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Grade 8 Classroom Strategies 1

Grade Eight

Classroom

Strategies

Book 2Book 2Book 2Book 2Book 2


Grade 8 Classroom Strategies2



The learner will understand andcompute with real numbers.

1.01 Identify subsets of the real numbersystem.

A. Transparency for number set structure (Blackline Master I - 1)Teachers may use this transparency to help students understand which realnumber sets are subsets of each other and which numbers belong to eachsubset.

B Real Number Race (Blackline Masters I - 2 through I - 3)Materials: Game board, spinner, one marker per student (up to 6 players)Each player chooses a side of the board from which to start. On eachplayer’s first turn, he will spin the spinner and get a number set. He thenmoves his marker to any circle on his side of the board that contains anumber from that set. Play continues with the next player. Once the playeris on the board, on his next turn he can move only to a circle adjacent to hisposition that contains a number from the number set he spins. Players maynot occupy the same space at the same time. Each player must pass throughthe zero ring in the center of the board as he moves across the board. If aplayer moves to an incorrect circle, the opponents may challenge him; awrong move has a penalty of being moved back on the board. If the playerhasn’t passed the zero ring, he is moved back to his starting position. If hehas passed through the zero ring, then he is moved to one of the circlessurrounding the zero ring. The winner is the first player to get across theboard to the opposite side.Note: Instead of using the spinner provided, students may roll a fair numbercube (each number on the cube would correspond to a section of thespinner). Additional notes are given on the student spinner page to helpstudents remember which types of numbers belong to each set.

C Subsets of Real Numbers Triangle Puzzle (BlacklineMaster I - 4) Students cut out the triangle puzzle pieces and reassemblethem so that touching edges match a number to its equivalent expression.Students can self-check by creating the shape on the puzzle sheet. This is acooperative activity which encourages students to use their mathematicsvocabulary.

1111111111Notes and textbook

references



1.02 Estimate and compute with rationalnumbers.

A. While introducing integers, discuss pairs of words in whichdirections might be useful, such as right/left, hotter/colder, east/west, north/south, spend/save, loss/gain, win/lose, before/after, deposit/withdraw, increase/decrease, positive/negative, above/below. Use the number line to show that additive inverses are the same distancefrom zero but opposite in direction. Indicate that negative numbers can beuseful in modeling real world situations in which quantities with oppositedirections are involved. Also discuss neutral words such as break even, tie, neutral. Relate theseto zero and the fact that zero is neither positive nor negative.

B Rational Number Review Triangle Puzzle (BlacklineMaster I - 5) Students should complete this puzzle in small groups. At thebeginning of the puzzle, each student should have some pieces of the puzzlein his/her possession. Students can share mental math strategies as theywork together to complete the puzzle. Students can self-check by creatingthe shape on the puzzle sheet.

C. Modeling Signed Numbers with Heaps and Holes Thisactivity is based on a few lines from the movie, Stand and Deliver. In themovie Jaime Escalante is trying to get his students to understand hownegative numbers work by filling in holes in the sand. Explain to yourstudents that +1 is like a pile (or heap) of sand on a level beach. A hole ofequivalent size dug into the beach represents -1. This model explainspositives as a surplus and negatives as a deficit.

First, convince your students that there are many ways to model zero. Someare shown below.

1 -1



Also show them various ways to model other numbers such as (+1) and (-1).

The following illustration shows the addition of 4 + (-2). To simplifythe drawing, a flat line can represent zero. Half-ovals above the linerepresent positive numbers (heaps) and half -ovals below the line representnegatives (holes). Here is the line drawing for 4 + (-2).

When the bottom and top ovals are lined up, a positive and a negativeform something that looks like zero, and the result is displayed more clearly.

1 1 1

-1 -1 -1

4 + (-2) = 2



More addition problems using heaps and holes diagrams:

In subtracting with heaps and holes, it is convenient to think ofsubtraction as take away. 3 - 4 means 3 with 4 taken away. 1- (-2) means -2 taken away from 1. -3 - (-5) means -5 taken away from -3. -2 - 7 means 7 taken away from -2.To do the problem “3 - 4,” start with 3.

