Unit 2: Whole Numbers, Factors, and Primes 11

Grade 6 Mathematics Grade 6 Mathematics

Table of Contents

Unit 1: Data and Decisions...... 1

Unit 2: Whole Numbers, Factors, and Primes...... 11

Unit 3: Fractions, Decimals, and Parts...... 22

Unit 4: Operating with Fractions and Decimals...... 32

Unit 5: Geometry, Perimeter, Area, and Measurement...... 41

Unit 6: Taking a Chance...... 53

Unit 7: Strengthening Whole Number Multiplication and Division...... 61

Unit 8: Integers, Patterns, and Algebra...... 67 Louisiana Comprehensive Curriculum, Revised 2008 Course Introduction

The Louisiana Department of Education issued the Comprehensive Curriculum in 2005. The curriculum has been revised based on teacher feedback, an external review by a team of content experts from outside the state, and input from course writers. As in the first edition, the Louisiana Comprehensive Curriculum, revised 2008 is aligned with state content standards, as defined by Grade-Level Expectations (GLEs), and organized into coherent, time-bound units with sample activities and classroom assessments to guide teaching and learning. The order of the units ensures that all GLEs to be tested are addressed prior to the administration of iLEAP assessments.

District Implementation Guidelines Local districts are responsible for implementation and monitoring of the Louisiana Comprehensive Curriculum and have been delegated the responsibility to decide if  units are to be taught in the order presented  substitutions of equivalent activities are allowed  GLES can be adequately addressed using fewer activities than presented  permitted changes are to be made at the district, school, or teacher level Districts have been requested to inform teachers of decisions made.

Implementation of Activities in the Classroom Incorporation of activities into lesson plans is critical to the successful implementation of the Louisiana 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 Grade-Level Expectations 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.

New 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.

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 each 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. The Access Guide will be piloted during the 2008-2009 school year in Grades 4 and 8, with other grades to be added over time. Click on the Access Guide icon found on the first page of each unit or by going directly to the url http://mconn.doe.state.la.us/accessguide/default.aspx. Louisiana Comprehensive Curriculum, Revised 2008

Grade 6 Mathematics Unit 1: Data and Decisions

Time Frame: Approximately four weeks

Unit Description

This unit examines the selection and use of appropriate statistical methods to analyze data in numerical and graphical ways, including use of an input-output table. Venn diagrams are used to solve problems involving counts of objects classified in multiple ways.

Student Understandings

Students can display data using frequency tables, stem-and-leaf plots, and scatter plots and discuss the patterns seen in each type of display. This representation of data can be described by using measures of central tendency. In addition, students can use Venn diagrams as an appropriate method of solving problems.

Guiding Questions

1. Can students organize and display data using frequency tables, stem-and-leaf plots, and scatter plots? 2. Can students use two-circle Venn diagrams to solve problems? 3. Can students use trends and patterns to describe given data? 4. Can students calculate measures of central tendency and range for a set of data? 5. Can students make informed decisions about which graph(s) might best be used to represent given data?

Unit 1 Grade-Level Expectations (GLEs)

GLE # GLE Text and Benchmarks Data Analysis, Probability, and Discrete Math 29. Collect, organize, label, display, and interpret data in frequency tables, stem- and-leaf plots, and scatter plots and discuss patterns in the data verbally and in writing (D-1-M) (D-2-M) (A-3-M) 30. Describe and analyze trends and patterns observed in graphic displays (D-2-M) 32. Calculate and discuss mean, median, mode, and range of a set of discrete data to solve real-life problems (D-2-M)

Grade 6 MathematicsUnit 1Data and Decisions 1 Louisiana Comprehensive Curriculum, Revised 2008

33. Create and use Venn diagrams with two overlapping categories to solve counting logic problems (D-3-M) Patterns, Relations, and Functions 37. Describe, complete, and apply a pattern of differences found in an input-output table (P-1-M) (P-2-M) (P-3-M)

Sample Activities

Activity 1: Frequency Tables (GLE: 29, 32)

Materials List: data set, Frequency Tables BLM, Vocabulary Card BLM, index cards, paper, pencil, one metal ring per student

Write a set of test scores or daily number grades, with an odd number of grades in the set, on the overhead or blackboard. Have students use the grades to construct a frequency table, discuss the role of intervals, and build a related line plot from a frequency table. Once the line plot is complete, probe students to question their interpretation of the shape of the data. Ask questions such as these:  What is the range of the data?  Are there any gaps in the grades?  Is there one grade that occurs more frequently than others? (Discuss mode)  Is there one grade that is set apart from the others? (Discuss outliers)  What might someone say if asked to describe a typical grade form the set of data? (Listen for many opinions- may hear answers referring to mean, median, mode, or average.)

Ask students to compare and contrast the differences when categorizing the grades by numerical values and by letter grades. Have students use tally marks to show the number of students earning each letter grade. Provide opportunities for students to make similar analyses using different sets of data. For additional practice, have students complete the Frequency Tables BLM. The students will collect data, create a frequency table, and analyze the data.

Below are several websites to provide additional data or to use instead of using grade data. NFL Stats http://www.nfl.com/stats/team NBA Stats http://www.nba.com/statistics/ Major League Baseball Stats http://mlb.mlb.com/NASApp/mlb/mlb/stats/index.jsp 100 Largest School Districts Stats http://nces.ed.gov/programs/digest/d05/tables/dt05_090.asp

To develop students’ knowledge of key vocabulary, have them create a vocabulary card (view literacy strategy descriptions) to define frequency table. Distribute 3x5 or 5x7 inch index cards to each student, and ask them to follow your directions in creating the sample card. Make a transparency of the Vocabulary Card BLM to demonstrate what the card should look like. Have the students place the targeted word in the middle of the card, as in the example below. Have the students work together in groups to define the term and then discuss the definitions as a class and select the one that best defines the word. Each student should then write the definition in the appropriate space. Next, have students list the characteristics or description, give one or two examples, and illustrate the term.

Throughout the unit as students come across key terms, have them create vocabulary cards for each term. Have students punch a hole in each card and use a metal ring to hold them together throughout the year. Allow time for students to review their cards and quiz a partner on the terms to hold them accountable for accurate information on the cards.

Definition Characteristics

A display to show Lists items and uses how often items, tally marks to record numbers, or a range and show the number of numbers occur. of times they occur

Frequency Examples Table Illustration

A survey was taken on Rural St. Cars Tally Frequency In each of 10 homes, people 1 III 3 were asked how many cars they 2 IIII 4 had at their household. Here 3 II 2 are the results: 4 I 1

1, 2, 3, 4, 2, 1, 3, 2, 1, 2

Activity 2: Stem-and-Leaf Plots (GLEs: 29, 30)

Materials List: data to analyze, Stem-and-Leaf Graph BLM, paper, pencil

Using a set of data from Activity 1, introduce students to a stem-and-leaf plot. Here the emphasis should be on seeing the relationship between choosing stem size and the resulting shape and nature of the plot. Make distinctions between ordered and non- ordered listings in the leaves. Distribute the Stem-and-Leaf Graph BLM. Have students work in small groups to choose a data set and construct a stem-and-leaf description plot

using different data sets, such as data from an almanac or book of facts. Ask student to write a short interpretation of the patterns they see.

Activity 3: Measures of Central Tendency (GLEs: 29, 30, 32)

Materials List: data to analyze; Mean, Median, Mode Word Grid BLM; paper; pencil

Using a set of data from Activity 1, have students find the mean, median, mode(s), and range for the data set. Have students repeat this activity with other data sets. A stem-and- leaf plot will aid in finding the modes and medians. Make sure that students have a clear understanding that the terms mean, median, and mode are all measures of “average” or central tendency and have them create vocabulary cards (view literacy strategy descriptions) for each of these terms.

As an extension, discuss when it is more appropriate to use one description - mean, median, or mode - over another one. On the board or a piece of chart paper draw a word grid (view literacy strategy descriptions) like the one below or use the Mean, Median, Mode Word Grid BLM on the overhead. In the first column are situations in which one of the central tendencies would be necessary. With the students’ participation, fill in the word grid by placing a “+” in the space corresponding with the central tendency that would be most appropriate for that situation.

Situation Mean Median Mode Calculating your + grade for a class Ordering jeans for + the Gap The average age of people in a 6th grade + class when the teacher is included

As a class, come up with additional situations to add to the word grid. Once the grid is complete, provide opportunities for students to quiz each other over information from the grid and use the grid to prepare for quizzes.

Teacher note: Students are only required to master the calculation and meaning of each measure of central tendency. The idea of most appropriate is an introduction to be mastered in the eighth grade.

Activity 4: Comparing Data (GLEs: 29, 30, 32)

Materials List: one inch square of paper for each student, one yard of masking tape, paper, pencil

Have students select and create a data set from the total number of letters in the first and last names of students in the class. Ask students to find a way to organize the data so that they can describe the length of a typical name. Create a frequency table to represent the data. The value that occurs most frequently is the mode of the set. Upon further examination, have students describe the range as the lowest value to the highest value in the set (i.e., 15 to 38). To find the median, instruct each student to write the length of his/her name on a one-inch square of paper. Evenly place the squares in order from smallest to largest along a yard of masking tape so that the squares are attached to the tape. Cut off excess tape. Fold the train in half to discover the median. If working with an even number data set, the median will fall between two. Ask students to look for patterns and make summary statements about the data focusing on comparing means, medians, and modes of the data sets.

Activity 5: Looking at Data (GLEs: 29, 30, 32)

Materials List: data to analyze, paper, pencil

Direct students to display a data set in a variety of ways—frequency charts, stem-and-leaf plots, or through data description using mean, median, and mode. Such data sets may be drawn from sporting results, local events, or personal data. Have students record any trends and patterns observed in their math learning log (view literacy strategy descriptions) and then share their observations with the class. This learning log should be kept in a small notebook used only for recording math understanding. Explain to the students that they will use the math learning log all year to record new understandings, explain math processes, pose and solve problems, make and check predictions, and reflect on what has been learned. Invite students to personalize their math log covers with their names, illustrations, and/or pictures from magazines.

Activity 6: Input-Output Table (GLE: 37)

Materials List: Input-Output Tables BLM, paper, pencil

Students should understand that data could be given, perhaps in an incomplete format, where decisions and interpretations need to be made. By examining the data and identifying patterns, it is possible to find missing pieces of data and to determine the rule for the given situation. Provide students with a variety of sample input-output tables. Guide them through finding patterns so that they can fill in missing data and determine the rule for the table. Help students make the connection to ordered pairs that can be graphed on a coordinate plane.

input output 0 8 1 11 2 14 3 4 5 50 158 Solution:

input output 0 8 1 11 2 14 3 17 4 20 5 23 50 158

Distribute the Input-Output Tables BLM to the students. Have them work in pairs to complete the problem. Discuss solutions as a class. These websites provide additional practice for students with input-output tables. http://www.globalclassroom.org/2005/mctm/function_machine.xls http://www.shodor.org/interactivate/activities/WholeNumberCruncher/? version=1.5.0_07&browser=MSIE&vendor=Sun_Microsystems_Inc. http://www.amblesideprimary.com/ambleweb/mentalmaths/functionmachines.html

Activity 7: What’s My Rule? (GLE: 37)

Materials List: What’s My Pattern? BLM, pencil

Distribute the What’s My Pattern? BLM to the students. Have students work in pairs to examine the table, identify the pattern and find the missing data for each input-output table. Discuss the solutions as a class.

Activity 8: Scatter Plots (GLEs: 29, 30)

Materials List: uncooked spaghetti noodles, Scatter Plots Data Sheet BLM, Graph Paper BLM, pencil, computers

Have students work in pairs to measure each other’s height, and record the measurements on the Scatter Plots Data Sheet BLM. Have each pair collect and record the data from other student pairs until they have data from the entire class. Possible measures could be their heights and heights of their waists from the ground. Ask students to place the height data on the x-axis and the heights of their waists from the ground data on the y-axis. Have students draw an estimate of the line of best fit. Trends or patterns in the data should be 6 identified and compared with the golden ratio, where y10 x .

Another set of measures could be the length of the arm from the bend of elbow to the tip of the pointer finger (in inches) and length of leg from back of knee to floor (in inches). Provide instruction and monitor student work to clarify any questions that may evolve.

If computers are available, have students enter the data into a spreadsheet to create the graph electronically. The following site can be used to create a scatter plot online: http://nces.ed.gov/nceskids/createagraph/default.aspx

Teacher Note: Students can use a piece of uncoooked spaghetti to place on their scatter plots to approximate a line of best fit. Recall that the line of best fit is that line which is as close to all the data as possible. Usually this will be a line that “splits” the data into two approximately equal groups above and below the line.

Activity 9: Venn Diagrams (GLEs: 30, 33)

Materials List: Venn Diagram BLM, paper, pencil

Begin discussion by asking students “What do you think when you hear the term Venn diagram?” Have the students write and reflect on their ideas in their math learning log (view literacy strategy descriptions). Have students share their ideas followed by conveying that Venn diagrams can be used to display information and to solve problems.

Draw two overlapping circles to use as a two-circle Venn diagram, and distribute the Venn Diagram BLM to the students. Ask students, “Who plays sports after school?” Make a list of the students who play sports. Ask, “Who works on homework after school?” Make a separate list of these students. Use the lists of after-school activities to create a two-circle Venn diagram. Include the Universe in the drawing. The Universe takes into account those that would not appear within either circle. Lead a discussion that considers questions such as these:  Can a student appear more than once on the diagram?  Does every student appear on the diagram?  What does the student’s position on the diagram tell about him or her?

 What is true about a student whose name doesn’t appear in one of the circles?  What is true about the students whose names appear in the intersection of the two circles?  How many students do homework after school?  How many students play sports after school?  How many students only do homework after school?  How many students only play sports after school?

Provide additional practice with Venn diagrams. Offer completed diagrams for class discussion purposes, or post various questions around the room for the students to answer using a Venn diagram. Then assign each Venn diagram to a different group of students to analyze and present their findings. Have students determine the total number of “items” belonging to each of the two categories depicted by the Venn diagram. Make sure students do not double count those items that lie in the intersection of the two circles. Continue with provided diagrams until students feel comfortable interpreting the information.

Activity 10: Using Venn Diagrams to solve problems (GLE: 33)

Materials List: Venn Diagram BLM, Venn Diagram Story Chain BLM, paper, pencil

Lead discussion of the significance of Venn diagrams in solving problems. Provide students with the two-circle Venn diagram BLM from Activity 10 and simple problems with two sets of values, such as these: In a class of 25 students, ten play sports, four take music lessons, and two participate in both sports and music. How many students in the class are not enrolled in either sports or music class? The answer is 13.

Thirty students participated in the school talent show. Ten were singers and 6 were dancers. Two of them were singers and dancers. How many students are involved in either singing or dancing? The answer is 14.

Encourage students to use the diagrams to record the numerical information then use it to solve the problem.

After discussing the solutions to the problems above, have the students create a story chain (view literacy strategy descriptions). Put students in groups of four. On a sheet of paper, ask the first student to write the opening sentence for the math story chain:

There are forty students in the 6th grade.

The student then passes the paper to the student sitting to the right, and that student writes the next sentence in the story.

14 are taking art and 29 are taking band.

The paper is passed again to the right to the next student who writes the third sentence of the story.

If five students are taking both classes, how many students are just taking art?

The paper is now passed to the fourth student who must use a Venn diagram to solve the problem and write out the answer. The other three group members review the answer for accuracy.

Answer: Nine students are just taking art.

