Section 3: Demonstrates Conceptual Understanding of
Mathematical Operations
NECAP: M(N&O) – X – 3
Vermont: MX: 3
Note: Appendix A of the Grade-Level Expectations contains the information
related
to GLEs for grades K – 2.
N&0 – 29 Inverse relationships in operations:
Addition and subtraction are inverse
ope rations of each other because addition annuls subtraction and subtraction
annuls
addition. Similarly, multiplication and division are inverse operations of each
other . The
chart below shows a few examples
| Operation |
Inverse operation |
 |
Be advised that there are other inverse operations beyond
those shown in the chart above.
N&0 – 30 Relationship between repeated addition and
multiplication of whole
numbers: Multiplication of whole numbers is the same as repeated addition.
Example 30.1: 4 × 3 = 12
N&0 – 31 Relationship between repeated subtraction and
division of whole
numbers: Division of whole numbers is the same as repeated subtraction.
Example 31.1: 8 ÷ 4 = 2
N&0 – 32 Meaning of remainders with respect to division
of whole numbers: In
problem situations involving division of whole numbers, students must decide how
to
inter pret the remainder and then defend their interpretations.
Example 32.1:
One hundred thirty-four students will board school buses
to go on a field trip. Each
school bus has seats for 60 students. What is the fewest number of buses needed
to
seat all the students? Explain your answer.
Answer: Three buses will be needed to seat all the
students. If you divide 134 students by 60 students per
bus, you obtain 2 buses with a remainder of 14 students without seats. So, a
third bus is needed to seat
(accommodate) the remaining 14 students.
Example 32.2:
Ms. Thompson has saved $134 to buy some new shoes. She
will buy as many pairs
of shoes as she can. Each pair of shoes costs $60. How many pairs of shoes can
she buy? Explain your answer.
Answer: Ms. Thompson can buy 2 pairs of shoes. If you
divide $134 by $60 per pair of shoes, you obtain 2
pairs of shoes with a remainder of $14. Since she can not buy a third pair of
shoes with $14, she is restricted
to buying only 2 pairs of shoes.
N&0 – 33 Describing or illustrating the meaning of a
power: Given a base number
and an exponent , students will explain how the base number and the exponent
(power)
are related. (See N&0 – 24.)
N&0 – 34 Effect on magnitude of a whole number when
multiplying or dividing by a
whole number, fraction, or decimal: To de termine the effect on the magnitude
of a
whole number when multiplying it by a fraction or decimal means to determine
whether
the magnitude of the whole number increases, decreases, or stays the same by
considering the magnitude of the fraction or decimal.
Example 34.1: The diagram be low shows four paths
from the Start to the End. You
are given 100 points at the start. Without making any calculations describe
which
path will result in the largest number of points at the end. Explain your
reasoning.
When a whole number is multiplied
by a fraction or decimal whose value
is between 0 and 1, the value of the
resulting product is less than the
original whole number. |
|
When a whole number is divided by a
fraction or decimal whose value is
between 0 and 1, the value of the
resulting product is more than the
original whole number. |
| |
 |
|
| |
|
Multiplying by 3 and dividing
by  have the
same affect on
the magnitude of a number. |
Answer: The route “Divide by
and then divide by
” is the route that
would result in the largest number
of points at the end. Since dividing by
and multiplying by 3
have the same affect on the magnitude of a
number, the path that would result in the greatest number can be decided on the
first move. In addition,
division by a fraction between 0 and 1 results in a larger number than
multiplication by a fraction between 0
and 1. Therefore, one of the paths with division by a fraction will result in a
larger number than the path with
multiplication by a fraction. Since
is less than
, there are more
fourths in 100 than halves. (There are
400 fourths in 100, and 200 halves in 100.) Therefore, the path that includes
“divide by and then
divide
by ” will result in
the larger number.
