Teaching Mathematics with Manipulatives. Course 2

Transcription

1 Teaching Mathematics with Manipulatives Course 2

2 Manipulatives Glencoe offers three types of kits to enhance the use of manipulatives in your Middle School Mathematics classroom. The Glencoe Mathematics Overhead Manipulative Resources contains translucent manipulatives designed for use with an overhead projector. The Glencoe Mathematics Classroom Manipulative Kit contains classroom sets of frequently used manipulatives in algebra, geometry, measurement, probability, and statistics. The Glencoe Mathematics Student Manipulative Kit contains an individual set of manipulatives often used in Student Edition activities. The manipulatives contained in each of these kits are listed on page vi of this booklet. Each of these kits can be ordered from Glencoe by calling (800) Glencoe Mathematics Overhead Manipulative Kit Glencoe Mathematics Classroom Manipulative Kit Glencoe Mathematics Student Manipulative Kit Copyright by The McGraw-Hill Companies, Inc. All rights reserved. Permission is granted to reproduce the material contained herein on the condition that such materials be reproduced only for classroom use; be provided to students, teachers, and families without charge; and be used solely in conjunction with the Glencoe Mathematics: Applications and Concepts, Course 2, program. Any other reproduction, for sale or other use, is expressly prohibited. Send all inquiries to: Glencoe/McGraw-Hill 8787 Orion Place Columbus, OH ISBN: Teaching Mathematics with Manipulatives Printed in the United States of America

3 Contents Easy-to-Make Manipulatives Page Base-Ten Models Decimal Models Fraction Models: Bars Fraction Models: Circles Counters Integer Counters Pattern for Cup Integer Mat Equation Mat Quarter-Inch Grid Centimeter Grid Square Dot Paper Isometric Dot Paper Tangram Number Lines First Quadrant Grids Coordinate Planes Percent Models Spinners Number Cube Patterns Protractors Rectangular Prism Pattern Cube Pattern Cylinder Pattern Cone Pattern Pyramid Pattern Pattern Blocks Circle Graph Template Problem-Solving Guide Activities Page CHAPTER Teaching Notes and Overview Mini-Project: Modeling Powers and Exponents Using Overhead Manipulatives: Variables and Expressions b Hands-On Lab Recording Sheet CHAPTER 2 Teaching Notes and Overview Mini-Project: Story Graph b Hands-On Lab Recording Sheet Using Overhead Manipulatives: Quartiles Using Overhead Manipulatives: How Much is a Handful? CHAPTER 3 Teaching Notes and Overview Mini Project: Coordinate Plane Puzzle a Hands-On Lab Recording Sheet a Hands-On Lab Recording Sheet Using Overhead Manipulatives: Multiplying Integers CHAPTER 4 Teaching Notes and Overview a Hands-On Lab Recording Sheet Mini-Project: Solving Multiplication Equations Using Overhead Manipulatives: Solving Two-Step Equations a Hands-On Lab Recording Sheet Using Overhead Manipulatives: A Function of Time CHAPTER 5 Teaching Notes and Overview a Hands-On Lab Recording Sheet Using Overhead Manipulatives: Percent CHAPTER 6 Teaching Notes and Overview Using Overhead Manipulatives: Multiplying Fractions and Mixed Numbers Mini-Project: Perimeter a Hands-On Lab Recording Sheet CHAPTER 7 Teaching Notes and Overview Using Overhead Manipulatives: Equal Ratios b Hands-On Lab Recording Sheet b Hands-On Lab Recording Sheet Mini-Project: Scale Drawings a Hands-On Lab Recording Sheet iii

4 CHAPTER 8 Teaching Notes and Overview Mini-Project: Percent and Estimation a Hands-On Lab Recording Sheet Using Overhead Manipulatives: Percent of Change CHAPTER 9 Teaching Notes and Overview Using Overhead Manipulatives: Exploring Permutations a Hands-On Lab Recording Sheet Mini-Project: Theoretical and Experimental Probability b Hands-On Lab Recording Sheet CHAPTER 0 Teaching Notes and Overview a Hands-On Lab Recording Sheet b Hands-On Lab Recording Sheet Using Overhead Manipulatives: Jelly Bean Statistics b Hands-On Lab Recording Sheet b Hands-On Lab Recording Sheet Using Overhead Manipulatives: Investigating Triangles and Quadrilaterals Mini-Project: Quadrilateral Tesselations Using Overhead Manipulatives: Angles of a Polygon Using Overhead Manipulatives: Inscribed Polygons b Using Overhead Manipulatives: Dilations b Hands-On Lab Recording Sheet CHAPTER Teaching Notes and Overview a Hands-On Lab Recording Sheet Using Overhead Manipulatives: Area a Hands-On Lab Recording Sheet Mini-Project: Areas of Circles, Rectangles, and Squares Using Overhead Manipulatives: Probability and Area Models....2 CHAPTER 2 Teaching Notes and Overview a Hands-On Lab Recording Sheet Using Overhead Manipulatives: Volume of Pyramids Mini-Project: Volume of Solids a Hands-On Lab Recording Sheet b Hands-On Lab Recording Sheet....2 iv

