Grade 8 Mathematics
Grade 8 Mathematics Table of Contents Unit 1: Real Numbers, Measures, and Models... 1-1 Unit 2: Transformations and the Pythagorean Theorem... 2-1 Unit 3: Transversals, Surface Area and Volume... 3-1 Unit 4: Expressions and Equations in Algebra... 4-1 Unit 5: Functions, Growth and Patterns, Part 1... 5-1 Unit 6: Functions, Growth and Patterns Part 2... 6-1 Unit 7: Data and Lines of Best Fit... 7-1 Unit 8: Enhancing Understanding and Fluency... 8-1
2012 Louisiana Transitional Comprehensive Curriculum Course Introduction The Louisiana Department of Education issued the first version of the Comprehensive Curriculum in 2005. The 2012 Louisiana Transitional Comprehensive Curriculum is aligned with Grade-Level Expectations (GLEs) and Common Core State Standards (CCSS) as outlined in the 2012-13 and 2013-14 Curriculum and Assessment Summaries posted at http://www.louisianaschools.net/topics/gle.html. The Louisiana Transitional Comprehensive Curriculum is designed to assist with the transition from using GLEs to full implementation of the CCSS beginning the school year 2014-15. Organizational Structure The curriculum is organized into coherent, time-bound units with sample activities and classroom assessments to guide teaching and learning. Unless otherwise indicated, activities in the curriculum are to be taught in 2012-13 and continued through 2013-14. Activities labeled as 2013-14 align with new CCSS content that are to be implemented in 2013-14 and may be skipped in 2012-13 without interrupting the flow or sequence of the activities within a unit. New CCSS to be implemented in 2014-15 are not included in activities in this document. Implementation of Activities in the Classroom Incorporation of activities into lesson plans is critical to the successful implementation of the Louisiana Transitional Comprehensive Curriculum. Lesson plans should be designed to introduce students to one or more of the activities, to provide background information and follow-up, and to prepare students for success in mastering the CCSS associated with the activities. Lesson plans should address individual needs of students and should include processes for re-teaching concepts or skills for students who need additional instruction. Appropriate accommodations must be made for students with disabilities. Features Content Area Literacy Strategies are an integral part of approximately one-third of the activities. Strategy names are italicized. The link (view literacy strategy descriptions) opens a document containing detailed descriptions and examples of the literacy strategies. This document can also be accessed directly at http://www.louisianaschools.net/lde/uploads/11056.doc. Underlined standard numbers on the title line of an activity indicate that the content of the standards is a focus in the activity. Other standards listed are included, but not the primary content emphasis. A Materials List is provided for each activity and Blackline Masters (BLMs) are provided to assist in the delivery of activities or to assess student learning. A separate Blackline Master document is provided for the course. The Access Guide to the Comprehensive Curriculum is an online database of suggested strategies, accommodations, assistive technology, and assessment options that may provide greater access to the curriculum activities. This guide is currently being updated to align with the CCSS. Click on the Access Guide icon found on the first page of each unit or access the guide directly at http://sda.doe.louisiana.gov/accessguide.
