Curriculum Design Project with Virtual Manipulatives. Gwenanne Salkind. George Mason University EDCI 856. Dr. Patricia Moyer-Packenham

Transcription

1 Curriculum Design Project with Virtual Manipulatives Gwenanne Salkind George Mason University EDCI 856 Dr. Patricia Moyer-Packenham Spring 2006

2 Curriculum Design Project with Virtual Manipulatives Table of Contents Lesson 1: Which is More Likely?...3 Implementation...6 External Review...6 Lesson 2: Is It Fair?...8 Implementation...11 External Review...11 Lesson 3: Problem Solving with Spinners...13 Student Work Sample...20 Implementation...22 External Review...22 Lesson 4: Secret Spinners...23 Student Work Samples...33 Implementation...39 External Review...39 Lesson 5: Space Blocks...40 Student Work...47 Implementation...51 External Review...51

3 3 Lesson 1 Title: Which Is More Likely? This lesson was adapted from Hide & Spin lessons in Data, Chance, and Probability (Jones & Thornton, 1992). Contributor s Name: Gwenanne Salkind Grade Level Band: K-2 NCTM Mathematics Standard: Data Analysis and Probability Students should discuss events related to students experiences as likely or unlikely. Lesson Objectives: Conduct simple probability experiments using spinners Identify events that are likely or unlikely Mathematics Vocabulary: probability, possible, impossible, likely, unlikely, outcome Virtual Manipulative Web Site: National Library of Virtual Manipulatives for Interactive Mathematics Grade Band Pre-K-2, Data Analysis & Probability, Spinners Materials: A computer with internet access and a presentation station Computers with internet access (one for each pair of students) Red and blue unifix cubes Discussion of the Mathematics: This lesson helps students recognize that some events are more likely than others. The warm up activity involves situations that occur in everyday life. In the rest of the lesson, students explore a spinner that has a greater chance of spinning blue than red. Approximate Duration of Lesson: 30 minutes Preparation: Prepare a spinner like this one on each computer that will be used in the lesson. (Or you can teach students how to modify the virtual spinner on the website.)

4 4 Procedures: 1. Warm up. Ask children to stand to show their vote for the following questions: On a rainy school day, which is more likely? Students will have umbrellas./students will wear swimsuits. On a cold day, which is more likely? Students will order hot lunches./students will bring their lunches. Ask a few students to explain why they voted for each outcome. Focus on the vocabulary: possible, impossible, certain, likely, and unlikely. Might students wear swimsuits to school? Is it possible? Is it likely? 2. Show the virtual spinner to the whole class. Ask: Which color has a better chance if you spin the spinner one time? (Thumbs up for red, thumbs down for blue.) Ask a few students to explain why they chose the color they did. 3. Divide the class into pairs. Tell the students that they will work with their partner to explore this spinner. When they go to their computers they will spin the spinner to see what color they get. Each time they spin the spinner, they will take a red or blue cube to show what color the spinner landed on. One child will collect the blue cubes. The other child will collect the red cubes. (Designate a student in each pair to collect red cubes and a student to collect blue cubes. Have a container of red and blue unifix cubes available for each pair of students.) They will keep spinning the spinner and collecting cubes until you tell them to stop. Tell the students to make sticks with their unifix cubes. This will make it easier for the students to carry their cubes back to the discussion area. Tell the students to go to the computers and begin working. 4. After about 5 minutes of spinning the spinners. Ask students to stop and bring the cubes they have collected to the discussion circle. (You will need to watch students and judge when to stop. Stop when most students have spun the spinner about 10 times. Some pairs will have spun more times, others less. You don t want students to get bored with the activity. Nor do you want too much data to deal with in the group discussion.) 5. Lead a whole group discussion. Ask: Did your spins turn out the way you thought? Why or why not? 6. Have the students group all the red cubes and all the blue cubes together. You may want to count the red and the blue cubes to see how many of each the class collected. Ask: Why did we collect more blue cubes than red cubes? If we did this activity again tomorrow with the same spinner, do you think we d collect more red or blue cubes? Why? Teacher Notes: When discussing the spinner, be sure that the students understand that there are two possible outcomes (red and blue). You may want to discuss the possible outcomes of the spinner before discussing which outcome is more likely. You could ask: What will always happen when we spin the spinner? What will sometimes happen when we spin the spinner? What will never happen when we spin the spinner? You may also want to ask: Is it possible to spin red? Is it likely to spin red? It is more likely that the spinner will land on blue than red. (There is a 3 to 1 ratio of blue to red. Students in grades K-2 do not need to identify or explain the 3 to 1 ratio. They should be able to verbalize that blue is more likely than red because there is more blue showing on the spinner.)