5 Teacher s Guide to Using Teaching Mathematics with Manipulatives The book contains two sections of masters Easy-to-Make Manipulatives and activities for Middle School Mathematics. Tabs help you locate the activities for each chapter. A complete list of manipulatives available in each of the three types of Glencoe Mathematics Manipulative Kits appears on the next page. Easy-to-Make Manipulatives The first section of this book contains masters for making your own manipulatives. To make more durable manipulatives, consider using card stock. To make algebra tiles similar to those shown in the Student Edition, have students use markers to color the tiles appropriately or use colored card stock. You can also make transparencies of frequently used items such as grid paper and number lines. Activity Masters Each chapter begins with Teaching Notes and Overview that summarizes the activities for the chapter and includes sample answers. There are three types of masters. Mini-Projects are short projects that enable students to independently investigate mathematical concepts. Using Overhead Manipulatives provides instructions for the teacher to demonstrate an alternate approach to the concepts of the lesson by using manipulatives on the overhead projector. Student Recording Sheets accompany the Hands-On Lab Activities found in the Student Edition. Students can easily record the results of the activity on prepared grids, charts, and figures. v

6 Glencoe Mathematics Manipulatives Glencoe Mathematics Overhead Manipulative Resources ISBN: Transparencies Overhead Manipulatives integer mat centimeter grid algebra tiles equation mat number lines spinners product mat lined paper two-dimensional cups inequality mat regular polygons red and yellow counters dot paper polynomial models decimal models (base-ten blocks) isometric dot paper integer models compass coordinate grids equation models protractor geoboard/geobands geometric shapes transparency pens in 4 colors Glencoe Mathematics Classroom Manipulative Kit ISBN: Measurement, Probability, Algebra and Statistics Geometry algebra tiles base-ten models compasses counters marbles geoboards cups measuring cups geobands centimeter cubes number cubes geomirrors equation mat/product mat protractors isometric dot grid stamp coordinate grid stamp and rulers pattern blocks ink pad scissors tangrams spinners stopwatches tape measures Glencoe Mathematics Student Manipulative Kit ISBN: algebra tiles red and yellow counters cups equation /product mat compass/ruler protractor scissors geoboard geobands tape measure vi

7 NAME Base-Ten Models Glencoe/McGraw-Hill Mathematics: Applications and Concepts, Course 2

8 NAME Decimal Models Glencoe/McGraw-Hill 2 Mathematics: Applications and Concepts, Course 2

9 Glencoe/McGraw-Hill 3 Mathematics: Applications and Concepts, Course 2 NAME Fraction Models: Bars

10 NAME Fraction Models: Circles Glencoe/McGraw-Hill 4 Mathematics: Applications and Concepts, Course 2

11 NAME Counters Glencoe/McGraw-Hill 5 Mathematics: Applications and Concepts, Course 2

12 NAME Integer Counters Glencoe/McGraw-Hill 6 Mathematics: Applications and Concepts, Course 2

13 NAME Pattern for Cup Glencoe/McGraw-Hill 7 Mathematics: Applications and Concepts, Course 2

14 NAME Integer Mat Glencoe/McGraw-Hill 8 Mathematics: Applications and Concepts, Course 2

15 NAME Equation Mat = Glencoe/McGraw-Hill 9 Mathematics: Applications and Concepts, Course 2

16 NAME Quarter-Inch Grid Glencoe/McGraw-Hill 0 Mathematics: Applications and Concepts, Course 2

17 NAME Centimeter Grid Glencoe/McGraw-Hill Mathematics: Applications and Concepts, Course 2

18 NAME Square Dot Paper Glencoe/McGraw-Hill 2 Mathematics: Applications and Concepts, Course 2

19 NAME Isometric Dot Paper Glencoe/McGraw-Hill 3 Mathematics: Applications and Concepts, Course 2

20 NAME Tangram Glencoe/McGraw-Hill 4 Mathematics: Applications and Concepts, Course 2

21 NAME Number Lines Glencoe/McGraw-Hill 5 Mathematics: Applications and Concepts, Course 2

22 NAME First-Quadrant Grids O O Glencoe/McGraw-Hill 6 Mathematics: Applications and Concepts, Course 2

23 NAME Coordinate Planes Glencoe/McGraw-Hill 7 Mathematics: Applications and Concepts, Course 2

24 NAME Percent Models Glencoe/McGraw-Hill 8 Mathematics: Applications and Concepts, Course 2

25 NAME Spinners Glencoe/McGraw-Hill 9 Mathematics: Applications and Concepts, Course 2

26 NAME Number Cube Patterns Glencoe/McGraw-Hill 20 Mathematics: Applications and Concepts, Course 2

27 Glencoe/McGraw-Hill 2 Mathematics: Applications and Concepts, Course 2 NAME Protractors

28 NAME Rectangular Prism Pattern Glencoe/McGraw-Hill 22 Mathematics: Applications and Concepts, Course 2

29 NAME Cube Pattern Glencoe/McGraw-Hill 23 Mathematics: Applications and Concepts, Course 2

30 NAME Cylinder Pattern Glencoe/McGraw-Hill 24 Mathematics: Applications and Concepts, Course 2

31 NAME Cone Pattern Glencoe/McGraw-Hill 25 Mathematics: Applications and Concepts, Course 2

32 NAME Pyramid Pattern Glencoe/McGraw-Hill 26 Mathematics: Applications and Concepts, Course 2

33 NAME Pattern Blocks Glencoe/McGraw-Hill 27 Mathematics: Applications and Concepts, Course 2

34 NAME Circle Graph Template 90% 95% 0% 5% 0% 85% 5% 0 80% % 75% 4 25% 70% % 65% 35% 60% 55% 50% 45% 40% Glencoe/McGraw-Hill 28 Mathematics: Applications and Concepts, Course 2

35 NAME Problem Solving Guide Problem: Explore Plan Solve These steps can help you solve problems. Examine Glencoe/McGraw-Hill 29 Mathematics: Applications and Concepts, Course 2

