Edited By. Drs. Babette M. Benken and Laura Henriques. with. Ms. Andrea Johnson. California State University, Long Beach

Transcription

1 Expanding Learning Time in After-School and Summer Enrichment Programs: Science, Math, and Integrated STEM Activities for Middle and High School Students Edited By Drs. Babette M. Benken and Laura Henriques with Ms. Andrea Johnson California State University, Long Beach

2 Expanding Learning Time in After- School and Summer Enrichment Programs: Science, Math, and Integrated STEM Activities for Middle and High School Students

3 Expanding Learning Time in After-School and Summer Enrichment Programs: Science, Math, and Integrated STEM Activities for Middle and High School Students Spring 2011 saw large numbers of elementary teachers being laid off due to state and district budget cuts. In an effort to help experienced teachers gain skills and new certifications, a program was developed by California State University, Long Beach faculty to help laid-off elementary teachers earn a middle school mathematics or science teaching credential. Generously funded by the S.D. Bechtel Jr. Foundation, the David and Lucille Packard Foundation, and the Southbay Workforce Initiative Board, this project (Fall 2011 Summer 2012) provided content courses and a secondary mathematics or science methodology course for approximately 50 laid-off elementary teachers from the Long Beach Unified School District (Southern California). As part of this program, teacher-participants developed and field-tested math, science, and integrated STEM lessons. While most of these lessons would work well in a traditional classroom setting, participants developed them for the after-school setting. Most of these lessons explore content relative to, and could be adapted for, multiple grades in middle school through high school. Post development, the activities were taught to middle school children in afterschool settings in Norwalk-La Mirada, CA. After field-testing, the activities were further revised numerous times. In some cases, the lessons were implemented yet again. In the final step of the process, we (the three editors for this project) made further additions, edits and revisions to the activities. This compilation of STEM activities is the result of that effort. Each activity has a 1-2 page cover sheet, which includes an overview of the activity, suggested grade ranges, approximate time to complete the activity, relevant mathematics and/or science standards, lesson objectives, a materials list, safety considerations (if any), and the list of contributing authors. In some cases multiple teacher-participants taught an early version of the lesson and provided lesson plans and field testing notes. For each activity, there is a detailed summary for implementation that includes assumed prior knowledge, relevant facilitator questions, and recommendations for how to assess understanding and scaffold students independent practice. Lessons also include suggestions for differentiation and/or extension activities. Where appropriate, we have added additional STEM connections so that you and your students are able to see how science, technology, engineering and mathematics intertwine. We hope that you find these lessons useful, engaging and thought provoking. Drs. Babette Benken & Laura Henriques, Co-Editors & Directors of the FLM/FLGS Programs CSU Long Beach

4 1. Use Your Shoe! Table of Contents Mean, median, and mode 2. Transformations and Illusions Rotations, translations, and reflections of figures 3. The Soundinator Sound waves 4. Drop of Doom! Quantitative and graphical representations of functions Velocity 5. Reflections on Light Properties and reflections of light Geometric angle constructions 6. How Much Water Fits on a Penny? Properties of water Mean, median, and mode 7. Snack Time! Representing data in bar graphs, pie charts, and box and whisker plots 8. Stretching It Hooke s Law Linear relationships 9. Run! Distance versus time graphs Rates of change and slope 10. Peas in a Pod Collecting and analyzing data Scatterplot graphs Modeling with functions 11. Happy Birthday to You! Number patterns and number sense

5 Use Your Shoe! 1 Each student will contribute to the class shoe size data, collect data for the entire class, and analyze the data by determining the mean, median and mode. Students will use their analyses to make inferences about the average shoe size of larger populations. Suggested Grade Range: 6-8 Approximate Time: 2 hours State of California Content Standards: Mathematics Content Standards Grade 8: Probability and Statistics 10.0 Students know the definitions of the mean, median, and mode of distribution of data and can compute each of them in particular situations. Science Content Standards Grades 6-8: Investigation and Experimentation 7. b. Students will use appropriate tools and technology to perform tests, collect data, and display data. 9. b. Students will evaluate the reproducibility of data. Relevant National Content Standards: Mathematics Common Core State Standard: 7.SP 2. Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. Gauge how far off the estimate or prediction might be. Lesson Content Objectives: Distinguish among and calculate the mean, median and mode of a data set. Collect and organize data from the group. Calculate the mean, median and mode of the group data. Draw inferences about a population after examining the group s sample. Practice calculating the mean, median and mode for data from different contexts. Materials Needed: One copy per student of the Mean, Median, and Mode notes sheet, Use Your Shoe! activity sheet, and Independent Practice sheet (included) One shoe card per student to record their shoe size (included) Tape 1 An early version of this lesson was adapted and field-tested by Alex Chao and Thy Pech, participants in the California State University, Long Beach Foundational Level Mathematics/General Science Credential Program. Use Your Shoe! 1-1

