Transcript What Do Arithmetic Computation and “Real World” Math Have

What Do Arithmetic
Computation and “Real
World” Math Have to Do with
Algebra or Algebraic
Thinking?
Johnny W. Lott
[email protected]
What ties if any does
arithmetic have to algebra?
 A different way to put this is the
following:
Is everything that we teach in algebra
new?
What should we think about if we talk
about algebraic thinking?
Arithmetic Computation?
What do you need to know?
Place value
Algorithms
Arithmetic Computation?
 Would your students say that
35 = 8 or
47 = 11?
 What would they say about
3 + 5 or
5 + 3?
Place value?
 Would your students say that
310 + 5 = 8
 What would they say about
3 + 5 or
5 + 3?
How are place value and algebraic symbolism
related?
Look at worksheet 1.
 Do the arithmetic as directed.
What happens when you look
at decimals?
 What is the meaning of 431.25?
 What would this look like in expanded
form?
Do worksheet 2.
How are decimals related to algebra?
Look at growing patterns.
 Use Exploring Houses.
 Use Building with Toothpicks.
 Use Tile Patterns.
Considering Patterns
 Will more than one pattern work?
 How many does it take to decide a
pattern?
 Can you prove your answer?
How Tall Are the Cups?
2 inches
7 inches
How tall is a stack of 100 cups?
What are your favorite
problems to solve?
Locker Problem
Squares on a Checkerboard
Problem
Tying the String to Get Married
Problem
Lockers all open.
1
11
21
31
41
51
61
71
81
91
2
12
22
32
42
52
62
72
82
92
3
13
23
33
43
53
63
73
83
93
4
14
24
34
44
54
64
74
84
94
5
15
25
35
45
55
65
75
85
95
6
16
26
36
46
56
66
76
86
96
7
17
27
37
47
57
67
77
87
97
8
18
28
38
48
58
68
78
88
98
9
19
29
39
49
59
69
79
89
99
10
20
30
40
50
60
70
80
90
100
Second student goes through
1
11
21
31
41
51
61
71
81
91
C
C
C
C
C
C
C
C
C
C
3
13
23
33
43
53
63
73
83
93
C
C
C
C
C
C
C
C
C
C
5
15
25
35
45
55
65
75
85
95
C
C
C
C
C
C
C
C
C
C
7
17
27
37
47
57
67
77
87
97
C
C
C
C
C
C
C
C
C
C
9
19
29
39
49
59
69
79
89
99
C
C
C
C
C
C
C
C
C
C
Third student goes through
1
11
C
31
41
C
61
71
C
91
C
12
C
C
42
C
C
72
C
C
C
13
23
C
43
53
C
73
83
C
C
C
24
C
C
54
C
C
84
C
5
C
25
35
C
55
65
C
85
95
6
C
C
36
C
C
66
C
C
96
7
17
C
37
47
C
67
77
C
97
C
18
C
C
48
C
C
78
C
C
C
19
29
C
49
59
C
79
89
C
C
C
30
C
C
60
C
C
90
C
Fourth student goes through
1
11
C
31
41
C
61
71
C
91
C
C
C
32
42
52
C
C
C
92
C
13
23
C
43
53
C
73
83
C
4
C
C
C
44
54
64
C
C
C
5
C
25
35
C
55
65
C
85
95
6
16
C
C
C
56
66
76
C
C
7
17
C
37
47
C
67
77
C
97
8
18
28
C
C
C
68
78
88
C
C
19
29
C
49
59
C
79
89
C
C
20
30
40
C
C
C
80
90
100
Questions to ask
 If 1000 students go through the school
and change the state of doors, how
many times is door 72 touched?
 What is the final state of door 432?
 Who touched door 46 last?
 What is the relation of the door
number and the number of factors?
Squares on a Checkerboard
Problem
 Give me one grain of wheat for the
first square.
 Give me two grains for the second
square.
 Give me four grains for the third
square and continue.
 How many grains in all when the board
is filled?
Questions to ask
 Would you take only the grains on the
64th square or would you take all the
grains on the first 63 squares if given
the option?
 How many grains are on the 15th
square?
Tying the String to Get
Married Problem
Six strings in my hand
Tie ends on top two at a time.
Tie ends on bottom two at a time
If a full loop is obtained, I can get
a marriage license. How likely?
Questions to ask
 Is a person’s chance of getting a license
more than 50% in the first year?
 Does the probability of getting the marriage
license change in a second year if the license
is not obtained in the first year?
 Suppose there are only five strings. Is the
probability more or less? Four strings?
Three strings? Two strings? One string?
Twist old problems
Locker problem gave perfect
squares.
Try the pig problem--even with
young kids.
Pig Problem
A farmer sold n cows for n dollars
each. With the proceeds, she
bought an odd number of sheep at
$10 each, and a pig for less than
$10. How much did the pig cost?
Think perfect squares.
 Why?
 Think of the ones digit of the
proceeds.
 Think of the tens digit of the proceeds.
 Look at a table.
 What is your answer?
 Can you prove it?
Checkerboard/Grains of Rice
Substitute the
“Would You Work for Me?”
Problem.
Would you?
Day
My Scheme G's Scheme
1
$1.00
$1.00
2
$0.50
$0.50
3
Algebra
 Algebra is a civil right. Robert Moses
 What types of formulas are used in spreadsheets?
 Teachers, what types of formulas are used in your
retirement packages?
 Students, how can you tell how long medication
stays in your blood stream?
 How do you decide on pricing for concert tickets?
Algebra Continued
 How do you learn multiplication facts?
 Why do you learn multiplication tables?
Yet More Algebra!
 Consider addition and all the pairs that
add to 12; now that add to 18; now
that add to 0. What do they have in
common?
 Try the same with multiplication.