36 Decimal Patterns and Algebra Teaching Notes and Overview Mini-Project Modeling Powers and Exponents (p. 3 of this booklet) Use With Lesson -2. Objective Use centimeter cubes to model powers and exponents. centimeter cubes grid paper Students use centimeter cubes to model powers and exponents. On grid paper, they sketch the top and side view of each model. Answers. top side 2. top side 3. top side Using Overhead Manipulatives Variables and Expressions (pp of this booklet) Use With Lesson -4. counters* cups* integer mat transparency* * = available in Overhead Manipulative Resources Kit Students use cups and counters to model and solve algebraic expressions. An Extension activity asks students to model and solve an algebraic expression using different values for the variable. Answers Answers appear on the teacher demonstration instructions on pages Hands-On Lab Recording Sheet Exploring Sequences (p. 34 of this booklet) Use With Lesson -7b. This corresponds to the activity on page 37 in the Student Edition. Objective Explore patterns in sequences using paper folding. calculator piece of paper colored pencils A table is provided for students to record their data. Space is also given for students to write and explain their answers. Answers See Teacher Wraparound Edition p. 37. Objective Use cups and counters to model algebraic expressions. Glencoe/McGraw-Hill 30 Mathematics: Applications and Concepts, Course 2

37 NAME DATE PERIOD _ Mini-Project (Use with Lesson -2) Modeling Powers and Exponents Chapter centimeter cubes, grid paper You can build a square or a cube using centimeter cubes. 3 2 is a three-by-three square and is centimeter cube high. You need 9 centimeter cubes to build 3 2. The top view of 3 2 is shown at the left below. The side view of 3 2 is shown at the right below. You need 64 centimeter cubes to build 4 3. The top view is shown at the left below. The side view is shown at the right below. Use centimeter cubes to build each square or cube. On grid paper, sketch the top view and the side view top view top view top view side view side view side view Glencoe/McGraw-Hill 3 Mathematics: Applications and Concepts, Course 2

38 Using Overhead Manipulatives (Use with Lesson -4) Variables and Expressions Objective Use cups and counters to model algebraic expressions. counters* cups* integer mat transparency* * = available in Overhead Manipulative Resources Kit Teacher Demonstration Tell students that the phrase the sum of two and some number is an algebraic expression. The number 2 in this phrase is a constant, a number that you know, and some number is an unknown value. The phrase can be represented by a cup for the unknown value and 2 counters for the number 2. Place a cup and 2 counters on the integer mat. Place 6 counters in the cup. The cup now has a value of 6. Remove the counters from the cup and count all the counters. There are a total of 8 counters. Thus, the expression has a value of 8. Ask students what the value of the expression would be if 4 counters were placed in the cup and if no counters were placed in the cup. Model the correct answers if necessary. (6, 2) Clear the mat. Glencoe/McGraw-Hill 32 Mathematics: Applications and Concepts, Course 2

39 Using Overhead Manipulatives Tell students that the phrase three times some number is also an algebraic expression. Ask students how this phrase can be expressed mathematically. (Use three cups.) Stress that the same number of counters must be placed in each cup. Place 2 counters in each cup. Empty the cups and count all the counters. There are 6. The expression has a value of 6. Ask students what the value of the expression is if counter is placed in each cup; if 5 counters are placed in each cup. Demonstrate the correct answers if necessary. (3, 5) Have students complete Exercises 5. Model each phrase with cups and counters. Then put four counters in each cup. How many counters are there in all? Record your answers by drawing pictures of your models.. the sum of 3 and a number (7) 2. 3 times a number (2) 3. 5 more than a number (9) 4. twice a number plus (9) 5. Write a sentence to describe what the cup represents. (Sample answer: The cup represents the variable or unknown quantity.) Extension Model the phrase four more than three times some number as a third example. Use 3 cups to represent three times some number, and use 4 counters to represent the number 4. Place 5 counters in each cup. Empty the cups and count all the counters. There are 9. The expression has a value of 9. Ask students what the value of the expression is if counter is placed in each cup: if 3 counters are in each cup. (7, 3) Glencoe/McGraw-Hill 33 Mathematics: Applications and Concepts, Course 2

40 NAME DATE PERIOD _ Hands-On Lab Recording Sheet (Use with the activity on page 37 in Lesson -7b of the Student Edition) Exploring Sequences calculator, piece of paper, colored pencils Investigate Work with a partner. Complete the table. Number Layers Fraction of Paper of Folds of Paper Not Shaded Writing Math Work with a partner.. Examine the sequence of numbers in the Layers column of your table. Is the sequence arithmetic or geometric? Then write a rule to find the next three terms. 2. Examine the sequence of numbers in the Fraction column of your table. Is the sequence arithmetic or geometric? Then write a rule to find the next three terms. (Hint: Write each fraction as a decimal.) 3. Assume you could continue the paper-folding process indefinitely. Suppose your unfolded piece of paper is inch thick. Add a column to your table and find the thickness of the paper for the first five folds. 4. How many folds would it take until the paper is as tall as you? 5. Explain the relationship between the number of layers and the fraction of the shaded region. Glencoe/McGraw-Hill 34 Mathematics: Applications and Concepts, Course 2

