There is a Difference: Histograms vs. Bar Graphs



Resource FileBrandon BowerCI 4040 TR 10:00-11:50Dr. Lynch-DavisSingle RunnerActivity: Single Runner Source: "Understanding Distance, Speed, and Time Relationships."Illuminations. (2000-2012): n. page. Web. 26 Mar. 2012. < Concept:SimulationsGrade Level: 3-5Material needed:Runner simulation toolRunners Take Your Mark activity sheetDetailed Description:Students are expected to use simulation software to simulate a race of one runner. The data will be examined, observed, and analyzed then put into a chart. Once they have gathered the results from the simulated data they can compare and contrast it to the predictions they made about the runnerSingle RunnerIn this activity, students use a software simulation of one runner along a track. Students control the speed and starting point of the runner, watch the race, examine a graph, and analyze the time-versus-distance relationship. This activity helps students understand, describe, and compare situations involving constant rates of change.Learning Objectives?Students will:identify and describe situations with constant rates of change and compare themmake and test predictions about step sizes and finish timesMaterials?Runners, Take Your Mark! (One Runner) Activity Sheet?Runner SimulationInstructional PlanTo introduce this activity, ask two student volunteers to stand in front of the classroom to physically demonstrate and discuss the results of each of the following scenarios:Scenario 1. Two students start from the same position at one end of the classroom. One student takes giant-steps while the other takes baby-steps.?Each student takes one step per second.?Who gets to the other end of the classroom first? How many steps are taken? Discuss the results.Scenario 2.?One student starts behind the other at the same end of the classroom, both walking with equal stride and pace.?Each student takes one step per second.?Who gets to the other end of the classroom first? How many steps does each student take? Discuss the results. Ask students to predict the effect of changing the length of stride.Place students into teams of two and distribute a?Runners, Take Your Mark! (Single Runner)?activity sheet to each group.Runners, Take Your Mark! (Single Runner) Activity SheetStudents should open the?Runner Simulation?tool.?Runner Simulation ToolWorking together, partners share the responsibility of "Mouse Driver" and "Reader/Recorder". The "Reader/Recorder" will read the directions from the activity sheet and record observations while guiding the activity. The "Mouse Driver" controls the action of the mouse and movement on the computer screen. Partners should switch roles until all have moved the runner.Be sure to tell students about two key assumptions used in this activity.(a) The runner always takes?one step per second?(no matter how big the step size is).(b) We will?measure time in seconds, even though the actual movement in the simulation will probably be much faster.??To begin, the students select either the male or female runner. To do this the student "clicks" upon the male or female icon in the box next to the graph of the runner they DON’T want to use. This will cause that runner to temporarily vanish from the running line and graph. Next the students set the runner to zero by dragging the icon along the track and clicking until the runner is facing the direction of running from left to right.The students should take out their?Runners, Take Your Mark! (Single Runner)?activity sheet, record the step size of "1", and set the step size on the interactive applet to "1".??The students then select the?Slow Run Button??and with each "click" (at least 10 times), results are recorded on the graph.The students then select the?Play Button?to run the simulation. After the runner is completely done, the stop button resets the simulation.Next, the students set the runner’s step size to 2, select the?Slow Run Button?and record the results on the graph.Repeat this with the step sizes of 4 and 5 and record the results. Students may vary the runner’s step size all the way up to 15.Teacher Note: In this race simulation software, the finish time is rounded up to the nearest whole number. Thus, for example, if a runner starts at 0 with step size 3, the finish time shown will be 34, rather than 33 1/3. Students may notice this and comment that 34?×?3 does not equal 100. They may notice that with step size of 3, and one step per second, the finish time should be 33 1/3 seconds. Please be aware of this limitation of the software as you teach the lesson.The closing should be structured so that students can review and pull together what they have learned. Include questions or tasks that encourage students to reflect on their work. For example, have students consider the?Questions for Students?(below). In so doing they will consolidate what they have learned. Furthermore, this will provide an opportunity for you and the students to assess what they have learned and what they still want or need to understand. This will give you ideas for further instruction.There is a Difference: Histograms vs. Bar GraphsActivity:There is a Difference: Histograms vs. Bar GraphsSource: ?"There is a Difference: Histograms vs. Bar Graphs."Illuminations. National Council of Teachers of Mathematics, 2000-2012. Web. 27 Mar 2012. < Concept:HistogramsGrade Level:3-8Material needed:ComputerCharting the Difference Activity SheetThe Life Cycle Activity SheetDetailed Description Students are expected to look up the party and ages at the inauguration of US Presidents. The data found will be placed in a histogram and bar graph. Next students will be able to categorize what bar graphs contain and do not contain as well as with histograms. There is a Difference: Histograms vs. Bar Graphs??Using data from the Internet, students summarize information about party affiliation and ages at inauguration of Presidents of the United States in frequency tables and graphs. This leads to a discussion about categorical data (party affiliations) vs. numerical data (inauguration ages) and histograms vs bar graphs.Learning Objectives?The students will:Summarize raw data in a frequency tableDetermine the category of the data as categorical or numericalCreate the appropriate chart (histogram or bar chart)Analyze data using histograms and bar chartsMaterials?Computer with Internet connectionCharting the Difference Activity SheetCharting the Difference Answer KeyThe Life Cycle Activity Sheet and Answer Key?(optional)Instructional PlanStart the lesson by engaging students in a discussion about the Presidents of the United States. You might ask: What factors influenced our last presidential race? Are there typical characteristics of a person running in any Presidential race? There are many answers to these questions. If no students suggest party affiliation and age at the time the person enters office, bring these characteristics into the discussion.Discuss the data provided for each president with the class. Use questions to structure the discussion. What is the difference between raw data and summarized data? [Raw data is the individual values while summarized data is presented in a way for analyzing the data.] If I wanted to find the number of Presidents in each party, how could I summarize the data? [frequency table, bar chart, line chart, histogram] What would be the title for the columns of the frequency table? [Political Party, Frequency]While distributing the?Charting the Difference?activity sheet, divide students into pairs a computer for each pair to share. If fewer computers are available, groups of 3 can be used with assign roles, such as announcer, recorder, and checker. However, avoid groups with more than 3, as there will be students with nothing to do.Charting the Difference Activity SheetHave students access at least one of these web sites for data collection:Presidents — Infoplease(recommended)Internet Public Library: PotusAbout the White House ? PresidentsRemind students on how to record data in frequency tables. Have them record the frequencies in groups of five tally marks.As students begin their bar graphs for the political party data, check the graphs being created for correctness. Ask questions to help struggling students. How is the frequency scaling determined? [by the range of the frequency column] What should the frequency scaling be based on the lowest and highest frequency? [0-20 with marks at 5, 10, 15, 20] Since the political parties are separate categories, makes sure students' bars do not touch each other but are of the same width.Circulate through the room as students work on the inauguration ages frequency table, asking questions like the following examples to help students.What is different about the data being collected in this frequency table?[The data is numerical, not categorical like the previous table.]Are we going to list every age at inauguration to summarize the data?[No, there would be too many rows.]What intervals of numbers might work better? How many intervals will there be?[Decades might work, but there would only be 3 intervals, which would not make the graph very informative. Intervals of 5 years would be better because then there would be 6 categories.]Did the order of the political parties matter in the previous table? Does it matter in this table?[The categories in the first table are not related so they can be in any order. In this table, it makes sense for the categories to be in ascending or descending order since the categories are intervals of numbers.]After all groups have finished, have the groups compare answers in a class discussion. Call on students to give their answers to the questions. As they do this, point out in ages at inauguration frequency table that the categories are numeric intervals in an increasing order with no gaps between the numbers. The graph also has no gaps between the bars. This is because there is no gap between the numbers for the ages. For example, one bar ends at 44 and then next starts at 45. This is a difference between the two frequency tables and this affects the graphs. To check for understanding, ask students: Why is it the bars in bar graphs do not touch and the bars in histograms do touch? [Bar graph bars do not touch because the data is categorical; histogram bars do touch because the categories intervals of continuous numbers.]While students share their generalizations from Question?5, use questions such as those in?Questions for Students?to show that some answers cannot be determined by the graphs but may have to be researched using the raw data or using other sources.Summarize Question?6, draw a table on the board with a column for characteristics of bar graphs and a second column for characteristics of histograms. A representative from each group can then add an observation to either or both columns, comparing and contrasting the attributes of the different graphs. Allow students to add statements one group at a time without repeat a prior statement. This will become more difficult as more groups go, so try to choose groups with longer answers to Question?6 after groups with shorter answers. A partially complete table may look like this:BAR GRAPHSHISTOGRAMSgraph title and labeled axesgraph title and labeled axesBars do not touch.Bars do touch.Vertical scale is frequency.Vertical scale is frequency.categorical datanumerical dataJumping Jack MathActivity:Jumping Jack MathSource: "Jumping Jack Math."