1. No category

GRADE SAMPLE PROBLEM/TASK STANDARD Mail Man: Students work in pairs and are given a box (represents a delivery truck) and manipulatives. They are also given card sets that represent several situations to model. Mail truck is empty. Truck goes to stop one and picks up 37 letters, the truck then goes to stop 2 and picks up 33 letters, at stop 3 she collects 17 letters. How many letters are on the truck now? As student one reads the problem he will stop after each transaction and model. After stop one the student will get out 37 bundling straws to represent the letters and put them in the truck. After stop two the student will get 43 to represent the other letters in the truck, etc. Student 2 must record and solve for the number of letters on the truck. Students may then access the mail truck to check. Problems should also include subtraction. Truck has 48 letters and dropped off 23 letters. Count* Around Activity: Students stand in a circle. The first student starts says a number in the range of 20 to 100, as selected by the teacher, and the students go around the circle with each student saying the next number in the sequence until they reach the "stop" number, also selected by the teacher. The student who says the "stop" number sits down and the next child begins the count again starting at the same number as before. Kinsley had some chicken nuggets on her plate. Kinsley went to play a game. Her little brother snuck and put 3 of his nuggets on Kinsley’s plate. When Kinsley came back there were 9 nuggets on her plate. Can you help figure out how many nuggets Kinsley was supposed to have? Students solve and write an equation given the following task. There will be 5 children at the birthday party. I have 3 party hats. How many more hats do I need? Write a multiplication problem that represents the picture below. Given a picture, ask students to color all of the rectangles blue, triangles green, etc. Match the fraction with the correct picture. ¼ Given a variety of shapes in different sizes have students sort the shapes into groups. Ask follow up questions related to these groups. How many shapes are in each group? How many more shapes are in “group A” than “group B”? Go Fish: Each player is dealt five cards. The rest of the cards are placed in a stack face down in the center of the table. If the students have any pairs of cards that total 10, they place them down in front of them and replace those cards with cards from the deck. Students take turns. On each turn, a player asks another player for a card that will go with a card in the player's hand to make 10. If he/she receives a card that makes a pair, the pair is placed on the table. This completes a turn. If the player does not get a card that makes a pair that totals 10, he/she takes the top card from the deck. If the card drawn from the deck makes a pair with a card in the player's hand, the pair is placed on the table. This completes the turn. If there are no cards left in a player's hand but still cards in the deck, that player takes two cards from the deck and continues playing. The game is over when there are no more cards left in the deck. At the end of the game each player writes a list of the number pairs he/she made.. ___tens ___ones ___ones ___tens ___ones ___ones Using linking cubes. Work with a partner to determine if the following numbers are odd or even. Draw a picture to record your work and be prepared to explain. 7 12 15 18 Our school is collecting quarters to help raise money for a new slide. Yesterday we our school brought 389 quarters to school. Today our school collected 437 quarters. How many quarters do we have total? Write a number sentence and solve. How far is it from our classroom to the gym? Students will randomly select three arrow cards. One from the hundreds, one from the tens, and one from the ones. They will then use these to build their number. Students will decide which one is greater/less and will then record their comparison statement on a sheet of paper. Replace the boxes with values from 1 to 6 to make each problem true. You can use each number as often as you want. You CANNOT use 7, 8, 9, 0. + + + .
Check the students work below to see if it is correct or not. 5 10 1 4 7 + 3 8 7 5 6 0 -­‐5 8 0 2 2 8 + 5 3 8 1 8 3 -­‐ 2 7 6 4 Students work in a small group. One student flips over a numeral card that has a number 0 to 10 on it. The other students in the group will write the number that goes with it to make ten. Example: I flipped over a 6…on your paper you must write the number 4. Given a blank hundred chart, students write the numbers to fill in the missing spaces. The first student will roll a dice with the numerals 5, 6, 7, 8, 9, 10 to find a “target” number. Then first student
chooses from a pile of ten frame cards that represent the numbers 0, 1, 2, 3, 4, 5 to be the first addend.
Example: Student rolls a seven as the target number then selects the card below
The other student chooses a ten frame card that will go with the first addend to make the target number.
Example: Since the target number rolled was seven. Student two should select the ten frame that represents3.
4+3=7
Timmy was asked to share a candy bar fairly between three kids. Draw a picture that shows how Timmy should have shared the candy bar. Label each piece of the candy bar with the correct fraction. Congruency of Assessment Items/Tasks Work in vertical grade level teams and determine in which grade level you think the item would be used. Then find the specific standard that is congruent to each item. Did your initial prediction match? Where there any surprises? SAMPLE PROBLEM/TASK GRADE STANDARD 1(10,000) + 2(1,000) + 4(100) + 3(10) + 2(1) + 5(1/10) + 3(1/100). Which number below is one-­‐tenth of the expanded form above? A 12422.53 B 1243.253 C 12432.53 D 12432.43 What number is shown by the model? When rounding to the nearest ten: a. What is the smallest whole number that will round to 50? b. What is the largest whole number that will round to 50? c. How many different whole numbers will round to 50? When rounding to the nearest hundred: d. What is the smallest whole number that will round to 500? e. What is the largest whole number that will round to 500? f. How many different whole numbers will round to 500? A group of 4 friends was at a restaurant. They each ordered an $8 meal. Then the group ordered a $6 dessert to share. Write an expression that represents this situation. 1 Congruency of Assessment Items/Tasks A. Arrange these numbers in order, beginning with the smallest.
2400 4002 2040 420 2004
B. Arrange these numbers in order, beginning with the greatest.
1470 847 710 1047 147
Write a multiplication problem that represents the picture below. Students will randomly select three arrow cards. One from the hundreds, one from the tens, and one from the ones. They will then use these to build their number. Students will decide which one is greater/less and will then record their comparison statement on a sheet of paper. Match the fraction with the correct picture. ¼ 2 Congruency of Assessment Items/Tasks What is the greatest common factor (GCF) of 25 and 35? a.
Helen raised $12 for the food bank last year and she raised 6 times as much money this
year. How much money did she raise this year?
b. Sandra raised $15 for the PTA and Nita raised $45. How many times as much money did
Nita raise as compared to Sandra?
c. Nita raised $45 for the PTA, which was 3 times as much money as Sandra raised. How
much money did Sandra raise?
Mail Man: Students work in pairs and are given a box (represents a delivery truck) and manipulatives. They are also given card sets that represent several situations to model. Mail truck is empty. Truck goes to stop one and picks up 37 letters, the truck then goes to stop 2 and picks up 33 letters, at stop 3 she collects 17 letters. How many letters are on the truck now? As student one reads the problem he will stop after each transaction and model. After stop one the student will get out 37 bundling straws to represent the letters and put them in the truck. After stop two the student will get 43 to represent the other letters in the truck, etc. Student 2 must record and solve for the number of letters on the truck. Students may then access the mail truck to check. Problems should also include subtraction. Truck has 48 letters and dropped off 23 letters. Karl's rectangular vegetable garden is 20 feet by 45 feet, and Makenna's is 25 feet by 40
feet. Whose garden is larger in area? 3 Congruency of Assessment Items/Tasks Alex is training for his school's Jog-A-Thon and needs to run at least 1 mile per day. If Alex
runs to his grandma's house, which is of a mile away, and then to his friend Justin's house,
which is of a mile away, will he have trained enough for the day? Imagine that each square in the picture measures one centimeter on each side. What is the
area of each letter? Try to work it out without counting each square individually.
Which number is closest to 12?
a. 1/8
b. 3/8
c. 7/8
d. 9/8
Two groups of students from Douglas Elementary School were walking to the library when it began to rain. The 7 students in Mr. Stem’s group shared the 3 large umbrellas they had with Ms. Thorn’s group of 11 students. If the same number of students were under each umbrella, how many students were under each umbrella? 4 Congruency of Assessment Items/Tasks A grocery store sign indicates that bananas are 6 for $1.50, and a sign by the oranges indicates that they are 5 for $3.00. Find the total cost of buying 2 bananas and 2 oranges. Mrs. Sawyer had 82 pictures of birds she gave to her 3 children. She gave each child as
many pictures as possible. Each child received the same number of pictures and Mrs.
Sawyer kept the rest. How many pictures did each child receive? How many times greater is the value of the digit 5 in 583,607 than the value of the digit 5 in 362,501? A 10 times B 100 times C 1,000 times D 10,000 times Students from three classes at Hudson Valley Elementary School are planning a boat trip. On the trip, there will be 20 students from each class, along with 11 teachers and 13 parents. Part A: Write an equation that can be used to determine the number of boats, b, they will need on their trip if 10 people ride in each boat. Equation: b =___________ Part B: How many boats will be needed for the trip if 10 people ride in each boat? Show your work. Answer: __________ boats Part C: It will cost $35 to rent each boat used for the trip. How much will it cost to rent all the boats needed for the trip? Show your work. 5 Congruency of Assessment Items/Tasks GRADE SAMPLE PROBLEM/TASK STANDARD Sierra walks her dog Pepper twice a day. Her evening walk is two and a half times
as far as her morning walk. At the end of the week she tells her mom,
“I walked Pepper for 30 miles this week!”
How long is her morning walk?
a. Amy wants to build a cube with 3 cm sides using 1 cm cubes. How many cubes
does she need?
b. How many 1 cm cubes would she need to build a cube with 6 cm sides?
