dscyfeducation.wikispaces.com 3.docx... · Web viewA.REI.1: Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step,

DSCYF EDUCATION UNIT

K-U-D (Know, Understand, Do) ChartGrade/Course _Algebra I__________ Unit Title ___Solving Inequalities_______________________________________

Content Standards:A.REI.1: Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution.

A.REI.3: Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

A.SSE.1: Interpret expressions that represent a quantity in terms of its context.

A.SSE.1.b: Interpret complicated expressions by viewing one or more of their parts as a single entity.

A.CED.1: Create equations and inequalities in one variable and use them to solve problems.

Know Understand Do(Note: concepts, facts, formulas, key vocabulary)

Symbols involved in inequalities: Greater than >, Less than <, Greater than or equal to ≥, Less than or equal to ≤

Steps to solve an inequality

The meaning of Absolute Value and the absolute value sign

Compound inequalities using “and” and using “or”

Intersection and union of sets and set notation

(Big idea, large concept, declarative statement of an enduring understanding)

Real world situations can be written and solved as algebraic equations and inequalities that can be used to guide decision making

Some solutions require the number be from within a given set of numbers while others can be from either one of two different sets of numbers.

Algebraic solutions with one variable can be graphed on a number line.

(Skills, competencies)

Write a real-world situation as an inequality

Solve an inequality including those with absolute values.

Graph solutions of inequalities with one variable on a number line

Solve and graph compound inequalities using “and” and those using “or”

Find the intersection and union of two or more distinct sets. Write the solutions on number lines and with set notation.

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Unit Essential Question: How can real-world situations be written as inequalities and solved to make decisions?

Key Learning: Solving Inequalities

Lesson Essential Question 1 Lesson 1 Vocabulary

How are inequalities represented on a number line?

Less than

Less than or equal to

Greater than

Greater than or equal to


What happens to an inequality when the same number is added or subtracted to both sides?

Equivalent inequalities


What is the difference between multiplying and dividing by positive and negative numbers?

Positive numbers

Negative numbers

Reverse


Why is it preferred to add and subtract before multiplying and dividing?

Distributive Property




How do you know whether the parts of the graph will intersect or go in opposite directions?

Compound inequalities

Inclusive


When solving compound inequalities, how do you know when to use “and” and when to use “or”?

Inclusive


How do you solve inequalities with absolute values in them?

Absolute value


What is the difference between the union and intersection of two sets?

Union

Intersection

Disjoint



Major Unit Assignment: Class Shirts

The Student Council has decided to order school T-shirts for the students to purchase.

Tees-R-Us charges $250 set up fee and $9 per shirt.

We Have Shirts has no set up fee and charges $10 per shirt.

Write the equation for each company. Graph the equations.

Write an inequality for each company to show for what number of shirts each company is the better deal.

Write a letter to the Student Council Officers reporting your findings and recommendation for which company they should use to purchase the T-shirts.



Student Assessments(How students will indicate learning and understanding of the concepts in the unit.

Note: Can have multiple assessments, one on each page.)

Unit Topic: Solving Inequalities

Title Performance TaskDescription TASK 1

An architect is designing a new lakefront cottage. He mustconsider the building codes in the design. The rectangular loton which the cottage will be built is 220 ft long and has 110 ft offrontage on the lake.a. The building code states that a building must be nocloser than 25 ft to the lot line. Write and solve aninequality to determine how long the cottage frontfacing the lake may be.b. The buyers expect that the cost of the cottage will beabout $125,000. They are able to deviate from this costby no more than $12,500. Write and solve an absolutevalue inequality to find the range of costs. Explain youranswer.

TASK 2a. Suppose you play third base for your high school softball team. In the last four games, you had a total of 6 hits in 15 at-bats. Assuming you will have 4 at-bats in tomorrow’s game, how many hits will you need in order to have a batting average greater than .450? Write and solve an inequality to find the number of hits you need.b. Write two inequalities that require that you to use the addition, subtraction, multiplication, and division properties in order to solve them. Solve the inequalities you wrote and graph the solutions on a number line.

TASK 3a. A camping supply store usually prices tents from $68 to $119. What is the range of possible prices if the stores is advertising all tents 10 to 25% off? Show your work.

