Probability, Percent, and Rational Number Equivalence Math A Honors
Module #1 Student Edition 2016-2017
Created in collaboration with Utah Middle School Math Project A University of Utah Partnership Project
San Dieguito Union High School District
Table of Contents MODULE 1: PROBABILITY, PERCENT, RATIONAL NUMBER EQUIVALENCE ......................................................... 3 STANDARDS FOR MATHEMATICAL PRACTICE: A GUIDE FOR STUDENTS AND PARENTS .................................... 5 SECTION 1.1: INVESTIGATE CHANCE PROCESSES. DEVELOP/USE PROBABILITY MODELS.* ................................. 6 1.0A ANCHOR PROBLEM: AMERICAN FOOTBALL.............................................................................................. 7 1.0B LESSON: REVIEW FROM EARLIER GRADES* ............................................................................................ 9 1.1A LESSON: PROBABILITY PREDICTIONS* .................................................................................................. 14 1.1A EXTRA PRACTICE: PROBABLY PROBABILITY .......................................................................................... 18 1.1B LESSON: PROBABILITY – RACE TO THE TOP* ........................................................................................ 19 1.1C LESSON: MORE PROBABILITY PRACTICE* ............................................................................................. 24 SECTION 1.2: UNDERSTAND/APPLY EQUIVALENCE IN RATIONAL NUMBER FORMS. CONVERT BETWEEN FORMS (FRACTION, DECIMAL, PERCENT).* ................................................................................................................. 26 1.2A LESSON: BAR MODELS WITH FRACTIONS AND DECIMALS* ..................................................................... 27 1.2B LESSON: RATIONAL NUMBER ORDERING AND ESTIMATION*................................................................... 28 1.2C LESSON: PROBABILITY, FRACTIONS, PERCENTAGE, & RATIO* ............................................................... 32 1.2D LESSON: RATIONAL NUMBERS IN APPLICATIONS WITH MODELS* ........................................................... 37 SECTION 1.3: SOLVE PERCENT PROBLEMS INCLUDING DISCOUNTS, INTEREST, TAXES, TIPS, AND PERCENT INCREASE OR DECREASE.* ............................................................................................................................ 40 1.3A LESSON: MODEL PERCENT AND FRACTION PROBLEMS* ........................................................................ 41 1.3B LESSON: TRANSITION TO NUMERIC EXPRESSIONS* ............................................................................... 43 * Denotes a lesson that was adapted from Utah Middle School Math Project. © Utah Middle School Math Project and University of Utah Partnership http://utahmiddleschoolmath.org/ This work is licensed under a Creative Commons Attribution-NonCommercial 2.5 Generic License http://creativecommons.org/licenses/by-nc/2.5/ This work is licensed under a Creative Commons Attribution-NonCommercial 3.0 Unported License http://creativecommons.org/licenses/by-nc/3.0/legalcode
SDUHSD Math A Honors Module #1 – STUDENT EDITION 2016-2017
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Module 1: Probability, Percent, Rational Number Equivalence Online support for this module can be found at http://goo.gl/s2vDtM (case sensitive) or using the QR code below. This website includes copies of student classwork, homework, and instructional videos for common core standards.
Curriculum Support Website
Common Core Standard(s) Number Sense: 1. Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats. 7.NS.2d 2. Solve real-world and mathematical problems involving the four operations with rational numbers. 7.NS.3 Probability and Statistics: 1. Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around ½ indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event. 7.SP.5 2. Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times. 7.SP.6 Equations and Expressions: 1. Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation. 7.EE.3 Module 1 Summary: Module 1 begins with a brief introduction to probability as a means of reviewing and applying arithmetic with whole numbers and fractions. In addition to covering basic counting techniques and listing outcomes in a sample space, students distinguish theoretical probabilities from experimental approaches to estimate probabilities. Another reason for starting the year with probability activities is to develop a culture of investigation, discussion and collaboration in the classroom. Throughout the module students are provided with opportunities to review and build fluency with fractions, percents, and decimals from previous grades. Students should understand that fractions, percent and decimals are all relative to a whole. Students will also compare and order fractions (both positive and negative.) This module concludes with a section specifically about solving percent and fraction problems, including those involving discounts, interest, taxes, tips, and percent increase or decrease. Vocabulary: chance, decimal, fraction, frequency, experimental probability, percent, probability, ratio, theoretical probability, probability statement, outcomes, sample space
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Connections to Content: Prior Knowledge In previous coursework, students developed the concept of a ratio. Thus students should be familiar with the idea of part:whole, part:part, and whole:whole relationships. It is important to emphasize that in this module only part:whole relationships are discussed (probability is a part:whole relationship as are fractions, decimals and percents.) Later in 7th grade students will discuss “odds” which are part:part relationships. Students have used all four operations (addition, subtraction, multiplication, and division) when working with fractions and decimals in prior grades. They should have used both number line and bar models to represent fractions, percent and decimals. In 6th grade students placed both positive and negative numbers on a number line, however they do not operate on negative numbers until 7th grade (this will take place in Module 2). Future Knowledge As students move through this module, they will begin by studying probability (this module is only an introduction to probability; students will work more with probability in Module 7.) The concepts learned in 7th grade around chance processes and theoretical and experimental probabilities will be extended in later courses when students study conditional probability, compound events, evaluate outcomes of decisions, use probabilities to make fair decisions, etc. While studying probability, students will continue their study of rational numbers. They will convert rational numbers to decimals and percents and will look at their placement on the number line. This lays the foundation for 8th grade where students study irrational numbers to complete the Real Number system.