Ocean water freezes at about −2 1∘ C. Fresh water freezes at 0∘ C. Antifreeze, a iquid
used to cool most car engines, freezes at −64∘ C.
Imagine that the temperature is exactly at the freezing point for ocean water. How
many degrees must the temperature drop for the antifreeze to turn to ice?
Coffee costs $18.96 for 3 pounds.
a. What is the cost per pound of coffee?
b. Let x be the number of pounds of coffee and y be the total cost of x pounds. Draw
a graph of the proportional relationship between the number of pounds of coffee
and the total cost.
c. Where can you see the cost per pound of coffee in the graph? What is it?
1 Congruency of Assessment Items/Tasks On the number line above, the numbers a and b are the same distance from 0.
What is a + b?
Explain how you know.
In triangle ΔABC, point M is the point of intersection of the bisectors of angles
BAC, ABC, and
ACB. The measure of
BAC is 64°. What is the measure of
ABC is 42°, and the measure of
BMC?
For each pair of numbers, decide which is larger without using a calculator.
Explain
your
choices.
This task
adapted
from a problem published by the Russian Ministry of
Education.
a. π2 or 9
b. √50 or √51
c. √50 or 8
d. −2π or −6
asked to identify their favorite
All the students at a middle school were
academic subject and whether they were in 7th grade or 8th grade. Here are
the results:
Favorite Subject by Grade
Grade
English
History
Math/Science
Other
Totals
7th Grade
38
36
28
14
116
8th Grade
47
45
72
18
182
Totals
85
81
100
32
298
Is there an association between favorite academic subject and grade for
students at this school? Support your answer by calculating appropriate relative
frequencies using the given data.
2 Congruency of Assessment Items/Tasks Below are the 25 birth weights, in ounces, of all the Labrador Retriever puppies
born at Kingston Kennels in the last six months.
13 14 15 15 16 16 16 16 17 17 17 17 17 17 17 18 18 18 18 18 18 18 18 19 20
a. Use an appropriate graph to summarize these birth weights.
b. Describe the distribution of birth weights for puppies born at Kingston
Kennels in the last six months. Be sure to describe shape,center and
variability.
c. What is a typical birth weight for puppies born at Kingston Kennels in the
last six months? Explain why you chose this value. The ratio of the number of boys to the number of girls at school is 4:5.
a. What fraction of the students are boys?
b. If there are 120 boys, how many students are there altogether? The students in Mr. Sanchez's class are converting distances measured in
miles to kilometers.
To estimate the number of kilometers, Abby takes the number of miles,
doubles it, then subtracts 20% of the result.
Renato first divides the number of miles by 5, then multiplies the result by 8.
a. Write an algebraic expression for each method.
b. Use your answer to part (a) to decide if the two methods give the same
answer. 3 Congruency of Assessment Items/Tasks Medhavi suspects that there is a relationship between the number of text
messages high school students send and their academic achievement. To
explore this, she asks each student in a random sample of 52 students from
her school how many text messages he or she sent yesterday and what his or
her grade point average (GPA) was during the most recent marking period.
The data are summarized in the scatter plot of number of text messages sent
versus GPA shown below.
Describe the relationship between number of text messages sent and GPA.
Discuss both the overall pattern and any deviations from the pattern.
The students in Ms. Baca’s art class were mixing yellow and blue paint. She
told them that two mixtures will be the same shade of green if the blue and
yellow paint are in the same ratio.
The table below shows the different mixtures of paint that the students made.
Yellow
Blue
A
1 part
2 part
B
2 parts
3 parts
C
3 parts
6 parts
D
4 parts
6 parts
E
6 parts
9 parts
a. How many different shades of paint did the students make?
b. Some of the shades of paint were bluer than others. Which mixture(s)
were the bluest? Show work or explain how you know.
4 Congruency of Assessment Items/Tasks 2
POSSIBLE COURSE SAMPLE PROBLEM/TASK* 1.
A team of farm-­‐workers was assigned the task of harvesting two fields, one twice the size of the other. They worked for the first half of the day on the larger field. Then the team split into two groups of equal number. The first group continued working in the larger field and finished it by evening. The second group harvested the smaller field, but did not finish by evening. The next day one farm-­‐
worker finished the smaller field in a single day's work. How many farm-­‐workers were on the team? 2. Consider the following algebraic expressions: 2
2
(n + 2) − 4 and n + 4n. a. Use the figures below to illustrate why the following expressions are equivalent
b.
Eric is playing a video game. At a certain point in the game, he has 31500 points. Then the following events happen, in order: He earns 2450 additional points. He loses 3310 points. The game ends, and his score doubles. a. Write an expression for the number of points Eric has at the end of the game. Do not evaluate the expression. The expression should keep track of what happens in each step listed above. b. Eric's sister Leila plays the same game. When she is finished playing, her score is given by the expression 3(24500 + 3610) − 6780. Describe a sequence of events that might have led to Leila earning this score. 4. Suppose P and Q give the sizes of two different animal populations, where Q>P. In (a)–(d), say which of the given pair of expressions is larger. Briefly explain your reasoning in terms of the two populations. a. P+Q and 2P c.
d.
5.
P +Q
P
and P +Q
2
(Q−P)/2 and Q−P/2 P+50t and Q+50t Judy is working at a retail store over summer break. A customer buys a $50 shirt that is on sale for 20% off. Judy computes the discount, then adds sales tax of 10%, and tells the customer how much he owes. The customer insists that Judy first add the sales tax and then apply the discount. He is convinced that this way he will save more money because the discount amount will be larger. a. Is the customer right? b. Does your answer to part (a) depend on the numbers used or would it work for any percentage discount and any sales tax percentage? Find a convincing argument using algebraic expressions and/or diagrams for this more general scenario. Kentucky Department of Education
Find some algebraic deductions of the same result. 3.
b.
STANDARD/ CLUSTER *Illustrativemathematics.org 1 Congruency of Assessment Items/Tasks 2
6.
The students in Mr. Sanchez's class are converting distances measured in miles to kilometers. To estimate the number of kilometers, Abby takes the number of miles, doubles it then subtracts 20% of the result. Renato first divides the number of miles by 5 and then multiplies the result by 8. a. Write an algebraic expression for each method. b. Use your answer to part (a) to decide if the two methods give the same answer. 7.
Consider the expression R1 + R2
R1R2
where R1 and R2 are positive. Suppose we increase the value of R1 while keeping R2 constant. Find an equivalent expression whose structure makes clear whether the value of the expression increases, decreases, or stays the same. 8.
Susan has an ear infection. The doctor prescribes a course of antibiotics. Susan is told to take 250 mg doses of the antibiotic regularly every 12 hours for 10 days. Susan is curious and wants to know how much of the drug will be in her body over the course of the 10 days. She does some research online and finds out that at the end of 12 hours, about 4% of the drug is still in the body. a. What quantity of the drug is in the body right after the first dose, the second dose, the third dose, the fourth dose? b. When will the total amount of the antibiotic in Susan’s body be the highest? What is that amount? c. Answer Susan's original question: Describe how much of the drug will be in her body at various points over the course of the 10 days. 9.
Consider the polynomial function 4
3
2
P(x) = x − 3x + ax − 6x + 14, where a is an unknown real number. If (x−2) is a factor of this polynomial, what is the value of a? 10. Suppose f is a quadratic function given by the equation 2
f(x) = ax + bx + c where a, b, c are real numbers and we will assume that a is non-­‐
zero. a. If 0 is a root of f show that c = 0 or, in other words, show that 2
ax + bx + c is evenly divisible by x. 2
b. If 1 is a root of f show that ax + bx + c is evenly divisible by x − 1. c. Suppose r is a real number. If r is a root of f show that ax2 + bx + c is evenly divisible by x − r. 11. In the equations (a)–(d), the solution x to the equation depends on the constant a. Assuming a is positive, what is the effect of increasing a on the solution? Does it increase, decrease, or remain unchanged? Give a reason for your answer that can be understood without solving the equation. a. x – a = 0 b. ax = 1 c. ax = a d.
x
= 1 a
12. If we multiply x/2 + 3/4 by 4, we get 2x + 3. Is 2x + 3 an equivalent algebraic expression to x/2 + ¾? Kentucky Department of Education
*Illustrativemathematics.org 2 Congruency of Assessment Items/Tasks 2
13. Solve the following two equations by isolating the radical on one side and squaring both sides: 2x + 1 − 5 = −2 2x + 1 + 5 = 2 i.
a.
ii.
Be sure to check your solutions. If we raise both sides of an equation a power, we sometimes obtain an equation which has more solutions than the original one. (Sometimes the extra solutions are called extraneous solutions.) Which of the following equations result in extraneous solutions when you raise both sides to the indicated power? Explain. i.
ii.
iii.
