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 𝒚𝒚𝒙𝒙

)

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 coinsfrom other parts of the world. After sorting the coins, she comes to the conclusion that six out of every ten of thecoins 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 inher 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 thenumber 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 ofproportionality.

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

𝟐𝟐 𝟏𝟏 𝟏𝟏𝟐𝟐

𝟑𝟑𝟓𝟓 𝟐𝟐

𝟓𝟓 𝟑𝟑𝟐𝟐

𝟏𝟏𝟑𝟑𝟏𝟏 𝟗𝟗𝟐𝟐

𝟏𝟏𝟐𝟐𝟏𝟏 𝟑𝟑𝟏𝟏𝟏𝟏

𝑬𝑬 =𝟑𝟑𝟓𝟓 𝒕𝒕

𝑵𝑵 =𝟐𝟐𝟓𝟓 𝒕𝒕

The graph for part f (European), because the change in y (rise) is greater for the same amount of change in x (run)

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 𝒚𝒚 of buying 𝒙𝒙 pounds of fruit. One graph showsthe cost of buying 𝒙𝒙 pounds of peaches, and the other shows the cost ofbuying 𝒙𝒙 pounds of plums.

2

𝒚𝒚𝟐𝟐

𝒚𝒚𝟏𝟏 𝟗𝟗

𝟏𝟏𝟏𝟏

𝟏𝟏𝟏𝟏𝟏𝟏

𝟏𝟏𝟑𝟑

𝟐𝟐𝟑𝟑

𝒚𝒚𝟐𝟐 𝒚𝒚𝟏𝟏

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

Yes, both graphs intersect the origin.

b. Which fruit costs more per pound? Explain.

Peaches, the line is steeper .

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 𝑥𝑥 pounds of bananas.

Draw a line in between the other two graphs.

5. Kristina and her sister, Tracee, are paintingrooms in their house. The graph belowrepresents the rate at which Kristina paints,and the table below shows how many squarefeet Tracee painted for given amounts oftime. Both sisters paint at a constant pace.

a. Who paints at the faster rate? Justify youranswer. (talk about unit rate/constant ofproportionality)

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

6-8: REVIEW:

6. In the diagram, ∆𝑨𝑨𝑨𝑨𝑨𝑨 ~ ∆𝑿𝑿𝑿𝑿𝑿𝑿. State the corresponding sidesand angles. Then find the scale factor that was used to dilate∆𝑨𝑨𝑨𝑨𝑨𝑨 to ∆𝑿𝑿𝑿𝑿𝑿𝑿.

Corresponding Sides Corresponding Angles Scale Factor

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

a. Explain why ∆𝐵𝐵𝐵𝐵𝐵𝐵 ~ ∆𝐴𝐴𝐴𝐴𝐵𝐵.

b. Determine the length of 𝐵𝐵𝐴𝐴����.

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

Kristina, she paints 4 sq ft/min, whichis greater than Tracee’s 3.75 sq ft/min

Kristina: 𝒚𝒚 = 𝟒𝟒𝒙𝒙 Tracee: 𝒚𝒚 = 𝟑𝟑. 𝟕𝟕𝟓𝟓𝒙𝒙 or 𝒚𝒚 = 𝟏𝟏𝟓𝟓𝟒𝟒

𝒙𝒙

𝐗𝐗𝐗𝐗���� and 𝐀𝐀𝐀𝐀���� ∠A and ∠X 𝐗𝐗𝐘𝐘���� and 𝐀𝐀𝐁𝐁���� ∠B and ∠Y 𝐗𝐗𝐘𝐘���� and 𝐀𝐀𝐁𝐁���� ∠C and ∠Z

𝟏𝟏𝟑𝟑

Angle-Angle Similarity, because∠ACE ≅ ∠BCD∠CEA ≅ ∠CDB

𝐃𝐃𝐃𝐃���� = 𝟒𝟒 𝐜𝐜𝐜𝐜

Translate 5 right Reflect over the y-axis

9. Asia is training for a 10K road race that will take place in October. She starts her training in July and runs everyweekend. On her first run, she ran at a constant pace and covered 5.5 miles in 60 minutes. On her last run beforethe 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 earnedevery 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

5.5 mi/hr 6.8 mi/hr

𝒚𝒚 = 𝟓𝟓. 𝟓𝟓𝒙𝒙 𝒚𝒚 = 𝟔𝟔. 𝟏𝟏𝒙𝒙

Last run First run

Yes, both graphs intersect the origin (0, 0).

Kelly, she makes $8.40/hr.

Kelly, $14 more.

Kelly, she has the greater constant of proportionality.

Kelly, $8.40 more.

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

a. Write a linear equation that represents the number ofbuckets, 𝑦𝑦, that each pipe can fill in 𝑥𝑥 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 isconsidering buying a new machine that will fill 78 jars of salsa every 3minutes. To support his decision, he wants to compare the rate of the newmachine to the rate of the old machine that is currently in the factory. Thegraph shows the number of jars of salsa filled over time with the oldmachine.