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Standards for Mathematical Practice: A Guide for Students and Parents The Standards for Mathematical Practices are central to the Common Core. These practices build fluency and help students become better decision-makers and problem solvers. The practices reflect the most advanced and innovative thinking on how students should interact with math content. Students and parents will develop skill with these standards by asking some of these questions: Make Sense of Problems and Persevere in Solving Them. • What is the problem that you are solving for? • Can you think of a problem that you recently solved that is similar to this one? • How will you go about solving the problem? (i.e. What’s your plan?) • Are you progressing towards a solution? How do you know? Should you try a different solution plan? • How can you check your solution using a different method? Construct Viable Arguments and Critique the Reasoning of Others. • Can you write or recall an expression or equation to match the problem situation? • What do the numbers or variables in the equation refer to? • What’s the connection among the numbers and variables in the equation? Reason Abstractly and Quantitatively. • Tell me what your answers(s) mean(s) • How do you know that your answer is correct? • If I told you I think the answer should be (a wrong answer), how would you explain to me why I’m wrong? Model with Mathematics. • Do you know a formula or relationship that fits this problem situation? • What’s the connection among the numbers in the problem? • Is your answer reasonable? How do you know? • What do(es) the number(s) in your solution refer to? Use Appropriate Tools Strategically. • What tools could you use to solve this problem? How can each one help you? • Which tool is most useful for this problem? Explain your choice. • Why is this tool (the one selected) better to use than (another tool mentioned)? • Before you solve the problem, can you estimate the solution? Attend to Precision. • What do the symbols that you used mean? • What units of measure are you using (for measurement problems) • Explain to me what (term from the lesson) means. Look For and Make Use of Structure. • What do you notice about the answers to the exercises you’ve just completed? • What do different parts of the expression or equation you are using tell you about possible correct answers? Look for and Express Regularity in Repeated Reasoning. • What shortcut can you think of that will always work for these kinds of problems? • What pattern(s) do you see? Can you make a generalization?
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Section 1.1: Investigate chance processes. Develop/use probability models.* Section Overview: This is students’ first formal introduction to probability. In this section students will study chance processes, which concern experiments or situations where they know which outcomes are possible, but they do not know precisely which outcome will occur at a given time. They will look at probabilities as ratios expressed as fractions, decimals, or percents (part:whole). Probabilities will be determined by considering the results or outcomes of experiments. They will learn that the set of all possible outcomes for an experiment is a sample space. They will recognize that the probability of any single event can be expressed in terms of impossible, unlikely, equally likely, likely, certain, or as a number between 0 and 1, inclusive. Students will focus on two concepts in probability of an event: experimental and theoretical. They will understand the commonalities and differences between experimental and theoretical probability in given situations. Concepts and Skills to be Mastered (from standards) 1. 2. 3. 4.
Understand and apply likelihood of a chance event as between 0 and 1. Approximate probability by collecting data on a chance process (experimental probability). Calculate theoretical probabilities on a chance process. Given the probabilities (different scenarios in a chance process), predict the approximate frequencies for those scenarios (if experimenting on a chance process). 5. Use appropriate fractions, decimals and percents to express the probabilities.
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1.0A Anchor Problem: American Football Name:
Period:
Mr. Hill grew up in San Diego, California – home of the legendary San Diego Chargers. On weekends, he and his friends would play football in the neighborhood park. To simplify keeping track of the score, they would only count touchdowns and field goals. One day, they decided to use the following scoring system: touchdowns = 5 points
field goals = 3 points
After they were done playing football for the day, Mr. Hill and his friends were hanging out wondering what scores were possible with their unique scoring system. After they spent a few minutes talking about it, Bridget said that after a certain number, all scores were possible. This sounded crazy – how could this be true? Bridget said that after this “highest impossible score” every score was possible all the way to infinity. For this anchor problem, answer the following questions and make Bridget proud! 1. On your own and thinking about the scoring system of touchdowns worth 5 points and field goals worth 3 points: a. What do you think the highest impossible score is? Explain.
b. Discuss with your teammates your response to part a. Decide as a group what you think the highest impossible score is. Explain. 2. With your group, use the hundreds chart (on next page) and the scoring system of touchdowns worth 5 points and field goals worth 3 points to help you determine the highest impossible score. Show your reasoning.