6 Summary of Lesson Sequence Provide the Warm Up activity sheet (included) for students to practice using the correct order of operations to evaluate multi-step expressions. Introduce the lesson by allowing students to find out the shoe size of five other students in the class and discussing how that data could be used to make a guess about the shoe size for the whole class. Lead students through the Mean, Median, and Mode: Guided Notes (included) by modeling how to find these measures of central tendency. Allow all students to contribute their shoe sizes by taping each student s shoe card to the board on a frequency chart. Guide students through their own practice of finding the mean, median and mode for the entire class using the data on the board and the Use Your Shoe!: Guided Practice activity sheet (included). Check for students understanding by asking the key questions provided while students are working on the guided practice. Close the lesson with a discussion of their findings and how their findings could be extended to larger populations. Provide the Independent Practice sheet for students to practice finding the mean, median and mode for data in different contexts. Assumed Prior Knowledge Prior to this lesson students should be able to use the correct order of operations to evaluate multi-step expressions. Classroom Set Up Students will be asked to participate in discussions and work in small groups for portions of this lesson. Lesson Description Introduction Provide students with the Warm Up activity, allowing them time to practice using the correct order of operations to evaluate multi-step expressions. Students should be able to complete the warm up on their own. Provide every student with a pre-cut square with a picture of a shoe on it. Students should write their shoe size on the square. Allow students a couple of minutes to move around the room and share shoe sizes with four other students, (each student write down their own size and the shoe size of four peers so all will have 5 data points). They can record the other students shoe sizes on the back of their own shoe card. After about two minutes, ask students to find their seats again to begin a brief discussion. Ask: Based on the data you collected from four other students and yourself, what do you think is the most common shoe size in the classroom? What shoe size do you think is right in the middle of the biggest and smallest? Use Your Shoe! 1-2

7 Can you hypothesize from the five shoe sizes that you know what the average shoe size for the whole class might be? Tell students that they will be learning how to analyze data using three measures of central tendency: mean, median and mode. Make sure your discussion arrives at classroom understanding of working definitions for these terms. Input and Model Provide students with the Mean, Median, and Mode: Guided Notes sheet and review the definitions for mean, median and mode: Mean: average Median: middle Mode: most The mean is the sum of all values divided by the number of values in the data set. The median is the middle value of the data set when the data set is ordered least to greatest. The mode is the most frequently occurring value of the data set. Demonstrate for students the method for calculating the mean by adding up all the shoe sizes of the sample set and dividing by the number of students in the sample set. Encourage students to find the mean of their sample of five students shoe sizes that they collected. Allow students to notice that their answers may vary because they collected data from different students. This can lead to a conversation about larger sample sizes being more representative of the entire group. Model for students the method for finding the median of the sample data by organizing the sample set as a list from least to greatest and finding the middle shoe size. Encourage students to find the median of their sample of five students shoe sizes that they collected. Discuss how they would find the median if their data had an even number of data points (e.g., 6 shoe sizes). Model for students the method for finding the mode of the sample data by observing which shoe size occurs most frequently in the sample set. Explain that data sets might not have a mode or might have more than one mode; ask volunteers to provide examples of data sets that have no or multiple modes. Students should recognize whether there is a mode for the sample of five shoe sizes and identify the mode if there is one. Show students how to create a frequency chart for the sample set of data. Model for students the second sample set of data for the number of text messages sent. Use Your Shoe! 1-3

8 Guide Students Through Their Practice Draw an outline of a frequency chart on the board for the whole class shoe size data; an example frequency chart outline is on the students Guided Notes. Allow students to come to the board and tape their shoe card above the number value for their shoe size, vertically stacking repeated values to create a bar graph. Students should use the data on the board to complete the Use Your Shoe!: Guided Practice activity sheet. Students may work together to complete the guided practice. Check for Understanding Check for students understanding while they are working on the guided practice by asking the following key questions: Are the mean, median and mode for the whole class different from the mean median and mode you found from the set of five that you collected? Why might they be the same or different? Each measure of central tendency tells us something different about the data. What does the mean tell us? The median? Mode? Which measure of central tendency do you think is helpful for understanding this particular shoe data? Why? Independent Practice Provide students with the Independent Practice sheet to complete on their own. Closure To close the lesson, allow students to share their findings about the mean, median, and mode for the whole class. Ask students: Was the sample of five shoe sizes helpful for guessing the mean, median, and mode for the whole class? Do you think we could use our findings for the whole class to guess the shoe size of a larger population? [Other classes of the same age, the whole school, all children of the same age around the world, all Americans, etc.] To what extent is the data we collected today reproducible by another class here at our school? Use Your Shoe! 1-4

9 Suggestions for Differentiation and Extension This activity may be extended by allowing students to develop their own survey, collect data outside of the classroom, and summarize and analyze the data using the measures of central tendency they learned in this lesson. If working with the same group of students over a period of time, students could be asked to explain what they learned in the days following the activity in a journal, or as a warmup. Students may conduct research using the Internet to determine whether the shoe size data for the classroom is representative of larger populations. Depending on the class, students heights may be collected instead of shoe size to provide more variability. Students may use graphing calculators to calculate the mean, median and mode for the shoe size data, as well as for the independent practice problems. Students may also use graphing calculators to create frequency tables for the sets of data. Use Your Shoe! 1-5