TASK 4Find the value of x such that the area of the figures satisfies the given conditions:a. A rectangle with one side = to 3, and the other side x such that the area of the rectangle is less than or equal to 72 square unitsb. A triangle with a base of 20 and a height of x+2 such that the area is greater than 30 square units.



TASK 5You have a rectangular photograph 12 inches wide and 18 inches long. You surround the photograph with a mat x inches wide. You have an 80 inch piece of wood to make a frame for the matted photograph. What are the dimensions of the frame that will enclose the greatest area?

Time (In Days) 1Differentiation Allow students to select one (or more) of the tasks. Allow

students to work with a partner.

Revise/Review

Resources & Materials Graph paper, calculator

Performance Task Rubric

Student Name: ____________________________________________________

CATEGORY 3 2 1 0Developing

Autonomy- The student

persevered to complete the project without help

completed most of the problem without help

needed key hints to solve the problem

needed extensive guidance to work the problem

The Solution Process – The students work

showed

a complete and appropriate solution process

an appropriate solution process that is almost complete

an appropriate process that is partially complete

an inappropriate process or no evidence of a process

The Conclusion/Answer

– The student’s answer is an

accurate conclusion, supported by valid evidence and reasons, appropriate to this problem and context

inaccurate but logical conclusion, supported by evidence and reasoning but incorrect due to minor factual error (in details of problem, in computation, recall of formula, etc.) or minor mistake in reasoning

inaccurate but logical conclusion that overlooks or gets wrong significant (about the problem, the rule, computation, etc.).

inappropriate conclusion, not supported by facts and logic, or there is no conclusion



Research: Performance Task (Essay) Students’ Names: ________________________________________

CATEGORY 5-4 3 2 1 Quality of Information (Double weighting)

Information clearly relates to the main topic and answers the question. It includes salient examples, lucid analysis and clear links to the question.

Information relates to the main topic and answers the question. It includes some salient examples, analysis and links to the question.

Information has a tenuous link to the main topic.Some details and/or examples are given, but might be irrelevant to the question.

Information has little or nothing to do with the main topic.

Organization (Half weighting)

Information is very organized with well-constructed paragraphs and very clear main points.

Information is organized with well-constructed paragraphs and clear main point.

Information is organized, but paragraphs are not well-constructed, and the main point is unclear.

The information appears to be disorganized.

Introduction The introduction consists of a very good argument, and outlines briefly the factors to be examined, and is very consistent with the essay.

The introduction consists of a good argument, and outlines briefly the factors to be examined, and is consistent with the essay.

The introduction consists of a rather weak argument, and outlines briefly the factors to be examined, but is not very consistent with the essay.

The introduction does not have an argument, and does not outline the factors to be examined.

Conclusion The conclusion deals fully with the requirements of the question, and is very consistent with the essay.

The conclusion deals with the requirements of the question, and is consistent with the essay.

The conclusion deals partially with the requirements of the question, but is not very consistent with the essay.

The conclusion does not deal with the requirements of the question, and is not consistent with the essay.

Mechanics (Half weighting)

No grammatical, spelling or punctuation errors.

Almost no grammatical, spelling or punctuation errors

A few grammatical, spelling, or punctuation errors.

Many grammatical, spelling, or punctuation errors.

Sources All sources (information and graphics) are accurately documented in the desired format.

All sources (information and graphics) are accurately documented, but a few are not in the desired format.

All sources (information and graphics) are accurately documented, but many are not in the desired format.

Some sources are not accurately documented.

Total marks: _________/ 30



Learning Goals for this Lesson:An inequality is a mathematical sentence that uses an inequality symbol to compare the values of two expressions. The solutions of inequalities can be represented on a number line.

Standards:A.REI.3: Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

Students Will Know:How to write inequalities from symbols to words, from words to symbols, and how to represent both on a number line.

Students Will Be Able To:Write inequalities in wordsWrite inequalities as symbolsGraph inequalities on a number line.

Lesson Essential QuestionHow are inequalities represented on a number line?