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

b. The manager is about to fill an order for 765 jars of salsa. How long would ittake 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 lesssteep than the old machine? Explain.

13. The corner market sells rice by the pound using the equation 𝒚𝒚 = 𝟏𝟏. 𝟐𝟐𝟓𝟓𝒙𝒙, where 𝒚𝒚 represents the total cost for 𝒙𝒙 number of pounds. The localgrocery store also sells rice by the pound. The relationship between costand 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 cornermarket, how would the graph compare to the graph at the grocery store? Explain.

5

𝒚𝒚 =𝟑𝟑

𝟏𝟏𝟒𝟒 𝒙𝒙 𝒚𝒚 =𝟏𝟏𝟓𝟓 𝒙𝒙

Pipe A, because using pipe B would take 15 minutes to fill up 3 buckets.

Old: 30 jars/min New: 26 jars/min

Old: 25.5 mins New: 29.4 mins

No, the old machine fills jars at a faster rate.

Less steep, because the constant of proportionality is less.

The corner market, because it only costs $1.25/lbinstead of $1.50/lb at the grocery store.

The line would be less steep because the constant ofproportionality is less.

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6

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 unitrate means in terms of the situation.

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

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

2. Three staircases are shown below.

a. Order the staircases from least steep tomost steep. Explain how you know, orshow 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 steepnesscompared to the others. Explain.

Time, t (minutes)

Num

ber o

f Cra

bs, c

7

Yes, at 0 minutes he would have cleaned0 crabs.

𝟒𝟒𝟏𝟏 4 crabs are cleaned every 1 minute.

𝒄𝒄 = 𝟒𝟒𝒕𝒕

The graph passes through the origin.

A, C, B.

Between A and C

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 𝑡𝑡, 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

𝟓𝟓. 𝟓𝟓𝟏𝟏

It costs $5.50 for 1 pound of sliced turkey.

𝒚𝒚 = 𝟓𝟓. 𝟓𝟓𝒙𝒙

Yes, because the equation follows the format 𝒚𝒚 = 𝒌𝒌𝒙𝒙 and it costs $0 for 0 lbs of turkey.

𝟑𝟑. 𝟓𝟓𝟏𝟏

The 3.5 gallons of water flows out of the faucet every minute.

𝒗𝒗 = 𝟑𝟑. 𝟓𝟓𝒕𝒕

0 gallons, 7 gallons, 17.5 gallons

Yes, because at 0 minutes there are 0 gallons. The graph will be a straight line passing through the origin.

𝒙𝒙 = −𝟏𝟏

𝟏𝟏𝟐𝟐𝟔𝟔° 𝟓𝟓𝟒𝟒°

𝟓𝟓𝟒𝟒°

𝟓𝟓𝟒𝟒° 𝟏𝟏𝟐𝟐𝟔𝟔°

𝟏𝟏𝟐𝟐𝟔𝟔°

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 = 2.5𝑥𝑥 b. 𝑦𝑦 = 𝑥𝑥 − 4 c. 𝑦𝑦 = 5𝑥𝑥 + 2 d. 𝑦𝑦 = −6𝑥𝑥

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

∠𝟒𝟒 and ∠𝟔𝟔, ∠𝟑𝟑 and ∠𝟓𝟓

∠𝟒𝟒 and ∠𝟓𝟓

∠𝟐𝟐 and ∠𝟒𝟒

∠𝟐𝟐 and ∠𝟔𝟔

∠𝟏𝟏 and ∠𝟒𝟒

∠𝟐𝟐 and ∠𝟏𝟏

∠𝟏𝟏 and ∠𝟏𝟏

Yes, 𝒚𝒚 = 𝒌𝒌𝒙𝒙 Yes, 𝒚𝒚 = 𝒌𝒌𝒙𝒙 No, not 𝒚𝒚 = 𝒌𝒌𝒙𝒙 No, not 𝒚𝒚 = 𝒌𝒌𝒙𝒙

Translation 3 right and 6 up, congruent and similar

Dilation by a scale factor of 4 using point A as the center of dilation, similar

They are proportional.

same 𝟐𝟐𝟏𝟏

𝟔𝟔𝟑𝟑

=𝟐𝟐𝟏𝟏

They are similar by Angle-Angle Similarity.

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10

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, ½, 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

11

𝒚𝒚 = −𝟒𝟒𝟏𝟏𝒙𝒙 + 𝟒𝟒𝟏𝟏𝟏𝟏 −𝟒𝟒𝟏𝟏

Tickets remaining decrease by 40 every minute since the theatre opened

𝒚𝒚 = 𝟓𝟓𝟏𝟏𝒙𝒙 𝟓𝟓𝟏𝟏

Dollars owed increases by 50 for each hour of labor.

