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NYS COMMON CORE MATHEMATICS CURRICULUM 6β’4 Lesson 30
Lesson 30: One-Step Problems in the Real World
Student Outcomes
Students calculate missing angle measures by writing and solving equations.
Lesson Notes
This is an application lesson based on understandings developed in Grade 4. The three standards applied in this lesson
include:
4.MD.C.5 Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and
understand concepts of angle measurement:
a. An angle is measured with reference to a circle with its center at the common endpoint of the rays, by
considering the fraction of the circular arc between the points where the two rays intersect the circle.
An angle that turns through 1/360 of a circle is called a βone-degree angle,β and can be used to
measure angles.
b. An angle that turns through n one-degree angles is said to have an angle measure of n degrees.
4.MD.C.6 Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure.
4.MD.C.7 Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts, the angle
measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction
problems to find unknown angles on a diagram in real-world and mathematical problems, e.g., by using an
equation with a symbol for the unknown angle measure.
This lesson focuses, in particular, on 4.MD.C.7.
Fluency Exercise (5 minutes)
Subtraction of Decimals Sprint
Classwork
Opening Exercise (3 minutes)
Students start the lesson with a review of key angle terms from Grade 4.
Opening Exercise
Draw an example of each term and write a brief description.
Acute
Less than ππΒ°
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NYS COMMON CORE MATHEMATICS CURRICULUM 6β’4 Lesson 30
Obtuse
Between ππΒ° and πππΒ°
Right
Exactly ππΒ°
Straight
Exactly πππΒ°
Reflex
Between πππΒ° and πππΒ°
Example 1 (3 minutes)
Example 1
β π¨π©πͺ measures ππΒ°. The angle has been separated into two angles. If one angle measures ππΒ°, what is the measure of
the other angle?
In this lesson we will be using algebra to help us determine unknown measures of angles.
How are these two angles related?
The two angles have a sum of ππΒ°.
What equation could we use to solve for π.
πΒ° + ππΒ° = ππΒ°
Now letβs solve.
πΒ° + ππΒ° β ππΒ° = ππΒ° β ππΒ° πΒ° = ππΒ°
The unknown angle is ππΒ°
MP.4
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NYS COMMON CORE MATHEMATICS CURRICULUM 6β’4 Lesson 30
Example 2 (3 minutes)
Example 2
Michelle is designing a parking lot. She has determined that one of the angles should be πππΒ°. What is the measure of
angle π and angle π?
How is angle π related to the πππΒ° angle?
The two angles form a straight line. Therefore they should add up to πππΒ°.
What equation would we use to show this?
πΒ° + πππΒ° = πππΒ°
How would you solve this equation?
πππ was added to the π, so I will take away πππ to get back to just π.
πΒ° + πππΒ°β πππΒ° = πππΒ°β πππΒ° πΒ° = ππΒ°
The angle next to πππΒ°, labeled with an π is equal to ππΒ°.
How is angle π related the angle that measures πππΒ°?
These two angles also form a straight line and must add up to πππΒ°.
Therefore, π and π must both be equal to ππΒ°.
Example 3 (3 minutes)
Example 3
A beam of light is reflected off of a mirror. Below is a diagram of the reflected beam. Determine the missing angle
measure.
How are the angles in this question related?
There are three angles that when all placed together form a straight line. This means that the three angles have a sum of
πππΒ°.
What equation could we write to represent the situation?
ππΒ° + πΒ° + ππΒ° = πππΒ°
πππΒ° πΒ°
πΒ°
MP.4
πΒ°
ππΒ° ππΒ°
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NYS COMMON CORE MATHEMATICS CURRICULUM 6β’4 Lesson 30
ππΒ° πΒ°
ππΒ°
How would you solve an equation like this?
We can combine the two angles that we do know.
ππΒ°+ ππΒ°+ πΒ° = πππΒ° πππΒ°+ πΒ° = πππΒ°
πππΒ°β πππΒ°+ πΒ° = πππΒ°β πππΒ°
πΒ° = ππΒ°
The angle of the bounce is ππΒ°.