Activating Strategy: Each student is to write his/her age on a post-it note and place it on a number line on the board. Ask students to summarize the information. Keep probing until statements are generated similar to: “All students in our class are greater than or equal to age 13,” and “All students in our class are less than or equal to age 18.”Key vocabulary to preview and vocabulary strategyLess than, Less than or equal to, Greater than, Greater than or equal to

Lesson Instruction: Pearson: Mathematics, Algebra I Common CoreLearning Activity 1: With the class work through the situation on page 167, to determine the maximum height allowed for a building on Pennsylvania Ave, D.C. http://www.pearsonsuccessnet.com/snpapp/learn/navigateIDP.do?method=vlo&internalId=130912100000039&isHtml5Sco=false

Review problem 1. Discuss the examples.

Assessment Prompt for Learning Activity 1: Think-Pair-Share on Questions 8-11 on page 168

Graphic Organizer Attached

Learning Activity 2: Review problem 2, which asks the students to substitute a number into an inequality to determine if it is a solution. Work through the “Got It?” 2 and 2b.

Assessment Prompt for Learning Activity 2: Do Now on Questions 12-16 on page 168. Popsicle sticks for random responses. Thumbs up or down to agree or disagree with response.

Assignment: Remedial 3.1 page 1 (page 2 optional)Learning Activity 3: Review problem 3, the steps to graphing an inequality. Review the

“Got It?” problems 3a-c. Monitor to see how the students graphed the inequalities. Ask for volunteers to write the answers on the board.

Assessment Prompt for Learning Activity 3: Think-Pair-Share on Questions 21-24 on page 168

Learning Activity 4: Review problem 3, on page 166 which demonstrates the process of



using a graph to write a corresponding inequality. Review the “Got It?” problem 4a and 4b.

Assessment Prompt for Learning Activity 4: and Exit Slip Write the inequalities represented by the equations on page 168 Questions 29-31.

Summarizing Strategy: Exit Slip: p168, #29-31



Graphic Organizer: Lesson 3.1

GREATER THAN

Symbol: >Meaning: larger than the number (but not that number)Number line: Open circle and goes to the right (the way the arrow points)

GREATER THAN OR EQUAL TO

Symbol: ≥Meaning: that number and all those larger than that numberNumber line: Closed/Solid circle and goes to the right (the way the arrow points)

LESS THAN

Symbol: <Meaning: smaller than the number (but not that number)Number line: Open circle and goes to the left (the way the arrow points)

LESS THAN OR EQUAL TO

Symbol: ≤Meaning: that number and all those smaller than that numberNumber line: Closed/solid circle and goes to the left (the way the arrow points)



Learning Goals for this Lesson

Properties of numbers are equality can be used to transform as equation (or inequality) into simpler equivalent equations (or inequalities) in order to find solutions.

Standards

A.REI.3: Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.


Students Will Know

The steps to solve an inequality using addition or subtraction

Students Will Be Able To

Add or subtract to solve inequalities

Lesson Essential Question

What happens to an inequality when the same number is added or subtracted to both sides?

Activating Strategy: p.171 Getting Ready: In a U.S. Presidential election, a candidate must win at least 270 out of 538 electoral votes to be declared the winner. Suppose a candidate has earned 238 electoral votes in states outside the southeastern U.S. What is the least number of states in the southeastern US that the candidate could win and still become president? What are these states? Justify your reasoning.

Key vocabulary to preview and vocabulary strategy

Equivalent inequalities

Lesson Instruction

Learning Activity 1: Work through the example with the students on p 172, Problem 1: What are the solutions to the inequality x – 15 > - 12 ? Then graph the solution.

Assessment Prompt for Learning Activity 1: What are the solutions of n – 5 < - ? Graph the solutions.

Learning Activity 2: p 172, Problem 2: What are the solutions of 10 ≥ x – 3? Graph and check the solutions.

Assessment Prompt for Learning Activity 2: What are the solutions of m - 11 ≥ – 2? Graph and check the solutions.

Graphic Organizer

Complete sections for addition and subtraction



Learning Activity 3: Review key concepts on p 171 and 173 addition and subtraction properties of inequality. Problem 3: What are the solutions of t + 6 > - 10 ? Graph the solutions.

Assessment Prompt for Learning Activity 3: What are the solutions of -1 ≥ y + 12 ? Graph the solutions.

Learning Activity 4: Guide the students through writing and solving an inequality with the example in Problem 4 on page 173.

