Jossey-Bass Teacher - Math Wise Phần 10 ppsx

Students will learn to mentally manipulate number sums inorder to determine possible game board solutions.. Ask students what scores are possible when, for example, 3 darts are thrown at

3 1/2-Phase M¨obius Strip: (a) The line will meet itself; there is just

1 line, and therefore this M¨obius strip has only side (b) Cuttingdown the middle results in a single ‘‘new’’ strip twice as long and1/2 as wide; further, the new strip is non-M¨obius (c) When cutdown the middle, the new non-M¨obius strip splits into 2 stripsthat are linked together

4 1/3-Phase M¨obius Strip: (a) The continuous line will miss itself on

the ‘‘first pass,’’ which is when you have gone all the way aroundthe paper But it will meet itself on the ‘‘second pass.’’ (b) When cut1/3 of the way in, the result will be a small, ‘‘fat’’ loop interlinkedwith a longer, ‘‘narrow’’ loop The narrow loop is non-M¨obius andthe fat loop is M¨obius Further, the fat loop is the center of theoriginal M¨obius strip, and the narrow one is its outside edge

5 Extension 1: M¨obius strips are in common use as conveyor (and

other) belts, because they will, theoretically, last twice as long asregular belts The reasoning for this is that the wear is distributedevenly to all portions of a M¨obius belt, whereas a regular beltwears only on one side

This project allows students to explore regular tessellations

and then create M C Escher–type tessellations of their own

M C Escher, a Dutch artist who lived from 1898 to 1972,

created drawings of interlocking geometric patterns (or

tes-sellations)

You Will Need:

locations Such everyday tessellations are most often composed ofregular polygons, including squares, triangles, and hexagons (forexample, ceramic tile patterns on bathroom floors, brick walls, orchain-link fences).

2 Next, explore some examples of the Escher-type tessellations and

tell the students that they will be learning the logical proceduresfor developing similar tessellations of their own When it is time

to construct the tessellations, it is suggested that the class worktogether as they create their first tessellations; that is, the students,even though they will probably use different designs, shouldcomplete together the steps outlined in the Example

3 After students have completed their first tessellations, engage them

in a discussion of the ‘‘motion’’ geometry they accomplished—inthis instance, the cutting out of segments and the subsequent

‘‘slide’’ motion to move these to their new locations (In otherinstances of motion geometry, such cut-out segments might be

‘‘flipped,’’ ‘‘turned,’’ ‘‘stretched,’’ or ‘‘shrunk.’’) This discussion,involving the logic of creating tessellations, should include such

par-ticipants to try out some of their ideas as they attempt the creation

of more tessellations

Step 1: Label your tagboard square

with the vertices A, B, C, and D as

shown above

Step 2: Draw a continuous line that

connects vertex B with vertex Cand cut along that line to get acut-out piece

Step 3: Slide the cut cut-out piece

around to the opposite side, place

the straight edge BC against AD,

and tape them together

Step 4: Draw a continuous line that

connects vertex D with vertex Cand cut along that line

1 The students might create tessellations to depict holidays or other

important events

2 Participants can create tessellation book covers; laminate or protect

them with clear, self-stick vinyl; and mount them on their personalbooks or schoolbooks

3 As a class, create a large tessellation (beginning perhaps with a

2-1/2-foot piece of cardboard) and cover an entire wall

Students will enhance their logical-thinking and mental-math

skills while enjoying some mathematical tricks

You Will Need:

Collect a series of Problem Puzzlers (see samples here) You

may wish to duplicate some of the selections for individual

or small group use

3 Some months have 30 days, some 31; how many have 28?

4 If your doctor gave you three pills and said to take one every

half hour, how long would they last?

5 There are two U.S coins that total 55¢ One of the coins is

not a nickel What are the two coins?

6 A farmer had 17 sheep All but 9 died How many does the

farmer have left?

7 Divide 30 by one-half and add 10 What is the answer?

8 How much dirt may be removed from a hole that is 3 feet

deep, 2 feet wide, and 2 feet long?

9 There are 12 one-cent stamps in a dozen, but how many

two-cent stamps are in a dozen?

10 Do they have a fourth of July in England?

11 A ribbon is 30 inches long If you cut it with a pair of scissors

into one-inch strips, how many snips would it take?

12 How long would it take a train one-mile long to pass

com-pletely through a mile-long tunnel if the train was going

60 miles per hour?

B Some other Problem Puzzlers that are lengthier or that require

pencil-and-paper computations are cited below Their solutionsare given at the end of the activity

1 Suppose you have a 9- by 12-foot carpet with a 1- by 8-foot

hole in the center, as shown in the drawing Can you cut thecarpet into two pieces so they will fit together to make a 10- by10- foot carpet with no hole?

When her paws would reach the top.

3 Horse trading: There was a sheik in Arabia who had three sons.

Upon his death, and the reading of the will, there came aboutthis problem He had 17 horses One-half (1/2) of the horses arewilled to his first son One-third (1/3) are willed to his secondson, and one-ninth (1/9) are willed to his third son How manyhorses will each son receive?