5 5 It is possible to make a virtual spinner with the same color in separate sections, but it is tricky. Suppose you want to make a spinner that looks like the one below. In the spinner site, click on Change Spinner. Type purple, red, green, blue, red, yellow in the text boxes. To set the colors, put your curser in the box you want to set. Click on the color you want. Here s the trick, you must set the second red section first. If you set the first red section first, it will not allow you to set the second section to red. Note: when you use this type of spinner on the site, the graph records the two red sections separately. Student Assessment: Are students able to predict which color the spinner will land on? Can students explain the reasons for their predictions? Do students use the words likely and unlikely to describe events? Student Work: In the warm up activity, student votes for the first question should be more one sided than the second question. It is much more likely that students bring umbrellas than wear swimsuits. The vote for the second question should be closer because both answers are reasonable. The cold weather may not affect the lunch count. Students may think that the spinner will land on red because it is their favorite color or because it landed on blue the last time and now it is red s turn. Students might also think that the spinner is deliberately creating a pattern. Help students to consider the reasonableness of their thinking. Be aware that the concepts involved in this lesson develop naturally over time. Students need many experiences and opportunities to discuss those experiences. If students have difficulty understanding, they need more experiences with these ideas. Extensions: Use the same lesson format to explore other spinners such as the ones shown below. References: Jones, G. A., & Thornton, C. A. (1992). Data, chance & probability: Grades 1-2 activity book. Vernon Hills, Illinois: Learning Resources, Inc. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.

6 6 Sheffield, L. J., Cavanagh, M., Dacey, L., Findell, C. R., Greenes, C. E., & Small, M. (2002). Spin it. In C. E. Greenes & P. A. House (Eds.), Navigating through data analysis and probability in prekindergarten--grade 2 (pp ). Reston, VA: National Council of Teachers of Mathematics. Implementation I implemented this lesson with a small group of first graders. They did not have any trouble understanding the math content of the lesson. One student could articulate that there was a better chance of spinning blue on the spinner, but still thought the spinner might be making a pattern when the spinner was spun. I added this idea to the student work section of the lesson. The lesson went quickly with the group, but I think it will take more time with a whole class (about a half hour). I adjusted the time frame of the partner work on the computers from 10 minutes to 5 minutes because I think 5 minutes is a better estimate of the time needed. External Review The external reviewer wrote suggestions directly on my lesson plan. These suggestions are addressed below. The external reviewer suggested that I talk about outcome in the discussion of the mathematics part of the lesson because outcome might be new vocabulary. Since vocabulary development seemed to be a concern for many of the reviewers, I added a new section to the lesson plan, called Mathematics Vocabulary. I listed the key mathematics vocabulary in this section. The external reviewer suggested that I combine procedures number 2 and 3 to make the procedures clear. I did this. The combined procedure is procedure # 3. The external reviewer suggested that I provide a container for the students to use to carry their cubes to the whole group discussion. I decided that having containers for each child would be too difficult to manage, but students could make sticks of ten with their unifix cubes. These sticks could easily be carried to the whole group discussion. I added this idea to the lesson procedures. The external reviewer also wrote some general thoughts and ideas about all the K-2 probability lessons. This text is shown on page 7. I addressed some of the comments as follows: I added a list of mathematics vocabulary to the lesson. I put the list of vocabulary at the beginning of the lesson plan so the teachers can introduce the words when they feel it is appropriate. I think the words could be introduced during the discussion part of the lesson. I added a time frame for the lesson.

7 8 Lesson 2 Title: Is It Fair? This lesson was adapted from Spin It in Navigating through Data Analysis and Probability in Prekindergarten Grade 2 (Sheffield et al., 2002). Contributor s Name: Gwenanne Salkind Grade Level Band: K-2 NCTM Mathematics Standard: Data Analysis and Probability Students should discuss events related to students experiences as likely or unlikely. Lesson Objectives: Conduct simple probability experiments using spinners Predict the results of probability experiments and test the predictions Identify events that are certain, possible, or impossible Identify events that are likely or unlikely Mathematics Vocabulary: probability, always, sometimes, never, likely, unlikely, possible, impossible, outcome Virtual Manipulative Web Site: National Library of Virtual Manipulatives for Interactive Mathematics Grade Band Pre-K-2, Data Analysis & Probability, Spinners Materials: A computer with internet access and a presentation station Computers with internet access (one for each pair of students) Red and blue unifix cubes (about 10 per student) Chart paper and markers Discussion of the Mathematics: In this lesson, students will have beginning experiences with probability. They will play games with spinners and discuss whether they think the games are fair or not. Students will be encouraged to play the games and discuss the results. They will also be asked to make predictions about the results of games with different spinners. Approximate Duration of Lesson: 45 minutes