Use the sample Venn Diagram Story Chain BLM to explain the activity to the students and model what their completed story chain will look like. Cover the story and reveal one line at a time to show an example of what each student would write. This activity allows students to use their writing, reading, and speaking skills while learning and reviewing important math concepts.

Sample Assessments

General Assessments

 The student will provide a written analysis of trends or patterns found in the scatter plots. Then facilitate/guide a whole-group discussion concerning the student’s findings.  Whenever possible, create extensions to an activity by increasing the difficulty or by asking “what if” questions.  The student will create portfolios containing samples of his/her experiments and activities.  The student will be afforded opportunities to reflect on the data represented by each graph, chart, and/or table. Throughout this unit, the students will be encouraged to write about the shape of the data and why a particular type of graph is a good depiction of the data.  Facilitate a small group discussion to determine student misconceptions, understandings, use of correct terminology, and reasoning abilities. Appropriate questions to ask might include these: o What you have done so far? Is there anything else to do? o What made you decide to use this method? o Is there another method that might have worked? o Have all the possibilities been explored? How can you tell? o What do you think about what ___ said? o Do you agree with your group’s answer? Why or why not?

o How would you convince the rest of us that your answer makes sense?  Create a performance task assessment to evaluate understanding at the end of the unit. One way to do this is to provide a small box of raisins for each student in the class, and have each student notice that the weight is given on the box but not the number of raisins. Have each student record an estimate of the number of raisins in the box on a sticky-note (no benchmarks). Instruct the student to open the box and look at the number of raisins on the top layer. Allow students to change estimates at this time, if desired. Place the sticky notes on the board or in clear view of all students. Next, have the students actually count the raisins in the box and record and display the number in a similar fashion. The data is then available for all students to use individually as they each create frequency tables, line plots, back-to-back stem and leaf plots. Student end products should be evaluated as well as written reflections of the data.

Activity-Specific Assessments

 Activity 8: The students will estimate the weights of their backpacks and then find actual weights. The student will collect and record information from all classmates in a table or t-chart. Using the collected data, the student will represent the data in an appropriately constructed scatter plot with the actual weight recorded on the x- axis and the estimate recorded on the y-axis.

 Activity 9: Ask the student to develop a Venn diagram from a given set of classification data.

 Activity 10: Allow small groups of students to formulate their own logic problems that can be solved using Venn diagrams. Have each group make an answer key with explanations for its problem. Have students distribute the problems to another group and check the work of the second group.

Grade 6 Mathematics Unit 2: Whole Numbers, Factors, and Primes

Time Frame: Approximately five weeks

Unit Description

This unit focuses on fundamental skills that are basic to operations with fractions and decimals. It provides opportunities for modeling and identifying perfect squares and working with operations involving powers of 10.

Students use various strategies to identify important foundation skills such as prime numbers, prime factorization, common factors, common multiples, least common multiple, and greatest common factor. They can mentally multiply and divide by 10 and powers of 10; it is actually inefficient to write down such computations or to use a calculator.

Guiding Questions

1. Can students factor whole numbers into primes? 2. Can students use number sense and factorization to find the greatest common factor (GCF) and least common multiple (LCM) of a pair of positive whole numbers? 3. Can students use different strategies to multiply and divide by powers of 10? 4. Can students model and identify perfect squares? 5. Can students describe patterns of growth and relationships?

GLE # GLE Text and Benchmarks Number and Number Relations 1. Factor whole numbers into primes (N-1-M) 2. Determine common factors and common multiples for pairs of whole numbers (N-1-M) 3. Find the greatest common factor (GCF) and least common multiple (LCM) for whole numbers in the context of problem-solving (N-1-M) 25 11. Mentally multiply and divide by powers of 10 (e.g., 10 = 2.5; 12.56 x 100 = 1,256) (N-6-M)

Grade 6 MathematicsUnit 2Whole Numbers, Factors, and Primes 11 Louisiana Comprehensive Curriculum, Revised 2008

GLE # GLE Text and Benchmarks Algebra 14. Model and identify perfect squares up to 144 (A-1-M) Patterns, Relations, and Functions 38. Describe patterns in sequences of arithmetic and geometric growth and now- next relationships (i.e., growth patterns where the next term is dependent on the present term) with numbers and figures (P-3-M) (A-4-M)

Sample Activities

Activity 1: Game Time - Working with Factors (GLE: 1)

Materials List: Factor Game BLM, First Moves BLM, Discussion Questions BLM, different colored pens, index cards, pencil

Review with students the concept of factor and compare it to the term divisor.

Distribute the Factor Game BLM, one per group of two students. Using an overhead projector, model the process by playing a game against a student by using the rules below. Use an appropriate number of games to clarify questions. Discussion should not include strategies at this time.

1) Form pairs and give each of the two students in the pair a different colored pen 2) Round 1:  Player 1 will choose and circle a number on the Factor Game board and then write this number in the correct score column.  Player 2 will circle all of the factors of the number that player 1 chose. The recorded score for Player 2 is the sum of the factors circled by Player 2.  Once a number is circled, it cannot be used again (it is out of play). 3) Round 2 – this is a role reversal of Round 1  Player 2 will choose and circle a number on the board and then write this number in the correct score column.  Player 1 will circle all of the factors of the number that player 2 chose unless the factor has been previously circled. The score for Player 1 is the sum of all the factors that Player 1 was able to circle. 4) If one player circles a number with all its factors already circled, that player loses his/her turn and does not receive points for that turn. 5) Repeat play until no numbers remain on the board with uncircled factors. When this occurs, the scores for each player are added and the player with the highest point total WINS.

Allow students time to play the game several times and fully understand the rules.

Distribute the First Moves BLM. Have the students work in pairs to list all of the factors for each number. After the students have made the list of factors, distribute the Discussion Questions BLM and give groups time to analyze the list of factors to answer the questions.

Have students create vocabulary cards (view literacy strategy descriptions) for factor and each new term throughout this unit. The students should add the new cards to their collection of cards from unit 1.

A number that is Numbers whose only multiplied by another factors are one and itself number to find a are prime numbers. product.

Factor

Examples Illustration

The factors of 12 are 1 x 12 = 12 1, 2, 3, 4, 6, 12 2 x 6 = 12 3x 4 = 12

This activity is adapted from Connected Mathematics Project, G. Lappan, J. Fey, W Fitzgerald, S. Friel and E. Phillips, Dale Seymour Publications, (1996) pp.17-25 by NCTM (National Council Teachers of Mathematics) in lessons published on their free Illuminations website. http://illuminations.nctm.org/LessonDetail.aspx?ID=L620

Activity 2: Game Time - Working with Products (GLE: 2)

Materials List: Product Game BLM, paper clips, colored markers or chips

Help students develop an understanding of factors and multiples and relationships between the two as they play The Product Game. This game is similar to The Factor Game.

Distribute the Product Game BLM. Using an overhead projector, model the process by playing a game against a student by using the rules below. Use an appropriate number of games to clarify questions. Two paper clips and colored markers or colored chips that are two different colors are needed as game pieces.

1) Form pairs and give each of the two students a paper clip and a different colored pen or chip. 2) Playing the game:  Player 1 will choose a number in the factor list and mark it by placing a paper clip on it.  Player 2 uses the other paper clip to mark a second factor and then places one of his/her color chips on the product of those two factors.  Player 1 then moves one of the paper clips to a different factor and uses one of his/her color chips to cover the new product. 3) If the product of a pair of factors has been covered already, the player gets to make no marks on the board. 4) Play continues in this manner until one player marks four squares in a row - up and down, horizontally, or diagonally.

The first few times the students play, they play for fun. Then students decide if it is best to go first or second in order to win the game. Let students discuss strategies while conversing about multiples, products, and factors.

This activity is adapted from Connected Mathematics Project, G. Lappan, J. Fey, W Fitzgerald, S. Friel and E. Phillips, Dale Seymour Publications, (1996) pp.17-25 by NCTM (National Council Teachers of Mathematics) in lessons published on their free Illuminations website. http://illuminations.nctm.org/LessonDetail.aspx?ID=L272

Activity 3: Prime Arrays (GLEs: 1, 14)

Materials List: Factors Table BLM, square tiles, Grid Paper BLM, pencil

Distribute the Factors Table BLM. Have students create rectangular arrays of various dimensions using square tiles to discover the concepts of prime and composite through visual representation. The dimensions will be the factors of the number given. For example, using the number 15, have students create all possible arrays: 3 by 5 and 1 by 15. For a prime number such as 7, students will be able to create only one array: 1 7 . By repeating this activity several times, students should see that the prime numbers have only two factors and therefore, only one array representation. Have students record their information on the Factors Table BLM for numbers 1 through 25. They will not fill out the prime factorization column at this time.

Have students identify those arrays that form squares. They will note that the numbers 1, 4, 9, 16, 25, and so forth have that attribute. Class discussion and further modeling can present this through the number 100 and beyond.

To ensure that students can apply this technique to numbers with several factors, be sure to use composite numbers such as 24. Students will be able to make connections from their arrays to create the “factor tree” of numbers as with 24: 2 2  2  3 . After creating arrays with color tiles, students transfer the information to Grid Paper BLM where they

Grade 6 MathematicsUnit 2Whole Numbers, Factors, and Primes 14 Louisiana Comprehensive Curriculum, Revised 2008 will draw the arrays and label the dimensions. Working in groups they can represent all the numbers from 1 to 25. Have students evaluate the arrays for each number using the labeled dimensions. If the dimension of an array is not a prime number, the student should continue to divide the rectangle until only prime numbers are left. For example, eight squares can be arranged as a 2 x 4 array. However, 2 is a prime number, 4 is not. The 4 can be broken into two 2s. Therefore, the array is now labeled with a 2 on one dimension and two 2s on the other. The prime factorization of 8 is 2创 2 2. Students will make factor trees of other numbers. Have students record their results in the prime factorization column on the Factors Table BLM.

Have students respond to the following prompt in the math learning log (view literacy strategy descriptions). Why is 2 the only even number that is prime? Explain your thinking in your math learning log. Use as many of the following terms correctly as you explain this question: prime, composite, multiple, divisor, factor. The math learning log is a notebook that students can use to record ideas, questions, reactions, and new understandings.

Activity 4: Factors (GLE: 2)

Materials List: Divisibility Rules BLM, Common Factors BLM, pencil

Review discovered concepts for factoring numbers, such as all even numbers have a factor of 2; any number ending with zero has a factor of 10 (and 2 and 5); and any number ending in five has a factor of 5. Extend this to revisit the divisibility rules, see Divisibility Rules BLM, for the numbers 3, 4, 6, 8, and 9. Distribute the Common Factors BLM to students and have students work in groups of two or three to find common factors. Discuss that the largest common factor is called the Greatest Common Factor. Have the students circle the greatest common factor for each pair of numbers. Provide students with a final pair of numbers. Have the students independently list the factors for each number and identify the greatest common factor. Discuss the solution as a class.

Activity 5: Greatest Common Factor (GLE: 2, 3)

Materials List: Greatest Common Factors BLM, pencil, square tiles, Grid Paper BLM

Have students create all possible rectangular arrays for each whole number in a given pair of whole numbers using square tiles or on Grid Paper BLM from Activity 3. Help students determine the greatest common factor (GCF) of the pair of whole numbers by comparing the arrays and selecting the two largest common dimensions. For example, for the pair of whole numbers 24 and 36, students will select the arrays 2 by 12 and 3 by 12 because they have the largest dimension of 12 in common. Distribute the Greatest Common Factors BLM. Have students work together or independently to practice finding the GCF. Discuss the answers as a class.

For additional practice with finding the greatest common factors, the students can visit the following website: http://www.aaamath.com/g72b-grt-com-fac.html#section3

Activity 6: Least Common Multiple (GLE: 3)

Materials List: Least Common Multiple BLM, pencil

Write 5 and 8 on the board or overhead. Have the students make a list of multiples for each number. 5 – 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85 8 – 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88

Have a student go to the board and circle all the numbers that are in both lists. Discuss that the smallest number of these common multiples is called the Least Common Multiple. Distribute the Least Common Multiple BLM. Have the students work together to solve the problems. Discuss the solutions as a class.

Activity 7: Prime factorization revisited (GLE: 1)

Materials List: Prime Factorization Examples BLM, Prime Factorization BLM, pencil

Once students have used concrete methods to discover prime factorization used to compute GCF and LCM, lead students through an efficient method of acquiring the factorization. Begin by providing a number.. Divide it by the smallest possible prime factor until the quotient is 1. For example, given the number 36:36 2 , 18 2 , 9 3 , and 3 3 . The prime factorization in taken from the divisors: 2, 2, 3, 3 or 2创 2 3 3. An example with a larger number, 1250:1250 2 ;625 5 ;125 5 ; 25 5, 5 5 or 2创 5 5 创 5 5 or 2 54 .

Use the Prime Factorization Examples BLM to demonstrate more examples if needed. Then distribute the Prime Factorization BLM and have the students work independently or in groups to solve the problems. Discuss the answers as a class.

Once the class has completed the Prime Factorization BLM discussion, ask them to demonstrate their understanding by completing a RAFT (view literacy strategy descriptions) writing assignment in groups of 2 or 3.

This form of writing gives students the freedom to project themselves into unique roles and look at content from unique perspectives. From these roles and perspectives, RAFT writing is used to explain processes, describe a point of view, envision a potential job or assignment, or solve a problem. It’s the kind of writing that when crafted appropriately should be creative and informative.

R - Role Mystery Number

A- Audience The class

F- Form Poem, song

T- Topic Prime Factorization

The students will write a song or a poem to share with the class describing but not revealing their mystery number. The song or poem must include the prime factorization of the mystery number and any other characteristics of the number the group wants to add. The small groups will present their song or poem to the class and the class will try to guess the mystery number. The group will then reveal their number.

Activity 8: Powers of 10 (GLE: 11)

Materials List: calculator, pencil, paper, Spinner BLM, Powers of 10 BLM, paper clip, index cards

Distribute calculators and give the following directions for groups of students to follow using the calculator: a. Enter the number 27 b. Multiply by 10 and record your answer c. Multiply by 10 and record your answer d. Multiply by 10 and record your answer e. Make observations from recorded answers about what happens when you multiply by 10 f. Discuss with your group members g. Enter the number 8.45 h. Multiply by 10 and record your answer i. Multiply by 10 and record your answer j. Multiply by 10 and record your answer k. Make observations from recorded answers about what happens when you multiply by 10 l. Discuss with your group members Have the class compare observations from the two activities and develop a general conjecture about multiplying by 10. Ask: What power of 10 would result in the same answer as multiplying by 10 threes as you did with the numbers above? m. Enter the number 27 n. Divide by 10 and record your answer o. Divide by 10 and record your answer

p. Divide by 10 and record your answer q. Make observations from recorded answers about what happens when you divide by 10 r. Discuss with your group members s. Enter the number 8.45 t. Divide by 10 and record your answer u. Divide by 10 and record your answer v. Divide by 10 and record your answer w. Make observations from recorded answers about what happens when you divide by 10 x. Discuss with your group members Have the class compare observations from the two activities and develop a general conjecture about dividing by 10. Ask: What power of 10 would result in the same answer as dividing by 10 three times as you did with the numbers above? Is this different than when we multiply?

Distribute Spinner BLM and a paper clip to each group and a Powers of 10 BLM to each student. Explain that the game is played by spinning the spinner and mentally moving the decimal on the starting number. The starting number is on the Powers of 10 BLM. The new number will then be used for the second spin. After each student spins four times and writes his/her new number numerically and in words, the group will compare the number of each group member after spin #4 to determine who has the largest number. Have groups write the number they end with after the 4th spin on index cards. The students in each group should then order the numbers from largest to smallest and tape their order to the front board. Students should make observations from all group’s cards and determine the largest number and smallest number anyone in the class ended with.