4 Rivers to cross: There is an old story about a man who had a

goat, a wolf, and a basket of cabbage Of course, he could notleave the wolf alone with the goat, for the wolf would kill thegoat And he could not leave the goat alone with the cabbage,for the goat would eat the cabbage

In his travels the man came to a narrow footbridge, which

he had to cross He could take only one thing at a time acrossthe bridge How did he get the goat, the wolf, and the basket

of cabbage across the stream safely?

5 Jars to fill: Mary was sent to the store to buy 2 gallons of vinegar.

The storekeeper had a large barrel of vinegar, but he did nothave any empty 2-gallon bottles Looking around, he found an8-gallon jar and a 5-gallon jar With these 2 jars he was able tomeasure out exactly 2 gallons of vinegar for Mary How?

11 Twenty-nine snips The last two inches are divided by one snip.

12 Two minutes From the time the front of the train enters the

tunnel to the time the back of the train leaves the tunnel,the train must travel two miles At 60 miles per hour, the train

is going a mile a minute

B Answers to the longer Problem Puzzlers that often require

pencil-and-paper computations

1 The original carpet might be cut as shown below Then slide

the top portion to the left 1 foot and down 2 feet The resultwill be a 10- by 10-foot carpet that can be sewn together orglued down

2´

1´

10´

10´

2 Johnson’s cat: Each day, the cat went up 11 feet and came

down 7 So she moved up 4 feet per day In 13 days the cat

the top, because 52 + 11 = 63

3 Horse trading:

4 Rivers to cross: Takes goat across; returns Takes wolf across;

brings back goat Takes cabbage across; returns Takes goatacross

5 Jars to fill: Call 8-gallon jar A and 5 gallon jar B Fill B; empty

B into A Fill B Fill A from B There are 2 gallons left in B

6 A vanishing dollar:

Students will learn to mentally manipulate number sums in

order to determine possible game board solutions

You Will Need:

Dartboards (a reproducible page of dartboards is provided at

the end of activity, or students can sketch them) with several

different number scoring patterns are required, as well as

pencils

How To Do It:

Place a dartboard in front of the classroom (or sketch one

on the chalkboard or overhead projector) and play a

prac-tice game Ask students what scores are possible when, for

example, 3 darts are thrown at the target shown here They

should start by making a list of possible number combinations

when throwing 3 darts and then add up the different

combi-nations to create a list of possible scores The students should

then find the greatest and least scores to see the range of

pos-sible scores Finally, have the students discuss what scores

within the range cannot be achieved by tossing 3 darts

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Next have the students consider the same target again, but this timewith 4 darts They may work in cooperative groups or individually todetermine the new possibilities; when they are finished, discuss theirfindings as a class Students should also consider the possible outcomesfor 2 darts, 5 darts, and 6 darts This could lead to a discussion of patternsfound for even or odd numbers of dart.

Example:

These students are considering the feasible scores for 5 darts using the

4+ 4

1 Have students use the dartboard shown below to answer the

following questions, assuming that 6 darts are thrown unlessotherwise noted: (1) Which of these scores are possible: 4, 19,

28, 58, 29, and 35? (2) What is the range of possible scores (theleast and greatest)? (3) List all possible scores and demonstrate

achievable scores for 6 darts? Why? (5) If 5 darts instead of 6 werethrown, what scores would be possible and why?

2 Have the students design their own targets and specify the

num-ber of darts to be thrown Allow them to try out the proposeddartboards on one another, but also have discussions about theanswers to such questions as those asked in Extension 1

Chapter 103

Angelica’s Bean Logic

Students will enhance their logical-thinking skills as they look

for solution patterns

You Will Need:

About 30 beans per player (or group) are needed, plus pencils

and paper

How To Do It:

Read the ‘‘Angelica’s Bean Logic’’ problem to the group,

provide beans, and allow the class to work together or

individ-ually to seek solutions When a solution is found, it should be

displayed (perhaps with an overhead projector) and recorded

The solver or solvers should share their solution procedures

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Angelica’s grandfather always liked to tell her stories or haveher try to solve puzzles One day he said, ‘‘If you will count out

24 beans for me, I will show you some interesting tricks I’m going

to arrange the beans around a square with 3 in each group [seebelow] That way you have 9 beans along each side of the square.Now what you must do is take 4 beans away and rearrange the rest

so that you will still have 9 beans along each side of the square Let

me see if you can do it!’’

