fielder’s snakes and ladders - nzmaths home · fielder’s snakes and ladders ... algebra...
TRANSCRIPT
Fielder’s Snakes and Ladders
We are practicing number and algebra. We are exploring the properties of numbers.
Exercise 1 – EAdders Snakes and Ladders You will need a 0-9 die and some markers. The game is better with more than one person. To begin, throw the die and the highest number starts. Take turns throwing the die and moving. If you land on a square with a formula use that calculation on your NEXT turn, where x is the number you throw. Slide down snakes when you land on them and climb up ladders. To win you must land on the 100 or highest number wins if you are short of time.
100
99
x+x 98
97
96
95
2x 94
x-1 93
92
91
81
82
83
84
85
Goto 25 86
87
3+x 88
89
90
x+2 80
79
x-4 78
77
76
75
74
73
72
71
61
62
63
64
65
-x 66
67
68
69
x+5 70
x+4 60
59
58
57
56
55
54
53
52
51
41
42
43
44
x+2 45
46
47
48
49
x-2 50
40
39
9-x 38
37
x+6 36
35
34
33
32
31
x+1 21
22
23
3times x 24
25
26
27
28
29
x-5 30
x-5 20
19
18
17
x+8 16
15
14
2times x 13
12
11
1
2
x+3 3
4
x-4 5
6
7
8
x+1 9
10
What effect is the expression –x in square 66?
AC
EA
AA
AM
AP
Exercise 2 – AAdders Snakes and Ladders You will need a 0-9 die and some markers. The game is better with more than one person. To begin, throw the die and the highest number starts. Take turns throwing the die and moving. If you land on a square with a formula use that calculation on your NEXT turn, where x is the number you throw. Slide down snakes when you land on them and climb up ladders. To win you must land on the 100 or highest number wins if you are short of time.
100
99
x-2x 98
97
96
95
x+6 94
x-20 93
92
91
81
82
83
84
85
Go to 25 86
87
x+3x 88
89
90
3x-9 80
79
x-12 78
77
76
75
74
73
72
71
61
62
63
64
65
4x-x 66
67
68
69
3x-x 70
x+2x 60
59
58
57
56
55
54
53
52
51
41
42
43
44
4x-4 45
46
47
48
49
1+x 50
40
39
3-2x 38
37
2x+3 36
35
34
33
32
31
5x 21
22
23
8-x 24
25
26
27
28
29
4x-3 30
9-2x 20
19
18
17
3-x 16
15
14
3x-3 13
12
11
1
2
x-1 3
4
x+4 5
6
7
8
2x+1 9
10
Simplify the expression in squares 60, 66, 70, 88 and 98
What does the expression in square 98 mean?
Exercise 3 – AMpliers Snakes and Ladders You will need a 0-9 die and some markers. The game is better with more than one person. To begin, throw the die and the highest number starts. Take turns throwing the die and moving. If you land on a square with a formula use that calculation on your NEXT turn, where x is the number you throw. Slide down snakes when you land on them and climb up ladders. To win you must land on the 100 or highest number wins if you are short of time.
100
99
1(x-2)
98 97
96
95
-2x
94
3x-5x
93 92
91
81
82
83
84
85
Goto 99 86
87
x+x+x
88 89
90
2(2x)
80 79
4(x-2)
78 77
76
75
74
73
72
71
61
62
63
64
65
7x-6x
66 67
68
69
2x-3x
70 x2+x2
60 59
58
57
56
55
54
53
52
51
41
42
43
44
2(x+8)
45 46
47
48
49
9(x-1)
50
40
39
9x-9
38 37
x2
36 35
34
33
32
31
x(3-x)
21 22
23
3(2x)
24 25
26
27
28
29
3x-3
30 3(x-5)
20 19
18
17
2(3x)
16 15
14
x(x-1)
13 12
11
1
2
2(x+1)
3
4
2(x-1)
5 6
7
8
2(3-x)
9 10
Simplify the expression in squares 60, 66, 70, 88 and 93
Write in words what the expression in square 94 means.
