Jossey-Bass Teacher - Math Wise Phần 4 doc

This activity involves the use of file folders, pockets, and cards to help students review basic mathematicalconcepts or to test their knowledge in basic mathemat-ical facts.. If student

Dot Paper for 1,000 and More

This activity gives students practice with basic facts,

com-putation, and problem solving by matching It encourages

students to work as individuals or in small groups and to

self-check their answers It can also be used as an alternate

way for the teacher to test the students

You Will Need:

Students will need manila file folders (amount will vary

depending on what you feel needs reviewing), glue, library

pocket envelopes (pockets made out of construction paper or

4- by 6-inch index cards will also work), 3- by 5-inch index

cards, and marking pens

How To Do It:

1 This activity involves the use of file folders, pockets,

and cards to help students review basic mathematicalconcepts or to test their knowledge in basic mathemat-ical facts

Glue the library pockets inside the file folders Thenumber of pockets depends on what is needed forthe review or test For example, if students need toreview their multiplication facts, then their folder could

119

look like the one pictured in Example 1 Use the marking pens

to write problems on the pockets and their answers on individualindex cards Prior to starting, hide an answer key to all the pockets

in the Answer Cards pocket

2 To begin, each student (or pair of students) must take all the

individual answer cards and place them in the matching problempockets If students need to make pencil-and-paper computationsfor a given problem, they should use a small piece of scratch paperand place it in the same pocket as the answer card Solutionsshould be checked against the answer key or by another player(sometimes with a calculator) If you choose to use the file folder

as a test of a student’s knowledge, then the activity could be timed,with you checking the solutions

Examples:

1 Using the file folder below,

a student must match tiplication facts with thecorresponding answers

mul-2 This file folder requires

stu-dents to practice telling timefrom a clock face Studentsmatch digital times to those

on ‘‘regular’’ clock faces

Multiply

6 × 5 7 × 3 8 × 6 9 × 5

4 × 9 36

Time

Answer Cards 5:15 6:00

9:00 3:00

Extensions:

1 Devise file folders for any area in which practice is needed The

players, after seeing how the folders are constructed, should make

a variety of folders for one another These file folders can be madefor matching numerals with pictured amounts; practicing basicfacts; exploring fractions, decimals, measurement, and geometricidentification; working with the concepts of time and money; andsolving short story problems, among other things

2 Older students can construct and use these file folders to provide

special help for younger learners Especially useful are foldersdealing with numerals, number sense, place value, and basic mathfacts The folders could also be at a math station in the classroom,and as students finish their regular work, they could go to thestation to practice certain skills

3 To increase the difficulty level for this activity and promote careful

thinking, include more than a single correct answer card for certainproblems (for example, a pocket reading ‘‘Find two numbers whoseproduct is 36’’ might be answered with 4× 9 and 6 × 6) or include

a few wrong answers that do not correspond to any of the folderproblems One or both of these tactics can be used to turn thisactivity into a math quiz

Students will practice basic facts and mental math This

activity will help students become more proficient in recalling

basic mathematical facts, and with this ability they will have

fewer problems with more complicated mathematics

You Will Need:

At least one calculator is required If Beat the Calculator is to

be a large group or whole class activity, the teacher can use a

calculator placed on an overhead projection device, or even

a virtual calculator found online For small groups of two to

five participants, only one calculator is necessary

How To Do It:

1 In a small group, with one calculator and three people,

the following procedure works well: Student 1 calls out

a math problem, such as 6× 7 Student 2 uses a lator to solve the problem and state the answer At thesame time, Student 3 solves it mentally and says theanswer The first to give the correct answer (Student 2

calcu-or Student 3) wins The players’ roles should ally be rotated To make the activity more competitive,have students tally the number of wins for each student

eventu-(Note: Students will soon discover that if they have practiced their

basic facts, they will be able to ‘‘beat the calculator’’ nearly everytime.)