41 Statistics: Analyzing Data Teaching Notes and Overview Mini-Project Story Graph (p. 37 of this booklet) Use With Lesson 2-2. Objective Interpret and describe a line graph. none Students interpret a given line graph and describe it by writing a story that fits the data shown. Sample Answer I left my house at 4:5 P.M. to go to the store for Mom. She asked me to buy some milk and tortillas for dinner. I walked 5 minutes before meeting my friend, Lina. We talked for 5 minutes. Then I continued walking to the store at the same pace I had been walking. I arrived at the store 0 minutes later. It took me 0 minutes to buy milk and tortillas. On the way home, I walked a little faster because Mom wanted the groceries by 5 P.M. I made it home in 0 minutes. I was early. markers ruler Students work in groups to collect data, create a frequency table, and create an appropriate graph to display their data. Students are asked to explain why they chose the type of graph they did for their data. Answers See Teacher Wraparound Edition p. 73. Using Overhead Manipulatives Quartiles (pp of this booklet) Use With Lesson 2-6. Objective Graph quartiles and determine the interquartile range. blank transparencies, prepared as described below transparency pens* * = available in Overhead Manipulative Resources Kit Chapter 2 Hands-On Lab Recording Sheet Are You Average? (p. 38 of this booklet) Use With Lesson 2-4b. This corresponds to the activity on page 73 in the Student Edition. Objective Use mean, median, mode, and range to describe a set of data. Students find and graph the upper and lower quartiles and the interquartile range of a given set of data. An Extension activity asks students to find and graph the upper and lower quartiles and the interquartile range of a second set of data, then compare the two sets of data. Answers Answers appear on the teacher demonstration instructions on pages Glencoe/McGraw-Hill 35 Mathematics: Applications and Concepts, Course 2

42 Chapter 2 Statistics: Analyzing Data Using Overhead Manipulatives How Much is a Handful? (p. 4 of this booklet) Use With Lesson 2-8. Objective Use data to make predictions. transparency pens* blank transparencies 40 counters* * = available in Overhead Manipulative Resources Kit Students collect data by tracing their hands on a transparency and counting the number of counters it takes to cover the shape. Based on the data they have gathered, students then make predictions about the number of counters needed to cover the hands of other students and adults. Answers Answers appear on the teacher demonstration instructions on page 4. Glencoe/McGraw-Hill 36 Mathematics: Applications and Concepts, Course 2

43 NAME DATE PERIOD _ Mini-Project (Use with Lesson 2-2) Story Graph Make up a story to fit this graph. The graph describes the distance you are from your house with respect to time. The story should include where you begin, end, and stop along the way. Make sure you also mention how long it takes you to travel from place to place and how long you stay at each place. To help you understand what is happening, you may want to act out the graph. Use seconds instead of minutes and ignore the distance scale Glencoe/McGraw-Hill 37 Mathematics: Applications and Concepts, Course 2

44 NAME DATE PERIOD _ Hands-On Lab Recording Sheet (Use with the activity on page 73 in Lesson 2-4b of the Student Edition) Are You Average? markers, ruler Investigate Work in groups of four. Use the space provided below for your frequency table. Find the mean, median, mode, and range of the data for each question if appropriate. Which measure best describes each set of data? Justify your answer. Create appropriate graphs to display your data from all ten questions. Each group will present their graphs with a description of what the average student in your classroom is like. Writing Math. Explain why results may vary if you survey another class in your school. 2. Explain why your group selected the graph that you did to display your data. Could you have used another type of graph? Explain. 3. The word bias means to influence. Describe any factors that could have unfairly influenced the responses given by your classmates. Is there a way to limit bias in a survey? Glencoe/McGraw-Hill 38 Mathematics: Applications and Concepts, Course 2

45 Quartiles Using Overhead Manipulatives (Use with Lesson 2-6) Objective Graph quartiles and determine the interquartile range. blank transparencies, prepared as described below transparency pens* * = available in Overhead Manipulative Resources Kit Teacher Demonstration Prepare blank transparencies with copies of the tables shown below and in the Extension. Show students the first table. Tell them that the table shows various suspension bridges in the United States and the length of their main span. Tell students that in a large set of data, it is helpful to separate the data into four equal parts called quartiles. Tell them you are going to find and graph the quartiles for these data. Ask students to list the data in order from least to greatest. (533, 549, 564, 60, 655, 655, 70, 704, 853,,067,,58,,280,,298) Write the data in order below the table. Ask students how to find the median of the data. (Find the data number that is in the middle when the data are arranged in order from least to greatest.) Then have them find the median. (70) Circle the median and point out that it separates the data into two equal groups. On the list of data, place an arrow between 564 and 60 and Length of Name, Location between,58 and,067 as Main Span (m) shown. Find their means and tell Ambassador International, Detroit 564 students that these numbers represent the median of the lower Bronx-Whitestone, New York City 70 Benjamin Franklin, Philadelphia 533 half, the lower quartile, and the Delaware Memorial, Wilmington, Delaware 655 median of the upper half, the George Washington, New York City,067 upper quartile, of the data. Golden Gate, San Francisco,280 Mackinac Straits, Michigan,58 Tacoma Narrows II, Tacoma, Washington 853 San Francisco-Oakland Bay, San Francisco 704 Seaway Skyway, Ogdensburg, New York 655 Throgs Neck, New York City 549 Varrazano-Narrows, New York,298 Walt Whitman, Philadelphia 60 Source: Federal Highway Administration Glencoe/McGraw-Hill 39 Mathematics: Applications and Concepts, Course 2