3
x = 5 , square both sides x = −5 , square both sides x = 5, cube both sides x = −5 , cube both sides 3
iv.
b. Create a square root equation similar to the one in part (a) that has an extraneous solution. Show the algebraic steps you would follow to look for a solution, and indicate where the extraneous solution arises. 14. Solve the quadratic equation, 2
2
x = (2x − 9) , using as many different methods as possible. 2
15. Suppose h(t) = −5t + 10t + 3 is an expression giving the height of a diver above the water, in meters, t seconds after the diver leaves the springboard. a. How high above the water is the springboard? Explain how you know. b. When does the diver hit the water? c. At what time on the diver's descent toward the water is the diver again at the same height as the springboard? d. When does the diver reach the peak of the dive? 16. Nola was selling tickets at the high school dance. At the end of the evening, she picked up the cash box and noticed a dollar lying on the floor next to it. She said, I wonder whether the dollar belongs inside the cash box or not. The price of tickets for the dance was 1 ticket for $5 (for individuals) or 2 tickets for $8 (for couples). She looked inside the cash box and found $200 and ticket stubs for the 47 students in attendance. Does the dollar belong inside the cash box or not? 17. Lauren keeps records of the distances she travels in a taxi and what she pays: Distance, d, in miles Fare, F, in dollars 3 8.25 5 12.75 11 26.25 a. If you graph the ordered pairs (d, F) from the table, they lie on a line. How can you tell this without graphing them? b. Show that the linear function in part (a) has equation F = 2.25d + 1.5. c. What do the 2.25 and the 1.5 in the equation represent in terms of taxi rides? 18. You have $100 to spend on a barbeque where you want to serve chicken and steak. Chicken costs $1.29 per pound and steak costs $3.49 per pound. a. Find a function that relates the amount of chicken and the amount of steak you can buy. b. Graph the function. What is the meaning of each intercept in this context? What is the meaning of the slope in this context? Use this (and any other information represented by the equation or graph) to discuss what your options are for the amounts of chicken and amount of steak you can buy for the barbeque.
Kentucky Department of Education
*Illustrativemathematics.org 3 Congruency of Assessment Items/Tasks 2
19. The population of a country is initially 2 million people and is increasing at 4% per year. The country's annual food supply is initially adequate for 4 million people and is increasing at a constant rate adequate for an additional 0.5 million people per year. a. Based on these assumptions, in approximately what year will this country first experience shortages of food? b. If the country doubled its initial food supply and maintained a constant rate of increase in the supply adequate for an additional 0.5 million people per year, would shortages still occur? In approximately which year? c. If the country doubled the rate at which its food supply increases, in addition to doubling its initial food supply, would shortages still occur? 20. Mike likes to canoe. He can paddle 150 feet per minute. He is planning a river trip that will take him to a destination about 30,000 feet upstream (that is, against the current). The speed of the current will work against the speed that he can paddle. a. Let s be the speed of the current in feet per minute. Write an expression for r(s), the speed at which Mike is moving relative to the river bank, in terms of s. b. Mike wants to know how long it will take him to travel the 30,000 feet upstream. Write an expression for T(s), the time in minutes it will take, in terms of s. c. What is the vertical intercept of T? What does this point represent in terms of Mike’s canoe trip? d. At what value of s does the graph have a vertical asymptote? Explain why this makes sense in the situation. e. For what values of s does T(s) make sense in the context of the problem? 21. You work for a video streaming company that has two monthly plans to choose from: Plan 1: A flat rate of $7 per month plus $2.50 per video viewed Plan 2: $4 per video viewed a. What type of functions model this situation? Explain how you know. b. Define variables that make sense in the context, and then write an equation with cost as a function of videos viewed, representing each monthly plan. c. How much would 3 videos in a month cost for each plan? 5 videos? d. Compare the two plans and explain what advice you would give to a customer trying to decide which plan is best for them, based on their viewing habits.
22. Consider the equation 5x−2y=3. If possible, find a second linear equation to create a system of equations that has: a. Exactly 1 solution. b. Exactly 2 solutions. c. No solutions. d. Infinitely many solutions. 23. Some of the students at Kahlo Middle School like to ride their bikes to and from school. They always ride unless it rains. Let d be the distance in miles from a student's home to the school. Write two different expressions that represent how far a student travels by bike in a four week period if there is one rainy day each week. Kentucky Department of Education
*Illustrativemathematics.org 4 Congruency of Assessment Items/Tasks 2
24. John makes DVDs of his friend’s shows. He has realized that, because of his fixed costs, his average cost per DVD depends on the number of DVDs he produces. The cost of producing x DVDs is given by C(x) = 2500 + 1.25x. a. John wants to figure out how much to charge his friend for the DVDs. He’s not trying to make any money on the venture, but he wants to cover his costs. Suppose John made 100 DVDs. What is the cost of producing this many DVDs? How much is this per DVD? b. John is hoping to make many more than 100 DVDs for his friends. Complete the table showing his costs at different levels of production. # of DVDs 0 10 100 1,000 10,000 100,000 1,000,000 Total Cost Cost per DVD c.
d.
e.
f.
25.
a.
b.
c.
a.
b.
c.
d.
27.
Explain why the average cost per DVD levels off. Find an equation for the average cost per DVD of producing x DVDs. Find the domain of the average cost function. Using the data points from your table above, sketch the graph of the average cost function. How does the graph reflect that the average cost levels off? A certain business keeps a database of information about its customers. Let C be the rule which assigns to each customer shown in the table his or her home phone number. Is C a function? Explain your reasoning. Customer Name Home Phone Number Heather Baker 3105100091 Mike London 3105200256 Sue Green 3234132598 Bruce Swift 3234132598 Michelle Metz 2138061124 Let P be the rule which assigns to each phone number in the table above, the customer name(s) associated with it. Is P a function? Explain your reasoning. Explain why a business would want to use a person's social security number as a way to identify a particular customer instead of their phone number. 26. A downtown city parking lot charges $0.50 for each 30 minutes you park, or fraction thereof, up to a daily maximum charge of $10.00. Let C assign to each length of time you park, t (in hours), the cost of parking in the lot, C(t) (in dollars). Complete the table below. t (in hours) C(t) (in dollars) 0 1/4 1/3 9/16 1 1/4 29/12 Sketch a graph of C for 0 ≤ t ≤ 8. Is C a function of t? Explain your reasoning. Is t a function of C? Explain your reasoning. Antonio and Juan are in a 4-­‐mile bike race. The graph below shows the distance of each racer (in miles) as a function of time (in minutes). Kentucky Department of Education
*Illustrativemathematics.org 5 Congruency of Assessment Items/Tasks 2
a.
Who wins the race? How do you know? b. Imagine you were watching the race and had to announce it over the radio, write a little story describing the race.
28. Imagine Scott stood at zero on a life-­‐sized number line. His friend flipped a coin 50 times. When the coin came up heads, he moved one unit to the right. When the coin came up tails, he moved one unit to the left. After each flip of the coin, Scott's friend recorded his position on the number line. Let f assign to the whole number n, when 1≤n≤50, Scott's position on the number line after the nth coin flip. a. If f(6) = 6 what can you conclude about the outcomes of the first 6 coin tosses? Explain. What if f(6) = −4? b. Is it possible that f(7) = 2? Explain. c. Find all integers m so that the probability that f(50) = m is zero. 29. In order to gain popularity among students, a new pizza place near school plans to offer a special promotion. The cost of a large pizza (in dollars) at the pizza place as a function of time (measured in days since February 10th) may be described as 0≤t <3
⎧ 9,
⎪
C (t ) = ⎨9 + t , 3 ≤ t < 8 ⎪ 20, 8 < t < 28
⎩
(Assume t only takes whole number values.) a. If you want to give their pizza a try, on what date(s) should you buy a large pizza in order to get the best price? th
b. How much will a large pizza cost on February 18 ? c. On what date, if any, will a large pizza cost 13 dollars? d. Write an expression that describes the sentence "The cost of a large pizza is at least A dollars B days into the promotion," using function notation and mathematical symbols only. e. Calculate C(9) − C(8) and interpret its meaning in the context of the problem. f. On average, the cost of a large pizza goes up about 85 cents per day during the first two weeks of the promotion period. Which of the following equations best describes this statement? •
•
C (13) + C (0 )
= 0.85
2
C (13) − C (0 )
= 0.85
13
Kentucky Department of Education
*Illustrativemathematics.org 6 Congruency of Assessment Items/Tasks 2
•
•
C (13)
= 0.85
13
C (February 23) − C (February 10 )
= 0.85 13
30. Given below are three graphs that show solar radiation, S, in watts per square meter, as a function of time, t, in hours since midnight. We can think about this quantity as the maximum amount of power that a solar panel can absorb, which tells us how intense the sunshine is at any given time. Match each graph to the corresponding description of the weather during the day. a. It was a beautifully sunny day from sunrise to sunset – not a cloud in the sky. b. The day started off foggy but eventually the fog lifted and it was sunny the rest of the day. c. It was a pretty gloomy day. The morning fog never really lifted. 1.
2.
3.
All three graphs show solar radiation measured in Santa Rosa, a city in northern California. What other information can you get from the graph? 31. An epidemic of influenza spreads through a city. The figure below is the graph of I = f(w), where I is the number of individuals (in thousands) infected w weeks after the epidemic begins. Kentucky Department of Education
*Illustrativemathematics.org 7 Congruency of Assessment Items/Tasks 2
Estimate f(2) and explain its meaning in terms of the epidemic. Approximately how many people were infected at the height of the epidemic? When did that occur? Write your answer in the form f(a)=b. c. For approximately which w is f(w) = 4.5; explain what the estimates mean in terms of the epidemic. d. An equation for the function used to plot the image above is f(w) = 6w(1.3) − w. Use the graph to estimate the solution of the inequality 6w(1.3) – w ≥ 6. Explain what the solution means in terms of the epidemic. (Task from Functions Modeling Change: A Preparation for Calculus, Connally et al., Wiley 2010.) 32. Consider the following four functions a.
b.