2 Beat the Calculator may also be played as a whole class activity In

this case, you or a leader operates the calculator, while studentssimultaneously do the mental math and call out answers; a chosenjudge calls out the problems and determines the winner of eachround The object of the activity is therefore to determine whichmethod is faster and more efficient for obtaining the solutions tobasic fact problems: using a calculator or just memorizing the fact

Example:

In the small group situation above, Sean had called out 5× 9 Susan hasbeen attempting to solve the problem with a calculator before Randycould do so mentally However, because Randy had mastered his 5smultiplication facts, he was able to beat the calculator

Extensions:

1 Young students might try counting with a calculator by entering a

number (try 1), an operation (try+), and pressing the equal buttonmultiple times (= = = =) to make the calculator count by 1s.They can also start with any number, like 20, and enter 20+ 1 =

= = = They might further try counting forward (addition) orbackward (subtraction) by any multiple; for example, they can

enter 3+ 3 = = = = and see what happens (Note: Learners should

use a calculator that has an automatic constant feature built in;most basic calculators do To test this, simply try the calculator; if

it ‘‘counts’’ as noted above, it has the needed constant.)

2 Students might use a calculator that has an automatic constant

to individually practice basic multiplication facts For example, topractice the 4s facts, they can enter 4× =, which the calculatorholds in its memory Then any number entered will be multiplied

by 4 when the = key is pressed Thus the student might enter 8,mentally think what the answer should be, press =, and see thatthe answer displayed is 32

3 Advanced students can work in pairs, taking turns trying to beat

the calculator with such tasks as 2× 12 ÷ 8 + 3 − 5 = or

7+ 32 ÷ 4 − 5 × 2 = (Note: In these cases, be certain that the

players use the proper order of operations The mnemonic ‘‘PleaseExcuse My Dear Aunt Sally’’ sometimes helps students rememberthe order: parentheses, exponents, multiply or divide from left toright, and add or subtract from left to right Students are frequentlyunclear about this concept, so even if the calculator has a built-inorder of operations feature, the students should be taught to putparentheses as shown here: 7+ (32 ÷ 4) − (5 × 2) The answers tothe two problems in this Extension are 1 and 5, respectively

Students will physically act out computation and

mathemati-cal problem-solving situations

You Will Need:

This activity requires a walk-on number line, which can be

constructed using soft chalk, tape, and number cards, or a

large roll of paper with a marking pen

How To Do It:

1 To construct the number line, write large numerals

about 1 foot apart either on the playground or floorusing soft chalk, or on a large roll of paper using a mark-ing pen If the number line will be used more than once,

it can be made by taping number cards to the floor orusing some more permanent method on the playgroundsurface Problems at the beginning will likely make use

of the numerals 0 through 10, but as the work becomesmore difficult, the number line can be expanded to 100

125

or more If signed numbers are to be used, it should also beextended from 0 to−1, −2, −3, and so on.

2 Students will solve math problems using this walk-on number

line The examples below show how to begin Once studentsunderstand the procedure, have them try some of the problems

in the Extensions section or other problems created by you orthe students Students should be ready both to explain how they

‘‘walked out’’ each problem and to use pencils and paper to showthe same solution

Examples:

1 For 4+ 3, students begin at 0 and

take 4 steps to the number 4 Then

they take 3 more forward steps and

check the number on which they

are now standing; it should be 7

(See the solid arrow in the

illus-tration below.) Finally, students

should keep a record by writing

4+ 3 = 7 in their notebooks

2 For such a problem as 9÷ 4, dents begin at the 9 and movetoward the 0, taking 4 steps at atime and holding up a finger foreach time Beginning at the 9, theystep to 8, 7, 6, and 5 and hold

stu-up 1 finger; then they step to 4,

3, 2, and 1 and hold up 2 gers They have therefore taken

fin-4 steps 2 times, but still need toget to 0; this will require 1 morestep Thus 9÷ 4 requires 2 sets of

4 steps with 1 step remaining, so

9÷ 4 = 2, with a remainder of 1

Have students use the floor number line to help solve either the followingproblems or others that students need to solve from a workbook or text.Students can also make up several of their own problems and have theirclassmates use the number line to figure out the solutions

1 8+ 3 2 7− 4 3 4× 6

4 20 ÷ 5 *5 −2 + −4 *6 −3 + 4

*(Hint: Face in the direction of the first signed number, and then

change direction every time the sign of the number changes.)

