Squarrows
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Squarrows An Elementary Introduction to Algebra (draft 1) Oliver Steele 2007
Squarrows An elementary introduction to abstract algebra. INTRODUCING SQUARROWS
1
+1
2
+10
5
+2
Squarrow Chains 2
3
!2
4
+2
3
+4
5
!2
More Squarrows +1
2
+3
4
3
!2
2
6
2
4
10
-1
11
÷2
Squircles
5
-1
1
-1
4
Find numbers that make a circle. Are any of these impossible? Do any have more than one answer?
÷2
+1 3
2
+4
Squarrow Tails
6
5
2
3
8
+2
15
7
÷2
4
-1
+3 -1
-1
+5
÷2
-10
x3
+10
x2
2 ÷2
+6
!3
6
9
-1
2 3 3
+1
4
+2
+1
+2
+1
+2
+2
+1
6
+1
+3
5
+1
+1
+3 -1
+1
÷2
-9
-1
+1
-10
x2
÷2
9 -1
3
3 2
3
3
+1
!2
+1
!2
+7
8
+7 +10 -1
+6
+3
÷2
+3
-1
+6
-10
10
÷2
-18
+5
÷2
+7 !2
+1
-1
3
÷2
-1
Squarrows is a workbook for teaching algebraic concepts at the elementary school level. Squarrows is independent of the formalisms of algebra (such as variables, and the syntax of equations), but can be used to develop many of its concepts. Background I used Squarrows in teaching a group of fourth graders at the Wildwood Elementary School in Amherst, MA, in fall 2007. We met once a week for 45 minutes; we used a number of different problems worksheets, but used Squarrows as one of two “running problem sets”, with a new page every week or two. I learned a lot from the students – how to think about math, but also about the degrees of abstraction, reflection, and sophistication that even very young minds are capable of. Class was more like a graduate-school seminar; I got more ideas from them than they from me.
©2007 by Oliver Steele • email: [email protected] • web: http://osteele.com!
PART I: BASIC SQUARROWS
©2007 by Oliver Steele • email: [email protected] • web: http://osteele.com!
2
3
+2
+1
4
+2
+1
3
+4
5
+2
1
+1
2 5
+1
4
Squarrows
3
+2
+2
6
+2 1
3
2
5
0
3
3
9
5
+2
8
+3
9
2
3
!2
+10
4
+2
+2
3
+4
5
!2
1
+1
2 5
+3
4
Squarrow Chains
3
+4
+6
6
4
3
+1
3
+1
+2
+1
+2
+2
+1
3 3
3
+2
+1
!2
+1
!2
!2
+1
6
9
10
6
!2
8
!3
9
More Squarrows +1
2
2
6
2
4
+1 3
2
3
2
+2
7
+1
+3
5
+1
+3 +1
2
+1
+7
8 +6
+3 +7
+7
+6
PART II: A DIVERSION – NUMBER PAIRS
Oliver Steele • email: [email protected] • web: http://osteele.com!
Fruit Facts
A B C D
5
+3
!
=8
3
+5
!
= ____
7
+2
!
= ____
2
+7
!
= ____
5
+3
!
=8
5
+3
!
=8
5
+3
!
=5
5
+3
!
= ____
5
+3
!
= ____
4
+7
!
= ____
+ ____
4
+7
!
= ____
+ ____
5
+2
+3
+4
! = 8
+6
5
+2
+3
+4
! = 9
+ 5
2
+1
+1
!
= ____
+ ____
2
+1
+1
!
= ____
+ ____
1
+3
+ 2
+8
= ____
+ ____
2
+5
+ 9
+4
= ____
+ ____
7
+2
+ 6
+4
= ____
+ ____
3
+1
+ 2
+1
= ____
+ ____
+3 + ____
E F G H
1
+4
+ 6
+2
= ____
+ ____
6
+2
+ 1
+4
= ____
+ ____
2
+9
+ 7
+3
= ____
+ ____
7
+3
+ 2
+9
= ____
+ ____
2
+3
+ 2
+3
= ____
+ ____
4
+6
+ 4
+6
= ____
+ ____
7
+4
+ 7
+4
= ____
+ ____
1
+3
+ 2
+5
= ____
+ ____
3
+1
+ 5
+2
= ____
+ ____
6
+4
+ 1
+9
= ____
+ ____
4
+6
+ 9
+1
= ____
+ ____
3
+8
+ 2
+1
= ____
+ ____
5
+7
+ ___
2
+3
+ 3
+2
= ____
+ ____
4
+6
+ 6
+4
= ____
+ ____
2
+5
+ ___
___
+ ___
+ ___
+ ___
+ ___
= ____
= ____
+ ___
Can you make up a problem that matches?
