NUMBER SENSE. 8 th Grade Activities. Jennifer McLachlan Little Falls Middle School

Transcription

1 NUMBER SENSE 8 th Grade Activities Jennifer McLachlan Little Falls Middle School jmclachlan@lfalls.k12.mn.us Maureen Wilke Heritage Middle School wilkem@isd197.k12.mn.us

2 Overview: Number sense an intuitive feel for numbers and their relationships develops when children solve problems for themselves. Source: Univ. of North Carolina, School of Education Number Sense is not something that can be taught during a short period of time. Because a student s number sense slowly grows over time, it must be part of student s daily experiences. The lessons we have gathered allow students to further develop and strengthen their understanding of numbers and operations. We will be using these lessons throughout the school year. The lessons are designed in such a way that you can pick one lesson that fits with material that you are already covering in the classroom. The lessons range from 20-minute games to reinforce learned concepts, to weekly challenge problems that the students will explore and discuss during the week. Contents: 1. Multiplying and Dividing Exponents (pages 5-10) 2. Exponential Growth (pages 11-15) 3. Multiplying and Dividing Scientific Notation (pages 16-21) 4. Modeling our Solar System (pages 22-29) 5. Ordering Rational and Irrational Numbers (pages 30-31) 6. Irrational Numbers can In-Spiral You (pages 32-38) 7. Fantastic Four (Order of Operations) (page 39) 8. Magic Number (pages 40-42) 9. Barbie Bungee (pages 43-48) 10. How High? (page 49) 2

3 This unit covers the following Minnesota Standards: Number & Operation Classify real numbers as rational or irrational. Know that when a square root of a positive integer is not an integer, then it is irrational. Know that the sum of a rational number and an irrational number is irrational, and the product of a non-zero rational number and an irrational number is irrational Compare real numbers; locate real numbers on a number line. Identify the square root of a positive integer as an integer, or if it is not an integer, locate it as a real number between two consecutive positive integers Determine rational approximations for solutions to problems involving real numbers Know and apply the properties of positive and negative integer exponents to generate equivalent numerical expressions. Express approximations of very large and very small numbers using scientific notation; understand how calculators display numbers in scientific notation. Multiply and divide numbers expressed in scientific notation, express the answer in scientific notation, using the correct number of significant digits when physical measurements are involved Algebra Understand that a function is a relationship between an independent variable and a dependent variable in which the value of the independent variable determines the value of the dependent variable. Use functional notation, such as f(x), to represent such relationships Use linear functions to represent relationships in which changing the input variable by some amount leads to a change in the output variable that is a constant times that amount. Represent linear functions with tables, verbal descriptions, symbols, equations and graphs; translate from one representation to another. Identify graphical properties of linear functions including slopes and intercepts. Know that the slope equals the rate of change, and that the y-intercept is zero when the function represents a proportional relationship. Justify steps in generating equivalent expressions by identifying the properties used, including the properties of algebra. Properties include the associative, commutative and distributive laws, and the order of operations, including grouping symbols. Solve multi-step equations in one variable. Solve for one variable in a multi-variable equation in terms of the other variables. Justify the steps by identifying the properties of equalities used Use linear inequalities to represent relationships in various contexts. Geometry 3

4 Make reasonable estimates and judgments about the accuracy of values resulting from calculations involving measurements. 4

5 Multiplying and Dividing Exponents Minnesota Standard: Know and apply the properties of positive and negative integer exponents to generate equivalent numerical expressions. Objective: Students will be able to multiply and divide positive and negative exponents by using a daily warm up to practice their new learned skill. Materials: Transparencies of warm up Launch: Students will enter the room and see the transparency on the overhead. They will get out their homework and notebook and start the warm up in a civil manner. Explore: For one week students will complete one chart daily in their notebooks. Share: Randomly choose students to come to the overhead to solve the chart. Pick two of the problems and ask students how they found their answer. Summarize: When multiplying exponents you need to add the exponents. When dividing exponents you need to subtract the exponents. 5