-5 + 3

-5 + 3 = -2

-4 + (-1)

-4 + (-1) = -5

4 + (-3) 4 + (-3) = 1



There aren’t enough heaps to take away 4, so we remedy the situation byadding a zero. The total hasn’t changed since we added a (+1) and a (-1),but now we can take away four.

More subtraction examples:

-2 -1 Start with -2.

There is no positive one to subtract, so we add a Heap and a Hole pair (a zero).

Now subtract 1.

-2 –1 = -3-2 - 1 = -3

-2 - 1

a heap and a hole pair (a zero).

3 – 4 = -1



3 - (-2) Start with 3

3 - (-2) = 5

There is no (-2) to subtract so we add 2 heaps and 2 holes (2 zeros).

Now subtract (-2).

-5 - (-3) is read “Negative 5 subtract negative 3.”

-5 -5We have enough to take away (–3).

-5 - (-3) = -2



Multiplying with heaps and holes line notation is easier if you think ofmultiplying as repeated addition if the first factor is positive or repeatedsubtraction if the first factor is negative. 3 x (-2) means add in 3 sets of negative 2. -2 x 3 means take away two sets of 3. -2 x (-3) means take away two sets of negative 3.Examples:

(-4).

( )

(-4)3 x -4 means add in 3 sets of -4.

3 x –4 = -12

-2 x 3 means take away 2 sets of 3. If we start with 0, there is noway to take away anything. But we can add additional symbolsthat still represent 0. Zero may be added to any number withoutchanging the total.

Now we can take away 2 sets of 3.

-2 x 3 = -6

3 x (-4) means add in 3 sets of (-4).

3 x (-4) = -12



D. Students who have trouble remembering the algorithm foradding fractions can be shown how to do that with diagrams. To use thistechnique, students should realize that to understand a fraction, one mustknow what the “whole” is. Is it one pizza, one rectangle, one circle, or onecandy bar? Also, the student should know how to simplify fractions.

Example:

These fractions can be added easily if we have a convenient diagram of thewhole. If graph paper is available, use a rectangle that is 3 x 5. If no graphpaper is available, make a dot matrix that is 3 dots wide and 5 dots long.Either of these can easily be divided into thirds or fifths. This model willrepresent the “whole” or “one.” Point out that in each model, one cell ordot is equal to .

35

13

+

1 15

.

-2 x (-4) = 8

Now we can take away 2 sets of (-4).

-2 x -4 means take away 2 sets of -4. If we start with 0, there is noway to take away anything. But we can add additional symbolsthat still represent 0. Zero may be added to any number withoutchanging the total.

Now we can take away 2 sets of -4.

-2 x –4 = 8

-2 x (-4) means take away 2 sets of (-4). If we start with 0, there is no



Note that this technique does not always give the answer in lowest terms.

E. Building Rectangles from Cubes (Blackline Master I - 6)Materials: 8-12 color cubes of each color (green, blue, red, yellow) foreach group. If you do not have color cubes in your classroom, studentsmay use grid paper and colored pencils to draw the rectangles.Students should work in groups of two or three. These tasks help students with their understanding of fractions such asthe concept that the same fraction can have different names and thenecessity for a common denominator when adding fractions.

F. Rational Math Bingo (Blackline Masters I - 7 through I -11) Each student is asked to make a bingo card with numbers asspecified on the card. Each column must contain different numbers fromthe indicated range, but the numbers can be placed in the column in anyorder. It is recommended that students work in pairs to discuss and checkwith each other. When the game is played, the teacher will put questions on the overheadprojector. Students work the problems mentally and look for the answer ontheir cards. The first student (or pair) to complete a line is the winner.

3/3 5/5

3/5 + 1/3 3/5 + 1/39/15 + 5/15 = 14/15 9/15 + 5/15 = 14/15

33

55

3 5

1 3

515

+

+

915

3 5

+

+ == 1415

1 3

9 15

5 15

1415



G. Four in a Row (Blackline Master I - 12) The class isdivided into two teams. To start play, the teacher puts an algebraicexpression on the overhead. On a team’s turn, they will give coordinates fora point they wish to capture. That point is circled. If the team can thengive the correct answer for substituting the coordinates into the expression,the team captures that point, and the circle is filled in with the team’s color.If the team gives an incorrect solution, the opposing team gets to try to fill itin. Teams alternate playing until one team has captured four points in a roweither horizontally, diagonally, or vertically. If the leader wishes to direct students toward negative numbers, he/shemay circle a point in the 2nd, 3rd, or 4th quadrant that may be used by eitherteam as a free spot. Each round of the game lasts only a few minutes, thusmaking this game an excellent time filler. You may wish to play severalrounds with your students to determine a winner.

