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P ROG RAMMABL E LOG I C
CON T RO L L E RS
7 -‐ E L EMEN T MATHEMAT I C S &
S C I E N C E L E S S ON
MANUFACTURING TECHNOLOGY AND ROBOTICS
Jessica L. Campbell
Ben Franklin Career and Technical Center Dunbar, WV
Table of Contents
SUMMARY ........................................................................................................................................... 3
ELEMENT ONE: IDENTIFY LEARNING GOALS ......................................................................................... 4 CAREER AND TECHNICAL STANDARDS: .............................................................................................................. 4 21ST CENTURY SKILLS: ................................................................................................................................... 4 ENGLISH LANGUAGE ARTS STANDARDS ............................................................................................................ 4 MATHEMATICS STANDARDS ........................................................................................................................... 5 SCIENCE STANDARDS ..................................................................................................................................... 5
ELEMENT TWO: PRE-‐ASSESSMENT ........................................................................................................ 6 ELEMENT TWO: PRE-‐ASSESSMENT RUBRIC ....................................................................................................... 8
ELEMENT THREE: EMBEDDED CONTEXTUAL EXAMPLES ..................................................................... 11 ELEMENT THREE: EMBEDDED CONTEXTUAL EXAMPLES -‐ SOLUTIONS ................................................................... 13
ELEMENT FOUR: RELATED CONTEXTUAL EXAMPLES ........................................................................... 15 ELEMENT FOUR: RELATED CONTEXTUAL EXAMPLES -‐ SOLUTIONS ........................................................................ 17
ELEMENT FIVE: TRADITIONAL EXAMPLES ........................................................................................... 20 ELEMENT FIVE: TRADITIONAL EXAMPLES-‐ SOLUTIONS ...................................................................................... 22
ELEMENT SIX: DEMONSTRATE UNDERSTANDING ................................................................................ 24
ELEMENT SEVEN: POST-‐ASSESSMENT ................................................................................................ 25 ELEMENT SEVEN: POST-‐ASSESSMENT – ANSWER KEY ...................................................................................... 27
Summary This cross-‐curricular lesson uses project-‐based learning to meet mathematics and science standards with applications to Programmable Logic Controllers as used in industrial processes. The lesson follows the seven element format. Element One – Identify Learning Goals Refer to the standards listed at the beginning of the lesson. Throughout the project, the following questions guide student learning:
• What are programmable logic controllers and how are they used in industrial applications?
• How can I define and evaluate logical statements? • How can I use mathematical modeling to solve real world problems? • What scientific principles can be applied to guide design and reach solutions?
Element Two – Pre-‐Assess Learning Students individually complete the Pre-‐Assessment, which serves as a formative assessment to identify common obstacles, misconceptions, and gaps in learning. The instructor may use the pre-‐assessment rubric to offer feedback aimed to elevate student understanding. Group students based on pre-‐assessment responses for team activities. Element Three – Embedded Contextual Examples Students model real-‐world problems in teams assigned homogeneously based on pre-‐assessment responses. Student teams complete a “gallery walk” to observe and offer feedback to each other. Instructor closes with plenary discussion. Element Four – Related Contextual Examples Students de-‐contextualize then re-‐contextualize to analyze and solve problem scenarios related to the original problem. Element Five – Traditional Examples Students encounter and solve problems as they would classically appear on standardized tests. Element Six – Students Demonstrate Understanding Students demonstrate understanding of key ideas through hands-‐on project based learning activities. Element Seven – Formal Post-‐Assessment Assessment items requiring mastery of standards addressed through project are completed by each student individually.
