Transcript The Empty Number Line: A Model For Thinking

The Empty Number
Line:
A Model For Thinking
Math Alliance
DeAnn Huinker & Beth Schefelker
April 27, 2010
Representations:
Models for Thinking
Strengthening the ability
to move between and
among representations
improves the growth of
children’s conceptual
understanding.
Vandewalle, J. Elementary and Middle School Mathematics
Teaching Developmentally. Pearson Education, 2007.
.
pictures
manipulative
models
written
symbols
oral
language
Real-world
situations
Models for thinking
A model for thinking about a mathematical concept
refers to any object, picture, or drawing that
represents the concept.

To see a concept in a model you must have some
relationship in your mind to impose on the model.

Models give children something to think about,
explore with, talk about, and reason with.
What is Cosette thinking?
The book cart has an assortment of things to read for
indoor recess. Out of the 70 books, 37 are comic books.
How many are not comic books?
Solve it (no standard algorithms allowed). Then turn &
share.
Cosette’s Model

What mathematical understanding is Cosette
demonstrating in her thinking?

How is she thinking about 70 – 37 = ?

How would she solve 80 – 52 = ?
The empty number line: A useful tool or
just another procedure?
Bobis, J. Teaching Children Mathematics. April 2007

What are the benefits for using an open number line
for students?

What are the benefits for using an open number line
for teachers?

What needs to be considered when introducing the
open number line to students?
Try the strategies in the article
Each person selects one strategy. Practice it. Use an empty
number line, if appropriate. Then teach it to your table partners.
A. Counting by 10
 23 + 40
 67 + 20
C. Jump Strategy
 46 – 39
 72 – 51
E. Split Strategy
 35 – 21
 56 + 32
B. Bridging to 10
 7+8
 38 + 5
D. Compensation Strategy
 45 – 29
 57 + 39
What connections are you finding to the strategies we
have explored and discussed in class?
Looking At Student Work


Pass out the four pieces of student work.
Analyze what the student is doing.

How are students keeping track of their thinking?

What relationships are students demonstrating as
they use the empty number line as a model for
their thinking?

How is the work similar? How is it different?

Can you connect the work to any of the strategies
from the article?
The book cart has an assortment of things to read for
indoor recess. Out of the 70, books 37 are comic
books. How many are not comic books?
How would you explain why you “add” the 1?
Big ideas behind the
Empty Number Line

Numbers can be decomposed and the subunits or smaller
amounts can be added or subtracted in varying orders, yet
still be equivalent.

25 +17 = 25 + 10 + 7
25 + 10 + 5 + 2
25 + 5 + 10 + 2

Place value patterns occur when adding on groups of ten.
 27, 37, 47, 57

Unitizing
 1 jump of 10 is the same as counting 10 units
Study of Sam
How does Sam use his understanding of
number and relationships as he works on the
math problems?

3rd Grader at Elm Creative Arts
26 + 39 = ____ + 27
clip starts at 2:10

Solve the problem.

What mathematical understandings are
evidenced in this work?
SAM

What does Sam know about the math?

What is Sam struggling with as he works?

What instructional moves were made by the
teacher?
71 – 69 = ?
clip starts at 5:29

Solve the problem.

Share your thinking with a neighbor.
SAM

What does Sam know about the math?

What is Sam struggling with as he works?

What instructional moves were made by the
teacher?
14 – 6 = ?

Turn and talk

Describe Sam’s method.
SAM

What does Sam know about the math?

What instructional moves were made by
the teacher?
Meet Grant
30:13
How does Grant approach solving…
25 + 37
How did the model help Grant explain
his thinking?
What are you walking
away with. . . ?

What thoughts are you walking away with as
you thought about the instructional moves in
the video clips?

In what ways do models for thinking support
the development of computational fluency?