Mathematics 7 Unit 1: Integers - Nova Scotia

Mathematics 7 Unit 1: Integers

N06

Yearly Plan Unit 1: GCO N06

SCO N06: Students will be expected to demonstrate an understanding of addition and subtraction of

integers, concretely, pictorially, and symbolically.

[C, CN, PS, R, V]

[C] Communication [PS] Problem Solving

[CN] Connections

[ME] Mental Mathematics and Estimation

[T] Technology

[V] Visualization

[R] Reasoning

Performance Indicators

Use the following set of indicators to determine whether students have achieved the corresponding specific curriculum outcome.

N06.01 N06.02 N06.03 N06.04 N06.05 N06.06

Explain, using concrete materials such as integer tiles and diagrams, that the sum of opposite integers is zero. Illustrate, using a number line, the results of adding or subtracting negative and positive integers. Add two given integers, using concrete materials and/or pictorial representations, and record the process symbolically. Subtract two given integers, using concrete materials and/or pictorial representations, and record the process symbolically. Illustrate the relationship between adding integers and subtracting integers. Solve a given problem involving the addition and subtraction of integers.

Scope and Sequence

Mathematics 6

N07 Students will be expected to demonstrate an understanding of integers contextually, concretely, pictorially, and symbolically.

Mathematics 7

N06 Students will be expected to demonstrate an understanding of addition and subtraction of integers, concretely, pictorially, and symbolically.

Mathematics 8

N07 Students will be expected to demonstrate an understanding of multiplication and division of integers, concretely, pictorially, and symbolically.

Background

Integers are the set of numbers consisting of the natural numbers (1, 2, 3, ... ), their opposites (?1, ? 2, ?3, ...), and zero. They are also referred to as the whole numbers and their opposites. Integers indicate both a quantity (magnitude) and a direction from zero. Positive integers are greater than zero and are located to the right of zero on the number line. They are represented by a positive symbol (+) before the integer, such as (+5). Negative integers are less than zero and are located to the left of zero on the number line. They are represented by a negative symbol (?) before the integer, such as (?3). There are two common notations for integers. The symbols are written with the + or ? sign preceding the integer, as in ?5 or +3 or enclosed within parentheses, as in (+5), (?3). The parentheses are commonly used in student materials to avoid any confusion between the integer sign and the notations for addition and subtraction. In the expression (+5) ? (?3) the parentheses indicate the numbers inside are integers and distinguish the integer symbols from the subtraction symbol.

Understanding and working with integers is important in daily life. Integers are regularly encountered in contexts such as finances (net worth, balance sheets, and profit or loss),

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Yearly Plan Unit 1: GCO N06

investments, temperatures, elevations, time relevant to events, and sports. Provide students with contextualized problems of this nature. Proficiency with integers is necessary when evaluating algebraic expressions and solving equations. It allows students to graph relations using all four quadrants. Work with integers will be applied to future study of rational expressions, and extended to irrational and real numbers. It continues to build number sense, preparing students for a wide range of problem-solving activities. Operations with integers build on operations with whole numbers.

The balance of positive and negative values is known as the zero principle, and it is the foundation for many computations involving integers. Emphasis must be placed on the zero principle and its application in addition and subtraction situations. For example, (?1) + (+1) = 0, (?3) + (+3) = 0, (? 17) + (+17) = 0. The sum of any pair of opposite integers is zero. Based on this principle, zero and other integers can be expressed in multiple ways.

A tile or counter representing (+1) and one representing (?1) form a zero pair. When combined, these tiles model the number zero. A given integer can be modelled in many different ways. For example, one way to represent ?3 using integer tiles is shown below.

Using integer tiles or counters and the zero principle to model the various ways a number can be represented will help students conclude that adding a zero pair does not change the value of the integer being modelled. Work with zero pairs will be important as students concretely add and subtract integers. Students will progress to adding and subtracting integers symbolically. They will generalize and apply rules for these integer operations.