Have the students brainstorm (view literacy strategy descriptions) how they could use powers of 10 to solve 4 x 12. Discuss with students that numbers that are not powers of 10 can be broken down into simpler problems using powers of 10. For example, 4 x 12 could be 4 x10 and 4 x 2. Then the students would add the two products to find the solution. Provide students with several problems to practice using this method and discuss the process used and solution as a class.

Have the students explain in their math learning log (view literacy strategy descriptions) how they could use the powers of 10 to solve 55 x 110. Students should explain the process in words and provide a solution. Have the students share their entry with the class to check for accuracy.

Activity 9: Applications of GCF and LCM (GLE: 3)

Materials List: paper, pencil

Have students apply the GCF and LCM in real-life situations. The following are two examples: Ms. Nguyen’s class purchased 45 pencils and 30 erasers to put in care packages. They want each package to contain the same number of pencils and the same number of erasers. What is the greatest number of packages they can make? 15 packages

The class is having a party. The students have voted to have hot dogs that come in packages of 10, with buns that come in packages of 8, and ice cream cups that come in packs of 24. What is the least number of packages of hot dogs, buns, and ice cream that you would need to buy to have the same number of each? 120 of each item: 12 packages of hot dogs, 15 packages of buns, and 5 packages of ice cream cups.

Have students make up similar problems to find the greatest common factor and least common multiple.

Activity 10: Sequences and Expressions (GLE: 38)

Materials List: Sequences BLM, pencil

Have students work in small groups to explore the following sequences to discover how they were generated.

1) 3, 8, 13, 18, 23, . . . 2) 1, 2, 4, 8, 16, . . .

They should look to see if the sequence shows an arithmetic sequence (common difference between terms in the sequence of terms) or a geometric sequence (product of a common number). Give samples of both types of sequences and some that have no pattern. Have students supply the next three terms in the sequence and explain their choices.

Example: 2, 6, 10,… 10, 40, 160, 640, … 2, 2, 4, 6, 10, 16,… arithmetic geometric neither 14, 18, 22 2560, 10240, 40960 26, 42, 68

Distribute the Sequences BLM. Have the students work independently or in groups to complete the sequences. Discuss the answers as a class.

Have students use their math learning logs (view literacy strategy descriptions) to describe in their own words the difference between an arithmetic and a geometric sequence and give an example of each.

Sample Assessments

General Assessments

 The student will create portfolios containing samples of experiments and activities.  Whenever possible, create extensions to an activity by increasing the difficulty or by asking “what if” questions.  The student will create and demonstrate original math problems by acting them out or using manipulatives to provide solutions on the board or overhead.  The student will create his/her own questions/problems and solve them.  Provide the student with a set of factor arrays and have him/her determine what number the array represents.  Divide the students into two teams in a “race to calculate” with powers of 10. One team will be given a calculator and the other team will be required to use mental math. Read a problem from a set of computation problems that involves multiplication or division by a power of 10. The team with the quickest correct answer earns a point. Repeat this assessment with teams, reversing their computational roles.  The student will model a prime factor array for a given prime number.  The student will take a given a pair of whole numbers and the GCF of the two and will determine the LCM.  The student will use concrete materials (manipulatives) to enhance understanding.  Facilitate during small group discussion to determine misconceptions, understandings, use of correct terminology, and reasoning abilities. Appropriate questions to ask to promote problem-solving may include: o What have you done so far? Is there anything else to do? o What made you decide to use this method? o Is there another method that might have worked? o Have all the possibilities been explored? How can you tell? o What do you think about what ___ said? o Do you agree with your group’s answer? Why or why not? o How would you convince the rest of us that your answer makes sense?  The student will create journal writings using such topics as these: o Write a letter to a friend giving advice on how to play The Factor Game. o Discuss the meaning of LCM or GCF. o Tell what the easiest thing was in today’s lesson.

 General Unit Assessment: Create performance tasks to evaluate understanding at the end of the unit. A scenario: Western Middle School and Taylor Middle School both start playing their “round the clock” volleyball games to benefit a charity at 7:00 P.M. on Friday night. Western takes 75 minutes to play each game, while Taylor takes 90 minutes to play each game. Some friends from another local school want to watch two complete games, one at each school. If the friends start their night at 7 P.M. by watching Taylor, at what time should they meet their friends at Western in order to see a complete game there? Assume that the games are played continuously through the night. If the friends started at Western, what time would they need to meet friends at Taylor?

 Activity 3: Give students a set of numbers. The students will determine whether each number is prime or composite and will justify each answer. Justification should be a combination of words and pictures or diagrams.

 Activity 8: Assign each student a multiplication or division problem. The student will show at least three different ways to compute the answer and explain his/her reasoning.

 Activity 9: Students will work in small groups to formulate their own problems with GCF and LCM and create an answer key with explanations for their problem. Distribute the problems to another group and have them solve the problem. The original groups should check student work.

Resources

Sample questions in the Assessment section are similar to questions from a free question card created, published, and distributed by TeacherLineSM at PBS.org/teacherline.

Grade 6 Mathematics Unit 3: Fractions, Decimals, and Parts

Unit Description

The focus of this unit is on concepts and basic relationships of fractions and decimals. There is an emphasis on estimating outcomes prior to developing the computation algorithms that give the exact answers. Focus is also given to writing fractions in lowest terms. The development of the concept of rate, ratio, proportion, and percent continues by representing and working with miles/hour, dollar/pound, miles/gallon, and other derived rates and percents.

Students can compare fractions, decimals, and integers by placement on a number line and/or by the use of symbols. They can solve ratio, proportion, and percent problems with models and pictures. Students can identify place value to the ten-thousandths place. They can use rates to solve real-life problems.

Guiding Questions

1. Can students represent and interpret values for decimals through ten- thousandths? 2. Can students generate equivalent forms of fractions and decimals? 3. Can students predict reasonable outcomes for the addition and subtraction of fractions and decimals? 4. Can students utilize various strategies to work with rates and ratios such as mph, mpg, and dollar/pound?

GLE # GLE Text and Benchmarks Number and Number Relations 4. Recognize and compute equivalent representations of fractions and decimals (i.e., halves, thirds, fourths, fifths, eighths, tenths, hundredths) (N-1-M) (N-3- M)

Grade 6 MathematicsUnit 3Fractions, Decimals, and Parts 22 Louisiana Comprehensive Curriculum, Revised 2008

GLE # GLE Text and Benchmarks 5. Decide which representation (i.e., fraction or decimal) of a positive number is appropriate in a real-life situation (N-1-M) (N-5-M) 6. Compare positive fractions, decimals, and positive and negative integers using symbols (i.e., <, =, >) and number lines (N-2-M) 7. Read and write numerals and words for decimals through ten-thousandths (N- 3-M) 8. Demonstrate the meaning of positive and negative numbers and their opposites in real-life situations (N-3-M) (N-5-M) 10. Use and explain estimation strategies to predict computational results with positive fractions and decimals (N-6-M) 13. Use models and pictures to explain concepts or solve problems involving ratio, proportion, and percent with whole numbers (N-8-M) Measurement 18. Measure length and read linear measurements to the nearest sixteenth-inch and millimeter (mm) (M-1-M) 20. Calculate, interpret, and compare rates such as $/lb., mpg, and mph (M-1-M) (A-5-M) Data Analysis, Probability, and Discrete Math 31. Demonstrate an understanding of precision, accuracy, and error in measurement (D-2-M)

Sample Activities

Activity 1: Magnified Inch (GLE: 18)

Materials List: 8.5 x 14 paper, ruler, pencil

Distribute one sheet of paper to each student. Give the students the following directions: (1) Fold the paper in half end to end with the fold parallel to the 8.5 inch side. (How many sections does your paper have?) (2) Draw a line along the fold about 3 inches long. (3) Write ½ under your line to show that it is half of the way along the paper.

(4) Fold your paper in half again. (5) Draw a line about 2 inches long on each new fold. (6) The first line is 1/4 of the way along the paper. Write 1/4 under it. (7) The next line is 2/4 or of the way along. (8) The third line is 3/4 of the way along. Write 3/4 under this line.

1/4 ½ 3/4 2/4

Continue the process with more folds to indicate the 8th and 16th markings.

Have students compare their magnified inch to an inch on the ruler. The students should notice that their magnified inch has the same marks as an inch on a ruler. The only difference is the magnified inch is an enlarged replica of the actual inch. At the end of the process, the student should have a magnified inch to aid them in reading fractional measurements.

Activity 2: A Measuring We Go… (GLEs: 5, 18, 31)

Materials List: Measurement BLM, ruler, paper, pencil

Have students work in groups of two or three. Provide students with measuring instruments that measure in both English and metric systems. Distribute the Measurement BLM and ask students in each group to find items in the classroom to measure. The measurement should be recorded using both systems. English system measurement should be recorded in fractional increments while the metric should be written in decimal numbers. Instruct students to find the most precise measurement offered by their instruments (sixteenths of an inch/ millimeters). Once the measurements are taken and recorded, focus discussion on the measurements, how they are represented, what measurement tool was used, and how accurate each measurement is. Discuss precision of instruments, error of the measurer (not reading from 0 on the ruler, misreading marks), and other situations that may cause the measurement to not be exact. Discuss how there errors can affect the accuracy of the measurement. Offer different types of tools to vary the outcomes and in turn promote further discussion: plastic rulers vs. wooden rulers; flexible meter sticks vs. rigid wooden ones; measuring tapes.

Have students respond to the following prompt in the math learning log (view literacy strategy descriptions). Is a quarter-inch or an eighth-inch bigger? Explain your reasoning.

Allow students time to share their responses with a partner or the class. Students should listen for accuracy and logic.

Activity 3: Reading and Writing Decimals (GLE: 7)

Materials List: Writing Decimals BLM, I Have Who Has BLM, Decimal Cards BLM, pencil

In the classroom, display a place value chart that students can refer to for this activity and throughout the year. The chart should range from billions to hundred-thousandths. Distribute the Writing Decimals BLM to the students. Have students work independently writing the decimals. Discuss the correct answers as a class. Cut apart and shuffle the cards on the I Have Who Has BLM and distribute all of the cards. Some students may have to take more than one card depending on the size of the class. One student will start by reading his/her card. “I have three and two hundredths. Who has one ten- thousandth?” The student who has the card with 0.0001 on it would respond, “I have one ten-thousandth. Who has forty-two and one tenth?” It does not matter which card starts the process. As long as all cards are used, the last card read will be answered by the first card read. Have the students try to read the decimals in words.

The professor know-it-all strategy (view literacy strategy descriptions) can be used to check for understanding. Cut apart the Decimal Cards BLM and place the cards in a box. Instead of forming groups, an individual student is chosen to come to the front of the class. The student will pull a decimal card from the box and read it aloud. After the student reads the decimal, the class will ask him a variety of questions. For instance: Can you write your decimal in words? What fraction represents your decimal? What digit is in the tenths place? The class should hold the student accountable for her/his answer. After a couple of questions, a new student should be asked to be a know-it-all and go to the front of the class to repeat the process.

Activity 4: Fraction Strips (GLEs: 4, 6, 18, 31)

Materials List: different colored cardstock, scissors, pencil, paper

Have students work in groups of 2 and give each member of the group a sheet of different colored cardstock. Instruct each student to make fraction strips by cutting the cardstock into 6 one-inch wide strips. Remind students to use a ruler to make appropriate measurement. Ask students to fold each strip into one of the following fractions: halves, fourths, eighths, thirds, sixths, and tenths and then write the fraction on each piece (each 1 of the thirds will have 3 on it). Direct students to use a ruler to check their work to determine how accurate the lengths of the strips are to the nearest half, fourth, eighth, and sixteenth of an inch. During discussion, address the accuracy of the measurement. What could cause the measurement not to be precise? (Tools used, quality of measurement materials, or student error.)

1 23 4 5 In the groups, use the strips to find equivalent fractions, 2= 4 = 6 = 8 = 10 . Ask students to compare their fraction strips and write down comparisons using symbols of =, <, and >. Write fractions in lowest terms. Check the work of student groups.

Activity 5: Same or Different? (GLEs: 4, 6)

Materials List: index cards, paper, pencil

Use the SQPL strategy (view literacy strategy descriptions) to challenge the students to further explore fraction and decimal equivalency. SQPL stands for Student Questions for Purposeful Learning and involves presenting the students with a statement that provokes interest and curiosity. Put the following statement on the board or overhead for students to read. “If I ate 3/5 of a pizza and you ate 0.55 of a pizza, then I ate more pizza.” Have students work with a partner and brainstorm 2-3 questions that would have to be answered to prove or disprove the statement. As a whole class, have each pair of students present one of their questions and write this question on chart paper or the board. Give the class time to read each of the questions presented. Give pairs of students time to select the ideas that they would use to prove or disprove the statement.

Divide the class into two groups. One group will be the “fraction” group and the other the “decimal” group. Ask the “fraction” group to provide a fraction expressed in halves, thirds, fourths, fifths, eighths, tenths, or hundredths to the “decimal” group. Give the “decimal” group a specified time limit to provide an equivalent decimal representation of 8 1 25 1 the fraction. Note that the fraction group could give 16 instead of 2 or 100 instead of 4 , Have the two groups decide if the decimal expression provided is equivalent to the original fraction. If the decimal is equivalent, then the decimal group earns 1 point. If not, the fraction group earns 1 point. Next, have the “decimal” group provide a decimal representation of a fraction to the “fraction” group. Allow the “fraction” group a specified time limit to provide an equivalent fraction in lowest terms. Again, ask the two groups to decide if the fraction is equivalent. If correct, the “fraction” group earns a point; if not, then the “decimal” group earns a point.

If an answer is determined to be correct, ask students to record each answer on an index card. Once complete, shuffle the index cards and then pass them out to individual students. Have students then form a “human number line” across the classroom. Order the fractions and decimals with the fraction/decimal equivalents standing one in front of the other. As a whole group, discuss the placement of students. Note that students will be paired across the number line. Place the cards on the wall so that students can view them and their placement at the end of the activity. Keep the cards for future use.

Reread the opening statement and the questions the students generated. As a class, answer each question and decide if the statement is true.

Activity 6: Box Scores (GLE: 6, 20)

Materials List: sports data, index cards

Depending on the season, use box scores from the newspaper’s sports section or the websites listed below to get data to order decimals. For example, show the average yardage for rushes by different football players, rebounds for basketball players, on-base percentage or batting averages for baseball, times for track stats—dashes and pole vaults. Write each statistic or number on an index card. As a class, order the stats on a number line. An example of batting averages would be .222, .234, .245, .255, .266, .289, and so on. Help students understand how these data can be interpreted as rates. For example, a baseball player batting .250 gets a hit once every four times he bats, on average. After exploring the decimals, have students mix these index cards with those from the previous activity. Repeat the “human number line” with the additional cards. Because the number of cards may now surpass the number of students, discuss with students how to solve this problem: i.e. pair the equivalent fractions and decimals, and eliminate one-half of them.

New Orleans Saints Statistics http://www.nfl.com/teams/statistics?team=NO

New Orleans Zephyrs Statistics http://web.minorleaguebaseball.com/milb/stats/stats.jsp? t=t_ibp&cid=588&stn=true&sid=t588

New Orleans Hornets Statistics http://www.nba.com/hornets/stats/

Activity 7: Rolling for Decimals (GLE: 7)

Materials List: three number cubes per group, Rolling for Decimals BLM, index cards, pencil

Have students work in small groups of no more than 4 for this activity. Provide 3 number cubes per group, and distribute the Rolling for Decimals BLM to each student. Ask students to take turns rolling the cubes. Then using the digits rolled, students will independently make the largest and smallest decimal number possible. All numbers should be less than one.