Assessment Prompt for Learning Activity 4: Have the students work with a partner to solve the Got It? 4a. and b. on p174.

Assignment:

P174 #s 13-20 and 33-38

Summarizing Strategy:

Exit Slip: Lesson Check on p174: Students need to answer #1, 3, 5 for their exit slip.



Learning Goals for this Lesson


Standards:

A.REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.


Students Will Know

The steps to solve an inequality using multiplication or division.


Multiply or divide to solve inequalities

Lesson Essential Question: What is the difference between multiplying and dividing by positive and negative numbers?

Activating Strategy: p.178 Getting Ready: Copy and complete each of the following statements by replacing each □ with a < or >. What happens to the inequality symbol when you multiply each side by a positive number? What happens to the inequality symbol when you multiply each side by a negative number? Justify your reasoning.

4 X 3 □ 1 X 3 4 X -1 □ 1 X -1

4 X 2 □ 1 X 2 4 X -2 □ 4 X -2

4 X 1 □ 1 X 1 4 X -3 □ 4 X -3

Key vocabulary to preview and vocabulary strategy: Positive numbers, Negative numbers, Reverse

Lesson Instruction

Learning Activity 1: Project the Key Concept from p.178 on to the screen. Ask students to review it, and copy it into their notes. Then ask for volunteers to explain what it shows. Ask for various examples to prove it works for any numbers.

With the students read the top of p. 179. Work through the example with the students on p 179, Problem 1: What are the solutions to the inequality x/3 <-2 ? Then graph the solution.

Graphic Organizer

Complete sections for multiplication and division



Assessment Prompt for Learning Activity 1: What are the solutions of c/8 < 1/4 ? Graph the solutions.

Learning Activity 2: Work through problem 2 from page 179 with the students. What are the solutions to -3/4 w≥ 3 . Graph and check the solution.

Assessment Prompt for Learning Activity 2: What are the solutions to -n/3 < -1? Graph and check the solutions.

Assignment: p.181 Complete problems #1-6.

#7 -10, # 19 - 22Learning Activity 3: Work through problem 3 from p 180. What are the solutions to 4.50 d ≥ 75 ? Graph and check the solutions.

Assessment Prompt for Learning Activity 3: What are the solutions to 3a. on p.180. Then, b., why is the answer rounded to the greatest whole number?

…………………………………………………………………………………………………………………………

Learning g Activity 4: Work through problem 4 on p. 181 What are the solutions to -9 y ≤ 63 ? Graph and check the solutions.

Assessment prompt for Learning Activity 4: What are the solutions to -5x > -10 ? Graph and check the solutions.

Summarizing Strategy: Exit Slip: 1. If a < b, then 2a would be < or > than 2b? Why?

2. If a < b, then -3a would be < or > than -3b? Why?



Learning Goals for this Lesson:


Standards:

A.REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

A.CED.1 Create equations and inequalities in one variable and use them to solve problems.

Students Will Know

How to solve multi-step in equalities


Solve multi-step inequalities

Lesson Essential Question: Why is it preferred to add and subtract before multiplying and dividing?

Activating Strategy: p186 Getting Ready: Math Club members are selling Pi Day T-shirts for $7.50 each. The goal is to raise $500 by Friday. The figure at the right shows how much they have raised by Wednesday ($337.50). What is the minimum number of t-shirts they must still sell in order to reach their goal? Explain your reasoning.

Key vocabulary to preview and vocabulary strategy: Distributive Property

Lesson Instruction: Show 2 minute video clip from www.youtube.com reviewing Order of Operations http://www.youtube.com/watch?v=4xsyXu9J8CA There are several others but this is short, catchy and to the point.

Learning Activity 1: Work through problem 1: 9 + 4t > 21 First subtracting 9 from both sides. Then, dividing both sides by 4. Explain that you are following the Order of Operations (PEMDAS)… but in reverse, as we do in algebra. Lead the students through the checking to make sure their answer is correct.