3. Make up ONE other scoring system that has a highest impossible score. Discuss with your teammates and decide as a group what you think the highest impossible score is for your scoring system. Explain how you arrived at your conclusion.
4. Make up ONE scoring system that does not have a highest impossible score – meaning the scores continue forever. Discuss with your teammates and decide as a group why you think this system does not have a highest impossible score. Explain how you arrived at your conclusion.
5. Look for a rule for the scoring systems that have a highest impossible score. The rule should easily calculate the high impossible score using the points associated with a touchdown and a field goal.
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1.0B Lesson: Review From Earlier Grades* Name:
Period:
Fraction
Decimal
Percent
Bar Model
A.
B.
C.
D.
E.
F.
1. Use a bar model to represent
2. Use a bar model to represent
3. Use a bar model to represent
! !
! !
of a whole.
of a whole.
! !"
of a whole.
4. What do you notice about the fractions in #1-3?
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Draw a diagram to match the description of the bars. Write the equivalent values for the following parts of the bar. Divide the bar on the left into two equal parts. 1 part =_____(fraction) = _____ (decimal) =_____ (percent) 2 parts =____(fraction) = _____ (decimal) =_____ (percent)
Divide the bar on the left into three equal parts. 1 part =_____(fraction) = _____(decimal) =_____(percent) 2 parts =_____(fraction) =_____(decimal) =_____ (percent)
Divide the bar on the left into four equal parts. 1 part =_____(fraction) = _____(decimal) =_____(percent) 2 parts =_____(fraction) =_____(decimal) =_____ (percent) 3 parts =_____(fraction) =_____(decimal) =_____ (percent)
Divide the bar on the left into five equal parts. 1 part =_____(fraction) = _____(decimal) =_____(percent) 3 parts =_____(fraction) =_____(decimal) =_____ (percent)
Divide the bar on the left into six equal parts. 1 part =_____(fraction) = _____(decimal) =_____(percent) 2 parts =_____(fraction) =_____(decimal) =_____ (percent) 4 parts =_____(fraction) =_____(decimal) =_____ (percent)
Divide the bar on the left into eight equal parts. 2 parts =_____(fraction) = _____(decimal) =_____(percent) 7 parts =_____(fraction) =_____(decimal) =_____ (percent)
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5. To the left is a 10 × 10 Grid. Why do you think it is called a 10 × 10 grid?
6. How could you use the 10 × 10 grid to show the fraction
7. Shade
! !""
6 ? Explain. 100
in the grid to the left.
8. Shade the given decimal in each grid below. Then write the fraction and percent equivalent in simplest form. a. 0.35
b. 0. .4
c. 0.125
fraction: ____
fraction: ____
fraction: ____
percent: ____
percent: ____
percent: ____
9. Shade the given fraction in each grid below. Then write the decimal and percent equivalent for each. a.
! !
b.
! !"
c.
! !
decimal: ____
decimal: ____
decimal: ____
percent: ____
percent: ____
percent: ____
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10. Shade the fractional part of each grid. Then write the fraction as a decimal and a percent below.
a.
! !
b.
!
c.
!
! !
decimal: ____
decimal: ____
decimal: ____
percent: ____
percent: ____
percent: ____
11. On the Bolts football team, one-fourth of the players walk to practice and 30% are driven by their parents. The remaining players go by bus. If there are 60 players on the team who go to practice, how many players take the bus?
12. In the past, changes in stock prices were given either as fractions or mixed numbers. A company ! reported the change in stock price on Tuesday as: −3 !"
Convert this to a decimal. Explain what this means in terms of the stock price.
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13. Which of the following is NOT equivalent to the other three? Explain.
A. 0.15
B.
!" !"
C. 75%
D. 0.75
14. In a class of 178 students, 49 tried out for the class play. What percent of the class did NOT try out for the class play? Round to the nearest percent.
15. You have taken a part-time job at a nearby shopping mall. Your assignment is to hand out 500 advertising leaflets. You have decided to work in a pattern. You will hand out 1 leaflet on the first day, 2 on the second day, 4 on the third day, 8 on the fourth day, and so on. How many days will it take to complete your assignment? Show your work.
