Integers

Expectations: B1.2, B1.3, E1.3

B1.2 read and represent integers, using a variety of tools and strategies, including horizontal and vertical number lines

B1.3 compare and order integers, decimal numbers, and fractions, separately and in combination, in various contexts

E1.3 plot and read coordinates in all four quadrants of a Cartesian plane, and describe the translations that move a point from one coordinate to another

Key Concepts

1. The negative integers are the “opposites” of the whole numbers. Each integer is the reflection of its opposite across a line perpendicular to and cutting the number line in 0.

2. Integer operations are based upon the zero principle, the fact that (-1) + (+1)=0

3. The meanings for the operations that apply to the whole numbers, fractions and decimals also apply to negative integers. Each meaning can be represented by a model, although some models suit some meanings better than others.

Social emotional learning Focus

Focus # 5: Develop self-awareness and sense of identity. Make connections among mathematical concepts, procedures, and representations, and relate mathematical ideas to other contexts (e.g. sports). See themselves as capable math learners, and strengthen their sense of ownership of their learning, as part of their emerging sense of identity and belonging.

Where do you see or hear positive and negative numbers being used?

Search in Google for Images of “Real Life Integers” (as a class or student led)

Discuss with a partner or as a class, one of the images and explain how the integers work. For example with temperature, when it is below zero or in the “minuses”, water freezes. When it is above zero in the “pluses” it is above freezing.

• Temperatures

• Floors below or above a main floor

• Being below or above sea level or ground level

• Golf scores below or above par

• Being in debt or not

Teacher note: When students are first starting to learn integers, it is encouraged to write a raised + or - sign or brackets around the number to write an integer, to make the distinction between these symbols and operation symbols for addition and subtraction. Source: Small, M. (2013) Making Math Meaningful to Canadian Students, K-8. Nelson Education (pp.324-325)

Provide students with for and against hockey scores for various teams. NHL Website They can use positive numbers for goals scored and negative numbers for goals against and rank the teams based on these scores.

Teacher Notes:

Principles for Comparing Integers

• Any negative integer is less than any positive integer.

• Every negative value is to the left of 0. Every positive value is to the right. Since the number line is built so that greater numbers are to the right, any positive integer must be greater than any negative number .A positive integer closer to 0 is always less than a positive integer farther away from 0; for example, +1 < +3

• 0 is to the left of all the positive integers. So a positive integer closer to 0 is farther left than one farther from 0. Since it is farther left, it is less.

• A negative integer closer to 0 is always greater than a negative integer father away from 0; for example, -7 > -10

Zero Principle

Mathematicians have defined (-1) as the number that you add to +1 to result in 0; that is, by definition, (-1) + (+1) = 0. This is referred to as the zero principle and is the foundation for computations involving negative numbers. As a consequence of this definition, any number can be added to its opposite to result in a value of 0.