Have students record the numbers on the BLM and read the numbers correctly to compare answers within the groups. After repeating this five or six times, have students write the numbers on note cards, turn the cards over and shuffle the cards. Instruct groups to turn the cards over and place them in order - largest to smallest. Repeat the process, this time from smallest to largest. To check, have the students read the correct word name for each number.

Activity 8: Grocery Math (GLEs: 10, 13, 20)

Materials List: grocery ads, Grocery Ad BLM, pencil, paper, calculators

Provide students with grocery ads from a current newspaper or the Grocery Ad BLM. Have students work in pairs with a specified task: Given $20 to go to the grocery store, buy a variety of fruits and vegetables.

Direct students to select a combination of at least four different fruits and/or vegetables, estimate the cost, estimate the tax at 10% and record the estimate total. Then have the students calculate the total cost and present their findings to the class. The findings should include the estimated cost and the final cost. Sketch a rectangle to represent the $20. Divide the rectangle into approximate parts to show prices (i.e. if $5.00 is spent on 1 apples, 4 of the rectangle should be marked and labeled “apples”). Use the rectangle as a visual representation of the information.

Considering the purchases, have students respond to the following: Do you have enough money to buy 3 lbs. of each? 4 lbs. of each? 5 lbs. of each? Use a calculator to determine the cost of each fruit/vegetable at 3, 4, and 5 lbs. If one item can be bought in three- pound bags for $2.99, select one item and decide if it is cheaper to buy by the bag or by the pound? Instruct students to record answers as a rate: 1lb. for $.997.

Teacher Note: When computing 10% tax, mental math should be encouraged. Students should be able to explain how they arrived at the answer. Accept answers that show a clear understanding that “moving the decimal 1 place to the left” is because they are multiplying by .1 (one tenth or 10- hundredths).

Activity 9: Tangram Ratio (GLE: 13)

Materials List: Tangrams BLM, one paper square for each student, scissors, pencil

A tangram puzzle is made up of seven pieces: 2 large triangles, 1 medium triangle, 2 small triangles, 1 parallelogram, and 1 square. A large square can be formed using all 7 tangram pieces. Have each student make his/her own set of tangrams so that he/she can have a direct understanding of the relationships between the parts and whole and among the pieces to one another. Simple directions for creating the tangrams are given below. Directions with visual representations are on the Tangrams BLM.

Fold and cut a square sheet of paper by following these instructions:

1. Fold the square in half diagonally, unfold, and cut along the crease into two congruent triangles. 2. Take one of these triangles. Fold in half, unfold, and cut along the crease. Set both of these triangles aside.

3. Take the other large triangle. Lightly crease to find the midpoint of the longest side. Fold so that the vertex of the right angle touches that midpoint, unfold and cut along the crease. You will have formed a middle-sized triangle and a trapezoid. Set the middle-sized triangle aside with the two large-size triangles. 4. Fold the trapezoid in half, unfold, and cut. To create a square and a small-sized triangle from the other trapezoid half, fold the acute base angle to the adjacent right base angle and cut on the crease. Place these two shapes aside. 5. To create a parallelogram and a small-sized triangle, take one of the trapezoid halves. Fold the right base angle to the opposite obtuse angle, crease, unfold, and cut. 6. You should have the 7 tangram pieces: 2 large congruent triangles 1 middle-sized triangle 2 small congruent triangles 1 parallelogram 1 square 7. The pieces may now be arranged in many shapes. Try recreating the original square.

After a quick review of the terms area and ratio, have students determine the ratio of the area of each piece to that of the other pieces by comparing the sizes of the pieces. For example, students should determine the ratio of a small triangle’s area compared to the medium triangle. Next, ask students to write the ratios of each of the tangram pieces to the whole (the completed puzzle). As an example, students should find that the ratio of a large triangle to the large square (the completed puzzle) to be 2 to 4, which reduces to a ratio of 1 to 2. This ratio compares the area of a large triangle to the area of the square (the completed puzzle). Have the students use the ratios to write proportions. When students complete the activity, have them rewrite the ratios of area of single pieces to the whole as fractions and then as percents.

Activity 10: Are You Positive? (GLE: 8)

Materials List: thermometer, Are You Positive? BLM, pencil

Show the students a Fahrenheit thermometer and ask: “What is the temperature on a hot, sunny day?” Have a student point out the degree on the thermometer. Ask a few other questions similar to this making sure the answer would be a positive number. Next, ask them what it means for the temperature to be 5 degrees below zero and show where this is on the thermometer. Explain that a negative sign is needed to write this number since it is below zero. Next, ask them where 20 degrees below zero would be on the thermometer, and then ask them whether it is hotter or colder than 5 degrees below zero. Since 20 degrees below zero is colder than 5 degrees below zero, negative 20 is less than negative 5. Discuss with the students that positive and negative whole numbers are called integers. Distribute the Are You Positive BLM to the students. Have the students complete the activity in groups or independently. Discuss the answers as a class. Have

Grade 6 MathematicsUnit 3Fractions, Decimals, and Parts 29 Louisiana Comprehensive Curriculum, Revised 2008 students respond to the following prompt in the math learning log (view literacy strategy descriptions). Name and give examples of 4 real life situations in which integers can be used.

Activity 11: Vacation Math (GLE: 20)

Materials List: Internet access, maps, or atlases, paper, pencil

We’re going on vacation! Allow students, in groups of 2, to make use of the Internet, maps, or atlases to locate the distance from home to a destination of their choice. Have the students predict how long it will take to drive at the posted speed limit. This distance with a variety of speeds will be used to determine trip length. Class discussion should focus on the distance formula with students discovering the formula instead of having it given to them. Questions student should explore include the following: If we are going to drive to visit our location, how long will it take to get there if we drive 60 mph? If the car we’re using gets 30 miles to the gallon, how much gas will we use to get there and back? If the price of gas is $2.50 per gallon, how much will it cost to go on our trip? Have each group make a presentation to the class sharing information.

Sample Assessments

General Assessments

 Observe individual and group work throughout the unit.  The student will create portfolios containing samples of experiments and activities.  Facilitate small group discussions to determine misconceptions, understandings, use of correct terminology, and reasoning abilities. Appropriate questions to ask might be: o How did you get your answer? o What are the key points or big ideas in this lesson? o How would you prove that? o What do you think about what ___ said? o Do you agree with your group’s answer? Why or why not? o How would you convince the rest of us that your answer makes sense?  The student will create learning logs (view literacy strategy descriptions) using such topics as: o The most important thing I learned in math this week was… o Explain today’s lesson to a student who was absent today.  The student will submit a written reflection to the following two questions as

a Performance Task Assessment of the unit: o How can you decide whether a fraction is closest to 0, ½, or 1? o When comparing two decimals such as 0.36 and 0.349, how can you decide which decimal represents the larger number?

 Activity 3: The students will complete an assessment where the student matches decimal numbers to the written equivalent.

 Activity 4: The students will write a story using fractions and decimals appropriately, making at least five comparisons about the numbers by using the symbols <, >, or =.

 Activity 5: The student will bring a copy of a recipe from home. He/she will write the amount of each ingredient in fraction, whole number, and or mixed number formats and convert the values to decimal numbers.

 Activity 9: The students will create a poster with original definitions and/or pictures to convey the meanings in order to demonstrate understanding of the math terms covered- ratio, area, triangle, trapezoid.

Grade 6 Mathematics Unit 4: Operating with Fractions and Decimals

Unit Description

This unit focuses on the refinement of understandings of addition and subtraction as students use concrete materials to model these operations with fractions and decimals. Activities provide opportunities to develop an understanding of fraction transformations, common denominators, and lowest terms in terms of equivalences. Precision is explored in real-life situations and compared to the meaning of accuracy in measurements..

Students use various strategies for addition and subtraction of fractions and decimals to solve real-world problems. Students relate the precision of a measurement to the unit of measurement chosen.

Guiding Questions

1. Can students use a variety of strategies to add and subtract fractions, leaving answers in lowest terms when specified? 2. Can students add and subtract decimals, leaving answers rounded to specified places? 3. Can students apply addition and subtraction of fractions and decimals to money and measurement problems? 4. Can students determine the precision of a measurement based on the measuring tool used? 5. Can students define accuracy as being the measurement that which is closest to the real measure (actual length, weight)?

GLE # GLE Text and Benchmarks Number and Number Relations 3. Find the greatest common factor (GCF) and least common multiple (LCM) for whole numbers in the context of problem-solving (N-1-M) 5. Decide which representation (i.e., fraction or decimal) of a positive number is appropriate in a real-life situation (N-1-M) (N-5-M)

Grade 6 MathematicsUnit 4 Operating with Fractions and Decimals 32 Louisiana Comprehensive Curriculum, Revised 2008

GLE # GLE Text and Benchmarks 9. Add and subtract fractions and decimals in real-life situations (N-5-M) GLE # GLE Text and Benchmarks Measurement 18. Measure length and read linear measurements to the nearest sixteenth-inch and millimeter (mm) (M-1-M) Data Analysis, Probability, and Discrete Math 31. Demonstrate an understanding of precision, accuracy, and error in measurement (D-2-M)

Sample Activities

Activity 1: Fraction Vocabulary Awareness (GLE: 3, 5)

Materials List: Fractions BLM, pencil

Before beginning the unit, have students complete a vocabulary self awareness chart (view literacy strategy descriptions). Provide students with the Fractions BLM. Do not give students definitions or examples at this point.

Word + Ö – Example Definition Greatest Common Factor (GCF) Least Common Multiple (LCM) Dimensions

Denominator Numerator Equivalent Fractions

Ask students to rate their understanding of each word with either a “+” (understands well), a “Ö” (some understanding), or a “–” (don’t know). During and after completing the activities throughout this unit, students should return to the chart to fill in examples and definitions in their own words. The goal is to have all plus signs at the end of the unit. After all the students have completed the chart, have them share their examples and definitions with each other to check for accuracy.

Activity 2: Mental Giants (GLEs: 9)

Materials List: Fraction Table BLM, Fraction Operations BLM, pattern blocks, pencil

Adding and subtracting fractions with unlike denominators should be a natural progression from work done in Unit 3. Distribute the Fraction Table BLM and review with students the process of finding equivalent fractions. Discuss the connections and application this has to finding GCF and LCM of numbers.

Distribute the Fraction Operations BLM and start with fractions with common denominators by having the students complete problems 1 – 3 in small groups. Discuss the answers as a class. Then give them simple, related fractions to add or subtract, such 1 3 5 2 as 8+ 4 or 6- 3 . Have students use the Fraction Table BLM or manipulatives to help find equivalent fractions. Use of pattern blocks with a focus on replacement would be helpful.

Have students complete the Fraction Operations BLM and discuss the process they are using in their groups.

In their math learning logs (view literacy strategy descriptions), have the students explain the process they used to add and subtract the fractions. This sometimes helps students to understand that adding and subtracting fractions requires a common “name” or denominator. As students become more proficient, have them add the fractions mentally. Include mixed numbers. Let students exchange log entries with a partner to check for logic and accuracy.

Activity 3: Adding and Subtracting Unlike Denominators (GLEs: 3, 9)

Materials List: Clock Face BLM, Ad Space BLM, pencil, paper

To facilitate understanding of the process of adding and/or subtracting fractions with unlike denominators, give students opportunities to discover the algorithm. Revisit application of GCF and LCM in class discussions. Give students problems that reflect real life problems which students can easily solve by drawing models and then transfer to relevant algorithms. Distribute the Clock Face BLM and Ad Space BLM for students to use with the following examples:

 Mary walked for 10 minutes and ran for 15 minutes. What fraction of an hour did she exercise? Solve the problem by using a clock face to model, write the related problem, and explain your answer. (Shade 10 minutes on the clock face from 12 to 2, indicating that each 5 1 2 1 minute period is 12 of an hour or that the 10 minute period is 12 or 6 hour. Continue to shade the next 15 minutes from 2 to 5 and indicate the 3 1 2 3 5 fraction of an hour there. ( 12 or 4 ) .The related problem is 12+ 12 = 12 1 1 5 (lowest terms 6+ 4 = 12 ))

 John always buys advertisement space in a local magazine. In the past, he has bought an ad the size of A. This year he would like to buy one the size of C + D + E. What fractional part of the page would he buy? How much more ad space did he buy this year than last year? Explain your work and write the problems you solved.

Teacher Note: A & B are equal in size; C & F are equal in size; D & E together are the same size as C; and the height of A and C are equal. 1 1 1 1 1 1 Students should determine that A=4 ; B = 4 ; C = 6 ; D = 12 ; E = 12 ; F= 6 ; 1 1 1 4 1 1 1 1 therefore, the first problem is 12+ 12 + 6 = 12 = 3 . The second is 3- 4 = 12 . To solve the problem, students will have to discover the need to subdivide the entire page into twelve equal parts.

Below are websites to provide additional practice adding and subtracting fractions. http://www.aaamath.com/fra.html http://www.funbrain.com/fractop/index.html http://math.rice.edu/~lanius/fractions/frac4.html

Activity 4: Numbers in the News (GLE: 9)

Materials List: newspapers or Internet access, paper, pencil

The newspaper and the Internet are good resources for applications of fractions and decimals. It may even be possible to find problems that relate to content students are covering in other subjects. Following are a few examples:  The weekly movie box office totals are given in decimals (for example, $27.2M for $27,200,000) in either the local paper, USA Today, or at http://www.usatoday.com/life/movies/news/box-office.htm. Have the chart made on a transparency or put on the board. Have students answer and make

up problems about the top ten movies of the week. Examples: How much more did the #1 rated movie make over the #2 or #10? How much did all ten movies make in one week?  Have students research various stocks and follow the changes in price for a period of one week (or longer). Have them calculate the new closing price each day. http://markets.usatoday.com/custom/usatoday-com/html- markets.asp  Have students gather rainfall information for various cities at www.weather.com . These are usually reported as fractions. Have students find the totals for the week or month.

Activity 5: Real-World Decimals and Fractions (GLEs: 5, 9)

Divide students into small groups. Have them brainstorm (view literacy strategy descriptions) a variety of situations where decimals or fractions would be required. Examples for decimals: money; timing at track meets, swim meets; converting liters to quarts/ounces; measuring height, or depth; gasoline; batting averages; stock prices; football statistics. Examples for fractions: dividing/cutting food—pizza, apples, oranges; weighing food.

After discussing the examples above, have the students create a math story chain (view literacy strategy descriptions) using real-life decimals and/or fractions. Put students in groups of four. On a sheet of paper, ask the first student to write the opening sentence for the math story chain similar to the example below. Mary is baking a cake for a friend’s birthday party. The student then passes the paper to the student sitting to the right, and that student writes the next sentence in the story. A recipe calls for ¼ cup of granulated sugar and ⅔cup of brown sugar. The paper is passed again to the right to the next student who writes the third sentence of the story. What is the total amount of sugar needed? The paper is now passed to the fourth student who must solve the problem and write out the answer: (11/12 of a cup of sugar) The other three group members review the answer for accuracy.

This activity allows students to use their writing, reading, and speaking skills while learning and reviewing the concept of adding and subtracting fractions and decimals.

