lesson 30: one-step problems in the real world...lesson 30: one-step problems in the real world exit...

NYS COMMON CORE MATHEMATICS CURRICULUM 6•4 Lesson 30

Lesson 30: One-Step Problems in the Real World

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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 the following:

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 “one-degree angle,” 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.

Classwork

Fluency Exercise (5 minutes): Subtraction of Decimals

Sprint: Refer to Sprints and Sprint Delivery Script sections in the Module Overview for directions on how to administer a

Sprint.

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 𝟗𝟎°

http://creativecommons.org/licenses/by-nc-sa/3.0/deed.en_US






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Obtuse

Between 𝟗𝟎° and 𝟏𝟖𝟎°

Right

Exactly 𝟗𝟎°

Straight

Exactly 𝟏𝟖𝟎°

Reflex

Between 𝟏𝟖𝟎° and 𝟑𝟔𝟎°

Example 1 (3 minutes)

Example 1

∠𝑨𝑩𝑪 measures 𝟗𝟎°. The angle has been separated into two angles. If one angle measures 𝟓𝟕°, 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 𝟗𝟎°.

What equation could we use to solve for 𝒙?

𝒙° + 𝟓𝟕° = 𝟗𝟎°

Now, let’s solve.

𝒙° + 𝟓𝟕° − 𝟓𝟕° = 𝟗𝟎° − 𝟓𝟕°

𝒙° = 𝟑𝟑°

The measure of the unknown angle is 𝟑𝟑°.

MP.4







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Example 2

Michelle is designing a parking lot. She has determined that one of the angles should be 𝟏𝟏𝟓°. What is the measure of

angle 𝒙 and angle 𝒚?

How is angle 𝒙 related to the 𝟏𝟏𝟓° angle?

The two angles form a straight line. Therefore, they should add up to 𝟏𝟖𝟎°.

What equation would we use to show this?

𝒙° + 𝟏𝟏𝟓° = 𝟏𝟖𝟎°

How would you solve this equation?

𝟏𝟏𝟓° was added to angle 𝒙, so I will take away 𝟏𝟏𝟓° to get back to angle 𝒙.

𝒙° + 𝟏𝟏𝟓° − 𝟏𝟏𝟓° = 𝟏𝟖𝟎° − 𝟏𝟏𝟓°

𝒙° = 𝟔𝟓°

The angle next to 𝟏𝟏𝟓°, labeled with an 𝒙, is equal to 𝟔𝟓°.

How is angle 𝒚 related to the angle that measures 𝟏𝟏𝟓°?

These two angles also form a straight line and must add up to 𝟏𝟖𝟎°.

Therefore, angles 𝒙 and 𝒚 must both be equal to 𝟔𝟓°.


Example 3

A beam of light is reflected off 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 𝟏𝟖𝟎°.

What equation could we write to represent the situation?

𝟓𝟓° + 𝒙° + 𝟓𝟓° = 𝟏𝟖𝟎°

𝟏𝟏𝟓° 𝒙°

𝒚°

MP.4

𝒙°

𝟓𝟓° 𝟓𝟓°







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How would you solve an equation like this?

We can combine the two angles that we do know.

𝟓𝟓° + 𝟓𝟓° + 𝒙° = 𝟏𝟖𝟎°

𝟏𝟏𝟎° + 𝒙° = 𝟏𝟖𝟎°

𝟏𝟏𝟎° − 𝟏𝟏𝟎° + 𝒙° = 𝟏𝟖𝟎° − 𝟏𝟏𝟎°

𝒙° = 𝟕𝟎°

The angle of the bounce is 𝟕𝟎°.


Example 4

∠𝑨𝑩𝑪 measures 𝟗𝟎°. It has been split into two angles, ∠𝑨𝑩𝑫 and ∠𝑫𝑩𝑪. The measure of ∠𝑨𝑩𝑫 and ∠𝑫𝑩𝑪 is in a ratio

of 𝟒: 𝟏. What are the measures of each angle?

Use a tape diagram to represent the ratio 𝟒: 𝟏.

What is the measure of each angle?

𝟓 units = 𝟗𝟎°

𝟏 unit = 𝟏𝟖°

𝟒 units = 𝟕𝟐°

∠𝑨𝑩𝑫 is 𝟕𝟐°. ∠𝑫𝑩𝑪 is 𝟏𝟖°.