Concrete models commonly used for representing integers are coloured counters, alge-tiles, and number lines. Digital models also exist and simulations allow students to work with representations of positive and negative counters. Students should be exposed to multiple models. Parallel development, using both models at the same time, may be the most conceptual approach. Integers involve two concepts, "quantity" and "opposite." Quantity is modelled by the number of counters or length of the arrows in a vector model. Opposite is represented as different colours or different directions. An example of adding integers using each model is provided.

(+4) + (?1) = +3

John was a +4 in his first game of a hockey tournament. He was a ?1 in the second game of the tournament. What is his plus/minus at the end of the second game?

Bill was +4 at the end of the first 9 holes of golf. In the next 9 holes, his score was ?1. What is his score at the end of 18 holes of golf?

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Yearly Plan Unit 1: GCO N06

After a heavy rain, the Sackville River was 4 metres above its usual level. After one day, the river dropped 1 m. What was the level of the river on the second day?

Start at 0. Move 4 units to the right to represent +4. From there, move 1 unit to the left to represent ?1.

It was a dry summer in Shubenacadie. The river was 2 m below its normal level. During August, there was no rain, and the water level went down another metre.

How far is the river below the normal level now? (?2) + (?1) = (?3)

The distance an integer is from zero represents the magnitude of the integer, and the direction from zero represents whether the integer is positive or negative. While teachers may model correct use of absolute value (the distance from an integer to zero on a number line), it is not an expectation for students to know the term. On a vertical number line, the distance above zero represents positive integers. Distances below zero represent negative integers. Numbers increase in value above zero on a vertical number line, and decrease in value below zero on a vertical number line. On a horizontal number line, numbers increase in value to the right of zero, and decrease in value to the left of zero. Values always increase from left to right, and decrease from right to left.

The use of models helps students develop a conceptual understanding of the following principles for adding integers.

1. The sum of two positive integers is positive. 2. The sum of two negative integers is negative. 3. The sum of a negative integer and a positive integer can be negative or positive. The sum has the

sign of the number that is further from zero.

As with addition, it is important for students to have a conceptual understanding of integer subtraction. Students should model subtraction of integers using coloured counters and number lines. One possibility when subtracting integers is to use a "take away" meaning. This is easily manageable when the integers have the same sign, and the subtrahend is closer to zero than the minuend, or starting amount. For example, (?4) ? (?3) can be modelled with counters as follows.

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Yearly Plan Unit 1: GCO N06

When taking away is not immediately possible, it is necessary to add zero pairs to re-represent the minuend and allow the student to be able to remove the tiles that represent the subtrahend. Guide this developmental understanding through the use of good questions. To solve (?3) ? (+1) ask students how they could represent (?3) that would allow them to take away (+1).

To subtract integers, you can also use a "think addition" meaning. To determine (+1) ? (?2), ask

"How much must be added to (?2) to get to (+1)?" Using a number line, begin at (?2) and draw an arrow to (+1). It has a length of 3 pointing right. (+1) ? (?2) = +3.

While the rule "to subtract an integer, add the opposite" allows students to reach the correct answer, it does not promote conceptual understanding. Students should be led to this conclusion through the use of models. A good example would involve using the number line to subtract a negative from a positive. Such a situation should make it easier for students to see why you can add the opposite to subtract. From the previous example, (+1) ? (?2) tells what to add to (?2) to get to (+1). To go from (?2) to (+1), move 2 to the right to get to 0, and then another 1 to the right to get to (+1). The total amount to be added is (+2) + (+1) or, since addition is commutative, (+1) + (+2). Students should now see that (+1) ? (?2) = (+1) + (+2). Patterning can be used to develop this as well. Ask students to study a pattern such as the one given here.

(+4) ? (+2) = 2 (+4) ? (+1) = 3 (+4) ? (0) = 4

(+4) + (?2) = 2 (+4) + (?1) = 3 (+4) + (0) = 4

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