name: per: m2 topic 1 homework packet m2-t1-l1 hw

14
Name: _____________________________________ Per: ________ M2-T1-L1 HW: Proportional Relationships 1. Determine the constant of proportionality represented in each graph. (remember that the constant of proportionality can be found by the ratio ) a. Line A: __________ b. Line B: __________ c. Line C: __________ 2. Melanie collects coins from all over the world. She is reorganizing her collection into coins from Europe and coins from other parts of the world. After sorting the coins, she comes to the conclusion that six out of every ten of the coins in her collection come from Europe. a. Write a ratio for the number of European coins to the total number of coins__________ b. Write a ratio for the number of non-European coins to the total number of coins __________ c. Write a ratio for the number of European coins to the number of non-European coins. __________ d. Melanie has 230 coins in her collection. Determine the number of European and non-European coins that she has in her collection. # of European coins _______________ # of non- European coins _______________ e. Melanie adds to her collection while keeping the same ratio of coins and now has 180 European coins. Determine the number of non-European coins and the total number of coins in her collection. Total # of coins ___________________ # of non- European coins ________________ f. Write an equation to determine the number of European coins, E, if Melanie has t total coins. Identify the constant of proportionality. g. Write an equation to determine the number of non-European coins, N, if Melanie has t total coins. Identify the constant of proportionality. h. If you graphed the equations from parts f and g, which line would be steeper? Explain how you know. A B C M2 TOPIC 1 HOMEWORK PACKET 1

Upload: others

Post on 13-May-2022

2 views

Category:

Documents


0 download

TRANSCRIPT

Page 1: Name: Per: M2 TOPIC 1 HOMEWORK PACKET M2-T1-L1 HW

Name: _____________________________________ Per: ________

M2-T1-L1 HW: Proportional Relationships 1. Determine the constant of proportionality represented in each graph.

(remember that the constant of proportionality can be found by the ratio π’šπ’šπ’™π’™

)

a. Line A: __________ b. Line B: __________ c. Line C: __________

2. Melanie collects coins from all over the world. She is reorganizing her collection into coins from Europe and coins from other parts of the world. After sorting the coins, she comes to the conclusion that six out of every ten of the coins in her collection come from Europe.

a. Write a ratio for the number of European coins to the total number of coins__________

b. Write a ratio for the number of non-European coins to the total number of coins __________

c. Write a ratio for the number of European coins to the number of non-European coins. __________

d. Melanie has 230 coins in her collection. Determine the number of European and non-European coins that she has in her collection.

# of European coins _______________ # of non- European coins _______________ e. Melanie adds to her collection while keeping the same ratio of coins and now has 180 European coins. Determine the

number of non-European coins and the total number of coins in her collection.

Total # of coins ___________________ # of non- European coins ________________ f. Write an equation to determine the number of European coins, E, if Melanie has t total coins. Identify the constant of

proportionality.

g. Write an equation to determine the number of non-European coins, N, if Melanie has t total coins. Identify the constant of proportionality.

h. If you graphed the equations from parts f and g, which line would be steeper? Explain how you know.

A B

C

M2 TOPIC 1 HOMEWORK PACKET

1

Page 2: Name: Per: M2 TOPIC 1 HOMEWORK PACKET M2-T1-L1 HW

3. Analyze each scenario and graph provided below to answer the given question.

Graph on the left: The number of students who do not sing soprano is represented by line ______. I know this is correct

because this line has a constant of proportionality of __________ , and the other line has a constant of proportionality of________. The steeper line corresponds to the greater constant of proportionality, so the line for the students who do

sing soprano is________.

Graph on the right: The number of vehicles that are not trucks is represented by line _______, and the number of

vehicles that are trucks is represented by line _______. The constant of proportionality for y2 is __________, and the constant of proportionality for y1 is _____________.

4. The graphs below show the cost π’šπ’š of buying 𝒙𝒙 pounds of fruit. One graph shows the cost of buying 𝒙𝒙 pounds of peaches, and the other shows the cost of buying 𝒙𝒙 pounds of plums.

a. Are both fruits examples of proportional relationships? Explain.

b. Which fruit costs more per pound? Explain.

c. Bananas cost less per pound than peaches, but more than plums. Draw a line alongside the other graphs that might represent the cost, 𝑦𝑦, of buying π‘₯π‘₯ pounds of bananas.

2

Page 3: Name: Per: M2 TOPIC 1 HOMEWORK PACKET M2-T1-L1 HW

5. Kristina and her sister, Tracee, are painting rooms in their house. The graph below represents the rate at which Kristina paints, and the table below shows how many square feet Tracee painted for given amounts of time. Both sisters paint at a constant pace.

a. Who paints at the faster rate? Justify your answer. (talk about unit rate/constant of proportionality)

b. Write an equation for the area painted versus time for each sister.