•
•
•
•
3
1 + e −3 x
e−x
(
)
g x =1−
2
ex
h(x ) = −2 + 2
3
k (x ) =
1 + e3x
f (x ) =
Below are four graphs of functions shown for −2 ≤ x ≤2. Match each function with its graph and explain your choice: 33. Which of the following equations could describe the function whose graph is shown below? Explain. Kentucky Department of Education
*Illustrativemathematics.org 8 Congruency of Assessment Items/Tasks 2
2
f1(x) = (x + 12) + 4 f5(x) = −4(x + 2)(x + 3) 2
f2(x) = −(x − 2) − 1 f6(x) = (x + 4)(x − 6) 2
f3(x) = (x + 18) − 40 f7(x) = (x − 12)(−x + 18) 2
f4(x) = (x − 10) − 15 f8(x) = (20 − x)(30 − x) 34. Without using the square root button on your calculator, estimate √ (800) , square root of 800, as accurately as possible to 2 decimal places. 35. How many cubes are needed to build this tower? How many cubes are needed to build a tower like this, but 12 cubes high? Justify your reasoning. c. How would you calculate the number of cubes needed for a tower n cubes high? 27. Using the graphs below, sketch a graph of the function s(x) = f(x) + g(x). a.
b.
28. According to the U.S. Energy Information Administration, a barrel of crude oil produces approximately 20 gallons of gasoline. EPA mileage estimates indicate a 2011 Ford Focus averages 28 miles per gallon of gasoline. a. Write an expression for g(x), the number of gallons of gasoline produced by x Kentucky Department of Education
*Illustrativemathematics.org 9 Congruency of Assessment Items/Tasks 2
b.
c.
d.
29.
a.
b.
c.
barrels of crude oil. Write an expression for M(x), the number of miles on average that a 2011 Ford Focus can drive on x gallons of gasoline. Write an expression for M(g(x)). What does M(g(x)) represent in terms of the context? One estimate (from www.oilvoice.com) claimed that the 2010 Deepwater Horizon disaster in the Gulf of Mexico spilled 4.9 million barrels of crude oil. How many miles of Ford Focus driving would this spilled oil fuel? 2
Let f be the function defined by f(x) = 2x + 4x − 16. Let g be the function defined 2
by g(x) = 2(x + 1) − 18. Verify that f(x) = g(x) for all x. 2
In what ways do the equivalent expressions 2x + 4x − 16 and 2
2(x + 1) − 18 help to understand the function f? Consider the functions h, l, m, and n given by h(x )
= x2
l (x ) = x + 1
m(x ) = x − 9
n(x )
d.
30.
a.
b.
c.
31.
a.
b.
Show that f(x) is a composition, in some order, of the functions h, l, m, and n. How do you determine the order of composition? Explain the impact each of the functions l, m, and n has on the graph of the composition. City Bank pays a simple interest rate of 3% per year, meaning that each year the balance increases by 3% of the initial deposit. National Bank pays an compound interest rate of 2.6% per year, compounded monthly, meaning that each month the balance increases by one twelfth of 2.6% of the previous month's balance. Which bank will provide the largest balance if you plan to invest $10,000 for 10 years? For 15 years? Write an expression for C(y), the City Bank balance, y years after a deposit is left in the account. Write an expression for N(m), the National Bank balance, m months after a deposit is left in the account. Create a table of values indicating the balances in the two bank accounts from year 1 to year 15. For which years is City Bank a better investment, and for which years is National Bank a better investment? Algae blooms routinely threaten the health of the Chesapeake Bay. Phosphate compounds supply a rich source of nutrients for the algae, Prorocentrum minimum, responsible for particularly harmful spring blooms known as mahogany tides. These compounds are found in fertilizers used by farmers and find their way into the Bay with run-­‐offs resulting from rainstorms. Favorable conditions result in rapid algae growth ranging anywhere from 0.144 to 2.885 cell divisions per day. Algae concentrations are measured and reported in terms of cells per milliliter (cells/ml). Concentrations in excess of 3,000 cells/ml constitute a bloom. Suppose that heavy spring rains followed by sunny days create conditions that support 1 cell division per day and that prior to the rains Prorocentrum minimum concentrations measured just 10 cells/ml. Write an equation for a function that models the relationship between the algae concentration and the number of days since the algae began to divide at the rate of 1 cell division per day. Assuming this rate of cell divison is sustained for 10 days, present the resulting algae concentrations over that period in a table. Did these conditions result in a bloom? Kentucky Department of Education
= 2x
*Illustrativemathematics.org 10 Congruency of Assessment Items/Tasks 2
c.
Concentrations in excess of 200,000 cells/ml have been reported in the Bay. If conditions support 2 cell divisions per day, when will these conditions result in a bloom? When will concentrations exceed 200,000 cells/ml? 32. A cup of hot coffee will, over time, cool down to room temperature. The principle of physics governing the process is Newton's Law of Cooling. Experiments with a covered cup of coffee show that the temperature (in degrees Fahrenheit) of the coffee can be modeled by the following equation −0.08t
+ 75. f(t) = 110e
Here the time t is measured in minutes after the coffee was poured into the cup. −0.08t
+ 75, why the coffee a. Explain, using the structure of the expression 110e
temperature decreases as time elapses. b. What is the temperature of the coffee at the beginning of the experiment? c. After how many minutes is the coffee 140 degrees? After how many minutes is the coffee 100 degrees? 33. A car is traveling down a long, steep hill. The elevation, E, above sea level (in feet) of the car when it is d miles from the top of the hill is given by E=7500–250d, where d can be any number from 0 to 6. Find the slope and intercepts of the graph of this function and explain what they mean in the context of the moving car. 34.
Given below is a table that gives the populations of foxes and rabbits in a national park over a 12 month period. Note that each value of t corresponds to the beginning of the month.
a.
b.
According to the data in the table, is F a function of R? Is R a function of F? Is either R or F functions of t? Explain your reasoning. This task is adapted from "Functions Modeling Change", Connally et al, Wiley 2007. 35. Stephanie is helping her band collect money to fund a field trip. The band decided to sell boxes of chocolate bars. Each bar sells for $1.50 and each box contains 20 bars. Below is a partial table of monies collected for different numbers of boxes sold. (imagine shrunk for formatting) a.
b.
c.
d.
e.
Complete the table above for values of m. Write an equation for the amount of money, m, that will be collected if b boxes of chocolate bars are sold. Which is the independent variable and which is the dependent variable? Graph the equation using the ordered pairs from the table above. Calculate how much money will be collected if 100 boxes of chocolate bars are sold. The band collected $1530.00 from chocolate bar sales. How many boxes did they sell? Kentucky Department of Education
*Illustrativemathematics.org 11 Congruency of Assessment Items/Tasks 2
36. You have been asked to place a fire hydrant so that it is an equal distance form three locations indicated on the following map. a.
Show how to fold your paper to physically construct this point as an intersection of two creases. b. Explain why the above construction works, and in particular why you only needed to make two creases. 37. The figure below is composed of eight circles, seven small circles and one large circle containing them all. Neighboring circles only share one point, and two regions between the smaller circles have been shaded. Each small circle has a radius of 5 cm. Calculate: a. The area of the large circle. b. The area of the shaded part of the figure. 38. You have been asked to place a warehouse so that it is an equal distance from the three roads indicated on the following map. Find this location and show your work. a.
b.
Show how to fold your paper to physically construct this point as an intersection of two creases. Explain why the above construction works, and in particular why you only needed to make two creases. Kentucky Department of Education
*Illustrativemathematics.org 12 GRADE SAMPLE PROBLEM/TASK STANDARD Mail Man: Students work in pairs and are given a box (represents a delivery truck) and manipulatives. They are also given card sets that represent several situations to model. Mail truck is empty. Truck goes to stop one and picks up 37 letters, the truck then goes to stop 2 and picks up 33 letters, at stop 3 she collects 17 letters. How many letters are on the truck now? As student one reads the problem he will stop after each transaction and model. After stop one the student will get out 37 bundling straws to represent the letters and put them in the truck. After stop two the student will get 43 to represent the other letters in the truck, etc. 2.NBT.6 Student 2 must record and solve for the number of letters on the truck. Students may then access the mail truck to check. Problems should also include subtraction. Truck has 48 letters and dropped off 23 letters. Count* Around Activity: Students stand in a circle. The first student starts says a number in the range of 20 to 100, as selected by the teacher, and the students go around the circle with each student saying the next number in the sequence until they reach the "stop" number, also selected by the teacher. The student who says the "stop" number sits down and the next child begins the count again starting at the same number as before. Kinsley had some chicken nuggets on her plate. Kinsley went to play a game. Her little brother snuck and put 3 of his nuggets on Kinsley’s plate. When Kinsley came back there were 9 nuggets on her plate. Can you help figure out how many nuggets Kinsley was supposed to have? Students solve and write an equation given the following task. There will be 5 children at the birthday party. I have 3 party hats. How many more hats do I need? K.CC.2 1.OA.1 K.OA.5 Write a multiplication problem that represents the picture below. 3.OA.3 Given a picture, ask students to color all of the rectangles blue, triangles green, etc. K.G.1 Match the fraction with the correct picture. ¼ 3.NF.1 Given a variety of shapes in different sizes have students sort the shapes into groups. Ask follow up questions related to these groups. How many shapes are in each group? How many more shapes are in “group A” than “group B”? 1.MD.4 Go Fish: Each player is dealt five cards. The rest of the cards are placed in a stack face down in the center of the table. If the students have any pairs of cards that total 10, they place them down in front of them and replace those cards with cards from the deck. Students take turns. On each turn, a player asks another player for a card that will go with a card in the player's hand to make 10. If he/she receives a card that makes a pair, the pair is placed on the table. This completes a turn. If the player does not get a card that makes a pair that totals 10, he/she takes the top card from the deck. If the card drawn from the deck makes a pair with a card in the player's hand, the pair is placed on the table. This completes the turn. If there are no cards left in a player's hand but still cards in the deck, that player takes two cards from the deck and continues playing. The game is over when there are no more cards left in the deck. At the end of the game each player writes a list of the number pairs he/she made.. K.OA.3 Write the number that is represented by the picture. 1.NBT.2 ___________ __________ ____________ ___________ Using linking cubes. Work with a partner to determine if the following numbers are odd or even. Draw a picture to record your work and be prepared to explain. 7 12 15 18 Our school is collecting quarters to help raise money for a new slide. Yesterday we our school brought 389 quarters to school. Today our school collected 437 quarters. How many quarters do we have total? Write a number sentence and solve. 2.OA.3 3.NBT.2 How far is it from our classroom to the gym? ?? Students will randomly select three arrow cards. One from the hundreds, one from the tens, and one from the ones. They will then use these to build their number. Students will decide which one is greater/less and will then record their comparison statement on a sheet of paper. 2.NBT.3 2.NBT.4 Replace the boxes with values from 1 to 6 to make each problem true. You can use each number as often as you want. You CANNOT use 7, 8, 9, 0. + + + 2.NBT.5 .