8 9 Procedures: 1. Ask students to tell something about school that happens always, sometimes, and never. Choose two of the events they say and ask them which is more likely to happen in school tomorrow. Continue the discussion in this manner for 5-10 minutes. 2. Using the virtual spinner, create a spinner that looks like Spinner #1. Ask students to tell you about the spinner. What happens always (The arrow Spinner #1 lands on red or blue.), sometimes (sometimes red, sometimes blue), and never (It never lands on green, purple, etc.)? 3. Ask: What color do you think the arrow will land on if I spin the spinner? Why do you think that? 4. Tell students that you are going to play a game with the spinner. Divide the class into two teams. Half the class will be the red team. The other half will be the blue team. Have containers of red and blue cubes within students reach. Each time the spinner lands on red, the students on the red team take a red cube. Each time the spinner lands on blue, the students on the blue team take a blue cube. The first team to collect ten cubes wins. 5. Spin the spinner one spin at a time. Each time you spin, make sure the students on the appropriate team take a cube. Stop when a team has won (collected 10 cubes). 6. Show students the graph that has been created by the computer program. Discuss the graph. Ask: Which color did the spinner land on the most? How many red spins were there? How many blue spins were there? 7. Show the class how to create Spinner #1 on the website. 8. Divide the class into pairs. Designate a student in each pair to be red and a student to be blue. Each pair will create their own virtual spinner and play the game. After they have finished playing, ask the students to return to the discussion area for a whole class discussion. They should bring the unifix cubes they won during the game with them. Ask each pair of students to tell the class who won their game (red or blue?). Use tallies to record the results on a chart like the one below. (After recording the number of games that red/blue won, be sure to record the number of spins in each game.) Ask if the results of the game were what they expected. Why or why not? Do they think this is a fair game? Why or why not? Ask students to predict the results of the game if they played it again. (It is more likely that red would win.) Results of the Games Using Spinner #1 Number of games that blue won Number of games that red won Number of blue spins Number of red spins Total number of spins 9. Create a virtual spinner like Spinner #2. (Be sure that students are paying attention to how you create the spinner so they can do it themselves in a moment.) Ask the students what they think will happen if they play the game with this spinner. Why? Ask the Spinner #2

9 10 same pairs of students to create Virtual Spinner #2 and play the game as before. 10. When students are finished playing, lead a whole class discussion. Record the results of the games on a similar chart. Ask the students: How are the two games alike? How are they different? Is Game #2 a fair game? Why or why not? Teacher Notes: It is possible to make a virtual spinner with the same color in separate sections, but it is tricky. Suppose you want to make a spinner that looks like the one below. In the spinner site, click on Change Spinner. Type purple, red, green, blue, red, yellow in the text boxes. To set the colors, put your curser in the box you want to set. Click on the color you want. Here s the trick, you must set the second red section first. If you set the first red section first, it will not allow you to set the second section to red. Note: when you use this type of spinner on the site, the graph records the two red sections separately. Student Assessment: Are students able to predict which color the spinner will land on? Can students explain the reasons for their predictions? Can students say whether they think a game is fair or not? Can students explain why they think a game is fair or not? Student Work: Students may think that the spinner will land on red (or blue) because it is their favorite color or because it landed on blue the last time and now it is red s turn. Students may also think that the spinner is deliberately creating a pattern. Help students to consider the reasonableness of their thinking. Be aware that the concepts involved in this lesson develop naturally over time. Students need many experiences and opportunities to discuss those experiences. If students have difficulty understanding, they need more experiences with these ideas. Extensions: Have students play the game with Spinner #3. (See teacher notes for directions how to create this type of spinner. It may be necessary for the Spinner #3 teacher to create this spinner for the students to use.) Ask students to predict the results of the game before they play it. Ask if the results of this game would be like Game #1 or Game #2. Why do they think so? Ask the students to create a virtual spinner that would help blue win. Have them play the game with the spinner they created to see whether blue wins. Ask students to design spinners with more than two colors. For each spinner they design, discuss whether the game would be fair.

10 11 References: Jones, G. A., & Thornton, C. A. (1992). Data, chance & probability: Grades 1-2 activity book. Vernon Hills, Illinois: Learning Resources, Inc. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author. Sheffield, L. J., Cavanagh, M., Dacey, L., Findell, C. R., Greenes, C. E., & Small, M. (2002). Spin it. In C. E. Greenes & P. A. House (Eds.), Navigating through data analysis and probability in prekindergarten--grade 2 (pp ). Reston, VA: National Council of Teachers of Mathematics. Implementation I implemented this lesson with a small group of first graders. They did not have any trouble understanding the math content of the lesson. One student could articulate that there was a better chance of spinning red on Spinner #1 and an equal chance of spinning blue or red on Spinner #2, but still thought the spinners might be making a pattern when the spinners were spun. I added this idea to the student work section of the lesson. The lesson went pretty quickly with the small group, so I estimated how much time it would take with a full class (about 45 minutes) and added this time frame to the lesson plan. I also did the extension lesson (Spinner #3) with the students. They understood that this spinner was like Spinner #2 and created a fair game. They were able to articulate that the red and blue occupied equal space on the spinner so it was equally likely to spin red or blue. External Review The external reviewer made one suggestion on a post-it note on my lesson and listed comments and suggestions for all of the K-2 probability lessons in an . These notes are shown on page 12. I addressed these suggestions in the following ways. I added a list of mathematics vocabulary to the lesson. I put the list of vocabulary at the beginning of the lesson plan so the teachers can introduce the words when they feel it is appropriate. I added a time frame for the lesson. I added instructions to the extension lesson to help teachers deal with the difficulty of making the spinner. I suggested that the teacher make the spinner for the students to use since it is a difficult spinner to make.