How can we represent this situation with an equation?

𝟒𝒙° + 𝒙° = 𝟗𝟎°

Solve the equation to determine the measure of each angle.

𝟒𝒙° + 𝒙° = 𝟗𝟎°

𝟓𝒙° = 𝟗𝟎°

𝟓𝒙° ÷ 𝟓 = 𝟗𝟎° ÷ 𝟓

𝒙° = 𝟏𝟖°

𝟒𝒙° = 𝟒(𝟏𝟖°) = 𝟕𝟐°

The measure of ∠𝑫𝑩𝑪 is 𝟏𝟖° and the measure of ∠𝑨𝑩𝑫 is 𝟕𝟐°.

∠𝑫𝑩𝑪

∠𝑨𝑩𝑫

MP.4







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𝟔𝟒° 𝒙°

𝟑𝟕°

Exercises (15 minutes)

Students work independently.

Exercises

Write and solve an equation in each of the problems.

1. ∠𝑨𝑩𝑪 measures 𝟗𝟎°. It has been split into two angles, ∠𝑨𝑩𝑫 and ∠𝑫𝑩𝑪. The measure of the two angles is in a

ratio of 𝟐: 𝟏. What are the measures of each angle?

𝒙° + 𝟐𝒙° = 𝟗𝟎⁰

𝟑𝒙° = 𝟗𝟎°

𝟑𝒙°

𝟑=

𝟗𝟎°

𝟑

𝒙° = 𝟑𝟎°

One of the angles measures 𝟑𝟎°, and the other measures 𝟔𝟎°.

2. Solve for 𝒙.

𝒙° + 𝟔𝟒° + 𝟑𝟕° = 𝟏𝟖𝟎°

𝒙° + 𝟏𝟎𝟏° = 𝟏𝟖𝟎°

𝒙° + 𝟏𝟎𝟏° − 𝟏𝟎𝟏° = 𝟏𝟖𝟎° − 𝟏𝟎𝟏°

𝒙° = 𝟕𝟗°

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.

𝒙°

𝟒𝟗° 𝒙° + 𝟒𝟗° = 𝟗𝟎°

𝒙° + 𝟒𝟗° − 𝟒𝟗° = 𝟗𝟎° − 𝟒𝟗°

𝒙° = 𝟒𝟏°







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𝟑𝟖° 𝒙°

𝟑𝟖°

𝟑𝟖˚

𝟐𝟕˚

𝒙˚

4. Rashid hit a hockey puck against the wall at a 𝟑𝟖° angle. The puck hit the wall and traveled in a new direction.

Determine the missing angle in the diagram.

𝟑𝟖° + 𝒙° + 𝟑𝟖° = 𝟏𝟖𝟎°

𝟕𝟔° + 𝒙° = 𝟏𝟖𝟎°

𝟕𝟔° − 𝟕𝟔° + 𝒙° = 𝟏𝟖𝟎° − 𝟕𝟔°

𝒙° = 𝟏𝟎𝟒°

The measure of the missing angle is 𝟏𝟎𝟒°.

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 𝟑𝟖° and is placed in the corner. A second piece has an angle measuring 𝟐𝟕° and 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.

𝒙° + 𝟑𝟖° + 𝟐𝟕° = 𝟗𝟎°

𝒙° + 𝟔𝟓° = 𝟗𝟎°

𝒙° + 𝟔𝟓° − 𝟔𝟓° = 𝟗𝟎° − 𝟔𝟓°

𝒙° = 𝟐𝟓°

The measure of the unknown angle is 𝟐𝟓°.

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 “the sum of the angles,” I knew to

add the measures together.

I also used my knowledge of angles to know the total angle measure. For example, I know a straight

angle has a measure of 180°, and a right angle or a corner has a measure of 90°.

Exit Ticket (7 minutes)







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Name Date


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 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 90°. One piece of the tile will have a measure of 38°. Write an equation, and use

it to determine the measure of the unknown angle.

𝑥°

38°

𝑥°

40° 30°







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𝒙˚

𝟓𝟐˚

Exit Ticket Sample Solutions

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.