Exercise 4 – APortional Snakes and Ladders You will need a 0-9 die and some markers. The game is better with more than one person. To begin, throw the die and the highest number starts. Take turns throwing the die and moving. If you land on a square with a formula use that calculation on your NEXT turn, where x is the number you throw. Slide down snakes when you land on them and climb up ladders. To win you must land on the 100 or highest number wins if you are short of time.
100
99
100% of 2x 98
97
96
95
150% of 2x 94
50% of 2x 93
92
91
81
82
83
84
85
Goto 74 86
87
100% of x 88
89
90
10% of 20x 80
79
40% of 5x 78
77
76
75
74
73
72
71
61
62
63
64
65
x 2 66
67
68
69
1% of 100x 70
x2+x2
60 59
58
57
56
55
54
53
52
51
41
42
43
44
300% of x 45
46
47
48
49
200% of x 50
40
39
0.25 of 8x 38
37
x2 36
35 34
33
32
31
0.75 of 4x 21
22
23
0.2 of 10x 24
25
26
27
28
29
100% of x 30
4x 20
19
18
17
100% of 2x 16
15
14
x(x+1)
13 12
11
1
2
x3
3 4
x3/x2
5 6
7
8
x4/x2
9 10
Simplify the expression in squares 20, 21, 70, 80 and 94
Write an expression which is the same as 2x.
Exercise 5 – Mystery Card Snakes and Ladders You will need a 0-9 die and some markers. The game is better with more than one person. To begin, throw the die and the highest number starts. Take turns throwing the die and moving. Slide down snakes when you land on them and climb up ladders. To win you must land on the 100 or highest number wins if you are short of time. This version of snakes and ladders has ? mystery squares. Select and use one of the sets of cards on the following pages. When you land on a mystery square take the card on top of the pile. This card has an instruction or question that must be answered by you before your next turn. It may be a calculation that is used with your next throw. Return the card to the bottom of the pile.
100
99
? 98
97
96
95
? 94
? 93
92
91
81
82
83
84
85
? 86
87
? 88
89
90
? 80
79
? 78
77
76
75
74
73
72
71
61
62
63
64
65
? 66
67
68
69
? 70
? 60
59
58
57
56
55
54
53
52
51
41
42
43
44
? 45
46
47
48
49
? 50
40
39
? 38
37
? 36
35
34
33
32
31
? 21
22
23
? 24
25
26
27
28
29
? 30
? 20
19
18
17
? 16
15
14
? 13
12
11
1
2
? 3
4
? 5
6
7
8
? 9
10
Your challenge is to make a new set of 20 cards. Choose a topic with the help of your teacher that focuses on one type of problem. Examples would be area or volume; algebra substitution; angle properties; adding integers; the solar system; animals.
Sample Mystery Cards A for Exercise 5. Print and cut out. Topic is Decimals AA
Name a number
between 1.5 and 2.5
Name a number
between 2.3 and 2.7
Name two numbers
bigger that 4.3
What decimal is the
same as
(a) ½;
(b) ¼?
Which is bigger?
2.3 or 3.2
Which is smaller?
1.0 or 0.1
What number is the same as 20 tenths?
What number is the same as 15 tenths?
0.5 + = 1
1 - = 0.2
– 0.5 = 0.3
2 - 0.5 =
+ = 1
+ = 5
+ = 3
+ = 9
Put in order of
smallest to biggest 0.6
0.85 0.234
Put in order of
smallest to biggest 0.4
0.04 0.444
Which is biggest?
0.21
0.212 0.2
Which is smallest?
0.1
0.01 1.001
Sample Mystery Cards B for Exercise 5. Print and cut out. Topic is Factors AM
Name the factors
of 3.
Name the factors
of 12.
Name the factors
of 18.
Name the factors
of 11.
How many factors do prime numbers
have?
What number has only one factor?
What whole number
is not a factor of any number?
What is the
smallest whole number with three
prime factors?
Name a common factor of 24 and 9.
Name a common factor of 38 and 32.
Name a common factor of
100 and 81.