46 Using Overhead Manipulatives ,067,58,280, median,067,58, Point out that half of the data numbers, the middle half, lie between the lower and upper quartiles, 587 and,2.5. Tell them that the range of the middle half of the data is called the interquartile range. Ask students to find the interquartile range of these data. (525.5) Below the data, draw a number line from 500 to,300. Graph the median of the data above the number line. Then show students as you graph the least value, the greatest value, the upper quartile, and the lower quartile above the number line. LQ M 70 UQ,2.5, ,00,300 Point out that the numbers graphed separate the data into four groups. Ask how many of the numbers in the data set fall in each group. (3) Extension Show students the table below. Tell them that the table shows various suspension bridges internationally and the lengths of their main spans. Have students find the median, upper quartile, lower quartile, least value, and greatest value for the data. (,043.5;,377; 988; 668;,990) Then have them graph those values. Finally, compare the number lines for the two sets of data. Ask students what conclusions they can make from the data. (Sample answer: The middle half of the data for international bridges is less spread out. International bridges generally have longer spans than those in the United States.) Name, Location Length of Main Span (m) Akashi Kaiko, Japan,990 First Bosporus, Istanbul, Turkey,074 Forth Road, Queensferry, Scotland,006 Humber, Hull, Britain,40 Kita Bisan-Seto, Japan 990 Minami Bisan-Seto, Japan,00 Ohnaruto, Japan 876 Pierre Laporte, Quebec, Canada 668 Ponte 25 de Abril, Lisbon, Portugal,03 Second Bosporus, Istanbul, Turkey,090 Severn, Beachley, England 988 Shimotsui Straits, Japan 940 Storebelt, Denmark,624 Tsing Ma Bridge, Hong Kong,377 Source: Federal Highway Administration Glencoe/McGraw-Hill 40 Mathematics: Applications and Concepts, Course 2

47 Using Overhead Manipulatives (Use with Lesson 2-8) How Much is a Handful? Objective Use data to make predictions. transparency pens* blank transparencies 40 counters* * = available in Overhead Manipulative Resources Kit Teacher Demonstration Ask four volunteers to trace the outline of their hand on a transparency. Place one of the outlines on the overhead and cover the shape with counters. Count the counters used and record on a fifth transparency. Repeat the procedure for each of the three outlines. Find the mode, median, and mean of the data. Ask the students which of these averages is most representative of the data. Discuss the most effective way to present the data. Alternatives include the frequency table, bar graph, line plot, or stem-and-leaf plot. Use the sixth blank transparency to prepare the data in the agreed-upon manner. Ask students to predict how many counters would be needed to cover a hand outline for a student in the same grade but not in the class. Have students randomly choose four students who are not in your class and trace the outline of their hands on a transparency. Record the counters needed to cover the shape. Have students compare the number of counters with their prediction. Trace your own hand on a transparency and find the number of counters needed to cover the shape. Ask how this number compares with the other data. (In most cases, it will be greater than the average of the student data.) Ask whether you could use the information about the teacher s hand to predict about how many counters would cover other adults hands. (Data from one adult would not be sufficient to predict for other adults.) Glencoe/McGraw-Hill 4 Mathematics: Applications and Concepts, Course 2

48 Algebra: Integers Teaching Notes and Overview Mini-Project Coordinate Plane Puzzle (p. 44 of this booklet) Use With Lesson 3-3. Objective Locate points for ordered pairs on a coordinate plane none Given ordered pairs, students locate the appropriate points labeled on a coordinate plane. Students solve the puzzle by matching each point with the correct ordered pair. Answer WE ARE COORDINATED! Hands-On Lab Recording Sheet Adding Integers (p. 45 of this booklet) Use With Lesson 3-4a. This corresponds to the activity on pages 8 9 in the Student Edition. Objective Use counters to model the addition of integers. counters integer mat Students use counters to add integers. Space is provided for students to explain their work. They will write their own addition sentences and draw conclusions on how to add integers. Answers See Teacher Wraparound Edition pp Hands-On Lab Recording Sheet Subtracting Integers (p. 46 of this booklet) Use With Lesson 3-5a. This corresponds to the activity on pages in the Student Edition. Objective Use counters to model the subtraction of integers. counters integer mat Students use counters to subtract integers. Space is provided for students to explain their work. They will write their own subtraction sentences and draw conclusions on how to subtract integers. Answers See Teacher Wraparound Edition pp Glencoe/McGraw-Hill 42 Mathematics: Applications and Concepts, Course 2

49 Chapter 3 Algebra: Integers Using Overhead Manipulatives Multiplying Integers (pp of this booklet) Use With Lesson 3-6. Objective Multiply integers by using models. counters* integer mat transparency* transparency pen* * = available in Overhead Manipulative Resources Kit This demonstration contains two activities. Demonstration shows how to model the multiplication of two positive integers or a positive and a negative integer. Demonstration 2 shows how to model the multiplication of two negative integers. Extension questions ask students to model and solve integer multiplication problems independently. Chapter 3 Answers Answers appear on the teacher demonstration instructions on pages Glencoe/McGraw-Hill 43 Mathematics: Applications and Concepts, Course 2

50 NAME DATE PERIOD _ Mini-Project (Use with Lesson 3-3) Coordinate Plane Puzzle Use the coordinate plane to find the point for each ordered pair below. Then write the letter for each point under its ordered pair at the bottom of the page. When you have filled in all the letters, you will know what two points on a grid said to each other. -9 y J O 2 H N O R R C I -2 E O E D A J B T E W A D G 9 x (4, 5) ( 5, 6) (5, 6) ( 4, 2) (2, 4) (0, 2) (2, 2) ( 7, 7) (6, 0) ( 4, 6) (5, ) (, 0) ( 2, 4) (2, 7) ( 8, 4) (8, 6) Glencoe/McGraw-Hill 44 Mathematics: Applications and Concepts, Course 2