10 Use Your Shoe! Warm Up 1. Simplify: = 6 2. Simplify: 2(4) + 4(3) + 2(7) = 3. (4 x 6) + (2 x 3) = x x x 2 = = Use Your Shoe! 1-6

11 Mean, Median, and Mode Guided Notes 1. Create a frequency chart for this sample set of shoes sizes: 6, 10, 9, 8, 8, 6, 12, 14, 9, and 8. Student # Shoe Size A 6 B 10 C 9 D 8 E 8 F 6 G 12 H 14 I 9 J 8 Find: Shoe Size A. Mean: add all shoe sizes and divide by total number of students. _ Sum of all shoe sizes Total number of students = = = B. Median: write all shoe sizes in order from least to greatest and find the middle shoe size. If there are two middle shoe sizes, add those two sizes and divide by = C. Mode: find the shoe size that occurs the most. Use Your Shoe! 1-7

12 2. Second sample: Last week, Jane sent 34 text messages, John sent 25, Sofia sent 41, Priscilla sent 12, Cisco sent 33, Fred sent 24, and Riley sent 25. Name # Of messages sent Total Number of Texts Find: A. Mean: add all text messages and divide by the number of students. Total text messages Total number of students = = = B. Median: write all numbers of messages sent from least amount to greatest and find the middle data point. If there are two data points in the middle, add them and divide by = C. Mode: find the amount of messages sent that occurs most frequently. Use Your Shoe! 1-8

13 Use Your Shoe! Guided Practice Review vocabulary: mean, median, mode. A. Mean is the sum of all values divided by the number of values in the data set. Another term used for mean is average. B. Median is the middle value in the data set. If there are two middle values, add them and divide by two. C. Mode is the most frequently occurring value data point of the set. Student Shoe Size Gender Shoe Size Mean (add all shoe sizes / total # of students) = Add all shoe sizes = = Total # of students Median (write all shoe sizes in order from least to greatest, find the middle shoe size. If there are two middle shoe sizes, add them up and divide by 2) = Mode (find the shoe size that occurs most frequently) = Use Your Shoe! 1-9

14 Mean, Median, and Mode Independent Practice Practice finding the mean of a data set by calculating the Grade Point Average (GPA) using the sample report card provided. First read the Report Card Information, then use the data from the Sample Student Report Card to find the GPA. Report Card Information Honor Roll Honor Roll is determined by grade point average (GPA). Honor Roll: High Honor Roll: Highest Honors: Grades (with Numerical Value) High schools use the table below for determining Grade Point Averages and Rank in Class: A+ = 4.3 B+ = 3.3 C+ = 2.3 A = 4.0 B = 3.0 C = 2.0 D = 1.0 A- = 3.7 B- = 2.7 C- = 1.7 F = 0.0 Grade points corresponding to letter grades for students in AP Courses (who take the AP Exam) are 0.3 units higher. ***For example a student in an AP Course would receive 3.6 points for a B+ instead of 3.3. An F for these students remains at 0 points. ***When calculating the GPA, the letter grade's numerical value is multiplied by the units to determine the weighted grade points. These are totaled and divided by the total units in that marking period. The following example illustrates the calculation of a GPA for a student with no AP Courses using the grade point table above: Use Your Shoe! 1-10

15 Sample Student Report Card Units Course Letter Grade Numerical Value Grade Points 1 A.P. English B Math A Science A Social Studies C Computer D 1 1 *.50 PE P Orchestra A 4 2 *PE is not counted in the divisor because of the Pass/Fail status of the course. Therefore, you DO NOT include it in your calculations of GPA. **All courses that are graded pass/fail are not considered in GPA (ex. College Seminar, Teaching Assistant, etc.). 1. Use the sample report card above to calculate the weighted GPA. 2. Based on the GPA you calculated, would a student with this report card earn any honors? Use Your Shoe! 1-11

16 Use Your Shoe! 1-12

17 Transformations and Illusions 1 Students will have the opportunity to explore geometric transformations in a piece of M.C. Escher s artwork after learning to identify and draw reflections, translations, and rotations of figures. Suggested Grade Range: 7-12 Approximate Time: 1 hour State of California Content Standards: Mathematics Content Standards Grades 8-12: Geometry 2.2 Know the effect of rigid motion on figures in the coordinate plane and space, including rotations, translations, and reflections. Relevant National Content Standards: Mathematics Common Core State Standard: 8.G 1. Verify experimentally the properties of rotations, reflections, and translations. Lesson Content Objectives: Identify and draw rotations, reflections and translations of angles, polygons, and other figures. Investigate rotations, reflections and translations of figures in a drawing by M.C. Escher. Describe transformations using correct mathematical vocabulary. Materials Needed: Lined or white paper Rulers Three colored pencils per student or pair One copy per student of M.C. Escher s Angels and Devils drawing (included) One copy per student of the Transformation Practice sheet (included) Mirrors (optional) One copy per student of the Transformation STEM Extension (included) 1 An early version of this lesson was adapted and field-tested by Monica D. Williams-Davis and Layla Nourbakhsh, participants in the California State University, Long Beach Foundational Level Mathematics/General Science Credential Program. Transformations and Illusions 2-1