?Illuminations. National Council of Teachers of Mathematics, 2000-2012. Web. 27 Mar 2012. < Concept:Mean, Median, and ModeGrade Level:3-5Material needed:CalculatorsTimerJumping Jack Math Activity SheetJumpalot Data Activity SheetTo Jumpalot and Beyond Activity SheetDetailed Description This activity will require students to use student-generated data to look at the different values of mean, median, and mode. Discuss and ask the students why everyone seems to have different values for the mean, median, and mode. Jumping Jack Math??In this lesson, students prepare jumping jack data to send to officials on the planet Jumpalot. Students record how many jumping jacks they can do in ten seconds and use their knowledge of time conversions to figure out how many jumping jacks they could complete in a minute all the way to a year if they never tired. Students then organize class data and explore mean, median, and mode and the effects extreme values have on these measures. Students then brainstorm the advantages and disadvantages each measure offers.Learning Objectives?Students will:Use student-created data to calculate mean, median, and modePractice time conversion (seconds, minutes, hours, days, weeks, year)Develop number senseDiscover the effects of extreme values on the meanAnalyze the advantages and disadvantages of using mean, median, and modeMaterials?CalculatorsTimer or clockJumping Jack Math Activity SheetJumpalot Data Activity SheetTo Jumpalot and Beyond Activity Sheet?(optional)Instructional Planright0Pass out the?Jumping Jack Math?and? HYPERLINK "" \t "_blank" Jumpalot Data?activity sheets. Read the introductory paragraph to students and explain that students will be developing a jumping jack data set which they will use to discuss mean, median, and mode. Students should be familiar with or have at least been introduced to mean, median, and mode before beginning this activity to get the most out of this lesson.Jumping Jack Math Activity SheetJumpalot Data Activity SheetRead the introduction of the Jumping Jack Math activity sheet and have students complete the time conversion chart in Question?1. Next, students should complete Question?2. Explain that students need to count how many jumping jacks they can complete in 10 seconds, and then write that number in the first row of their chart. An easy way to organize this is to have every other student stand up in the room and spread out. If your room is particularly small you could have every third or fourth student stand up at a time. Have the students sitting help count for the students jumping so everyone is engaged. Tell students that you will be the official timer, and then using a timer or a clock with a second hand, tell students, "Go!" and then "Stop!" when the time is up. Continue until all students have collected their data.Next, have students pair up; you may want to encourage students with similar numbers to work together Students who have completed the same number of jumping jacks will have identical charts for Question?2. Have students complete Question?2 together. You may either allow students to use calculators the entire time, or have students complete the chart using paper and pencil and then allow them to check their answers with a calculator. The chart goes up to one year is to give students experience with very large numbers and to help develop number sense.During this time, circulate and have students explain the math behind the time conversions when you come to them. Some students may want to multiply by ten to get from ten seconds to a minute, instead of multiplying by six, which would make the rest of their data chart incorrect.Groups who finish Question?2 should move on to Questions?3 and?4 after checking with the teacher. Encourage students who find they have a vastly different number from the other students at the hour mark to go back and check their work. Many students will correct themselves when they are collecting data in Question?3 from classmates, but you may have to point it out for some students.After students have completed Questions?3 and?4, bring the group together and have students share their answers to Question?4. Ask students:Why do you all seem to have different values for the mean, median, and mode?[Because everyone used data sets with information from ten different people, rather than the whole class]Next, explain Questions?5–8 and have students go back to their partner or group to work on completing Questions?5 through?8.After students have completed Questions?7 and?8 bring the students together to discuss their answers. Students should say that Jumpalot School District should admit Speedy because his jumping jack value increases the value of the mean which would mean more energy production for the school. For Question?8, students should find that the extreme values affect the mean with an extremely low value making the mean lower and an extremely high value making the mean higher, but have little effect on the median or mode.?Summary ActivityHave students brainstorm the best times to use mean, median, or mode. To complete this, you have a few options:Students can complete it with the partner or group they are working with then you can have a class discussionYou could break the class into groups and give each group a different measure to focus on. Then you could have each group write their results on a poster to share with the class. You could have a class discussion about them in which you wrote student responses on the board, interactive whiteboard, or overhead.Expected ValueActivity:Expected ValueSource: . "Expected Value."?Shodor. N.p., n.d. Web. 27 Mar 2012. < Concept:Expected ValueGrade Level:5-8Material needed:ComputerPencil and PaperStock Exchange WorksheetDetailed Description In this lesson students will be introduced, develop, discuss, and learn the definition of what expected value is. Describe what the stock market is and how expected value is related. They will pull information from the stock market to help answer questions. Expected ValueAbstractThis lesson's activity and discussion introduce and develop the idea of expected value. The discussion helps students investigate the definition and formula of expected value.ObjectivesUpon completion of this lesson, students will:have learned about expected valuehave been introduced to the concept of varying payoffshave used a computer simulation of a "real world" example where expected value is usedStandardsThe activities and discussions in this lesson address the following?NCTM Standards:Number and OperationsUnderstand numbers, ways of representing numbers, relationships among numbers, and number systemswork flexibly with fractions, decimals, and percents to solve problemsCompute fluently and make reasonable estimatesdevelop and analyze algorithms for computing with fractions, decimals, and integers and develop fluency in their usedevelop and use strategies to estimate the results of rational-number computations and judge the reasonability of the resultsData Analysis and ProbabilityUnderstand and apply basic concepts of probabilityuse proportionality and a basic understanding of probability to make and test conjectures about the results of experiments and simulationsLinks to other?standards.Student PrerequisitesArithmetic: Students must be able to:Calculate with decimalsInterpret data in decimal formTechnological: Students must be able to:perform basic mouse manipulations such as point, click and draguse a browser such as Netscape for experimenting with the activitiesTeacher PreparationStudents will need:Access to a browserPencil and PaperStock Exchange WorksheetKey TermsThis lesson introduces students to the following terms through the included discussions:Expected ValueProbabilityLesson OutlineFocus and ReviewRemind students of what they learned in previous lessons that will be pertinent to this lesson and/or have them begin to think about the words and ideas of this lesson:Ask students to recall what probability is.Ask the class if they know what the Stock Market is or ask if any students own stock. Briefly discuss the Stock Market.ObjectivesLet the students know what they will be doing and learning today.Lead a discussion on what the stock market is and why someone would risk their money to purchase stock. Discuss this in terms of probability.Ask such questions as, "Why would someone choose to purchase stock from one company rather than another?" This will lead into a discussion of expected value.Teacher InputLead a class discussion using the?Expected Value?discussion that will formalize and develop the concepts introduced by the?Stock Exchange?activity.Guided PracticeDescribe the?Stock Exchange?activity which introduces the idea of varying payoffs, bringing probability and payoffs together and developing the concept of expected value.Independent PracticeHave groups of students experiment with the activity to generate discussion ideas to use during the closing discussion.ClosureLead a class discussion using the?Expected Value?discussion that will formalize and develop the concepts introduced by the?Stock Exchange?activity.Alternate OutlinesThis lesson can be rearranged in several ways.Introduce the discussion before having students work with the?Stock Exchange?activity.Or use the material in the discussion to prepare a "live" discussion.Suggested Follow-upAfter this discussion and activity, the students will have learned about expected value and payoff. At this point, if students still have questions about probability, choose some activities and discussions from earlier lessons that may have been omitted and use them to reinforce what students already know. If needed, refer to the?Crazy Choices Game?and draw comparisons with the Stock Exchange activity for students.Stock Exchange Game SuggestionsSeveral games can be based on this applet. Possible math goals of each game are indicated in parenthesis.Game 1(idea of varying payoffs and its relationship with game fairness): Enter the probability of 0.1 for Player 1, and the probability of 0.3 for Player 2. Enter the payoff of 5 (points) for Player 1, and the payoff of 3 for Player 2. Run the program for many times (1000 or more) and observe which player wins more. Why does it happen?Game 2(expected value):? Without changing the probabilities, change payoffs in such a way that the game is fair. Do it in several different ways. Run the program many times to see if it is fair indeed.Game 3(expected value; theoretical and experimental ways of finding expected value): Reset the payoffs and the probabilities in your own way. What average payoff per game do you expect for each player? This number is called?expected value. Use Question #1 and Question #2 forms below the applet to have the computer check your expected payoffs. Run the program many (1000 or more) times to see if the average payoff per game gets close to your prediction.Candy FractionsActivity:Candy FractionsSource: ?"Candy Fractions."