51 NAME DATE PERIOD _ Hands-On Lab Recording Sheet (Use with the activity on pages 8 9 in Lesson 3-4a of the Student Edition) Adding Integers counters, integer mat Your Turn Use counters to find each sum. a b. 3 ( 5) c. 5 ( 4) d. 7 3 e. 2 ( 5) f. 8 ( 6) g. 6 5 h. 3 ( 6) i. 2 7 j. 8 ( 3) k. 9 l. 4 0 Writing Math. Write two addition sentences where the sum is positive. In each sentence, one addend should be positive and the other negative. 2. Write two addition sentences where the sum is negative. In each sentence, one addend should be positive and the other negative. 3. MAKE A CONJECTURE Write a rule that will help you determine the sign when finding the sum of integers. Glencoe/McGraw-Hill 45 Mathematics: Applications and Concepts, Course 2

52 NAME DATE PERIOD _ Hands-On Lab Recording Sheet (Use with the activity on pages in Lesson 3-5a of the Student Edition) Subtracting Integers counters, integer mat Your Turn Use counters to find each difference. a. 7 6 b. 5 ( 3) c. 6 ( 3) d. 5 8 e. 6 ( 3) f. 7 3 g. 5 ( 7) Writing Math Work with a partner.. Write two subtraction sentences where the difference is positive. Make sure you use a combination of positive and negative integers. 2. Write two subtraction sentences where the difference is negative. Make sure you use a combination of positive and negative integers. 3. MAKE A CONJECTURE Write a rule that will help you determine the sign of the difference of two integers. Glencoe/McGraw-Hill 46 Mathematics: Applications and Concepts, Course 2

53 Using Overhead Manipulatives (Use with Lesson 3-6) Multiplying Integers Objective Multiply integers by using models. counters* integer mat transparency* transparency pen* * = available in Overhead Manipulative Resources Kit Teacher Demonstration for Activity Review with students the meaning of the two colors of counters. If necessary, mark the yellow counters to show they are positive counters and the red counters to show they are negative counters. Remind students that 3 2 means three sets of two items. Tell students that modeling 3 2 means to place 3 sets of 2 positive counters on the mat. Model 3 2. Ask students to complete: 3 2?. (6) Place 3 sets of 2 negative counters on the mat. Ask students to state the multiplication sentence that has been modeled. (3 ( 2) 6) Teacher Demonstration for Activity 2 Clear the mat. Write 3 2 at the base of the mat. Tell students that since 3 is the opposite of 3, 3 2 means to remove 3 sets of 2 positive counters. Place 6 zero pairs on the mat. Ask students to state the value of the mat. (0) Glencoe/McGraw-Hill 47 Mathematics: Applications and Concepts, Course 2

54 Using Overhead Manipulatives Remove 3 sets of 2 positive counters. Ask students to state the value of the mat. ( 6) Ask students to complete the sentence 3 2 =?. ( 6) Clear the mat. Write 3 ( 2) at the base of the mat. Ask students how many zero pairs must be placed on the mat to model 3 ( 2). (6) Place 6 zero pairs on the mat. Ask students whether positive or negative counters should be removed to model 3 ( 2). (negative) Remove 3 sets of 2 red counter. Ask students the value of the mat. (6) Ask students to complete the sentence 3 ( 2) =?. (6) Have students complete Exercises 7. Use counters to find each product. Use your result to write a multiplication sentence (2 3 6) 2. 2 ( 4) (2 ( 4) 8) 3. 2 ( 3) ( 2 ( 3) 6) 4. 3 ( 4) (3 ( 4) 2) (4 0 0) 6. ( 5) ( ( 5) 5) 7. Find 9 ( 3) without using models. ( 27) Extension Ask students to model each multiplication and then write a multiplication sentence. 4 3 (4 3 2) 4 ( 3) ( 4 ( 3) 2) 4 ( 3) (4 ( 3) 2) 4 3 ( 4 3 2) Glencoe/McGraw-Hill 48 Mathematics: Applications and Concepts, Course 2

55 Algebra: Linear Equations and Functions Teaching Notes and Overview Hands-On Lab Recording Sheet Solving Equations Using Models (p. 5 of this booklet) Use With Lesson 4-2a. This corresponds to the activity on pages in the Student Edition. Objective Solve equations using algebra tiles cups and counters equation mat Students use algebra tiles to model and solve addition and subtraction equations. Space is provided for students to explain their answers and draw conclusions. Answers See Teacher Wraparound Edition pp Mini-Project Solving Multiplication Equations (p. 52 of this booklet) Use With Lesson Using Overhead Manipulatives Solving Two-Step Equations (pp of this booklet) Use With Lesson 4-4. Objective Solve two-step equations by using models. Chapter 4 Objective Model and solve multiplication equations. cups and counters equation mat Students use cups and counters to model and solve multiplication equations. Space is provided for students to sketch their models and write their answers. cups* counters* equation mat transparency* * = available in Overhead Manipulative Resources Kit Students solve a two-step equation as it is modeled on the overhead. An Extension activity asks students to explain in writing how the modeling equations uses the work backward strategy. Answers 2. Answers will vary. Answers Answers appear on the teacher demonstration instructions on pages Glencoe/McGraw-Hill 49 Mathematics: Applications and Concepts, Course 2