18 Summary of Lesson Sequence Introduce the lesson by providing a copy of M.C. Escher s Angels and Devils drawing (included) for students to view and discuss. Lead students through compiling Cornell Notes on the vocabulary for the lesson including transformation, reflection, translation, and rotation. Model writing letters and their transformations, providing an example of each. Guide students through their own practice drawing transformations of letters. Check for students understanding by allowing students to identify and color figures from Escher s Angels and Devils that are reflections, translations, and rotations. Allow groups to develop a way to teach the rest of the class how to reflect, rotate, and translate a letter or polygon. To close the lesson, students may discuss in small groups examples of transformations that may be observed inside the classroom or outside, then report to the whole class. For independent practice, students will complete an included activity sheet. Provide the optional Transformation Extension activity sheet and allow students to find designs where transformations of shapes are used. Assumed Prior Knowledge Prior to this lesson students should know how to identify, draw, and measure shapes and angles, and be familiar with the quadrants of the coordinate plane. Classroom Set Up Students will be asked to work in small groups for portions of this lesson. Lesson Description Introduction Provide students with a copy of M.C. Escher s Angels and Devils (included) and allow students to view and discuss the piece. Generate discussion by using probing questions: What elements of the piece are repeated to form a pattern? Are there angles that are congruent? What is different about the angles (direction and position)? Input Lead students through taking notes of the key vocabulary for the lesson: transformation, reflection, translation, and rotation: Transformation: a change. A transformation changes the position of a shape on a coordinate plane. The shape moves from one place to another when it is transformed. Transformations and Illusions 2-2

19 Reflection: a flip. Translation: a slide. A reflection takes place when a shape is flipped across a line and faces the opposite direction. A shape and its reflection are mirror images of each other. An object is translated when it moves in one direction from the starting point to the end. Demonstrate a reflection using an object in the classroom and a mirror. Rotation: a turn. An object that is rotated turns on a point to face another direction like the hand turning on the face of a clock. Demonstrate a translation by sliding an object in the classroom across a table. Demonstrate a rotation with an object in the classroom or by directing students attention to a clock with rotating arms. Model Demonstrate a reflection, translation, and rotation of the letter E. Provide another example if necessary without taking from students opportunity to explore transformations during the guided practice. Be careful to choose a letter, which has some asymmetry or students will not easily see that it has been changed. Guide Students Through Their Practice Move around the classroom checking students progress and allowing students to assist each other after assigning the following transformations: Reflect: p, m, T, nap, and their name. Translate: S, V, e, and a square. Rotate: L, w, C, and a triangle. Check for Understanding Using their copy of M.C. Escher s Angels and Devils, ask students to color one of the figures in the picture (an angel or devil). Have them find a reflection of that shape and color it in as well. Have students hold up their papers to check their understanding. Transformations and Illusions 2-3

20 Ask students to use a different color to indicate another figure and its translation. Using a third color, ask students to color a third figure and its rotation. Student Team Teach Allow students time to discuss a method for teaching how to perform a transformation on a polygon. Assign a polygon and a transformation to each group, and after allowing time to prepare, call on representatives of each group to teach the class how to perform the transformation on their polygon. Independent Practice Provide students with the Transformation Practice activity sheet (included) to complete independently. Closure Allow students to discuss in small groups examples of transformations that they observed in the classroom or outside of the classroom and share with the whole class. Students may be permitted to show with their own body a reflection, translation or rotation. Ask students: Can you think of a situation where the result of a reflection may be the same as the result of a rotation? (A 180 rotation yields the same result as a reflection; you can point out that the term 180 is used in skateboarding, etc.). Suggestions for Differentiation and Extension Explain to students that symmetry is commonly found in engineering and architectural designs and allow students to share examples of designs they are familiar with that have an element of symmetry. Provide students with the Transformation Extension activity sheet (included) so that they may explore symmetry in designs on their own. Encourage students to use a digital camera or cellular phone to take a photograph of a symmetric design they encounter so that they may share it in class. Transformations and Illusions 2-4

21 Transformation Practice 1. A translation... flips a shape slides a shape turns a shape 2. A reflection... flips a shape slides a shape turns a shape 3. What rotation would turn the capital letter Z into a capital N? 360 degrees 90 degrees 180 degrees 4. A triangle on a grid is rotated 90 degrees about the centre of the grid. The distance between the tip of the triangle and the centre of the grid... increases stays the same decreases 5. A drawing of a stick-man is reflected in a mirror line. The eye of the reflected man is... the same distance from the mirror line as the original shape. further from the mirror line than the original shape. nearer the mirror line than the original shape. 6. A dot at position (4,3) on a grid is translated four squares to the right. What are its new coordinates? (0,3) (4,3) (8,3) 7. A dot at position (0,0) on a grid is translated one square to the right and two squares up. What are its new coordinates? (0,0) (2,1) (1,2) 8. Which of these techniques can transform the letter b into the letter d? Reflection Rotation Translation 9. Which of these techniques can NOT transform the letter M into the letter W? Reflection Rotation Translation Transformations and Illusions 2-5