1.03 Compare, order, and convert amongfractions, decimals (terminating and non-terminating), and percents.

A. Patterns for Repeating Decimals (Blackline Master I - 13)This worksheet enables students to discover for themselves how somerepeating decimals can be changed into rational numbers.

B. Show students an algebraic way to convert repeatingdecimals to ratios.Example 1: x = 0.44444…

Use the equations: 10x = 4.44444… x = 0.44444…and now subtract 9x = 4 x =

Example 2: x = 0.233333….

Use equations: 100x = 23.33333… 10x = 2.33333…and now subtract 90x = 21 x = =

By the 8th gradestudents have beenexposed to all therational arithmeticoperations. Push themto underscore theirunderstanding throughactivities that requiremental math.

49

2190

730



C. Play fraction card games. (Blackline Masters I-57 throughI-63) Pairs of students can play Concentration. Deal all cards face down infive rows of 14. Players take turns turning over two cards at a time. If thefractions are equivalent, the student keeps the pair. The winner is the personwith the most cards when all have been taken. Play Go Fishing. Deal five cards to each player. Stack the remainderface down in the middle of the table. The object is to get books of twoequivalent fractions. At each turn players may ask others in the group for acertain fraction. As long as someone gives the person a card, the player maykeep asking. When no one has an equivalent fraction to give the player, theperson “goes fishing” by drawing from the deck. At the end of the game,the player with the most books wins. Adapt other card games to your equivalent fraction deck.

1.04 Solve problems involving percent ofincrease and percent of decrease.

A. Thousand-Mile Race (Blackline Masters I - 14 throughI - 20)Materials: A transparency of the playing mat, transparencies of the gamecards which have been cut apart and placed in a paper bagObject of the game: Be the first team to reach exactly 1,000 milesCards: The deck has mile cards worth 50, 100, 150 or 200 miles. There arealso GO, STOP, and CHASE cards. GO cards start the teams rolling afterthey have been stopped or put in a chase situation. STOP and CHASE cardsare played by one team against an opponent to slow down their trip.Directions: Divide the class into three teams. Place 3 cards on the playingmat. On a team’s turn, they may choose one of the cards displayed on themat as cards in play. If they choose a mile card, the team must give thecorrect answer in order to gain the mile points. The points are recorded inthe top rectangle on the mat. If a STOP or CHASE card is showing, theteam may choose to use one of these to play against an opposing team. If ateam has a STOP card played on it, they may not gain more points until theyfind a GO card to get them rolling again. If a team has a CHASE cardplayed against it, they may only use the 150 or 200 mile point cards untilthey find a GO card to remedy the CHASE situation. If team 1 decides toplay a hazard card on team 2, the hazard card is displayed on the playingmat in the lowest rectangle under the corresponding team. It is removedwhen a GO card is used. If a GO card is chosen on a team’s turn, they mayimmediately play it to remove a hazard, or they may stockpile it to use at alater time. If none of the three cards can be used by a team, then the teamhas to pass its turn with no play made. After a team has chosen a card toplay, the leader removes that card and replaces it with a new one drawn fromthe bag so that each team playing has three cards to choose from.



B. Four’s A Winner (Blackline Master I - 21)Materials: Gameboard, two paper clips, two different colored sets ofmarkersNumber of players: Two players or two teamsDirections: Player one places one paper clip on a percent expression, thesecond paper clip on a number, and a marker on the correct answer to thepercent change of the number. The second player moves one paper clip onlyand places a marker on the corresponding correct answer. Play continues inthis manner. The winner of the game is the first player to get four in a rowvertically, horizontally, or diagonally.Note: The gameboard could be put on a transparency and this could be usedby two teams of students.