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1.1A Lesson: Probability Predictions* Name:
Period:
1. In your own words, what do you think these terms mean? a) “Experimental” probability: b) “Theoretical” probability: 2. We will examine experimental and theoretical probability in this activity. Your bag contains a total of 12 green and blue tiles. Without looking in the bag, you will be making a guess as to how many GREEN tiles the bag contains. **DO NOT LOOK IN THE BAG UNTIL YOU ARE TOLD TO DO SO!** a. Draw a marble/tile (without looking in the bag), record the color, replace it, redraw, record, replace. Do this 6 times. Draw # 1 2 3 4 5 6 Color Based on your experiment, what portion of the 6 draws were GREEN? ________ Based on your experiment of 6 draws, how many green tiles do you think are in the bag?______ b. Repeat the experiment in “a”, but this time do it 12 times. Draw # Color
1
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Based on your experiment, what portion of the 12 draws were GREEN? ________ Based on your experiment of 12 draws, how many green tiles do you think are in the bag?_____ c. Repeat the experiment in “a” but this time do it 24 times. Draw # Color Draw # Color
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Based on your experiment, what portion of the 24 draws were GREEN? ________ Based on your experiment of 24 draws, how many green tiles do you think are in the bag?_____ d. Repeat the experiment in “a”, but this time do it 30 times. Draw # Color Draw # Color
1
2
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Based on your experiment, what portion of the 30 draws were GREEN? ________ Based on your experiment of 30 draws, how many green tiles do you think are in the bag?_____ e. How many TOTAL trials did you perform in a through d above? ________ Based on your experiment of the TOTAL trials above, what portion of the total trials were GREEN? ________ Based on your experiment of TOTAL trials above, how many green tiles do you think are in the bag?_____ SDUHSD Math A Honors – Module #1 - STUDENT EDITION 2016-2017
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Probability has standard notation. In this situation, we understand P(G) to means the probability of drawing a green tile. To determine P(G) we need to know the “observed frequency” and the “total number of trials.” Experimental Probability is the ratio of the observed frequency to the total number of trials. The following is an example of how to write a complete probability statement using probability notation.
P(G) =
observed frequency total number of trials
For each of your data sets in #2, write your group’s P(G) as a simplified fraction and as a decimal. 2a) P(G) =
__________
__________
2d) P(G) =
__________
__________
2b) P(G) =
__________
__________
2e) P(G) =
__________
__________
2c) P(G) =
__________
__________
3. Explain why the probability of drawing a GREEN tiles must be a fraction between 0 and 1.
4. How does knowing the probability of drawing a GREEN tiles help you know the probability of drawing a BLUE tile?
5. Record a final prediction of how many of the 12 tiles in your bag are GREEN. Justify your answer.
Theoretical Probability:
Experimental Probability:
6. Explain the formulas for both Theoretical and Experimental Probabilities in words. How would you explain the difference in the denominators?
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7. Theoretical Probability is the ratio of the number of outcomes favoring the event to the total number of possible outcomes. NOW you can look in your bag! Count how many blue and green tiles are actually in your bag. Based on this information, what is the “theoretical” probability of drawing a green tile from your bag?
8. How did your group’s “experimental” P(G) compare with the “theoretical” P(G)?
As groups discuss: 9. Using your answer to #3 to guide you, show where the following probabilities belong on the number line below: certain, impossible, likely as not, unlikely, likely
10. Using your answer to #9 to guide you, how would you classify your group’s probability of getting a GREEN outcome? Justify your answer.
11. You’re a teacher in a 7th grade math class and you want to create an experiment for your class with red, yellow and purple marbles in a bag. You want the theoretical probability of drawing a red marble ! ! to be , the theoretical probability of drawing a yellow to be , and the theoretical probability of drawing !
!
!
a purple to be . If you want a total of 120 marbles in the bag: !
a. How many red marbles should you put in the bag? __________ b. How many yellow marbles should you put in the bag? __________ c. How many purple marbles should you put in the bag? __________ d. What do the theoretical probabilities add up to? __________
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12. You’ve decided you want to make the marble experiment a little more difficult. You want to use 400 marbles and you want six different colors – blue, red, green, yellow, purple, and pink. You also do not want any two colors to have the same probability. State the number of each color you are going to put in the bag and what the theoretical probability of drawing the colors will be. a.
Blue: Actual number of blue ______________________ and P(B)
_________________
b.
Red: Actual number of red _______________________ and P(R)
_________________
c.
Green: Actual number of green ___________________ and P(G)
_________________
d.
Yellow: Actual number of yellow ___________________ and P(Y)
_________________
e.
Purple: Actual number of purple ___________________ and P(P)
_________________
f.
Pink: Actual number of pink ______________________ and P(P)
_________________
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1.1A Extra Practice: Probably Probability Name:
Period:
Find the probability of each outcome if the spinner is spun once. Write each answer as a simplified fraction, decimal and percent. 1. P (9) 2. P (multiple of 2) 3. P (even number) 4. P (prime number) 5. P (number < 8) 6. P (factor of 8) !
7. Give an example of a spin that has a probability of . Explain. !
A laundry basket contains 3 red socks, 5 orange socks, 4 blue socks, and 8 black socks. Without looking, choose a sock. What is the probability for each event? Write each answer as a simplified fraction, decimal and percent. 8. P (orange) 9. P (red) 10. P (not red) 11. P (white) 12. What do you notice about numbers 9 and 10? Explain.