6-8: REVIEW:

6. In the diagram, βˆ†π‘¨π‘¨π‘¨π‘¨π‘¨π‘¨ ~ βˆ†π‘Ώπ‘Ώπ‘Ώπ‘Ώπ‘Ώπ‘Ώ. State the corresponding sides and angles. Then find the scale factor that was used to dilate βˆ†π‘¨π‘¨π‘¨π‘¨π‘¨π‘¨ to βˆ†π‘Ώπ‘Ώπ‘Ώπ‘Ώπ‘Ώπ‘Ώ.

Corresponding Sides Corresponding Angles Scale Factor

7. In the diagram, 𝑨𝑨𝑩𝑩����� || 𝑨𝑨𝑨𝑨����.

a. Explain why βˆ†π΅π΅π΅π΅π΅π΅ ~ βˆ†π΄π΄π΄π΄π΅π΅.

b. Determine the length of 𝐡𝐡𝐴𝐴����.

8. Describe the sequence of transformations to generate line segment A’B’ from original line segment AB.

Tracee’s Painting

Time (minutes)

Area Painted (sq ft)

5 18.75

8 30

12 45

20 75

3

Page 4: Name: Per: M2 TOPIC 1 HOMEWORK PACKET M2-T1-L1 HW

9. Asia is training for a 10K road race that will take place in October. She starts her training in July and runs every weekend. On her first run, she ran at a constant pace and covered 5.5 miles in 60 minutes. On her last run before the race, she ran at a constant pace and covered 10.2 miles in 1.5 hours.

a. What were Asia’s speeds, in miles per hour, on her first and last runs?

First: Last:

b. Write an equation to represent Asia’s distance ran as a function of time for her first run and for her last run.

First: Last:

c. Determine which graph below represents Asia’s first run and her last run.

d. Does each graph represent a proportional relationship? Why or why not?

10. Kelly works at an after-school program at an elementary school. The table shows how much money was earned every day last week.

Mariko has a job mowing lawns that pays $7 per hour.

a. Who makes more money for working 1 hour? Explain.

b. Who would make more after working for 10 hours? How much more?

c. If you graphed each person’s job on a graph, whose line would be steeper? Explain.

d. Determine who makes more for 6 hours of work and approximately how much more.

4

4

Page 5: Name: Per: M2 TOPIC 1 HOMEWORK PACKET M2-T1-L1 HW

11. Water flows out of Pipe A at a constant rate. Pipe A can fill 3 buckets of the same size in 14 minutes. The graph below represents the rate at which Pipe B can fill the same-sized buckets.

a. Write a linear equation that represents the number of buckets, 𝑦𝑦, that each pipe can fill in π‘₯π‘₯ minutes.

Pipe A: Pipe B:

b. Which pipe fills buckets faster? Justify your answer.

12. At a factory, a machine fills jars with salsa. The manager of the factory is considering buying a new machine that will fill 78 jars of salsa every 3 minutes. To support his decision, he wants to compare the rate of the new machine to the rate of the old machine that is currently in the factory. The graph shows the number of jars of salsa filled over time with the old machine.

a. What is the unit rate, or constant of proportionality for the old machine and for the new machine?

b. The manager is about to fill an order for 765 jars of salsa. How long would it take to fill this order on each machine?

c. Should the manager replace the old machine with the new one? Explain.

d. If you were to graph a line that represented the jars filled on the new machine over time, would it be steeper or less steep than the old machine? Explain.

13. The corner market sells rice by the pound using the equation π’šπ’š = 𝟏𝟏.πŸπŸπŸπŸπ’™π’™, where π’šπ’š represents the total cost for 𝒙𝒙 number of pounds. The local grocery store also sells rice by the pound. The relationship between cost and weight at the grocery story is shown in the graph.

a. Which store offers a better deal on rice? Explain your answer.

b. If you were to graph the relationship of cost and weight of rice at the corner market, how would the graph compare to the graph at the grocery store? Explain.

5

Page 6: Name: Per: M2 TOPIC 1 HOMEWORK PACKET M2-T1-L1 HW

PAGE INTENTIONALLY LEFT BLANK

6

Page 7: Name: Per: M2 TOPIC 1 HOMEWORK PACKET M2-T1-L1 HW

Name: _____________________________________ Per: ________

M2-T1-L2 HW: Similar Triangles & Steepness of Line 1. Maximilian is cleaning crabs. He cleans 4 crabs every minute. Use time, 𝒕𝒕, in minutes as the independent quantity

and the number of crabs, 𝒄𝒄, as the dependent quantity.

a. Is the relationship proportional? Explain how you know.

b. Identify the unit rate of this relationship. Explain what the unit rate means in terms of the situation.

c. Write an equation that determines the number of crabs cleaned given any time.

d. Create a graph of the relationship. How does the graph show you that the relationship between time and number of crabs is proportional?