+ ____
+ ____
= ____
+ ____
Fruits Facts 3: Fruits versus Decimals 2
+7
+ 3
+1
= ____
+ ____
1
+3
+ 2
+4
= ____
+ ____
3
+5
+ ___
27
+
31
= ____
13
+
24
= ____
35
+
32
= ____
1
+8
+ 2
+5
= ____
+ ____
3
+7
+ 1
+4
= ____
+ ____
18
+
25
= ____
37
+
14
= ____
+ ___
= ____
+ ____
Fruit Pears Pairs Let's use «5, 9» to mean 5 + 9 . The first number is always the number of apples, and the second number is always the number of pears. Can you fill in the blanks in these fruit facts?
«1, 6» + «2, 4» = «3, 10» «1, 6» + «4, 2» = «5, 8»
=
«7, 9» + «5, 11» = «___, ___»
«3, 7» + «10, 5» = «13, ___» «2, 14» + «9, 4» = «11, ___» «12, 10» + «12, 7» = «___, 17» «12, 14» + «18, 30» = «___, ___» «19, 75» + «12, 8» = «___, ___» «101, 3» + «30, 73» = «___, ___»
«33, 18» + «12, ___» = «45, 24» «40, 30» + «___, ___» = «61, 72» «12, 14» + «___, 8» = «23, ___» «___, 30» + «18, ___» = «59, 42»
1
+6
+ 2
+4
= 3
+ 10
1
+6
+ 4
+2
= 5
+8
7
+9
+ 5
+11
= ___
+ ___
You don't need to draw apples and pears here, but you can if you want.
Arrowbic Exercise You can add two arrows if one ends where the next starts. The sum is a new arrow, from the start of the first arrow to the end of the second arrow. One arrow in each triangle below is the sum of the other two. 16,5
5,3 11,2
5,3
11,2
5,-2
5,-2
11,4
11,4 16,2
1. What do the labels mean? (Hint: try counting lines of the grid.) 2. Add each pair of arrows below, and label all the arrows (the ones you're adding, and the result). 3. Look for patterns! A
B
C
D
F
G
E
F
H
Arrowbic Exercise: Part 2 You can add two arrows if one ends where the next starts. The sum is a new arrow, from the start of the first arrow to the end of the second arrow. One arrow in each triangle below is the sum of the other two. 16,5
5,3 11,2
5,3
11,2
5,-2
5,-2
11,4
11,4 16,2
1. Add each pair of arrows below, and label all the arrows (the ones you're adding, and the result). 2. Add the fruits, and the number pairs. 3. Look for patterns!
A
D
5
+1
+ 3
+4
= ___
+ ___
7
+2
+ 2
+6
= ___
+ ___
5
+4
+ 5
+1
= ___
+ ___
B
C
E
«5, 1» + «3, 4» = «____, ____» «7, 2» + «2, 6» = «____, ____» «5, 4» + «5, 1» = «____, ____»
«5, 3» + «1, 4» = «____, ____»
PART III: MORE SQUARROWS
Oliver Steele • email: [email protected] • web: http://osteele.com!
Sparrows (Squarrows with Pairs) +1
«2, 10»
«3, 11»
«3,2»
x2
+2 «2, 4»
«4, 6»
«2, 7»
«4, 8»
«5, ___»
«6, ___»
!2 «2, 4»
«2, 3»
!2
«4, 5»
!2
+3
+3 !2
«8, 1»
«15, 6»
4!
+7
x2
«2, 7» !3
4!
+7
!3
Squarrow Tails 10
-1
11
5
÷2
4
-1
÷2 -1
1 15
-1
÷2
2
-1
÷2 -1
-1
÷2
-1
÷2 -1
3
-1
÷2
-1
÷2 -1
3
÷2
-1
Squircles Find numbers that make a circle. Are any of these impossible? Do any have more than one answer?