Have a discussion about why one representation is more appropriate than the other in their math story chains. Ask questions like, would it make more sense to use fractions or decimals in this situation? or Can you think of a situation where using fractions clarifies 1 the solution process? Give examples. Would you ask for .25 of a pizza or 4 ? Why?

Have students record the following SPAWN (view literacy strategy descriptions) prompt in their math learning logs (view literacy strategy descriptions) and give them time to

Grade 6 MathematicsUnit 4 Operating with Fractions and Decimals 36 Louisiana Comprehensive Curriculum, Revised 2008 respond. SPAWN is an acronym that stands for five categories of writing options (Special Powers, Problem Solving, Alternative Viewpoints, What If?, and Next). These prompts should require considered and critical written responses by students. This prompt could be one using the ‘P’ or Problem Solving category from SPAWN: Jill has 1 quarter, 3 dimes, and 5 nickels. What part of a dollar does she have? Represent your answer as a fraction and a decimal. Which representation would you use if you were writing the amount of money you have? Explain why?

Activity 6: Landscaping (GLE: 3)

Materials List: Landscaping BLM, graph paper, colored pencils, paper, pencil

Give students the Landscaping BLM. Have them work in groups to solve the problems. Students can also use graph paper and colored pencils to show the different possible combinations of plants and to represent the various fractional parts. Ask students to make up similar problems using flowers, trees, or other plants. Review GCF and LCM and how they apply to solving these problems.

Activity 7: Adding and Subtracting Decimals (GLE: 9)

Materials List: calculators, Decimal Operations BLM, pencil

Have students use the calculator to solve addition and subtraction problems with decimals. Ask them to look for patterns they see with the problems so that they can develop their own algorithms for solving these types of problems. After several examples have been explored and the students can correctly state and explain an algorithm, distribute the Decimal Operations BLM to solve without using calculators. Stress the importance of first solving by mental math to arrive at an estimated answer. When students have correctly mastered use of the algorithm of adding and subtracting (set-up problems by lining up decimals and solving as with whole numbers), continue by giving them real-life situation problems to solve.

Below are websites to provide additional practice adding and subtracting decimals. http://www.math.com/school/subject1/practice/S1U1L4/S1U1L4Pract.html http://www.shoreline.edu/callab/GED/Decaddsub1.htm http://www.funbrain.com/football/index.html http://www.quia.com/rr/31090.html

Activity 8: Let’s Be Precise or At Least Accurate (GLEs: 9, 18, 31)

Materials List: Ruler BLM, Measurement BLM, pencil

Ask students what it means to be accurate. Most will probably say that if one is accurate, he/she is correct. Indicate to students that in order to determine if a measurement is accurate (correct), it is necessary to know the actual or real measurement of the item being measured. Indicate to students that there are standards for measuring. It would be beneficial to have students research the development of the length of a “foot” throughout history with the understanding that this measurement has been standardized. Others could research how a manufacturer would know if the meter stick the company is making is truly a meter. Indicate to students that when they measure something in class, the standard is determined by the teacher and, in most cases, they will be given some leeway because of varying factors that affect measuring.

Discuss with students that precision has a different meaning than accuracy when measuring. One definition of precision is based on the how many subdivisions are in one unit on the measuring tool being used. A ruler that has been subdivided into sixteen sections per inch allows the measure to be more precise than a ruler that has been subdivided into four units per inch. To promote student discussion, have student groups of two to three use various measurement tools that are calibrated differently to measure common objects found in the classroom. Distribute the Ruler BLM and Measurement BLM to students. Have the students carefully cut out the rulers and complete the Measurement BLM. Ask students to report their measurements as precisely as their tools allow. In the case of the rulers marked in inches, they should report their measurements only in inches. If the rulers are marked in eighths of an inch, the measurement should be reported to the nearest eighth of an inch. Students should be able to see that there is less estimation of the exact measurement when there are more sections in a unit of the measuring tool.

Next, ask students if their measurements are accurate. Remind students that accuracy can be determined by how closely their measurements are to the “true” or “real” lengths, but other barriers impact accuracy. Any measurement made with a measuring device is 1 approximate. For example, using the ruler marked in half inches and a ruler marked in 4 1 3 inches, suppose a book’s length is reported as 11 2 inches and 11 4 inches. The two measurements here may not be the same. Students should be aware that this does not reflect a “mistake in measurement.” It shows the uncertainty in measurement. This difference in measurement might also be indicated by the use of the terms “rounding” or “to the nearest.” Have students measure the height of the door to the nearest inch. Let them discuss how accurate that measure is. If they state it is 83 in. in height, it could actually be somewhere between 82.5” and 83.5”. The actual measurement is within .5 inches. Look for other examples of this discrepancy and discuss possible solutions (e.g., measure to the nearest sixteenth of an inch). Point out that an error in measurement is generally an error on the part of the measurer. Forgetting to align the starting point of a segment to be measured with the zero point on the ruler or miscounting the number of spaces in a unit could be two reasons for an error in measuring.

An excellent website for the teacher and students can be found at http://regentsprep.org/Regents/math/error/TError.htm (lab sheet on error in measurement with link to homepage with further background information for teacher) or http://regentsprep.org/Regents/math/error/PracErr.htm (interactive practice page for students).

Activity 9: Precision Instruments (GLE: 31)

Materials List: various measurement instruments, paper, pencil

Offer opportunities for students to look at various measurement instruments and select the one that is “most precise.” Have students explain why they selected the instrument in 1 1 their math learning log (view literacy strategy descriptions). (Various rulers - 32 in., 16 1 1 in., 8 in.; measuring cups- 1 cup with no markings, 2 cups with markings to 4 cup; clocks- some with quarter-hour markings, five minute marking, one minute markings, and clocks with seconds-hand.) Have students discuss with a partner which instrument they selected and explain why. Students should listen for appropriateness and logic.

Sample Assessments

General Assessments

 The student will create a portfolio containing samples of experiments and activities.  The student will give real-life examples of using fractions and/or decimals (purchase of gasoline shopping for loose vegetables and fruit, and keeping and interpreting sports statistics).  The student will add or subtract several fractions with unlike denominators.  Provide the student with decimal answers and fraction expressions. The student will determine which fraction expressions produce a given decimal answer.  Create extensions to an activity by increasing the difficulty or by asking “what if” questions, whenever possible.  The student will create and demonstrate original math problems by acting them out or using manipulatives to provide solutions on the board or overhead.  Facilitate during small group discussion to determine misconceptions, understandings, use of correct terminology, and reasoning abilities. Appropriate questions to ask to encourage reflection as students develop algorithms for addition/subtraction with unlike denominators are o How did you get your answer? o Does your answer seem reasonable? Why or why not? o Can you describe your method to us all? Can you explain why it works?

o What if you had started with… rather than…? o What if you could only use…? o What have you learned or found out today? o Did you use or learn any new words today? What do they mean? How do you spell them? Ask probing questions to help students build confidence in their reasoning by asking … o Why is that true? o How did you reach that conclusion? o Does that make sense? o Can you make a model to show that?  The student will create journal writings using such topics as: o The most important thing I learned in math this week was… o The easiest thing about today’s lesson was…

 Activity 2: The student will thoroughly answer the following: In adding two fractions, is it better to use a common denominator or the LCD to write the equivalent fractions? Why?

 Activity 3: The student will determine which fractions will have to be renamed before adding or subtracting and explain his/her reasoning. 1 2 7 5 7 1 5 1 a. 4+ 3 b. 8- 8 c. 12- 9 d. 6+ 6

 Activity 3: Supply the student with different sets of fraction bars such as one 1 1 that has three 4 s and another that has five 6 s. The student will write a word problem that can be solved with these two bars.

1 1 1 1 6 6 6 6

1 1 1 ? 4 4 4

1 Possible answer: Johnny got 5 brownies; each one was 6 of a pan. His friends Keisha, Donna, and Harold each wanted pieces that were one-fourth the size of the original pan. If he gave them the amount they wanted, how 1 much would be left for him? (A piece that is 12 the size of the original pan.)

 Activity 3: The student will explain how drawing a diagram can help to add and subtract fractions.

 Activity 8: The student will explain what can be done to get a more precise measurement when measuring an object

Grade 6 Mathematics Unit 5: Geometry, Perimeter, Area, and Measurement

Time Frame: Approximately five weeks

Unit Description

This unit focuses on analyzing characteristics and properties of 2- and 3-dimensional shapes and developing mathematical arguments about geometric relationships. It provides students opportunities to make and test predictions regarding tessellations and to extend their work to plotting points in all four quadrants of the coordinate plane. Area formulas are extended and include giving reasonable estimates of objects in the classroom or everyday real-life environments. Evaluating simple and two-step algebraic equations, modeling of squares up to 144, and the matching of equations and expressions to their verbal statements are addressed as they relate to geometry and measurement.

Students use correct terminology to adequately describe 2- and 3-dimensional figures and their attributes. They should be able to discuss the measures of angles and the role the measures play in defining figures. In and out of the classroom, students should be able to recognize geometric shapes and be able to give reasonable estimates of their approximate measures.

Guiding Questions

1. Can students recognize and discuss the features of solids? 2. Can students identify the properties of 2 and 3 dimensional figures, angles, and geometric relationships? 3. Can students make predictions regarding tessellations? 4. Can students use various strategies to find the areas of triangles, parallelograms, and trapezoids? 5. Can students use various strategies to find reasonable estimates for measures of geometric objects in the world around them?

Grade 6 MathematicsUnit 5 Geometry, Perimeter, Area and Measurement 42 Louisiana Comprehensive Curriculum, Revised 2008

GLE # GLE Text and Benchmarks Algebra 14. Model and identify perfect squares up to 144 (A-1-M) GLE # GLE Text and Benchmarks 15. Match algebraic equations and expressions with verbal statements and vice versa (A-1-M) (A-3-M) (A-5-M) (P-2-M) 16. Evaluate simple algebraic expressions using substitution (A-2-M) 17. Find solutions to 2-step equations with positive integer solutions (e.g., 3x  5  13 , 2x 3 x  20) (A-2-M) Measurement 18. Measure length and read linear measurements to the nearest sixteenth-inch and millimeter (mm) (M-1-M) 19. Calculate perimeter and area of triangles, parallelograms, and trapezoids (M-1- M) 21. Demonstrate an intuitive sense of relative sizes of common units for length and area of familiar objects in real-life problems (e.g., estimate the area of a desktop in square feet, the average adult is between 1.5 and 2 meters tall) (M- 2-M) (G-1-M) 22. Estimate perimeter and area of any 2-dimensional figure (regular and irregular) using standard units (M-2-M) 23. Identify and select appropriate units to measure area (M-3-M) Geometry 24. Use mathematical terms to describe the basic properties of 3-dimensional objects (edges, vertices, faces, base, etc.) (G-2-M) 25. Relate polyhedra to their 2-dimensional shapes by drawing or sketching their faces (G-2-M) (G-4-M) 26. Apply concepts, properties, and relationships of points, lines, line segments, rays, diagonals, circles, and right, acute, and obtuse angles and triangles in real- life situations, including estimating sizes of angles (G-2-M) (G-5-M) (G-1-M) 27. Make and test predictions regarding tessellations with geometric shapes (G-3- M) 28. Use a rectangular grid and ordered pairs to plot simple shapes and find horizontal and vertical lengths and area (G-6-M)

Sample Activities

Activity 1: Vocabulary Cards (GLE: 24, 26)

Materials List: index cards, pencil

To develop students’ knowledge of key vocabulary, have them create vocabulary cards (view literacy strategy descriptions) for terms related to geometric properties of figures, formulas, and measurement. Distribute 3x5 or 5x7 inch index cards to each student and ask them to follow your directions in creating the sample card. On the board, place a targeted word in the middle of the card, as in the example below. Ask the students to provide a definition. It is best if a word can be defined in students’ own words, but make sure the definitions are correct and complete. Write the definition in the appropriate space. Next, have students list the characteristics or description, give one or two examples, and illustrate the term. Have students create cards for the following terms: line, line segment, ray, parallel, perpendicular, intersect, angle, right, obtuse, acute, area, perimeter, edge, vertices, face, and base.

Allow time for students to review their cards and quiz a partner on the terms to hold them accountable for accurate information on the cards.

An angle of less than  less than 90° 90°  more than 0°

Acute Angle

Examples Illustration

Activity 2: Body Measurement (GLE: 21)

Materials List: rulers or tape measures, deck of cards, football, calibrating wheel or meter stick, pencil

Teacher Note: Students will find that they can always use a common object as a measuring tool—whether it’s a deck of cards or a football. Once a student has a picture in his/her mind of a deck of cards and knows the measurement, he/she will be able to estimate an unknown size by comparing the picture in his/her brain. Students would not want to use something that is going to change in size, but a standard-sized item will always be a visual tool. Check out the following website for an activity similar to this one: http://www.learner.org/channel/courses/learningmath/measurement/session3/part_b/.

Give students rulers or tape measures and have them measure parts of their bodies or common items (pencil, deck of cards, football). Have them find an inch measurement (a finger joint) and a centimeter (fingernail) so they will always have a “measurement unit” on them. Measure the desktops, the door, and the length of the room so that two meters (the door height), one meter (desk width), and books (one foot), as well as other basic sizes are things they can envision. Most students have been to a football game—100 yards is a visual. The length of a particular car (one that all the students know) or a school bus is another measurement that becomes real (and visual). Have students estimate the distance between the classroom and the main office or the cafeteria and use a calibrating wheel or meter stick/yardstick to check for accuracy. Use the same process for other units of length, such as miles, or kilometers. Provide students with two familiar landmarks that are about a mile apart.

Have students record the following SPAWN (view literacy strategy descriptions) prompt in their math learning logs (view literacy strategy descriptions) and give them time to respond. This prompt could be one using the ‘W’ or What if? category from SPAWN: What if standard units of measurement had not been established? How could we measure items or distances? Why do you think there is a need for standard units of measurement? Allow students time to share their responses with a partner or the class. Students should listen for accuracy of the response.

Activity 3: String Lengths (GLE: 21)

Materials List: string, newsprint, paper, pencil

Begin this lesson with the literacy strategy student questions for purposeful learning (SQPL) (view literacy strategy descriptions) by writing the statement, “With just a ruler, I can tell you the total distance around the Earth.” This strategy is used to encourage the students to generate questions that they would need to answer to verify the statement. Have students in groups of four brainstorm (view literacy strategy descriptions) different questions that they might need to answer to determine whether this statement is true or

Grade 6 MathematicsUnit 5 Geometry, Perimeter, Area and Measurement 45 Louisiana Comprehensive Curriculum, Revised 2008 false. Have each group of students highlight 2 or 3 questions that their group wrote for use with whole class discussion. Write these questions on a sheet of newsprint for use as closure when the students have completed this activity.

Give each student a one-yard or one-meter and/or one-foot length of heavy string or rope. Have him/her use the string to find items in his/her daily lives that are exactly one length for the unit they were provided. The students will find that there are many objects in their homes that can be measured with the string lengths. In class, have students share their findings. Then have students estimate the lengths of various objects or distances without using the string. Check estimates with string length. Discuss accuracy of estimates. When students are comfortable with finding objects that are one given unit in length, have them use three identical additional string lengths to create the outline of a square unit. Whenever possible, lay the outline over an area that can be measured. For example, if a table is one meter long, lay a one square meter outline over the table so students can make connections between the linear measure and the related area.

Once the entire activity is completed, have the students reread the list of questions that were generated at the beginning of the activity. As a class, discuss whether or not these questions were answered as they completed the activity. Have the students take out their math learning log (view literacy strategy descriptions) and write any ways the SQPL strategy helped them with the problem solving involved in the activity.