Check the students work below to see if it is correct or not. 5 10 1 4 7 + 3 8 7 5 6 0 -­‐5 8 0 2 2 8 + 5 3 8 1 8 3 -­‐ 2 7 6 4 4.NBT.4 Students work in a small group. One student flips over a numeral card that has a number 0 to 10 on it. The other students in the group will write the number that goes with it to make ten. Example: I flipped over a 6…on your paper you must write the number 4. Given a blank hundred chart, students write the numbers to fill in the missing spaces. 1.OA.6 1.NBT.1 The first student will roll a dice with the numerals 5, 6, 7, 8, 9, 10 to find a “target” number. Then first student
chooses from a pile of ten frame cards that represent the numbers 0, 1, 2, 3, 4, 5 to be the first addend.
Example: Student rolls a seven as the target number then selects the card below
The other student chooses a ten frame card that will go with the first addend to make the target number.
Example: Since the target number rolled was seven. Student two should select the ten frame that represents3.
4+3=7
Timmy was asked to share a candy bar fairly between three kids. Draw a picture that shows how Timmy should have shared the candy bar. Label each piece of the candy bar with the correct fraction. K.OA.2 3.NF.1 Congruency of Assessment Items/Tasks Work in vertical grade level teams and determine in which grade level you think the item would be used. Then find the specific standard that is congruent to each item. Did your initial prediction match? Where there any surprises? SAMPLE PROBLEM/TASK GRADE STANDARD 1(10,000) + 2(1,000) + 4(100) + 3(10) + 2(1) + 5(1/10) + 3(1/100). Which number below is one-­‐tenth of the expanded form above? A 12422.53 B 1243.253 C 12432.53 D 12432.43 What number is shown by the model? 5.NBT.2 5.NBT.3a New York sample item Coach Book says 3.NBT.1 but it is actually 2.NBT.7 When rounding to the nearest ten: a. What is the smallest whole number that will round to 50? b. What is the largest whole number that will round to 50? c. How many different whole numbers will round to 50? When rounding to the nearest hundred: 3.NBT.1 Illustrative Math.org d. What is the smallest whole number that will round to 500? e. What is the largest whole number that will round to 500? f. How many different whole numbers will round to 500? A group of 4 friends was at a restaurant. They each ordered an $8 meal. Then the group ordered a $6 dessert to share. Write an expression that represents this situation. 5.OA.2 1 Congruency of Assessment Items/Tasks A. Arrange these numbers in order, beginning with the smallest.
2400 4002 2040 420 2004
B. Arrange these numbers in order, beginning with the greatest.
1470 847 710 1047 147
4.NBT.2 Illustrative Math.org Write a multiplication problem that represents the picture below. 3.OA.3 Students will randomly select three arrow cards. One from the hundreds, one from the tens, and one from the ones. They will then use these to build their number. Students will decide which one is greater/less and will then record their comparison statement on a sheet of paper. 2.NBT.3 2.NBT.4 Match the fraction with the correct picture. ¼ 3.NF.1 2 Congruency of Assessment Items/Tasks What is the greatest common factor (GCF) of 25 and 35? 6.NS.4 4.OA.2 a.
Helen raised $12 for the food bank last year and she raised 6 times as much money this
year. How much money did she raise this year?
b. Sandra raised $15 for the PTA and Nita raised $45. How many times as much money did
Nita raise as compared to Sandra?
c. Nita raised $45 for the PTA, which was 3 times as much money as Sandra raised. How
much money did Sandra raise?
These problems
involve
multiplicative
comparison,
which is not
included in
3.OA.3 and is
strictly excluded
from 3.MD.2,
making this a
th
4 rather than
rd
3 grade
problem. Mail Man: Students work in pairs and are given a box (represents a delivery truck) and manipulatives. They are also given card sets that represent several situations to model. Mail truck is empty. Truck goes to stop one and picks up 37 letters, the truck then goes to stop 2 and picks up 33 letters, at stop 3 she collects 17 letters. How many letters are on the truck now? As student one reads the problem he will stop after each transaction and model. After stop one the student will get out 37 bundling straws to represent the letters and put them in the truck. After stop two the student will get 43 to represent the other letters in the truck, etc. 2.NBT.6 Student 2 must record and solve for the number of letters on the truck. Students may then access the mail truck to check. Problems should also include subtraction. Truck has 48 letters and dropped off 23 letters. Karl's rectangular vegetable garden is 20 feet by 45 feet, and Makenna's is 25 feet by 40
feet. Whose garden is larger in area? 4.OA.3 4.MD.3 Illustrative Math.org 3 Congruency of Assessment Items/Tasks Alex is training for his school's Jog-A-Thon and needs to run at least 1 mile per day. If Alex
runs to his grandma's house, which is of a mile away, and then to his friend Justin's house,
which is of a mile away, will he have trained enough for the day? 5.NF.1 Illustrative Math.org Imagine that each square in the picture measures one centimeter on each side. What is the
area of each letter? Try to work it out without counting each square individually.
3.MD. Illustrative Math.org Which number is closest to 12?
3.NF.2 Illustrative Math.org a. 1/8
b. 3/8
c. 7/8
d. 9/8
Two groups of students from Douglas Elementary School were walking to the library when it began to rain. The 7 students in Mr. Stem’s group shared the 3 large umbrellas they had with Ms. Thorn’s group of 11 students. If the same number of students were under each umbrella, how many students were under each umbrella? 3.OA.8 3.OA.2 4 Congruency of Assessment Items/Tasks A grocery store sign indicates that bananas are 6 for $1.50, and a sign by the oranges indicates that they are 5 for $3.00. Find the total cost of buying 2 bananas and 2 oranges. 6.RP.3b 6.RP.2 New York sample item Mrs. Sawyer had 82 pictures of birds she gave to her 3 children. She gave each child as
many pictures as possible. Each child received the same number of pictures and Mrs.
Sawyer kept the rest. How many pictures did each child receive? 4.OA.2 CIITS How many times greater is the value of the digit 5 in 583,607 than the value of the digit 5 in 362,501? A 10 times B 100 times C 1,000 times D 10,000 times Students from three classes at Hudson Valley Elementary School are planning a boat trip. On the trip, there will be 20 students from each class, along with 11 teachers and 13 parents. Part A: Write an equation that can be used to determine the number of boats, b, they will need on their trip if 10 people ride in each boat. Equation: b =___________ Part B: How many boats will be needed for the trip if 10 people ride in each boat? Show your work. Answer: __________ boats Part C: It will cost $35 to rent each boat used for the trip. How much will it cost to rent all the boats needed for the trip? Show your work. 5.NBT.1 New York sample item 4.OA.3 New York sample item 5 Congruency of Assessment Items/Tasks 2
POSSIBLE COURSE SAMPLE PROBLEM/TASK* 1.
A team of farm-­‐workers was assigned the task of harvesting two fields, one twice the size of the other. They worked for the first half of the day on the larger field. Then the team split into two groups of equal number. The first group continued working in the larger field and finished it by evening. The second group harvested the smaller field, but did not finish by evening. The next day one farm-­‐
worker finished the smaller field in a single day's work. How many farm-­‐workers were on the team? 2. Consider the following algebraic expressions: 2
2
(n + 2) − 4 and n + 4n. a. Use the figures below to illustrate why the following expressions are equivalent
b.
3.
