Chapter 4: Place Value on the Horizon

First Lets Review Number Sense:

Match the term with the right definition.

1. Magnitude *2. Cardinality *3. Hierarchical Inclusion *

A. Know that numbers are included in other numbers (that 1 is actually part of 2, and that 1 and 2 are part of 3).

B. Knowing which group has more; size

C. Number tells quantity; means amount

Understand how students’ developing understandings about our number system parallels the historical development of number systems through the ages.

Understand that children’s symbol usage moves from tallies, to groups, to numerals – from additive systems to multiplicative systems.

Understand the importance of zero and unitizing – two big ideas.

After reading and discussing this chapter , we will……

How many?How many?

Our eye can perceive 5 or fewer. We can not see larger quantities – instead we must operate on large amounts to determine how many. Humans have constructed ways to organize amounts and represent them symbolically.

• 30,000 years ago: cavemen made tallies carved into walls. They used 1:1 correspondence.• Fifth Century BC systems were refined and symbols were created to represent different quantities. Units were represented by tallies, stones, or knots.• Almost all early number systems used ones, fives, tens and twenties – early counting was usually done with fingers and toes.

How does this match your students’ development? How does this match your students’ development?

Lets read Pg 52 – 54. The Invention of Numerals, Why Did Place Value Take So Long? & Helping Children Develop Mathematical Notation.

Pay attention to: Positional Notation, Zero & Unitizing.

NOW, THINK LIKE A CHILD – HOW HARD IS THIS? Understanding zero? What can 2 represent? 2, or 202, or 2,222? Cardinality of Units? (amount the unit

represents) Unitizing?

Look at the representations of different amounts made by children on pgs 55 - 61. Think about what the representations tell us about the students understandings.

Figure 4.7 was done in response to the following question – “We have a class of 28 students, how many cookies will be needed if each child got 5?” Read the first paragraph on pg. 62. Discuss. Share the importance of this critical landmark.

I have seen many of our students get stuck and continue to represent objects using 1:1 correspondence. We wonder, how do we move them forward?

Read the second paragraph on page 63 – think about fostering the development of organization and a system of tens in your classroom.

How important is this to later understandings about multiplication?

Early numerical systems were examples of additive numeration. The operation of addition is employed; the value of the number is equal to the sum of the values of the symbols. (IX = 9, XI = 11) l always equaled 1 whether it was placed before or after the X.

The positional notation that characterizes our number system was a big idea in the evolution of number systems, because it employed the operation of multiplication. 2 in the second column stands for 2 units of 10.

The Invention of Place Value – pg. 53

http://www.youtube.com/watch?v=_ofQ_WnQiZ4&NR=1 (Math Solutions: Cena)

http://www.youtube.com/watch?v=1puQxclB2aw (Math Solutions: Jonathon)

Discuss and share with the group how students’ developing understandings about our number system parallels the historical development of number systems through the ages.

Discuss and share with group the importance of zero and unitizing – two big ideas.

Discuss and share how students' representations of large groups in units helps to facilitate the move from an additive system to multiplicative system.