6 Warm up Book 3 Exponents X

7 Warm up Book 3 Exponents X

8 Warm up Book 3 Exponents X a 5 a a 8 a 1 a 6 a 7 8

9 Warm up Book 3 Exponents

10 Warm up Book 3 Exponents d 2 d 3 d 4 d 8 d 11 d 18 10

11 Minnesota Standard: Exponential Growth Know and apply the properties of positive and negative integer exponents to generate equivalent numerical expressions. Objective: Students will gain an intuitive understanding of basic exponential growth patterns. Students will begin to recognize exponential patterns in tables. Students will solve problems involving exponential growth and express a number in exponential notation and standard notation. Materials: 40 pieces of construction paper. Scissors of each group (8) 2 transparencies for selected group of students to share with class Launch: The computers broke down at school today and we have to vote for school president and vice president. We are going to have to go back to the old way and make ballots for everyone in the school. Each group will get a piece of paper and a scissors. Your job is to make equal size ballots out of this piece of paper. This idea was from Connected Mathematics Growing, growing, growing Explore: Put students into groups of two. Students will cut the same piece of paper in half until they can no longer cut the pieces in half. They will make a table keeping track of the number of cuts to the number of pieces of paper. Follow the attached lesson. Share: Have the two groups share their results. Compare with others. Are they the same? Use the attached lesson and start at problem 1.1 follow up. Questions (1-6). Summarize: When you are multiplying by the same number multiple times, it is better to write it in exponential form. Using a calculator for large exponents it is faster to use the exponent key then multiplying the same number multiple times. 11

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16 Multiplying and Dividing Scientific Notation Minnesota Standard: Express approximations of very large and very small numbers using scientific notation; understand how calculators display numbers in scientific notation. Multiply and divide numbers expressed in scientific notation, express the answer in scientific notation, using the correct number of significant digits when physical measurements are involved. Objective: Students will be able to multiply and divide numbers in scientific notation by using a daily warm up to practice their new learned skill. Materials: Transparencies of warm up Launch: Students will enter the room and see the transparency on the overhead. They will get out their homework and notebook and start the warm up in a civil manner. Explore: For one week students will complete one chart daily in their notebooks. Share: Randomly choose students to come to the overhead to solve the chart. Pick two of the problems and ask students how they found their answer. Summarize: When multiplying numbers in scientific notation you need to multiply the coefficients and add the exponents. When dividing numbers in scientific notation you need to divide the coefficients and subtract the exponents. 16

17 Warm up Book 3 Scientific Notation X

18 Warm up Book 3 Scientific Notation X

19 Warm up Book 3 Scientific Notation X

20 Warm up Book 3 Scientific Notation

21 Warm up Book 3 Scientific Notation

22 Modeling our Solar System Minnesota Standard: Express approximations of very large and very small numbers using scientific notation; understand how calculators display numbers in scientific notation. Multiply and divide numbers expressed in scientific notation, express the answer in scientific notation, using the correct number of significant digits when physical measurements are involved. Objective: Students will engage in a class activity that will use their knowledge of scientific notation to represent relative distances between objects in our solar system. Materials: Find a location in the commons for students to represent the planets Masking tape for marking distances of the planets. 40 sheets of butcher paper Markers Ruler Copy master 10 Camera (optional) This idea came from Impact Mathematics Course Launch: Show the table of the average distance of the planets from the sun. Imagine lining up the planets with the sun on one end and Pluto on the other end so each planet is at its average distance from the sun. How would the planets be spaced? Here is one student s prediction: Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Pluto Now make your own prediction. Without using your calculator sketch a scale version of the planets lined up in a straight line from the sun. Don t worry about the sizes of the planets. Explore: Students will work together to put the planets in order based on their distance from the sun. They will be able to look at large numbers in scientific notation and they can compare the distances of the planets from the sun. See attached lesson starting at Create a Model. Share: On attached lesson go through questions 7-15 (skip 13) with students. Have students write the scientific notation to standard notation. Summarize: When working with very small and very large numbers it is easier to compare numbers using scientific notation instead of standard notation. 22