22 Transformation Extension Engineers, architects, and inventors often design objects that make use of transformed shapes. Find a building, a machine, or any other man made object for which you can identify a transformed shape in the design. 1. If possible, take a digital photograph of the object you found so that you may refer to it to answer the next questions and so that you may share your findings with the class. 2. Draw a simple diagram of the object you found. Label the shape and its transformation or transformations. 3. Explain how the transformed shape contributes to the design of the object. Transformations and Illusions 2-6

23 The Soundinator 1 Students construct an apparatus to make sound using plastic cups, string, and hangers and investigate what is required to experience sound. Students will use the string to understand the relationship between the length of the string and the frequency of the sound. Suggested Grade Range: 6-8 Approximate Time: 1 hour State of California Content Standards: Science Content Standards Grade 6: Physical Sciences Students know energy can be carried from one place to another by heat flow or by waves, including water, light and sound waves, or by moving objects. Science Content Standards Grade 7: Investigation and Experimentation Scientific progress is made by asking meaningful questions and conducting careful investigations. As a basis for understanding this concept and addressing the content in the other three strands, students should develop their own questions and perform investigations. Students will: d. Construct scale models, maps, and appropriately labeled diagrams to communicate scientific knowledge (e.g., motion of Earth s plates and cell structure). Science Content Standards High School: Physics 4. Waves have characteristic properties that do not depend on the type of wave. As a basis for understanding this concept: a. Students know waves carry energy from one place to another. b. Students know how to identify transverse and longitudinal waves in mechanical media, such as springs and ropes, and on the earth (seismic waves). d. Students know sound is a longitudinal wave whose speed depends on the properties of the medium in which it propagates. Relevant National Content Standards: Next Generation Science Standards: Middle School Physical Science MS-PS4-1. Use mathematical representations to describe a simple model for waves that includes how the amplitude of a wave is related to the energy in a wave. [Clarification Statement: Emphasis is on describing waves with both qualitative and quantitative thinking.] [Assessment Boundary: Assessment does not include electromagnetic waves and is limited to standard repeating waves.] 1 An early version of this lesson was adapted and field-tested by Emily Sanders, Steven Richardt, Diane Jackson, Krissy Cuevas, Janie Oetken, Mireya Valenzuela, and Angela Lytle, participants in the California State University, Long Beach Foundational Level Mathematics/General Science Credential Program. The Soundinator 3-1

24 MS-PS4-2. Develop and use a model to describe that waves are reflected, absorbed, or transmitted through various materials. [Clarification Statement: Emphasis is on both light and mechanical waves. Examples of models could include drawings, simulations, and written descriptions.] [Assessment Boundary: Assessment is limited to qualitative applications pertaining to light and mechanical waves.] Lesson Content Objectives: Construct a device to experiment with creating and experiencing sound, called a Soundinator. Investigate and understand the necessary components for experiencing sound including the vibrating source, the medium, and the receiver. Draw and appropriately label a diagram of a Soundinator to indicate the vibrating source, the medium, and the receiver. Materials Needed: 1-2 feet of string per pair of students One metal hanger per pair of students Wire cutters (one or two per class) Optional o One plastic cup per pair of students with a hole punched in the bottom (do not use hard plastic cups because there needs to be some give in order to punch a hole without cracking the cup) o One paper clip per pair of students The Soundinator 3-2

25 Summary of Lesson Sequence Introduce the lesson by leading students in a discussion about their experience with sound. Guide students through building and experimenting with their sound experience apparatus, the Soundinator. Check for students understanding by asking the key questions provided while students are experimenting with the Soundinator. Allow students to practice their understanding of the three necessary components of sound experience by drawing and labeling a model of the Soundinator. To close the lesson, have students verbally describe other devices for which they can distinguish the vibrating source, medium, and receiver. Assumed Prior Knowledge Prior to this lesson students should have a basic understanding that energy can be carried from one place to another by sound waves. Classroom Set Up Students should work in groups of two or three to construct and experiment with their Soundinator. Lesson Description Introduction Lead students in a discussion of their different experiences with sound by asking: What experiences or situations have you been in when sound was different than normal? Under what contexts is sound decreased? Have you ever heard sounds under water? How is that different than hearing sounds in the air? How might we make different sounds from the same object? Does anyone in the class play an instrument? How do you create different sounds with that instrument? When students have had an opportunity to share their experiences and knowledge about sound, tell them: Today we will determine the necessary components to experience sound and we will be constructing a device with those components to experiment with. The Soundinator 3-3