If you roll a die and toss a dime, what is the probability of each event occurring? Start by listing out all the possible outcomes. Write each answer as a simplified fraction, decimal and percent. 13. List the possible outcomes for this event: 14. P (1, head) 15. P (2, tail) 16. P (6, heads or tails) 17. P (even, tail) 18. P ( odd, heads or tails) 19. P (even, heads and tails) SDUHSD Math A Honors – Module #1 - STUDENT EDITION 2016-2017
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1.1B Lesson: Probability – Race to the Top* Name:
Period
1. Predict which horse you think will win (Horse 2 through 12). 2. In your group, take turns rolling two dice and give each horse their win by shading one box on the histogram as they earn it. The race is over after 30 rolls of the dice.
2
3
4
5
6
7
8
9
10
11
12
3. Which horse won (had the most rolls) in your group?______ 4. List your group’s experimental probability for each outcome. Remember to simplify, if possible: a.
P(2)=
b.
P(3)=
c.
P(4)=
d.
P(5)=
e.
P(6)=
f.
P(7)=
g.
P(8)=
h.
P(9)=
i.
P(10)=
j.
P(11)=
k.
P(12)=
5. Create a class histogram that combines all the winning horses from each group. What do you notice about the histogram?
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12
6. Which horse won the most often for all the groups? Why?
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7. Do you think that this game is fair? Why or why not?
8. What are all the possible outcomes when you roll two dice? In your group, organize these possible outcomes on the given chart.
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3
4
5
6
7
8
9
10
11
12
9. How many total outcomes did you get? Explain the system you used to get all those outcomes.
10. Use the above information to determine the theoretical probability for each outcome. Write as a simplified fraction. P(1)=________
P(7)= ________
P(2)= ________
P(8)= ________
P(3)= ________
P(9)= ________
P(4)= ________
P(10)= ________
P(5)= ________
P(11)= ________
P(6)= ________
P(12)= ________
11. Why is P(1) impossible? Discuss another event that would also be impossible within this “sum of two dice” context. Be sure to justify your choice.
12. Emily has two dice (standard number cube), one red and one white. a. Find the probability Emily rolls a 1 on the red die and a 6 on the white die. P(Red 1,White 6) = b.
Find the probability Emily rolls a sum of 7. P(sum of 7) =
c.
Explain why are (a) and (b) different.
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Tree Diagrams:
a.
Draw a tree diagram that represents rolling two six-sided dice.
b.
Draw a tree diagram that represents flipping 3 coins.
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Tables:
c.
Draw a table that represents rolling two dice.
d.
Draw a table that represents sums for rolling two dice.
Sample Space: a.
What are the possible combinations in flipping three coins?
Fundamental Counting Principle:
a.
How many possible outcomes are there in flipping three coins?
b.
At the sandwich shop you can choose from three types of bread, two types of meat, 3 types of cheese. You must pick only one of each. How many different types of sandwiches can be made?
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12. Say you rolled 3 dice: a. Would you use a table, tree diagram, or the Fundamental Counting Principle to find how many possible outcomes could you have? Justify your answer. b.
How many possible outcomes could you have?
13. Given 3 dice, find P(sum of 15). List the possible combinations.
14. Given three dice, what would your minimum sum be? Explain your answer.
15. Given three dice, what would your maximum sum be? Explain your answer.
16. What is the probability of rolling the minimum sum when rolling three dice?
17. What is the probability of rolling the maximum sum when rolling three dice?
18. What would the most likely sum be when rolling three dice? Hint: look at your chart from #8, and explain your thinking.
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1.1C Lesson: More Probability Practice* Name:
Period:
Use the data from Problem #1 on Homework 1.1.B to answer the following: 1. What is the probability of getting exactly one HEAD on the first flip? ____________ 2. What is the probability of getting HEADS on the first and second flips? ____________ 3. What is the probability of getting HEADS on the first, second and third flips? ____________ 4. What is the probability of getting HEADS on all four flips? ____________ 5. What pattern do you see emerging?
6. What is the probability of getting HEADS for n flips? ____________ 7. After two flips of the coin, what is the probability of getting at least one HEAD? ____________ 8. After three flips of the coin, what is the probability of getting at least one HEAD? ____________ 9. After four flips of the coin, what is the probability of getting at least one HEAD? ____________ 10. What do you notice about the probabilities in questions 7-9?
11. What is the probability of getting at least one HEAD on n flips? ____________
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Traveling with Probability: A travel agent plans trips for tourists travelling from Chicago to Miami. He gives them three options to get from town to town: airplane, bus, train. Once the tourists arrive, there are two ways to get to the hotel: hotel van or taxi. The cost of each type of transportation is given in the table below. Transportation Type Airplane Bus Train Hotel Van Taxi
Cost $350 $150 $225 $60 $40
12.
Draw a tree diagram to illustrate the possible choices for the tourists. Determine the cost for each outcome.
13.
If these six outcomes are chosen equally by tourists, what is the probability that a randomly selected tourist travels on a bus?
14.
What is the probability that a person’s trip cost less than $300?
15.
What is the probability that a person’s trip costs more than $350?
16.
The subway would be a third way to get to the hotel. How would this change the number of outcomes? Use the Fundamental Counting Principle to explain your answer.