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30 Ordering Rational and Irrational Numbers MINNESOTA Standards: Compare real numbers; locate real numbers on a number line. Identify the square root of a positive integer as an integer, or if it is not an integer, locate it as a real number between two consecutive positive integers. Objective: Students will compare and order real numbers including numbers expressed in scientific notation, in words, as powers of 10, as radicals, or in standard form. Materials: Student Worksheet Tag Board (for numbers) Magnetic Tape / Squares (to place on the back of the tag board) This Idea came from Britannica Math in Context Launch: Teacher will provide a line on the board. The number line will only have zero marked. Provide magnetic numbers for students to place on the number line. Invite students to the board to order the numbers when finished with assignments through out the week. The student exploration will begin after students have received instruction on scientific notation, exponents and square roots. Explore: Students will work with a partner to order the numbers from least to greatest. Students should be allowed to use calculators to help them compare numbers. Share: What did you notice about some of the numbers? Have students discuss which numbers had the same value but were written differently. Is zero larger or smaller than a decimal number? Have students offer solutions to the worksheet. Summarize: Even though numbers look different, they can have the same value. Another way to look at it is that 100 and 10 2 occupy the place on the number line. 30

31 Each circle contains a number. The numbers are expressed in scientific notation, in words, as powers of 10, as radicals, or in standard form. 1) Draw lines to connect the numbers that are the same. 2) In the column below, write the numbers in order from largest to smallest , One hundred thousand x 10 5 Half a Million 10-4 Irrational Numbers can In-Spiral You Minnesota Standard: 500,

32 Classify real numbers as rational or irrational. Know that when a square root of a positive integer is not an integer, then it is irrational. Know that the sum of a rational number and an irrational number is irrational, and the product of a non-zero rational number and an irrational number is irrational. Compare real numbers; locate real numbers on a number line. Identify the square root of a positive integer as an integer, or if it is not an integer, locate it as a real number between two consecutive positive integers Determine rational approximations for solutions to problems involving real numbers. Objective: To investigate the difference between rational and irrational numbers by visualizing, creating, and discussing a project that displays rational and irrational numbers. To be able to determine the approximations of irrational numbers. To be able to compare, locate, and identify positive and negative square roots on a number line. Materials: White paper (one for pair of partners) Index card (one for pair of partners) Markers Number line -20 to 20 The idea for this lesson came from Mathematics Teaching in the Middle School, April Launch: Go around the room and select students and decide who will get homework this evening and who won t. Students who don t get homework I could say things such as I like your shoes, I like the color of your hair, I like how you sit up in your desk, etc. The students who get homework I could say things such as I don t like you tapping your pencil, I don t like the desk you are in, I don t like that you came in late, etc. Ask the students in the class what they are thinking about my behavior. Write the words on the board. Could we think of other words? If they can t come up with irrational, eventually say it. Discuss the difference between rational and irrational numbers. Explore: Students will create a right triangle and put them together to create a wheel. Students will be solving radical numbers on the hypotenuse to see if there is a pattern to the numbers and the wheel. See attached lesson. Extension: After students complete their Wheel of Theodorus they need to find the square roots of all the hypotenuse numbers. On a number line from -20 to 20 put the hypotenuse numbers on the number line. Introduce negative square root and ask the students to put them in order on the number line. Share: Groups will pair share with another group to compare their Wheel of Theordorus and calculations. They will also compare their 32

33 rational and irrational numbers on the number line. Students may notice that as the number under the radical sign grows so does the value of the square root. May also notice the pattern of perfect squares between 1 and 4 are 3, between 4 and 9 are 5, between 9 and 16 are 7, (consecutive odds). Students may look at odds, size of the wheel, and evens or other patterns as well. Looking at our Wheel of Theordorus can you explain the difference between a rational and an irrational number? When looking the hypotenuse sides, what can you say about the square roots of those numbers? Summarize: Display students work on walls throughout the school area (pod or house). Looking at the wheel a rational number is a perfect square because it ends or repeats. It can be written as a fraction. An irrational number is a number that is nonrepeating and nonterminating. You cannot write it as a fraction. 33