Eric is playing a video game. At a certain point in the game, he has 31500 points. Then the following events happen, in order: He earns 2450 additional points. He loses 3310 points. The game ends, and his score doubles. a. Write an expression for the number of points Eric has at the end of the game. Do not evaluate the expression. The expression should keep track of what happens in each step listed above. b. Eric's sister Leila plays the same game. When she is finished playing, her score is given by the expression 3(24500 + 3610) − 6780. Describe a sequence of events that might have led to Leila earning this score. 4. Suppose P and Q give the sizes of two different animal populations, where Q>P. In (a)–(d), say which of the given pair of expressions is larger. Briefly explain your reasoning in terms of the two populations. a. P+Q and 2P b.
c.
d.
Find some algebraic deductions of the same result. 5.
N-­‐Q.A.1 A-­‐SSE.B.3.a, A-­‐
SSE.B.3.b, A-­‐
SSE.A 5.OA.A.2 A-­‐SSE.A.1, A-­‐
SSE.A.2 P +Q
P
and P +Q
2
(Q−P)/2 and Q−P/2 P+50t and Q+50t Judy is working at a retail store over summer break. A customer buys a $50 shirt that is on sale for 20% off. Judy computes the discount, then adds sales tax of 10%, and tells the customer how much he owes. The customer insists that Judy first add the sales tax and then apply the discount. He is convinced that this way he will save more money because the discount amount will be larger. a. Is the customer right? b. Does your answer to part (a) depend on the numbers used or would it work for any percentage discount and any sales tax percentage? Find a convincing argument using algebraic expressions and/or diagrams for this more general scenario. Kentucky Department of Education
STANDARD/ CLUSTER *Illustrativemathematics.org A-­‐SSE.B 1 Congruency of Assessment Items/Tasks 2
The students in Mr. Sanchez's class are converting distances measured in miles to kilometers. To estimate the number of kilometers, Abby takes the number of miles, doubles it then subtracts 20% of the result. Renato first divides the number of miles by 5 and then multiplies the result by 8. a. Write an algebraic expression for each method. b. Use your answer to part (a) to decide if the two methods give the same answer. 7.EE.A 7.
A-­‐SSE.B.3 6.
Consider the expression R1 + R2
R1R2
where R1 and R2 are positive. Suppose we increase the value of R1 while keeping R2 constant. Find an equivalent expression whose structure makes clear whether the value of the expression increases, decreases, or stays the same. 8.
Susan has an ear infection. The doctor prescribes a course of antibiotics. Susan is A-­‐SSE.B.4 told to take 250 mg doses of the antibiotic regularly every 12 hours for 10 days. Susan is curious and wants to know how much of the drug will be in her body over the course of the 10 days. She does some research online and finds out that at the end of 12 hours, about 4% of the drug is still in the body. a. What quantity of the drug is in the body right after the first dose, the second dose, the third dose, the fourth dose? b. When will the total amount of the antibiotic in Susan’s body be the highest? What is that amount? c. Answer Susan's original question: Describe how much of the drug will be in her body at various points over the course of the 10 days. 9.
Consider the polynomial function 4
3
2
P(x) = x − 3x + ax − 6x + 14, where a is an unknown real number. If (x−2) is a factor of this polynomial, what is the value of a? A-­‐APR.B.2 10. Suppose f is a quadratic function given by the equation 2
f(x) = ax + bx + c where a, b, c are real numbers and we will assume that a is non-­‐
zero. a. If 0 is a root of f show that c = 0 or, in other words, show that 2
ax + bx + c is evenly divisible by x. 2
b. If 1 is a root of f show that ax + bx + c is evenly divisible by x − 1. c. Suppose r is a real number. If r is a root of f show that ax2 + bx + c is evenly divisible by x − r. A-­‐APR.B.2 11. In the equations (a)–(d), the solution x to the equation depends on the constant A-­‐REI.A a. Assuming a is positive, what is the effect of increasing a on the solution? Does it increase, decrease, or remain unchanged? Give a reason for your answer that can be understood without solving the equation. a. x – a = 0 b. ax = 1 c. ax = a d.
x
= 1 a
12. If we multiply x/2 + 3/4 by 4, we get 2x + 3. Is 2x + 3 an equivalent algebraic expression to x/2 + ¾? Kentucky Department of Education
*Illustrativemathematics.org 7.EE.A 2 Congruency of Assessment Items/Tasks 2
13. Solve the following two equations by isolating the radical on one side and squaring both sides: 2x + 1 − 5 = −2 2x + 1 + 5 = 2 i.
a.
ii.
Be sure to check your solutions. If we raise both sides of an equation a power, we sometimes obtain an equation which has more solutions than the original one. (Sometimes the extra solutions are called extraneous solutions.) Which of the following equations result in extraneous solutions when you raise both sides to the indicated power? Explain. i.
ii.
iii.
3
x = 5 , square both sides x = −5 , square both sides x = 5, cube both sides x = −5 , cube both sides 3
iv.
b. Create a square root equation similar to the one in part (a) that has an extraneous solution. Show the algebraic steps you would follow to look for a solution, and indicate where the extraneous solution arises. 14. Solve the quadratic equation, 2
2
x = (2x − 9) , using as many different methods as possible. 2
15. Suppose h(t) = −5t + 10t + 3 is an expression giving the height of a diver above the water, in meters, t seconds after the diver leaves the springboard. a. How high above the water is the springboard? Explain how you know. b. When does the diver hit the water? c. At what time on the diver's descent toward the water is the diver again at the same height as the springboard? d. When does the diver reach the peak of the dive? 16. Nola was selling tickets at the high school dance. At the end of the evening, she picked up the cash box and noticed a dollar lying on the floor next to it. She said, I wonder whether the dollar belongs inside the cash box or not. The price of tickets for the dance was 1 ticket for $5 (for individuals) or 2 tickets for $8 (for couples). She looked inside the cash box and found $200 and ticket stubs for the 47 students in attendance. Does the dollar belong inside the cash box or not? 17. Lauren keeps records of the distances she travels in a taxi and what she pays: Distance, d, in miles Fare, F, in dollars 3 8.25 5 12.75 11 26.25 a. If you graph the ordered pairs (d, F) from the table, they lie on a line. How can you tell this without graphing them? b. Show that the linear function in part (a) has equation F = 2.25d + 1.5. c. What do the 2.25 and the 1.5 in the equation represent in terms of taxi rides? 18. You have $100 to spend on a barbeque where you want to serve chicken and steak. Chicken costs $1.29 per pound and steak costs $3.49 per pound. a. Find a function that relates the amount of chicken and the amount of steak you can buy. b. Graph the function. What is the meaning of each intercept in this context? What is the meaning of the slope in this context? Use this (and any other information represented by the equation or graph) to discuss what your options are for the amounts of chicken and amount of steak you can buy for the barbeque.
Kentucky Department of Education
A-­‐REI.A.2 *Illustrativemathematics.org A-­‐REI.B.4 A-­‐REI.D.11 F-­‐IF.C.8.a A-­‐REI.B.4.b A-­‐REI.C.6 A-­‐REI.D.10 F-­‐LE.B.5 8.F.B.4 3 Congruency of Assessment Items/Tasks 2
19. The population of a country is initially 2 million people and is increasing at 4% per year. The country's annual food supply is initially adequate for 4 million people and is increasing at a constant rate adequate for an additional 0.5 million people per year. a. Based on these assumptions, in approximately what year will this country first experience shortages of food? b. If the country doubled its initial food supply and maintained a constant rate of increase in the supply adequate for an additional 0.5 million people per year, would shortages still occur? In approximately which year? c. If the country doubled the rate at which its food supply increases, in addition to doubling its initial food supply, would shortages still occur? 20. Mike likes to canoe. He can paddle 150 feet per minute. He is planning a river trip that will take him to a destination about 30,000 feet upstream (that is, against the current). The speed of the current will work against the speed that he can paddle. a. Let s be the speed of the current in feet per minute. Write an expression for r(s), the speed at which Mike is moving relative to the river bank, in terms of s. b. Mike wants to know how long it will take him to travel the 30,000 feet upstream. Write an expression for T(s), the time in minutes it will take, in terms of s. c. What is the vertical intercept of T? What does this point represent in terms of Mike’s canoe trip? d. At what value of s does the graph have a vertical asymptote? Explain why this makes sense in the situation. e. For what values of s does T(s) make sense in the context of the problem? 21. You work for a video streaming company that has two monthly plans to choose from: Plan 1: A flat rate of $7 per month plus $2.50 per video viewed Plan 2: $4 per video viewed a. What type of functions model this situation? Explain how you know. b. Define variables that make sense in the context, and then write an equation with cost as a function of videos viewed, representing each monthly plan. c. How much would 3 videos in a month cost for each plan? 5 videos? d. Compare the two plans and explain what advice you would give to a customer trying to decide which plan is best for them, based on their viewing habits.
22. Consider the equation 5x−2y=3. If possible, find a second linear equation to create a system of equations that has: a. Exactly 1 solution. b. Exactly 2 solutions. c. No solutions. d. Infinitely many solutions. 23. Some of the students at Kahlo Middle School like to ride their bikes to and from school. They always ride unless it rains. F-­‐LE.A.2 F-­‐LE.A.3 A-­‐REI.D.11 F-­‐BF.A.1.a F-­‐IF.B.4 F-­‐IF.B.5 8.F.B.4 8.EE.C.8 6.EE.A.2 Let d be the distance in miles from a student's home to the school. Write two different expressions that represent how far a student travels by bike in a four week period if there is one rainy day each week. Kentucky Department of Education
*Illustrativemathematics.org 4 Congruency of Assessment Items/Tasks 2
24. John makes DVDs of his friend’s shows. He has realized that, because of his fixed costs, his average cost per DVD depends on the number of DVDs he produces. The cost of producing x DVDs is given by C(x) = 2500 + 1.25x. a. John wants to figure out how much to charge his friend for the DVDs. He’s not trying to make any money on the venture, but he wants to cover his costs. Suppose John made 100 DVDs. What is the cost of producing this many DVDs? How much is this per DVD? b. John is hoping to make many more than 100 DVDs for his friends. Complete the table showing his costs at different levels of production. # of DVDs 0 10 100 1,000 10,000 100,000 1,000,000 Total Cost Cost per DVD c.
d.
e.
f.