Activity 4: Room Measurement (GLEs: 22, 23)

Materials List: meter stick or calibrating wheel, graph transparency, various measuring tools, list of objects to measure, paper, pencil

If a door is two meters high, ask students to estimate how many doors (if laid down end- to-end) would go across the room’s width and length. Check accuracy with a meter stick or calibrating wheel. Use the now known measures of length and width to show students how many square feet/yards/meters are in the room’s floor. Using a transparency of a grid, demonstrate how to figure the perimeter of the floor. Give students various measuring tools and a list of objects to measure. Provide opportunities in which the students consider estimations of perimeter and area as well. Ask students to find the measures of other items (top of the desks, bulletin boards, other rooms in the school, sidewalks.) and use those measurements to determine the perimeter and area using the appropriate unit of measure. It is important to include irregular shaped objects to measure.

Activity 5: Measure It! (GLE: 18)

Materials List: rulers, objects to measure, Measure It BLM, pencil

Have students work in pairs. Provide each pair of students with two rulers, one marked in millimeters and the other marked in sixteenths of an inch. Distribute the Measure It BLM. Give students a list of linear objects to measure, using each tool. Have students estimate the length of each item and then measure and record their measurements to the nearest millimeter or sixteenth of an inch. After students have completed their measurements, have students share their findings with the class. Comparisons of the measurements of the same items by different students should take place. Questions about why the measurements are only accurate to the millimeter or sixteenth of an inch should be discussed. Repeat this activity several times with different linear objects to measure.

Activity 6: Calculating Perimeter and Area of Triangles, Parallelograms, and Trapezoids (GLEs: 14, 15, 16, 17, 19)

Materials List: Parallelogram BLM, iLEAP Reference Sheet BLM, Area and Perimeter BLM, scissors, pencil

Have students review how to calculate the area of a rectangle.

Next, provide students with the Parallelogram BLM. Have students cut off the triangular part formed when a line segment is drawn from a corner of the parallelogram perpendicular to the opposite side. This cut is along the height of the parallelogram. Next, have them flip this triangular shape over and reattach it to the opposite end of the parallelogram to form a rectangle (see second diagram below). Ask students to indicate how they could find the area of the newly formed rectangle, and thus the parallelogram.

Examine the isosceles trapezoid. Have students find the midpoints of the legs, cut them forming the midsegment, cut along the midsegment, and then rotate the top around one of the midpoints to make the trapezoid into a parallelogram. Lead a discussion about the formula for finding the area of the parallelogram and the trapezoid.

Discuss with the students that length x width in a rectangle is the same as base x height. Students should see that base x height can be used for all parallelograms as long as they are sure that the base is one side of the parallelogram and the height is the perpendicular distance between the base and the side opposite the base.

Next, have students cut a rectangle along one of its diagonals to form two right triangles. Lead a discussion about how they could determine the area of the resulting triangles.

Finish by asking questions about finding the perimeter of these shapes. Distribute the iLEAP Reference Sheet BLM and the Area and Perimeter BLM. Have students use the area and perimeter formulas on the iLEAP Reference Sheet to complete the problems. Discuss the solutions as a class.

Activity 7: Estimating Area and Perimeter of 2-D Shapes (GLEs: 21, 22, 23)

Materials List: 2-D Shapes BLM, Grid Paper BLM, string, pencil

Distribute the 2-D Shapes BLM, have the students brainstorm (view literacy strategy descriptions) ways to determine the lengths of segments which do not follow a grid line. Have the students estimate the area and perimeter of each of the shapes. Have the students guess the shape that has the largest area. To explore irregular shapes, have students trace one of their hands onto the Grid Paper BLM and estimate the area of their hands by counting the squares and partial squares and record their estimates. Finally, have students estimate the perimeter of their hand by using the sides and diagonals of the squares on the grid paper. Students may decide to use the string used earlier that was cut to 1 yard to find the perimeter of their hand.

Activity 8: 3-D Figures (GLEs: 24, 25)

Materials List: three-dimensional objects, paper, pencil

Have students observe three-dimensional objects and record their properties in a split- page notetaking (view literacy strategy descriptions) format. Model the approach by placing on the board or overhead an example of how to set up their paper for split-page notetaking similar to the example below. Explain the value of taking notes in this format by saying it logically organizes information and ideas; it helps separate big ideas from supporting details; it allows inductive and deductive prompting for rehearsing and remembering the information.

Tell students to draw a vertical line approximately 2 to 3 inches from the left edge on a sheet of notepaper. They should try to split the page into one third and two thirds. Have students count the number of faces, edges, vertices and determine the shape of each face for each of the following three-dimensional figures: triangular pyramids, square pyramids, rectangular pyramids, triangular prisms, cubes, and rectangular prisms. After

the students examine the three-dimensional figures and record their observations in the 2/3 column of the notes page, have students compare their notes with a partner. Show them how they can prompt recall by bending the sheet of notes so that information in the right or left columns is covered. Continue to periodically model and guide students as they use split-page notetaking and increase their effectiveness with this technique. Assessments should include information that students recorded in their split-page notetaking. In this way, they will see the connection between the taking notes in this format and achievement on quizzes and tests.

Example: Date: Topic: Three-Dimensional Figures Period:

Rectangular Prism --Number of faces: --Number of edges: --Number of vertices: --Shape of faces:

Triangular Prism --Number of faces: --Number of edges: --Number of vertices: --Shape of faces:

Activity 9: 3-D Construction (GLE: 25)

Materials List: toothpicks or straws, clay balls, miniature marshmallows or gumdrops, paper, pencil

Once students show an understanding of the polygon faces, the edges and vertices of various polyhedra, have students create individual models. Students may work alone or in pairs to plan and implement the construction of the solid figures using toothpicks or straws, and clay balls, miniature marshmallows or gumdrops. Then have students draw 2- D sketches of the faces of the figures with labels to indicate the number of vertices (clay balls, miniature marshmallows or gumdrops) and edges (toothpicks or straws) needed in construction. The labels should also indicate the names of the shapes of each face. Figures to be constructed by the class should include but not limited to: triangular pyramids, square pyramids, rectangular pyramids, triangular prisms, cubes, and rectangular prisms.

Activity 10: Sliding Shapes (GLE: 27)

Materials List: Pattern Blocks BLM, paper, pencil, soccer ball

Before the activity, copy the Pattern Blocks BLM so that each group can have two of each shape. These should be copied on cardstock or construction paper. Number the corners of the blocks with a fine-tip marker. Give each group of students two of each of the common pattern blocks (or cardstock shapes).

On paper, trace the shape and write the numbers in each corner. Have students test a variety of pattern blocks to determine which will tessellate based on translations or reflections. Ask students to make predictions and sketch their predictions before they start.

Slide the shape so the shape shares a common side with the drawn one. Trace again and write the numbers in the corners. Students will see that sliding the shapes will not alter the placement of the numbers.

Using the reflection, have students again trace and add the numbered corners. Check to see that the students find the pattern in the numbers (where the corners meet).

Using a soccer ball, have students identify the two shapes that make up the pattern on the soccer ball (pentagons and hexagons).

Have students make their own “block quilt” with two of the shapes and predict and sketch what the pattern will look like before they begin.

Lead a discussion about which shapes tessellated and which did not. Ask: Can you tessellate an equilateral triangle and a square? Give time for exploration. Discuss.

Activity 11: Ray Time (GLE: 26)

Materials List: compass, paper, pencil

Have students use a compass to draw a circle and construct a “clock” using two rays emanating from the center of the circle. Discuss with students how to determine the location of the hour numbers (i.e., 12, 1, 2 . . . 11) on a twelve-hour clock. Once students have constructed their clocks, they should determine the kind of angle made by the hour hand and the minute hand at various times specified by the teacher. For example, what angle does the clock’s hands make when it is 2:15 (acute angle), 2:45 (obtuse angle), 3:00 (right angle)?

Help students connect to the real world by reminding them of the built-in protractor they always carry. Have them spread the fingers on one hand as far as possible. What kinds of angle are formed between each finger? (Acute angles or angles less than 90˚) When only

Grade 6 MathematicsUnit 5 Geometry, Perimeter, Area and Measurement 50 Louisiana Comprehensive Curriculum, Revised 2008 stretching the thumb and the forefinger, what type of angle is formed? (Right angle or 90˚angle) What type of angle would be formed from the thumb and any other finger except the forefinger? (Obtuse angle or angle larger than 90˚) Have students examine angles formed in various shapes seen in the classroom. Objects to explore include overhead projector mirror, floor and wall, or the doorframe with a door ajar.

Activity 12: Picture It! (GLEs: 24, 25, 26)

Materials List: magazines, digital camera (optional)

Assign small groups of students a 2 or 3 dimensional figure. Have groups find a picture or create a display of their assigned figure. This could be done in a variety of ways. Students could use a digital camera to take pictures and display them using a slide show, or students could create a collage of magazine pictures on a poster. Have students identify and explain the shapes and their properties, including such things as the name of the shape and its faces, parallel and perpendicular lines, types of angles, and properties of the figures. Once students have acquired their information, ask them to demonstrate their understanding of the figure by completing a RAFT writing (view literacy strategy descriptions). This form of writing gives students the freedom to project themselves into unique roles and look at content from unique perspectives. From these roles and perspectives, RAFT writing is used to explain processes, describe a point of view, envision a potential job or assignment, or solve a problem. It’s the kind of writing that when crafted appropriately should be creative and informative.

R - Role Two or three dimensional figure

A- Audience The class

F- Form Poem, song

T- Topic Properties of two or three dimensional figures

In their RAFTed song or poem, students must include the properties to describe their figure but not name the figure. The small groups will present their song or poem to the class and the class will identify the described figure. The group will then reveal its figure and share the picture or display of their figure.

Activity 13: The Fly On The Ceiling (GLE: 28)

Materials List: The Fly on the Ceiling by Julie Glass, Coordinate Points BLM, pencil

The book The Fly on the Ceiling by Julie Glass, Random House Books for Young Readers (1998) offers delightful insight into the mathematician Rene Descartes and to the Cartesian coordinate system. Read the story to motivate the students and let them become familiar with point plotting.

For an introduction to the coordinate plane, The Math Forum offers a graphic representation that presents an understanding of points on a line, finding and graphing points in a plane, and estimating points. The Math Forum is located at http://mathforum.org/cgraph/cplane/.

Distribute the Coordinate Points BLM. Have the students work with a partner to answer the questions. Discuss the answers as a class.

The history of the coordinate plane, how it was invented, and who named it is at http://mathforum.org/cgraph/history/.

Activity 14: Plotting Shapes (GLEs: 19, 28)

Materials List: Grid Paper BLM, ruler, pencil

With the overhead or board, use a grid and four (pre-determined) ordered pairs to demonstrate plotting simple shapes. For rectangles, students should see that the x and y- coordinates are the same on two of each of the pairs. Distribute the Grid Paper BLM, then have students plot other sets of ordered pairs to find the shapes. Using the coordinates, students can figure out the length and area. Have students draw a linear picture (simple sailboat) using a ruler and then figure out the coordinates for each point. Ask students to give their lists of points to other students to plot to find the shape. A variation is to have students play the Battleship® game.

Sample Assessments

General Assessments

 The student will create portfolios containing samples of his/her work.  Facilitate during small group discussions to determine misconceptions, understandings, use of correct terminology, and reasoning abilities. Appropriate questions to ask to promote problem solving might be: o What do you need to find out? o What information do you have? o What strategies are you going to use?

 Ask probing questions to help students learn to reason mathematically. Possible questions to ask may be… o Is that true for all cases? Explain o Can you think of a counterexample? o How would you prove that? o What assumptions are you making? o How would you convince the rest of us that your answer makes sense?  The student will create learning logs (view literacy strategy descriptions) using such topics as these: o The most important thing I learned in math this week was… o The easiest thing about today’s lesson… o The opening of Tessy Triangle’s new Tessellating Art Show is seen as…

 Activity 5: Give the student a copy of the Broken Ruler BLM. Provide a list of items of less than ten inches in length and readily accessible in the classroom for the students to measure. The student will measure and record the length or width in both centimeters and inches. After recording the information, the student will respond to the following questions: Are you sure your answers are accurate? What made this task difficult? How would you explain your method of measurement to someone that doesn’t know how to read a ruler? Did you find metric measurement easier, more difficult, or same as English measure? Why?

 Activity 7: Draw an irregular figure on a grid and present it to the student with the following task: How would you estimate the area and perimeter of an irregular figure such as the one given. Could you find the actual perimeter? If so, what method would you use? Explain your method thoroughly. What is a reasonable estimate for the area of the figure? Explain your reasoning.

 Activity 12: Collect street maps from cities in the United States or around the world, ones that have interesting patterns to the layout of the streets. The student will choose one of the maps, perhaps from a city the student might like to visit. Ask him/her to measure and classify the angles formed by different pairs of intersecting streets. The student must be able to name the streets that formed each angle and find as many examples possible of each type of angle: acute, right, and obtuse. The student will indicate the street intersection that formed the largest angle, the smallest, and the closest to a particular amount, perhaps 75˚.

Grade 6 Mathematics Unit 6: Taking a Chance

Time Frame: Approximately two weeks

Unit Description

The focus of this unit is to further the development of the fundamental counting principle. Opportunities are provided for students to organize and list possible outcomes to solve real-life situations. The extension of probability settings includes complementary events and recognition of equally likely (equally probable) events in experiments.

Students can apply the counting principle to find the total number of possibilities or choices for certain selections. They know that to find the total number of possible occurrences, one multiplies the number of different ways each choice can occur.

Guiding Questions

1. Can students use the fundamental counting principles to find the number in a sample space? 2. Can students use various strategies when working with complementary events to find probability? 3. Can students recognize and use equally likely/probable events when working with real-life problems?

GLE # GLE Text and Benchmarks Data Analysis, Probability, and Discrete Math 34. Use lists, tree diagrams, and tables to determine the possible combinations from two disjoint sets when choosing one item from each set (D-4-M) 35. Illustrate and apply the concept of complementary events (D-5-M) 36. Apply the meaning of equally likely and equally probable to real-life situations (D-5-M) (D-6-M)

Grade 6 MathematicsUnit 6Taking a Chance 54 Louisiana Comprehensive Curriculum, Revised 2008

Sample Activities

Activity 1: Probability Vocabulary Awareness (GLE: 35, 36)

Materials List: Probability BLM, pencil

Before beginning the unit, have students complete a vocabulary self awareness chart (view literacy strategy descriptions). Provide students with the Probability BLM. Do not give students definitions or examples at this point.

Word + Ö – Example Definition Probability Equally Probable Tree diagram

Multiples Outcomes Ordered pair Complementary Events Experimental Probability Theoretical Probability

Ask students to rate their understanding of each word with either a “+” (understands well), a “Ö” (some understanding), or a “–” (don’t know). During and after completing the activities throughout this unit, students should return to the chart to fill in examples and definitions in their own words. The goal is to have all plus signs at the end of the unit. After all the students have completed the chart, have them share their examples and definitions with each other to check for accuracy.

Activity 2: Probability (GLEs: 34, 36)

Materials List: Spinner BLM, paper clips, Results Table BLM, newsprint, paper, pencil

Begin this lesson with the literacy strategy student questions for purposeful learning (SQPL) (view literacy strategy descriptions) by writing the statement “If a spinner is divided into 3 sections, then the probability of landing on each section is ⅓.” This strategy is used to encourage the students to generate questions that they would need to answer to verify the statement. Have students in groups of four, brainstorm (view literacy strategy descriptions) different questions that they might need to answer to determine whether this statement is true or false. Have each group of students highlight 2

Grade 6 MathematicsUnit 6Taking a Chance 55 Louisiana Comprehensive Curriculum, Revised 2008 or 3 questions that their group has come up with for use with whole class discussion. Write these questions on a sheet of newsprint for use as closure when the students have completed this activity.