11 Notes from External Reviewer: 12

12 13 Lesson 3 Title: Problem Solving with Spinners Contributor s Name: Gwenanne Salkind Grade Level Band: 3-5 NCTM Mathematics Standards: Data Analysis and Probability In grades 3-5 all students should: collect data using observations, surveys, and experiments; represent data using tables; describe events as likely or unlikely and discuss the degree of likelihood using such words as certain, equally likely, and impossible; predict the probability of outcomes of simple experiments ant test the predictions; understand that the measure of the likelihood of an event can be represented by a number from 0 to 1. Lesson Objectives: Design a spinner based on given criteria Predict the outcomes of simple experiments using spinners and test the predictions Determine the probability of a given simple event Write probability as ratios Mathematical Vocabulary: probability, outcome, chance, fraction, ratio, likely, unlikely, equally likely, possible, impossible, likelihood Virtual Manipulative Web Site: National Library of Virtual Manipulatives for Interactive Mathematics ( Grade Band 3-5, Data Analysis and Probability, Spinner ( Materials: A computer with internet connection for every two students Problem Solving with Spinners Record Sheet (one for every two students) One transparency of the Problem Solving with Spinners Record Sheet A paper lunch bag (one for every two students) A set of Spinner Cards (one set for every two students) Discussion of the Mathematics: In this lesson, students will explore probability by creating virtual spinners and conducting probability experiments. They will think and reason mathematically as they use given criteria to design spinners. They will list possible outcomes, write probabilities as fractions, make predictions, test their predictions, and record actual results. Additionally, students will explain why or why not their results represent their spinners.

13 14 Preparation: 1. Copy of set of Spinner Cards for each pair of students. 2. Cut the cards apart and put them in a paper lunch bag. (Students can do this as part of the activity.) Approximate Duration of Lesson: 60 minutes Procedures: 1. Warm up discussion. Use the virtual spinner to create a spinner that is half red and half blue. Ask the students What is the likelihood that the arrow will land on red when I spin this spinner? (Students should explain that there is an equal chance of getting red or blue.) Ask: What is a fraction that would describe the probability of landing on red? Ask, How could I design a spinner that has a greater chance of landing on red than blue? Elicit student responses; then create a spinner based upon the recommendation of one of the students. Ask: What is the probability of landing on red? Write the probability in the form of a fraction. Explore spinning the spinner. Spin the spinner 5 times. Before each spin, ask the students to show you where they think the arrow will land (thumbs up red, thumbs down blue). Make a record (tallies) of the outcomes on the blackboard or chart paper. After five spins, discuss the results. Did it land on red more times? Show the students the graph by clicking on the Record Results button. Spin the spinner 20 more times by typing 20 in the Spins box. Discuss the results. 2. Model the activity. Tell the students they will be working with a partner to design virtual spinners and conduct probability experiments. Draw a card from a bag of Spinner Cards. Read the card to the students and ask them how you might design a spinner that meets the conditions on the card. Use one of the student s suggestions to design a virtual spinner. Model on the overhead how to record on the record sheet. Involve students in deciding what to record, asking them to justify their thinking as you work through the activity. Draw the spinner. You can color the sections with crayons or markers or just record the first letter of each color. List the possible outcomes of the spinner. List the probability of each outcome as a fraction. Conduct two probability experiments, each time decide how many times to spin, make a prediction, spin the appropriate number of times, and record the actual results. Be sure to model how to use the spin box, so that students don t try to complete the experiments by spinning one spin at a time. Ask the students if the data you just collected represents the spinner you designed. Why or why not? 3. Divide the students into pairs. Give each pair of students a copy of the Record Sheet. Students will design four spinners by choosing spinner cards from the bag. They will conduct two probability experiments with each spinner. After they have done that, they will choose one of the spinners about which to write. They will answer the questions: Do your experimental results represent your spinner? Why or why not?

14 15 4. As the students work, circulate around the room. Ask: How did you decide how to design this spinner? What is the probability that the spinner will land on blue? How do you know? How many times do you think the spinner will land on blue if you spin it 30 times? Why do you think that? Do your results represent your spinner? Why or why not? etc. 5. After students have completed the activity, lead a whole group discussion. Ask: What did you find out about probability from doing this activity? Did your results represent your spinners? How do you know? Did anything unexpected happen as you were conducting your probability experiments? What did you notice about the number of spins in your trials? Is it better to have a large number of spins or a small number? Why? Teacher Notes: It is possible to make a virtual spinner with the same color in separate sections, but it is tricky. Suppose you want to make a spinner that looks like this one. In the spinner site, click on Change Spinner. Type purple, red, green, blue, red, yellow in the text boxes. To set the colors, put your curser in the box you want to set. Click on the color you want. Here s the trick, you must set the second red section first. If you set the first red section first, it will not allow you to set the second section to red. Note: when you use this type of spinner on the site, the graph records the two red sections separately. The probability of an event occurring is the ratio of desired outcomes to the total number of possible outcomes. When a probability experiment has very few trials, the results can be misleading. The more times an experiment is done (i.e., the spinner is spun), the closer the experimental probability comes to the theoretical probability. Student Assessment: Can the student create a spinner based upon given criteria? Can the student tell all the possible outcomes that could result from spinning a spinner? Does the student accurately record probabilities as fractions? Does the student make reasonably predictions? Can the student explain the reasons for his/her predictions? Can the student explain why or why not the experimental results represent the spinner? Are the student s justifications reasonable? Does the student use appropriate vocabulary such as outcome, fraction, likely, unlikely, and chance? References: National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.