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Section 1.2: Understand/apply equivalence in rational number forms. Convert between forms (fraction, decimal, percent).* Section Overview: In this section students solidify and practice rational number sense through the careful review of fractions, decimals and percents. The two key objectives of this section are a) students should be confidently able to articulate with words, models and symbols the relationship among equivalent fractions, decimals, and percent and b) students should understand and use models to find portions of different wholes. The concept of equivalent fractions naturally leads students to the issues of ordering and estimation. Ordering positive and negative fractions will be connected to the number line. It is important that students develop estimation skills in conjunction with both ordering and operating on positive and negative rational numbers. Lastly, students look at percent as being a fraction with a denominator of 100. Percent and fraction contexts in this section should be approached intuitively with models. In 1.3 students will begin to transition to writing numeric expressions. Concepts and Skills to be Mastered (from standards) 1. 2. 3. 4. 5. 6. 7. 8.
Express probability using appropriate fractions, ratios, decimals, and percents. Find the percent of a quantity using a model. Express and convert between rational numbers in different forms. Express fractions, decimals and percents as related to area models. Draw models to show equivalence among fractions and rational numbers. Solve problems with rational numbers using models. Solve problems with rational numbers using estimation. Compare rational numbers in different forms.
SDUHSD Math A Honors – Module #1 - STUDENT EDITION 2016-2017
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1.2A Lesson: Bar Models with Fractions and Decimals* Name: Period: Reduce/simplify each fraction. Draw a bar model to show the equivalence between the original fraction and the reduced one: !
1.
!"
2.
!" !"
3.
! !"
Find an equivalent fraction for each. Draw a bar model to show the equivalence between the original fraction and the new one. 4.
1 ? = 3 9
5.
3 6 = 7 ?
6.
4 ? = 5 25
Change each improper fraction to a mixed number. Draw a bar model to show the equivalence between the original fraction and the new one. 7.
! !
8.
!" !
SDUHSD Math A Honors – Module #1 - STUDENT EDITION 2016-2017
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!" !
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1.2B Lesson: Rational Number Ordering and Estimation* Name:
Period:
1. Wanda measured the distance across her father’s farm four times and got four different lengths. Her measurements are shown in the table. Distance Across Farm (km) 58 7. 3 8
7.749
7
3 6
a. To estimate the actual length, Wanda first approximated each distance to the nearest hundredth. Then she averaged the four numbers. Find Wanda’s estimate.
b. Wanda’s father estimated the distance across his farm to be 7.483 km. How does this distance compare to Wanda’s estimate?
2. Four people found the distance in kilometers across Nowhere Canyon using different methods. Their results are given in the table. Plot and label them on the number line below. Distance Across Nowhere Canyon (km) Laurie Rachel 23 5. 5 4
Joana 5.25
Jett 1 5 2
3. Joe says that 12. 6 is less than 12.63. Do you agree? Explain why or why not.
4. Plot each fraction on the number line below. a.
1 1 3 2 , , , 2 3 4 5
1 2
1 3
b. − , − , −
3 2 , − 4 5
c. Compare the fractions using <, >, or =.
1 1 ____ 2 3
−
1 1 _____ − 3 2
2 3 _____ 5 4
SDUHSD Math A Honors – Module #1 - STUDENT EDITION 2016-2017
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3 2 _____ − 4 5
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d. What differences do you observe when comparing the fractions in part c?
e. How does the number line help you determine which number is greater?
5. Classify these fractions as close to 0, close to
1 , or close to 1. 2
1 1 5 5 7 1 5 3 1 , , , , , , , , 2 9 8 6 8 5 9 8 7
Close to
Close to 0
1 2
Close to 1
6. Order the fractions from least to greatest. Fractions: ______, ______, ______, ______, ______, ______, ______, ______, ______
7. Classify these fractions as close to 0, close to −
1 , close to -1. 2
1 4 3 2 5 2 4 7 − , − , − , − , − , − , − , − 2 28 5 7 14 11 5 9
Close to −1
Close to -
1 2
Close to 0
8. Order the above numbers from least to greatest. Fractions: ______, ______, ______, ______, ______, ______, ______, ______
SDUHSD Math A Honors – Module #1 - STUDENT EDITION 2016-2017
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Plot and label each fraction on the number line. Fill in the blank with a <, >, or =. 9. 4
2 3 _____ 4 7 10
10. −4
3 2 _____ −4 10 7
Plot and label each fraction on the number line. Fill in the blank with <, >, or =. How do you know your answer is correct? Justify your answer. 11. -2.15 ___ -2.13 Justification:
12. 0.15 ___
3 20
Justification:
13. -0.3 _____ −
1 3
Justification: Answer each of the following questions in a sentence. Show your work. 14. It takes Allen three-fifths of an hour to complete his math homework, five-sixths of an hour to complete his reading homework, and two-thirds of an hour to complete his science homework. Order the subjects by the time from least to greatest. How many hours did Allen spend total on his homework?
