Lesson Notes By Weeks and Term v5 - Grade 7

Lesson Video

This page supports the lesson note with a companion video and a short classroom-ready summary.

For class groups and homework, share this lesson page so learners also get the summary, objectives, and full lesson context.

Performance objectives

• Order and compare integers on a number line.
• Add and subtract integers.
• Multiply and divide integers.
• Solve real-world problems using integers.
• Identify and use additive and multiplicative inverses.

Lesson summary

This week, we delve deeper into whole numbers and integers, building on the basic understanding you developed in previous grades. Mastering this topic is crucial because whole numbers and integers form the foundation for almost all other mathematical concepts. From calculating your savings in rands to understanding temperature changes or even managing your tuckshop inventory, the applications of this knowledge are vast and relevant to your everyday life in South Africa. Imagine you're planning a braai with friends – you need to calculate the cost of the meat, drinks, and snacks, and ensure you have enough money. This involves working with whole numbers and integers.

Lesson notes

2.1 Integers: Integers are whole numbers (no fractions or decimals) that can be positive, negative, or zero.

The set of integers is represented as: {..., -3, -2, -1, 0, 1, 2, 3, ...}.

Positive Integers: Whole numbers greater than zero (1, 2, 3, ...). These are used for counting objects, representing money earned, or measuring distances above sea level.

Negative Integers: Whole numbers less than zero (-1, -2, -3, ...). These can represent debts, temperatures below zero degrees Celsius (important in some parts of South Africa, especially during winter), or distances below sea level.

Zero: Zero is an integer that is neither positive nor negative. 2.2 Number Line: The number line is a visual representation of numbers, with integers placed at equal intervals along a horizontal line. Zero is at the center, with positive integers to the right and negative integers to the left. The number line helps in visualizing the order and relationship between integers.

Example: Draw a number line and plot the integers -3, 0, 2, and -

1. Explanation: Draw a horizontal line. Mark a point as zero. Mark points equally spaced to the right of zero as 1, 2, 3,... and to the left of zero as -1, -2, -3,... Now, locate and mark the given integers on the number line. 2.3 Ordering and Comparing Integers: Integers are ordered based on their position on the number line. Numbers to the right are greater than numbers to the left.

Examples: 5 > 2 (5 is greater than 2) -1 -2 (0 is greater than -2) Real-life

Example: Imagine the temperature in Johannesburg is 25°C and in Sutherland it's -5°

C. Johannesburg is warmer because 25 > -5. 2.4 Addition and Subtraction of Integers: Adding Integers with the Same Sign: Add the absolute values of the numbers and keep the same sign.

Example: (-3) + (-2) = -5 (Think: You owe R3 and then borrow another R2, so you owe R5)

Example: (+4) + (+5) = +9 (Think: You have R4 and earn another R5, so you have R9)

Adding Integers with Different Signs: Subtract the smaller absolute value from the larger absolute value and keep the sign of the number with the larger absolute value.

Example: (-7) + (+3) = -4 (Think: You owe R7 and pay back R3, so you still owe R4)

Example: (+8) + (-2) = +6 (Think: You have R8 and spend R2, so you have R6 left)

Subtracting Integers: Change the subtraction to addition and change the sign of the number being subtracted. Then follow the rules for addition.

Example: 5 - 3 = 5 + (-3) = 2

Example: 2 - 5 = 2 + (-5) = -3

Example: 4 - (-2) = 4 + 2 = 6 (Subtracting a negative is the same as adding a positive)

Example: -3 - (-1) = -3 + 1 = -2 2.5 Multiplication and Division of Integers: Multiplying/Dividing Integers with the Same Sign: The result is always positive.

Example: (-2) x (-3) = 6

Example: (4) x (2) = 8

Example: (-6) ÷ (-2) = 3

Example: (8) ÷ (2) = 4 Multiplying/Dividing Integers with Different Signs: The result is always negative.

Example: (-4) x (3) = -12

Example: (5) x (-2) = -10

Example: (-9) ÷ (3) = -3

Example: (10) ÷ (-2) = -5 Mnemonic: "Same sign, positive; different signs, negative." 2.6 Additive and Multiplicative Inverses Additive Inverse (Opposite): The additive inverse of a number is the number that, when added to the original number, results in zero. For example, the additive inverse of 5 is -5, because 5 + (-5) =

0. Similarly, the additive inverse of -3 is 3, because -3 + 3 =

0. Multiplicative Inverse (Reciprocal): The multiplicative inverse of a number is the number that, when multiplied by the original number, results in one. For example, the multiplicative inverse of 2 is 1/2, because 2 (1/2) =

1. The multiplicative inverse of -4 is -1/4, because -4 * (-1/4) =

1. Note that zero does not have a multiplicative inverse. Guided Practice (With Solutions)

Question 1: Order the following integers from smallest to largest: -5, 2, -1, 0, 4, -

3. Solution: -5, -3, -1, 0, 2,

4. We can visualize this on a number line to confirm the order.

Commentary: We are ordering from left to right on the number line. Numbers further to the left are smaller.

Question 2: Calculate: (-8) + (5) - (-2)

Solution: (-8) + (5) - (-2) = (-8) + (5) + (2) (Change subtraction to addition and change the sign) = (-8) + (7) = -1

Commentary: Break down the problem into smaller steps. First, change the subtraction to addition. Then, add the numbers, remembering to consider the signs.

Question 3: Calculate: (-4) x (3) ÷ (-2)

Solution: (-4) x (3) ÷ (-2) = (-12) ÷ (-2) (Multiply first, -4 x 3 = -12) = 6 (Divide, and since both numbers are negative, the result is positive)

Commentary: Remember the order of operations (multiplication and division from left to right). Pay close attention to the signs of the numbers.

Question 4: What is the additive inverse (opposite) of -12?

Solution: The additive inverse of -12 is

1

2. Commentary: The additive inverse is the number that, when added to -12, gives zero. -12 + 12 =

0. Question 5: What is the multiplicative inverse (reciprocal) of 5?