Grade 8 Mathematics Unit 1: Real Numbers, Measures, and Models Time Frame: Approximately four weeks Unit Description This unit focuses on number theory and the comparison of rational and irrational numbers. Comparing the size of these numbers to each other and zero is the focus of the contents of the unit. Writing very large and very small numbers in scientific notation is also a part of this unit. Student Understandings The student will determine the relative size of rational numbers, comparing fractions, integers, decimals and percents. The student will compare rational and irrational numbers and discuss the differences. Students will determine which two whole numbers that radicals are located between. Guiding Questions 1. Can students compare rational numbers using symbolic notation as well as use position on a number line? 2. Can students recognize, interpret, and evaluate problem-solving contexts with rational numbers? 3. Can students determine approximate value of non-square radicals? 4. Can students group numbers into categories of rational and irrational numbers? 5. Can students perform operations with numbers written in scientific notation? Unit 1 Grade-Level Expectations (GLEs) and Common Core State Standards (CCSS) Grade-Level Expectations GLE # GLE Text and Benchmarks Number and Number Relations 1. Compare rational numbers using symbols (i.e., <, <, =, >, >) and position on a number line (N-1-M) (N-2-M) 2. Use whole number exponents (0-3) in problem-solving contexts (N- 1-M) (N-6-M) CCSS# CCSS Text 8.NS.1 Know that numbers that are not rational are called irrational, and approximate them by rational numbers. Understand informally that Grade 8 Mathematics Unit 1 Real Numbers, Measures and Models 1-1
8.NS.2 8.EE.1 8.EE.2 8.EE.3 8.EE.4 every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number. Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π2). For example, by truncating the decimal expansion of 2, show that 2 is between 1.4 and 1.5, and explain how to continue to get better approximations. Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 3 2 x 3-5 = 1 1 3 3 = 27 Use square root and cube root symbols to represent solutions to equations of the form x 2 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that 2 is irrational. Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology. Sample Activities Activity 1: Compare and Order (GLE: 1, 2) Materials List: Rational Number Line Cards - student 1 BLM, Rational Number Line Cards - student 2 BLM, Rational Number BLM, Compare and Order Word Grid BLM, calculators, paper, pencil Have students work in pairs. Provide a number line showing only the integers 1, 0, and 1. Give each student a deck of cards containing rational numbers including some negative rational numbers. Use the Rational Number Line Cards - Student 1 BLM and the Rational Number Line Cards Student 2 BLM to make both sets of cards for each pair of students. Student 1 should get a deck of rational numbers in fraction form, made by using Rational Number Line Cards - Student 1 BLM, and Student 2 should get a deck of rational numbers in decimal form, made using Rational Number Line Cards - Student 2 BLM. Have each student select a card from his/her deck and compare the cards. The comparison can be done using a calculator, mental math, or paper/pencil. Ask students to correctly place both rational numbers on the number line Grade 8 Mathematics Unit 1 Real Numbers, Measures and Models 1-2
and then write a correct comparison statement using symbols. For example, if the two rational numbers were 1 and 0.05, they would place a mark at the 1 2 2 point and the.05 point on their number line; then they would write a correct statement like 0.05< 1 or 0.05 1 2 2. Continue the activity having students place these on the number line. Distribute the Rational Number Line BLM to students for additional practice with comparing and ordering rational numbers. Once the students have placed the numbers correctly on the number line and discussions ensure that students understand the size of the various numbers. Put 4 0 on the board and ask the students to discuss with their partner where this might be located along the number line. This concept can be taught by making a table of values for 4 as shown in the table at the right. Ask the students to determine a method of determining the value of 4 0. If students need a hint to get started, ask them to think of how each of the values changes as the exponents decrease. Students will determine that to get from one to the next, the number is divided by four. Thus, 4 divided by 4 is 1. Ask the students to try another base and raise the power to four and decrease the exponent by one each time until the base is raised to the zero power to determine if the method works every time. A second table has been drawn at the right to illustrate negative bases raised to different powers. This also demonstrates that dividing by the base gives the next value or if going up, multiplying by the base; these are done going in a downward direction to illustrate that the Power of zero yields 1. Once this has been established, explain to the students that any number raised to the zero power results in 1. The division by the base rule (-3) 0 1 discussed above will work for integers except 0 because any number divided by 0 is undefined. We accept that 0 0 will