56 Chapter 4 Algebra: Linear Equations and Functions Hands-On Lab Recording Sheet Functions and Graphs (p. 55 of this booklet) Use With Lesson 4-6a. This corresponds to the activity on page 76 in the Student Edition. Objective Graph a function on a scatter plot. stop watch uncooked spaghetti Students will perform the wave with various numbers of students in the class. They are provided with a graph on which they can record their data. They use the graph to find a pattern, make predictions, and explain their reasoning. Answers See Teacher Wraparound Edition p. 76. Using Overhead Manipulatives A Function of Time (pp of this booklet) Use With Lesson 4-6. Objective Use a function rule to find the output of a function. coordinate grid transparency* transparency pen* transparency prepared with the two tables shown in the demonstration * = available in Overhead Manipulative Resources Kit Students use given data for the input values and the function rule to find the output values of a function. Students then use the graph of the function to make predictions. An Extension activity asks students to give examples of other functions of time. Answers Answers appear on the teacher demonstration instructions on pages Glencoe/McGraw-Hill 50 Mathematics: Applications and Concepts, Course 2

57 NAME DATE PERIOD _ Hands-On Lab Recording Sheet (Use with the activity on pages in Lesson 4-2a of the Student Edition) Solving Equations Using Models cups and counters, equation mat Investigate The scale at the right is balanced, and the bag contains a certain number of blocks.. Suppose you cannot look in the bag. How can you find the number of blocks in the bag? 2. In what way is a balanced scale like an equation? 3. What does it mean to solve an equation? Your Turn Solve each equation using models. a. x 3 b. x 3 7 c. x 4 4 d. x 3 2 e. x 4 f. 2 x g. x 3 2 h. x 3 i. 4 x 2 Writing Math. How is solving an equation similar to keeping a scale in balance. 2. For any equation, how can you determine how many counters to add or subtract from each side? 3. Identify the property of numbers that is illustrated by a zero pair. 4. Identify the property of numbers that allows you to add or subtract zero without changing the value of a number. 5. MAKE A CONJECTURE Write a rule that you can use to solve an equation like x 3 2 without using models. Glencoe/McGraw-Hill 5 Mathematics: Applications and Concepts, Course 2

58 NAME DATE PERIOD _ Mini-Project (Use with Lesson 4-3) Solving Multiplication Equations Write the equation modeled by the cups and counters.. 2. Solve each equation using cups and counters. Sketch the arrangement in the boxes. 3. 4x 2 x 4. 2x 4 x 5. 3x 5 x Solve without using models. 6. 5x 0 x Glencoe/McGraw-Hill 52 Mathematics: Applications and Concepts, Course 2

59 Using Overhead Manipulatives (Use with Lesson 4-4) Solving Two-Step Equations Objective Solve two-step equations by using models. cups* counters* equation mat transparency* * = available in Overhead Manipulative Resources Kit Teacher Demonstration Remind students that, in modeling, red counters represent negative integers, yellow counters represent positive integers, and a cup represents x. Review the use of zero pairs. Ask students what happens to an equation if a zero pair is added to or subtracted from a side. (An equivalent equation results.) Place 3 cups and 2 yellow counters on the left side of an equation mat and place 4 red counters on the right side. Ask students what equation is modeled. (3x 2 4) + + Add 2 red counters to each side of the equation mat. Point out that you do this to create zero pairs on the left side. Remove the zero pairs. Ask students why you can remove the zero pairs. (Their value is 0.) + + Glencoe/McGraw-Hill 53 Mathematics: Applications and Concepts, Course 2

60 Using Overhead Manipulatives Ask students what equation is represented on the mat. (3x 6) Pair up an equal number of counters with each cup. Ask students what the solution of the equation 3x 2 4 is. ( 2) Extension Have students write a few sentences explaining how the modeling demonstrated in this lab uses the work backward strategy presented in Lesson 4-4a. (Sample answer: Modeling the solution to two-step equations uses the reverse of the order of operations.) Glencoe/McGraw-Hill 54 Mathematics: Applications and Concepts, Course 2

61 NAME DATE PERIOD _ Hands-On Lab Recording Sheet (Use with the activity on page 76 in Lesson 4-6a of the Student Edition) Functions and Graphs stop watch, uncooked spaghetti Writing Math Work with a partner.. Graph the ordered pairs (number of students, time) on the coordinate grid at the right. 2. Describe how the points appear on your graph. 3. Place one piece of uncooked spaghetti on your graph so that it covers as many of the points as possible. Predict how long it would take 30 students to complete the wave. Make a prediction for 50 students. Time (seconds) Students 4. Find a pattern in the data and use the pattern to predict how long it would take the students in your school to complete the wave. Explain your reasoning. 5. A function describes the relationship between two quantities. In a function, one quantity depends on the other. Complete the sentence: The time it takes to do the wave depends on?. Glencoe/McGraw-Hill 55 Mathematics: Applications and Concepts, Course 2

62 Using Overhead Manipulatives (Use with Lesson 4-6) A Function of Time Objective Use a function rule to find the output of a function. coordinate grid transparency* transparency pen* transparency prepared with the two tables shown below * = available in Overhead Manipulative Resources Kit Teacher Demonstration Tell students that a certain secretary can type an average of 55 words per minute. Ask students to use this information to complete the table. Minutes Minutes Average Number of Words Total per Minute Words? (55) 2 2? (0) 3 3? (65) 4 4? (220) 5 5? (275) Tell students that in this example the minutes are called the input values, the total words processed are called output values, and the middle column contains the function rule. Graph the ordered pairs (minutes, words) on the coordinate grid transparency. Draw a line passing through the points. Ask students how they could use the graph to estimate the number of words typed in 8 minutes. (Extend the line; 440 words.) Say, Let t be the time in minutes. What is the expression for the total number of words typed in t minutes? (55t) Glencoe/McGraw-Hill 56 Mathematics: Applications and Concepts, Course 2