Exercises 1β5 (20 minutes)
Students will work independently.
Exercises 1β5
Write and solve an equation in each of the problems.
1. β π¨π©πͺ measures ππΒ°. It has been split into two angles, β π¨π©π« and β π«π©πͺ. The measure of the two angles is in a
ratio of π:π. What are the measures of each angle?
πΒ° + ππΒ° = ππβ°
ππΒ° = ππΒ°
ππΒ°
π =
ππΒ°
π
πΒ° = ππΒ°
One of the angles measures ππΒ°, and the other measures ππΒ°.
2. Solve for π.
3. Candice is building a rectangular piece of a fence according to the plans her boss gave her. One of the angles is not labeled. Write an equation and use it to determine
the measure of the unknown angle.
πΒ°
ππΒ°
πΒ° + ππΒ° = ππΒ° πΒ° + ππΒ° β ππΒ° = ππΒ° β ππΒ°
πΒ° = ππΒ°
MP.4
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NYS COMMON CORE MATHEMATICS CURRICULUM 6β’4 Lesson 30
ππΒ° πΒ°
ππΒ°
ππ Μ
ππΛ
πΛ
4. Rashid hit a hockey puck against the wall at a ππΒ° angle. The puck hit the wall and traveled in a new direction. Determine the missing angle in the diagram.
ππΒ° + πΒ° + ππΒ° = πππΒ°
5. Jaxon is creating a mosaic design on a rectangular table. He
has added two pieces to one of the corners. The first piece has an angle measuring ππΒ° that is placed in the corner. A second piece has an angle measuring ππΒ° that is also placed in the corner. Draw a diagram to model the situation.
Then, write an equation and use it to determine the measure of the unknown angle in a third piece that could be
added to the corner of the table.
πΒ° + ππΒ° + ππΒ° = ππΒ° πΒ° + ππΒ° = ππΒ°
πΒ° + ππΒ° β ππΒ° = ππΒ° β ππΒ° πΒ° = ππΒ°
Closing (3 minutes)
Explain how you determined the equation you used to solve for the missing angle or variable.
I used the descriptions in the word problems. For example, if it said βthe sum of the angles,β I knew to
add the measures together.
Exit Ticket (7 minutes)
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NYS COMMON CORE MATHEMATICS CURRICULUM 6β’4 Lesson 30
Name Date
Lesson 30: One-Step Problems in the Real World
Exit Ticket
Write an equation and solve for the missing angle in each question.
1. Alejandro is repairing a stained glass window. He needs to take it apart to repair it. Before taking it apart he makes
a sketch with angle measures to put it back together.
Write an equation and use it to determine the measure of the
unknown angle.
2. Hannah is putting in a ti le floor. She needs to determine the angles that should be cut in the tiles to fit in the corner .
The angle in the corner measures 90Β° . One piece of the tile will have a measure of 38Β° . Write an equation and use
it to determine the measure of the unknown angle.
π₯Β°
38Β°
π₯Β° 40Β° 30Β°
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NYS COMMON CORE MATHEMATICS CURRICULUM 6β’4 Lesson 30
πΛ
ππ Μ
Exit Ticket Sample Solutions
1. Alejandro is repairing a stained glass window. He needs to take it apart to repair it. Before taking it apart he makes
a sketch with angle measures to put it back together.
Write an equation and use it to determine the measure of the
unknown angle.
ππΒ°+ πΒ° + ππΒ° = πππΒ° πΒ° + ππΒ°+ ππΒ° = πππΒ°
πΒ° + ππΒ° = πππΒ° πΒ° + ππΒ°β ππΒ° = πππΒ° β ππΒ°
πΒ° = πππΒ°
The missing angle measures πππΒ°.
2. Hannah is putting in a tile floor. She needs to determine the angles that should be cut in the tiles to fit in the corner.
The angle in the corner measures ππΒ°. One piece of the tile will have a measure of ππΒ°. Write an equation and use it to determine the measure of the unknown angle.