also have a value of 1. Have a student make a rational number card with a zero exponent and place the card on the number line. Students will place this on their individual number line. A modified word grid (view literacy strategy descriptions) will be used to encourage higher order thinking through comparing and contrasting mathematical characteristics of numbers. The purpose of the Compare and Order Word Grid BLM is to develop an understanding of the relative size of a number when using the four operations as they make comparisons of the numbers. The Compare and Order Word Grid BLM can be given as a homework assignment and a modified word grid (0.5)(-11) 0.5/0.25 4 4 256 4 3 64 4 2 16 4 1 4 4 0 1 (-3) 4 81 (-3) 3-27 (-3) 2 9 (-3) 1-3 =2 >3 <1 (view literacy strategy descriptions) will be used to encourage higher order thinking through comparing and contrasting mathematical characteristics of numbers. This grid is modified to help the students apply their understanding of number relationships and how the function performed affects the outcome, whereas a word grid is used to help students relate terms and concepts. Use a simple modified word grid like the one shown and model how it is constructed and what they will need to do to complete each cell. The example has the student apply what he/she knows about multiplying and dividing decimals. Students should check each cell and determine whether the given operation in the vertical column will produce an outcome given in the horizontal row. If it is determined that this will be true, the student will check the corresponding cell, if not, leave the cell blank. Have the student justify the choice by showing Grade 8 Mathematics Unit 1 Real Numbers, Measures and Models 1-3
proof. The purpose of the Compare and Order Word Grid BLM is to develop an understanding of the relative size of a number when using the four operations as students make comparisons of the numbers using inequality symbols. The Compare and Order Word Grid BLM can be given as a homework assignment and returned the following day for discussion. A question such as the followings can lead to rich classroom discourse and should be responded to in their math learning logs (view literacy strategy descriptions): Is the rule you discovered the same for any two numbers? Why or why not? (Encourage students to think of cases in which they can challenge the answer). A learning log is a notebook or some other tool used to record reflections or understandings that have been experienced with mathematics. Explain to the students that their math log will be used all year to record new learning, and they should write questions that they want to understand through math class. This math learning log should be kept either in a separate notebook or a section in the binder used for reflection of mathematical throughout the year. Activity 2: Fraction or Not? (CCSS: 8.NS.1; 8.NS.2; 8.EE.2) Materials List: Real Number BLM cards for each pair of students, Squares and Square Root BLM, grid paper, unlined paper, pencil, calculator Teacher information: This is the first introduction to irrational numbers for the students. Explore the meaning of square root and discover that there are some radical numbers that do not have an exact square root. The activity is an exploration activity and the formal definitions will be discussed in the next activity. Rational numbers are all of the numbers that can be written as fractions and do not have a denominator of zero. Rational numbers include natural numbers, whole numbers, integers, fractions and decimals that repeat or terminate. Students will find that some decimals do not repeat and, therefore, they do not end these are irrational numbers. Pi is an irrational number and the approximation of 7 22 represents the approximate value of pi. It is critical to stress that whatever is used for pi (22/7, 3.14) is all approximation only; they are not, in fact, pi. Begin the class by using SQPL (view literacy strategy descriptions). This strategy, Student Questions for Purposeful Learning, begins with a statement or question that pertains to the content. Write the statement on the board or read it to the students, and then pair them up. Begin with the statement, Every number can be written as a fraction (students have explored repeating and terminating decimals in earlier grades and have written whole numbers as fractions in early grades). Have the students pair up and generate 2 3 questions that they would like to have answered about the statement. Give the groups 2 3 minutes to generate their questions. Students will then share their questions with the whole class, and the teacher will record the questions as the students share. After all questions have been shared, look over the list and add any questions that need to be answered during the lesson. Tell the students that another class had these questions. Possible questions: What can you say about repeating decimals? Grade 8 Mathematics Unit 1 Real Numbers, Measures and Models 1-4