15 Name 16 Problem Solving with Spinners Record Sheet Draw a spinner card from the paper bag. Design a virtual spinner that meets the conditions on the card. For each spinner you create: 1) draw the spinner, 2) list the possible outcomes, and 3) list the probability of each outcome as a fraction. You will conduct two probability experiments using each spinner you create. For each experiment: 1) decide how many times you will spin the spinner, 2) predict how many times the spinner will land on each color, 3) spin the spinner the correct number of times, and 4) record the number of times the spinner lands on each color. Spinner Possible Outcomes Probability (fraction) Trial #1 ( spins) Trial #2 ( spins) Predict Actual Predict Actual Draw your spinner. Spinner Possible Outcomes Probability (fraction) Trial #1 ( spins) Trial #2 ( spins) Predict Actual Predict Actual Draw your spinner.

16 17 Spinner Possible Outcomes Probability (fraction) Trial #1 ( spins) Trial #2 ( spins) Predict Actual Predict Actual Draw your spinner. Spinner Possible Outcomes Probability (fraction) Trial #1 ( spins) Trial #2 ( spins) Predict Actual Predict Actual Draw your spinner. Choose your favorite spinner. Spinner Do your experimental results represent your spinner? Why or why not?

17 Spinner Cards 18 A B C Design a spinner that has an equal chance of landing on red or blue. D Design a spinner that has no chance of spinning red. G E Design a twocolor spinner that has a slightly better chance of landing on red. Design a fourcolor spinner that has an equal chance for all four colors. H Design a spinner that has a much greater chance of landing on blue than red. F Design a four-color spinner that has a much greater chance of getting one of the colors than any other color. I Design a spinner that has a 3/8 probability of getting blue. Design a spinner that is three times more likely to spin green than blue. Design a spinner that has a ½ probability of spinning red and a ¼ probability of spinning blue.

18 Spinner Cards 19 J K L Design a spinner that has a 5/6 chance of NOT spinning green. M Design a threecolor spinner that is twice as likely to spin one color than the other two. N Design a spinner that has a ¼ chance of spinning blue, a 1/8 chance of spinning green, and a ½ chance of spinning red. O Design your own spinner. Design your own spinner. Design your own spinner. P Q R Design a spinner that works like flipping a coin. Design a spinner that works like rolling a die. Design a spinner that has a 1/3 chance of spinning red.

19 22 Implementation I implemented this lesson with a small group of fifth graders. During the course of the lesson I noticed that there was no way for the teacher to tell if the students had created appropriate spinners based on the given criteria because the spinner cards were not labeled. I decided to label the spinner cards with letters and have the students record the letter of the card above the picture of the spinner they created on the worksheet. This will allow the teacher to assess whether the students can create a spinner based on given criteria. I also added this assessment idea to the student assessment part of the lesson. External Review The notes from the external reviewer are listed below. I addressed the reviewer s suggestions in the following ways: I revised the lesson objectives to make them more specific and measurable. I removed a vague objective: Reason mathematically about probability. I added a timeframe for the lesson. Since many of the reviewers seemed concerned about mathematics vocabulary, I added a mathematics vocabulary section to the lesson plan.

20 23 Lesson 4 Title: Secret Spinners Contributor: Gwenanne Salkind Grade Level Band: 3-5 NCTM Mathematics Standard: Data Analysis and Probability In grades 3-5 all students should: collect data using observations, surveys, and experiments; represent data using tables and graphs such as bar graphs; describe events as likely or unlikely and discuss the degree of likelihood using such words as certain, equally likely, and impossible; predict the probability of outcomes of simple experiments ant test the predictions; understand that the measure of the likelihood of an event can be represented by a number from 0 to 1. Lesson Objectives: Predict the probability of outcomes of simple experiments using spinners and test the predictions Represent and interpret data using bar graphs Mathematical Vocabulary: probability, outcome, chance, multiple, factor, likely, unlikely, equally likely, possible, impossible, likelihood Virtual Manipulative Web Site: National Library of Virtual Manipulatives for Interactive Mathematics ( Grade Band 3-5, Data Analysis and Probability, Spinner ( Materials: 1 copy of Spinner Cards (cut apart) 9 x 12 envelopes (one for each group) Secret Spinner Record Sheets (one for each group) Bar Graph Template (one for each group) Spinner Template (one for each group) Markers (set of 8 classic colors for each group) Secret Spinner Class Summary Page (one for each student) Discussion of the Mathematics: In this lesson, students work in small groups. Each group creates a different spinner using the virtual spinner in the Library of Virtual Manipulatives. The groups use the spinners they created to conduct probability experiments. They create bar graphs of the results which are posted for all students to see. The secret spinners are also revealed. The students use their