5
4
-1
÷2
+3
+5
-10
x2
2
+1
-10
x3
-9
+10
+10
-10
x2
+3
÷2
-18
+5
Squarrows An elementary introduction to abstract algebra. INTRODUCING SQUARROWS
1
+1
2
+10
5
+2
Squarrow Chains 2
3
!2
4
+2
3
+4
5
!2
More Squarrows +1
2
+3
4
3
!2
2
6
2
4
10
-1
11
÷2
Squircles
5
-1
1
-1
4
Find numbers that make a circle. Are any of these impossible? Do any have more than one answer?
÷2
+1 3
2
+4
Squarrow Tails
6
5
2
3
8
+2
15
7
÷2
4
-1
+3 -1
-1
+5
÷2
-10
x3
+10
x2
2 ÷2
+6
!3
6
9
-1
2 3 3
+1
4
+2
+1
+2
+1
+2
+2
+1
6
+1
+3
5
+1
+1
+3 -1
+1
÷2
-9
-1
+1
-10
x2
÷2
9 -1
3
3 2
3
3
+1
!2
+1
!2
+7
8
+7 +10 -1
+6
+3
÷2
+3
-1
+6
-10
10
÷2
-18
+5
÷2
+7 !2
+1
-1
3
÷2
-1
Squarrows is a workbook for teaching algebraic concepts at the elementary school level. Squarrows is independent of the formalisms of algebra (such as variables, and the syntax of equations), but can be used to develop many of its concepts. Background I used Squarrows in teaching a group of fourth graders at the Wildwood Elementary School in Amherst, MA, in fall 2007. We met once a week for 45 minutes; we used a number of different problems worksheets, but used Squarrows as one of two “running problem sets”, with a new page every week or two. I learned a lot from the students – how to think about math, but also about the degrees of abstraction, reflection, and sophistication that even very young minds are capable of. Class was more like a graduate-school seminar; I got more ideas from them than they from me.
©2007 by Oliver Steele • email: [email protected] • web: http://osteele.com!
PART I: BASIC SQUARROWS
©2007 by Oliver Steele • email: [email protected] • web: http://osteele.com!
2
3
+2
+1
4
+2
+1
3
+4
5
+2
1
+1
2 5
+1
4
Squarrows
3
+2
+2
6
+2 1
3
2
5
0
3
3
9
5
+2
8
+3
9
2
3
!2
+10
4
+2
+2
3
+4
5
!2
1
+1
2 5
+3
4
Squarrow Chains
3
+4
+6
6
4
3
+1
3
+1
+2
+1
+2
+2
+1
3 3
3
+2
+1
!2
+1
!2
!2
+1
6
9
10
6
!2
8
!3
9
More Squarrows +1
2
2
6
2
4
+1 3
2
3
2
+2
7
+1
+3
5
+1
+3 +1
2
+1
+7
8 +6
+3 +7
+7
+6
PART II: A DIVERSION – NUMBER PAIRS
Oliver Steele • email: [email protected] • web: http://osteele.com!
Fruit Facts
A B C D
5
+3
!
=8
3
+5
!
= ____
7
+2
!
= ____
2
+7
!
= ____
5
+3
!
=8
5
+3
!
=8
5
+3
!
=5
5
+3
!
= ____
5
+3
!
= ____
4
+7
!
= ____
+ ____
4
+7
!
= ____
+ ____
5
+2
+3
+4
! = 8
+6
5
+2
+3
+4
! = 9
+ 5
2
+1
+1
!
= ____
+ ____
2
+1
+1
!
= ____
+ ____
1
+3
+ 2
+8
= ____
+ ____
2
+5
+ 9
+4
= ____
+ ____
7
+2
+ 6
+4
= ____
+ ____
3
+1
+ 2
+1
= ____
+ ____
+3 + ____
E F G H
1
+4
+ 6
+2
= ____
+ ____
6
+2
+ 1
+4
= ____
+ ____
2
+9
+ 7
+3
= ____
+ ____
7
+3
+ 2
+9
= ____
+ ____
2
+3
+ 2
+3
= ____
+ ____
4
+6
+ 4
+6
= ____
+ ____
7
+4
+ 7
+4
= ____
+ ____
1
+3
+ 2
+5
= ____
+ ____
3
+1
+ 5
+2
= ____
+ ____
6
+4
+ 1
+9
= ____
+ ____
4
+6
+ 9
+1
= ____
+ ____
3
+8
+ 2
+1
= ____
+ ____
5
+7
+ ___
2
+3
+ 3
+2
= ____
+ ____
4
+6
+ 6
+4
= ____
+ ____
2
+5
+ ___
___
+ ___
+ ___
+ ___
+ ___
= ____
= ____
+ ___
Can you make up a problem that matches?