Distribute the Spinner BLM and a paper clip to each student. Ask the students to think of the likelihood of each of the numbers appearing in a spin. (2 out of 4 chance to spin a 1, 1 out of 4 chance of spinning a 2 or 3.) Discuss which number is more likely to occur on each spin and/or which number will occur on each spin. They should speculate if anything could affect the outcome such as where we start the spinner in a spin. Also discuss percents and fractions that may describe possible outcomes.

Have students spin the spinner twenty times and record the results on the Results Table BLM. Evaluate the results. Discuss how experimental and theoretical probability are related and compare results..

Once the entire activity is completed, have the students reread the list of questions that were generated at the beginning of the activity. As a class, discuss whether or not these questions were answered as they completed the activity.

Activity 3: On a Roll! (GLE: 34)

Materials List: dice, Rolling Number Cubes BLM, paper, pencil

Give each group a pair of dice—each die of a different color to ease data collection. Have the students brainstorm (view literacy strategy descriptions) to decide how to determine all the combinations they could get with a roll of the dice. Students should make a list of these combinations. For example, they could create a list of ordered pairs where the first number is the first die and the second number is from the second die, for example, (1, 1) or (6, 2). When finished, they should have listed all 36 possible outcomes for rolling two dice. Distribute the Rolling Number Cubes BLM and have students conduct an experiment of rolling the number cubes 36 times, keeping track of the outcome of each roll. Have students compare their results (experimental probability) to the list of combinations they created earlier in class (theoretical probability).

Activity 4: A Lunch Combo (GLE: 34)

Materials List: Tree Diagrams BLM, paper, pencil

Distribute the Tree Diagrams BLM to the students. Have students use a tree diagram to solve the problems. After discussing the solutions to the problems, have the students create a math story chain (view literacy strategy descriptions). Put students in groups of four. On a sheet of paper, ask the first student to write an opening sentence for the math story chain similar to the following example:

Susie is trying to pick an outfit for the school dance.

She has a jean skirt or a pink skirt, and a white shirt, plaid shirt or striped shirt.

The paper is passed again to the right to the next student who writes the third sentence of the story.

If she selects one skirt and one shirt, how many outfits does she have to choose from?

The paper is now passed to the fourth student who must solve the problem and write out the answer. The other three group members review the answer for accuracy.

Answer: Six outfits

This activity allows students to use their writing, reading, and speaking skills while learning and reviewing the concept of tree diagrams.

As an extension to the lesson, give students ownership in the process by creating a product. A variation could include having the students explore the different varieties of a trail mix they could create. They choose between three different kinds of dry cereal, two different candy-coated chocolates, and dry-roasted and honey roasted peanuts. Once the students have explored all possible combinations, allow the students to choose a type and make their own trail mix.

Activity 5: Paper Pull (GLE: 35)

Materials List: paper bag, paper numbered 1 to 100, Multiples Table BLM, pencil

Provide students with a bag containing one hundred slips of paper numbered from 1 to 100. Instruct them to draw one slip at random. Have students determine the probability of drawing a multiple of 8, a multiple of 6, a number that is not a multiple of 6, and/or a number that is a multiple of both 6 and 8. With the students working in small groups, let them draw slips and record the results on the Multiples Table BLM. Use the results to promote a discussion using correct terminology. Use this opportunity to explore that the events - pulling a slip that is a multiple of a number and pulling a slip that is not a multiple of a given number - are complementary events. Complementary events are two events from the same sample space whose probabilities add up to 1.

Activity 6: How Likely? (GLEs: 35, 36)

Materials List: Color Spinner BLM, Spinner Probability BLM, paper clips, paper, pencil

Distribute the Color Spinner BLM, Spinner Probability BLM, and paper clips. Have the students use the spinner to answer questions. Students should work in pairs or small groups and describe the complementary events. (If the spinner has a one out of eight chance of landing on blue, then there is a seven out of eight chance it’s not going to land on blue.) Use the following questions for class discussion after the activity is complete: If you have only one chance to spin, how likely is it that you will land on the blue? (1 out of 8) What other color is it equally likely that you could land on? (pink) What color on the spinner is most likely to be landed on? (red) How likely is it that you will land on a non-blue part? (7 out of 8)

Activity 7: Figure This! (GLEs: 34, 35, 36)

Materials List: computer with Internet access, paper, pencil

Have the students put to use their knowledge of probability by visiting the Figure This!® website at http://www.figurethis.org/challenges/math_index.htm. Activities that are appropriate to this unit are:  Applying probability 19) Two Points 26) I Win! 52) Capture-Recapture 63) Matching Birthday 69) Misaddressed

 Complementary/Mutually exclusive events 63) Matching Birthdays 68) Bones.

This website is appropriate for classroom use but is also an excellent source to allow students to work at home as a family unit to explore rich mathematical activities. If computers are not available to students at home, print the problem from the Internet and send paper copies home with the students. If the Internet is not available, a Figure This!® CD is available. Have students write an explanation of all problems worked at home or school in their math learning log (view literacy strategy descriptions). Have the students share their learning log responses with the class. Students should listen for logic and accuracy of responses.

Activity 8: Probability and Literature (GLEs 34, 35, 36)

Materials List: Do You Wanna Bet? Your Chance to Find Out About Probability by Jean Cushman, Socrates and the Three Little Pigs by Tuyosi Mori

Provide students with the experience of seeing the mathematics of probability presented in a different context: literature. The two books listed below are grade appropriate and offer connections to recording combinations, complementary events, and equally likely/probable outcomes. Any one of these books will promote classroom discussion and activities related to the unit. Choose from one of the following:  Do You Wanna Bet? Your Chance to Find Out About Probability by Jean Cushman; illustrated by Martha Weston. Clarion Books/Houghton Mifflin, New York, 1991.

Two boys discover that everyday events such as sports scores, weather predictions, and coin flips are dependent on many factors of chance and probability. The book includes bibliographical references and index.

 Socrates and the Three Little Pigs (also published under the title Anno’s Three Little Pigs) by Tuyosi Mori; illustrated by Mitsumasa Anno. G.G. Putnam, New York, 1986. Note: this book is currently out of print but many school libraries have copies of the book. It might be obtained through a used bookstore or online.

Socrates, a wolf, is challenged to catch one of the three little pigs for his wife’s dinner. The pigs collectively own five houses. With the help of his frog friend, the mathematician Pythagoras, Socrates tries to determine the possible cottages the pigs might be in. The book takes simple combinations and walks the reader through combinations and permutations in quite a succinct manner. This book helps showcase the art of problem solving in a high-interest format.

After reading the stories, have the students work in pairs to create their own story involving probability. These scenarios will be used as the student pairs present their scenario using a modified questioning the author (QtA) (view literacy strategy descriptions). In this case, instead of students questioning the author of a text, they will question each other as authors of these scenarios on probability. As the authors of the scenario, the pair will be involved in a collaborative process of building understanding with probability situations. Students should develop a model of their probability situation and represent the situation mathematically. Once the student pairs have developed their scenario, the pair begins answering the questions from classmates about its scenario and solution. The teacher strives to elicit students’ thinking while keeping them focused in their discussion. The pair will answer questions that the class asks about its scenario and justify the mathematics involved in the solution.

Sample Assessments

General Assessments

 The student will create a portfolio containing samples of experiments and activities.  The student will give real-life examples of combinations such as shirts and pants, flowers for garden, choices when buying a car, getting cable channels, and eating at a buffet.  Create extensions to an activity by increasing the difficulty or by asking “what if” questions, whenever possible.  The student will create and demonstrate original math problems by acting them out or using manipulatives to provide solutions on the board or overhead.  The students will create his/her own questions.  Provide the use of concrete materials (manipulatives) to enhance student understanding.  Facilitate discussion during small group work to determine misconceptions, understandings, use of correct terminology, and reasoning abilities. Appropriate questions to ask to promote problem-solving: o What do you need to find out? o What strategies are you going to use? o What do you think the result will be? To check for understanding: o Did you use or learn any new words today? What do they mean? How do you spell them? To help students when they get stuck, ask… o Would it help to create a diagram? Make a table? To promote thinking in making conjectures, ask: o What would happen if …? What if not? o What are some possibilities here? o Can you predict the next one?  The students will create journal writings using such topics as o Why is a tree diagram a good way to account for all possibilities? o What was the easiest thing about today’s lesson? o Discuss the meaning of the expression “chances are 50-50.”

 Activity 3: The student will show all possible three-letter combinations using the letters A, R, and T and will use a tree diagram to solve them. The student will determine how many three-letter combinations there are. (6) Do any combinations form real words? (Yes) If so, which ones? (ART, TAR, RAT)

 Activity 4: The student will give a probability that would indicate a highly probable event, and explain his/her reasoning, doing the same for an unlikely 8 event. Sample answer for highly probable might state: 9 means that there is an eight out of nine chance. This is very near 1, so it indicates a very likely 1 event. Likewise, 9 is very close to 0, therefore an unlikely event.

 Activity 5: The student will work with a partner and be given mixtures of an item. He/she will find the likelihood of selecting items from the mixture. o Supply mixtures for each pair of students such as bags of different colored candies, animal crackers, colored plastic chips, or bags of marbles. o The students will list each type item in the mixture. o The students will count out the items and record numbers with a tally chart for each item. o The students will use information to decide if each item has the same chance of being randomly selected. o The students will show the mixture and the data to the class and will explain the results.

 Activity 6: The student will respond to the following: An event has a probability of .75. What words could you use to describe that event? The student is being checked to see if he/she can understand or interpret numerical statements of probability. In his/her answers, the student will include words such as good chance, very likely, or better than 50/50.

Grade 6 Mathematics Unit 7: Strengthening Whole Number Multiplication and Division

Time Frame: Approximately two weeks

Unit Description

This unit focuses on reviewing multiplication and division algorithms for whole numbers that involves various representations of the remainder. The scope of problems extends to dividing a 4-digit by a 2-digit number.

Students can divide with a 2-digit number and can select the appropriate representation of the remainder in real-life situations. Students understand the essential role that estimation plays in solving division problems.

Guiding Questions

1. Can students explain the reasonableness of an answer in a division problem? 2. Can students use different methods to solve division problems? 3. Can students handle the choice of representations for quotients? 4. Can students solve multi-step problems involving multiplication and division of whole numbers?

GLE # GLE Text and Benchmarks Number and Number Relations 12. Divide 4-digit numbers by 2-digit numbers with the quotient written as a mixed number or a decimal (N-7-M)

Grade 6 MathematicsUnit 7 Strengthening Whole Number 62 Louisiana Comprehensive Curriculum, Revised 2008

Sample Activities

Activity 1: Understanding Division (GLE: 12)

Materials List: Dividing BLM, pencil

Provide students with several scenarios that require a solution obtained through division. These scenarios should depict situations that help students see division as repeated subtraction and as sharing. The algorithm for division with a 2-digit divisor is the simple process of repeated subtraction, but starting with multiples of 10.

Example: There are 357 sixth graders registered for Southfork Middle School. The principal wants to know the least number of teachers he has to hire if the classes can be no larger than 27. Ten classes of students would total 270, but 20 teachers could teach 540. Subtracting 270 is a good place to start. 357- 270 = 87 . Ask students to estimate how many more teachers will be needed. How would the estimation be checked? Students may say that 87 is close to 90 and 27 is close to 30 and30 3  90 . Check the answer, 3 27  81. That leaves 6 students without a teacher! Discuss with students the best way to deal with a remainder in this situation.

Distribute the Dividing BLM and have them work in groups to solve the problems. These problems will help students better understand the concept of division as well as the process. They will begin to understand that the division algorithm is a shortcut for using repeated subtraction. Have students write the remainders as fractions and as decimals. Discuss the answers as a class.

Activity 2: Remainder Game (GLE: 12)

Materials List: Remainder Game BLM, pencil

Provide the students with the opportunity to test various divisors and the impact of those divisors on the quotient and remainder. This activity is similar to The Product Game and is played in pairs. Distribute the Remainder Game BLM to the students. The BLM has the numbers 1 to 20 across the top. As they are each used, they will be crossed out and will not be used again.

Begin with the number 100 as the starting number. The first student chooses a divisor from the list and divides the number 100 by that divisor. The remainder is awarded to that player as his/her score. The other student records the division sentence and marks the problem as belonging to the first player. Both players subtract the remainder from the starting number to get the new dividend. The second player now chooses a divisor for the new dividend, divides the problem, and takes the remainder as a score. The first player is

Grade 6 MathematicsUnit 7 Strengthening Whole Number 63 Louisiana Comprehensive Curriculum, Revised 2008 charged with recording the division sentence for the second player. Play continues in this fashion with the two players alternating until the starting number is 0 or it is no longer possible for either player to score. The person with the highest score wins. Emphasize to students that points come from remainders so students should choose divisors carefully to yield the largest remainder possible. If a remainder of 0 is given, the student receives no points, and the opponent repeats the previous starting number but chooses a different divisor.

The Remainder Game 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Starting # 100 100 ÷ 18 = 5 R 10 John 90 90 ÷ 19 = 4 R 14 Mary 76 76 ÷ 13= 5 R 11 John 65 65 ÷ 12 = 4 R 5 Mary 60 60 ÷ 17 = 3 R 9 John 51 51 ÷ 14 = 3 R 9 Mary 42 42 ÷ 15 = 2 R 12 John 30 30 ÷ 16 = 1 R 14 Mary 16 16 ÷ 9 = 1 R 7 John 9 9 ÷ 5 = 1 R 4 Mary 5 5 ÷ 3= 1 R 2 John 3 3 ÷ 2 = 1 R 1 Mary 2 2 ÷ 1 = 2 R 0 John John’s score: 51 Mary’s score: 47

Allow students to play several times until they feel comfortable with the game and have begun developing a strategy to win the game. Ask questions like, “What is the best first move?” “Why?” or “What strategies did you use to decide your best move?” “What would happen if the starting number was a lower number? A higher number?”

After the students have thoroughly explored this number, give them a new starting number that is larger than 100. In their math learning log (view literacy strategy descriptions), have students explain how the changes of the divisor affect the size of the quotient. Give a real world example to illustrate your answer. Allow students to work in pairs to review their log entries for accuracy.

Activity 3: Dealing with Remainders (GLE: 12)

Materials List: calculator, Remainders BLM, pencil

Allow students to use calculators to solve these division problems where the focus will be on how to handle the remainders. As a class solve the following problems and discuss how to represent the remainder.

 Dropped remainders - If it takes 8 yards of ribbon to make one bow, how many bows can be made with 22 yards of ribbon? (Division gives 2.75. Since you can’t make 0.75 bow, the correct answer is 2 bows.)  Remainders written as fractions- 25 mini-pizzas were delivered to school. If there 1 were 20 students in the class, what part would each student receive? (1 4 )  Remainders written as decimals- Mary bought 22 cans of soup. The total cost was $14.96. How much did one can of soup cost? ($0.68)  Remainders that dictate an increase in the quotient- 150 scouts are going on a camping trip. Each tent will sleep no more than 12 scouts. How many tents will they need? (12.5 - Since you need a whole tent instead of one-half a tent, the answer would be “at least 13 tents”.)