𝟒𝟎° + 𝒙° + 𝟑𝟎° = 𝟏𝟖𝟎°

𝒙° + 𝟒𝟎° + 𝟑𝟎° = 𝟏𝟖𝟎°

𝒙° + 𝟕𝟎° = 𝟏𝟖𝟎°

𝒙° + 𝟕𝟎° − 𝟕𝟎° = 𝟏𝟖𝟎° − 𝟕𝟎°

𝒙° = 𝟏𝟏𝟎°

The missing angle measures 𝟏𝟏𝟎°.

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 𝟗𝟎°. One piece of the tile will have a measure of 𝟑𝟖°. Write an equation, and use


𝒙° + 𝟑𝟖° = 𝟗𝟎°

𝒙° + 𝟑𝟖° − 𝟑𝟖° = 𝟗𝟎° − 𝟑𝟖°

𝒙° = 𝟓𝟐°

The measure of the unknown angle is 𝟓𝟐°.

Problem Set Sample Solutions

Write and solve an equation for each problem.

1. Solve for 𝒙.

𝒙° + 𝟓𝟐° = 𝟗𝟎°

𝒙° + 𝟓𝟐° − 𝟓𝟐° = 𝟗𝟎° − 𝟓𝟐°

𝒙° = 𝟑𝟖°

The measure of the missing angle is 𝟑𝟖°.

𝒙°

𝟑𝟖°

𝒙° 𝟒𝟎° 𝟑𝟎°







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𝟏𝟎𝟓° 𝟔𝟐°

𝒙°

2. ∠𝑩𝑨𝑬 measures 𝟗𝟎°. Solve for 𝒙.

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 𝟗𝟎°. One piece of the tile will have a measure of 𝟐𝟒°. Write an equation, and use


𝒙° + 𝟐𝟒° = 𝟗𝟎°

𝒙° + 𝟐𝟒° − 𝟐𝟒° = 𝟗𝟎° − 𝟐𝟒°

𝒙° = 𝟔𝟔°

The measure of the unknown angle is 𝟔𝟔°.

4. Solve for 𝒙.

𝒙° + 𝟏𝟎𝟓° + 𝟔𝟐° = 𝟏𝟖𝟎°

𝒙° + 𝟏𝟔𝟕° = 𝟏𝟖𝟎°

𝒙° + 𝟏𝟔𝟕° − 𝟏𝟔𝟕° = 𝟏𝟖𝟎° − 𝟏𝟔𝟕°

𝒙° = 𝟏𝟑°

The measure of the missing angle is 𝟏𝟑°.

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.

𝟓𝟐° + 𝒙° + 𝟔𝟖° = 𝟏𝟖𝟎°

𝟏𝟐𝟎° + 𝒙° = 𝟏𝟖𝟎°

𝟏𝟐𝟎° + 𝒙° − 𝟏𝟐𝟎° = 𝟏𝟖𝟎° − 𝟏𝟐𝟎°

𝒙° = 𝟔𝟎°

The measure of the missing angle is 𝟔𝟎°.

6. The measures of two angles have a sum of 𝟗𝟎°. The measures of the angles are in a ratio of 𝟐: 𝟏. Determine the

measures of both angles.

𝟐𝒙° + 𝒙° = 𝟗𝟎°

𝟑𝒙° = 𝟗𝟎°

𝟑𝒙°

𝟑=

𝟗𝟎°

𝟑

𝒙° = 𝟑𝟎°

The angles measure 𝟑𝟎° and 𝟔𝟎°.

𝟏𝟓° + 𝒙° + 𝟐𝟓° = 𝟗𝟎°

𝟏𝟓° + 𝟐𝟓° + 𝒙° = 𝟗𝟎°

𝟒𝟎° + 𝒙° = 𝟗𝟎°

𝟒𝟎° − 𝟒𝟎° + 𝒙° = 𝟗𝟎° − 𝟒𝟎°

𝒙° = 𝟓𝟎°

𝟓𝟐°

𝟔𝟖°

𝒙°







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7. The measures of two angles have a sum of 𝟏𝟖𝟎°. The measures of the angles are in a ratio of 𝟓: 𝟏. Determine the

measures of both angles.