2. Three staircases are shown below.

a. Order the staircases from least steep to most steep. Explain how you know, or show calculations to prove you are correct.

b. Staircase D climbs 8 feet over a horizontal distance of 10 feet. Where would this staircase fall in order of steepness compared to the others. Explain.

Time, t (minutes) N

umbe

r of C

rabs

, c

7

M2 TOPIC 1 HOMEWORK PACKET

Page 8: Name: Per: M2 TOPIC 1 HOMEWORK PACKET M2-T1-L1 HW

3. At the grocery store, sliced turkey deli meat is on sale for $22 for 4 pounds.

a. What is the unit rate of this situation? Explain what the unit rate represents in this scenario.

b. Write an equation to represent the cost, y, of sliced turkey deli meat measured in x pounds.

c. Is this relationship proportional? How do you know?

4. Water flows at a constant rate out of a faucet. Suppose the volume of water that comes out in three minutes is 10.5 gallons. Use time, 𝒕𝒕, as the independent variable and Volume of water, 𝒗𝒗, as the dependent variable.

a. What is unit rate of this scenario? What does it mean in this scenario?

b. Write a linear equation to represent the volume of water, 𝑣𝑣, that comes out of the faucet in 𝑑𝑑, minutes.

c. Find the volume of water out of the faucet after 0 minutes, 2 minutes, and 5 minutes.

d. Is this scenario a proportional relationship? Explain. What would the graph look like?

5. In the diagram shown, line 𝒔𝒔 and line 𝒕𝒕 are parallel. Determine the value of 𝒙𝒙, and then determine the measure of each angle. Mark all angles on the figure.

8

Page 9: Name: Per: M2 TOPIC 1 HOMEWORK PACKET M2-T1-L1 HW

6. Use the figure to answer the following questions.

a. Name 2 pairs of alternate interior angles.

b. Name 2 pairs of same side interior angles.

c. Name 4 pairs of vertical angles.

d. Name 2 pairs of corresponding angles.

e. Name 4 sets of linear pairs.

f. Name 2 pairs of alternate exterior angles.

g. Name 2 pairs of same side exterior angles.

7. Determine whether each equation represents a proportional relationship. Explain how you know.

a. 𝑦𝑦 = 2.5π‘₯π‘₯ b. 𝑦𝑦 = π‘₯π‘₯ βˆ’ 4 c. 𝑦𝑦 = 5π‘₯π‘₯ + 2 d. 𝑦𝑦 = βˆ’6π‘₯π‘₯

8. A line is drawn through the points A, C, E, and G as shown in the graph below.

a. What transformation takes triangle ABC to triangle EFG? Be specific. Are the triangles congruent, similar, or neither?

b. What transformation takes triangle ABC to triangle ADG? Be specific. Are the triangles congruent, similar, or neither?

c. If two triangles are similar, then what do you know about their corresponding side lengths?

d. The slope of any two points on a line will always be the _________.

What is the slope of the line through points A and C? __________Points C and G? __________ Points A and E? ___________

e. Triangle ABC and triangle ADG share two angles (angle A and angle G). What do you know about two triangles with two congruent angles?

9

Page 10: Name: Per: M2 TOPIC 1 HOMEWORK PACKET M2-T1-L1 HW

REMEMBER: Slope is another name for the rate of change of a linear relationship graphed as a line.

β€’ The equation for a proportional linear relationship is π’šπ’š = π’Žπ’Žπ’™π’™, where m is the slope.

β€’ An equation for a non-proportional linear relationship is π’šπ’š = π’Žπ’Žπ’™π’™+ 𝒃𝒃, where m is the slope and b is the

y-coordinate of the point where the graph crosses the y-axis (b represents the starting point of the line)

9. Consider each graph shown. Answer the questions below each graph. Remember, for part b, write an equation in the

form π’šπ’š = π’Žπ’Žπ’™π’™ (proportional) or π’šπ’š = π’Žπ’Žπ’™π’™+ 𝒃𝒃 (non- proportional) to represent the relationship. Slope of each graph can be

found by ( π’„π’„π’„π’„π’„π’„π’π’π’ˆπ’ˆπ’ˆπ’ˆ π’Šπ’Šπ’π’ π’—π’—π’ˆπ’ˆπ’—π’—π’•π’•π’Šπ’Šπ’„π’„π’„π’„π’—π’—π’„π’„π’„π’„π’„π’„π’π’π’ˆπ’ˆπ’ˆπ’ˆ π’Šπ’Šπ’π’ π’„π’„π’‰π’‰π’—π’—π’Šπ’Šπ’‰π’‰π’‰π’‰π’π’π’•π’•π’„π’„π’—π’—