Element One: Identify Learning Goals Career and Technical Standards: MA 1870 Industrial Equipment Maintenance 1871.32 Explain the purpose of Programmable Logic Controllers (PLC) and their applications in industrial locations. 1871.33 Identify the component parts of a PLC system. 1871.34 Explain how to program and trouble shoot PLC’s. MA 1880 Industrial Technology 1808.1 Students will demonstrate a knowledge of the purpose, function, and application of programmable logic controllers (PLCs) & electronic controls. 1808.7 Students will define a logical truth table. 1808.9 Students will evaluate logical statements. 21st Century Skills: 21C.O.5-‐8.1.LS.2 Student interprets abstract visuals and creates products (e.g. digital storytelling) that reflect a growing understanding of visual language and require the effective use of tools (e.g. cropped photos, original charts and graphs, well-‐chosen images from databases, video clips). 21C.O.5-‐8.2.LS.1 Student engages in a critical thinking process that supports synthesis and conducts evaluations by applying comprehensive criteria. 21C.O.5-‐8.3.LS.2 Student is flexible in approach to solving problems and completing tasks, considers alternative methods, solutions and perspectives, abandons strategies that do not work, and reallocates time and resources as priorities change. 21C.O.5-‐8.3.LS.6 Student maintains focus on larger project goal, frames appropriate questions, reflects on possible courses of action and their likely consequences, develops and initiates a plan of action with appropriate smaller objectives and benchmarks, and submits the completed project when due. English Language Arts Standards ELA.6.10 Determine the meaning of words and phrases as they are used in an informational text, including figurative, connotative, and technical meanings. ELA.6.15 Integrate information presented in different media or formats (e.g., visually and/or quantitatively) and in words to develop a coherent understanding of a topic or issue. ELA.6.31 Interpret information presented in diverse media and formats (e.g., visually, quantitatively, and/or orally) and explain how it contributes to a topic, text, or issue under study.
Mathematics Standards 7.EE.B.4 Use variables to represent quantities in a real-‐world mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities. BF.A.1.A Determine an explicit expression, recursive process, or steps for calculation from a context. LE.B.5 Interpret parameters in a linear or exponential function in terms of a context. Science Standards S.6-‐8.L.4 Students will determine the meaning of symbols, key terms, and other domain-‐specific words and phrases as they are used in a specific scientific or technical context relevant to grades 6–8 texts and topics. S.6-‐8.L.7 Integrate quantitative or technical information expressed in words in a text with a version of that information expressed visually (e.g., in a flowchart, diagram, model, graph, or table). S.6-‐8.ETS.1 Define the criteria and constraints of a design problem with sufficient precision to ensure a successful solution, taking into account relevant scientific principles and potential impacts on people and the natural environment that may limit possible solutions.
Element Two: Pre-‐Assessment 1. Describe the scenario depicted by the logic ladder below:
2. Explain the process the flow chart below illustrates.
3. Create a truth table for each statement below: A. Bi-‐conditional statement: x = 12 iff x * 0.25 = 3 B. Converse: x * 0.25 = 3 iff x = 12 C. Inverse: x ≠ 12 iff x * 0.25 ≠ 3 D. Contrapositive: x * 0.25 ≠ 3 iff x ≠ 12
4. Describe the actions performed by a robot moving on two wheels given the script
below. AUX 1 represents the left wheel motor and AUX 2 represents the right wheel motor. Time is measured in one-‐tenth second increments.
5. Complete the logic table for p XOR q (p⨁q)
p q (p⨁q)
Element Two: Pre-‐Assessment Rubric Assessment Item Solution Common
Misconceptions Recommended Feedback
1. Describe the scenario depicted by the logic ladder.
If switch 1 is on and switch 2 is off, the irrigation system will turn on. If switch 2 is on and switch 1 is off, the irrigation system will turn on.
Students are unsure about symbolic representation. Students are confused about logic of process.
How do symbols represent actions? Which symbols can be used to represent “off” or “on?” What variables and symbols are encountered when you trace from input to output?
2. Explain the process the flow chart illustrates.
The programming interface is used to program conditions for automated actions. The CPU receives the program and exchanges information with the I/O section. Sensor input is received by the I/O section. The I/O section outputs action commands based on the program and information received from the sensors.
Student misunderstands direction of flow chart arrows. Student misunderstands function of one or more components.
Which arrows represent data being sent? Which represent data being received? Which components receive data, and from where? Which components send data, and where do they send it to? What context clues or known vocabulary can be used to infer the function of each component?
3. Create a truth table for each statement below:
A. Bi-‐conditional statement: x = 12 iff x * 0.25 = 3
B. Converse: x * 0.25 = 3 iff x = 12
C. Inverse: x ≠ 12 iff x * 0.25 ≠ 3
A, B, C, D. p q p ⟺
q T T T T F F F T F F F T
Student is unsure about notation. Student is unsure how to construct a truth table. Student incorrectly evaluates truth values.
What does the notation used in the problem mean? What are the column headings of a truth table? How many possibilities are there for the truth values of p and q? Write the statement for each truth value of p and q. When
D. Contrapositive: x * 0.25 ≠ 3 iff x ≠ 12
you read the statements as complete sentences, how can you determine whether each is true or false?
4. Describe the actions performed by a robot moving on two wheels given the script below. AUX 1 represents the left wheel motor and AUX 2 represents the right wheel motor.