?Instructor Web. Instructor Web, 2005. Web. 27 Mar 2012. < Concept:Area ModelGrade Level:3-5Material needed:Candy Fractions WorksheetCookie, some coins, and a rulerSmall paper cupsSmall unwrapped candiesPencils, crayons, colored tapeDetailed Description This activity is for students who already have learned or have prior knowledge of fraction models. With this activity students will be able to analyze a discrete model to identify fractions by using candy given to them by the teacher. CANDY FRACTIONSContent AreaElementary Math - FractionsContent TargetsIdentifying parts of a whole, percentages, working with fractions as discrete models, graphing data using bar graphs, recording data in a table, classifying and sorting objects, number senseLearning ObjectiveAfter learning about the different types of fraction models, students will be able to analyze a discrete model in order to identify fractions (as fractions, decimals and percentages) by working with a candy sample to fill out a table and draw a bar graph.?Grades3rd Grade - 4th Grade -5th gradeCANDY FRACTIONS LESSON PLAN?Academic Language Focus? How to distinguish between an area model, discrete model and number line? How to sort and categorize candies by color? What it means to record data? What it means to show the same quantity as a fraction, decimal and percentage?Key VocabularyArea model, discrete model, numerator, denominator, percentage, fraction, decimalLesson Materials??Candy Fractions worksheets? A cookie, some coins and a ruler? Small paper cups? A bag of small, unwrapped candies? Pencils, crayons, colored tape.IntroductionWrite the fraction ? on the board. Have students identify the numerator (part) and denominator (whole). Explain to students that there are other ways to write this fraction. Show students how to show ? as a decimal and as a percentage. Write one or two more fractions and have students convert or tell you how to convert them in to decimals and percentages.?Erase everything on the board except ?. Ask a student to show the class how to draw this as a model. Most likely, the student will make a pie chart and shade in one fourth. Show the students a cookie and with colored tape, mark off ?. Tell students that this is an area model. Remind students that an area model can be a square, rectangle or any other plane. Explain that there are two other models that we could use to show ?. Pull out a ruler and tell students that this is an example of a number line.?Explain that 12 inches is the whole and ask students what they think ? of the 12 in. is. Mark off three inches with the tape to show ?. Next, tell students there is yet another way to show ?. Set out some coins (pennies and nickels) and ask a student volunteer to show that ? of a group of coins are pennies. A student can show 1 penny and 3 nickels, 2 pennies and 6 nickels, etc. Tell students that this is a discrete model, since the parts of the whole are separate objects.?Display the three models of ? in the classroom for students to refer to.BodyAsk 3 students with fair hair to stand up in front of the class. Ask 2 students with dark hair to stand up in front of the class.Ask students what is the whole (5).Ask student students what fraction of the students has fair hair (3/5).What fraction has brown hair? (2/5)What type of model is this? (discrete)Pass out small cups of candy to students (10-20 pieces per cup) and the Candy Fraction packet (see? HYPERLINK "" \l "LESSON_PRINTABLES" printables). Tell students that today we are going to be working with a discrete model- pieces of candy. We are going to divide our candy samples by color to find out what colors we have as fractions of our whole sample.?Read through the whole packet with students and do steps 1 and 2 together as a class. Chose a color to do together as an example and answer any questions that come up (fill in the table and bar graph for that one color as a class). Remind students that they will have different answers since they all have different samples. Students can then fill in their packet with the rest of the colors.ClosureTo review the major concepts, have students choose one fraction they found and draw sketches of it as an area model, discrete model and number line.Have students share their findings with a partner and post their tables and bar graphs on a bulletin board.?ActivityName: Directions: Work with a partner! Look at your pack of Smarties candy to answer the following questions. When you are finished with question 9, you may eat your candy! ?1. How many Smarties does it take to make one pack? ___2. What is the fraction name for one piece of this candy? ___3. How many different colors are there in your pack? ___4. Tell the fraction name for each color: Blue___ Pink___Yellow___ Orange___ Green___ Purple___ White___5. What is the largest fraction you wrote in question 4? ___6. What is the smallest fraction you wrote in question 4? ___7. What is the fraction name for the whole pack of Smarties? ____ This is another name forwhat number? ____8. You may share the pack of Smarties with your partner, but first tell what the fraction name is for how much you will get to eat! ____(Do question 9 before you eat your candy!)9. On the back of this paper, make a graph to show what colors were in your pack. Then write three questions for the teacher to answer about your graph. For example: Howmany more green Smarties are there than purple Smarties Scatter PlotsActivity:Putting Scatter Plots to UseSource: Reeves, Kristin. "Putting Scatter Plots to Use."?Lesson Plans Page. Hot Chalk, 2011. Web. 27 Mar 2012. < Concept:ScatterplotsGrade Level:6-7Material needed:Basketball Scatter WorksheetDetailed Description Students will create a student-generated scatter plot from the data they gather from the shots they make/do not make with the basketball. Once they have the data on a scatter plot they will make observations and assumptions about the classes data from hoops made and not made. Putting Scatter Plots to UseSubject(s): Computers & Internet,?MathGrades(s): Grades 6-7Title – Putting Scatter Plots to UseBy – Kristin Reeves?Primary Subject – Math?Grade Level – 7Multimedia Graphing Unit Contents:Lesson 1: YouTube Scatter Plot InstructionsLesson 2: Using Scatter Plots?(below)Lesson 3: Types of Graphs PowerPointLesson 4: Create Excel Chart and Kidspiration PresentationLesson 5: Mean, Median, Mode and M&MsContent:Putting our new knowledge of scatter plots to use.Benchmarks:D.AN.07.04 Create and interpret scatter plots and find the line of best fit; use and estimated line to answer questions about the data.4.b.1. Students create a project (e.g., presentation, web page, newsletter, information brochure) using a variety of media and formats (e.g., graphs, charts, audio, graphics, video) to present content information to an audience.Learning Resources and Materials:Worksheet with a blank chart and a blank scatter plot.Basketball and Basketball CourtDevelopment of Lesson:Introduction:The previous lesson “Making Scatter Plots” is the introduction to this lesson.Methods/Procedures:The class will be divided into groups.If available, we will go into the gym or to an outside basketball court. Each group will be assigned to a basketball hoop. The number of groups will correspond to the number of hoops available.Each group member will be given five turns to make a basket, each turn taking a step away from the basket. (These increments should be marked on the floor to make everyone equal.)The groups will record in their chart, how many people make baskets at each distance.Then, back in the classroom, the groups will input their data into Microsoft Excel and turn it into a scatter plot.Observing their data, they will make observations about the trend(s).Finally, we will compare our results as a class.Accommodations/Adaptations:For my two students with Down Syndrome, I will work with them personally to record their results. They will get a smaller net and a chance for one-on-one learning.Closure:In our final test on graphing, there will be a section on scatter plots. Performances on this test will prove how well the students have absorbed this material and whether or not these lessons have been effective.WorksheetName _________________________________ Hour ___________ Date __________________Group Number _______ Group Members ____________________________________________?BASKETBALL SCATTERDirections:Record each group member’s name the spaces provided below.Record who made a basket at each trial plete the scatter plot below with the appropriate labels.NameTrial OneTrial TwoTrial ThreeTrial FourTrial Five???????????????????????????????????????????__________________________________Tree DiagramsActivity:Tree Diagrams/Fundamental Counting PrincipalSource: Ivory, Kaiulani. "Tree Diagrams/Fundamental Counting Pricnipal."?Better Lesson. Better Lesson, 2012. Web. 27 Mar 2012. < Concept:Tree diagramsGrade Level:6Material needed:Activity WorksheetDetailed Description Discuss and explore what tree diagrams do, how they look, and how to set one up correctly. Students should understand upon completion of this assignment what a tree diagram does and how to properly set one up. ObjectiveUse a tree diagram, chart or list to find all possible outcomes of two or more events Use the fundamental counting principal to identify all possible outcomes from two or more events.Lesson Plan?Aim/Objective:Key PointsUse a tree diagram, chart or list to find all possible outcomes of two or more events?Use the fundamental counting principal to identify all possible outcomes from two or more events·??Assessment:Madison plans to sew one button and one ribbon on a clown costume. She has one each of the following colors of buttons in her pocket:blackgreenredwhiteAll the buttons are the same size and shape.??Madison will select one button from her pocket without looking.What is the probability that she will select a red button? Show or explain how you got your answer.Madison has one each of the following colors of ribbon in a bag:blackwhiteyellowAll the ribbons are the same size and material. She will select one button from her pocket and one ribbon from her bag without looking.Make an organized list or a tree diagram showing all the possible color combinations that Madison could select.What is the probability that Madison will select a black button and a black ribbon? Show or explain how you got your answer.What is the probability that Madison will select a button and a ribbon that are different colors from each other? Show or explain how you got your answer.?LessonIntro/Direct Instruction:·?We have discussed theoretical probability as being the ratio of desired outcomes to total possible outcomes·?Sometimes, however, we will have to look at events where all of the possible outcomes are not easy to calculate.?·?Let’s look at an example:?o????Mario is playing a game where he flips a two sided coin and rolls a dice.??He wins if he flips a tails and rolls a 5 or 6 – what is the probability that Mario will win?o????Walk through what we know – we know that probability is the ratio of desired outcomes, in this case tails and 5 or 6 to all possible outcomes.?However, we need to figure out all of the possible outcomes.??We can figure out the outcomes in one of three ways – we can make an organized list, we can make a chart or we can make a tree diagram.??Today we are going to walk through all of these different ways to figure out the examples·?Tree Diagramso????A tree diagram is one way to organize and figure out all possibilities for events occurringo????It is called a tree diagram because it looks like a tree and has brancheso????Tree diagrams can be written vertically from the top down or horizontally from left to write.o????The first step is to take on of the events and write all of the possibilities.??So let’s start with flipping a coin – what are all of the possibilities for flipping a coin – they are heads and tails – so I write heads and tails.?o????Next, I am going to think of the second event – in this case rolling a dice.??I am going to think through all of the possibilities for rolling a dice – a 1, 2, 3, 4, 5 or 6.?o????From each of the possibilities in the first event, I will write all of the possibilities for the second evento????Now, I have listedo????Lastly – I want to finish off each branch by writing the total possibility for each event.??So, I will write heads, 1; heads 2; heads 4…..tails 1, tails 2, tails 3…o????I can count each branch to know the total number of possible outcomes and the total number of desired outcomes·?Charto????Another way to write all of the possibilities to chart it.o????We can do this by thinking of the area model.??First we will draw a big rectangleo????Then since there are two possible outcomes for a coin flip we will divide it into two columns and write heads and tails above each columno????Then since their a six outcomes for rolling a dice we will divide it into 6 rows and write the outcomes next to each rowo????Now like a multiplication chart – we are going to combine each one·?Organized listo????The last way to figure out the total number of possible outcomes is to list them out in a systematic way.??Just like with tree diagrams, it is important to start with one category and to list all of the possibilities with that category before moving on to the next category.