1.05 Use scientific notation to express largenumbers and numbers less than one. Writein standard form numbers given in scientificnotation.

A. Scientific Notation Square Puzzle (Blackline Master I - 22)Students work in groups to rearrange the small squares back into a largesquare. Two touching edges must contain equivalent expressions. Note: Itwould be best to cut out the small squares and place them in an envelopebefore giving the puzzle to the students as the blackline gives the “answer”.This puzzle should be worked by pairs or small groups of students. Eachgroup member should be in possession of some of the puzzle pieces at thestart of the activity.

B. Scientific Notation Team Game (Blackline Master I - 23)Materials: Transparency or laminated sheet of the playing mat. Two colorsof dry erase markers or two objects with different shapes are used to markthe position of each team. A large paper clip is needed for the spinner.Directions: Divide the class into two teams, or let students play against eachother in teams. The leader begins the game by writing a number in scientificnotation in the top rectangle on the board. On a team’s turn, they spin andchange the number according to the instructions on the spinner. If they arecorrect, the team advances one square, and the number in play is changed tothe number the team just constructed. If they are incorrect, the number inplay remains the same and the team is moved backwards one square. Thewinner is the first team to reach the finish.



1.06 Use rules of exponents.

A. Rules of Exponents Triangle Puzzle (Blackline MasterI - 24) Let students work in groups to put the puzzle together. Each pair oftouching edges should show equivalent expressions. When the puzzle iscompleted correctly it will be in the shape shown in miniature on the page.

B. Power Bingo (Blackline Masters I - 25 through I - 30)Materials: A bingo board for each pair of players; questions copied ontotransparency film, cut apart, and placed in a paper bagDirections: Before play begins, students are to work in pairs to create/complete a bingo card. They are to add integer exponents (-5 through 5) toeach of the indicated base numbers. The same exponent should not berepeated in a given column, and the exponents may be used in any order in agiven column. When the game begins, the leader displays the questions onthe overhead one at a time. Students mark off answers to the questions if theanswer is on their card. The first pair of students to get a line (horizontally,vertically or diagonally) filled in wins.

C. Exponent Experts Game (Blackline Masters I - 31through I - 32)Materials: Each group needs a spinner and a set of cards that has been cutapart.Directions: Students play in groups of two to four students. The cards areshuffled and distributed among the students. On a player’s turn, he spins thespinner and gives the answer that results when substituting the spun numberfor the variable in the expression on one of the cards. One point is awardedfor each correct answer. At the end of the game, individual points and teampoints are totaled.



1.07 Estimate the square root of a numberbetween two consecutive integers; using acalculator, find the square root of a numberto the nearest tenth.

A. Students may estimate square roots to the nearest tenth by thefollowing technique.Example: Find 27We know that 27 falls between 5 and 6 because 27 falls between 25 and36. The difference between 25 and 36 is 11. The difference between 25 and27 is 2. The fraction 2/11 is a good approximation for the distance between 5and 27 . Since 2/11 is approximately 0.2, 5.2 is a good approximation for

27 . A more precise value is 5.196, but 5.2 is correct to the nearest tenth.

1.08 Solve problems involving exponents andscientific notation.

A. Cooperative Problem-Solving Cards - Exponents (BlacklineMasters I - 33 through I - 34) Let students work in groups of four to solvethese two problems. Give each person in the group one of the cards. Thestudents may share the information on the cards with the group, but theycannot give the card to anyone else. This gives each student something tocontribute to the group, and each student gets an opportunity to observe thethought processes of his peers.

B. A professor once promised his students an A if they could folda piece of paper in half seven times (each time doubling the thickness of thepaper). No student ever got an A that way. Why not? What would make thepaper folding easier? He then followed up with this question, “If you had asheet of paper as big as you needed, only 0.001” thick, and you had all thehelp you need, could you fold the paper in half 50 times? How high wouldthe stack be? NOTE: A good problem to add to this is to find the thickness ofa sheet of regular copy paper. It might help to look at a pack of 500 sheets.