Assessment Prompt for Learning Activity 1: Ask the students before they begin what is different about dividing by a negative number. p187, Got It 1. a. -6a – 7 ≤ 17

b. -4 < 5 – 3n

c. 50 > 0.8X + 30

Graphic Organizer: Order of Operations

P

parenthesesE

exponentsM D

multiply and divide in order from left to right

A S add and subtract in order

from left to right

http://www.youtube.com/watch?v=4xsyXu9J8CA



*Create as a “hop scotch” board with one, one , two and two squares. Explain that there are four “STEPS” and in solving algebraic equalities and inequalities we “reverse” the order to break down the problem to find the solution

Learning Activity 2: p187 Work through problem 2 discussing that when you have “at most” your equation must be “less than or equal to.”

Assessment Prompt for Learning Activity 2: Allow the students to find the possible widths of a rectangular banner 18 feet long, since you have only 48 feet of trim to go around it.

Assignment:

P189 #s 1 – 4, 9 through 21 odd, 44, 45

Learning Activity 3: Distribute means to pass out, or hand out , therefore the Distributive Property requires us to “Pass out” the number being multiplied to every number in the parentheses!! Missing someone at the dinner table when distributing dessert would be RUDE! Don’t be rude… by missing one of the numbers/terms in the parentheses!

Work through problem 3 on p188 with the students: 3(t + 1) – 4t ≥ -5 First distribute, and then combine like terms. Then continue with the two steps subtracting and dividing to get the correct answer.

Assessment Prompt for Learning Activity 3: Have the students solve Got It number 3 on p188:

15 ≤ 5 – 2(m + 7)

_______________________________________________________________

Learning Activity 4: Remind students how to deal with equations having variables on both sides of the equals sign…. And that working with inequalities with variables on both sides of the inequality is the same (a good practice is to move the smallest number, in most cases).

Lead the students through problem 4 on p188: 6n – 1 > 3n + 8. Once again checking the solution to make sure it is correct.

Assessment Prompt for Learning Activity 4: What are the solutions of

3b + 12 > 27 – 2b ? Did anyone add 2b for the first step? Did anyone subtract 3b for the first step? Check to see that they all arrived at the same answer.

______________________________________________________________



Learning Activity 5: Review the top of page 189 with the students explaining about the special solutions they may get when solving inequalities. Lead the students through both problems in Problem 5 p189: 10 – 8a ≥ 2(5 – 4a) and 6m – 5 > m + 7 - m

Assessment Prompt for Learning Activity 5: Have the students solve both of the inequalities: 9 + 5n ≤ 5n – 1 and 8 + 6x ≥ 7x – 2 - x

Summarizing Strategy: Write an explanation to a friend (in two or three sentences) who was absent from this lesson why is makes sense to add and subtract before multiplying and dividing to find the solution to a 2 step inequality problem.

Learning Goals for this Lesson ( 3-6) Standards

A.REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients




represented by letters.


Students Will Know:

How to solve and graph inequalities with the word “and”

How to solve and graph inequalities with the word “or”


Solve and graph inequalities with the word “and”

Solve and graph inequalities with the word “or”

Lesson Essential Question: How do you know whether the parts of the graph will intersect or go in opposite directions?

Activating Strategy: The diagram shows the number of boxes of oranges that an orange tree can produce in 1 year. An orange grower earns $9.50 for each box of oranges he sells. How much could the grower expect to earn in 1 year from 1 year? Explain your reasoning.

Key vocabulary to preview and vocabulary strategy:

Compound Inequalities: Discussion of the meaning of the two words separately, and then what they mean together. Read p200 with the students to make sure they understand a compound inequality can be “and” or “or”. Inclusive means “including…”

Lesson Instruction

Learning Activity 1 : Review the Essential Understanding section on p200 with the students having them make the graph for each inequality without looking at the text for the top 4 inequalities: x ≥ 3, x < -2, x ≤ 7, and x ≥ 1. Reinforce whether the graphs will have open or closed dots. Then ask the students how to graph x ≥ 3 AND x ≤ 7 on the same graph. Review the graphs, then ask them which numbers would be part of the solution, Make sure they include 3 and 7 since it is inclusive, as well as fractions and decimals. Next, ask them how they would graph x < -2 or x ≥1. Again check for open or closed dots. Now ask what number would be part of this solution. Is -2 one of the numbers (no)? What about 1 (yes)? Ask someone to explain why that is the case. Ask for more numbers…. Fractions decimals... Ask the students if any of the numbers in the solution are in both sections of the graph. No, because there are 2 separate sections. Therefore the answer must be OR, not AND.