·?Fundamental counting principleo????If you just need to know all of the possible outcomes – you can use the fundamental counting principle?àthe fundamental counting principle??tells us that we can just multiply the number of outcomes for each event to find out the total number of possible outcomes.Guided Practice·?Guide students through creating a tree diagram, chart, organized list and fundamental counting principle for a situation·?Allow students to choose which is the easies for them to make after they can proficiently make each one.Independent Practice·?Students will practice independently on a work page while teacher circulates.?Closing·?Review steps/exit ticket??Activity:Source: Mathematical ConceptGrade Level:Material needed:Detailed DescriptionActivity:Source: Mathematical ConceptGrade Level:Material needed:Detailed DescriptionArchimedes’ PuzzleBibliography:Zordak, Samuel. "Archimedes' Puzzle."?Illuminations. National Council of Teachers of Mathematics, 2011. Web. 25 Oct 2011.Mathematical concept: SymmetryTransformationsGrade Level:4-8Materials:Student SheetDescription:Students throughout the activity will investigate and use a tangram puzzle to learn about symmetric properties and learn how to use correct vocabulary when describing rotations and reflections. They are given the Stomachion, an ancient puzzle, to rearrange the pieces to form shapes and figures. Archimedes' Puzzle??|The?Stomachion?is an ancient tangram-type puzzle. Believed by some to have been created by Archimedes, it consists of 14?pieces cut from a square. The pieces can be rearranged to form other interesting shapes. In this lesson, students learn about the history of the?Stomachion, use the pieces to create other figures, learn about symmetry and transformations, and investigate the areas of the pieces.Learning ObjectivesBy the end of this lesson, students will:Investigate tangram-like puzzles with the?Stomachion?and arrange the pieces to create interesting shapesDescribe the symmetric properties of the puzzle piecesUse correct vocabulary to describe rotations and reflectionsCalculate the relative areas of the puzzle pieces (optional)Materials: Student sheetInstructional PlanThe?Stomachion?is an ancient puzzle that is at least 2,200?years old. It consists of 14?pieces that can be cut from a 12?×?12?square, as shown below left. As with its cousin the tangram, the object of the?Stomachion?is to rearrange the pieces to form interesting shapes. Some of the many shapes that can be formed are shown below right.??It is not known whether Archimedes developed the?Stomachion, though the puzzle was definitely known by the ancient Greeks. Because Archimedes wrote about the puzzle extensively, however, two of its alternative names are?Loculus of Archimedes?and?Archimedes' Puzzle.Prior to the lesson, copy the 14 pieces onto a transparency sheet and cut them out to use on an overhead projector.You may wish to present some of the above history to students to begin this lesson. Explain that you will allow them to play with the Stomachion in just a few moments.Explain that the?Stomachion?consists of 14?pieces, and display the pieces on the overhead projector. To get students thinking about symmetry, ask the following questions:Are any of the pieces congruent to one another? How do you know? [Yes. There are two pairs of congruent triangles. One pair share the center point of the square in the figure above; these triangles have a base of?6?units and a height of?2?units. The other congruent triangles have a base of?6?units and a height of?4?units, and they appear in the upper right and lower left corners of the square. To show that the pieces are congruent, lay one over top of the other to prove that they are the same size and shape.]Are any of the pieces similar to one another? How do you know? [Yes. The congruent triangles mentioned above are also similar, since congruence is a special type of similarity. To show that the pieces are similar, align the angles to show that they have the same measure.]Do any of the pieces have rotational or reflexive symmetry? [No.]What kinds of pieces appear in the?Stomachion? [All of the pieces have?3, 4, or?5?sides; that is, they are triangles, quadrilaterals, and pentagons.]After the warm-up discussion, distribute the?Archimedes' Puzzle?activity sheet. To begin, you may wish to distribute only the first two pages; the third page contains questions about the area of the pieces, which is an optional component for older or more advanced students.left0Archimedes' Puzzle Activity SheetAllow students some time to cut out the pieces of the?Stomachion. (To save class time, you can distribute the first page of the activity sheet the day before teaching this lesson and ask the students to cut out the pieces as homework.) Allow all students in the class to arrange the pieces to form the large right triangle shown at the bottom of page 1 of the activity sheet. Circulate among students to ensure that they are able to do so, offering assistance to those who need it. Then, tell students that you would like them to work in pairs to construct at least two of the shapes that appear on page 2. (Depending on time limitations, you can allow students to create many more shapes.) To ensure that students work together, specify that one student's set of pieces be used for one arrangement, the other student's set of pieces be used for the other arrangement, and no one is allowed to touch their partner's puzzle pieces.After they have constructed two shapes, students should answer Questions?1–3 on the activity sheet. For students who finish quickly, allow them to construct more shapes after answering the questions. When all students have answered the questions, conduct a class discussion on Questions?2 and?3. Students should be able to identify the center of rotation for those shapes with rotational symmetry, and they should be able to identify the line of symmetry for those shapes with reflexive symmetry. In the figure below, the red shapes have reflexive symmetry; the green shapes have rotational symmetry; and the blue shape has both reflexive and rotational symmetry.?<="" td="">?As a final part of the lesson, you may wish to have students compute the areas of the puzzle pieces. On page 3 of the?Archimedes' Puzzle?activity sheet, the 14?pieces are arranged in a 12?×?12?square configuration. When arranged as shown, all of the intersections occur on lattice points. Consequently, it is easy to calculate the area of each piece. Have students determine the area of each piece, and then discuss the results. (As a preliminary question for class discussion, you may wish to ask students to determine the area of the entire square.) In particular, students should notice thatThe area of every piece is an integer.More precisely, the area of every piece is a multiple of?3.Paper Quilts Lesson 3Bibliography:"Paper Quilts."?Illuminations. National Council of Teachers of Mathematics, 2011. Web. 25 Oct 2011. < concept: Reflections and TranslationsGrade Level: 3-5Materials:Paper Quilt BibliographyWhite, 3-inch Squares (4 per student)Crayons or markers6-inch square work mat, divided into 4 equal parts (1per student)Description:Students are to explore the result of sliding a square into a new position, the results of flipping a square, and naming the geometric transformation used after the product is finished. Students will grasp the concept of geometric translations by hands-on activities such as this one rotating and flipping squares given to them by the teacher. Since they get to create their own design they will focus their attention, more so, on the lesson then if they were just given an assignment. Paper QuiltsLesson 3Exploring Flips and SlidesThis lesson builds on the previous two lessons and encourages students to explore the geometric transformations of reflection and translation. Students create a design then, using flips and slides, make a four-part paper "mini-quilt" using that design as the basis. While the formal terms are reflection and translation, the more informal terms slide and flip are used at this stage. The experience focuses students’ attention on the changes these geometric transformations make in a student-designed quilt square.Learning Objectives?Students will:explore the results of sliding a square into a new positionexplore the results of flipping a squarename, using informal language, the geometric transformation used to create a given designMaterials?Paper Quilts Bibliography?White, 3-Inch Squares (4 per student)Crayons or Markers?6-Inch Square Work Mat, divided into 4 equal parts (1 per student)Instructional PlanTo set the stage for this lesson, you may wish to read another of the books listed in the?Paper Quilts Bibliography, such as?Sam Johnson and the Blue Ribbon Quilt, calling attention to quilt squares which show flips and slides. While students remain seated, give each child four white squares (or to save time, four?copies of the chosen quilt square) and crayons or markers. Then display a quilt square and ask them to copy it four?times in any color they wish, using the same colors each time. Some simple designs include:??To help focus discussion, you might display a model square on the chalkboard or overhead.Provide students with a 6"?×?6" square workmat divided into four equal parts. Have them label the small squares starting from the top left and going clockwise. The top positions, therefore, are numbered?1 and?2, position?3 will be under position?2, and position?4 will be under position?1.Now ask children to place one of the colored 3"?×?3" squares in position 1 and then to place an identical model square face up on top of it so that like parts are touching. Then have them slide the top square down to position?4 so that it is directly under the square in position 1. [The squares in positions?1 and?4 will look the same.] Now have them place a third identical square on top of the square in position?1 so that sections colored alike are touching, then slide it into position?2, directly to the right of the square in position?1. Encourage students to find as many ways as they can of how they can slide a square into position?3. Then have them compare the 4?squares. [They will all look alike.]Next ask children to place one of the colored 3"?