Practice · Small angles combine into larger angles · Sensim Math

Variant 1 easy

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.

Your answer

Show solution

1 · Understandwhat's really being asked

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.

2 · Planchoose the strategy

#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.

3 · Execute4 carry out the plan

1List the single small angles

#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.

2List angles made of two neighbors

#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.

3List the longer combinations and test them

#7 Identify Subproblems 4.MD.C.7

Runs of three or more small angles add up to larger openings, and any run that reaches 90 degrees or more is not acute. Of all 6 runs, 3 reach a right angle or beyond and are dropped; only the shorter runs stay acute.

6 - 3 = 3 \text{ non-acute }(\ge 90^\circ)

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

4Add the acute counts

#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

4 · Reviewdoes it hold up?

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.

Another way: 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.

Standardsmin 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.

💡Takeaway. 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 easy

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.

Your answer

Show solution

1 · Understandwhat's really being asked

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.

2 · Planchoose the strategy

#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.

3 · Execute4 carry out the plan

1List the single small angles

#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.

2List angles made of two neighbors

#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.

3List the longer combinations and test them

#7 Identify Subproblems 4.MD.C.7

Runs of three or more small angles add up to larger openings, and any run that reaches 90 degrees or more is not acute. Of all 6 runs, 3 reach a right angle or beyond and are dropped; only the shorter runs stay acute.

6 - 3 = 3 \text{ non-acute }(\ge 90^\circ)

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

4Add the acute counts

#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

4 · Reviewdoes it hold up?

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.

Another way: 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.

Standardsmin 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.

💡Takeaway. 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 medium

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.

Your answer

Show solution

1 · Understandwhat's really being asked

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.

2 · Planchoose the strategy

#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.

3 · Execute4 carry out the plan

1List the single small angles

#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.

2List angles made of two neighbors

#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.

3List the longer combinations and test them

#7 Identify Subproblems 4.MD.C.7

Runs of three or more small angles add up to larger openings, and any run that reaches 90 degrees or more is not acute. Of all 10 runs, 6 reach a right angle or beyond and are dropped; only the shorter runs stay acute.

10 - 4 = 6 \text{ non-acute }(\ge 90^\circ)

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

4Add the acute counts

#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

4 · Reviewdoes it hold up?

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.

Another way: 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.

Standardsmin 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.

💡Takeaway. 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 medium

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.

Your answer

Show solution

1 · Understandwhat's really being asked

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.

2 · Planchoose the strategy

#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.

3 · Execute4 carry out the plan

1List the single small angles

#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.

2List angles made of two neighbors

#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.

3List the longer combinations and test them

#7 Identify Subproblems 4.MD.C.7

Runs of three or more small angles add up to larger openings, and any run that reaches 90 degrees or more is not acute. Of all 10 runs, 6 reach a right angle or beyond and are dropped; only the shorter runs stay acute.

10 - 4 = 6 \text{ non-acute }(\ge 90^\circ)

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

4Add the acute counts

#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

4 · Reviewdoes it hold up?

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.

Another way: 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.

Standardsmin 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.

💡Takeaway. 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 easy

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.

Your answer

Show solution

1 · Understandwhat's really being asked

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.

2 · Planchoose the strategy

#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.

3 · Execute4 carry out the plan

1List the single small angles

#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.

2List angles made of two neighbors

#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.

3List the longer combinations and test them

#7 Identify Subproblems 4.MD.C.7

Runs of three or more small angles add up to larger openings, and any run that reaches 90 degrees or more is not acute. Of all 10 runs, 6 reach a right angle or beyond and are dropped; only the shorter runs stay acute.

10 - 4 = 6 \text{ non-acute }(\ge 90^\circ)

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

4Add the acute counts

#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

4 · Reviewdoes it hold up?

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.

Another way: 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.

Standardsmin 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.

💡Takeaway. 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 medium

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.

Your answer

Show solution

1 · Understandwhat's really being asked

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.

2 · Planchoose the strategy

#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.

3 · Execute4 carry out the plan

1List the single small angles

#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.

2List angles made of two neighbors

#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.

3List the longer combinations and test them

#7 Identify Subproblems 4.MD.C.7

Runs of three or more small angles add up to larger openings, and any run that reaches 90 degrees or more is not acute. Of all 15 runs, 6 reach a right angle or beyond and are dropped; only the shorter runs stay acute.

15 - 9 = 6 \text{ non-acute }(\ge 90^\circ)

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

4Add the acute counts

#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

4 · Reviewdoes it hold up?

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.

Another way: 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.

Standardsmin 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.

💡Takeaway. 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 medium

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.

Your answer

Show solution

1 · Understandwhat's really being asked

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.

2 · Planchoose the strategy

#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.

3 · Execute4 carry out the plan

1List the single small angles

#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.

2List angles made of two neighbors

#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.

3List the longer combinations and test them

#7 Identify Subproblems 4.MD.C.7

Runs of three or more small angles add up to larger openings, and any run that reaches 90 degrees or more is not acute. Of all 15 runs, 6 reach a right angle or beyond and are dropped; only the shorter runs stay acute.

15 - 9 = 6 \text{ non-acute }(\ge 90^\circ)

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

4Add the acute counts

#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

4 · Reviewdoes it hold up?

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.

Another way: 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.

Standardsmin 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.

💡Takeaway. 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 hard

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.

Your answer

Show solution

1 · Understandwhat's really being asked

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.

2 · Planchoose the strategy

#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.

3 · Execute4 carry out the plan

1List the single small angles

#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.

2List angles made of two neighbors

#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.

3List the longer combinations and test them

#7 Identify Subproblems 4.MD.C.7

Runs of three or more small angles add up to larger openings, and any run that reaches 90 degrees or more is not acute. Of all 21 runs, 10 reach a right angle or beyond and are dropped; only the shorter runs stay acute.

21 - 11 = 10 \text{ non-acute }(\ge 90^\circ)

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

4Add the acute counts

#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

4 · Reviewdoes it hold up?

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.

Another way: 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.

Standardsmin 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.

💡Takeaway. 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 hard

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.

Your answer

Show solution

1 · Understandwhat's really being asked

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.

2 · Planchoose the strategy

#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.

3 · Execute4 carry out the plan

1List the single small angles

#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.

2List angles made of two neighbors

#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.

3List the longer combinations and test them

#7 Identify Subproblems 4.MD.C.7

Runs of three or more small angles add up to larger openings, and any run that reaches 90 degrees or more is not acute. Of all 21 runs, 9 reach a right angle or beyond and are dropped; only the shorter runs stay acute.

21 - 12 = 9 \text{ non-acute }(\ge 90^\circ)

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

4Add the acute counts

#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

4 · Reviewdoes it hold up?

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.

Another way: 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.

Standardsmin 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.

💡Takeaway. 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 hard

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.

Your answer

Show solution

1 · Understandwhat's really being asked

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.

2 · Planchoose the strategy

#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.

3 · Execute4 carry out the plan

1List the single small angles

#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.

2List angles made of two neighbors

#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.

3List the longer combinations and test them

#7 Identify Subproblems 4.MD.C.7

Runs of three or more small angles add up to larger openings, and any run that reaches 90 degrees or more is not acute. Of all 28 runs, 10 reach a right angle or beyond and are dropped; only the shorter runs stay acute.

28 - 18 = 10 \text{ non-acute }(\ge 90^\circ)

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

4Add the acute counts

#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

4 · Reviewdoes it hold up?

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.

Another way: 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.

Standardsmin 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.

💡Takeaway. 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

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