Name a common factor of 66 and 22.
What is the highest common factor of
3 and 11?
What is the highest common factor of
14 and 49?
What is the highest common factor of
18 and 32?
What is the highest common factor of
84 and 72?
Odd numbers only have odd factors.
True or False?
Even numbers only have even factors.
True or false?
factor x factor
= product True or False?
What are the
factors of 3xy2?
Sample Mystery Cards C for Exercise 5. Print and cut out. Topic is Astronomy
Name our nearest
star.
Name the nearest
star to our sun.
Name the three
planets closest to the sun.
Name the two
largest planets in our solar system.
How many planets do we have in our
solar system?
Name the Moon.
Of which planet is
Io a moon?
How many moons
does Jupiter have?
There is about 6hrs between high and low tide on Earth.
True or False?
A comet is made mainly of ice.
True or False?
All water on Earth came from ancient comet collisions. True or False?
What is our galaxy called?
Describe how to use the Southern Cross
to find South.
Name a
constellation.
Pluto is not really a
planet.
True or False?
Night and day is caused by the
Earth’s rotation.
True or False?
Point in the
direction the earth is rotating.
Point to your zenith
point.
Point to North.
What is the speed
of light?
Blank Mystery Cards for Exercise 5. For use by Teachers and students.
Sample Mystery Cards D for Exercise 5. Print and cut out. Topic is Decimals AA/AM/AP P1 1. Name the shaded decimal.
2.
Write 103 as a decimal.
3. Name the shaded decimal.
4.
Write seven tenths as a decimal.
5. Name the shaded decimal.
6.
How many tenths in all of 2.3?
7.
1.2 ÷ 0.2 =
8.
0.8 + 0.4 =
Sample Mystery Cards D for Exercise 5. Print and cut out. Topic is Decimals P2 9.
0.58 + = 1
10.
How many hundredths in all of 0.73?
11.
What number is three tenths
less than 5?
12.
1.47 - = 1.07
13.
5.687 - = 5.68
14.
Name a decimal between 5.9 and 6.
15.
What number is one tenth more than 7.256?
16.
What number is half way between 3.8 and 4.3?
Sample Mystery Cards D for Exercise 5. Print and cut out. Topic is Decimals P3 17.
7 8
What number is marked by the arrow?
18.
2 3
What number is marked by the arrow?
19.
2.7 2.8
What number is marked by the arrow?
20.
4 5
What number is marked by the arrow?
21.
0.4 × 3 =
22. Name the shaded decimal.
23.
In 0.375 the 7 is the number of ….
24.
In 1.38 the 3 is the number of ….
Sample Mystery Cards D for Exercise 5. Print and cut out. Topic is Decimals P4 25.
What number is 10 more than 0.37?
26.
2.47 + = 2.478
27.
0.726 + = 0.73
28. What number is one hundredth
more than 0.39?
29.
Which is larger 0.37 or 0.4?
30.
Which is larger 0.92 or 0.916?
31.
Name a decimal between 0.37 and 0.38.
32.
0.7 ÷ 2 =
Sample Mystery Cards D for Exercise 5. Print and cut out. Topic is Decimals P5 33.
0.9 ÷ 0.1 =
34.
0.2 × 0.4 =
35.
Write thirteen tenths as a decimal.
36.
Write seven hundredths as a decimal.
37.
0.8 × 5 =
38.
0.9 × 0.6 =
39.
Which is larger 30 × 0.9 or 30 ÷ 0.9?
40.
0.7 × 10 =
Sample Mystery Cards E for Exercise 5. Print and cut out. Topic is AM/AP Algebra P1 1.
If n is any non-zero number
what is n ÷ n?
2.
What is the value of 3e + 2f if
e = 5 and f = 7?
3.
Write an expression for the perimeter of
w
w w
4.
Write an expression for the perimeter of
3 3
d
b
d
5.
What goes in the box if
36 + 48 = 39 + ?
6.
Find A and B if
A × B = 40
A + B = 13
7.
Find p and q if
p + q = 10
and p – q = 4
8.