πΒ° + ππΒ° = ππΒ° πΒ° + ππΒ°β ππΒ° = ππΒ°β ππΒ°
πΒ° = ππΒ°
The unknown angle is ππΒ°.
Problem Set Sample Solutions
Write and solve an equation for each problem.
1. Solve for π.
πΒ° + ππΒ° = ππΒ° πΒ° + ππΒ° β ππΒ° = ππΒ° β ππΒ°
πΒ° = ππΒ°
2. β π©π¨π¬ measures ππΒ°. Solve for π.
ππΒ°+ πΒ° + ππΒ° = ππΒ° ππΒ°+ ππΒ° + πΒ° = ππΒ°
ππΒ° + πΒ° = ππΒ° ππΒ°β ππΒ° + πΒ° = ππΒ°β ππΒ°
πΒ° = ππΒ°
πΒ°
ππΒ°
πΒ° ππΒ° ππΒ°
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NYS COMMON CORE MATHEMATICS CURRICULUM 6β’4 Lesson 30
πππΒ° ππΒ°
πΒ°
3. Thomas is putting in a tile floor. He needs to determine the angles that should be cut in the tiles to fit in the corner. The angle in the corner measures ππΒ°. One piece of the tile will have a measure of ππΒ°. Write an equation and use
it to determine the measure of the unknown angle.
πΒ° + ππΒ° = ππΒ° πΒ° + ππΒ° β ππΒ° = ππΒ° β ππΒ°
πΒ° = ππΒ°
The unknown angle is ππΒ°.
4. Solve for π.
5. Aram has been studying the
mathematics behind pinball machines. He made the
following diagram of one of his observations. Determine the measure of the missing angle.
6. The measures of two angles have a sum of ππΒ°. The measures of the angles are in a ratio of π:π. Determine the measures of both angles.
ππΒ° + πΒ° = ππΒ° ππΒ° = ππΒ° ππΒ°
π=
ππ
π
πΒ° = ππΒ°
The angles measure ππΒ° and ππΒ°.
7. The measures of two angles have a sum of πππΒ°. The measures of the angles are in a ratio of π:π. Determine the
measures of both angles.
ππΒ° + πΒ° = πππΛ ππΒ° = πππ ΜππΒ°
π=
πππ
π
πΒ° = ππΛ
The angles measure ππΒ° and πππΒ°.
ππΒ°
ππΒ°
πΒ°
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NYS COMMON CORE MATHEMATICS CURRICULUM 6β’4 Lesson 30
Subtraction of Decimals β Round 1
Directions: Subtract the decimals to determine the difference.
1. 9.4 β 4.1 16. 41.72 β 33.9
2. 7.4 β 3.2 17. 354 .65 β 67.5
3. 49.5 β 32.1 18. 448.9 β 329.18
4. 20.9 β 17.2 19. 8 β 5.38
5. 9.2 β 6.8 20. 94.21 β 8
6. 7.48 β 2.26 21. 134 .25 β 103.17
7. 58.8 β 43.72 22. 25.8 β 0.42
8. 38.99 β 24.74 23. 115 β 1.65
9. 116.32 β 42.07 24. 187.49 β 21
10. 46.83 β 35.6 25. 345 .77 β 248.69
11. 54.8 β 43.66 26. 108 β 54.7
12. 128.43 β 87.3 27. 336 .91 β 243.38
13. 144.54 β 42.09 28. 264 β 0.742
14. 105 .4 β 68.22 29. 174.38 β 5.9
15. 239.5 β 102.37 30. 323.2 β 38.74
Number Correct: ______
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NYS COMMON CORE MATHEMATICS CURRICULUM 6β’4 Lesson 30
Subtraction of Decimals β Round 1 [KEY]
Directions: Subtract the decimals to determine the difference.