Can the denominator be a zero? Distribute grid paper to each student. Instruct students to sketch a square with an area of 9 square units. Discuss the concept that each side of the square has a length of 9 units or 3 units in length. Have them draw squares with areas of 1, 4, 16, 25, 36 and 49 square units. Discuss the length of the sides in square root and whole numbers. Have the students take out their grid paper once more and make a sketch of a square with an area ½ square unit. Give students time to find the length of the side. On grid paper this will be easily seen. Explain to the students that 1 can also 4 1 be written as and students can find the square root of both 1 and 4 to get ½. Discuss the idea 4 of the relationship of ¼ of a square unit and that the length of the side of the square is ½ unit. Repeat this part of the activity by having students make a sketch of a square with the area of 1 square unit. A sketch can be made on a grid as shown in the diagram. 16 Distribute unlined paper and ask the students if it is possible to draw a square with the area of 2 square units. Have students draw a square with an area of 2 square units. Ask students what they know for sure about the length of the side of the square with an area of 2 square units. The students can relate that the 1 = 1and 4 = 2, justifying that the 2 is between 1and 2. Have them write the length of the side of the square as 2. Lead a discussion about the square root of two and have students use their calculators to find the square root of two. Encourage a discussion about how they know that the square root of 2 is closer to 1 than to 2 on the number line. Students can refer to the original comparison between 1 = 1and 4 = 2 and explain that 2 is closer to the square root of 1, so it will be less than half-way between 1 and 2 on the number line. Students will see that it is a decimal that does not terminate or repeat. Explain that the side length is the square root of 2 ( 2 ). Make a sketch of a number line on the board with only 0 placed and have students determine the position of the 2 on the number line. Ask students if this is the only place that the 2 can be correct. Discuss the position and if discussion does not elicit the idea that the unit has not been established, explain that the position of the 2 could be any position along the line. Place a 1 on the number line and have the students determine the position of 2 and 2. Have students place the 1 on the number line and discuss the placement. 4 Distribute the Exploring Squares and Square Root BLM. Give the students about 3 minutes to explore the questions in pairs. Discuss findings by having the groups share with the class. Grade 8 Mathematics Unit 1 Real Numbers, Measures and Models 1-5
Give each student pair a set of cards with Real Numbers, BLM. Some cards will have a rational number and others irrational. Give the students time to determine which of their numbers can be written as a fraction and which numbers do not repeat or terminate. Give the students time to explore. Some frustration is okay because they have to learn how to persevere in their explorations. Monitor groups, however, so that questions can be asked as needed. Once the students have completed their explorations, have them group the cards into those that can be written as fractions and those that cannot be written as fractions Have groups share their findings with the class. Have students place cards along a number line on the board. Students should justify the placement of each of their numbers. To ensure whole class participation, have students draw a number line on paper and place the points justified at the board on their paper. Refer to the questions that were generated at the beginning of the class and have the students determine which of these can now be answered. Have students respond in their learning logs (view literacy strategy descriptions) as they reflect on what they learned today about rational and irrational numbers. Activity 3: Real Numbers (CCSS: 8.NS.1) Materials List: Venn Diagram BLM, paper, pencil Discuss the different groupings of real numbers by placing a list of the following numbers on the board: -1, 0, ½, 0.122344456666377777..., 0.04, and -4. Tell the students that A rational number is a number that can be represented by b a, where a and b are integers and b 0. Rational numbers are sometimes referred to as rationals, which does not mean the same as when a person is referred to as being a rational person. It means that the 5 numbers represent a ratio. Some examples of rational numbers are, 1. 3, 7.5, -5, and 9. Ask 8 the students which of the numbers listed on the board will fit into this category (all but the 0.122344456666377777 will fit into the rational number category). Next, ask the students why 0.122344456666377777 will not fit this definition of a rational number (it cannot be written as a fraction). Put the following definitions on the board and ask the students to determine which of the rational numbers listed fit into each of these categories. Natural or counting numbers are the set of numbers used to count objects. {1, 2, 3, 4, 5,..} Whole numbers are natural numbers including zero. {0, 1, 2, 3, 4, 5...} Grade 8 Mathematics Unit 1 Real Numbers, Measures and Models 1-6