×?3" squares in position?1 and then to place an identical model square face up on top of it so that like parts are touching. Then have them flip the top square down to position?4 so that it is directly under the square in position?1. [The square in position?4 will be flipped over to the side without a design.] Now have them place a third identical square on top of the square in position?1 so that sections colored alike are touching, then flip it into position?2. [The square in position?2 will also be showing the side without a design.] Next have them align the fourth square with the square in position?1, then slide it down into position?4 then across into position?3. Now ask them to compare the four?squares. [Squares in positions?1 and?3 will look alike, as will the squares in positions?2 and?4.]Then ask the students to put the 4?squares into a pile, place one of the squares in position?1, then explore several ways they can slide and flip the four?squares to make Four Patch designs. You may wish the students to record one of the ways they found by gluing the four small squares on a sheet of paper and describing how the Four Patch square was created.When the children are ready, call them together to share designs and describe how each of the squares is related to the other three?squares. You may wish to reinforce the vocabulary they use by modeling it with demonstration squares. You may wish to ask the students to describe the results of the reflection and translations in written form.Unit OverviewPaper Quilts Lesson 4Bibliography:"Paper Quilts."?Illuminations. National Council of Teachers of Mathematics, 2011. Web. 25 Oct 2011. < concept: RotationsGrade Level:3-5Materials:4 black and white copies of selected quilt squares (per student)?Crayons or markers?6-inch by 6-inch work mat divided into 4 equal parts (one per student)?Glue?Large sheets of paper?Paper Quilts Bibliography?Square in a Square Quilt Block TemplateDescription:This lesson gives students a chance to learn about the geometric concept about rotations. Students make their own designs with the materials they are given to make a paper quilt that they can use to see how see what the geometric transformation of a rotation is. They will see the end results of turning a square and look at the sequence of turns they make. They should achieve a better understanding of how and what a rotation is through this activity. Paper Quilts Lesson 4Unit OverviewLesson 1Lesson 2Lesson 3Lesson 5Lesson 6Exploring TurnsThis lesson encourages students to explore the geometric transformation of rotation. Students create a design then, using turns, make a fourpart paper "mini-quilt" with that design as the basis. While the formal term is rotation, the more informal turn is used at this grade band. The experience focuses students’ attention on the changes the geometric transformations make in a student-designed quilt square.Learning Objectives?Students will:explore the results of turning a square into a new positionexplore the results of a sequence of turnsbe able to name the geometric transformations used to create a given design using informal languageMaterials?4 black and white copies of selected quilt squares (per student)?Crayons or markers?6-inch by 6-inch work mat divided into 4 equal parts (one per student)?Glue?Large sheets of paper?Paper Quilts Bibliography?Square in a Square Quilt Block TemplateInstructional PlanTo set the stage for this lesson, you may wish to read?Eight Hands Round, or another of the books listed in the?Paper Quilts Bibliography, calling attention to any quilt squares which show turns. Then display a quilt block such a "Square in a Square", one of the simpler designs which can be found in the template.Square in a Square Quilt Block Template.Give each child four black-and-white copies of the selected quilt square and crayons or markers. Then ask them to color one square using any colors they wish. Then have them color the other four?squares in the same way. To help focus discussion, you may wish to display a model on the chalkboard or overhead or bring in an actual quilt.As in the previous lesson, provide each child with a 6"?×?6" work mat which has been divided into four?equal parts. Ask the students to place one of their squares in position?1, next ask them to place an identical square face up on top of it. Caution students to place the top square so that the designs are in the same position (or orientation). When they are ready, have them turn the top square a half turn, then place it directly to the right of the first square, in position?2. Encourage students to find as many ways as they can of how the second square is like the first. Then ask them to discuss how the second square is different from the first.Continue by asking the children to place a third square face up on top of the square in position?2, being careful that the designs are aligned, and then to turn this top square one half turn and place it in position?3. Ask them to compare the squares in positions?1 and?3. [They will be in the same orientation.] Encourage students to find as many ways in which the third square is like the first. Then ask if they see any differences. Finally, ask them to align the fourth square face up on top of the square in position?3, rotate it a half turn and place it directly under square 1 to complete a large square with four?rotated sections. Ask students to share the likeness and differences they notice. [The squares in positions?2 and?4 will be identical.]If the students have been successful, ask them to put the 4?squares into a pile, place one of the squares in position?1, then explore what happens when they turn the squares only one-fourth of the way to make a different fourpart design. You may direct students to record one of the fourpart designs they found by gluing the squares on a large sheet of paper. [Squares in positions?1 and?3 will be flips of each other, as will squares in positions?2 and?4.]When the children are ready, call them together to share one of their designs and describe how each of the squares is related to the other three?squares. You may wish to reinforce the vocabulary word "turn" they use by modeling it with demonstration squares. You may wish to ask the students to describe the results of the rotations in a written form.Angle SumsBibliography:"Angle Sums."?Illuminations. National Council of Teachers of Mathematics, 2011. Web. 25 Oct 2011. < concept: AnglesGrade Level:6-8Materials:Access to computer (student observations and explorations will be made on the computer)Description:This is an activity where students will be able to explore the different angles in a triangle, quadrilateral, pentagon, hexagon, heptagon, and an octagon. Also they will have a chance to explore the relationships between these shapes such as the relationships between their sides and the sum of the interior angles. They will take each figure to make different shapes and sizes to make assumptions about each of the figures. Angle Sums??Examine the angles in a triangle, quadrilateral, pentagon, hexagon, heptagon or octagon. Can you find a relationship between the number of sides and the sum of the interior angles?Instructions?Choose a polygon, and reshape it by dragging the vertices to new locations. As the figure changes shape, the angle measures will automatically update.For triangles and quadrilaterals, you can play an animated clip by clicking the image in the lower right corner. This movie will provide a visual proof for the value of the angle sum.Exploration?Begin by exploring triangles and quadrilaterals. As you drag the vertices to reshape these polygons, occasionally play the animated clip in the lower right corner.Is there a different result for different shapes?Test your conjecture by creating a square, rhombus, parallelogram, and other quadrilaterals. For each type of shape, play the movie. How does the result change?Use the applet to examine pentagons, hexagons, heptagons and octagons, too. Do you notice a pattern? How does the sum of the angles change as the number of sides changes?What happens to the angle sum as you reshape the polygon by moving the original vertices?Find a formula that relates the number of sides?(n) to the sum of the interior angle measures???Constructing Rectangular PrismsBibliography:Alejandre, Suzanne. "Constructing Rectangular Prisms."?Math Forum. Drexel University, 1994-2011. Web. 25 Oct 2011. < concept: PolyhedraGrade Level:6-8Materials:2 sheets of lavender^ paper2 sheets of purple^ paper4 scissors and rulersTape^or any two contrasting colors of paperDescription:This activity is for students to begin to understand what a polyhedral is by constructing their own. They are to work in group and each group is to make three rectangular prisms and they are evaluated on how well their measurements are. This allows the students to see what a polyhedral is and what properties is possess. Constructing Rectangular PrismsStudents work in groups of 4.Materials:2 sheets of lavender* paper (construction paper or a similar weight)2 sheets of purple* paper (construction paper or a similar weight)4 pair scissors4 rulersScotch tape*or any two contrasting colors of paperProcedure:Each group is to make three rectangular prisms.Each group member measures and cuts out his or her part of the prism.?[Measurements for each prism are given below.]The 6 sides are taped together to form a rectangular prism.Evaluation:As each rectangular prism is finished, the group "checker" takes the product to?Quality Control?(the teacher, who has a ruler!) and the prism is measured.The product is marked perfect or imperfect.Groups are challenged to make at least?one?perfect product.This gets the students working together, thinking of face, edge, vertex, and three dimensions.?Rectangular prism ONEGroup member?1?makes a rectangle 4 inches by 2 inches using lavender paper.Group member?2?makes a rectangle 4 inches by 2 inches using lavender paper.Group member?3?makes two rectangles 4 inches by 6 inches using purple paper.Group member?4?makes two rectangles 2 inches by 6 inches using purple paper.Rectangular prism TWOGroup member?1?makes a rectangle 3 inches by 4 inches using lavender paper.Group member?2?makes a rectangle 3 inches by 4 inches using lavender paper.Group member?3?makes two rectangles 3 inches by 5 inches using purple paper.Group member?4?makes two rectangles 4 inches by 5 inches using purple paper.Rectangular prism THREEGroup member?1?makes a rectangle 2 inches by 3 inches using lavender paper.Group member?2?makes a rectangle 2 inches by 3 inches using lavender paper.Group member?3?makes two rectangles 2 inches by 4 inches using purple paper.Group member?4?makes two rectangles 3 inches by 4 inches using purple paper.Cube NetsBibliography:"Cube Nets."?Illuminations. National Council of Teachers of Mathematics, 2000-2011. Web. 25 Oct 2011. < concept: NetworksGrade Level:3-8Materials:Student sheet on cube nets or use the cube net activity on the web pageDescription:This activity is for students to see how a two-dimensional net can be folded into a three-dimensional cube. Students should also be aware of the properties such as the area of the net and the cube that was made. This activity also encompasses real world applications because building cardboard boxes is the same concept as nets. Cube NetsA?net?is a two-dimensional figure that can be folded into a three-dimensional object. Which of the nets below will form a cube?Instructions?Click on any net. Nets that are able to be folded into a cube will change color.If you click on a net that cannot be folded into a cube, an explanation will be provided.Exploration?The net below can be folded into a cube. Do you see it? In your mind, try to figure out how it happens. Then, click on the image to watch the net fold into a cube.There are exactly eleven nets that will form a cube. Which of the figures below can be folded into a cube? Click on the figures to find out. Those that form a cube will change colors. For the others, you will be told why they won't work.??Polygon CaptureBibliography: Carroll, William "Polygon Capture."