Distribute the Remainders BLM for the students to solve. Have the students work in groups of two to solve the problems. Discuss the answers as a class.

Activity 4: Division Problems (GLE: 12)

Materials List: Division BLM, calculator, pencil, paper

Distribute the Division BLM. Have students work the problems by hand. When they are finished, have them use a calculator to get the answer in decimals. Check for accuracy. Have students put the answers in both fractions (if solution has a remainder) and decimals.

After discussing the solutions to the problems above, have the students create a story chain (view literacy strategy descriptions). Put students in groups of four. On a sheet of paper, ask the first student to write the opening sentence for the math story chain:

Jen had to read a book with 434 pages.

She read the same number of pages for 14 days.

The paper is passed again to the right to the next student who writes the third sentence of the story. How many pages did she read each day?

The paper is now passed to the fourth student who must solve the problem and write out the answer. The other three group members review the answer for accuracy.

Answer: She read 31 pages each day.

This activity allows students to use their writing, reading, and speaking skills while learning and reviewing the concept of division.

Activity 5: I Have; Who Has? (GLE: 12)

Materials List: I Have Who Has BLM, card stock

Cut apart the I Have Who Has BLM cards and distribute them to the students. A few students may have to have more than one card depending on the number of students in the class. One student will start by reading their card. For example, the first card might read, “I have 420 divided by 24; who has the decimal answer?” and then the student with the answer will read his/her card. The process will continue until the last student reads his/ her card which should connect to the first card read.

Sample Assessments

General Assessments

 The student will create a portfolio containing samples of students work and game score sheets.  The students will create his/her own real-life division problems  Make observations to determine student understanding as he/she engages in the various activities.  Create extensions to an activity by increasing the difficulty or by asking “what if” questions, whenever possible.  The student will create and demonstrate original math problems by acting them out or using manipulatives to provide solutions on the board or overhead.  The student will create his/her own questions.  The student will show understanding by demonstrating mathematical principles with the use of concrete materials (manipulatives).  The students will perform divisions with 4-digit dividends and 2-digit divisors.

 The students will determine which problems from a given set should be solved using division.  The student will decide the appropriate handling of remainders in real-life problem situations.  The students will create an “I Have Who Has” division game using his/her own problems (either numerical or word problems).  Facilitate discussion during small group work to determine misconceptions, understandings, use of correct terminology, and reasoning abilities. Appropriate questions to ask to promote problem-solving: o Will a calculator help? o What strategies are you going to use? o What do you think the result will be? To help students when they get stuck, ask… o Could you try it with simpler numbers? o Would it help to draw a picture? o Can you guess and check? o Have you compared your work with anyone else?  The student will create journal writings using such topics as these: o Explain the mistakes made in this problem: 3 ÷ 4 = 12 o What was the easiest thing about today’s lesson? o Why is it important to know how to divide by two digit numbers?

 Activity 1: After reviewing the type of problems explored in Activity 1, the student will write his/her own problem, work the problem showing the thought process used, and present the problem and findings to the class.

 Activity 2: The student will check other students’ game boards in the Remainder Game for accuracy. Each student will check the mathematics within the board to make sure the division was worked correctly and to check that the final scores were accurate. Student evaluators will write a response explaining to the players of the game exactly what was incorrect about the game board or write a comment commending the game players on the accuracy of the work by citing examples of the math. Make up “fake” game boards that have particular errors. (It would be a good idea to keep boards year-to-year to acquire a variety of tools.)

 Activity 3: The student will give real life examples where the remainder for the problem 135 ÷ 18 can be represented in four different ways:

1 o 135 cookies shared by 18 children (7 2 cookies per child) o 135 pencils given to 18 students (7 pencils per student with 9 left over) o $135.00 dispersed among 18 people ($7.50 per person)

o 135 people placed in mini buses with no more than 18 per bus (at least 8 buses are needed.)

Grade 6 Mathematics Unit 8: Integers, Patterns, and Algebra

Unit Description

The focus of this unit is on working with integers, patterns, and variables. A number line is used to represent integers and inverses. Opportunities to represent, analyze, and generalize a variety of patterns with tables, graphs, words, and when possible, symbolic rules are provided.

Students should have a conceptual understanding of different uses of a variable. They know that symbolic algebra can be used to represent situations and to solve problems, especially those that involve linear relationships. Students use modeling as an appropriate strategy to solve math problems whether by drawing figures, using a number line, or other technique. They can model and identify perfect squares up to 144 and can match algebraic equations and expressions with verbal statements and vice versa.

Guiding Questions

1. Can students use different methods to interpret and represent integers on a number line? 2. Can students discuss the significance of additive inverses? 3. Can students recognize squares to 144? 4. Can students evaluate expressions for specified variable values? 5. Can students match or create stories to go with a given algebraic expression or equation? 6. Can students recognize patterns in tables, arithmetic sequences, geometric sequences, or now-next (recursive) patterns and discuss them or extend them?

GLE # GLE Text and Benchmarks Number and Number Relations 6. Compare positive fractions, decimals, and positive and negative integers using symbols (i.e., <, =, >) and number lines (N-2-M)

Grade 6 MathematicsUnit 8 Integers, Patterns, and Algebra 69 Louisiana Comprehensive Curriculum, Revised 2008

GLE # GLE Text and Benchmarks 8. Demonstrate the meaning of positive and negative numbers and their opposites in real-life situations (N-3-M) (N-5-M) Algebra 14. Model and identify perfect squares up to 144 (A-1-M) 15. Match algebraic equations and expressions with verbal statements and vice versa (A-1-M) (A-3-M) (A-5-M) (P-2-M) 16. Evaluate simple algebraic expressions using substitution (A-2-M) 17. Find solutions to 2-step equations with positive integer solutions (e.g., 3x  5  13 , 2x 3 x  20) (A-2-M) Patterns, Relations, and Functions 37. Describe, complete, and apply a pattern of differences found in an input-output table (P-1-M) (P-2-M) (P-3-M) 38. Describe patterns in sequences of arithmetic and geometric growth and now- next relationships (i.e., growth patterns where the next term is dependent on the present term) with numbers and figures (P-3-M) (A-4-M)

Sample Activities

Activity 1: Graphing Perfect Squares (GLE: 14)

Materials List: Graph Paper BLM, colored pencils or markers, scissors, paper, glue or tape, pencil

To give the students a visualization of perfect squares, distribute the Graph Paper BLM. Ask them to make twelve perfect squares—12 , 22 , and so on through 122 . Then ask students to outline or “frame” each square with a colored pencil or marker, cut out the square, write the name of each square in the frame (e.g.,122 = 12 12 ), write the area of the square (e.g.,122 = 144 ), and finally, attach each square in order by size on another piece of paper.

Remind students of previous work with perfect squares in Unit 2.

Activity 2: Below Zero Temperatures (GLE: 8)

Materials List: thermometer, safety pin, temperatures from major cities, Planets BLM, paper, pencil

Before class, obtain (or make) a model of a thermometer. The thermometer can be made with heavy poster board with two holes cut toward the top and bottom. The scale should reflect both positive and negative temperatures. A one-inch wide piece of elastic and a red marker will be used for the “mercury.” Safety pin the ends to make an elastic loop.

Demonstrate a thermometer reading by moving the colored part of the elastic up and down the scale. Have students make up problems using the thermometer—it was 32 degrees during the day and the temperature dropped to –13. How many degrees did the temperature fall? Wind chill temperatures can also be used. Have students check each other’s work for accuracy.

Distribute the Planets BLM. Have the students work together to complete. Discuss the answers as a class.

Activity 3: Newspaper Comparisons (GLEs: 6, 8)

Materials List: newspapers, paper, pencils

Teacher Note: GLE 8 is a part of this activity because of the use of negative fractions and decimals. However, no computations with these negative numbers are to be done in this activity. The goal of the activity is to help students appropriately use the symbols <, >, and =.

Have students use sections from the local newspaper to create scenarios comparing fractions, decimals or integers. Have them use the weather page for temperatures (+ and – integers), the food section (fractions), and the sports page (decimals). Have students choose one section and make up scenarios using the information found in the charts. For example: Show the relationship between yesterday’s highest and lowest recorded temperatures. In the recipes from the food section, the problems can reflect the relationship between amounts of one particular ingredient.

These scenarios will be used as the student pairs present their scenario using a modified questioning the author (QtA) (view literacy strategy descriptions). In this case, instead of students questioning the author of a text, they will question each other as authors of these scenarios. As the authors of the scenario, the pair will be involved in a collaborative process of building understanding with fraction, decimal and integer situations. Students should develop a model of their situation and represent the situation mathematically. Once the student pairs have developed their scenario, the pair begins answering the questions from classmates about their scenario and solution. Strive to elicit students’ thinking while keeping them focused in their discussion. The pair will answer questions that the class asks about their scenario and justify the mathematics involved in the solution.

Activity 4: Equal Concentration (GLE: 15, 16)

Materials List: Concentration BLM (one set per group of two students), Solutions BLM, pencil, card stock

Create a Concentration® type game using the Concentration BLM. Have the students match algebraic equations and expressions to equivalent verbal statements. Students will work in groups of two to play the game. The student with the largest number of matching pairs wins the game.

Using the Concentration game cards, have the students complete the Solutions BLM. The students will complete the table with the word phrases, the accompanying algebraic expressions and 3 possible replacement values for the variables. Students should then solve the expressions using the replacement values given in the table. Model the process for the students using the following two cards in the table below. Then have each pair of students divide the remaining cards between them. Each student should have 7 word phrases to complete on the table. The students may select their own replacement values.

Word Algebraic 1st Solution 2nd Solution 3rd Solution Phrase Expression Replacemen Replacemen Replacemen t Value t Value t Value 7 less x - 7 7 7– 7 = 0 8 8 – 7 = 1 9 9 – 7 = 2 than a number x 4 4 - x 0 4 – 0 = 4 1 4 – 1 = 3 2 4 – 1 = 3 decreased by a number x

Activity 5: Substituting Numbers (GLE: 16, 17)

Materials List: Solve It! BLM, number cubes, pencil

Distribute the Solve It! BLM and a number cube to each group of two students. The students will take turns solving and checking the problems. The first student rolls the number cube and the second student completes the problem substituting the rolled number in place of the variable. The student who rolled the number then checks the solution. The students switch rolls for each problem.

Activity 6: Two-Step Math (GLE: 17)

Materials List: Two-Step BLM, paper, pencil

Provide students with several scenarios that can be represented algebraically in a two-step problem. Have students solve these problems, and discuss the equations and solutions as a class. Use problems similar to this: If Randy subtracts 3 times his number from 25, he gets 4. What is Randy’s number? ( 25 - 3n = 4; n = 7 )

Liz had $10.00 in her piggy bank. Then she saved her weekly allowance for 4 weeks. At the end of that time, she had $30.00. How much did she get in allowance each week? (4n+10 = 30; n = 5 )

Distribute the Two-Step BLM. Have the students work in pairs to write an equation to represent each problem and then solve. After the students have completed the problems, discuss the solutions as a class.

Activity 7: Sequence Counting (GLE: 38)

Materials List: paper, pencil, newsprint

Begin this lesson with the literacy strategy student questions for purposeful learning (SQPL) (view literacy strategy descriptions) by writing the statement “All sequences have a constant difference between terms.” This strategy is used to encourage the students to generate questions that they would need to answer to verify the statement. Have students in groups of four brainstorm (view literacy strategy descriptions) different questions that they might need to answer to determine whether this statement is true or false. Have each group of students highlight 2 or 3 questions that their group has come up with for use with whole class discussion. Write these questions on a sheet of newsprint for use as closure when the students have completed this activity.

Have students work in pairs to generate an arithmetic or geometric sequence of numbers when a “rule” that generates the sequence is given. For example, provide student pairs with the rule “term number times 4 plus 1.” Have students generate ten terms of the sequence. Ask student pairs to provide an alternate rule for generating the terms of the sequence (in this example, they could say “add 4 to the previous term to get the next term”). Another example might be to generate the sequence for the rule “start with $1 and double your money each day.”

Once the entire activity is completed, have the students reread the list of questions that were generated at the beginning of the activity. As a class, discuss whether or not these questions were answered as they completed the activity. Students should determine that the SQPL statement was not accurate and only applies to arithmetic sequences.

Activity 8: Pumping Gas Rules! (GLE: 38)

Provide students with examples of arithmetic sequences that are used daily. For example, purchasing gasoline for a car can be thought of as an arithmetic sequence where the term number is the same as the number of gallons pumped (i.e., 1, 2, 3 . . .), and the value of a term is the price for that many gallons. Have students investigate this sequence and describe how the terms are generated.

Present the following problem to the students. Have them record their response in their math learning log (view literacy strategy descriptions).

Donnie was watching Gas Watch, a segment on the local news where the cheapest gas prices in town are identified, when it showed the following table for local gas prices:

Gallons of 5 6 7 8 Gas Total Cost for 10.45 12.54 14.63 16.72 Gas

Write an expression to find the total cost of any number of gallons of gasoline. Explain your thinking.

Have the students share their learning log responses with the class. Students should listen for logic and accuracy of responses.

Expression: 2.09g

Explanation: To find the total cost of gas, you would multiply the cost for one gallon times the number of gallons needed. The cost per gallon is determined by dividing the total cost by the number of gallons. In this case the gas is $2.09 per gallon.

Activity 9: Input-Output (GLE: 37)

Materials List: Input-Output Tables BLM, Graph Paper BLM, pencil

Distribute the Input-Output Tables BLM and Graph Paper BLM. Have students complete the input-output tables independently. After completing the table, have students find the differences between successive outputs. Ask students to work with a partner to check whether the differences are the same in their tables and determine if the differences are constant or varied. Have students plot the ordered pairs of (input, output) for each table

Grade 6 MathematicsUnit 8 Integers, Patterns, and Algebra 74 Louisiana Comprehensive Curriculum, Revised 2008 and compare. Students will find that constant differences are associated with a linear set of points.

Sample Assessments

General Assessments

 The student will generate several terms of a sequence (arithmetic, geometric).  The student will solve two-step equations with integer coefficients and solutions.  Have the student write a scenario that could be described using positive and negative numbers.  The student will create a portfolio containing samples of experiments and activities.  The students will create his/her own real-life examples of demonstrated problems.  Use observation to determine student understanding as he/she engages in the various activities.  Whenever possible, create extensions to an activity by increasing the difficulty or by asking “what if” questions.  The student will create and demonstrate original math problems by acting them out or using manipulatives to provide solutions on the board or overhead.  The students will create their own questions.  The student will use manipulatives to demonstrate understanding of various mathematical principles.  Facilitate during small group discussion to determine misconceptions, understandings, use of correct terminology, and reasoning abilities. Appropriate questions to ask to help students learn to reason mathematically: o Is that true for all cases? Explain. o How would you prove that? o What assumptions are you making? To help check student progress, ask: o Can you explain what you have done so far? What else is there to do? o Why did you decide to use this method? o What do you notice about the patterns in the flowers and fruits you examined? o Do you think this always holds true? o How could you check on this?

 Activity 2: Take specific information from the collected data about temperature and give the student the temperatures and the names of the related cities. The student will use the information to generate five to ten statements about the data.

 Activity 5: The student will create a set of five verbal statements and the algebraic expressions and equations that match them and defend his/her work to the class in a brief presentation- answering any questions that may arise. Once the problems are approved, the student will transfer the information to the back of index cards and use them to make a memory type of game. All cards will be collected and made into a class game.

 Activity 7: The student will supply the next figure in a sequence.

Grade 6 MathematicsUnit 8 Integers, Patterns, and Algebra 76