# Number and Operations

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.

 Definition Page Number Definition Number Describing or illustrating the meaning of a power 28 N&O – 33 Effect on magnitude of a whole number when multiplying or dividing by a whole number, fraction, or decimal 29 N&O – 34 Inverse relationships in operations 27 N&O – 29 Meaning of remainders with respect to division of whole numbers 28 N&O – 32 Relationship between repeated addition and multiplication of whole numbers 27 N&O – 30 Relationship between repeated subtraction and division of whole numbers 27 N&O – 31

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

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

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.

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