?Illuminations. National Council of Teachers of Mathematics, 2000-2011. Web. 25 Oct 2011. < Concept:Classifying polygonsGrade Level:6-8Materials:Polygon capture game rules, cards, and game polygonsDescription:Polygon capture is a game where students, after completion should be able to classify polygons better. This activity will get students actively engaged in classifying polygons because the students can compete while learning the material. Progress should be made from basic geometric properties to students understanding relationships between geometric properties.Polygon CaptureIn this lesson, students classify polygons according to more than one property at a time. In the context of a game, students move from a simple description of shapes to an analysis of how properties are related. This lesson was adapted from an article which appeared in the October, 1998 edition of?Mathematics Teaching in the Middle School.Learning Objectives?Students will:precisely describe, classify, and understand relationships among types of two- and three-dimensional objects using their defining propertiescreate and critique inductive arguments concerning geometric ideas and relationshipsprogress from description to analysis of geometric shapes and their propertiesMaterials?Polygon Capture Game Rules?Polygon Capture Game Cards, photocopied onto cardstock?Polygon Capture Game Polygons, photocopied onto cardstock?Instructional PlanThe purpose of this game is to motivate students to examine relationships among geometric properties. From the perspective of the Van Hiele model of geometry, the students move from recognition or description to analysis (Fuys 1988). Often, when asked to describe geometric figures, middle school students mention the sides ("The opposite sides are equal") or the angles ("It has four right angles"), but they rarely use more than one property or describe how two properties are related. For example, is it possible to have a four-sided figure with opposite sides not equal and four right angles? Or a triangle with three right angles? What geometric relationships make such figures possible or impossible? By having to choose figures according to a pair of properties, players go beyond simple recognition to an analysis of the properties and how they interrelate.Choosing all figures in the?Polygon Capture Game Polygons?sheet that have parallel opposite sides is relatively easy. Choosing all figures with parallel opposite sides?and?at least one obtuse angle requires reasoning, and a good analysis of such figures leads to the inference that all nonrectangular parallelograms have these two properties, as does the regular hexagon.??Another purpose of the game is to give students a format for using important geometric vocabulary-parallel, perpendicular, quadrilateral, acute, obtuse, and right angle-in a playful situation. The basic game is described below and is followed by warm-ups and extensions.To get ready for the game, distribute copies of?Polygon Capture Game Rules,?Polygon Capture Game Cards, and?Polygon Capture Game Polygons. You will need only one copy of each master for every two students. Before introducing the game, have the students cut out the polygons and the cards. They should also mark each card on the back to designate it as an "angle" or "side" card. The eight cards from the top of?? HYPERLINK "" \t "_blank" Polygon Capture Game Cards?sheet should be marked with an "A" for angle property; the eight cards from the bottom should be marked with an "S" for side property.Before the game, assess the students' familiarity with the vocabulary used in this game, such as?parallel, perpendicular, polygon,?and?acute angleby engaging students in a class discussion in which they define, illustrate, or find examples of the geometry terms.?Basic Rules of the GameHave the students read the rules on?Polygon Capture Game Rules sheet.left0Polygon Capture Game RulesTeachers have found it helpful to begin by playing the game together, the teacher against the class. You may want to do so a few times until the class is confident about the rules. For the first game, remove the Steal Card to simplify the game.To introduce the game as a whole-class activity, lay all twenty polygons in the center of the overhead projector. Students may lay out their shapes and follow along. An introductory game observed in one of the classrooms?(as shown in step 4, below)?proceeded as follows.The teacher draws the cards?All angles have the same measure?andAll sides have the same measure. She takes figures D, G, Q, and S, placing them in her pile and out of play.Students then pick the cards?At least two angles are acute?and?It is a quadrilateral. They choose figures I, J, K, M, N, O, and R.On her second turn, the teacher picks the cards?There is at least one right angle?and?No sides are parallel. She chooses figures A and C and then asks students to find a figure that she could have taken but forgot. One student points out that figure H has a right angle and no parallel sides. Other students are not sure that this polygon has a right angle, which leads to a discussion of how they might check.The students then proceed to take two new cards.?(a) Teacher selects cards.?Angle card: All angles have the same measure.?Side card: All sides have the same measure.(b) Students select cards.?Angle card: At least two angles are acute.?Side card: It is a quadrilateral.(c) Teacher selects cards.?Angle card: There is at least one right angle.?Side card: No sides are parallel.(d) Students capture piece that teacher missed.Sample Steps in a Game?When no polygons remain in play that match the two cards chosen, the player may turn over one additional card-either an angle or a side card. This move calls for some planning and analysis to determine whether an angle card or a side card is most likely to be useful in capturing the most polygons. If the player still cannot capture any polygons, play moves to the opponent. When all cards in a deck are used up before the end of the game, they are reshuffled. Play continues until two or fewer polygons remain. The player with the most polygons is the winner.When the "Wild Card" is selected, the player may name whatever side property he or she wishes; it need not be one of the properties listed on the cards. Again, a good strategy to capture the largest number of polygons requires an analysis of the figures that are still in play.Steal CardWhen the "Steal Card" comes up, a card from the deck is not drawn. Instead, the player has the opportunity to capture some of the opponent's polygons. The person who has chosen the Steal Card names two properties (one side and one angle) and "steals" the polygons with those properties from the opponent. The students may select their own properties, not necessarily those on the game cards. If the opponent has no polygons yet, the Steal Card is put back in the deck and a new card chosen.Teacher NotesOne interesting aspect of the game is the various strategies that students use. Some students go through the figures one at a time, using a trial-and-error method to match them to properties on the cards. Some students perform two sorts; they find the polygons that match the first card and, of this group, those that also match the second card. Others seem to analyze the properties and mentally visualize the polygons that are possible. In analyzing properties ("Is this angle acute?"), students quickly learn to use angles and sides in other figures as benchmarks, for example, using the right angle in a rectangle to check whether a triangle has a right angle. Generally classes play with no time limits, although students could choose a limit as an option.Pigging OutBibliography: "Pigging Out."?Illuminations. National Council of Teachers of Mathematics, 2000-2011. Web. 25 Oct 2011. < Concept:Similarities and DifferencesGrade Level:3-5Materials:Activity sheet, Venn diagram, any version of the three little pigs story, three 20ft pieces of yarn, scissors and tape for each group of fourDescription:This activity uses Venn diagrams for students to see similarities and differences with geometric objects. With the use of Venn diagrams students should be able to infer and draw conclusions easier in geometry. They will also get a chance to compare their work with their classmates to analyze what is similar or different about their maps. Pigging OutThis lesson uses the story of?The Three Little Pigs?to motivate students to think and reason mathematically in a number of ways. Students develop reasoning skills and identify similarities and differences through the use of Venn Diagram. Spatial reasoning is also emphasized in this lesson.Learning Objectives?Students will:develop reasoning skills through the use of Venn diagramsidentify similarities and differencesdraw and interpret a simple mapmeasure distances using concrete objectsMaterials?Pigging Out Activity Sheet?Pigging Out: Venn Diagram Overhead?Any version of the three-little-pigs story (see?Bibliography)?Three twenty-foot pieces of yarn?Scissors and tape for each group of four students?Multicultural version of the story (see?Bibliography)Instructional PlanAsk students to decide what each would use - straw, wood, brick, or a combination of two or all three - to build a house for themselves. Have each child record this decision by marking his or her initials in the region on the Venn diagram that they believe shows this preference on thePigging Out?activity sheet.Pigging Out Activity Sheet?On the classroom floor, form three large intersecting loops of yarn to match the Venn Diagram on the activity sheet. Have each child stand inside the loop or loops that he or she believes represents the preference stated above. Discuss the preferences of the class as a whole.Use the?Pigging Out: Venn Diagram?overhead transparency = to tabulate the results by putting each child's initials in the appropriate place.??Each child should then compare the location marked on the page with the place he or she was standing in the yarn circles. Discuss what it means to have a child in each of the seven regions. Where would a child stand who chooses none of the three materials? [Outside the three circles.]For an experience involving estimation and graphing, ask students to recall that the wolf "huffed and puffed" a number of times in the story. Have each student cut, fold, and tape the house pattern on the?Pigging Out Activity Sheet.Ask the students to estimate how far they can blow the house across the floor. Have groups of four record estimates and then conduct the experiment. What would happen if these houses were made with different materials, such as construction paper, newspaper, or interlocking blocks?To reinforce measurement and map skills, have students create a map within the boundary on the second page of the?Pigging Out Activity Sheet. Next they mark with an "sh," "wh," and "bh" - for straw house, wood house, and brick house, respectively - where they think the pigs in the story built their houses. They should also indicate with a "w" where they think the wolf might have lived. Identify a standard unit of measure, such as a centimeter cube, with which to measure distances on the map.??Have pairs of students compare their measurements and their maps to explore similarities and differences. For example, two students with similar-looking maps would have similar distances between houses, but it is possible that two students with similar distances between houses may have very different-looking maps. Ask pairs of children to sit back-to-back and have one child describe his or her map while the other student attempts to draw it next to his or her map on the second page of the?Pigging Out Activity Sheet. Have students answer the questions on the activity sheet and discuss their results with the class.Get the Turtle to the PondBibliography: "Get the