If 5 * 4 = 14
and 2 * 7 = 11
What is 8 * 3?
Sample Mystery Cards E for Exercise 5. Print and cut out. Topic is Algebra P2 9.
What comes next in the
pattern
s – 5, s – 2, s + 1, ….?
10.
Find s, t and u if
s × t × u = 30
s + t + u = 10
11.
If n is any number what is
n – n?
12.
What goes in the box if
9 × 16 = 18 × ?
13.
What goes in the box if
72 ÷ 4 = 36 ÷ ?
14.
What is the length of the heavy
line? 3b a
a a
15.
How many white squares will there be in the tenth design in the pattern?
16.
How many white squares will there be in the tenth design in the pattern?
Sample Mystery Cards E for Exercise 5. Print and cut out. Topic is Algebra P3 17. What rule has been used to complete this table?
In 3 8 4 15 Out 10 25 13 46
18.
What is the missing number in this table?
In 2 11 6 4 10 Out 7 25 15 11 ?
19.
What is the missing number in this table?
In 5 9 3 6 ? Out 8 16 4 10 20
20.
How many white squares will there be in the one hundredth design in the pattern?
21.
How many squares will there be in the fiftieth design in the pattern?
22.
How many circles will there be in the eighth design in the pattern?
23.
What number goes in the box if
+ + 7 = + 18?
24.
What is the 10th line in the pattern?
12 = 1 22 = 1 + 3 32 = 1 + 3 + 5 42 = 1 + 3 + 5 + 7
Sample Mystery Cards E for Exercise 5. Print and cut out. Topic is Algebra P3 25.
How many sticks will there be in the twentieth design in the pattern?
26.
What goes in the box if
95 - 38 = - 40?
27.
How many sticks will there be in the twentieth design in the pattern?
28. If n is any number, which is the odd one out?
n + n, n × 2, n + 2, 2n
29. What rule has been used to complete this table?
In 3 8 4 15 Out 10 15 11 22
30.
What goes in the box if
3 × 55 = 33 ×
31. What is the missing number in this table?
In 2 5 8 10 7 Out 5 26 65 101 ?
32.
Find S and T if
S × T = 24
S - T = 2
Sample Mystery Cards E for Exercise 5. Print and cut out. Topic is Algebra P4 33. What number goes in the box if
× + 1 = 17?
34. What number goes in the box if
+ + 1 = 17?
35.
What is the 10th line in the pattern?
1 × 2 = 2 2 × 3 = 2 + 4 3 × 4 = 2 + 4 + 6 4 × 5 = 2 + 4 + 6 + 8
36.
What is the 10th line in the pattern?
12 – 1 = 0 × 2 22 – 1 = 1 × 3 32 – 1 = 2 × 4 42 – 1 = 3 × 5
37.
Which weights need to be moved to balance the scales?
38. What rule has been used to complete this table?
In 3 8 5 13 Out 5 15 9 25
39.
Complete this magic square.
4x
2x+1
2 x+3
40.
What number goes in the box in this number pattern?
