears in the figure.

Givens

6 small angles in a row sit between neighboring rays above the base line.
Each of the small angles is acute (less than 90 degrees).
The small angles together fill the half-turn above the line, so they add up to 180 degrees.
Neighboring small angles can be combined into a single larger angle.

Unknowns

How many of the angles formed (single or combined) are acute.

Constraints

Only acute angles (strictly less than 90 degrees) are counted.
A combined angle must use neighboring small angles with no gaps.

Plan

#2 Make a Systematic List · also uses: #7 Identify Subproblems#1 Draw a Diagram

List every angle you can build from consecutive small angles in an orderly way (length 1, then 2, then 3...). Group them by how many small angles they contain, then keep only the ones small enough to stay acute.

Execute

#2 Make a Systematic List 4.MD.C.5

The 6 angles 1-6 each stand alone. All 6 are acute, so all 6 single angles count.

6 \text{ single acute angles}

An angle is the opening between two rays; each gap between neighboring rays is one such angle.

#2 Make a Systematic List 4.MD.C.7

Joining two neighboring small angles gives (1+2), (2+3), (3+4), (4+5), (5+6): 5 larger angles, of which 5 still open less than 90 degrees and so are acute.

(1+2), (2+3), (3+4), (4+5), (5+6) \rightarrow 5 \text{ acute angles}

Angle measure adds: two angles laid side by side make one angle whose size is their sum.

#7 Identify Subproblems 4.MD.C.7

Runs of three or more small angles add up to larger openings. Of all 21 possible runs, 0 of the longer ones reach 90 degrees or more, so they are not acute and are not counted.

21 - 6 - 5 = 10 \text{ non-acute or longer runs}

Keep adding pieces of the 180-degree half-turn and the angle eventually reaches a right angle and beyond, so it stops being acute.

#2 Make a Systematic List 4.MD.C.7

Acute angles are the 6 singles plus the 5 pairs (plus any acute longer runs).

6 + 5 = 11

Just total up the groups we kept.

Answer: 11 acute angles

Review

There are 21 angles in all (6 + 5 + 4 + 3 + 2 + 1). The longer combinations open toward and past a right angle, so it makes sense that only the 11 shorter ones stay acute. 11 is less than 21, as it must be.

Guess and check (tool 6): sketch the rays roughly equally spaced and estimate each small angle, then add neighbors until the running total first reaches 90 degrees to see where acute stops.

Standards · min grade 4

4.MD.C.5 Recognize angles as geometric shapes formed when two rays share an endpoint — Identifying each opening between neighboring rays as an angle.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Combining neighboring small angles and judging whether the total stays acute.

💡 List the angles by how many pieces they use, then keep only the ones still smaller than a right angle - that is Grade 4 angle sense you already have!

Variant 8 answer: 9 acute angles

Find how many acute angles, large and small, can be found in the figure.

From a single point on a straight line (the base), several rays extend upward, forming $5$ small angles labeled $1, 2, 3, 4, 5$ between neighboring rays. Each of the $5$ small angles is acute, and neighboring small angles can be combined to make larger angles. Count every acute angle (single or combined) in the figure.

Show solution

Understand

Several rays come out of one point on a straight line, making 5 small acute angles in a row labeled 1-5. By joining neighboring small angles we get bigger angles too. We must count every acute angle (small and large) that appears in the figure.

Givens

5 small angles in a row sit between neighboring rays above the base line.
Each of the small angles is acute (less than 90 degrees).
The small angles together fill the half-turn above the line, so they add up to 180 degrees.
Neighboring small angles can be combined into a single larger angle.

Unknowns

How many of the angles formed (single or combined) are acute.

Constraints

Only acute angles (strictly less than 90 degrees) are counted.
A combined angle must use neighboring small angles with no gaps.

Plan

#2 Make a Systematic List · also uses: #7 Identify Subproblems#1 Draw a Diagram

List every angle you can build from consecutive small angles in an orderly way (length 1, then 2, then 3...). Group them by how many small angles they contain, then keep only the ones small enough to stay acute.

Execute

#2 Make a Systematic List 4.MD.C.5

The 5 angles 1-5 each stand alone. All 5 are acute, so all 5 single angles count.

