MTH 492/592
Homework 1.2 Whole Number Properties
1. For each question, address whether or not each of the properties does or does
not hold as directed on the next two pages under the “does hold” and ‘does not
hold” category directions (and examples). Be sure to do all four parts DH-1, 2, 3
& 4 for properties that do hold and all three parts DNH-1, 2 & 3 for properties
that do not hold. There is not enough room to write on this page.
a. The Commutative Property for Whole Number Multiplication
b. Commutative Property for Whole Number Division
2. Do the following properties hold? If so, write the symbolic statement (DH1) and
briefly explain what the property means. If not, give a numerical
counterexample (DNH 2)
a. Associative Property for Whole Number Addition
b. Associative Property for Whole Number Subtraction
c. Associative Property for Whole Number Multiplication
d. Associative Property for Whole Number Division
e. Distributive Property for Whole Number Multiplication over Addition
f. Distributive Property for Whole Number Multiplication over Subtraction.
Topic 1.2 HW, page 1
PROPERTY DOES HOLD DIRECTIONS & EXAMPLES (DH-1, 2, 3, & 4)
DH-1
DH-2
DH-3
DH-4
Symbolic Statement
Write the symbolic statement of what it means for the property to hold.
Numerical Example
Show that at least one numerical example holds. Use words (verbal) and pictures
(visual). This should be suitable for a young learner, and also foreshadow the
general proof in DH-4.
Real World Example
Give a concrete example using a real world context that would illustrate the
property for young learners.
General Proof
Explain why the property holds—in general. Your explanation should be
analogous to DH-2 without switching to some new point of view. This is the “leap
to abstraction” and like DH-2, should be a general visual, verbal and symbolic
response for adult learners.
Example The Commutative Property for Whole Number Addition DOES HOLD
Symbolic example DH1
a + b = b + a for all a, b in W (or a + b = b + a ∀ a, b  W)
Numerical example DH2
The order in which you add whole numbers does not matter. For example, if I
start with 2 tiles and add 5 tiles, or if I start with 5 tiles and add 2 tiles, I see that
both groups have 7 tiles and the order in which I add the tiles does not matter.
Real world example DH3
Suppose Jane has 5 marbles and Joe has 2 marbles and they decide to put their
marbles together in a bag. It doesn’t matter if Jane puts in her 5 marbles first and
then Joe puts in his 2 marbles, or if Joe puts his 2 marbles in first and then Jane
puts in her 5 marbles. Either way there will be 7 marbles in the bag.
General proof example DH4
The order in which you add whole numbers does not matter. For any whole
numbers a and b, you can think about having one group of a tiles and another
group of b tiles – it doesn’t matter if you add the first group to the second group or
the second group to the first group, you still have the same total number of tiles
(the sum).
a + b is represented by
■ ■ ■ ■ ■ ■ ■ …■ ■ □ □ □ …□
b + a is represented by
□ □ □ …□ ■ ■ ■ ■ ■ ■ ■ …■ ■
Lined up in this way, we see the tiles match-up and the total number of tiles is the
same in either order. Therefore, a + b = b + a ∀ a, b  W.
Topic 1.2 HW, page 2
PROPERTY DOES NOT HOLD DIRECTIONS & EXAMPLES (DNH-1, 2 & 3)
DNH-1
DNH-2
DNH-3
Symbolic Statement
Write the symbolic statement of what it means for the property to not hold.
Numerical Counterexample
Give at least one numerical counterexample. Use words (verbal) and include
pictures (visual) if you think it would help a young learner.
Real World Example
Give a concrete example using a real world context that would illustrate how this is
not a property for young learners.
Example Symbolic example DNH1
a - b ≠ b - a for some a, b in W (or a - b ≠ b - a for some a, b  W)
Numerical counterexample DNH2
The order in which you subtract does matter. For example,
7 – 4 = 3, but 4 – 7 = -3. Since 3 ≠-3, the order in which you subtract whole numbers
does matter.
Real word example DNH3
Suppose Jane has $7 and gives her friend Joe $4. Jane now has $7 - $4 = $3.
Suppose Jane has $4 and wants to give her friend Joe $7, since $4 - $7 = -$3, Jane
is $3 short and she can’t do this.
The order in which subtract whole numbers does matter.
Topic 1.2 HW, page 3