1. Both wheel motors are on, therefore (2) the robot moves forward for 2.5 seconds (twenty-‐five one-‐tenth second increments).
3. Both wheel motors turn off, then (4) only the left wheel motor turns on for half a second.
4. The left wheel motor turns on causing the robot to turn right
5. The robot turns right for 0.5 seconds (five one-‐tenth second increments).
6. The left wheel motor turns off
7. Both wheel motors turn on
8. The robot moves forward for one second.
9. Both wheel motors turn off.
Student is unsure about what action occurs when one or both wheel motors are on. Student is confused about time increments
What happens when both wheel motors are on? What happens when the left wheel spins but the right wheel remains stationary? What happens when the right wheel spins but the left wheel remains stationary? What is 25 * 0.1? How many one-‐tenth second increments are in one second?
5. Complete the logic table for p XOR q (p⨁q) p q (p⨁
q)
p q (p⨁q)
T T F T F T F T T F F F
Student is unsure about meaning of notation Student is certain about meaning of logical statements
What does XOR mean? What does the symbol, ⨁, mean in the context of the logical statement? Is the statement true or false if both p and q are true? What if neither are true? What if exclusively one or the other is true?
Element Three: Embedded Contextual Examples Pair each card from set A to its best match from set B. Explain your reasoning and justify your selection.
Card Set A Card Set B
If (x,y) = (0,90) then turn 90o
If <Switch 1 AND NOT Switch 2> input, OR <Switch 2 AND NOT Switch 1> input
then output irrigation.
p q p → q T T T T F F F T T F F T
If the engine temperature is too hot, then the temperature gauge indicates
red. If the engine temperature is too hot, then the temperature gauge does not
indicate red. If the engine temperature is not too
hot, then it is possible the temperature gauge may indicate red.
If the engine temperature is not too hot, then the temperature gauge does
not indicate red.
If x = 8 then x * 0.75 = 6 If x * 0.75 = 6 then x = 8 If x ≠ 8 then x * 0.75 ≠ 6 If x * 0.75 ≠ 6 then x ≠ 8
Conditional Statement
Programmable Logic Controller
Conditional Statement If hypothesis (p) then conclusion (q)
Converse If q then p Inverse
If not p (~p) then not q (~q) Contrapositive If ~q then ~p
Create Your Own Create Your Own
Element Three: Embedded Contextual Examples -‐ Solutions Card Set A Card Set B
If (x,y) = (0,90) then turn 90o Conditional Statement
If <Switch 1 AND NOT Switch 2> input, OR <Switch 2 AND NOT Switch 1> input then output irrigation.
If the engine temperature is too hot, then the temperature gauge indicates red.
If the engine temperature is too hot, then the temperature gauge does not indicate
red. If the engine temperature is not too hot, then it is possible the temperature gauge
may indicate red. If the engine temperature is not too hot, then the temperature gauge does not
indicate red.
p q p → q T T T T F F F T T F F T
If x = 8 then x * 0.75 = 6 If x * 0.75 = 6 then x = 8 If x ≠ 8 then x * 0.75 ≠ 6 If x * 0.75 ≠ 6 then x ≠ 8
Conditional Statement If hypothesis (p) then conclusion (q)
Converse If q then p Inverse
If not p (~p) then not q (~q) Contrapositive If ~q then ~p
Programmable Logic Controller
Create Your Own Create Your Own
Element Four: Related Contextual Examples 1. Which script would program a robot to carry out the instructions below? Let Aux1 represent
the left wheel motor and Aux2 represent the right wheel motor of the robot. Explain your reasoning in complete sentences.
Instructions: The robot should move forward for 5 seconds, then make a right-‐hand (clockwise) turn for 4.5s. Next, the robot should move forward for 10s then perform a left-‐hand (counterclockwise) turn for 4.5s. It should finally drive forward 5s.
A. B. C. D. 2. Write the converse, inverse, and contrapositive of the conditional statement
If the soil moisture has dropped below the lower control limit then the irrigation system turns on.
3. Use logic symbols from the table below along with the variables p and q to represent each statement.
Logic Connective Operator Symbol Conditional if then à Biconditional if and only if ⇔ Negation NOT ~ Conjunction AND Λ Disjunction OR (either p, q, or both) V Exclusive or either…or but not both XOR ⊕
A. If I pay attention (p), then I achieve more (q). B. The robot turns (p) if and only if exactly one wheel is moving (q). C. The switch is open (~p) and the light is off (~q). D. Either the switch is closed or the button is depressed. E. Either the material is too hot or the material is too cold.