Irrational Numbers Integers are whole numbers and their opposites. {-2, -1, 0, 1, 2,.. } All rational and irrational numbers form the set of real numbers. Have the students organize the information on the BLM into a graphic organizer (view literacy strategy descriptions). Graphic organizers are used to assist students as they organize information to make it easier to understand. Since this is the first graphic organizer of the year, it might be best to suggest a type of organizer to use with the information they just read. A flow chart might be an example of a graphic organizer to use for this information. Any graphic organizer can be used that will help Real Numbers the student make sense of the information. Rational Numbers Once the students have completed a graphic Integers organizer, have them take the number cards from Activity 2. Tell the pairs of students to group their Whole Numbers cards as rational numbers or irrational numbers. This should take only about 2-3 minutes. Have Counting or Natural Numbers students share their groupings and justify why the number is either rational or irrational. Distribute the Venn Diagram BLM and using their graphic organizer, have the students take their rational numbers and place the numbers in the correct circle. Use the Venn Diagram as a formative assessment to monitor the pairs of students and observe their level of understanding. Activity 4: Yes or No (CCSS: 8.NS.1) Materials: Word Grid BLM, Paper, Pencil Have a volunteer share his or her method of organizing real numbers into a graphic organizer. Lead a discussion with students about how the numbers are related. The word grid (view literacy strategy descriptions) used in this activity provides students with an organized framework for comparing real numbers. Before beginning the activity, put a simple word grid on the board and discuss a concept of classifying real numbers. For students to understand the use of a word grid, students should complete the simple word grid. With the example, students should justify 2 5 why 2 is considered rational, an integer, a whole number and a rational natural number while the square root of 5 is simply an irrational number. Good vocabulary review with this example. irrational Distribute the word grid BLM with the unit vocabulary comparing real numbers. Have students work independently to complete the word grid. This can be used as a formative assessment of student understanding of classifying rational and irrational numbers. integer whole number natural number Grade 8 Mathematics Unit 1 Real Numbers, Measures and Models 1-7
Activity 5: Let s do Radicals! (CCSS: 8.NS.2) Materials: Real Number Cards BLM, Number Line BLM, paper, pencils, calculators Begin the lesson by asking students to make a list of the first ten perfect squares or square numbers. Review square roots by having the students write the square roots of the square numbers. Ask students to discuss with a partner how to estimate the square root of a number that is not a square number. Have the students share their ideas with another pair of students. Ask for volunteers to explain their ideas to the class. Ask the students to determine which two whole numbers it will fall between and to which of the two numbers it is closer. Have the class determine the position of 5 on a number line. Have students write the two square numbers closest to 5 and instruct students to find the square roots of these two numbers. ( 4 = 2 9 =3). Using this information, have students justify whether the 5 will be > 2.5 or < 2.5(it is less than 2.5 because 5 is closer to 4 than it is to 9). Distribute cards with real numbers, Real Number Cards BLM, to groups of four students. Explain to the students that they will work to put the numbers in order from least to greatest. If a group of students is stuck, guide them to look at their square root list that was prepared at the beginning of the lesson. After the students have had time to put the numbers in order, distribute number line BLM and have students place the numbers from the cards in correct position along the number line. Have students indicate which numbers on the number line are irrational numbers and have someone explain again the definition of an irrational number. Discuss results as a class. Challenge the students by asking the question: What do you think 3 27 symbolizes? This may be the first time the students have seen the cube root symbol. Have the students use a factor tree to find the prime factorization of 27 so that they will get the exponential representation of 3 3. This gives a hint without telling the students that it represents something with a cube. At this time, tell the students that when a number is a perfect cube, the result of finding the cube root will be a whole number, as was done with the square roots. Have the students make a list of the first five perfect cube numbers. (1, 8, 27, 64, 125) Tell the students to write equations to show 3 3 3 3 3 the cube root of each of these perfect cubes. ( 1 = 1; 8 = 2; 27 = 3; 64 = 4; 125 = 5 ) Activity 6: Radically! More or Less (CCSS: 8.NS.2) Materials: Paper, pencils Students should write the following radical numbers on their paper: 5, 12, 111, 67, 55, 99, 43, 67. Have the students work in groups of four to determine the two consecutive whole numbers that are on either side of the numbers and which of these whole numbers is closest to the radical number. Students should be able to justify their answers. Give the class time to complete the assignment. Grade 8 Mathematics Unit 1 Real Numbers, Measures and Models 1-8