Have students attempt the problems that follow

1 Place the 24 beans around the square in groups of 3, as was done

originally This time, however, add 4 beans and still show 9 beansalong each edge See the original configuration below

3 3 3

3 3

3 3 3

2 Try bean-square arrangements with 12 or 15 beans along each

edge and see what patterns are possible

5 5 5

5 5

5 5 5

4

12 beans per edge 15 beans per edge

4 4

4 4

4 4 4

Solutions:

1 Angelica’s Bean Logic problem (show 9 beans along each edge after

removing 4):

1 7 1

1 1

1 1 7

1 6 2

1 1

1 2 6

2 Extension 1 (add 4 beans and still show 9 on each edge):

5 2 2

5 5

5 2 2

5 1 3

5 5

5 3 1

Students will use their problem-solving and logical reasoning

skills to investigate the intersection points of lines in a plane

You Will Need:

Students will require ‘‘Line It Out’’ record-keeping sheets

considered as part of an answer.) Suggested solutions to the worksheetare provided at the end of the activity.

Examples:

Shown below are solutions for Problems A-0, C-1, and D-5 on the ‘‘Line

It Out’’ worksheet

Show 2 LINES with

NO INTERSECTIONS Show 4 LINES with1 INTERSECTION Show 5 LINES with5 INTERSECTIONS

Extensions:

1 Give students this challenge question: With 5 lines (as in Problem

Set D), is it possible to have more than 10 intersections?

2 Have able students create their own Problem Sets E through I for 6

through 10 lines

3 Another challenge question that students might attempt is as

follows: What solution or solutions might arise if these samequestions on the ‘‘Line It Out’’ worksheet were considered in a3-dimensional setting?

5—FIVE 6—SIX 7—SEVEN 8—EIGHT 9—NINE 10—TEN

0—NONE 1—ONE 2—TWO 3—THREE 4—FOUR

D FIVE LINES

INTERSECTION

POINTS:

4—FOUR 5—FIVE 6—SIX 7—SEVEN

0—NONE 1—ONE 2—TWO 3—THREE

C FOUR LINES

INTERSECTION POINTS:

Students will manipulate and compute duplicate digits in a

mathematical problem involving adding, subtracting,

multi-plying, or dividing They will use logical strategies in an effort

to determine whether multiple, single, or no solutions are

possible

2 After the students have shown and discussed all possible solutions

for the initial problem, present them with another that will stretchtheir logical-thinking abilities a bit further Begin by asking which

of the suggested solutions for DDD—111, 222, 333, 444, 555, and666—are not possible, and why

AAA

B B B +CCC DDD

3 Have the students also determine and list all of the possible

solutions Then have them consider whether the problem belowcan be solved

Extensions:

Have students consider the following problems

1 This problem involves subtracting duplicate digits Determine all

possible solutions (one example is provided)

AAA

−BBB CCC

999

−111888

2 This problem involves palindromic (reversible) products of

dupli-cate digits Two problems with palindromic outcomes are shownbelow What others are possible?

33

×113333363

222

×11122222222224642

3 Which division problems using duplicate digits have answers

(quotients) with no remainders? Two examples are shown below.Are there other possibilities? List and discuss them

4

222888

3

333999

4 Activities for students may be extended beyond problems with

duplicate digits to include such problems as the following (seeSolutions below) Participants may also wish to create problems oftheir own In this problem, each letter stands for a different digit,and students are to find numbers that work for the letters shown

DOG +CAT TOAD

Solutions:

Chapter 106

String Triangle Geometry

This hands-on activity allows students to construct geometric

figures and to use concepts such as congruence, similarity,

and parallelism to analyze the figures formed

You Will Need:

Each group of students requires about 3 yards (or meters) of

string; such measuring devices as rulers, yardsticks, or meter

sticks; masking tape; pencils; paper; and scissors

How To Do It:

In this activity, students will measure and manipulate string

and tape as they practice their geometric problem-solving

skills It also provides students with an opportunity to apply

their geometric identification and labeling skills

1 Organize the class into groups of two or four students

and give each group a piece of string approximately

3 yards long, about 10 inches of masking tape, a pencil,and scissors Then instruct students to cut off 1/3 to

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1/2 of their string and tape it

to some flat surface (a tabletop,the chalkboard, or the floor, forexample) in the form of any type

of triangle Each group should thenlabel their triangle’s vertices A, B,and C by writing in pencil on thetape (see the illustration)

A

D H K J G

E B

L I

2 Next, have the students locate the midpoints of each edge of

their triangles by any means that they desire (for example, bymeasuring or folding string) They should then connect thosepoints with string and label the new vertices D, E, and F Whenthis is completed, ask, ‘‘What shapes do you see now?’’

3 Students should then find the midpoints of the new triangle DEF

and use string to connect them, labeling these points G, H, and I.Students repeat this process one more time, labeling the resultingtriangle JKL Then proceed to ask leading questions (answers areprovided), such as

a How do triangles ADF and DBE compare?

b What can be said about triangles GHI and ABC?

c How many triangles the size of JKL are contained in triangle

ABC?

d If triangle JKL were subdivided one more time into triangle

MNO, how many triangles of that size would be included intriangle ABC?

e Can you determine a rule that will tell how many of the smaller

triangles will result from each successive subdivision?