Turtle to the Pond."Illuminations. National Council of Teachers of Mathematics, 2000-2011. Web. 25 Oct 2011. < Concept:Establishing angle measure and estimating lengthGrade Level:K-2Materials:Turtle Pond AppletDrawing PaperDescription:This activity is for younger children that have not developed a good understanding of length and angle measurement. Students use the turtle pond applet to create a path for the turtle and analyze their paths. Students will compare their path to others to see who achieved the destination in the shortest length. Get the Turtle to the PondThis activity provides opportunities for creative problem solving while encouraging young students to estimate length and angle measure. Using the?Turtle Pond Applet, students enter a sequence of commands to help the turtle get to the pond. Children can write their own solutions using LOGO commands and input them into the computer. The turtle will then move and leave a trail or path according to the instructions given.Learning Objectives?Students will:Use directional words, such as right, left, forward, and back, to create and explore pathsCreate a path using an appletCompare their paths with other students and determine the shortest path to a destinationMaterials?Turtle Pond Applet?Drawing paper?Instructional PlanBegin the lesson by reviewing the following directional words:Right?Left?Forward?Back?Place students in pairs. Give one student a piece of blank drawing paper. He or she will follow the directions given by the other student.The other student may give directions such as the following:Draw a turtle (or other picture) in the middle corner of your paper.Move the turtle forward (up) 3 inches (or any similar measurement your students are familiar with.)Make a right turn.Move the turtle forward 3 inches.Make a right turn.Move the turtle forward 3 inches.Make a right turn.Move the turtle forward 3 inches.In the example just given, students should notice that their turtle lands at the starting point.Within each pair, students should take turn giving and following directions for their turtles.Next, project the?Turtle Pond Applet?for the students.left0Turtle Pond AppletShow students Path 1, as demonstrated on the applet.In pairs, students can explore Path 2, as demonstrated on the applet.Students can try out either path by checking the box and pressing "draw". To erase a path, press "clear". To create a new path, adjust the length by using the blue sliders and then press he "forward", "back", or "turn" buttons.In an activity like the one described above, once children have found paths to the pond, they can share their programs with other students. Discussing their programs and the turtle paths with other students helps children reflect on their own method of solving the problem and on the relationships between distance and turtle movement and angle and turtle movement.Virtual Pattern BlockBibliography: Hargrove, Tracy. "Virtual Pattern Blocks."?Illuminations. National Council of Teachers of Mathematics, 2000-2011. Web. 25 Oct 2011. < Concept:Fraction RelationshipsGrade Level:3-5Materials:Patch tool (online)Student sheetDescription:Students will be using virtual pattern blocks to obtain a greater understanding about parts of a whole. By using these blocks they will be able to see the relationships between fractions and parts of a whole. With the patch tool online activity students will create their own observations and gain reasoning about the identity of fractions and their relationships.Virtual Pattern BlocksStudents use virtual pattern blocks to problem solve and reason with fractions. They investigate relationships between parts and wholes using another representation of a region model, virtual fractions. Students use conversation to explain their understandings in order to extend and clarify their mathematical content knowledge.Learning Objectives?Students will:demonstrate understanding that a fraction is part of a wholeidentify fraction relationshipsMaterials?Patch ToolRegion Relationships 1 Activity Sheet?(new, blank copy for each student)Instructional PlanExplain to students that they will the?Patch?tool to model part-whole relationships. The directions for using the tool should be reviewed prior to this lesson. The students may need to be guided in how to drag the pattern blocks to the work area. When making comparisons, the students can drag one pattern block on top of another one to compare the area of the region. If your technological resources are limited, you can also create and print pattern block activity sheets using the?Dynamic Paper?tool. Pattern blocks can be found under the Shapes tab.Patch ToolHave students work in pairs using the Web-based pattern blocks to explore relationships among the four shapes. Use the same Guiding Questions found in Lesson One. This will facilitate exploration with a new tool, help the students focus on the mathematical concepts demonstrated with this region model, and reinforce the content of the previous lesson,?Investigating with Pattern Blocks.The students should use the virtual pattern blocks to answer the questions. Provide students with a new, blank copy of the?Region Relationships?1?activity sheet and ask them to complete it.left0Region Relationships 1 Activity SheetEncourage students to discuss various options with their partner. This allows students to articulate their understandings, making them available for discussion, clarification, and extension.Using the Core Learning Goals: GeometryBibliography: "Using the Core Learning Goals: Geometry."?School Improvement in Maryland. N.p., 1997-2010. Web. 25 Oct 2011. < Concept:Similarity and CongruenceGrade Level:6-9Materials:Worksheets/Answer key, calculator paper ruler formula sheet, protractor, patty paperDescription:Students are to investigate figures that are given to them and prove that the two figures are similar or congruent by using geometric terms. Once the activity is complete students should be able to address how figures are alike and how they are different. GEOMETRY, MEASUREMENT, & REASONING?|Lesson Plan:?Lesson plans were written by Maryland mathematics educators and could be used when teaching the concepts.Goal 2?Geometry, Measurement, And ReasoningExpectation 2.1?The student will represent and analyze two- and three-dimensional figures using tools and technology when appropriate.Indicator 2.1.1?The student will analyze the properties of geometric figures.Lesson ContentSimilarity and CongruenceObjectiveStudents will know how to determine that two figures are similar or congruent by investigating figures that are similar and figures that are congruent. Then they will know how to prove that two figures are similar or congruent by using definitions, postulates, and theorems.Other Indicators Addressed2.2.1?The student will identify and/or verify congruent and similar figures and/or apply equality or proportionality of their corresponding parts.Approximate TimeOne 30 minute lesson for the discovery lesson and up to 60 minutes for the practice, depending on how well students remember the pre-requisite skills.Prerequisite Concepts NeededStudents will need to be able to measure segments using a ruler, measure angles using a protractor, and be familiar with the theorems and postulates used to prove figures similar and congruent.Materials NeededWorksheet/Answer Key:?Prediction GuideWorksheet/Answer Key:?Comparing Sizes of FiguresWorksheet/Answer Key:?Congruent and Similar Triangle Investigation Activity OneWorksheet/Answer Key:?Congruent and Similar Triangle Investigation Activity TwoWorksheet/Answer Key:?Congruent and Similar Triangle Investigation Activity ThreeWorksheet/Answer Key:?Congruent and Similar Triangle Investigation Activity FourWorksheet/Answer Key:?Practice with Congruent and Similar TrianglesWorksheet/Answer Key:?Applicationscalculatorpaperrulerformula sheetprotractorpatty paperLesson StructureWarm-Up/Opening ActivityUse the?Prediction Guide?to determine what students already know. Feel free to add/delete/change statements as they fit your class.?Accept students' answers and justifications and ask if the class thinks they are reasonable.Development of IdeasWorksheet:?Comparing Sizes of Figures?Ask students to work in groups of 2 or 3 to measure all of the segments listed using a ruler. It is helpful if all students measure using the same unit (suggest measuring in centimeters). It may be helpful to have the students change their ratios to a decimal (about 1.4) to show that the ratios are equal.?Explorations/Investigations?Present the concept and discover the SSS and SAS postulates of congruency and the SSS, SAS and AA postulates of similarity.?Activity One?Worksheet:?Congruent and Similar Triangle Investigation Activity One?Organize students into groups of two. Each student needs to cut out the three strips of paper. Place the strips together corner-to-corner to create a triangle. Compare your triangle to your partner's and to other triangles in the class. Are they congruent??Encourage the students to prove the triangles congruent using the definition of congruent triangles. Measure each side with a ruler and each angle with a protractor (measuring each side may not be necessary if students realize that everyone started with the same three strips of paper).?What was the minimum amount of information used to create these two congruent triangles? [three congruent sides- this demonstrates SSS theorem of congruency]?Can you replicate this process in the same manner? Explain. [Allow students to create another set of congruent triangles if they are not convinced. You can use the beginning of Activity Two to show this congruence.]?Activity Two?Worksheet:?Congruent and Similar Triangle Investigation Activity Two?Only ONE person in each group: Using the three strips of paper from Activity One, fold and cut each strip of paper in half. Create a triangle using the one-half pieces of each of the original strips. Compare this triangle to the first one. Describe any similarities and differences. [the two triangles are not congruent, but are the same shape. These two triangles are similar. You may want to emphasize that all of the NEW triangles are congruent to each other- have students prove this to you using SSS] Help students trace the new triangle and label the sides the same as in Activity One.?Students will justify the similarity of the old and new triangles. Have students measure each side of the triangle and place the measure in the space requested on the worksheet. Note that the ratios of the lengths of each pair of corresponding sides are proportional.?Have students measure each pair of corresponding angles. Note that the measures of each pair of corresponding angles are congruent. The answers in #4 and #5 show that the triangles are similar.?Activity Three?Worksheet:?Congruent and Similar Triangle Investigation Activity Three?Each student will copy the two sides and included angle using patty paper. Students should now draw segment AC to create a triangle. Lead a discussion about what parts of the triangle were given and how their measures compare with everyone else's [SAS].?Now, students will compare their triangle to the other triangles in the group. Have students justify that the triangles are congruent – or similar [by definition they are both. Encourage them to show that the corresponding three sides of the triangles are congruent and therefore the triangles are congruent.]