Number Sense Lessons And Supplemental Activities For Middle School Grades 7-8. By Greg Gearey Patrick McNallan George Taus

Transcription

1 Number Sense Lessons And Supplemental Activities For Middle School Grades 7-8 By Greg Gearey Patrick McNallan George Taus Number Sense/Number Theory Summer 2005

2 About Our Class This lesson can be found at Unit Overview In the following lesson, students participate in an activity in which they focus on the uses of numbers. The activity explores how students use numbers in school and every day settings as a way for students and the teacher to get to know each other at the beginning of the school year. The activity involves data collection and interpretation, and probability and statistics. This activity also emphasizes a growthand-development theme. Length: 2 periods Grades: 6-8 Learning Objectives Students will be able to: * collect and analyze data * determine the arithmetic mean * represent data using percent Materials * One "About Our Class" activity sheet for each student Instructional Plan Background Information In this activity, students collect and analyze data about a student's family and his or her personal interests. Students determine the arithmetic mean and represent data using percent. Preparing the Investigation Reproduce a copy of the activity sheet "About Our Class" for each student. Structuring the Investigation 1. Have the students complete survey questions a - g. Discuss the results as a class. Ask students to think of a way to represent this information. Consider having groups of students construct a table to display the class's data for all responses to questions a - g.

3 2. Have the students determine the mean number of people living in each house. (Find the sum of the people in all the houses and divide this sum by the number of houses.) Define range as the difference between the numbers of people in the smallest and largest household, then ask for the range of the household size. 3. Ask the class to report the total number and mean of the number of cars. Then ask how students could determine the percent of families that have at least one car. (Make a table showing the numbers of families with various numbers of cars, and then determine the percent of families with at least one car by dividing the number of families with one or more cars by the total number of families.) 4. Have the class discuss the mean number of calculators in the families and the percent of families that have at least one calculator in their home. Remind the students that if they have at least one, they could have more than one. 5. Ascertain the favorite activity for the class. Ask how this answer was determined. 6. Discuss the difference between the number of time the most common favorite and the second-most-common-favorite TV shows were mentioned. The greater the difference between the two numbers, the greater is the likelihood that the favorite will still be the favorite in another sample. 7. Discuss the completed graphs and questions from the class. Extensions Have students work in groups to create a way to organize the class's responses to each of the survey questions. Ask the students to display their data to the class. Standards and Expectations Data Analysis & Probability 6-8 This lesson covers the following Data Analysis & Probability Standard Expectations: * use proportionality and a basic understanding of probability to make and test conjectures about the results of experiments and simulations. References Dianne Bankard and Francis (Skip) Fennell. The Arithmetic Teacher. September, pp Curcio, Frances R. Developing Graph Comprehension: Elementary and Middle School Activities. Reston, Va.: National Council of Teachers of Mathematics, 1989.

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5 Travel in the Solar System This lesson can be found at Unit Overview In the following lesson, students participate in an activity in which they focus on the uses of very large numbers. Length: 2.5 periods Grades: 6-8 Learning Objectives Students will be able to: apply measurement and computation to gain insight into the large numbers associated with distances in space Materials Calculator Planets at a Glance data sheet Instructional Plan Mission To apply measurement and computation to gain insight into the large numbers associated with distances in space To plan a trip to a planet in the solar system This lesson focuses on human travel in space. One problem associated with traveling in the solar system is the distance from on planet to another. But other problems arise. A spacecraft needs fuel to make the long journeys in space, and humans need food and water throughout the long journeys. The effects of microgravity, that is, near zero gravity, on humans over time are unknown. The probability of collisions with asteroids is uncertain, and many other aspects of long manned flights make the task of space travel very complex. However, publicity about unmanned flights to the planets continues to raise the question of humans' traveling in space. Research is required to increase the probability that prolonged space travel for humans can be accomplished safely. One of the NASA projects that will move us closer to space travel is the International Space Station, which will serve as a platform for many research agendas associated with living and working in space for long periods of time.