1. 9.4 β 4.1 π. π 16. 41.72 β 33.9 π. ππ
2. 7.4 β 3.2 π. π 17. 354.65 β 67.5 πππ. ππ
3. 49.5 β 32.1 ππ. π 18. 448.9 β 329.18 πππ. ππ
4. 20.9 β 17.2 π. π 19. 8 β 5.38 π. ππ
5. 9.2 β 6.8 π. π 20. 94.21 β 8 ππ. ππ
6. 7.48 β 2.26 π. ππ 21. 134.25 β 103 .17 ππ. ππ
7. 58.8 β 43.72 ππ. ππ 22. 25.8 β 0.42 ππ. ππ
8. 38.99 β 24.74 ππ. ππ 23. 115 β 1.65 πππ. ππ
9. 116.32 β 42.07 ππ. ππ 24. 187.49 β 21 πππ. ππ
10. 46.83 β 35.6 ππ. ππ 25. 345.77 β 248 .69 ππ. ππ
11. 54.8 β 43.66 ππ. ππ 26. 108 β 54.7 ππ. π
12. 128.43 β 87.3 ππ. ππ 27. 336.91 β 243 .38 ππ. ππ
13. 144.54 β 42.09 πππ. ππ 28. 264 β 0.742 πππ. πππ
14. 105 .4 β 68.22 ππ. ππ 29. 174.38 β 5.9 πππ. ππ
15. 239.5 β 102.37 πππ. ππ 30. 323.2 β 38.74 πππ. ππ
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NYS COMMON CORE MATHEMATICS CURRICULUM 6β’4 Lesson 30
Subtraction of Decimals β Round 2
Directions: Subtract the decimals to determine the difference.
1. 8.4 β 5.4 16. 14 β 10.32
2. 5.6 β 3.1 17. 43.37 β 28
3. 9.7 β 7.2 18. 24.56 β 18.88
4. 14.3 β 12.1 19. 33.55 β 11.66
5. 34.5 β 13.2 20. 329 .56 β 284.49
6. 14.86 β 13.85 21. 574.3 β 342.18
7. 43.27 β 32.14 22. 154 β 128.63
8. 48.48 β 27.27 23. 247.1 β 138.57
9. 64.74 β 31.03 24. 12 β 3.547
10. 98.36 β 24.09 25. 1.415 β 0.877
11. 33.54 β 24.4 26. 185.774 β 154.86
12. 114.7 β 73.42 27. 65.251 β 36.9
13. 45.2 β 32.7 28. 144.2 β 95.471
14. 74.8 β 53.9 29. 2.11 β 1.949
15. 238.4 β 114.36 30. 100 β 34.746
Number Correct: ______
Improvement: ______
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NYS COMMON CORE MATHEMATICS CURRICULUM 6β’4 Lesson 30
Subtraction of Decimals β Round 2 [KEY]
Directions: Subtract the decimals to determine the difference.
1. 8.4 β 5.4 π. π 16. 14 β 10.32 π. ππ
2. 5.6 β 3.1 π. π 17. 43.37 β 28 ππ. ππ
3. 9.7 β 7.2 π. π 18. 24.56 β 18.88 π. ππ
4. 14.3 β 12.1 π. π 19. 33.55 β 11.66 ππ. ππ
5. 34.5 β 13.2 ππ. π 20. 329 .56 β 284.49 ππ. ππ
6. 14.86 β 13.85 π. ππ 21. 574.3 β 342.18 πππ. ππ
7. 43.27 β 32.14 ππ. ππ 22. 154 β 128.63 ππ. ππ
8. 48.48 β 27.27 ππ. ππ 23. 247.1 β 138.57 πππ. ππ
9. 64.74 β 31.03 ππ. ππ 24. 12 β 3.547 π. πππ
10. 98.36 β 24.09 ππ. ππ 25. 1.415 β 0.877 π. πππ
11. 33.54 β 24.4 π. ππ 26. 185.774 β 154.86 ππ. πππ
12. 114.7 β 73.42 ππ. ππ 27. 65.251 β 36.9 ππ. πππ
13. 45.2 β 32.7 ππ. π 28. 144.2 β 95.471 ππ. πππ
14. 74.8 β 53.9 ππ. π 29. 2.11 β 1.949 π. πππ
15. 238.4 β 114.36 πππ. ππ 30. 100 β 34.746 ππ. πππ