䡺× Visual/pictorial activity

䡺× Abstract procedure

Why Do It:

Egg Carton Math provides students with a fun way to

gen-erate problems and practice computation, and helps them

gain a better understanding of factors of a number, prime

factorization, and probability

You Will Need:

A 12-cell or 18-cell egg carton with its lid still attached is

required for every student or pair of students Also required

are paint and brushes, card stock, glue or tape, sticky dots

for labeling, crayons or markers, pencils, beans (larger beans

preferably, such as pinto or lima beans), and photocopies of

the ‘‘Egg Carton Probability’’ handouts (provided)

How To Do It:

There are many things you can do with an egg carton; here

are a few activities designed for various levels of math ability

1 In order to learn about probabilities, students will be

shaking an egg carton with a bean inside The cells ofthe egg carton will be painted different colors, and theprobability of the bean landing on a certain color isdiscussed

Begin with the students painting the egg carton cells with threedifferent colors in random order Either tell students how manycells will be painted red, how many green, and how many blue, orlet them decide for themselves The colors might also be chosenrandomly by tossing a die: if 1 or 4 appears, students paint thecell red; if 2 or 5 comes up, they paint the cell green; and if 3 or 6

is rolled, they paint the cell blue With this method, it is possiblethat students will never paint a cell red, which simply means thatthe probability of getting a red is zero Students then need to cut apiece of card stock the size of the top of the egg carton and glue

it to the top to block the holes through which beans might slip.They then place one bean inside the egg carton and close the lid.Students will use the ‘‘Egg Carton Probability’’ activity sheet toperform the probability experiment After recording their results,students can also draw a bar graph on the back of their activitysheets Younger students can fill in sections of the blank bar graphincluded at the end of this activity

2 Another way to use the egg carton is to write the numbers 1

through 12 or 1 through 18 (depending on the size of the eggcarton) on sticky dots, and put the dots in each cell such that thenumbers show Students can use two beans, three beans, or moredepending on their math ability level After the student shakes theegg carton, he or she opens it and writes down the numbers onwhich the beans have fallen If two beans land in the same cell,the student is to write down that number twice At this point, thestudent can add or multiply the numbers together It works best

if the students are in pairs, so that they can check each other’sanswers If there are just two beans in the egg carton, the activitycan help students review basic addition and multiplication facts.For example, one student, after shaking the egg carton, could open

it to find the beans have landed on 5 and 6 The student wouldthen ask his or her partner to find the sum of 5 and 6 Students canalso keep track of how many correct answers each player comes

up with in 10 or 20 shakes

3 The third activity is for more advanced students and helps enhance

students’ understanding of prime numbers (counting numberswith exactly two different factors), composite numbers (countingnumbers with more than two different factors), and prime fac-torization Students begin by writing the first twelve or eighteenprime numbers on the sticky dots (2, 3, 5, 7, 11, 13, 17, 19, 23,

29, 31, 37, 41, 43, 47, 53, 59, and 61) Then they place the dots

in individual cells of the egg carton Students can use two ormore beans, shake the egg carton, and use the prime numbers

on which the beans fall as prime factors of a composite number.Students can then find the composite number by multiplying the

prime numbers together They can also practice using exponentialnotation For example, a student’s five beans could land on 2, 2,

7, 5, and 13 The prime factorization is written 2× 2 × 5 × 7 × 13

or 22× 5 × 7 × 13 which is equal to the composite number 1,820.Once students have the prime factorization of a number, theycan find all the factors of that number For example, using theprime factorization of the number 36 (which is 22× 32), a table asshown below can be set up to find all the factors of 36 Becausethe exponent of 2 is 2, 0 through 2 will be used for the exponents

of 2 Similarly, because the exponent of 3 is 2, the exponents of 3will range from 0 to 2 Also remind students that any number with

an exponent of 0 is equal to 1 To fill out each square in the chart,students will fill in the power of 2 using the exponent in thesame row at the left and the power of 3 using the exponent inthe same column at the top

Extensions:

1 Whenever a game requires a random number generator, the egg

carton with numbers in it can be used, such as when doing the

Sticky Gooey Cereal Probability simulation with 12 prizes instead

of 6 (see p 214)

2 Place fractions in the cells and have students practice adding or

multiplying fractions Also, students can discuss the fractional part

of the whole that the colors red, green, or blue represent If eachstudent’s egg carton is different, students will be able to see themany fractions that develop from coloring an egg carton To extendthis idea further, students could make a list of all the fractions thatare possible when egg cartons are colored in different ways

3 For advanced students, put positive and negative numbers on the

sticky dots in the egg carton and have them add, subtract, andmultiply the signed numbers Algebraic expressions can also beplaced in the cells of the egg carton, such as −5x, 2x2, 8x, and

−4x2, and students can be asked to add, subtract, multiply, ordivide the expressions

If you were to do this experimentmany times, on which color doyou think the bean would landthe most?