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39 Fantastic Four MINNESOTA Standards: Justify steps in generating equivalent expressions by identifying the properties used, including the properties of algebra. Properties include the associative, commutative and distributive laws, and the order of operations, including grouping symbols. Objective: Students will apply properties of algebra and order of operations to generate equivalent expressions. Materials: Set of Overhead Cards Launch: Write 2, 3, 4, 5 on the board. Ask students to spend a minute trying to develop equations using all four numbers. The equations can be equal to any number. Encourage them to use parenthesis, exponents and radicals. Invite students to the board to share equations. Then ask students to see if they can arrange the numbers to equal 13. Ask for possible solutions x 3 +5 = 13 Explore: The teacher will turn over 5 cards. The sixth card will be a target number. Teams will be given a predetermined amount of time to work in their group to develop equations. Teams will earn points for each equation. An equation using 2 numbers = 2 pts, 3 numbers = 3 pts, etc. Equations with exponents or radicals get 2 extra points. Share: After each round students will share solutions that earned more than 3 pts. Class will discuss strategies and importance of order of operations. Summarize: Discuss how many of the equations are similar, but because of how numbers are grouped or the rules for order of operation, the solutions are the same. The teacher should conclude you can make new equations by using the associative and commutative properties. 39

40 Mystery Number MINNESOTA Standards: Solve multi-step equations in one variable. Solve for one variable in a multi-variable equation in terms of the other variables. Justify the steps by identifying the properties of equalities used. Objective: Students will formulate generalized patterns out of specific cases. Students will develop a deeper understanding that a variable represents a range of possible values Students will enhance their fluency in manipulating symbolic expressions. Materials: Worksheet for each student This idea came from Mathematics Teaching in the Middle School, March 2007 Launch: Say to the students I can read your minds... and I can prove it! Ask a volunteer to write down 3 consecutive numbers. Have the student add up the three numbers and tell you the total. (Teacher will divide the number by 3 to find the middle number in the series.) Repeat this process several times. Then tell the students that they TOO can be mind readers. Explore: Students will work together in teams to develop a set of steps to explain each scenario on the worksheet. Students should pick a number and run through the directions for each situation to make sure it works. Then they will identify what mathematical operation is changing their number. Finally, they will write a generalized rule or algebraic equation for each step. ** Students may need help with the last scenario. This case may require that they look at the arithmetic and algebraic equations first. They should see that when the answer is given after completing the steps, there are still a few simple calculations that the mind reader needs to do before the solution is obtained. 7) Take away 10 from the answer 8) Divide it by 10 9) Reveal the favorite day!! Share: Allow students to share strategies to each of the problems. Develop the rules/steps used to figure out the original situation (3 consecutive integers) as a whole class. Summarize: Conclude that variables help link simple arithmetic to algebra by reasoning about generalized patterns. Variables are place holders for a range of possible values. Justify each of the steps by identifying the properties of equalities were used. 40

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43 Barbie Bungee MINNESOTA Standards: Understand that a function is a relationship between an independent variable and a dependent variable in which the value of the independent variable determines the value of the dependent variable. Use functional notation, such as f(x), to represent such relationships Represent linear functions with tables, verbal descriptions, symbols, equations and graphs; translate from one representation to another. Identify graphical properties of linear functions including slopes and intercepts. Know that the slope equals the rate of change, and that the y-intercept is zero when the function represents a proportional relationship Use linear inequalities to represent relationships in various contexts. Objective: Students will explore relationship between tables and linear graphs, to explain mathematical relationships. Students will find the line of best fit and approximate and interpret rate of change from graphical and numerical data. Materials: Rubber bands (all the same size and type) Yardsticks or measuring tapes Masking tape Barbie dolls (or similar) Barbie Bungee Activity Sheet *This lesson was found on the National Council of Teachers of Mathematics website. Launch: Watch a video segment on bungee jumping. Do you think the length of the cord and size of the person matters when bungee jumping. Would it be smart to lie about your height and weight? Allow students to offer suggestions as to why an accurate estimate of height and weight would be important to conduct a safe bungee jump. Explore: After a brief introduction, set up the lesson by telling students that they will be creating a bungee jump for a Barbie doll. Their objective is to give Barbie the greatest thrill while still ensuring that she is safe. This means that she should come as close as possible to the ground without hitting the floor. Explain that students will conduct an experiment, collect data, and then use the data to predict the maximum number of rubber bands that should be used to give Barbie a safe jump from a height of 400 cm. (At the end of the lesson, students should test their conjectures by dropping Barbie from this height. If you school does not have a location that will allow such a drop, then you may wish to adjust the height for this prediction.) Hand-out the Barbie Bungee packet to each student. In addition, give each group of 3-4 students a Barbie doll, rubber bands, a large piece of paper, some tape, and a measuring tool. Before students begin, demonstrate how to create the double loop that 43