26 Input and Model Demonstrate for students how to construct the Soundinator. There are two different versions of the Soundinator. Version 1 Version 2 1. Each pair needs a cup, a piece of string, a paper clip, and a metal hanger. 2. Tie one end of the string to the paper clip. 3. Thread the other end of the string through the hole punched in the cup so that the paper clip is inside of the cup with the string coming out through the bottom of the cup. 4. Tie the metal hanger to the string or simply hang it over the string while holding the other end of the string. 5. Two students may listen at the same time if a second cup is connected to the other end of the string as shown. 6. Tap the hanger and listen! 1. Each pair needs a cup, a piece of string and a metal hanger. 2. Tie the metal hanger to the string. 3. Hold the string up to your ear and press against the bone outside your ear. 4. Tap the hanger while holding the string to your ear. Note: the hanger does not need to be cut/bent as shown in the picture above. It could be a complete hanger connected to two cups. Be sure to use a metal hanger (not plastic or one with a cardboard cover). Guide Students Through Their Practice Explain that each student will have the opportunity to be a listener and also a sound producer. Allow students to choose who will be first as the listener. Have students hold the cup so that the hanger is dangling from the cup in mid-air. Next, tap the hanger and observe what kinds of sound they hear, including whether the sound is loud or soft. Next, the listener should stand up and hold the cup to their ear so that the string and hanger may hang freely without touching anything. The producer will then produce sound by gently tapping the hanger with their pencil on different places. The listener should not tell the producer what they hear. Students should switch roles after some time. After both students have had a chance to the listener, have them remove the cup and hold the string with the hanger attached to their ear or jawbone. The producer should gently tap the hanger. Students should switch roles so both get a chance to listen. The Soundinator 3-4

27 Check for Understanding Check for students understanding while they are experimenting by asking the following key questions: What parts of your Soundinator help you to experience sounds when you are the listener? What was the purpose of the hanger? What was the purpose of the string? What was the purpose of the cup (if using version 1)? Do you need all the different parts of the Soundinator to experience sounds? What else do you need to experience sound? You need a vibrating source, a medium, and a receiver to experience sound. Can you identify these components as parts of your Soundinator? When students have identified the three components necessary for experiencing sound, and are experiencing sounds with the string, ask the following key questions: Did it sound different if you hit different parts of the hanger? Did it sound differently with and without the cup? Do you think it would sound differently if you had a short piece of string versus a long piece of string? Do you think it would sound differently if you had a smaller or larger hanger, or only part of the hanger? Students should recognize that the size of the hanger makes a difference in terms of what they hear. The vibrating hanger is the source. (If students have not figured this out, encourage them to try listening to a whole hanger versus a hanger piece that has been cut off. You will need wire cutters to do this investigation.) Students should recognize that the length of string does not make a difference in terms of what they hear. The string is the medium that carries or transmits the wave. Independent Practice After students have had the opportunity to use the Soundinator and have verbally identified the vibrating source, medium and receiver components, instruct them to create a drawing of the Soundinator. Students should label their drawing with the components of the device and their functions (students will need to also draw a receiver, their ear, but might need explicit direction to do so). The Soundinator 3-5

28 Closure To close the lesson, challenge students to think of other sound experiencing devices for which they can identify the vibrating source, medium, or receiver. It might be worth revisiting the pre-lesson questions to see if students are able to apply their understanding of sound. At this point, ask students about sound in space (or a vacuum). Can astronauts hear sounds in space? What is different in space than on Earth that changes how we experience sound? What does this tell us about the accuracy of movies or television shows that take place in outer space? (It is quiet in space. There is no atmosphere, so no medium to vibrate and carry the sound.) Suggestions for Differentiation and Extension Sound Bite: Have students slide a straw over a pencil or wooden dowel and place one end on a wall or table. Ask students to bite on the pencil or dowel while plugging their ears and observe what they hear without using their ears. (The straw is for sanitary purposes, it is not a required element.) PHET Sound Simulation: Visit the website and let students experience the ranges of frequencies which they are able to hear by selecting the Listen to a Single Source page. Enable the audio on the program to hear the sounds being generated and adjust the frequency and amplitude. Discuss the relationship between the frequency and wavelength and how those concepts are related to the length of the string from their Soundinator and the sounds they experienced. Visit the Listen with Varying Air Pressure page from the same PHET website to allow students to hear how the volume decreases and eventually disappears as air pressure decreases to a vacuous state. Straw Kazoo: Visit one of the websites below to get directions on how to make a kazoo from a straw. The pitch of the kazoo will change as the length of the straw changes. The straw is vibrating as the students blow through it. If the straw is cut the length of straw which vibrates is shorter. A shorter wavelength means higher frequency or pitch. You can see (and hear) the kazoo in action at The Soundinator 3-6