Why do you think this?

Experiment:

Shake the egg carton with the bean in it Now open the egg carton andlook at what color the bean has landed on Record the color in the chartbelow using an X Do this 20 times, recording your results each time

RedBlueGreen

3 Draw a bar graph to record your results.

Bar Graph for Egg Carton Probability

Chapter 36

Cross-Line Multiplication

You Will Need:

Students will need pencils and paper

How To Do It:

This activity requires that students use a pencil to draw

crossing lines that correspond to factors (numbers) in any

given multiplication problem If a student wants to solve

3× 7, he or she will draw three parallel horizontal lines and

seven parallel vertical lines crossing the horizontal lines Then

they will count the number of intersections (line crossings)

to find the answer to that specific problem In this case, the

number of intersections is 21 and that is the answer to 3× 7

See the figure for clarification

133

To solve the problem 6× 2, the student will need to

show 6 groups of 2 After drawing 6 horizontal lines

and crossing them with 2 vertical lines, the student

will then count 12 crossings, which is the answer to

the problem

(shows 6 × 2 = 12)

Also, by turning the drawing sideways, the problem

2× 6 or 2 groups of 6 can also be shown Thus, 2 ×

6= 12

(shows 2 × 6 = 12)

Students could solve the sample problems below

1 Draw crossing lines to show 3× 5 There are line crossings

1 8 3

Add 1 from the 18 to the 2 in 12. All middle line crossings

Add 2 from the

24 to the 2 in the 12

9 4

Chapter 37

Highlighting Multiplication

Each student will practice multiplication facts using a visual

procedure and will gain a better conceptual understanding of

these facts

You Will Need:

A supply of multiplication charts (reproducibles are provided)

and highlighter pens or colored pencils are required

How To Do It:

This activity will allow a student to find the answer to a

multiplication problem by shading a fact chart Have students

use a multiplication chart and highlighter pens or colored

pencils to shade areas that show the answers to multiplication

facts On a separate piece of paper, students then write the

problems and answers that the highlighted areas show

7 6 5 4 3 2 1

20

14 12 10 8 6 4 2

30

21 18 15 12 9 6 3

40

28 24 20 16 12 8 4

50

35 30 25 20 15 10 5

60

42 36 30 24 18 12 6

70

49 42 35 28 21 14 7

80

56 48 40 32 24 16 8

90

63 54 45 36 27 18 9

100

70 60 50 40 30 20 10

×

Extensions:

Have students complete the following sample problems

1 Shade in the area for 2× 3 How many spaces did you highlight?Thus 2× 3 = Explain how you have also shown the area for

3× 2

2 Highlight 3× 5 Thus 3 × 5 = and 5× 3 =

3 Show 5× 8 Thus 5 × 8 = and 8× 5 =

4 When you highlight 9× 3 or 3 × 9, the area equals

5 7× 9 or 9 × 7 = What do you notice, on the multiplicationchart, about the location of the answer number?

6 Complete a series of highlighted charts and post them next to

one another on a bulletin board For example, do the 7s facts byhighlighting 7× 1 on the first chart, 7 × 2 on the second, 7 × 3 onthe third, and so on

Highlighting Multiplication 137

7 6 5 4 3 2 1

20

14 12 10 8 6 4 2

30

21 18 15 12 9 6 3

40

28 24 20 16 12 8 4

50

35 30 25 20 15 10 5

60

42 36 30 24 18 12 6

70

49 42 35 28 21 14 7

80

56 48 40 32 24 16 8

90

63 54 45 36 27 18 9

100

70 60 50 40 30 20 10

10

7 6 5 4 3 2 1

20

14 12 10 8 6 4 2

30

21 18 15 12 9 6 3

40

28 24 20 16 12 8 4

50

35 30 25 20 15 10 5

60

42 36 30 24 18 12 6

70

49 42 35 28 21 14 7

80

56 48 40 32 24 16 8

90

63 54 45 36 27 18 9

100

70 60 50 40 30 20 10

×