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This is one
A Journey into math and math instruction
Models by Teachers for Students
 Often we desire a model to make math easier for
students. It is there that there are 2 initial mistakes.
1. Our job is to not make math easier. It is to allow it to make
more sense to students.
2. Handing someone a model we have worked to make
sense for ourselves is actually adding more for them to learn
unless the model is internalized conceptually by the student.
String Theory
 Each of you will receive a small length of string
 Each length may be different
 This will be the beginning for all the math we do
today
“This is One”
 Hold up your string so you are displaying its
whole length and say, “This is one!”
 Why is this one?
 “Because I said so!”
Using your “one” to show larger
numbers
 Say , “This is one” so this is two” (show what two
would be)
Extend to multiplication
 You have shown 2 and 3 with your length of 1.
 Show 2 X 3. Remember, all representations begin
with the “one” you set up from the beginning.
How about division
 Use your string to show 1÷2 (with result).
 Do this on your own first
 Compare your demonstration with a partner
Second division problem
 Use your string to show 1÷1/2 (with result).
 Do this on your own first
 Compare your demonstration with a partner
Issue to explore!
 How can I demonstrate division in one way that works for
1÷2 and 1÷ ½.
In other words, where dividend, divisor and quotient are
represented in the same consistent manner?
A Little History
 We first learn about division through whole numbers
 We extend that to other rational numbers such as
fractions and decimals
Primary students see division
two ways.
 These two ways are called measurement and partition.
 Young students do this naturally, but in math instruction
the distinction becomes fuzzy.
Partitive Division
(Divisor is number of sets)
 When dividing an amount by 2 we are taking the amount
and separating it into two equal sets. Think of separating
what you have into two bags:
Partitive Division
(Divisor is number of sets)
 Imagine you have 8 dots:
• • • • • • • •
When I divide by 2, I split that 8 into two equal
groups. Each group has 4:
• • • •
• • • •
8÷2 = 4
Measurement Division
(Divisor is size of units to count)
 Imagine you have 8 dots:
• • • • • • • •
This time, you are now counting sets of 2 dots
Measurement Division
(Divisor is size of units to count)
 Imagine you have 8 dots:
• • • • • • • •
1
This time, you are now counting sets of 2 dots
Measurement Division
(Divisor is size of units to count)
 Imagine you have 8 dots:
• • • • • • • •
1
2
This time, you are now counting sets of 2 dots
Measurement Division
(Divisor is size of units to count)
 Imagine you have 8 dots:
• • • • • • • •
1
2
3
This time, you are now counting sets of 2 dots
Measurement Division
(Divisor is size of units to count)
 Imagine you have 8 dots:
• • • • • • • •
1
2
3
4
This time, you are now counting sets of 2 dots
There are 4 sets of 2 in 8.
8÷2=4
1 ÷ 1/2
 1÷ ½
“How many one-halves in 1?”
 Answer: There are two one-halves in 1.
Dividing a number by 1/2
 1÷ 1/2
“How many one-halves in 1?” 1 ÷ ½ = 2
 2 ÷ 1/2
“How many one-halves in 2?” 2 ÷ ½ = 4
 4 ÷ 1/2
“How many one-halves in 4?” 4 ÷ ½ = 8
Dividing a number by 1/2
 1÷ 1/2
“How many one-halves in 1?” 1 ÷ ½ = 2
 2 ÷ 1/2
“How many one-halves in 2?” 2 ÷ ½ = 4
 4 ÷ 1/2
“How many one-halves in 4?” 4 ÷ ½ = 8
 So, what would 10 ÷ ½ be equal to?
What is the usual rule?
 To divide by a fraction, multiply by its reciprocal.
 “What?”
 Example --- the reciprocal of ½ is 2/1.
Divide 4 by 1/2 : 4÷1/2
=
4 1
¸
1 2
=
4 2
x
1 1
=
8
1
= 8
In short, we multiplied by 2 when dividing by 1/2.
The common algorithm
Divide 4 by 1/2 : 4÷1/2
4
1
= 1¸2
4 2
x
1 1
8
=
1
=
= 8
The shortcut algorithm works – why?
What is gained from conceptually understanding division by
a fraction?
Common Core and Division
Grade 5 Number and Operations- Fractions
 Cluster: Apply and extend previous understandings of
multiplication and division to multiply and divide
fractions.
 5.NF.7 Apply and extend previous understandings of
division to divide unit fractions by whole numbers and
whole numbers by unit fractions.
Common Core and Division
Grade 5 Number and Operations- Fractions
 Cluster: Apply and extend previous understandings of
multiplication and division to multiply and divide
fractions.
 5.NF.7 Apply and extend previous understandings of
division to divide unit fractions by whole numbers and
whole numbers by unit fractions.
What are these “Previous Understandings”?
Area Model- Whole numbers
 Use an area model (array) to show 3 X 4. Label factors
and product
Area Model- Whole numbers
 Use an area model (array) to show 3 X 4. Label factors
and product.
 Use your model to show the relationship between 3x4=12
and division with related facts to that equation.
Area Model- Where is one?
Use your model to show where one is in
the factors and in the product.
Area Model- Product “1” is sq. unit
1+1 +1 +1
Factors (length)
1
+
1
+
1
1
3 x 4 = 12
Factors are dimensions in length.
Product is area in square units
“1” is a 1x1 unit square.
Product (area)
Area Model- Fractions
 Use an area model (array) to show 1/2 X 4. Label factors
and product.
 Use your model to show the relationship between
½ x 4 = 2 and division with related facts to that equation.
Factors are ½ and 4
4
1
2
4 is 1+1+1+1 in length
4
1
2
1
+
1
+
1
+
1
Product is measured in area.
4
1
2
1
+
1
+
1
+
What is the area of the shaded region?
1
This is 4 regions 1 by 1/2
4
1
2
1
2
1
+
1
+
1
+
1
What is the area of the shaded region?
Each of the 4 regions is ½ x 1. Each has an area of ½ sq. units
Total area is the product
4
1
2
1
2
1
2
1
+
1
1
2
+
1
1
2
+
1
What is the area of the shaded region?
Each of the 4 regions is ½ x 1. Each has an area of ½ sq. units
Total area = 2.
½x4=2
Division as inverse of Multiplication
?
1
2
2
2÷½=?
What times ½ would give the product 2?
Division as inverse of Multiplication
1
2
1
2
1
2
4
Area (dividend) = 2
1
2
1
2
2÷½=?
“I need 4 halves to make 2 because 4 X ½ = 2”
2÷½=4
Making Models Powerful
 Models for instruction are to provide opportunities for
exploring concepts to build understanding. The power of
models such as arrays is not for solving problems.
 The first step to being able to use a model is being able to
describe what the parts of the model represent. From there,
talking about the mathematics being represented provides
a greater window into a student’s mathematical thinking.