5 \text{ single acute angles}

An angle is the opening between two rays; each gap between neighboring rays is one such angle.

#2 Make a Systematic List 4.MD.C.7

Joining two neighboring small angles gives (1+2), (2+3), (3+4), (4+5): 4 larger angles, of which 4 still open less than 90 degrees and so are acute.

(1+2), (2+3), (3+4), (4+5) \rightarrow 4 \text{ acute angles}

Angle measure adds: two angles laid side by side make one angle whose size is their sum.

#7 Identify Subproblems 4.MD.C.7

Runs of three or more small angles add up to larger openings. Of all 15 possible runs, 0 of the longer ones reach 90 degrees or more, so they are not acute and are not counted.

15 - 5 - 4 = 6 \text{ non-acute or longer runs}

Keep adding pieces of the 180-degree half-turn and the angle eventually reaches a right angle and beyond, so it stops being acute.

#2 Make a Systematic List 4.MD.C.7

Acute angles are the 5 singles plus the 4 pairs (plus any acute longer runs).

5 + 4 = 9

Just total up the groups we kept.

Answer: 9 acute angles

Review

There are 15 angles in all (5 + 4 + 3 + 2 + 1). The longer combinations open toward and past a right angle, so it makes sense that only the 9 shorter ones stay acute. 9 is less than 15, as it must be.

Guess and check (tool 6): sketch the rays roughly equally spaced and estimate each small angle, then add neighbors until the running total first reaches 90 degrees to see where acute stops.

Standards · min grade 4

4.MD.C.5 Recognize angles as geometric shapes formed when two rays share an endpoint — Identifying each opening between neighboring rays as an angle.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Combining neighboring small angles and judging whether the total stays acute.

💡 List the angles by how many pieces they use, then keep only the ones still smaller than a right angle - that is Grade 4 angle sense you already have!

Variant 9 answer: 3 acute angles

Find how many acute angles, large and small, can be found in the figure.

From a single point on a straight line (the base), several rays extend upward, forming $3$ small angles labeled $1, 2, 3$ between neighboring rays. Each of the $3$ small angles is acute, and neighboring small angles can be combined to make larger angles. Count every acute angle (single or combined) in the figure.

Show solution

Understand

Several rays come out of one point on a straight line, making 3 small acute angles in a row labeled 1-3. By joining neighboring small angles we get bigger angles too. We must count every acute angle (small and large) that appears in the figure.

Givens

3 small angles in a row sit between neighboring rays above the base line.
Each of the small angles is acute (less than 90 degrees).
The small angles together fill the half-turn above the line, so they add up to 180 degrees.
Neighboring small angles can be combined into a single larger angle.

Unknowns

How many of the angles formed (single or combined) are acute.

Constraints

Only acute angles (strictly less than 90 degrees) are counted.
A combined angle must use neighboring small angles with no gaps.

Plan

#2 Make a Systematic List · also uses: #7 Identify Subproblems#1 Draw a Diagram

List every angle you can build from consecutive small angles in an orderly way (length 1, then 2, then 3...). Group them by how many small angles they contain, then keep only the ones small enough to stay acute.

Execute

#2 Make a Systematic List 4.MD.C.5

The 3 angles 1-3 each stand alone. All 3 are acute, so all 3 single angles count.

3 \text{ single acute angles}

An angle is the opening between two rays; each gap between neighboring rays is one such angle.

#2 Make a Systematic List 4.MD.C.7

Joining two neighboring small angles gives (1+2), (2+3): 2 larger angles, of which 0 still open less than 90 degrees and so are acute.

(1+2), (2+3) \rightarrow 0 \text{ acute angles}

Angle measure adds: two angles laid side by side make one angle whose size is their sum.

#7 Identify Subproblems 4.MD.C.7

Runs of three or more small angles add up to larger openings. Of all 6 possible runs, 0 of the longer ones reach 90 degrees or more, so they are not acute and are not counted.

6 - 3 - 0 = 3 \text{ non-acute or longer runs}

Keep adding pieces of the 180-degree half-turn and the angle eventually reaches a right angle and beyond, so it stops being acute.

#2 Make a Systematic List 4.MD.C.7

Acute angles are the 3 singles plus the 0 pairs (plus any acute longer runs).

3 + 0 = 3

Just total up the groups we kept.