3 × 37 = 111 6 × 37 = 222 9 × 37 = 333
× 37 = 888
3kg
4kg
9kg
5kg
7kg
Fielder’s Snakes and Ladders Teacher’s Notes
These exercises, activities and games are designed for students to use independently or in small groups to practise number properties. Typically an exercise is a 10 to 15 minute activity. Exercise 5 is an open ended activity with sample activity cards. The blank sheet can be used by students and teachers alike to make up their own set of cards and add them to the class resources. Number Framework domain and stage: Multiplication and Division – EA-AM Curriculum reference: Number, Level 3-5 Materials: • 0-9 dice or others • counters • calculator Prior Knowledge. Students should be able to:
• have knowledge of the meaning of x as a variable • know 2x means 2 times x Prior knowledge needs will vary depending on numeracy stage. During these activities students will meet: • place value, factors, multiples, squares, square roots, variables, negative numbers Comments on these exercises The AC or counting students can be given the game of snakes and ladders without tricks to play. Be aware that this may reinforce counting as a good strategy. Year 9 counting AC students are an AT RISK group. Every effort should be to make them part-whole thinkers. A note on factors: All integers except zero can be factors and of course all variables and their powers. Zero cannot be a factor as 0x2, 0x3, 0x4… would all be factors of zero. Zero is chosen not to be or to have a factor. One is a factor of every number and expression. E.g. the factors of 2xy are 1, 2, x, y, 2x, 2y, xy, 2xy. This exercise gets even more interesting with powers. –x has 1, -x, -1, x as factors. This is not all that easy and is probably the domain of the proportional thinker. What happens if there are fractions and decimals? E.g. factorising 0.25x + 0.5x = 0.25(x+2) and many other expressions. If students have developed these advanced understandings then consider the learning to have been established. A note on decimals. The leading zero on 0.7111 is only there to help draw attention to the decimal point. “.7111” can be easily misinterpreted as “7111”. Exercise 1 Teaching lessons preceding this exercise could include:- • meaning of x as a variable
• meaning of 2x as 2 times x • -x as negative or go back x
This exercise aims to move EA students to AA. It has some tricky understanding such as 9-x and these may need to be explored with materials. Exercise 2 Teaching lessons preceding this exercise could include:- • revise meaning of x as a variable
• meaning of 3 – x when x >3 • -2x as negative or go back 2x
This exercise targets AA students moving to AM. Exercise 3 Teaching lessons preceding this exercise could include:- • revise meaning of x as a variable
• meaning of 3 – x when x >3 • -2x as negative or go back 2x • the meaning and use of brackets • the meaning of x2
This exercise targets AM students moving to AP. Exercise 4 Teaching lessons preceding this exercise could include:- • revise meaning of x as a variable
• meaning of 3 – x when x >3 • -2x as negative or go back 2x • the meaning and use of brackets • revise meaning of percentage and percentage of a quantity • the meaning of square root x and x squared.
This exercise targets AP students. Exercise 5 Teaching lessons preceding this exercise could include revising the meanings of the mathematical representations encountered in the cards supplied. See answers that follow.
Fielder’s Snakes and Ladders Answers
Exercise 1 Grid answers will vary and are not supplied. The effect of the expression –x is to reverse the direction of the move; i.e. backwards by x. Exercise 2 Grid answers will vary and are not supplied. The simplified expressions are 60: x + 2x = 3x; 66: 4x – x = 3x; 70: 3x – x = 2x; 88: x + 3x = 4x; 98: x – 2x = -x. The expression in 98 means go back x. Exercise 3 Grid answers will vary and are not supplied. The simplified expressions are 60: x2 + x2 = 2x2; 66: 7x – 6x = 1x = x; 70: 2x – 3x = -1x = -x; 88: x + x + x = 3x; 93: 3x – 5x = -2x. The expression in 94 means go back 2x. Exercise 4 Grid answers will vary and are not supplied. 20: √x4 = x2; 21: 0.75 of 4x = 3x; 70: 1% of 100x = 1x; 80: 10% of 20x = 2x; 94: 150% of 2x = 3x. Any expression which equals 2x is acceptable, e.g. 4x divided by 2 Exercise 5 Students are expected to create at least one new set of cards. Topic Decimals AA Answers, left to right and top to bottom. Various e.g. 1.9; Various e.g. 2.4; Various e.g. 4.4 and 4.5; (a) 0.5 (b) 0.25; 3.2 is bigger; 0.1 is smaller; 2 = 20 tenths; 1.5 = 15 tenths; 0.5; 0.8; 0.8; 1.5; 0.5 + 0.5; 2.5 + 2.5; 1.5 + 1.5; 4.5 + 4.5 (this assumes the numbers are equal), 0.234, 0.6, 0.85: 0.04, 0.4, 0.444; 0.212 is biggest; 0.01 is smallest. Topic Factors AM Answers, left to right and top to bottom. F3 = 1,3; F12 = 1, 2, 3, 4, 6, 12; F18 = 1, ,3, 6,18; F11 = 1,11; 2; 1; 0; 2 x 3 x 5 = 30; One of 1or 3; One of 1 or 2; 1; One of 1 or 2 or 11; HCF of 3 and 11 = 1; HCF of 14 and 49 = 7; HCF of 18 and 32 = 2; HCF of 84 and 72 = 12; True; False; True; factors = 1, 3, x, y, y2, 3x, 3y, 3y2, xy, xy2, 3xy, 3xy2. Topic Astronomy example to show use in other subject or context Answers, left to right and top to bottom. Sun; Alpha Centauri; Mercury, Venus, Earth; Saturn, Jupiter; 9 planets at the last count; Moon; Jupiter; 12; True; True; True; The Milky Way; Intersection of the perpendicular bisector of The Pointers and the main axis of Southern Cross projected vertically to the horizon is very close to true South; Scorpio (various); True, Pluto is more like a captured large meteorite; True with help of the Sun; Point East; Point above self; Point North; c = 300,000,000m/s.