Use the Pearson Interactive Learning to guide the students through the

Graphic Organizer

Additional Vocabulary Support 3-6



problems on p201 where they are asked what inequality represents the phrases: a. all real numbers that are greater than -2 and less than 6 and b. all real numbers that are less than 0 or greater than or equal to 5

Assessment Prompt for Learning Activity 1: Got It? P201 Write an inequality that represents each phrase and graph the solutions:

a. All real numbers that are greater than or equal to -4 or less than 6b. All real numbers that are less than or equal to 2 ½ or greater than 6c. What is the difference between “x is between -5 and 7” and “x is

between -5 and 7, inclusive”?

Learning Activity 2: Explain to the students that a solution of a compound inequality involving AND is any number that makes BOTH inequalities true

Work through the compound inequality with AND demonstrating how to split the inequalities and solve both sections. -3 ≤ m – 4 < -1 Then graph the parts and graph.

Assessment Prompt for Learning Activity 2: Find the solutions of

-2 < 3y – 4 < 14 Graph

Assignment:

p.204 #9-29 odd, 31-50

Learning Activity 3: Read the Test Average item on p. 202 with the students. Work through the example presented in the book. Remind students they are actually working two problems at the same time to get a maximum and a minimum score.

Assessment Prompt for Learning Activity 3: Suppose a student scored 78, 78,and 79 on the first 3 tests. Is it possible for the student to receive a B (average btw 85 and 93) for the course? Assume the highest test grade possible is 100, and there are only four tests for the marking period.

________________________________________________________________

Learning Activity 4: Explain that unlike an “and” equation, an “or equation only requires either one of the inequalities to be true. However, to solve an “or” inequality, you must split the inequality and solve BOTH resulting inequalities.

Guide the students through the example of p202 to solve

3t + 2 < -7 or -4t + 5 < 1, then graph the result

Assessment Prompt for Learning Activity 4: What are the solutions of -2y + 7 <



1 or 4y + 3 ≤ -5

Summarizing Strategy:

In a quick note to a friend who is absent today explain how to tell whether an equation for a compound inequality will result in a graph that goes together or in opposite ways.

Learning Goals for this Lesson (3-7)

Absolute value equations and inequalities can be solved by first isolating the absolute value expression, if necessary, then, writing an equivalent pair of equations or inequalities.

Standards


A.SSE.1: Interpret expressions that represent a quantity in terms of its context.



A.SSE.1.b: Interpret complicated expressions by viewing one or more of their parts as a single entity.

Students Will Know:

How to use and solve inequalities with absolute value


Solve inequalities containing absolute values.

Lesson Essential Question: How do you solve inequalities with absolute values in them?

Activating Strategy: p207 Getting Ready: Serena skates toward Darius and then passes by him. She skates at a constant speed of 20ft/s. At what time(s) is Serena 60ft. from Darius? Explain your reasoning.

The distance from Serena to Darius decreases then increases. You can use absolute value to model such changes.

Key vocabulary to preview and vocabulary strategy: absolute value

Remind students of absolute value - - - the distance a number is from zero… What number is 2 spaces from zero? Then ask, What other number is also 2 spaces from zero?

Lesson Instruction

Learning Activity 1: Explain to the students that the absolute variable must be isolated and then written as two separate I equalities to be solved. Work through Problem 1 on page 207 with the students. What are the solutions of │ x │ + 2 = 9 ?

Assessment Prompt for Learning Activity 1: What are the solutions for │ n │ - 5 = -2 ? Remind students to graph and check their solutions.

Graphic Organizer

Solving Absolute Value worksheet

Learning Activity 2: Have the students record the key concept in their notebooks “To solve an equation in the form of │ A │ = b, where A represents a variable expression and b > 0, solve A = b and A = -b.”

Work through the following problem with the students: Starting from 100 feet away, your friend skates toward you and then passes you. She skates at a constant speed of 20ft/sec. Her distance d from you in feet is after t seconds is given by d = │ 100 – 20t │. At what times is she 40 feet from you?