15. Maria jogged for two-fifths of a mile, Laura jogged for one-fourth of a mile, and Kaitlin jogged for three-tenths of a mile. Who jogged the greatest distance? How far did they jog altogether?
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16. One day Bob worked 8 hours. Janet said she worked a third of the day. Liz worked 30% of the day. Did they spend equivalent portions of the day working? Explain why or why not. Use numbers to support your explanation.
17. Lisa is a radio reporter and is allowed exactly 90 seconds to relate a story. What fraction of a 30minute broadcast is this?
18. The results of a survey of 100 families showed that 55 families only used the YMCA swimming pool, 20 families only used the YMCA tennis courts, and 15 families used both the pool and the courts. What fraction of the families surveyed used neither the pool nor the courts?
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1.2C Lesson: Probability, Fractions, Percentage, & Ratio* Name:
Period:
A bag contains 100 marbles. The table below shows how many red, blue, green and yellow marbles are in the bag. Use that data to complete the table below. The first row is completed for you.
Color Red Blue Green Yellow
Color of Marble
A. Red
Number of Marbles 16 24 45 15
Probability of drawing that marble color
Percentage of marble color in the bag
16 4 = 100 25
16%
B. Blue C. Green D. Yellow E. Orange F. Red or blue G. Red, blue, green, or yellow
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Modeling to find the part, whole or percent. Example 1: Kim wants to compute 60% of 145. 145 is the whole. She knows that 60% =
60 6 3 = = . 100 10 5
This means that 145 should be divided into either 10 or 5 parts. Dividing 145 into 10 parts would mean 14.5 in each part; dividing into 5 parts would mean 29 in each part. She decided to use 5 parts:
Sixty percent of 145 is the same as 3/5 of 145; thus 60% of 145 is 29 + 29 + 29 or 87. !
Kim could have also decided to show this by dividing 145 into 10 parts. of 145 is !" 14.5 + 14.5 + 14.5 + 14.5 + 14.5 + 14.5 or 87. Notice that the 60% represents the same value whether Kim used 5 parts or 10 parts.
Example 2: What percent of 195 is 78? 195 is the whole and 78 is the part. We know that 10% of 195 is 19.5, so one way to think about this is to cut 195 into ten parts:
Four out of the ten boxes make 78, thus 40% of 195 is 78 Above, we can see that two 10% portions make 39; thus four 10% portions make 78. So, 78 is 40% of 195.
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Example 3: 15% of what number is 45?
In this problem, the part is 45, and we are trying to find the whole. We know that percents are relationships of 100 and that if we can find 10%, we can compute anything fairly easily. If 45 is 15% of the whole, then 15 is 5% of the whole—15% divided into 3 is 5%, and 45 divided into 3 is 15. That means that 5% of the whole is 15, so 10% is 30:
If each 10% portion is 30, then the whole is 30 x 10 or 300. Use models like the ones shown previously to answer the following questions. Answer in a complete sentence. 1. Find 60% of 180.
2. Find 40% of 80.
3. Find 25% of 324.
4. What percent of 80 is 60?
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5. What percent of 120 is 48?
6. What percent of 95 is 19?
7. 12 is 10 percent of what number?
8. 30 is 25 percent of what number?
9. 200% of what number is 72?
For #10-15, solve using the method requested in each problem. Answer in a sentence. 10. A digital calculator is selling at a discount of 15%. The original price was $8.60. How much money will you save by buying the calculator at the sale price? Use a model to solve.
11. This year, 360 people ran in the town marathon. Only 315 people finished the race. What percent of the people finished the race? No model needed. Show your work.
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12. On Saturday, 78% of the breakfast customers at Millie’s Restaurant ordered orange juice. If 33 customers did not order orange juice, how many breakfast customers were there altogether? How many did order orange juice? Model is optional.
13. What percent of 25 is 75? Use a model.
14. A town with a population of 27,530 has voted to allot 40% of the entire budget to education. If the education budget is $11,012, what is the total budget? Model is optional.
!
15. The bike store has a bicycle regularly priced at $660. Tom negotiated a 33 % discount. How much ! money will Tom save buying the bike on sale? How much will he have to pay for the bike?
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1.2D Lesson: Rational Numbers in Applications with Models* Name:
Period:
Draw a bar model to answer the following questions. Answer in a complete sentence. 1. 60 is 40% of what number?
2. There are 36 students in a math class. After math,
3 of the students go to art class and the rest go to 4
social studies. How many students have art after math?
3. Bailey got 80% correct on a science quiz with 150 questions. How many questions did she get correct?
4. Stanley received a 20% raise. He now earns $7.92 an hour. How much money did Stanley earn before his raise?
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5. A bag of M&M’s boasts that it contains 10% more candy than the smaller bag. If the larger bag has 132 pieces of candy, how many pieces of candy does the smaller bag contain?