6 Purpose This lesson affords students the opportunity to think about two aspects of the time required to complete space travel within the solar system. First, students consider the amount of time that space travelers must spend on the journey. Second, students think about what kinds of events might occur on Earth while the space travelers are on their journey. Thinking about both situations improves students' concept of time and distance as well as improves their understanding of the solar system. Getting Started Begin the class by engaging students in a discussion about humans traveling through the universe. In the movies and on television, students encounter science-fiction stories about traveling at the speed of light and beyond to cross entire galaxies in a matter of seconds. The activities in lesson 2 have reminded us that most of our travel speeds are quite slow. We have not come close to traveling at the speed of light. Although radio signals can return from Mars in a short time, it takes much longer for a spacecraft from Earth to reach Mars. Developing the Activity Use the data about the Space Shuttle below to determine its speed in miles per hour. Remind students that the Shuttle is not designed for travel among the planets. It is designed for Earth-orbit tasks. However, its speed is helpful in judging the speeds for twentieth century spacecraft. After students have done the calculations, come to some agreement on an approximate speed for interplanetary travel. Assume that the agreement is about 50,000 miles per hour. This figure gives us a reasonable speed to use in thinking about space travel today. In the future, speeds will undoubtedly increase. Space Shuttle Data with Speed Computation Space Shuttle Data: Distance traveled by Shuttle = 4,164,183 miles Time to travel given distance = 9d 23h 30 min Speed Computation: Using the data sheet The Planets at a Glance, students can determine the distance from Earth to each of the other planets. This task is not trivial. The distances in the chart are given in millions of miles. To facilitate computation and estimation, students need to translate 67.2 million miles, the distance from the Sun to Venus, into its full numeric form, 67,200,000. Before thinking about traveling to Venus, students must remember that Earth is about 93,000,000 miles from the Sun. Use the mean distance from the Sun to specified planets to calculate each distance from Earth to the targeted planet. To make this exploration manageable for middle school students, we are assuming that the planets are aligned at their mean distances. Teachers should explain to students that in actuality, this greatly simplified situation is unlikely to occur.

7 The students' next task is to calculate the time required to travel to each planet on spacecraft that travel from 10,000 to 100,000 miles per hour. Group the students into their mission teams of four students. Ask them to complete a chart for travel to all the planets of the solar system at the speeds shown below. They should use the mean distance of each planet from Earth. In a typical class, students groan at the prospect of completing a chart with eighty entries. The groans provide the opportunity to challenge the teams to think of strategies for reducing the amount of calculations required to complete the chart. When patterns are used to complete entries, the teams should record them. All students in the teams should make the chart. Discuss the patterns that teams used to complete the chart. List the patterns on the chalkboard. After all team members have shared how their patterns helped reduce their workload, ask each team how many different patterns they used. The class now has a complete chart for travel to the solar system's planets computed in hours of travel. It is time to develop some better notions of what these times mean. Pose questions that require students to think about the practicality of space travel: If we travel at the approximate speed of the Space Shuttle, which planets can we reach in less than 10 years? How fast must we be able to travel to reach Jupiter in less than 10 years? Traveling to some of the planets at some of the indicated speeds would take more than a lifetime, which is about 75 years. Which planets are too far away to be reached in a lifetime? We would like to make round trips. Traveling at 10,000 per hour, to which planets could we make a round trip in our lifetime? These questions should be posed to the mission teams. The teams should discuss the questions and agree on a team response. After students begin thinking about time questions, have each mission team make up questions for the class to solve. Closing the Activity Pose selected mission team questions to the entire class. The quality of the questions and the responses should indicate which students are gaining an understanding about the time required for space travel. Standards and Expectations Number & Operations 6-8 This lesson covers the following Number & Operations Standard Expectations: understand and use ratios and proportions to represent quantitative relationships. develop an understanding of large numbers and recognize and appropriately use exponential, scientific, and calculator notation. understand the meaning and effects of arithmetic operations with fractions, decimals, and integers. References Adapted from Travel in the Solar System in Mission Mathematics, Linking Aerospace and the NCTM Standards, 5-8, a NASA/NCTM project, NCTM 1997.

8 Additional activities for this lesson can be found at

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10 Integer Product Game Objective: Materials: Students will practice multiplying integers and determining factors for numbers. Game sheet, 2 paper clips, handful of chips or beans or something to cover products on game sheet. Instructional Plan: -We suggest playing a practice game on the overhead against a student volunteer. -Hand out the rules for the game found in the book and make sure rules are clear. -Partner students up to play the game. -Hand out 2 paperclips per pair and some markers of some sort to cover products as kids play. -Encourage students to write down any strategies they are using or thinking about. -After students have played several times, have them discuss their strategies, and then you will likely want to re-pair them to play against a different opponent. Reference: This game is part of Lesson 4.3 from 7 th grade CMP, the book Accentuate the Negative, pages