44 attaches to Barbies feet. Also show how a slipknot can be used to add additional rubber bands. Then allow students enough time to complete the experiment and record the results in the data table for Question 2. After all groups have completed the table, ask them to check their data. They should look for numerical irregularities. If any numbers in the table do not seem to fit, they may need to re-do the experiment for JUST the numbers of rubber bands where the data appears to be abnormal. To create a graph of the data, you may wish to have the students use the Illuminations website. You may use the Line of Best fit Activity or allow them to enter the data in the Barbie Bungee Spreadsheet. At the end of the lesson, take students to a location where Barbie can be dropped from a significant height. Possibilities include a balcony, the top row of bleachers, or even standing on a ladder in an area with a high ceiling. Allow students to test their conjecture about the maximum number of centimeters that can be used for a jump of 400 centimeters. Share: How many rubber bands are needed for Barbie to safely jump from a height of 400 cm? [Answers will vary, but students should use the line of best fit and the regression equation to determine an answer.] What is the minimum height from which Barbie should jump if 25 rubber bands are used? [Answers will vary, but students should use the line of best fit and the regression equation to determine an answer.] How do you think the type and width of the rubber band might affect the results? Do you think age of the rubber bands would affect the results--that is, what would happen if you used older rubber bands? [Rubber bands lose their elasticity with age or when exposed to extreme temperatures. Students would probably choose not to jump from a bridge if the bungee cord were old and brittle.] If some weight were added to Barbie, would you need to use more or fewer rubber bands to achieve the same results? Conjecture a relationship between the amount of weight added and the change in the number of rubber bands needed. Summarize: Slope means the rate at which the line changes and y-intercept is the initial value. Describe slope and y-intercept as found within the context of this problem. Identify and define the dependent and independent variables. Examine tables of data and discuss its relationship to the linear equation. 44

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49 How High? MINNESOTA Standards: Make reasonable estimates and judgments about the accuracy of values resulting from calculations involving measurements Solve multi-step equations in one variable. Solve for one variable in a multi-variable equation in terms of the other variables. Justify the steps by identifying the properties of equalities used. Objective: Students will compare relationship among volumes of similar objects. Students will calculate the height of each object through algebraic equations. Students will develop their own conjecture about the volume of a cone in relation to the volume of a cylinder with the same base. Materials: Computer Lab National Library of Virtual Manipulatives Two cylinders of different height, the bases must be the same size. One cylinder that has a different size base. Launch: Partially fill one of the two similar cylinders with water. If I were to pour this water into this second cylinder, how high will the liquid be? Let students ponder for a few moments. Pour the liquid and discuss the results. Then ask them to think about how high the liquid would be in the cylinder that has a larger/smaller base. Give students more time to predict the out come. Pour the water and discuss the results. Explore: Students will go to the computer lab and access the website for the National Library of Virtual Manipulatives. Under the Geometry strand for 6-8 Grades the students will choose the activity How High?. Students should be sure to bring paper, pencil and a calculator to the lab. Students will calculate the height of the liquid before moving the slider to the appropriate height. The teacher needs to circulate through the lab to make sure that students are not just randomly guessing. As students begin to understand the relationships and can manipulate the formula for volume of a rectangular prism and a cylinder, challenge them to find a rule (formula) for the relationship of the cones volume to that of a cylinder with the same base. Share: Return to the classroom for large group discussion about the activity. Ask for student s rules or formula s describing the relationship of a cone and a cylinder with the same base. Summarize: 49

50 Successfully made reasonable estimates about volume from calculations. Conclude that the volume of a cone is one-third the volume of a cylinder with the same base. 50