𝟓𝒙° + 𝒙° = 𝟏𝟖𝟎°

𝟔𝒙° = 𝟏𝟖𝟎°

𝟔𝒙°

𝟔=

𝟏𝟖𝟎

𝟔°

𝒙° = 𝟑𝟎˚

The angles measure 𝟑𝟎° and 𝟏𝟓𝟎°.







342



Subtraction of Decimals—Round 1

Directions: Evaluate each expression.

1. 55 − 50 23. 9.9 − 5

2. 55 − 5 24. 9.9 − 0.5

3. 5.5 − 5 25. 0.99 − 0.5

4. 5.5 − 0.5 26. 0.99 − 0.05

5. 88 − 80 27. 4.7 − 2

6. 88 − 8 28. 4.7 − 0.2

7. 8.8 − 8 29. 0.47 − 0.2

8. 8.8 − 0.8 30. 0.47 − 0.02

9. 33 − 30 31. 8.4 − 1

10. 33 − 3 32. 8.4 − 0.1

11. 3.3 − 3 33. 0.84 − 0.1

12. 1 − 0.3 34. 7.2 − 5

13. 1 − 0.03 35. 7.2 − 0.5

14. 1 − 0.003 36. 0.72 − 0.5

15. 0.1 − 0.03 37. 0.72 − 0.05

16. 4 − 0.8 38. 8.6 − 7

17. 4 − 0.08 39. 8.6 − 0.7

18. 4 − 0.008 40. 0.86 − 0.7

19. 0.4 − 0.08 41. 0.86 − 0.07

20. 9 − 0.4 42. 5.1 − 4

21. 9 − 0.04 43. 5.1 − 0.4

22. 9 − 0.004 44. 0.51 − 0.4

Number Correct: ______







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Subtraction of Decimals—Round 1 [KEY]


