Introductory
Lesson on Quantity, Teacher Notes 2.1
R.MacDuff and R.A. Hewko, Copyright
Info Dynamics Applications, 2007
The main purpose of this unit is to develop an understanding of number and its role in the description of quantities. A quantity is, roughly speaking, something that has – or comes in – units, such as three horses or four and one-half inches or five dots. Students think of numbers as a stand-in for quantity, an abstraction of it. They will use this concept of number as a starting place for understanding quantity, and, as they move through this unit, finally arrive at a deeper understanding of number.
Each of the activities to follow involves dots, which can be used to represent any discrete quantity. Although the students will be using numbers, surprisingly they will not be seeing the numbers as a part of arithmetic. They will just see each numeral as a symbol for representing different quantities of dots or different quantities consisting of groups of dots. In fact when most students write down the numerical symbols these are not symbols to represent mathematical concepts, but rather symbols to represent discrete things. The students have not separated their concept of number from the concept of quantities. It is important that the students discover the link between their descriptions of objects, their own actions on objects and their concept of number.
This
diagram can be thought of as being composed of both dots and groups of
dots. The object can be described
as having two groups, where each group has three dots. However the same object can be
regrouped into three groups, each having two dots. For the student to understand the process by which they can
change the grouping structure and know and understand what stays the same and
what changes, they need conceptual tools. These tools allow students to construct ideas that are
relevant to the problem. The ability to represent their ideas requires representational
tools. These
representational tools provide ways of translating their ideas into a form that
can be considered and communicated.
In this section the students will learn to distinguish between dots and groups of dots (and groups of groups of dots). They will learn to pay attention to the way the dots are grouped, and will learn to represent these entities using numerals (the mathematical symbols for number: 1, 2, 3, 4, … etc.), the operation + and the equality (=) relation.
Conceptual Tools
The basic ideas are that objects can be grouped into groups or multiple groups and that the rearrangement of the groups or the number of objects within a group does not alter the total number of objects.
The importance of dots is that every quantity can be represented by dots, a group of dots or as multiple groups of dots. The students, by rearranging the same quantity of dots in multiple ways, will become aware that the group structure does not influence the total quantity of dots.
A description of the dots requires the student to pay attention to both dots and groups of dots. There are three things that the student must be aware of when examining different groups of dots:
1/ When are things “the same”? Things are the same when they have the same number of groups and each group contains the same number of dots.
2/ When are things “not the same”? Things are not the same when the total number of dots is not the same.
3/ When are things “the same but not the same”? Things are the same but not the same when the quantity of dots is the same but the number of groups is different.
Representational Tools
Students will learn to represent each dot diagram, where circles indicate groups of dots, in two different ways:
a) in words; -- two groups of four dots;
b) with a number sentence.
In
a number sentence, a group of dots is represented by brackets ( ), and the number of dots within the
group is indicated by a number placed within the group symbol, e.g. (4). A number placed in front of the group
symbol, e.g. 2(4), indicates the number
of groups. For example, the diagram
to the right can be described in the following way: 2(4) = 8, which can be read as “2 groups, where each
group is made up of 4 dots”.
Many students will want to read 2(4) as “two times
four”. This should be
discouraged initially, so that the student will focus on the meaning associated
with the mathematical phrase. (The
language “two times four” will be used later to describe the
process of constructing the symbol for an equivalent group.) The students will be surprised when
they eventually discover that the descriptions they have created can be
interpreted in terms of multiplication!
Students will be led to articulate the insight that the order followed in describing the grouping possibilities of a quantity – e.g. “three groups of two dots” or “two groups of three dots” – is important. However while these both represent the same quantity of dots they are nevertheless not the same thing, this is the concept of equivalence: the same but not the same.
The initial activities are designed to tap into the students’ creativity. Thus students are asked to be creative in writing as many different ways as possible of organizing a given set of dots and representing those organizations using mathematical symbols.
It is important in the beginning to keep the different representations separated, as each representation has its own syntax and rules. Mixing the representations can cause much confusion, as it makes it extremely difficult for the student to learn the different syntaxes associated with each language.
Consider the following example where there are eight dots. Starting out with eight dots in total the student is asked to regroup the dots in a variety of different ways, and is asked to represent what s/he has constructed using both symbols and words.
Representations
Dot Diagrams
a. Dots: (Eight dots, no groups)
b. Word sentence: “Two groups of four dots
is the equivalent to eight dots.”
Number sentence: 2(4) = 8
c. Word sentence: “Four groups of two dots is
equivalent to eight dots.”
Number sentence: 4(2) = 8
d. Word sentence: “One groups of eight dots is
equivalent to eight dots.”
Number sentence: 1(8) = 8
(Students could also write 4+ 4 = 8, 2 + 2 + 2 + 2 = 8, etc.)
Students should be led to articulate the insight that the order followed – e.g. three groups of two dots or two groups of three dots – is important. While they both represent the same number of dots and are therefore “equivalent”, they have different grouping structures, and are therefore “the same but not the same”. Some students will have a struggle with this. Direct their attention to the grouping structure as being important. Invite them to think of examples of things that are grouped differently but are equivalent. For example, a 12-pack, two six-packs and twelve singles of 12-ounce cans of Pepsi are equivalent – the same amount of Pepsi - but they won’t be mixed together on the shelves, they won’t be “rung up” as the same thing at the cash register and they may not cost the same amount.
Students will persist in using the word
“equals” to describe the relationship between two expressions that
contain the same number of dots.
Gently but persistently steer them back to using the word “equivalent”
since the focus to get students to see they are “the same but not the
same.”
Students will tend to skip writing the “word
sentences”. These are an absolutely essential part of the description
process and of the students’ development of their ability to think about
mathematical concepts using multiple representations. Keep reminding them not to skip the written part of an
answer.
Simple Grouping
– Activity 1 Name:
Objectives:
After completing this section the student will understand:
1/ a group can
be regrouped in multiple ways.
3(2) = 2(3) = 1(6) = 6(1)
2/ regrouping
does not affect the total number. 3(2) =6= 2(3)
Carefully consider each set of dots given
below. Place a circle around dots
to form groups in as many different ways as possible. Describe each regrouping using both mathematical symbols and
words. Pay close attention to the groups and the dots within each group.
a)
Various answers: Look for patterns in the answer, and
help students to see new arrangements such as (a) all groups having an
equal number of dots, (b) 10 groups of one dot, (c) 1 group of ten dots,
(d) one group of seven dots with 3 dots left over, (e) five groups of zero
dots and one group of 10 dots, (f) ½ group of 20 dots, (g) all
groups different, etc. Try to
help students to be creative and remove self-imposed blockages to learning.
c)
Change the following word sentences into number
sentences and dot diagrams.
d) two groups of five dots is equivalent to ten dots.
2(5) = 10
1) number sentence ----
2)
dot diagram
f) one group of two dots and two groups of one dot is equivalent to _______________
1) number sentence ----
2) dot diagram
Change the following number sentences into word
sentences and dot diagrams.
1(21)
h) 6(2) + 3(3) = ___________.
1) word sentence ----
Six groups of two plus three
groups of three is the same as one group of twenty one.
2)
dot diagram
i) 1(5) + 4(3) = _________.
1) word sentence ----
2) dot diagram