29 Drop of Doom! 4 Students will use the function for the height of a free falling object to explore functions and their graphs. Suggested Grade Range: 7-12 Approximate Time: 1 hour State of California Standards: Mathematics Standards Grade 7: Algebra and Functions 1.2: Use the correct order of operations to evaluate algebraic expressions. 1.5: Represent quantitative relationships graphically and interpret the meaning of a specific part of a graph in the situation represented by the graph. Mathematics Standards Grades 8-12: Algebra I 23.0: Students apply quadratic equations to physical problems, such as the motion of an object under the force of gravity. Science Content Standards Grade 8: Physical Sciences 1. The velocity of an object is the rate of change of its position. As a basis for understanding this concept: b. Students know that average speed is the total distance traveled divided by the total time elapsed and that the speed of an object along the path traveled can vary. d. Students know the velocity of an object must be described by specifying both the direction and the speed of the object. e. Students know changes in velocity may be due to changes in speed, direction, or both. f. Students know how to interpret graphs of position versus time and graphs of speed versus time for motion in a single direction. Science Content Standards Grades 9-12: Physics 1. Newton s laws predict the motion of most objects. As a basis for understanding this concept: e. Students know the relationship between the universal law of gravitation and the effect of gravity on an object at the surface of Earth. Relevant National Standards: Mathematics Common Core State Standards: 4 An early version of this lesson was adapted and field-tested by Katiria Hernandez and Gina Hryze, participants in the California State University, Long Beach Foundational Level Mathematics/General Science Credential Program. Drop of Doom! 4-1

30 8.EE Understand the connections between proportional relationships, lines, and linear equations. 8.F Define, compare, and evaluate functions. Next Generation Science Standards: 5-PS2-1. Support an argument that the gravitational force exerted by Earth on objects is directed down. [Clarification Statement: Down is a local description of the direction that points toward the center of the spherical Earth.] [Assessment Boundary: Assessment does not include mathematical representation of gravitational force.] HS-PS2-1. Analyze data to support the claim that Newton s second law of motion describes the mathematical relationship among the net force on a macroscopic object, its mass, and its acceleration. [Clarification Statement: Examples of data could include tables or graphs of position or velocity as a function of time for objects subject to a net unbalanced force, such as a falling object, an object rolling down a ramp, or a moving object being pulled by a constant force.] [Assessment Boundary: Assessment is limited to one-dimensional motion and to macroscopic objects moving at non-relativistic speeds.] Lesson Content Objectives: Evaluate functions using the correct order of operations. Describe and represent functions using tables and graphs. Make interpretations about an object s motion and position based on its function and graph. Materials Needed: Lined or graph paper One copy per student of the warm-up activity sheet and Drop of Doom! activity sheet (included) Adapted From: Rubin, K. (n.d.). Illuminations: Roller Coasting Through Functions. Retrieved from Drop of Doom! 4-2

31 Summary of Lesson Sequence To introduce the lesson, connect students work on graphing functions from the warm up (included) to their experiences with roller coaster rides by discussing how the motion of a roller coaster may be described using formulas or graphs. Lead students through compiling notes while modeling how to evaluate and graph a function that represents a falling object s motion. Guide students through their own practice of evaluating a function, completing a table of values, and graphing the function using the Drop of Doom! activity sheet (included). Check for students understanding by asking the key questions provided. To close the lesson, allow students to discuss in pairs how to find the time it takes for a roller coaster to reach the bottom of a drop. For independent practice, students may create and graph their own function to represent the motion of a roller coaster that they design. Assumed Prior Knowledge Prior to this lesson students should know how to use the correct order of operations to solve equations and be able to graph ordered pairs to represent a function. Classroom Set Up Students will be asked to participate in discussions and work in pairs for portions of this lesson. Lesson Description Introduction Provide students with the Warm Up: Preparing for the Drop of Doom (included) to focus their attention to working with functions. As students are finishing the warm up, ask students to discuss what mathematics is associated with roller coaster rides. Ask: How do you think roller coasters and math are related? [speed, height, formulas, etc.] Explain that engineers use functions to determine a roller coaster s height above ground after a certain amount of time. Input and Model Lead students through notes by modeling using the function describing the height of an object in free fall dropped above Earth. Explain the following: The height of an object that is dropped from above Earth can be determined using the formula h = f(t) = ½ (-32)t 2 + s, where s is the starting height of the object in feet. Notice that the height of the object is a function of time and does not depend on the object s mass. The coefficient -32 comes from the fact that an object s acceleration due to gravity as it falls to Earth is -32 feet per second squared. The formula may be simplified to f(t) = -16t 2 + s, where: t = time in seconds and s = initial height in feet. Drop of Doom! 4-3