1. 55 − 50 𝟓 23. 9.9 − 5 𝟒. 𝟗

2. 55 − 5 𝟓𝟎 24. 9.9 − 0.5 𝟗. 𝟒

3. 5.5 − 5 𝟎. 𝟓 25. 0.99 − 0.5 𝟎. 𝟒𝟗

4. 5.5 − 0.5 𝟓 26. 0.99 − 0.05 𝟎. 𝟗𝟒

5. 88 − 80 𝟖 27. 4.7 − 2 𝟐. 𝟕

6. 88 − 8 𝟖𝟎 28. 4.7 − 0.2 𝟒. 𝟓

7. 8.8 − 8 𝟎. 𝟖 29. 0.47 − 0.2 𝟎. 𝟐𝟕

8. 8.8 − 0.8 𝟖 30. 0.47 − 0.02 𝟎. 𝟒𝟓

9. 33 − 30 𝟑 31. 8.4 − 1 𝟕. 𝟒

10. 33 − 3 𝟑𝟎 32. 8.4 − 0.1 𝟖. 𝟑

11. 3.3 − 3 𝟎. 𝟑 33. 0.84 − 0.1 𝟎. 𝟕𝟒

12. 1 − 0.3 𝟎. 𝟕 34. 7.2 − 5 𝟐. 𝟐

13. 1 − 0.03 𝟎. 𝟗𝟕 35. 7.2 − 0.5 𝟔. 𝟕

14. 1 − 0.003 𝟎. 𝟗𝟗𝟕 36. 0.72 − 0.5 𝟎. 𝟐𝟐

15. 0.1 − 0.03 𝟎. 𝟎𝟕 37. 0.72 − 0.05 𝟎. 𝟔𝟕

16. 4 − 0.8 𝟑. 𝟐 38. 8.6 − 7 𝟏. 𝟔

17. 4 − 0.08 𝟑. 𝟗𝟐 39. 8.6 − 0.7 𝟕. 𝟗

18. 4 − 0.008 𝟑. 𝟗𝟗𝟐 40. 0.86 − 0.7 𝟎. 𝟏𝟔

19. 0.4 − 0.08 𝟎. 𝟑𝟐 41. 0.86 − 0.07 𝟎. 𝟕𝟗

20. 9 − 0.4 𝟖. 𝟔 42. 5.1 − 4 𝟏. 𝟏

21. 9 − 0.04 𝟖. 𝟗𝟔 43. 5.1 − 0.4 𝟒. 𝟕

22. 9 − 0.004 𝟖. 𝟗𝟗𝟔 44. 0.51 − 0.4 𝟎. 𝟏𝟏







344



Subtraction of Decimals—Round 2


1. 66 − 60 23. 6.8 − 4

2. 66 − 6 24. 6.8 − 0.4

3. 6.6 − 6 25. 0.68 − 0.4

4. 6.6 − 0.6 26. 0.68 − 0.04

5. 99 − 90 27. 7.3 − 1

6. 99 − 9 28. 7.3 − 0.1

7. 9.9 − 9 29. 0.73 − 0.1

8. 9.9 − 0.9 30. 0.73 − 0.01

9. 22 − 20 31. 9.5 − 2

10. 22 − 2 32. 9.5 − 0.2

11. 2.2 − 2 33. 0.95 − 0.2

12. 3 − 0.4 34. 8.3 − 5

13. 3 − 0.04 35. 8.3 − 0.5

14. 3 − 0.004 36. 0.83 − 0.5

15. 0.3 − 0.04 37. 0.83 − 0.05

16. 8 − 0.2 38. 7.2 − 4

17. 8 − 0.02 39. 7.2 − 0.4

18. 8 − 0.002 40. 0.72 − 0.4

19. 0.8 − 0.02 41. 0.72 − 0.04

20. 5 − 0.1 42. 9.3 − 7

21. 5 − 0.01 43. 9.3 − 0.7

22. 5 − 0.001 44. 0.93 − 0.7

Number Correct: ______

Improvement: ______







345



Subtraction of Decimals—Round 2 [KEY]


1. 66 − 60 𝟔 23. 6.8 − 4 𝟐. 𝟖

2. 66 − 6 𝟔𝟎 24. 6.8 − 0.4 𝟔. 𝟒

3. 6.6 − 6 𝟎. 𝟔 25. 0.68 − 0.4 𝟎. 𝟐𝟖

4. 6.6 − 0.6 𝟔 26. 0.68 − 0.04 𝟎. 𝟔𝟒

5. 99 − 90 𝟗 27. 7.3 − 1 𝟔. 𝟑

6. 99 − 9 𝟗𝟎 28. 7.3 − 0.1 𝟕. 𝟐

7. 9.9 − 9 𝟎. 𝟗 29. 0.73 − 0.1 𝟎. 𝟔𝟑

8. 9.9 − 0.9 𝟗 30. 0.73 − 0.01 𝟎. 𝟕𝟐

9. 22 − 20 𝟐 31. 9.5 − 2 𝟕. 𝟓

10. 22 − 2 𝟐𝟎 32. 9.5 − 0.2 𝟗. 𝟑

11. 2.2 − 2 𝟎. 𝟐 33. 0.95 − 0.2 𝟎. 𝟕𝟓

12. 3 − 0.4 𝟐. 𝟔 34. 8.3 − 5 𝟑. 𝟑

13. 3 − 0.04 𝟐. 𝟗𝟔 35. 8.3 − 0.5 𝟕. 𝟖

14. 3 − 0.004 𝟐. 𝟗𝟗𝟔 36. 0.83 − 0.5 𝟎. 𝟑𝟑

15. 0.3 − 0.04 𝟎. 𝟐𝟔 37. 0.83 − 0.05 𝟎. 𝟕𝟖

16. 8 − 0.2 𝟕. 𝟖 38. 7.2 − 4 𝟑. 𝟐

17. 8 − 0.02 𝟕. 𝟗𝟖 39. 7.2 − 0.4 𝟔. 𝟖

18. 8 − 0.002 𝟕. 𝟗𝟗𝟖 40. 0.72 − 0.4 𝟎. 𝟑𝟐

19. 0.8 − 0.02 𝟎. 𝟕𝟖 41. 0.72 − 0.04 𝟎. 𝟔𝟖

20. 5 − 0.1 𝟒. 𝟗 42. 9.3 − 7 𝟐. 𝟑

21. 5 − 0.01 𝟒. 𝟗𝟗 43. 9.3 − 0.7 𝟖. 𝟔

22. 5 − 0.001 𝟒. 𝟗𝟗𝟗 44. 0.93 − 0.7 𝟎. 𝟐𝟑





lesson 30: one-step problems in the real world...lesson 30: one-step problems in the real world exit...

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