Explain to the class that professor know-it-all (view literacy strategy descriptions) is a method of reviewing content. One group of four students will be in the front of the room and answer questions about determining the approximate value of radical numbers that are not square numbers. The group at the front will be the experts, and the class members can ask questions they have about approximating the value of radical numbers. The group at the front will huddle after a question is asked so that they can agree upon an answer. Since this is the first time they have used this strategy this year, it might help to have a list of questions prepared and distributed so that the students have an idea as to what type of questions will be beneficial for review. Questions such as 1) How do you determine where to start? 2) How do you decide if the radical is closer to one whole number or the other? 3) Is it possible to have a radical number that is halfway between two whole numbers? Activity 7: Computing Using Scientific Notation (CCSS:8.EE.3) Materials List: paper, pencil, Scientific Notation BLM, calculators Have the students write one hundred twenty-three billion (123,000,000,000). Lead a discussion about how numbers this large become cumbersome and not easy to record accurately. This is why scientific notation is used. This same number can be written as: 1.23 x 10 11 The first number 1.23 is called the coefficient. It must be greater than or equal to 1 and less than 10. The second number is called the base. It must always be 10 in scientific notation. The base number 10 is always written in exponent form. In the number 1.23 x 10 11, the number 11 is referred to as the exponent or power of ten. To write a number in scientific notation, place the decimal point in the original number so that a number >1 and <10 is created. Moving the decimal to the left is the same as dividing by a power of 10. To determine what power of 10 was used in the division, count the number of decimal places that the decimal would have to be moved to get back to the original number. Use this number of decimal places as the power of 10. The zeros to the right of 3 are no longer needed as they would be eliminated when the division was made. Example: 1.23000000000 x 10 11 becomes 1.23 x 10 11 after the zeros are dropped. There are also numbers that are extremely small. Suppose you were to measure the length of an ant. The ant is 0.0625 inch long. The decimal will be moved to the right so that a number >1 and <10 is created. Teacher Note: The negative exponents might be a new idea for the students; be sure the discussion clarifies any confusion. Point out that to multiply by products of 10, move the decimal to the right and the inverse would be to divide by products of 10 so move the decimal to the left, Grade 8 Mathematics Unit 1 Real Numbers, Measures and Models 1-9
and likewise, the exponent will be negative to indicate that the number in scientific notation is smaller than 1 but greater than 0. Example: 0.0625 would become 6.25 x 10-2, with the negative exponent indicating the original number was less than 1. Activity 8: Trip to Mars! (CCSS: 8.EE.4) Materials List: Trip to Mars BLM, paper, pencil, calculator Begin the class by using SQPL (view literacy strategy descriptions). The distance from Earth to Mars changes every minute with the difference between the closest and farthest distance being more than 300,000,000 kilometers. Distribute Scientific Notation BLM and give students time to complete the situations. The information to answer questions generated by the SQPL statement should be answered using the information on the BLM. After students have completed the BLM, refer to the SQPL questions at the beginning of the lesson and have students determine if the questions have been answered. Questions may arise that are more science related; challenge the students to research these and report their findings tomorrow and share with the class. 2013-2014 Activity 9: Powerful numbers (CCSS: 8.EE.1) Materials List: Powerful Numbers BLM, paper, pencil, calculator Have students use a calculator and the Powerful Numbers BLM to complete the chart with powers of 10 from 4 to 4. Discuss the patterns that are observed and the significance of negative exponents. Discuss the idea developed earlier when looking at the Power of zero, that as the exponent increases or decreases they are multiplying by the base or dividing by the base. Challenge the students by asking them to complete an exponent chart using the powers of 2 from -4 to 4. Ask the students to work with a partner and develop a conjecture describing the effect of the negative exponent on the value of the number. Ask the students what a number like 4.5 x 10-3 would be before having them complete the Powerful Numbers BLM. After the students have completed the scientific notation portion of the BLM, discuss any problems the students may have encountered. Provide students with real-life situations for which scientific notation may be necessary, such as the distance from the planets to the sun or the mass of a carbon atom. Have students investigate scientific notation using a calculator. Allow students to convert numbers from scientific notation to standard notation and vice versa. Relate the importance of scientific notation in the areas of physical science and chemistry. Grade 8 Mathematics Unit 1 Real Numbers, Measures and Models 1-10