32 The formula may be used to create a chart of values for the time and height of the object which may be graphed as ordered pairs (t, f(t)). For example, an object dropped from 144 feet above Earth, will have an initial height of 144 ft. (see table below). Copy the following table in your notes: Time in Height in Ordered Pair f(t) = -16(t) 2 + s seconds feet (t, f(t)) 0 f(0) = -16(0) 2 + s 144 (0, 144) 1 f(1) = -16(1) 2 + s 128 (1, 128) 2 f(2) = -16(2) 2 + s 80 (2, 80) 3 f(3) = -16(3) 2 + s 0 (3, 0) Ask students to discuss what happens 3 seconds into the roller coaster ride. Graph the resulting ordered pairs on a height vs. time coordinate plane. Guide Students Through Their Practice Distribute the Drop of Doom! activity sheet (included) and allow students to work independently or in pairs. Guide students by moving through the classroom assisting those who need it and checking that students are using the correct order of operations. Check for Understanding Check for students understanding by orally guiding them through the key questions below. These questions may be used to determine whether students are ready to do similar work on their own. 1. Why are all of the graphical representations in the first quadrant only? [Time is on the x-axis and we cannot have negative time. Height is on the y-axis, and roller coasters do not go below ground or 0. ] 2. The equation we are using does not take into account certain things that may have an effect on the roller coaster as it drops. What are some things that could affect the drop? [e.g., friction, weight of people in the car, weather] 3. If you saw the ordered pair (3,112) in your data table, what would it mean? [After 3 seconds, the roller coaster is 112 feet above ground.] 4. Let t represent the time in seconds and f(t) represent the height above ground of the roller coaster. What is a question that could represent the ordered pair (t, 60)? What is a question that could represent the ordered pair (4.5, f(t))?" [The ordered pair (t, 60) represents the question, "How long will it take the roller coaster to be 60 feet above ground?" The ordered pair (4.5, f(t)) represents the question: "How high above ground will the coaster be after 4.5 seconds?"] 5. Which of the following ordered pairs would be unreasonable if it appeared in the context of this problem?(1,144), (-2, 1000), (8, -124), (0,1)? Why? [The second, third, Drop of Doom! 4-4

33 and fourth are unreasonable ordered pairs. The second is unreasonable because time should not be negative. The third is unreasonable because roller coasters do not travel 124 feet below ground. The last is unreasonable because the initial drop is more than 1 foot tall.] Independent Practice Students may practice what they learned independently by designing a roller coaster drop, writing the function for it s height, finding values for the height at different times, and graphing the results. To design their roller coaster, students are choosing the height, s, of the drop in the formula: f(t) = -16t 2 + s. Closure Allow students to discuss a way to find out the time it takes for a roller coaster to reach the bottom of a drop in pairs and write their responses as a ticket out the door. Suggestions for Differentiation Students who have difficulty evaluating functions or graphing ordered pairs could be pulled into a small group to receive assistance while the class is working on their guided practice. To extend the activity, advanced students may be introduced to the formula for the motion of a roller coaster that has an initial velocity before it begins to fall: f(t) = 16t 2 + vt + s. Drop of Doom! 4-5

34 Warm Up Preparing for the Drop of Doom Directions: Complete each function table by finding the values for y. Show your work in the space provided. Graph the ordered pairs (x, f(x)) for each equation. 1. f(x) = x 5 Show work here: Graph: x f(x) f(x) = 2x Show work here: Graph: x f(x) -2 ½ Drop of Doom! 4-6

35 Drop of Doom! Name A new roller coaster ride at Six Flags Magic Mountain recently opened in Valencia, California. At the highest point of the ride, Drop of Doom drops thrill seekers from a record-breaking height! Use the formula f(t) = -16t to determine the height of the coaster at several times during the descent and use the data to determine how long it takes the coaster to reach the bottom. 1. Complete the table to determine how long it takes Drop of Doom to reach the bottom of its highest drop: Time t f(t) = -16t Height (ft) Ordered Pair (t, f(t)) (sec) Graph the ordered pairs below: Drop of Doom! 4-7

36 3. What is the height of the coaster before it begins to drop? How do you know? 4. After how many seconds does Drop of Doom reach the bottom? How do you know? 5. If you did not know the height of a specific drop on a roller coaster, how could you find out the height without measuring it directly? Write a plan for how you would collect the data you would need to determine the height of a drop on a roller coaster you were watching or riding. Drop of Doom! 4-8

37 Reflections on Light 5 Students will explore the general behavior of light when it is reflected and the behavior of light when it is reflected in specific angles. Suggested Grade Range: 7-12 Approximate Time: 1 hour State of California Content Standards: Mathematics Content Standards Grade 5: Measurement and Geometry 2.1 Measure, identify, and draw angles, perpendicular and parallel lines, rectangles, and triangles by using appropriate tools (e.g., straightedge, ruler, compass, protractor, drawing software). Science Content Standards Grade 7: Physical Science 6. Physical principles underlie biological structures and functions. As a basis for understanding this concept: c. Students know light travels in straight lines if the medium it travels through does not change. f. Students know light can be reflected, refracted, transmitted, and absorbed by matter. g. Students know the angle of reflection of a light beam is equal to the angle of incidence. Relevant National Standards: Mathematics Common Core State Standard: High School Geometry G-CO 12. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. Next Generation Science Standards: MS-PS4-2. Develop and use a model to describe that waves are reflected, absorbed, or transmitted through various materials. [Clarification Statement: Emphasis is on both light and mechanical waves. Examples of models could include drawings, simulations, and written descriptions.] [Assessment Boundary: Assessment is limited to qualitative applications pertaining to light and mechanical waves.] 5 An early version of this lesson was adapted and field-tested by Karen Hardy, a participant in the California State University, Long Beach Foundational Level Mathematics/General Science Credential Program. Reflections on Light 5-1