Answer: 3 acute angles

Review

There are 6 angles in all (3 + 2 + 1). The longer combinations open toward and past a right angle, so it makes sense that only the 3 shorter ones stay acute. 3 is less than 6, as it must be.

Guess and check (tool 6): sketch the rays roughly equally spaced and estimate each small angle, then add neighbors until the running total first reaches 90 degrees to see where acute stops.

Standards · min grade 4

4.MD.C.5 Recognize angles as geometric shapes formed when two rays share an endpoint — Identifying each opening between neighboring rays as an angle.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Combining neighboring small angles and judging whether the total stays acute.

💡 List the angles by how many pieces they use, then keep only the ones still smaller than a right angle - that is Grade 4 angle sense you already have!

Variant 10 answer: 12 acute angles

Find how many acute angles, large and small, can be found in the figure.

From a single point on a straight line (the base), several rays extend upward, forming $6$ small angles labeled $1, 2, 3, 4, 5, 6$ between neighboring rays. Each of the $6$ small angles is acute, and neighboring small angles can be combined to make larger angles. Count every acute angle (single or combined) in the figure.

Show solution

Understand

Several rays come out of one point on a straight line, making 6 small acute angles in a row labeled 1-6. By joining neighboring small angles we get bigger angles too. We must count every acute angle (small and large) that appears in the figure.

Givens

6 small angles in a row sit between neighboring rays above the base line.
Each of the small angles is acute (less than 90 degrees).
The small angles together fill the half-turn above the line, so they add up to 180 degrees.
Neighboring small angles can be combined into a single larger angle.

Unknowns

How many of the angles formed (single or combined) are acute.

Constraints

Only acute angles (strictly less than 90 degrees) are counted.
A combined angle must use neighboring small angles with no gaps.

Plan

#2 Make a Systematic List · also uses: #7 Identify Subproblems#1 Draw a Diagram

List every angle you can build from consecutive small angles in an orderly way (length 1, then 2, then 3...). Group them by how many small angles they contain, then keep only the ones small enough to stay acute.

Execute

#2 Make a Systematic List 4.MD.C.5

The 6 angles 1-6 each stand alone. All 6 are acute, so all 6 single angles count.

6 \text{ single acute angles}

An angle is the opening between two rays; each gap between neighboring rays is one such angle.

#2 Make a Systematic List 4.MD.C.7

Joining two neighboring small angles gives (1+2), (2+3), (3+4), (4+5), (5+6): 5 larger angles, of which 5 still open less than 90 degrees and so are acute.

(1+2), (2+3), (3+4), (4+5), (5+6) \rightarrow 5 \text{ acute angles}

Angle measure adds: two angles laid side by side make one angle whose size is their sum.

#7 Identify Subproblems 4.MD.C.7

Runs of three or more small angles add up to larger openings. Of all 21 possible runs, 1 of the longer ones reach 90 degrees or more, so they are not acute and are not counted.

21 - 6 - 5 = 10 \text{ non-acute or longer runs}

Keep adding pieces of the 180-degree half-turn and the angle eventually reaches a right angle and beyond, so it stops being acute.

#2 Make a Systematic List 4.MD.C.7

Acute angles are the 6 singles plus the 5 pairs (plus any acute longer runs).

6 + 5 + 1 = 12

Just total up the groups we kept.

Answer: 12 acute angles

Review

There are 21 angles in all (6 + 5 + 4 + 3 + 2 + 1). The longer combinations open toward and past a right angle, so it makes sense that only the 12 shorter ones stay acute. 12 is less than 21, as it must be.

Guess and check (tool 6): sketch the rays roughly equally spaced and estimate each small angle, then add neighbors until the running total first reaches 90 degrees to see where acute stops.

Standards · min grade 4

4.MD.C.5 Recognize angles as geometric shapes formed when two rays share an endpoint — Identifying each opening between neighboring rays as an angle.
4.MD.C.7 Recognize angle measure as additive and solve addition and subtraction problems — Combining neighboring small angles and judging whether the total stays acute.

💡 List the angles by how many pieces they use, then keep only the ones still smaller than a right angle - that is Grade 4 angle sense you already have!

Small angles combine into larger angles · 10 practice problems

Generated variants — 10

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