Assessment Prompt for Learning Activity 2: Got It: p 208 Another friend’s distance d from you (in feet) after t seconds is given by d = │ 80 – 5t │. At what times is she 60 feet from you? What does the 80 in the equation

Assignment:

p. 211 #4-47 odd and 49-76



represent? What does the 5 in the equation represent?

Learning Activity 3: Remind the students that absolute value represents the distance from 0 on a number line. Distance is always nonnegative (positive). So any equation that states the absolute value of an expression is negative has NO solutions.

Work through the next problem with the students beginning by subtracting 12 from both sides and then dividing by 3. 3│ 2z + 9 │ = 10

It will become apparent that there is no solution for the resulting equation: │ 2z + 9 │ = -2/3, since an absolute value can never be negative.

Assessment Prompt for Learning Activity 3: What are the solutions for │ 3x - 6 │ - 5 = -7 ?

________________________________________________________________

Learning Activity 4: Explain to the students that absolute value inequalities can be written as compound inequalities. │ n – 1 │ < 2 represents all numbers with a distance from one that is less than 2 units. So │ n – 1 │ < 2 means -2 < n – 1 < 2. Show the students the graph as shown on the top of p.209.

│ n – 1 │ > 2 represents all numbers with a distance from one that is greater than 2 units. So │ n – 1 │ > 2 means n – 1 < - 2 or n – 1 > 2. Show the students the graph as shown on the top of p.209.

Have the students record the key concept in their notebooks, “To solve an inequality in the form of │ A │ < b, where A is a variable expression and b > 0, solve the compound inequality – b < A < b. To solve an inequality in the form of │ A │ > b, where A is a variable expression and b > 0, solve the compound inequality A < -b and A > b. Similar rules are true for │ A │ ≤ b or │ A │ ≥ b.

Assessment Prompt for Learning Activity 4: Write the compound inequality to be solved for │ x + 2 │ < 5 and │ x – 3 │ > 7 ( answers: -5 <x+2<5 and x-3 > 7 or x-3 <-7)

________________________________________________________________

Learning Activity 5: Work through problem 4 on p.209 What are the solutions of │8n│ ≥ 24 ? Graph the solutions.

Assessment Prompt for Learning Activity 5: Got It p.209, What are the solutions of │ 2x + 4 │ ≥ 5, and graph the solutions.



________________________________________________________________

Learning Activity 6: Work through problem 5 on p.201 A company makes boxes of crackers that should weigh 213 g. A quality-control inspector randomly selects boxes to weigh. Any box that varies from the weight y more than 5 grams is sent back. What is the range of allowable weights for a box of crackers? Graph the solution.

Assessment Prompt for Learning Activity 6: Got It p.201 problems 5a. and b.

5a. A food manufacturer makes 32 ounce boxes of pasta. Not every box weighs exactly 32 ounces. The allowable difference from the ideal weight is at most0.05 oz. Write and solve an absolute value inequality to find the range of allowable weights.

5b. In problem 5, could you have solved the inequality│ w – 213 │ ≤ 5 by first adding 213 to each side? Explain your reasoning.

Summarizing Strategy: Explain why there are 2 inequalities written from a single absolute value inequality.

Learning Goals for this Lesson (3-8)

Students will understand that given 2 of more sets, the set of elements belonging to AT LEAST ONE of the sets is the union of the sets, and the set of elements belonging to ALL of the sets is the intersection of the sets.

Standards:


Students Will Know Students Will Be Able To



How to find the intersection of 2 or more sets, and the union of 2 or more sets.

Distinguish between union and intersection of sets, and correctly find each of them.

Lesson Essential Question: What is the difference between the union and intersection of two sets?

Activating Strategy: Getting Ready! P.214 Fifty students are polled about their after-school activities. All said they participate in one or more of three clubs: the Robotics Club (RC), the Student Council (SC), and the Theater Club (TC). How many students participate in the Theater Club only? Use the information in the table. Explain your reasoning.

Key vocabulary to preview and vocabulary strategy: apply meaning from other contexts to develop mathematical meaning

Intersection, union, disjoint

Lesson Instruction

Learning Activity 1 : Work through Problem 1 p.215 with the students. In your left pocket you have a quarter, a paper clip, and a key. In your right pocket, you have a penny, a quarter, a pencil, and a marble. What is a set that represents the different items in your pockets? Demonstrate the way to represent sets using proper notation. List the items from both sets to demonstrate the UNION of the two sets.