63 Using Overhead Manipulatives Tell students that another secretary can type an average of 70 words per minute. Repeat the activity using the information below. Minutes Minutes Average Number of Words Total per Minute Words? (70) 2 2? (40) 3 3? (20) 4 4? (280) 5 5? (350) Extension Have students give examples of other functions of time. (Sample answer: revolutions per minute) Glencoe/McGraw-Hill 57 Mathematics: Applications and Concepts, Course 2

64 Fractions, Decimals, and Percents Teaching Notes and Overview Hands-On Lab Recording Sheet Exploring Factors (p. 59 of this booklet) Use With Lesson 5-a. This corresponds to the activity on page 96 in the Student Edition. Objective Discover factors of whole numbers. 5 index cards cut in half markers Students work as a class to determine factors of whole numbers by identifying patterns. Space is provided for students to explain their answers and make predictions. Answers See Teacher Wraparound Edition p. 96. Using Overhead Manipulatives Percent (p. 60 of this booklet) Use With Lesson 5-5. Objective Illustrate the meaning of percent using models. centimeter grid transparency* transparency pens* * = available in Overhead Manipulative Resources Kit Using a model on the centimeter grid transparency, students write a ratio of the number of shaded squares to the total number of squares. Students make conclusions about the meaning of percent based on the model. Answers Answers appear on the teacher demonstration instructions on page 60. Glencoe/McGraw-Hill 58 Mathematics: Applications and Concepts, Course 2

65 NAME DATE PERIOD _ Hands-On Lab Recording Sheet (Use with the activity on page 96 in Lesson 5-a of the Student Edition) Exploring Factors 5 index cards cut in half, markers Writing Math Work as a class.. How many students are standing at the end of the activity? Which cards are they holding? 2. LOOK FOR A PATTERN Suppose there were 00 students holding index cards. Extend the pattern in Exercise to predict the numbers that would be held by students standing at the end of the activity. 3. Explain the relationship between the numbers on the front and the back of the cards. 4. Separate the cards into two groups: one group with exactly two numbers on the back of the card and one group with more than two numbers. Describe any special characteristics of each group. Chapter 5 Glencoe/McGraw-Hill 59 Mathematics: Applications and Concepts, Course 2

66 Percent Using Overhead Manipulatives (Use with Lesson 5-5) Objective Illustrate the meaning of percent using models. centimeter grid transparency* transparency pens* * = available in Overhead Manipulative Resources Kit Teacher Demonstration On the centimeter grid transparency, outline a 0-by-0 square. Using a different colored pen, shade 25 of the squares as shown. Ask, How many small squares are in the model? (00) Ask, How many small squares are shaded? (25) Ask students to write a ratio of shaded squares to squares in the model Tell students that the model represents 25 percent. Ask them to make a conjecture about the meaning of the word percent. (Sample answer: Percent is a ratio comparing a number to 00.) Glencoe/McGraw-Hill 60 Mathematics: Applications and Concepts, Course 2

67 Applying Fractions Teaching Notes and Overview Using Overhead Manipulatives Multiplying Fractions and Mixed Numbers (pp of this booklet) Use With Lesson 6-4. Objective Use models to multiply fractions and mixed numbers. blank transparency ruler* transparency pens* * = available in Overhead Manipulative Resources Kit This demonstration contains three activities. Demonstration shows the multiplication of two fractions. Demonstration 2 shows the multiplication of a whole number and a fraction. Demonstration 3 shows the multiplication of a fraction and a mixed number. Students are asked to find products of fractions and mixed numbers independently. An Extension activity asks students to model the multiplication of two mixed numbers. Answers Answers appear on the teacher demonstration instructions on pages Given several figures, students measure the sides, label them, and then find the perimeter. Answers. P in. 2. P in. 3. P in. 4. P in. 5. P in. 6. P in. Hands-On Lab Recording Sheet Circumference (p. 65 of this booklet) Use With Lesson 6-9a. This corresponds to the activity on page 274 in the Student Edition. Objective Find a relationship between circumference and diameter. ruler measuring tape circular objects Mini-Project Perimeter (p. 64 of this booklet) Use With Lesson 6-8. Objective Measure the sides of figures and find the perimeter. ruler Students find the diameter and circumference of various circular objects and record their measurements in a table. A graph is provided for students to graph their data, find the slope of the line, and discover a relationship between circumference and diameter. Answers See Teacher Wraparound Edition p Chapter 6 Glencoe/McGraw-Hill 6 Mathematics: Applications and Concepts, Course 2

68 Using Overhead Manipulatives (Use with Lesson 6-4) Multiplying Fractions and Mixed Numbers Objective Use models to multiply fractions and mixed numbers. blank transparency ruler* transparency pens* * = available in Overhead Manipulative Resources Kit Teacher Demonstration for Activity Use the ruler and a black transparency pen to draw a square like the one shown. Divide the square vertically into halves and horizontally into fourths. Tell students you are going to model 2 4. Color one half of the square blue. Color one fourth of the square red. Count the small rectangles that are shaded both colors. Ask students what number is represented. 8 Write 2 4 below the model. 8 Teacher Demonstration for Activity 2 Use the ruler and a black transparency pen to draw 2 squares side-by-side. Divide them horizontally into fifths. Tell students you are going to model Color the 2 squares blue. Color three-fifths of the squares red. Count the small rectangles that are shaded both colors. Point out that 6 small rectangles are shaded and each has an area of 5 of a unit square. Ask students to add the areas. 6 5 or 5 Write or below the 5 model. Blue Red Purple Red Purple Blue Glencoe/McGraw-Hill 62 Mathematics: Applications and Concepts, Course 2