) of any two points on line.

a) proportional or non- proportional? (circle one)

b) Equation? ___________________________

c) Slope? _________

d) What does slope represent in the situation?

a) proportional or non- proportional? (circle one)

b) Equation? ___________________________

c) Slope? _________

d) What does slope represent in the situation?

a) proportional or non- proportional? (circle one)

b) Equation? ___________________________

c) Slope? _________

d) What does slope represent in the situation?

a) proportional or non- proportional? (circle one)

b) Equation? ___________________________

c) Slope? _________

d) What does slope represent in the situation?

10

Page 11: Name: Per: M2 TOPIC 1 HOMEWORK PACKET M2-T1-L1 HW

10. Find the slope of each line. Remember to find the slope, choose two easy points, and then find the

( π’„π’„π’„π’„π’„π’„π’π’π’ˆπ’ˆπ’ˆπ’ˆ π’Šπ’Šπ’π’ π’—π’—π’ˆπ’ˆπ’—π’—π’•π’•π’Šπ’Šπ’„π’„π’„π’„π’—π’—π’„π’„π’„π’„π’„π’„π’π’π’ˆπ’ˆπ’ˆπ’ˆ π’Šπ’Šπ’π’ π’„π’„π’‰π’‰π’—π’—π’Šπ’Šπ’‰π’‰π’‰π’‰π’π’π’•π’•π’„π’„π’—π’—

). Don’t forget that if the line is going down, the slope is negative. Simplify if necessary.

a. Slope = m = _____________ b. Slope = m = _____________ c. Slope = m = _____________

d. Slope = m = _____________ e. Slope = m = _____________

11. Write the equation of each line in π’šπ’š = π’Žπ’Žπ’™π’™ + 𝒃𝒃 form or π’šπ’š = π’Žπ’Žπ’™π’™ form.

a. Equation: ___________________ b. Equation: _____________________ c. Equation: _____________________

d. Equation: ___________________ e. Equation: ___________________ f. Equation: _____________________

11

Page 12: Name: Per: M2 TOPIC 1 HOMEWORK PACKET M2-T1-L1 HW

PAGE INTENTIONALLY LEFT BLANK

12

Page 13: Name: Per: M2 TOPIC 1 HOMEWORK PACKET M2-T1-L1 HW

Name: _____________________________________ Per: ________

M2-T1-L3 HW: Slopes with Similar Triangles 1. Use the graphs below to answer the questions that follow.

2. Draw at least 4 right triangles using the points provided on the line. Show how no matter which two points you choose, the slope will always be the same.

3. Rank the slopes from steepest to flattest. (Be careful…the 2nd one is tricky)

a. 4, 2, Β½, 7, 3.5, 34 b. 5, – 2, 8, – 4, 1, – 10

0 1 2 3 4 5 6 7 8 9 10

Minutes Since Movie Theatre Opened

1000

900 800 700 600 500 400 300 200 100

Ti

cket

s Rem

aini

ng

a) proportional or non- proportional? (circle one)

b) Equation? ___________________________

c) Slope? _________

d) What does slope represent in the situation?

a) proportional or non- proportional? (circle one)

b) Equation? ___________________________

c) Slope? _________

d) What does slope represent in the situation?

A

D

C B

13

M2 TOPIC 1 HOMEWORK PACKET

Page 14: Name: Per: M2 TOPIC 1 HOMEWORK PACKET M2-T1-L1 HW

4. Consider the graph of the equation y = 2x + 3 shown to the right.

a. The points on the line were used to create triangles. Describe the relationship between the two triangles.

b. Use the two provided similar triangles to determine the slope between any two points on the line. (be sure to pay attention to the scale of the graph- each line counts as 2)

5. Consider each graph shown. Determine the slope of each line and then use similar triangles to justify that the slope is the same between any two points.

6. Determine the unknown angle measure for each triangle.

a. m∠A = 46°, m∠B = 90°, m∠C = ________ b. m∠P = _______, m∠Q = 10°, m∠R = 110°

7. Consider the graph of lines a, b, c, and d.

a. Which line(s) have positive slope? _____________

b. Which line(s) have negative slope? ______________

c. What is the slope of line c? ___________

8. Solve for the unknown angle measure given that f βˆ₯ g.

14