21 24 knowledge of probability to match the spinners to the bar graphs. As students match the spinners to the graphs they justify and explain their decisions based upon their knowledge of probability. Students also have opportunities to discuss the difference between theoretical and experimental probability as some of the class experiments may not yield the expected results. They should notice that the experimental probability approaches the theoretical probability as the number of spins increases. Preparation: 1. Divide the class into 6-12 groups of 2-4 students. Each group will need a computer with internet access. 2. Prepare a Secret Spinner envelope for each group. Each envelope should contain: a. A Spinner Card (a different card for each group) b. A Secret Spinner Record Sheet c. A Bar Graph Template d. A Spinner Template Approximate Duration of Lesson: 60 minutes Procedures: 1. If students have never used the virtual spinner before, demonstrate how to use it. 2. Explain/model the activity to the class using a spinner that is half red and half blue. Model creating the spinner, making predictions for the first four trials, and conducting Trials #1, #2, and #3. Conduct Trials #1 and #2 using single spins and tally marks. Conduct Trial #3 using the multiple spin option. Be sure to model how to spin the spinner multiply times and how to clear the graph between trials. 3. Each group will: a. Find the Secret Spinner Card in their envelope. Follow the directions on the card to create a specific virtual spinner. b. List the possible outcomes of the spinner on their Secret Spinner Record Sheet. c. Predict the outcomes of the first four trials (Trial #1 6 spins, Trial #2 12 spins, Trial #3 60 spins, Trial #4 120 spins). d. Conduct the first four trials by spinning the virtual spinner. Record the results on the Secret Spinner Record Sheet. e. Use the data from Trial #4 to create a bar graph using the Bar Graph Template. f. Use the Spinner Template to create a paper copy of their virtual spinner. g. Conduct Trails #5-#8. Students can decide how many times to spin the spinner for each of these trials. They should make a prediction for each experiment before conducting it. h. Draw their spinner on the Secret Spinner Record Sheet. i. Discuss and write an answer to the following questions: Do you experimental results represent your spinner? What happens to your results as you increase the number of spins in each trial? j. Post their graph in a central location. Give their paper spinner to the teacher. 4. After all the bar graphs have been posted, post the spinners together in a group. Give each student a copy of the Secret Spinner Class Summary Page. Allow 5-10 minutes for students to individually record their predictions on the Secret Spinner Class Summary Page.

22 25 5. Ask students to share their predictions with a partner. Students should discuss the matches they made and justify their decisions to their partner. 6. Lead a whole group activity to physically match the paper spinners to the bar graphs. Before starting, tell students they are not allowed to match the spinner and bar graph that their group made. Call on one student at a time to chose a spinner and match it to a graph. The student must tell why he/she thinks the spinner matches the graph. Have the other students show a thumbs up if they agree. If students disagree, they should tell why. Continue having students match spinners to graphs until all the matches are made. Teacher Notes: It is possible to make a virtual spinner with the same color in separate sections, but it is tricky. Suppose you want to make a spinner that looks like this one. In the spinner site, click on Change Spinner. Type purple, red, green, blue, red, yellow in the text boxes. To set the colors, put your curser in the box you want to set. Click on the color you want. Here s the trick, you must set the second red section first. If you set the first red section first, it will not allow you to set the second section to red. Note: when you use this type of spinner on the site, the graph records the two red sections separately. The Bar graph template is designed to be used with many different results. The students will need to decide the increments that they should use to create their bar graph. Equal increments should be used. There are 30 spaces in each bar. If the spinner landed on each color 30 or less times, students can use increments of one. In this case they would label the y-axis from 1 to 30. If the spinner landed on a color more than 30 times, the students will need to use increments of 2 or more. They might label the y-axis from 2 to 60 or from 3 to 90. See examples of student work for clarification. Student Assessment: Can the student tell all the possible outcomes that could result from spinning a spinner? Can the student predict the results of spinning a spinner a certain number of times? Can the student explain the reasons for his/her prediction? Does the bar graph accurately portray the results of the probability experiment? Is the student able to explain why the experimental results represent or do not represent the spinner? Is the student using appropriate vocabulary such as outcome, likely, unlikely, and chance? Student Work: See student work samples. Blackline Masters: BLM 1 & 2 Secret Spinner Cards BLM 3 Secret Spinner Record Sheet BLM 4 Bar Graph Template BLM 5 Spinner Template BLM 6 Secret Spinner Class Summary Page

23 26 References: National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.