6. Juan earned money by creating a webpage for a local business. He used for new shoes and
1 of the money he earned 2
2 of the remainder for music. He has $20 left. How much money did he earn for his 3
work?
7. Lydia volunteers with an organization that helps older citizens take care of their yards. 75% of the volunteers in the organization are 20-30 years old. Of the remaining portion, 75% are over 30 and 25% are under 20. If there are 15 people under 20, how many people are in the organization?
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8. There are 360 seventh grade students at Eisenhower Middle School. One-fourth of the students went to Clermont Elementary. Of the rest, half went to Central Elementary and the others came from a variety of other elementary schools. How many students came from Central Elementary?
9. A snowboard at a local shop normally costs $450. Over Labor Day weekend, the snowboard is on sale for 50% off. Customers who make their purchase before 8:00 AM earn an additional 10% off of the sale price. If Mia buys the snowboard before 8:00 AM, how much will she pay?
10. A local business took out a loan. At the end of one year, they found that they spent !
!
! !
of the loan on
payroll; of the remaining amount to purchase inventory; and of the money that was left after that ! ! went to paying off their original loan. They have $100,000 left. What was the original amount of the loan?
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Section 1.3: Solve Percent Problems Including Discounts, Interest, Taxes, Tips, and Percent Increase or Decrease.* Section Overview: In this section, students continue to solve contextual problems with fractions, decimals and percent but begin to transition from relying solely on models to writing numeric expressions. In future modules, students will extend their understanding by writing equations and proportions using variables. Concepts and Skills to be Mastered (from standards) 1. 2. 3. 4.
Use models to solve problems involving percent and fractions. Solve percent problems involving discounts, interest, taxes, tips, etc. Solve percent problems involving percent increase and decrease. Develop algebraic expressions and equations from percent and fraction models.
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1.3A Lesson: Model Percent and Fraction Problems* Name:
Period:
Use a model to solve each of the following multi-step problems. Then write a number sentence that reflects your model and answer. Answer in a complete sentence. 1. George has a piece of rope that’s 12 feet long. a.
He cuts off 25% of the rope. How long is the rope now?
b. Jane has a rope that is 25% longer than George’s 12-foot long rope. How long is Jane’s rope?
2. Marilyn invested $150. a. Marilyn earned 10% on her investment. How much money does she have now?
b. How much money would Marilyn have if she lost 10% on her investment?
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3. A big screen television costs $1200 wholesale. If the mark-up on the television is 20%, what is the new price?
4. Charlotte's resting heart rate is 50 beats per minute. Her target exercise rate is 150% of her resting rate. What is her target rate?
5. A pair of boots was originally priced at $200. The store put them on sale for 25% off. A month later, the boots were reduced an additional 50% off the previous sale price. What is the price now?
6. Marie went out for dinner with her friend. The dinner cost $25. Tax is 5% and Marie wants to leave a 15% tip on the pre-tax amount. How much will Marie pay all together for dinner?
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1.3B Lesson: Transition to Numeric Expressions* Name: 1.
Period: For each problem below, use a model to answer the question.
Context
Model
Fraction Change
Fraction of Original
Percent Change
Percent of Original
a) Pedro had $600 in his savings account. Five years later he has $800.
Write the fraction change expression and the percent change expression. Simplify.
b) It used to take Naya 10 minutes to walk to school. Now it takes her 8 minutes.
Write the fraction change expression and the percent change expression. Simplify.
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Write a number sentence and solve the following problems. You may also use a model if helpful. State your answer in a complete sentence. 2.
Li planted a 12 foot tall tree in her yard. 10 years later it is 18 feet tall. What is the tree’s percent of growth?
3.
If the price of a $65 purse increases 15%, what is the new price of the purse?
4.
Last year, Mrs. Howdy’s class size increased from 20 to 25 students. If the her class size increases by the same percent this year, how many students can she expect to have in class next year?
5.
The price of a camera will be 25% greater in 2016 than in 2015. In 2015 the price of the camera was 10% greater than in 2014. If the camera cost $300 in 2014, how much will it cost in 2016?
6.
Marcie went out for dinner with her friend. The dinner cost $25. Tax is 5% and Marcie wants to leave a 15% tip on the pre-tax amount. How much will Marcie pay all together for dinner?
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7. 100 is increased by 10%. The result is decreased by 10%. Is the final result 100? Explain your reasoning.
8. A jacket that originally cost $100 was discounted 20% for a 4th of July sale. After the sale was over, the jacket was marked-up 20%. How much does the jacket cost after the sale was over? Show your work.
9. A music store marks everything up 67%. A DVD costs the store $10.15. Find the markup. Then find how much the store will sell the DVD to its customers.
10. A surfboard is marked $700 at Hansen’s. The store is offering 20% off for their Fourth of July sale. They are offering an additional 20% off if you come early to shop between the hours of 8AM-9AM. How does this deal compare to 40% off the surfboard that they will offer for their Labor Day sale? When should I buy the surfboard?
11. How can you explain why the deals in #10 above are not the same?
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