C. How thick is a sheet of toilet paper? How can you find out?



D. In an episode of The Lucy Show, Lucy needs $5000 from herbanker to buy new furniture. He tells her that if she can start with a pennyon day 1, and then double the amount she has each following day, before amonth is over she will have enough money. How many days would itactually take her to get at least $5000? Lucy finds out that a bean company is offering “double your moneyback” if the beans are not the best the buyer ever tasted. Lucy knows herGrandmother’s baked beans are the best ever, so she sees this as a chance todouble her money. She starts buying beans and then returning them fordouble the money back. She uses that money to buy twice as many cans asthe day before. She plans to continue buying more cans and returning themfor double the amount until she has enough to buy the furniture. If the beanscost $0.50 a can, and she makes one buy-and-return transaction per day, howmany days would it take her to have enough for her furniture? Note: Lucyfinally tastes the beans and decides she can’t accept the money. However,the bean company owner decides to pay her for her testimonial, so there is ahappy ending for all.

E. The story is told that the King of Persia was so thrilled withthe game of chess that he offered the creator of the game anything hewished. The proud chessman asked for something seemingly simple. Heasked for one grain of rice to be placed on the first square of a chessboard,twice as much on the 2nd square, twice as much again on the 3rd square andso on until all 64 squares had been filled with each square having twice asmuch as the one before. The king was puzzled, but decided to grant the request. However, thisturns out to be enough rice to cover the country of Persia with a blanket ofrice one meter thick (or the state of California with a blanket of rice 1 footthick). We are not told what reward the chessman finally received.

F. The Last Digit (Blackline Master I - 35) Students can findthe last digits of these problems through number sense and looking atpatterns. If they have trouble figuring out the last digit of 31000, have themlook at the first several powers of 3. They should observe a pattern.



G. The Towers of Hanoi (Blackline Masters I - 36 through I -38) There is an ancient legend that in the great tower of Hanoi there arethree diamond spindles. On the middle one there is a stack of 64 disks ofdifferent sizes, each one smaller than the one below it. Monks in the templehave the task of moving the disks from one spindle to another, but they canmove only one disk at a time, and they can never place a larger disk on topof a smaller one. The legend says that when this task is complete, thetemple will disappear in a clap of thunder and the world will end. If themonks are very efficient and move these disks in the quickest way possiblewith each move lasting only one second, how long do we have until theworld ends? Models of such towers with seven disks can be purchased or made fromwooden blocks, nails, and washers. Computer graphics are also useful touse in solving the problem. A suggested strategy is to start with a smallernumber of disks and find the smallest number of moves to transfer all thedisks. Gradually increase the number of disks in the puzzle and look for apattern. The solution is 264 - 1 moves. If each move takes a second, this is well

over 500 billion years.

1.09 Determine the absolute value of anumber.

A. Absolute Value Triangle Puzzle (Blackline Master I - 39)Let students work in groups to put the puzzle together. Each pair oftouching edges should show equivalent expressions. When the puzzle iscompleted correctly it will be in the shape shown in miniature on the page.

1.10 Identify, explain and apply thecommutative, associative and distributiveproperties, inverses, and identities inalgebraic expressions.

A. Mental Math Using Properties (Blackline Master I - 40through I - 46) Make transparencies of each of these sheets. Tape paper onthe back of the transparency to cover up the answer and property side. Use

Sometimes a smallchange has a big payoff!Find activities that getyour students usingtheir math vocabulary.This increases theircomprehension,retention, and ability toread those terms.



other paper to mask all but the question in play. Divide the class into two or more teams. On a team’s turn, display thetop line only of one of the problems. The team’s task is to give the answerusing only mental math. If they need help, show the second line of theproblem. A team scores two points if they answer the problem without ahint, one point if they answer it with the hint showing. If you wish, you canalso add a third point if the student is able to tell you the property illustratedthat allows changing the first line to the second line. Note: In addition tosimplifying expressions, additional properties may be required to obtain thegiven answers.

B. Alien Math (Blackline Master I - 47) Allow students toexplore the addition and multiplication tables to answer the questions on theworksheet. NOTE: The math used in this example is Modulus 5 arithmetic.