Assessment Prompt for Learning Activity 1: Got It? P.215 Write sets P and Q below in roster form. What is P U Q?

b. What is true about the union of 2 distinct sets if one set is a subset of the other?

Graphic Organizer

Venn DiagramsTwo circles not overlapping = disjoint setsTwo circles with small overlapping area = overlapping area is intersection al 3 areas represent union (each element listed one time)

Learning Activity 2: Using the Venn diagrams on p.215 make sure students understand the difference between the intersection of sets and disjoint sets. Refer back to the after-school activities. Ask what information you could gather about the students if the representation of the data was a three non-overlapping circles. (disjoint)

Work through Problem 2 on p.216 with the students. If students are having trouble identifying the intersection, have them list the elements (or some of them) of each of the sets. Set X is the set of natural numbers less than 19.

Assignment: p. 218 # 11 – 31 odd, and 33-43



Set Y is the set of odd integers. And Set Z is the set of multiples of 6.

a. What is X ∩ Z ? b. What is Y ∩ Z ?

Make sure to remember to demonstrate the symbol for the empty set (aka disjoint sets)

Assessment Prompt for Learning Activity 2: Got It? Let A = { 2, 4, 6, 8} and B = {0, 2, 5, 7, 8}, and C = { n│ n is an odd whole number}.

a. What is A ∩ B ?b. What is A ∩ C ?c. What is C ∩ B ?

Venn diagrams can be used to solve problems involving relationships between sets.

Learning Activity 3: Work through problem 3 on p.216 with the students. Use technology to display the pictures of the three backpacks and their contents.

Have students create Venn diagrams to represent the union and intersection of the sets. Allow them to compare their diagrams with each other.

Assessment Prompt for Learning Activity 3: Got It? P.217 Let A = {x│ x is one of the first five letters of the alphabet }, B = {x │ x is a vowel }, and C = {x │ x is a letter in the word VEGETABLE }. Which letters are in all three sets? (The intersection)

Refer the students back to the original Getting ready activity which gave information about students after school activities. Allow the students to discuss whet the overlapping parts of the circles represent in the Venn diagram. Ask what is represented in the section where all 3 circles overlap. Explain this section of students involved in all 3 clubs is called the INTERSECTION of the sets. Intersection is represented by the up-side down U, which can be seen as capturing the items in “ALL” the sets. Listing the elements in at least one of the sets is called the UNION of the sets, or where the sets are united, listing all the elements one time. This is also a good time to tie the previous sections to this by reminding the students of absolute value inequalities. The solutions that are represented by OR are the UNION of the sets, and those with AND are the INTERSECTION of the sets.



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Learning Activity 4: Work through Problem 4 on p.217 Using a Venn Diagram to Show Number of Elements. POLLING: Of 500 commuters polled, some drive to work, some take public transportation, and some do both. Two hundred commuters drive to work and 125 use both types of transportation. How many commuters take public transportation?

Explain the process of listing what you already KNOW, what you NEED to know, and a PLAN to figure it out. Work through the steps listed on the page to find the solution.

Assessment Prompt for Learning Activity 4: Got It? P.217

Of 30 students in student government, 20 are honor students and 9 are officers and honor students, All of the students are officers, honor students, or both. How many are officers but not honor students?

Use a work - pair - share activity to work through the problem.

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Learning Activity 5: Work through problem 5 with the students. What are the solutions of │ 2x – 1 │ < 3? Write the solution as the intersection or union of two sets. This will result in an intersection of the two sets x > -1 and x < 2. Show the students how the solution can be written using set notation.

Assessment Prompt for Learning Activity 5: Got It? P.218

5. Solve each inequality. Write the solutions as either the intersection or union of two sets. a. 8 ≤ x + 5 < 11 b. │ 4x – 6 │> 14

Summarizing Strategy: Exit ticket - In a short note to a friend explain the difference between intersection and union of sets and how they relate to absolute value inequalities.


Documents

dscyfeducation.wikispaces.com 3.docx... · Web viewA.REI.1: Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step,