?Ask students to locate the midpoint of?AB?and of?BC?and label these points D and E, respectively. Draw segment DE. Compare this triangle to triangle ABC. [the two are similar]. Use the definition of similar triangles to justify that the triangles are similar.?Lead a discussion about what parts of the triangle were given and how this is a minimal amount of information needed in order to prove two triangles similar [SAS].?Activity Four?Worksheet:?Congruent and Similar Triangle Investigation Activity Four?Students will now trace two angles and use these to create two triangles. These directions may be difficult for students to follow- please try to demonstrate using the overhead and transparencies.?Have the students create two different triangles. Lead a discussion about why the two triangles are not congruent and that AA [or AAA] is not a way to prove two triangles congruent.?The two triangles they create will be similar. Give them time to measure the sides and show that the ratios of the corresponding sides are proportional. This will be a great revelation because the ratios of the corresponding sides will be different than the others they have seen in these activities (all of the former being 2:1). Also have the students show that the three angles are congruent to each other.?Lead a discussion about how the minimum amount of information needed to prove two triangles similar is that two corresponding angles must be congruent. [AA]?By the time they finish these investigations, they will be ready to believe that the ASA theorem of congruence works, or you can show this using a demonstrations. Discuss that ASA works for similarity as well and can really be the same as the AA theorem.?Worksheet:?Practice with Congruent and Similar Triangles.?Worksheet:?ApplicationsClosureGo back to the Prediction Guide. Allow students to make corrections to their Prediction Guides since they have worked through this lesson. Discuss any corrections and the correct answers.Measuring UpBibliography: Carbone, Katie. "Measuring Up."Illuminations. National Council of Teachers of Mathematics, 2000-2011. Web. 25 Oct 2011. < Concept:Ratio’sGrade Level:6-8Materials:Chart paper, markers, measuring tape, yardstick, stringDescription:Students after this activity should be able to understand the concept of what a ratio is and how they are in real life. Students will take measurements of them and apply it to what they have leaned about ratio’s and they will explore the Golden Ratio and how it’s related to many things around us.Measuring UpThe Golden RatioStudents learn about ratios, including the “Golden Ratio”, a ratio of length to width that can be found in art, architecture, and nature. Students examine different ratios to determine whether the Golden Ratio can be found in the human body.Learning Objectives?Students will:understand the concept of ratioexplore ratios dealing with the human body by taking measurements and calculating the ratiosexplore the Golden RatioMaterials?Chart paper?Markers?Measuring tape?Yardstick?StringInstructional PlanTo assess students’ prior knowledge, ask students to recall where they have heard of or used ratios before. Their answers may include the following:Ratios compare the relative sizes of two numbers.Ratios compare wins to losses in the World Series; for example, the Braves are ahead in the series three games to one.Ratios can describe the number of students to a teacher in a classroom or the number of boys to girls in a classroom.Record student answers on chart paper for future reference.Discuss the meaning of ratio in each of the examples from the class list. Explain that a fraction expresses a ratio when it is written as a quotient of one number divided by the other number, and that there are several ways to write ratios. Note the ways in which students have used fractions to express such anize the class into groups of three or four. Provide a measuring tape or yardstick and some string (about two yards or so) to each group. Encourage students to assist one another in measuring their total body heights with the string, as well as their heights from their feet to their navels. Have them record their measurements on a chart.Have students write their two measurements as a ratio. Explain the history of the Golden Ratio (The Golden Ratio is a ratio of the length to width and is approximately 1.618. This ratio not only appears in art and architecture, but also in natural structures including the human body). Then describe that the Golden Ratio is the ratio of a person’s total height to height from their feet to their navels. The students may have heard of this as well as other ratios said to exist in the human body.??See how the students “measure up” to the Golden Ratio.As an independent assignment, invite the students to measure other lengths, such as the distance from their waist to the floor and from the top of their head to their waist, to see whether a similar ratio exists between those measurements. This activity provides extra practice in writing ratios and in developing a better understanding of how ratios are used.Enter the class data on a chart. Graph the ratios by individual student and compare the different groups of students, such as girls to boys or students of one age to students of another.Covering the Plane with Rep-TilesBibliography: "Covering the Plane with Rep-Tiles."?Illuminations. National Council of Teachers of Mathematics, 2000-2011. Web. 25 Oct 2011. < Concept:Similarity of objects using transformationsGrade Level:6-8Materials:Students sheet, construction paper, scissors, pattern blocksDescription:Students will experiment will different shapes to gain a better visualization of what spatial sense is. After examining the similarity of objects using transformations students should be able to identify and create rep-tiles with polygons. Covering the Plane with RepTilesStudents discover and explore a special kind of tiling of the plane. Reptiles are geometric figures such that?n?copies can fit together to form a larger, similar figure. Students experiment with various shapes and values of?n. Spatial sense is encouraged by the need to visualize and perform transformations with the shapes involved. This lesson was adapted from an article by Linda Fosnaugh and Marvin Harrell, which appeared in the JanuaryFebruary 1996 edition of?Mathematics Teaching in the Middle School.Learning Objectives?Students will:precisely describe, classify, and understand relationships among types of two-dimensional objects using their defining propertiesexamine the similarity of objects using transformationsidentify and create rep-tiles using familiar polygonsMaterials?RepTiles Activity Sheet?Construction Paper and ScissorsPattern Blocks or Colored Transparency Sheets (optional)Instructional PlanActivitating Prior KnowledgeBegin the lesson by asking students where they have seen examples of tiling. Some common uses of tilings include the covering of walls, floors, ceilings, streets, sidewalks, and patios.?Background Information about RepTilesA tiling is a partitioning of the plane into regions, or tiles, as suggested by Grunbaum and Shephard (1987). It is commonly known that any triangle, quadrilateral, and regular hexagon will tile the plane. Golomb (1964) suggests an unusual type of tile, a reptile, that tiles the plane.A reptile is a geometric figure whose copies can fit together to form a larger similar figure. Another way that one can think of a rep-tile is as a puzzle piece, where a larger similar figure is the entire puzzle. For example, it is well known that four congruent squares fit together to form another square. The smaller square, the puzzle piece, is a reptile. In addition, four copies of the large square can be fitted together to form an even larger square. By repeating this process infinitely many times with still larger squares, we can tile the plane.Another example is the triangle. If one makes copies, or replicas, of any triangle, four of these copies can be fitted together to form a larger triangle similar to the original triangle. One could continue this process with the larger triangle and then again with still a larger similar triangle. Again, if this process is continued infinitely many times, the plane will be tiled. (a) Note that four congruent triangles create a similar figure; thus we call this tile a rep4 tile. Similarly, as can be seen in (b), four copies of any parallelogram fit together to create a similar parallelogram. Hence, any parallelogram is a rep4 tile.??(a)(b)Two Rep4 Tiles: a Triangle and a Parallelogram?In the figure below, additional reptiles are illustrated. The first image (a) shows a rep2 tile formed by joining two isosceles right triangles, whereas (b) illustrates a rep3 tile in which three 30o-60o-90o?triangles are fitted together to form a larger 30o-60o-90o?triangle. The sphinx shape (c) is made up of six equilateral triangles and is a rep4 tile. Nevada (d) is formed by attaching an isosceles right triangle to a square and repeating this shape four times, and in figure (e) we have a right triangle whose sides measure 1, 2, and the square root of 5. As one can see from the two figures above and the five figures directly below, although simple examples of reptiles exist, one quickly finds oneself working with more complex geometric figures.??(a) Rep2 tile(b) Rep3 tile(c) Rep4 tile(d) Rep4 tile(e) Rep5 tileAdditional Examples of RepTiles?It should be noted that a repntile exists for any natural number?n?>?1. In other words, for any natural number?n?>?1, a tile exists in which?n?copies can be fitted together to create a larger similar figure.It is interesting to note that the rep4 triangle previously discussed is also a rep9 triangle. We can show this fact by simply adding a row of five small congruent triangles to the base of the large triangle found in the initial figures, and also (a) below. One interesting fact about reptiles is that each of our rep4 tiles is also a rep9 tile. Conversely, each of the rep9 tiles is also a rep4 tile. The second and third images (b) and (c) below also illustrate this fact.??(a)(b)(c)Rep9?Tiles that were shown earlier to be Rep4?Tiles??Reptiles not only present an alternative way to tile the plane but also help students acquire spatial sense.?Introducing the ActivityAfter the class reviews such concepts as tilings (tessellations), congruent figures, and similar figures, the teacher can introduce reptiles using squares, equilateral triangles, and parallelograms. For example, pattern blocks on an overhead projector can be used to illustrate reptiles. Otherwise, cutouts from colored transparency sheets or cardboard figures work just as well. After having the students form small groups, the teacher can then distribute copies of theRepTiles?activity sheet.left0RepTiles Activity SheetAfter students have worked through the activity sheet, a class discussion summarizing their findings is beneficial. Students should realize that any of the similar figures, no matter their size, can be used to tile the plane.?ConclusionWith the introduction of reptiles in a classroom setting, students can visualize and represent geometric figures with special attention to developing spatial sense as well as explore transformation of geometric figures. Creating a tiling pattern requires such mental imagery as visualizing the possible rotations and placements of a tile in the tiling pattern, thus further developing spatial sense. In addition, the construction of a tiling with reptiles allows students to see how congruent copies of a polygon can be fitted together to form a larger similar copy of the polygon. After discovering the pattern, the students may see that each tile can be dissected into smaller, similar copies of itself, thus discovering how any of the similar figures can be used to generate an entire tiling. ................
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