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14 A Conceptual Model for Solving Percent Problems Length: 1.5 period(s) Grades: 6-8 Number & Operations This lesson is found at Illuminations on the NCTM website. In this grades 5-8 lesson, students will use a 10 x 10 grid, which is a common model for visualizing percents to solve various types of percent problems. This model offers a means of representing the given information as well as suggesting different approaches for finding a solution. This NCTM Publication-Based Lesson Plan is adapted from "A Conceptual Model for Solving Percent Problems" by Albert B. Bennett, Jr., and L. Ted Nelson. The article appeared in Mathematics Teaching in the Middle School Vol.1, No.1 (April 1994) Pp Learning Objectives Students will be able to: * shade a 10 x 10 grid to represent a given percent * determine the percent for a given shaded amount on a 10 x 10 grid * determine the value of one of the small squares in a 10 x 10 grid * determine the value of the 10 x 10 grid given the value of one of the small squares * use the 10 x 10 grid model to solve problems Materials * 10 x 10 Grids * Dry-erase, water-based, or grease markers. Instructional Plan Applications of percents are often taught by solving proportions that require variables and some familiarity with algebra. Such formal approaches to teaching percents have not been successful for many junior high and high school students (Wiebe 1986). This article presents an alternative method that focuses on the basic

15 concept of percent, that of "parts per hundred." A 10 x 10 grid, which is a common model for visualizing percents, is extended in the following examples to solve various types of percent problems. This model offers a means of representing the given information as well as suggesting different approaches for finding a solution. Standards and Expectations Number & Operations 6-8 * compare and order fractions, decimals, and percents efficiently and find their approximate locations on a number line. * develop meaning for integers and represent and compare quantities with them. * work flexibly with fractions, decimals, and percents to solve problems. References * Bennett, Albert B., and L. Ted Nelson. Mathematics for Elementary Teachers: An Activity Approach. 3d ed. Dubuque, Ia.: Wm. C. Brown Publishers, * Dye, David, chair of committee for Minnesota Department of Education. "Position Paper; Teaching and Learning Percent." St. Paul, Minn.: Department of Education, * Wiebe, James H. "Manipulating Percentages." Arithmetic Teacher 33 (January 1986): * Bennett, Albert B. and L. Ted Nelson. "A Conceptual Model for Solving Percent Problems." Mathematics Teaching in the Middle School 1 (April 1994): *, Mathematics for Elementary Teachers: A Conceptual Approach. 3d ed. Dubuque, Ia: Wm. C. Brown Publishers, * National Council of Teachers of Mathematics. Curriculum and Evaluation Standards for School Mathematics. Reston, Va.: The Council, 1989.

16 Having Fun with Baseball Statistics Length: 2 period(s) Grades: Communication Number Sense This lesson can be found at Illuminations on the NCTM website The following grades 5-8 activities allow students to explore statistics surrounding baseball. They are exposed to connections between various mathematical concepts and see where this mathematics is used in areas with which they are familiar. This lesson plan appeared in the May 1996 edition of Mathematics Teaching in the Middle School Journal. Learning Objectives Students will * develop skills in mathematical reasoning and computations and apply those skills to everyday life Materials * Game-Card Worksheets * Baseball cards * Calculators * Paper clips * Overhead projector and sample transparencies are helpful but are not necessary. Instructional Plan How often have middle school teachers had to ask students to put away their baseball cards? The lesson described in this article capitalizes on students' interest in sports while providing valuable instruction on a variety of appropriate mathematical topics. Specifically, this lesson meets the recommendations of the NCTM's Curriculum and Evaluation Standards for School Mathematics (1989) by integrating the study of fractions, decimals, percents, rounding, Cartesian coordinates, probability,

17 and statistics in a cooperative setting that allows students to have fun as they learn. Overview The lesson involves the following four phases: (1) students familiarize themselves with the meanings of the statistics found on baseball cards; (2) students complete the worksheet portion of the game-card worksheet, by converting information from their baseball cards into appropriate decimals, fractions, and percents; (3) students complete the game-card portion of the worksheet based on these data; (4) students play a simulated game of baseball. Approximately ninety minutes is required to complete this activity. It can, therefore, be scheduled in a single extended time block, often available in middle school schedules, or split over two forty-five-minute class periods. I have found that in the latter, phases 1 and 2 can be completed during the first class session. The teacher can then collect and check the students' worksheets before they use this information to create the 10-by-10 array on their game cards for phase 3. References * National Council of Teachers of Mathematics. Curriculum and Evaluation Standards for School Mathematics. Reston, Va.: The Council, * Written by Robert J. Quinn Mathematics Teaching in the Middle School Journal, May * The Topps Company. Baseball cards, licensed by Major League Baseball and the Major League Baseball Player s Association, 1993.