+ ____
+ ____
= ____
+ ____
Fruits Facts 3: Fruits versus Decimals 2
+7
+ 3
+1
= ____
+ ____
1
+3
+ 2
+4
= ____
+ ____
3
+5
+ ___
27
+
31
= ____
13
+
24
= ____
35
+
32
= ____
1
+8
+ 2
+5
= ____
+ ____
3
+7
+ 1
+4
= ____
+ ____
18
+
25
= ____
37
+
14
= ____
+ ___
= ____
+ ____
Fruit Pears Pairs Let's use «5, 9» to mean 5 + 9 . The first number is always the number of apples, and the second number is always the number of pears. Can you fill in the blanks in these fruit facts?
«1, 6» + «2, 4» = «3, 10» «1, 6» + «4, 2» = «5, 8»
=
«7, 9» + «5, 11» = «___, ___»
«3, 7» + «10, 5» = «13, ___» «2, 14» + «9, 4» = «11, ___» «12, 10» + «12, 7» = «___, 17» «12, 14» + «18, 30» = «___, ___» «19, 75» + «12, 8» = «___, ___» «101, 3» + «30, 73» = «___, ___»
«33, 18» + «12, ___» = «45, 24» «40, 30» + «___, ___» = «61, 72» «12, 14» + «___, 8» = «23, ___» «___, 30» + «18, ___» = «59, 42»
1
+6
+ 2
+4
= 3
+ 10
1
+6
+ 4
+2
= 5
+8
7
+9
+ 5
+11
= ___
+ ___
You don't need to draw apples and pears here, but you can if you want.
Arrowbic Exercise You can add two arrows if one ends where the next starts. The sum is a new arrow, from the start of the first arrow to the end of the second arrow. One arrow in each triangle below is the sum of the other two. 16,5
5,3 11,2
5,3
11,2
5,-2
5,-2
11,4
11,4 16,2
1. What do the labels mean? (Hint: try counting lines of the grid.) 2. Add each pair of arrows below, and label all the arrows (the ones you're adding, and the result). 3. Look for patterns! A
B
C
D
F
G
E
F
H
Arrowbic Exercise: Part 2 You can add two arrows if one ends where the next starts. The sum is a new arrow, from the start of the first arrow to the end of the second arrow. One arrow in each triangle below is the sum of the other two. 16,5
5,3 11,2
5,3
11,2
5,-2
5,-2
11,4
11,4 16,2
1. Add each pair of arrows below, and label all the arrows (the ones you're adding, and the result). 2. Add the fruits, and the number pairs. 3. Look for patterns!
A
D
5
+1
+ 3
+4
= ___
+ ___
7
+2
+ 2
+6
= ___
+ ___
5
+4
+ 5
+1
= ___
+ ___
B
C
E
«5, 1» + «3, 4» = «____, ____» «7, 2» + «2, 6» = «____, ____» «5, 4» + «5, 1» = «____, ____»
«5, 3» + «1, 4» = «____, ____»
PART III: MORE SQUARROWS
Oliver Steele • email: [email protected] • web: http://osteele.com!
Sparrows (Squarrows with Pairs) +1
«2, 10»
«3, 11»
«3,2»
x2
+2 «2, 4»
«4, 6»
«2, 7»
«4, 8»
«5, ___»
«6, ___»
!2 «2, 4»
«2, 3»
!2
«4, 5»
!2
+3
+3 !2
«8, 1»
«15, 6»
4!
+7
x2
«2, 7» !3
4!
+7
!3
Squarrow Tails 10
-1
11
5
÷2
4
-1
÷2 -1
1 15
-1
÷2
2
-1
÷2 -1
-1
÷2
-1
÷2 -1
3
-1
÷2
-1
÷2 -1
3
÷2
-1
Squircles Find numbers that make a circle. Are any of these impossible? Do any have more than one answer?
5
4
-1
÷2
+3
+5
-10
x2
2
+1
-10
x3
-9
+10
+10
-10
x2
+3
÷2
-18
+5
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