4. Complete the logic table for p AND q (p∧q)
p q (p∧q)
5. Interpret the logic ladder below and describe the process in complete sentences using logic
notation.
Element Four: Related Contextual Examples -‐ Solutions 1. Which script would program a robot to carry out the instructions below? Let Aux1
represent the left wheel and Aux2 represent the right wheel of the robot. Explain your reasoning in complete sentences.
Instructions: The robot should move forward for 5 seconds, then make a right-‐hand (clockwise) turn for 4.5s. Next, the robot should move forward for 10s then perform a left-‐hand (counterclockwise) turn for 4.5s. It should finally drive forward 5s.
B. B. C. D. B is the correct script for the instruction set given. Both wheel motors turn on for fifty one-‐tenth second increments, or 5 seconds. While both wheels are turning, the robot is moving forward. Both wheels stop then the left wheel turns for 4.5 seconds. While only the left wheel is spinning, the robot is turning right (clockwise). The left wheel motor turns off then both wheel motors turn on for ten seconds, meaning the robot moves forward for ten seconds. Next, both wheels turn off and only the right wheel spins for 4.5 seconds. While only the right wheel is spinning, the robot is turning left (counterclockwise). The right wheel turns off then both wheel motors turn on causing the robot to move forward for 5 seconds. The script given in A produces mirrored results; for example, the robot would be turning counterclockwise when it should be turning clockwise. Neither script C nor D would cause the robot to carry out the correct instruction set.
2. Write the converse, inverse, and contrapositive of the conditional statement If the soil moisture has dropped below the lower control limit then the irrigation system turns on. Converse: If the irrigation system turns on then the soil moisture has dropped below the lower control limit.
Inverse: If the soil moisture has not dropped below the lower control limit then the irrigation system does not turn on. Contrapositive: If the irrigation system does not turn on then the soil moisture has not dropped below the lower control limit.
3. Use logic symbols from the table below along with the variables p and q to represent each statement.
Logic Connective Operator Symbol Conditional if then à Biconditional if and only if ⇔ Negation NOT ~ Conjunction AND Λ Disjunction OR (either p, q, or both) V Exclusive or either…or but not both ⊕
A. If I pay attention (p) then I achieve more (q). B. The robot turns (p) if and only if exactly one wheel is moving (q). C. The switch is open (~p) and the light is off (~q). D. Either the switch is closed or the button is depressed. E. Either the material is too hot or the material is too cold. A. Paying attention à higher achievement. p à q B. Robot turns ⇔ exactly one wheel moving. p ⇔ q. C. Open switch ∧ light off. ~p ⋀ ~q. D. The switch is closed, the button is depressed, or both. p ∨ q. E. The material is too hot or the material is too cold, but it is not both too hot and too cold.
p ⊕ q.
4. Complete the logic table for p AND q (p⋀q)
p q (p∧q) T T T T F F F T F F F F
5. Interpret the logic ladder below and describe the process in complete sentences using
logic notation.
If either input A or input B is true, but not both, then output. The scenario can be represented using logic notation as A ⊕ B → Output.
Element Five: Traditional Examples
1. Write the given statement as a conditional statement, then write its converse, inverse, and contrapositive. Strong effort (p) is directly related to high achievement (q).
2. Create a flow chart for the geometric proof shown below:
Statements Reasons ABCD is a rectangle Given Line segment AB is congruent to line segment BD. 𝐴𝐵 ≅ 𝐶𝐷
Definition of a rectangle
Line segment AD is congruent to line segment BC. 𝐴𝐷 ≅ 𝐵𝐶
Definition of a rectangle
𝑚∠𝐴 = 𝑚∠𝐵 = 𝑚∠𝐶 = 𝑚∠𝐷 = 90° Definition of a rectangle ∠𝐴 ≅ ∠𝐵 ≅ ∠𝐶 ≅ ∠𝐷 Definition of congruence △ 𝐴𝐵𝐷 ≅ △ 𝐶𝐷𝐴 Side – Angle – Side (SAS) postulate
3. Create a truth table for the statement: An angle is acute if and only if (iff) the angle measure is less than 90°.
4. Write four of your own conditional logic statements.
5. Create a truth table for one of your conditional statements.
Element Five: Traditional Examples-‐ Solutions 1. Write the given statement as a conditional statement, then write its converse, inverse,
and contrapositive. Strong effort (p) is directly related to high achievement (q). Answers may vary. For example, Conditional statement: If strong effort is applied, then high achievement will be accomplished. Converse: If high achievement is accomplished, then strong effort has been applied. Inverse: If strong effort is not applied, then high achievement will not be accomplished. Contrapositive: If high achievement is not accomplished, then strong effort has not been applied.