24 Secret Spinners Go to the National Library of Virtual Manipulatives Click on the box in the matrix: 3-5, Data Analysis & Probability Click on Spinners Make a spinner that is one-third green one-third orange one-sixth blue one-sixth red Secret Spinners Go to the National Library of Virtual Manipulatives Click on the box in the matrix: 3-5, Data Analysis & Probability Click on Spinners Make a spinner that is one-half red one-sixth green one-sixth orange one-sixth blue Secret Spinners Go to the National Library of Virtual Manipulatives Click on the box in the matrix: 3-5, Data Analysis & Probability Click on Spinners Make a spinner that is five-sixth red one-sixth blue Secret Spinners Go to the National Library of Virtual Manipulatives Click on the box in the matrix: 3-5, Data Analysis & Probability Click on Spinners Make a spinner that is two-thirds green one-sixth blue one-sixth red Secret Spinners Go to the National Library of Virtual Manipulatives Click on the box in the matrix: 3-5, Data Analysis & Probability Click on Spinners Make a spinner that is one-half blue one-sixth yellow one-sixth red one-sixth green Secret Spinners Go to the National Library of Virtual Manipulatives Click on the box in the matrix: 3-5, Data Analysis & Probability Click on Spinners Make a spinner that is one-half red one-sixth yellow one-sixth green one-sixth blue Spinner Cards BLM 1

25 Secret Spinners Go to the National Library of Virtual Manipulatives Click on the box in the matrix: 3-5, Data Analysis & Probability Click on Spinners Make a spinner that is one-third red one-third blue one-third green Secret Spinners Go to the National Library of Virtual Manipulatives Click on the box in the matrix: 3-5, Data Analysis & Probability Click on Spinners Make a spinner that is two-thirds blue one-sixth red one-sixth green Secret Spinners Go to the National Library of Virtual Manipulatives Click on the box in the matrix: 3-5, Data Analysis & Probability Click on Spinners Make a spinner that is one-sixth green one-sixth yellow one-sixth orange one-sixth red one-sixth purple one-sixth blue Secret Spinners Go to the National Library of Virtual Manipulatives Click on the box in the matrix: 3-5, Data Analysis & Probability Click on Spinners Make a spinner that is one-sixth green one-third red one-sixth orange one-third blue Secret Spinners Go to the National Library of Virtual Manipulatives Click on the box in the matrix: 3-5, Data Analysis & Probability Click on Spinners Make a spinner that is two-thirds red one-third blue Secret Spinners Go to the National Library of Virtual Manipulatives Click on the box in the matrix: 3-5, Data Analysis & Probability Click on Spinners Make a spinner that is one-half blue one-half red Spinner Cards BLM 2

26 Names Secret Spinner Record Sheet List all the possible outcomes for your spinner. For each trial, record your prediction for how many times each outcome will occur. Spin the virtual spinner as many times as the trial calls for. Record the outcomes. Possible Outcomes Trial #1 (6 spins) Trial #2 (12 spins) Trial #3 (60 spins) Trial #4 (120 spins) Prediction Actual Prediction Actual Prediction Actual Prediction Actual Possible Outcomes Trial #5 ( spins) Trial #6 ( spins) Trial #7 ( spins) Trial #8 ( spins) Prediction Actual Prediction Actual Prediction Actual Prediction Actual How did you choose the number of spins in each trial? Draw your spinner. Do your experimental results represent your spinner? BLM 3

27 Make a bar graph to record your data from Trial #4. Group # Number of Spins Color BLM 4

28 BLM 5

29 39 Implementation I implemented this lesson with a small group of fifth graders and with about 40 Title I mathematics teachers at a workshop. Some of the teachers had trouble creating an appropriate scale on the bar graph, so I added a note in the teacher notes section to address this issue. Some of the students had trouble answering the question, What happens to your results as you increase the number of spins in each trial? Instead of understanding that the experimental results approach the theoretical results with increased number of spins, they were finding the opposite. I coached them into understanding the numbers proportionally. In analyzing the difficulty, I decided that the table did not help to illustrate this mathematical idea. Since the idea is not an important concept at grades 3-5, I decided to remove the question from the worksheet. I also decided to add a question, How did you choose the number of spins in each trial? The students answers to this question will help teachers to assess student understanding of multiples and factors. For example: if a student says he/she choose 36 spins in the trial because that number made it easier to predict the outcomes, then the teacher knows that the student is using multiples and factors to create probability ratios when making predictions. External Reviewer The notes from the external reviewer are listed below. I addressed the reviewer s suggestions in the following ways: I thought about the lesson objectives and decided that they were appropriate and measurable. I added a timeframe for the lesson. Since many of the reviewers seemed concerned about mathematics vocabulary, I added a mathematics vocabulary section to the lesson plan.