18 Dilution Learning Objectives: *Students will get to work with small numbers that apply to everyday life. Materials: *Red or blue food coloring ( 1 ounce per group);water (2 gallons per group); 10 ounce transparent, graduated measuring containers; stirring stick. Instructional Plan: *First pour 1 ounce of food coloring into a container and add 9 ounces of water to it. This will make a 10-ounce solution. Stir the solution. *Record the diluted solution into a chart in decimal form. *Next pour 1-ounce of the new solution into another container and add 9 ounces of water to it. *Record the new diluted solution. *Continue this process to the millionths place. *Leave all 6 solutions in the front of the room so students can see the process. *Students will describe what happens when a power of 10 is repeatedly divided by 10. *Students continue to solve problems about dilution that involve powers of 10 using negative exponents. Number of Dilutions Portion of Food Coloring Reference: Britannica, Mathematics in Context p

19 Number Portion of of Food Dilutions Coloring

20 Pick Up Sticks Objective: Develop the concepts of odd and even in game play strategy. Materials: Writing board, chalk or marker. May also be played with any 15 items such as sticks, beans, paperclips, etc Activity: - The goal is to force your opponent to pick up the last stick. - Students should pair up into teams. - There are 3 rows of five items, for a total of 15 items. I I I I I I I I I I I I I I I - You may pick 1, 2, 3, 4 or 5 sticks from any one row on your turn. - Alternate turns. - If you force your opponent to pick up the last stick you win. - Model the game with the students on the writing board. - Let the students play a few games to get a feel for the game. - Then challenge teams to develop a winning strategy and express it on paper. - Challenge the strategies to a real game and see if it wins. Winning strategy: Go first, leave odd number of sticks with odd number of rows or even number of matchsticks with even number of rows.

21 Four 4 s Project Objective: Students will improve their number sense and improve efficiency using correct order of operations. Materials: Four 4 s sheet (optional) Activity: -The instructor will begin the activity with a challenge for the students---to write a mathematical problem with an answer of 1 using exactly four fours, and no other numbers. Have selected students show answers on the board. (Hopefully they will generate several different solutions) -Discuss answers and proper order of operations. -Students will now work in groups to write 19 more mathematical equations using exactly four fours in it with answers of 2, 3, 4,..etc -Compile students answers after they have had enough time to work on all of the problems. -Have students correct their peers work on the board. Homework: Challenge -Students will use the numbers 1,2,3, and 4 only, to write math problems with the answers of 1,2,3,4,5,6,7,8,9,10,11, and 12. Encourage them to come up with more than one solution.

22 Bullseye Objectives: Students will get to know each other and they will be able to do the order of operations correctly. Materials: Bullseye sheets Activity: -You will need to assemble students into group of 3 or 4. -Students will collaborate as a group to come up with a set of numbers. -After groups have answered the set of questions they will use the numbers in an equation to hit a target answer of ten using mathematical operations. If students can not hit the number they are to get as close as they can. - They will continue the activity by using the same numbers in an equation to hit an answer of a hundred. - You could continue setting new target numbers for your students to challenge their ability using order of operations.

23 Timber Objective: Real life applications using division. Materials: Timber worksheet, rulers, and scrap paper. Activity: -Students will need to draw lines that are 120mm, 160mm, 200mm, 240mm and 280mm. - For each problem students will need to use a ruler to measure distances to find the most affective use of building materials from the lengths above.

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25 The Right Price Game Learning Objectives: Students will work on basic addition and subtraction facts either mentally or paper pencil. Materials: The right price Game sheet, or make up three items of your own and a total price : Instructional Plan *Find as many prices as you can that will work for each round, that is the sum of the three items adds up to the total. *You have 15 seconds per round. *After each game list all solutions the class came up with. *Play as many rounds as time permits.

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27 Observational adjustments: Number Sense Assessment We will use in class observations of student progress to make adjustments to our instructional plan when using game activities. With games like Pick Up Sticks, the game will only hold a student s interest until they figure out the winning strategy. After that, given the rules of the game, the student s will lose interest in the activity. As teachers we will either modify the rules or move on to the next activity when we observe that kids have mastered the winning strategy. Formal assessments: The lessons in our project that are more involved, and meet the state standards, fit into our curriculum in a number of different units. We will be assessing the material from these lessons by slightly modifying our current assessment tools.