2. Create a flow chart for the geometric proof shown below:
Statements Reasons ABCD is a rectangle Given Line segment AB is congruent to line segment BD. 𝐴𝐵 ≅ 𝐶𝐷
Definition of a rectangle
Line segment AD is congruent to line segment BC. 𝐴𝐷 ≅ 𝐵𝐶
Definition of a rectangle
𝑚∠𝐴 = 𝑚∠𝐵 = 𝑚∠𝐶 = 𝑚∠𝐷 = 90° Definition of a rectangle ∠𝐴 ≅ ∠𝐵 ≅ ∠𝐶 ≅ ∠𝐷 Definition of congruence △ 𝐴𝐵𝐷 ≅ △ 𝐶𝐷𝐴 Side – Angle – Side (SAS) postulate
3. Create a truth table for the statement: An angle is acute if and only if (iff) the angle
measure is less than 90°. p q p ⟺ q T T T T F F F T F F F T
4. Write four of your own conditional logic statements.
Answers will vary. A conditional statement has the form if (condition) then (conclusion). Examples: If I oversleep then I miss the bus. If I forget my homework, then I receive a zero. If I apply my best effort, then I will perform better. If I take advantage of opportunities to do well, then I will have more opportunities. If I meet challenges with a positive attitude, then I have a higher chance of persevering.
5. Create a truth table for one of your conditional statements. Answers will vary. For example, If I oversleep, then I will miss the bus.
p q p → q T T T T F F F T T F F T
Element Six: Demonstrate Understanding Students complete the STEMWorks Virtual Robotics Lab online at http://stem-‐works.com/subjects/1-‐robotics/activities/26 alternate link: http://www.mind.ilstu.edu/curriculum/virtual_robotics_lab/lab.html
Element Seven: Post-‐Assessment
1. Compose a script that will cause a two-‐wheeled robot to move forward for ten seconds, turn right for five seconds, move forward again for fifteen seconds, then turn left for 2.5 seconds. Finally, the robot should stop. Designate the left wheel motor as AUX 1 and the right wheel motor as AUX 2. Compute time in one-‐tenth increments of a second.
2. Explain the flow chart given below in a paragraph summary.
3. Complete the truth table for p OR q (p ∨ q).
p q p ∨ q
4. Write a conditional statement based on the statement given, then write its converse,
inverse, and contrapositive. Meeting challenges with a positive attitude increases the chance of perseverance.
5. Create a logic ladder for the process described. If either condition or both conditions are met, then unlock the mechanism.
Element Seven: Post-‐Assessment – Answer Key 1. Compose a script that will cause a two-‐wheeled robot to move forward for ten seconds,
turn right for five seconds, move forward again for fifteen seconds, then turn left for 2.5 seconds. Finally, the robot should stop. Designate the left wheel motor as AUX 1 and the right wheel motor as AUX 2. Compute time in one-‐tenth increments of a second. 1. AUX 1, 2 ON 2. Wait 100 3. AUX 1, 2 OFF 4. AUX 1 ON 5. WAIT 50 6. AUX 1 OFF 7. AUX 1, 2 ON 8. WAIT 150 9. AUX 2 ON 10. WAIT 25 11. AUX 2 OFF
2. Explain the flow chart given below in a paragraph summary.
It is given that ABCD is a rectangle. By the definition of a rectangle, opposite sides are congruent and the interior angles are right angles. All right angles are congruent. It is proven that interior triangles created by the diagonals are congruent by the Side-‐Angle-‐Side Postulate.
3. Complete the truth table for p OR q (p ∨ q).
p q p ∨ q T T T T F T F T T F F F
4. Write a conditional statement based on the statement given, then write its converse,
inverse, and contrapositive. Meeting challenges with a positive attitude increases the chance of perseverance. Conditional statement: If challenges are met with a positive attitude, then the chance of perseverance is increased. Converse: If the chance of perseverance is increased, challenges are met with a positive attitude. Inverse: If challenges are not met with a positive attitude, then the chance of perseverance is not increased. Contrapositive: If the chance of perseverance is not increased, then challenges are not met with a positive attitude.
5. Create a logic ladder for the process described. If either condition or both conditions are met, then unlock the mechanism.