30 40 Lesson 5 Title: Space Blocks Contributor: Gwenanne Salkind Grade Level Band: 6-8 NCTM Mathematics Standards: Geometry and Measurement In grades 6-8 all students should: use two-dimensional representations of three-dimensional objects to visualize and solve problems such as those involving surface area; understand, select, and use units of appropriate size and type to measure surface area; Lesson Objectives: Investigate and solve problems involving surface area of solid shapes. Use visualization, spatial reasoning, and geometric modeling to solve problems. Develop a procedure and formula for finding the surface area of a rectangular prism. Mathematics Vocabulary: surface area, minimum, maximum Virtual Manipulative Web Site: National Library of Virtual Manipulatives for Interactive Mathematics Grade Band 6-8, Geometry, Space Blocks Materials: Computers with internet connection (one for every two students) BLM 1 Space Blocks Worksheet (one per student) Centimeter or inch cubes Discussion of the Mathematics: In this lesson, students will use visualization and spatial reasoning to solve problems involving surface area of solid shapes. They will build geometric solids with cubes using Space Blocks (a virtual manipulative). I suggest that actual cubes are also available (centimeter cubes, 2-cm cubes, or inch cubes) as some students may have difficulty visualizing three-dimensional figures on a two-dimensional computer screen. There are three problems that students will be asked to solve in the lesson: 1. Connect 8 blocks to form a solid with minimum surface area. 2. Connect 8 blocks to form a solid with the largest possible surface area. 3. Connect 8 blocks to form a solid with surface area equal to 28 square units. Students will also be asked to write similar problems involving surface area. They will use isometric grid paper to draw their solutions. They will explain their strategies for solving the problems and finding surface area. Students will work with partners as they solve problems and share solution strategies.

31 41 While students may explore surface area of solid shapes that are not rectangular prisms, this lesson gives them an opportunity to explore surface areas of rectangular prisms. Encourage students to develop procedures and formulas for finding the surface area of a rectangular prism. Approximate Duration of Lesson: minutes (can be done over two class periods) Procedures: 1. Model the Space Blocks web site to students. Show them how to add blocks, turn blocks, and connect blocks. Give them time to explore the website. 2. Ask students to work in pairs. Give each student a Space Blocks Worksheet. 3. Explain the activity, showing students where the three problems are located on the website and how the computer can be asked to check their solutions. Review the concept of surface area by asking students to figure out the surface area of a shape that you have created on the Space Blocks web site. (Do not create a shape that solves one of the problems posed on the site.) Tell students that they may work together, but each student must fill out his/her own worksheet. 4. Allow students time to complete the worksheet. Students should work with a partner. Encourage partners to solve each others problems and discuss their solution strategies. As students are working, circulate. Ask: How do you know when you ve found the minimum surface area? Why do you think the shape you created has the largest surface area? What is your strategy for finding building a solid with a surface area of 28? 5. During the last 10 minutes of class, lead a whole class discussion. Ask students to share their solution strategies. Focus on strategies for finding surface area of rectangular prisms. What ideas do students have about finding the minimum and maximum surface areas with a given amount of blocks? Teacher Notes: The three problems posed in this lesson are on the Space Blocks web site. Go to Activities, then List Activities. Students will work on the third problem first, Minimizing Surface Area; then the second problem, Maximizing Surface Area; then the first problem, Constructing Figures with a Given Surface Area. Students will also create their own problems. When they create their own problems, they can use the web site to build solid shapes, but the computer will not check their work. Be sure they understand this aspect of the web site. The surface area of a rectangular prism is the sum of the areas of all six faces (SA = 2lw + 2lh + 2wh). Student Assessment: Does the student use appropriate vocabulary such as surface area, cube, rectangular prism, face, edge, and vertex? Can the student explain his/her strategies for building solid shapes with minimum, maximum, and given surface areas? (The shapes may or may not be rectangular prisms.) Can the student find the surface area of a given rectangular solid? Can the student explain his/her procedure for finding the surface area? Can the student articulate a formula for finding the surface area of a cube? Can the student articulate a formula for finding the surface area of a rectangular prism that is not a cube?

32 42 Student Work: See attached. Blackline Master: BLM 1 Space Blocks Worksheet References: National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.

33 Name: Space Blocks Minimizing Surface Area 1. Connect 8 blocks to form a solid with minimum surface area. Draw your solid shape below. 2. What is the surface area of this solid shape? 3. Write your own minimum surface area problem. Ask a friend to use Space Blocks to figure out the answer to your problem. Write your problem below. 4. Draw the answer to your problem here. What is the surface area of this solid shape? BLM 1

34 Maximum Surface Area 5. Connect 8 blocks to form a solid with the largest possible surface area. Draw your solid shape below. What is the surface area of this solid shape? 6. Write your own maximum surface area problem. Ask a friend to use Space Blocks to figure out the answer to your problem. Write your problem below. 7. Draw the answer to your problem here. What is the surface area of this solid shape? BLM 1

35 Constructing Figures with a Given Surface Area 8. Connect 8 blocks to form a solid with surface area equal to 28 square units. Draw your solid shape below. 9. Write your own surface area problem. Ask a friend to use Space Blocks to figure out the answer to your problem. Write your problem below. 10. Draw the answer to your problem here. BLM 1