Topic Decimals AA/AM/AP (1) 20 hundredths, 0.2; (2) 0.3; (3) 24 hundredths = 0.24; (4) 0.7; (5) 75 hundredths = 0.75; (6) 23 tenths; (7) 6; (8) 1.2; (9) 0.42; (10) 73 hundredths; (11) 4.7; (12) 0.4; (13) 0.007; (14) various, e.g. 5.91; (15) 7.356; (16) 4.05; (17) 7.8; (18) 2.05; (19) 2.73; (20) 4.6; (21) 1.2; (22) 0.07; (23) hundredths but this is inconsistent with “How many hundredths are there in 0.375?”; (24) tenths; (25) 10.37; (26) 0.008; (27) 0.004; (28) 0.391; (29) 0.4; (30) 0.92; (31) various, e.g. 3.71111111119; (32) 0.35; (33) 9; (34) 0.08 (model this); (35) 1.3; (36) 0.07; (37) 0.4; (38) 0.54; (39) It is not 30 x 0.9; (40) 7. Topic Algebra AM/AP (1) 1; (2) 29; (3) 3w; (4) d + d + 3 + b + 3 or equivalent; (5) 45; (6) A = 8 and B = 5 or vice versa; (7) p = 7 and q = 3; (8) 19 velly tricky! Hint is 5*4 = 2 x 5 + 4; (9) s + 4, add 3; (10) a whole number solution is 2, 3, 5; (11) zero; (12) 8, halve and double; (13) 2, halve; (14) 3b – a + 2a = 3b + a; (15) 3 x (10 + 1) – 10 = 23; (16) 10 x 10 – 10 or equivalent; (17) triple (In) and add 1 or 3n + 1; (18) rule is double and add three; 10 → 23; (19) rule is double take 2, reverse is subtract 2 and halve, 20 → 11; (20) rule is (n + 2) x (n + 2) – n x n so in hundredth design there are 102 x 102 – 100 x 100 = 404; (21) Rule is n + (n – 1) so 50th term is 50 + 49 = 99; (22) Pattern is triangular numbers, nth term = n(n + 1)/2 so eight design is 8 x 9/2 = 36; (23) 11; (24) 102 = 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19; (25) Rule is 3n + 1 so 20th is 60 + 1 = 61; (26) 97, add 2 to both; (27) Rule is 2n + 2 so 20th design is 42; (28) n + 2 because it is sometimes odd, or n + 2 because all the others are the same; (29) Rule is n + 7; (30) 5 does because 3 x 55 = 33 x 5; (31) Rule n x n + 1 so 7 → 50; (32) S = 6, T = 4; (33) 4; (34) 8; (35) 10th is 10 x 11 = 2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 + 18 + 20; (36) 102 – 1 = 9 x 11 this is difference of two squares n2 – 1 = (n – 1)(n + 1); (37) 9kg and 7kg making 14kg on each side; (38) rule is 2n – 1 or double and take 1; (39) 3x-1 4-x 4x 3x+2 2x+1 x 2 5x-2 x+3 (40) 24 x 37 = 888, investigate the triples of the multiples of 37!