2013-2014 Activity 10: Exponential Growth (CCSS: 8.EE.1) Materials List: paper, pencil, 1 sheet of computer or copy paper, Exponential Growth and Decay BLM 1 Explain to the students that will look at exponential decay. Give each student a sheet of 8 by 2 11 paper. Have students look at the number of regions on the unfolded paper. There is one region (8 ½ x 11 ). Have them make one fold in the paper so that the two areas or regions are equivalent. Ask students to determine the number of regions and the area of the smallest region using the sheet of paper as the unit. Number of Folds Number of Regions Area of Smallest Region 0 1 1 sheet of paper 1 1 2 1 2 sheet of paper 2 1 2 4 2 2 sheet of paper 4 1 3 8 3 2 sheet of paper 8......... N 2 n 1 n or 2 sheet of paper 2 n Distribute the Exponential Growth and Decay BLM and have students complete the table folding the paper until they can confidently complete the table information. Instruct students to fold the paper in half several times, but after each fold, they should stop and fill in a row of the table. After students have completed the BLM, have them discuss how they were able to identify the independent and dependent variables. The independent variable is the number of folds; the dependent variable is the number of regions. Ask, will the graph be graph linear? This is called an exponential growth pattern. The third column gives the area of the smallest region. When comparing the number of folds to the area of the smallest region, the pattern becomes one of exponential decay. Include the significance of integer exponents as exponential decay is discussed. 2013-2014 Activity 11: Properties of exponents (CCSS: 8.EE.1) Materials: Vocabulary Awareness BLM, Exponent BLM, pencil, paper, calculator Begin this activity by explaining that today s activity involves the literacy strategy called vocabulary self-awareness (view literacy strategy descriptions). Distribute the Vocabulary Awareness BLM. This chart has the vocabulary necessary already inserted. It can also be given to the students without vocabulary so that they can write the vocabulary needed themselves. Grade 8 Mathematics Unit 1 Real Numbers, Measures and Models 1-11
Have the students rate each vocabulary word (in this case the properties of exponents) according to their level of familiarity and understanding. A (+) sign indicates a high degree of comfort and knowledge, a check ( ) indicates uncertainty, and a minus sign (-) indicates the word is brand new to them. Students should also provide a definition and example for each word; some of these will be guesses. Over the course of the activity, students will develop an understanding of these properties. After students have had time to complete the Exponent BLM, have them take out their Vocabulary Awareness BLM and make additions or corrections to those vocabulary terms that were unfamiliar or were not completely understood. Have students get with a shoulder partner and discuss each of the properties. Ask students to share these properties with the class to further develop a complete understanding of the properties of exponents. Sample Assessments Performance assessments can be used to ascertain student achievement. General Assessments Give the student a list of about fifteen rational numbers including fractions, decimals and percents, making sure that some of the values are equivalent (i.e. 1 4 and 25%). The student will make a number line and place all fifteen rational numbers along the number line in the correct position. To complete the assessment, the student will write at least 10 inequality statements using the symbols <, >,, and. Give the student a list of real numbers and have him/her place the numbers in order from least to greatest. Challenge the student to write one scientific notation real-life problem by researching scientific facts that result in very large or very small numbers. Whenever possible, create extensions to an activity by increasing the difficulty or by asking what if questions. Have the student create a portfolio containing samples of the experiments and activities. Grade 8 Mathematics Unit 1 Real Numbers, Measures and Models 1-12
Activity-Specific Assessments Activity 3: The students will be given a list of real numbers, and they willclassify them using a graphic organizer. Activity 4: Provide students with a situation such as, Tell whether the following can be found between the numbers 4 and 5. If it is possible, give an example. Provide explanations for each answer. o A real number o A rational number o An irrational number o An integer Activity 6: Provide students with a list of radicals similar to the ones used in this activity. Have students give the whole number values on either side of the radical and state whether it is closer to one of them or the other. Example: 6 is between whole number square roots of 4 and 9, so the square root is between 2 and 3. 6 4 is 2 and 9 6 is 3, so the square root of 6 is closer to the square root of 4 than the square root of 9. Activity 7: Provide students with sets of real numbers and have them put them in order from least to greatest. Activity 8: Have students determine the maximum number of people that could travel on the trip to Mars if the food for one person will occupy 0.00098 cubic feet of the available cargo space and they can use no more than 0.05 cubic feet of the available cargo space for food. 0.05/0.00098 = 51.02 so 51 people could store enough food Grade 8 Mathematics Unit 1 Real Numbers, Measures and Models 1-13