𝑨𝑨𝑨𝑨 = −𝟑𝟑𝟔𝟔 = −

𝟏𝟏𝟐𝟐

𝑨𝑨𝑨𝑨 = −𝟐𝟐𝟒𝟒 = −

𝟏𝟏𝟐𝟐

𝑨𝑨𝑨𝑨 = −𝟏𝟏𝟐𝟐

𝑨𝑨𝑩𝑩 = −𝟔𝟔

𝟏𝟏𝟐𝟐 = −𝟏𝟏𝟐𝟐

𝟕𝟕, 𝟒𝟒, 𝟑𝟑. 𝟓𝟓, 𝟐𝟐, 𝟑𝟑𝟒𝟒

, 𝟏𝟏𝟐𝟐 −𝟏𝟏𝟏𝟏, 𝟏𝟏, 𝟓𝟓, −𝟒𝟒, −𝟐𝟐, 𝟏𝟏

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 ∥ g.

The triangles are similar.

𝟏𝟏𝟒𝟒

= 𝟐𝟐 𝟏𝟏𝟔𝟔𝟏𝟏

= 𝟐𝟐

𝒎𝒎 = −𝟒𝟒 −𝟒𝟒𝟏𝟏

= −𝟏𝟏𝟐𝟐

𝒎𝒎 = 𝟏𝟏𝟑𝟑 𝟏𝟏

𝟑𝟑= 𝟐𝟐

𝟔𝟔

𝟑𝟑𝟒𝟒° 𝟔𝟔𝟏𝟏°

a, b d

0

𝒙𝒙 = 𝟏𝟏𝟓𝟓°

12

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

• The equation for a proportional linear relationship is 𝒚𝒚 = 𝒎𝒎𝒙𝒙, where m is the slope.

• An equation for a non-proportional linear relationship is 𝒚𝒚 = 𝒎𝒎𝒙𝒙 + 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 𝒚𝒚 = 𝒎𝒎𝒙𝒙 (proportional) or 𝒚𝒚 = 𝒎𝒎𝒙𝒙 + 𝒃𝒃 (non- proportional) to represent the relationship. Slope of each graph can be

found by ( 𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄 𝒊𝒊𝒄𝒄 𝒗𝒗𝒄𝒄𝒗𝒗𝒕𝒕𝒊𝒊𝒄𝒄𝒄𝒄𝒗𝒗𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄 𝒊𝒊𝒄𝒄 𝒄𝒄𝒉𝒉𝒗𝒗𝒊𝒊𝒉𝒉𝒉𝒉𝒄𝒄𝒕𝒕𝒄𝒄𝒗𝒗

) 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?

13

𝒚𝒚 = 𝟏𝟏𝟏𝟏𝒙𝒙 𝒚𝒚 = 𝟐𝟐𝒙𝒙 + 𝟓𝟓

𝟏𝟏𝟏𝟏 𝟐𝟐

10 feet per minute $2 per item

𝒚𝒚 = −𝟑𝟑𝟒𝟒

𝒙𝒙 + 𝟏𝟏𝟐𝟐 𝒚𝒚 =𝟏𝟏𝟓𝟓𝟏𝟏

𝒙𝒙 𝟏𝟏𝟓𝟓𝟏𝟏

−𝟑𝟑𝟒𝟒

Decreases 3 cm every 4 days or

Decreases 𝟑𝟑𝟒𝟒 cm every 1 day

15 miles every 1 hour

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

( 𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄 𝒊𝒊𝒄𝒄 𝒗𝒗𝒄𝒄𝒗𝒗𝒕𝒕𝒊𝒊𝒄𝒄𝒄𝒄𝒗𝒗𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄 𝒊𝒊𝒄𝒄 𝒄𝒄𝒉𝒉𝒗𝒗𝒊𝒊𝒉𝒉𝒉𝒉𝒄𝒄𝒕𝒕𝒄𝒄𝒗𝒗

). 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 𝒚𝒚 = 𝒎𝒎𝒙𝒙 + 𝒃𝒃 form or 𝒚𝒚 = 𝒎𝒎𝒙𝒙 form.

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

d. Equation: ___________________ e. Equation: ___________________

14

𝟏𝟏𝟒𝟒

−𝟑𝟑𝟏𝟏

𝐨𝐨𝐨𝐨 − 𝟑𝟑 −𝟏𝟏𝟑𝟑

−𝟏𝟏𝟑𝟑

𝟑𝟑𝟐𝟐

𝒚𝒚 = −𝟏𝟏𝟐𝟐

𝒙𝒙 + 𝟒𝟒 𝒚𝒚 = −𝟏𝟏𝟒𝟒

𝒙𝒙 + 𝟐𝟐 𝒚𝒚 = −𝟑𝟑𝟓𝟓

𝒙𝒙 + 𝟏𝟏

𝒚𝒚 = −𝟑𝟑𝟐𝟐

𝒙𝒙 + 𝟑𝟑 𝒚𝒚 = −𝟏𝟏𝟐𝟐

𝒙𝒙