C. Matching Game (Blackline Masters I - 48 through I - 51)Materials: Each group needs a deck of cards.Directions: The dealer shuffles the deck and distributes eight cards to eachplayer. The remaining cards are placed face down in a draw pile. The topcard is turned over and placed beside the draw pile to start a discard pile.On a player’s turn, he may choose either the top discard or draw a card. Hethen discards one card into the discard pile. Play moves around the table.The game is over when a player can display two complete sets of matchingcards. A matching set contains three cards, one card with a property statedand two cards with illustrations of that property.

D. When explaining the commutative, associative, anddistributive properties, give examples of other uses of the words. Forexample, a “commuter” goes back and forth to work. “Commuting” ismoving from one place to another. Whom you “associate” with is the sameas saying who is in your group. When a teacher “distributes” papers, shegives one to each member of the class. The paperboy “distributes” a paperto every house on his route.



Tips for Problem-Solving in Your Class

• Set the expectation that everyone thinks! State a problem and then giveeveryone a moment to think about it.

• Use think-pair-share to jumpstart your students’ problem-solving processes.First they think over the question, then they talk it over in pairs, then eachpair shares with a larger group.

• Don’t let textbooks or other published supplementary materials thwart theproblem-solving process. Be wary of texts that give many drill problemswith one word problem that is solved the same way as the previous problems.

Also watch out for problem sets that are all basically identical.

• Incorporate group problem-solving into your lessons, so students have achance to observe their peers.

• Use problems from a variety of sources. Ask questions in a variety of ways.

• Ask a variety of questions from the same problem source data. Studentsbegin to anticipate what a question will be without having really read theproblem. Keep them flexible in their expectations.

• Expose students to problems in which the numbers they read in the problemare not necessarily the ones they will “crunch” to solve the problem. Useprice lists, menus and other materials so that students will search outmeaning and not just begin to crunch numbers.

1.11 Simplify algebraic expressions.

A. Heaps and Holes II (Blackline Masters I - 52 through I - 54) These sheets walk students through simplifying algebraicexpressions through the use of diagrams. Some of the examples touch onthe additive identity, multiplicative identity, and the distributive property.

B. Algebraic Expressions Square Puzzle (Blackline MasterI - 55) Students work to create a large square from the 16 small squares.Touching edges should contain equivalent expressions. Note: It would bebest to cut out the small squares and place them in an envelope beforegiving the puzzle to the students, as the blackline gives the “answer”.



1.12 Analyze problems to determine if thereis sufficient or extraneous data, selectappropriate strategies, and use anorganized approach to solve usingcalculators when appropriate.

A. Bottom Line Cards (Blackline Master I - 56) Teams offour play against each other. In one round, each team turns the cards in anydirection they choose to make a problem. The problem is created byreading the bottom line of each card. After the teams have decided on theproblem, each team solves the other team’s problem. A team gets 2 pointsfor solving a problem and 1 point for stumping the other team. Have students make their own bottom-line cards.

B. Have students draw a picture of a word problem. After it isdrawn, they must identify every number in the problem as it exists in thedrawing. For example, prices will go on price tags, distances can be shownalongside roads, etc. Once a student has developed this much understandingof a problem, chances are he will be much better prepared to solve it.

C. Divide your class into groups of four and give a group ofstudents a menu, catalog, train schedule, postage chart, payroll chart, etc.Have them write five questions from the information given. Tell them towrite questions that other groups might not be able to answer. When thequestions are written, have groups exchange problem sets. Each group earnsone point for solving a question correctly, and two points for writing aquestion that stumps the other group



Review Activities

A. I Have Who Has (Blackline Masters I-64 through I-66)Distribute the cards among your students so that each student has one ormore of the cards. Keep one card for yourself. Begin the game by readingyour card. When you ask, “Who has...”, the person with the answer will readhis card and so on until the question comes back to your card. This activityreviews many of the concepts from the Number Sense, Numeration, andNumerical Operations strand.

B. Miscellaneous Review (Blackline Masters I-67 through I-72)These sheets include a review of rational number operations, algebraicexpressions, exponents, and scientific notation.

Documents

Grade Eight Classroom Strategies - NCWiseOwlmath.ncwiseowl.org/UserFiles/Servers/Server_4507209/File...Grade Eight Classroom Strategies Book 2. Notes and textbook references ... Use