25.
a.
b.
c.
a.
b.
c.
d.
27.
Explain why the average cost per DVD levels off. Find an equation for the average cost per DVD of producing x DVDs. Find the domain of the average cost function. Using the data points from your table above, sketch the graph of the average cost function. How does the graph reflect that the average cost levels off? A certain business keeps a database of information about its customers. Let C be the rule which assigns to each customer shown in the table his or her home phone number. Is C a function? Explain your reasoning. Customer Name Home Phone Number Heather Baker 3105100091 Mike London 3105200256 Sue Green 3234132598 Bruce Swift 3234132598 Michelle Metz 2138061124 Let P be the rule which assigns to each phone number in the table above, the customer name(s) associated with it. Is P a function? Explain your reasoning. Explain why a business would want to use a person's social security number as a way to identify a particular customer instead of their phone number. 26. A downtown city parking lot charges $0.50 for each 30 minutes you park, or fraction thereof, up to a daily maximum charge of $10.00. Let C assign to each length of time you park, t (in hours), the cost of parking in the lot, C(t) (in dollars). Complete the table below. t (in hours) C(t) (in dollars) 0 1/4 1/3 9/16 1 1/4 29/12 Sketch a graph of C for 0 ≤ t ≤ 8. Is C a function of t? Explain your reasoning. Is t a function of C? Explain your reasoning. Antonio and Juan are in a 4-­‐mile bike race. The graph below shows the distance of each racer (in miles) as a function of time (in minutes). Kentucky Department of Education
*Illustrativemathematics.org F-­‐IF.B.4 F-­‐IF.B.5 F-­‐IF.A.1 F-­‐IF.A.1 8.F.B.5 5 Congruency of Assessment Items/Tasks 2
a.
Who wins the race? How do you know? b. Imagine you were watching the race and had to announce it over the radio, write a little story describing the race.
28. Imagine Scott stood at zero on a life-­‐sized number line. His friend flipped a coin 50 times. When the coin came up heads, he moved one unit to the right. When the coin came up tails, he moved one unit to the left. After each flip of the coin, Scott's friend recorded his position on the number line. Let f assign to the whole number n, when 1≤n≤50, Scott's position on the number line after the nth coin flip. a. If f(6) = 6 what can you conclude about the outcomes of the first 6 coin tosses? Explain. What if f(6) = −4? b. Is it possible that f(7) = 2? Explain. c. Find all integers m so that the probability that f(50) = m is zero. 29. In order to gain popularity among students, a new pizza place near school plans to offer a special promotion. The cost of a large pizza (in dollars) at the pizza place as a function of time (measured in days since February 10th) may be described as F-­‐IF.A.2 F-­‐IF.B 0≤t <3
⎧ 9,
⎪
C (t ) = ⎨9 + t , 3 ≤ t < 8 ⎪ 20, 8 < t < 28
⎩
(Assume t only takes whole number values.) a. If you want to give their pizza a try, on what date(s) should you buy a large pizza in order to get the best price? th
b. How much will a large pizza cost on February 18 ? c. On what date, if any, will a large pizza cost 13 dollars? d. Write an expression that describes the sentence "The cost of a large pizza is at least A dollars B days into the promotion," using function notation and mathematical symbols only. e. Calculate C(9) − C(8) and interpret its meaning in the context of the problem. f. On average, the cost of a large pizza goes up about 85 cents per day during the first two weeks of the promotion period. Which of the following equations best describes this statement? •
•
C (13) + C (0 )
= 0.85
2
C (13) − C (0 )
= 0.85
13
Kentucky Department of Education
*Illustrativemathematics.org 6 Congruency of Assessment Items/Tasks 2
•
•
C (13)
= 0.85
13
C (February 23) − C (February 10 )
= 0.85 13
30. Given below are three graphs that show solar radiation, S, in watts per square meter, as a function of time, t, in hours since midnight. We can think about this quantity as the maximum amount of power that a solar panel can absorb, which tells us how intense the sunshine is at any given time. Match each graph to the corresponding description of the weather during the day. a. It was a beautifully sunny day from sunrise to sunset – not a cloud in the sky. b. The day started off foggy but eventually the fog lifted and it was sunny the rest of the day. c. It was a pretty gloomy day. The morning fog never really lifted. 1.
2.
3.
All three graphs show solar radiation measured in Santa Rosa, a city in northern California. What other information can you get from the graph? 31. An epidemic of influenza spreads through a city. The figure below is the graph of F-­‐IF.B.4 I = f(w), where I is the number of individuals (in thousands) infected w weeks after the epidemic begins. Kentucky Department of Education
F-­‐IF.B.4 *Illustrativemathematics.org 7 Congruency of Assessment Items/Tasks 2
Estimate f(2) and explain its meaning in terms of the epidemic. Approximately how many people were infected at the height of the epidemic? When did that occur? Write your answer in the form f(a)=b. c. For approximately which w is f(w) = 4.5; explain what the estimates mean in terms of the epidemic. d. An equation for the function used to plot the image above is f(w) = 6w(1.3) − w. Use the graph to estimate the solution of the inequality 6w(1.3) – w ≥ 6. Explain what the solution means in terms of the epidemic. (Task from Functions Modeling Change: A Preparation for Calculus, Connally et al., Wiley 2010.) 32. Consider the following four functions a.
b.
•
•
•
•
F-­‐IF.C.7 3
1 + e −3 x
e−x
(
)
g x =1−
2
ex
h(x ) = −2 + 2
3
k (x ) =
1 + e3x
f (x ) =
Below are four graphs of functions shown for −2 ≤ x ≤2. Match each function with its graph and explain your choice: 33. Which of the following equations could describe the function whose graph is shown below? Explain. Kentucky Department of Education
*Illustrativemathematics.org F-­‐IF.C.8.a 8 Congruency of Assessment Items/Tasks 2
2
f1(x) = (x + 12) + 4 f5(x) = −4(x + 2)(x + 3) 2
f2(x) = −(x − 2) − 1 f6(x) = (x + 4)(x − 6) 2
f3(x) = (x + 18) − 40 f7(x) = (x − 12)(−x + 18) 2
f4(x) = (x − 10) − 15 f8(x) = (20 − x)(30 − x) 34. Without using the square root button on your calculator, estimate √ (800) , square root of 800, as accurately as possible to 2 decimal places. 8.NS.A F-­‐BF.A.1 35. How many cubes are needed to build this tower? How many cubes are needed to build a tower like this, but 12 cubes high? Justify your reasoning. c. How would you calculate the number of cubes needed for a tower n cubes high? 27. Using the graphs below, sketch a graph of the function s(x) = f(x) + g(x). F-­‐BF.A.1 a.
b.
28. According to the U.S. Energy Information Administration, a barrel of crude oil produces approximately 20 gallons of gasoline. EPA mileage estimates indicate a 2011 Ford Focus averages 28 miles per gallon of gasoline. a. Write an expression for g(x), the number of gallons of gasoline produced by x Kentucky Department of Education
*Illustrativemathematics.org F-­‐BF.A.1.c 9 Congruency of Assessment Items/Tasks 2
b.
c.
d.
29.
a.
b.
c.
barrels of crude oil. Write an expression for M(x), the number of miles on average that a 2011 Ford Focus can drive on x gallons of gasoline. Write an expression for M(g(x)). What does M(g(x)) represent in terms of the context? One estimate (from www.oilvoice.com) claimed that the 2010 Deepwater Horizon disaster in the Gulf of Mexico spilled 4.9 million barrels of crude oil. How many miles of Ford Focus driving would this spilled oil fuel? 2
Let f be the function defined by f(x) = 2x + 4x − 16. Let g be the function defined 2
by g(x) = 2(x + 1) − 18. Verify that f(x) = g(x) for all x. 2
In what ways do the equivalent expressions 2x + 4x − 16 and 2
2(x + 1) − 18 help to understand the function f? Consider the functions h, l, m, and n given by h(x )
F-­‐BF.B.3 = x2
l (x ) = x + 1
m(x ) = x − 9
n(x )
d.
30.
a.
b.
c.
31.
a.
b.
Show that f(x) is a composition, in some order, of the functions h, l, m, and n. How do you determine the order of composition? Explain the impact each of the functions l, m, and n has on the graph of the composition. City Bank pays a simple interest rate of 3% per year, meaning that each year the balance increases by 3% of the initial deposit. National Bank pays an compound interest rate of 2.6% per year, compounded monthly, meaning that each month the balance increases by one twelfth of 2.6% of the previous month's balance. Which bank will provide the largest balance if you plan to invest $10,000 for 10 years? For 15 years? Write an expression for C(y), the City Bank balance, y years after a deposit is left in the account. Write an expression for N(m), the National Bank balance, m months after a deposit is left in the account. Create a table of values indicating the balances in the two bank accounts from year 1 to year 15. For which years is City Bank a better investment, and for which years is National Bank a better investment? Algae blooms routinely threaten the health of the Chesapeake Bay. Phosphate compounds supply a rich source of nutrients for the algae, Prorocentrum minimum, responsible for particularly harmful spring blooms known as mahogany tides. These compounds are found in fertilizers used by farmers and find their way into the Bay with run-­‐offs resulting from rainstorms. Favorable conditions result in rapid algae growth ranging anywhere from 0.144 to 2.885 cell divisions per day. Algae concentrations are measured and reported in terms of cells per milliliter (cells/ml). Concentrations in excess of 3,000 cells/ml constitute a bloom. Suppose that heavy spring rains followed by sunny days create conditions that support 1 cell division per day and that prior to the rains Prorocentrum minimum concentrations measured just 10 cells/ml. Write an equation for a function that models the relationship between the algae concentration and the number of days since the algae began to divide at the rate of 1 cell division per day. Assuming this rate of cell divison is sustained for 10 days, present the resulting algae concentrations over that period in a table. Did these conditions result in a bloom? Kentucky Department of Education
= 2x
*Illustrativemathematics.org F-­‐LE.A.1 F-­‐LE.A.1.c F-­‐LE.A.2 F-­‐LE.A.4 10 Congruency of Assessment Items/Tasks 2
c.
Concentrations in excess of 200,000 cells/ml have been reported in the Bay. If conditions support 2 cell divisions per day, when will these conditions result in a bloom? When will concentrations exceed 200,000 cells/ml? 32. A cup of hot coffee will, over time, cool down to room temperature. The principle of physics governing the process is Newton's Law of Cooling. Experiments with a covered cup of coffee show that the temperature (in degrees Fahrenheit) of the coffee can be modeled by the following equation −0.08t
+ 75. f(t) = 110e
Here the time t is measured in minutes after the coffee was poured into the cup. −0.08t
+ 75, why the coffee a. Explain, using the structure of the expression 110e
temperature decreases as time elapses. b. What is the temperature of the coffee at the beginning of the experiment? c. After how many minutes is the coffee 140 degrees? After how many minutes is the coffee 100 degrees? 33. A car is traveling down a long, steep hill. The elevation, E, above sea level (in feet) of the car when it is d miles from the top of the hill is given by E=7500–250d, where d can be any number from 0 to 6. Find the slope and intercepts of the graph of this function and explain what they mean in the context of the moving car. 34.
Given below is a table that gives the populations of foxes and rabbits in a national park over a 12 month period. Note that each value of t corresponds to the beginning of the month.
a.
b.
c.
d.
e.
8.F.A.1 According to the data in the table, is F a function of R? Is R a function of F? Is either R or F functions of t? Explain your reasoning. 6.EE.C.9 Complete the table above for values of m. Write an equation for the amount of money, m, that will be collected if b boxes of chocolate bars are sold. Which is the independent variable and which is the dependent variable? Graph the equation using the ordered pairs from the table above. Calculate how much money will be collected if 100 boxes of chocolate bars are sold. The band collected $1530.00 from chocolate bar sales. How many boxes did they sell? Kentucky Department of Education
8.F.B.4 This task is adapted from "Functions Modeling Change", Connally et al, Wiley 2007. 35. Stephanie is helping her band collect money to fund a field trip. The band decided to sell boxes of chocolate bars. Each bar sells for $1.50 and each box contains 20 bars. Below is a partial table of monies collected for different numbers of boxes sold. (imagine shrunk for formatting) a.
b.
F-­‐LE.B.5 F-­‐LE.A.4 *Illustrativemathematics.org 11 Congruency of Assessment Items/Tasks 2
36. You have been asked to place a fire hydrant so that it is an equal distance form three locations indicated on the following map. a.
Show how to fold your paper to physically construct this point as an intersection of two creases. b. Explain why the above construction works, and in particular why you only needed to make two creases. 37. The figure below is composed of eight circles, seven small circles and one large circle containing them all. Neighboring circles only share one point, and two regions between the smaller circles have been shaded. Each small circle has a radius of 5 cm. Calculate: a. The area of the large circle. b. The area of the shaded part of the figure. 38. You have been asked to place a warehouse so that it is an equal distance from the three roads indicated on the following map. Find this location and show your work. a.
b.
G-­‐C.A.3 G-­‐CO.D 7.G.B.4 G-­‐C.A.3 G-­‐CO.D.12 Show how to fold your paper to physically construct this point as an intersection of two creases. Explain why the above construction works, and in particular why you only needed to make two creases. Kentucky Department of Education
*Illustrativemathematics.org 12 Congruency of Assessment Items/Tasks Answer Sheet GRADE SAMPLE PROBLEM/TASK STANDARD Sierra walks her dog Pepper twice a day. Her evening walk is two and a half times
as far as her morning walk. At the end of the week she tells her mom,
“I walked Pepper for 30 miles this week!”
6 6.EE.B.7 How long is her morning walk?
a. Amy wants to build a cube with 3 cm sides using 1 cm cubes. How many cubes
does she need?
6.G.A.2 6 b. How many 1 cm cubes would she need to build a cube with 6 cm sides?
Ocean water freezes at about −2 1∘ C. Fresh water freezes at 0∘ C. Antifreeze, a liquid used to 7 cool most car engines, freezes at −64∘ C. Imagine that the temperature is exactly at the freezing point for ocean water. How many degrees must the temperature drop for the antifreeze to turn to ice? 7.NS.A.1
Coffee costs $18.96 for 3 pounds.
a. What is the cost per pound of coffee?
7 b. Let x be the number of pounds of coffee and y be the total cost of x pounds.
Draw a graph of the proportional relationship between the number of pounds of
coffee and the total cost.
c. Where can you see the cost per pound of coffee in the graph? What is it?
7.RP.A.2
1 Congruency of Assessment Items/Tasks Answer Sheet 7 On the number line above, the numbers a and b are the same distance from 0.
7.NS.A.1
What is a + b?
Explain how you know.
In triangle ΔABC, point M is the point of intersection of the bisectors of angles
∘,
∠BAC, ∠ABC, and ∠ACB. The measure of ∠ABC is 42 and the measure of
∠BAC is 64∘ . What is the measure of ∠BMC?
8.G.A.5
8 For each pair of numbers, decide which is larger without using a calculator.
Explain
your
choices.
This task
adapted
from a problem published by the Russian Ministry of
Education.
a. π2 or 9
8 b. √50 or √51
8.NS.A.2
c. √50 or 8
d. −2π or −6
asked to identify their favorite
All the students at a middle school were
academic subject and whether they were in 7th grade or 8th grade. Here are
the results:
Favorite Subject by Grade
8 Grade
English
History
Math/Science
Other
Totals
7th Grade
38
36
28
14
116
8th Grade
47
45
72
18
182
Totals
85
81
100
32
298
Is there an association between favorite academic subject and grade for
students at this school? Support your answer by calculating appropriate relative
frequencies using the given data.
8.SP.A.4 2 Congruency of Assessment Items/Tasks Answer Sheet Below are the 25 birth weights, in ounces, of all the Labrador Retriever puppies
born at Kingston Kennels in the last six months.
13 14 15 15 16 16 16 16 17 17 17 17 17 17 17 18 18 18 18 18 18 18 18 19 20
6 a. Use an appropriate graph to summarize these birth weights.
b. Describe the distribution of birth weights for puppies born at Kingston
Kennels in the last six months. Be sure to describe shape,
center and variability.
c. What is a typical birth weight for puppies born at Kingston Kennels in the last
six months? Explain why you chose this value. 6.SP.A.2,
6.SP.B.4
The ratio of the number of boys to the number of girls at school is 4:5.
6 a. What fraction of the students are boys?
6.RP.A
b. If there are 120 boys, how many students are there altogether? The students in Mr. Sanchez's class are converting distances measured in
miles to kilometers.
To estimate the number of kilometers, Abby
takes the number of miles, doubles it, then subtracts 20% of the result.
7 Renato first divides the number of miles by 5, then multiplies the
result by 8.
7.EE.A
a. Write an algebraic expression for each method.
b. Use your answer to part (a) to decide if the two methods give the same
answer. 3 Congruency of Assessment Items/Tasks Answer Sheet Medhavi suspects that there is a relationship between the number of text
messages high school students send and their academic achievement. To
explore this, she asks each student in a random sample of 52 students from
her school how many text messages he or she sent yesterday and what his or
her grade point average (GPA) was during the most recent marking period.
The data are summarized in the scatter plot of number of text messages sent
versus GPA shown below.
8.SP.A.1 8 Describe the relationship between number of text messages sent and GPA.
Discuss both the overall pattern and any deviations from the pattern.
The students in Ms. Baca’s art class were mixing yellow and blue paint. She
told them that two mixtures will be the same shade of green if the blue and
yellow paint are in the same ratio.
The table below shows the different mixtures of paint that the students made.
Yellow
Blue
A
1 part
2 part
B
2 parts
3 parts
C
3 parts
6 parts
D
4 parts
6 parts
E
6 parts
9 parts
7.RP.A.2
a. How many different shades of paint did the students make?
b. Some of the shades of paint were bluer than others. Which mixture(s)
were the bluest? Show work or explain how you know.
4