Lesson Notes By Weeks and Term v5 - Grade 7

Lesson Video

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Performance objectives

• Add and subtract with positive and negative integers.
• Multiply and divide with positive and negative integers.
• Use BODMAS to solve problems with integers.
• Solve real-life problems using integers.
• Understand the properties of integers.

Lesson summary

This week, we delve deeper into the world of whole numbers and integers, focusing on operations involving integers and their application in solving problems. Understanding integers and how to work with them is crucial in many aspects of life, from managing your pocket money to interpreting weather reports and even understanding basic concepts in accounting. In South Africa, where we often deal with both hot and cold weather and have a diverse economic landscape, a strong grasp of integers is essential for informed decision-making. Imagine tracking your airtime balance (sometimes overspending leading to a negative balance!) or understanding changes in the price of petrol.

Lesson notes

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

Examples: ..., -3, -2, -1, 0, 1, 2, 3, ... The set of integers is represented by the symbol 'ℤ'. 2.2 Understanding Positive and Negative Signs A positive integer is greater than zero (e.g., +5, 20). Often, the "+" sign is omitted, so 5 is the same as +

5. A negative integer is less than zero (e.g., -3, -10). The "-" sign is always written. Zero is neither positive nor negative. 2.3 Addition and Subtraction of Integers Adding integers with the same sign: Add the absolute values (the number without the sign) and keep the original sign.

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

8. Think: You have 3 rands, and you earn 5 rands. You now have 8 rands.

Example: (-2) + (-4) = -

6. Think: You owe someone 2 rands, and then you borrow another 4 rands. You now owe 6 rands.

Adding integers with different signs: Subtract the smaller absolute value from the larger absolute value. The result takes the sign of the integer with the larger absolute value.

Example: (+7) + (-3) = +

4. Think: You have 7 rands, but you owe someone 3 rands. After paying them back, you have 4 rands left.

Example: (-9) + (+2) = -

7. Think: You owe someone 9 rands, and you have 2 rands to pay them back. You still owe 7 rands.

Subtraction of Integers: Subtraction can be thought of as adding the opposite. Change the subtraction sign to addition and change the sign of the integer being subtracted. Then, follow the rules for addition.

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

Example: (+3) - (-4) = (+3) + (+4) = +7

Example: (-6) - (+1) = (-6) + (-1) = -7

Example: (-2) - (-5) = (-2) + (+5) = +3 2.4 Multiplication and Division of Integers Multiplying/Dividing integers with the same sign: The result is positive. (+) x (+) = (+) (-) x (-) = (+) (+) ÷ (+) = (+) (-) ÷ (-) = (+)

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

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

Example: (+10) ÷ (+2) = +5

Example: (-15) ÷ (-3) = +5 Multiplying/Dividing integers with different signs: The result is negative. (+) x (-) = (-) (-) x (+) = (-) (+) ÷ (-) = (-) (-) ÷ (+) = (-)

Example: (+6) x (-2) = -12

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

Example: (+8) ÷ (-4) = -2

Example: (-20) ÷ (+5) = -4 2.5 Order of Operations (BODMAS/PEMDAS) BODMAS/PEMDAS is an acronym to help remember the order in which to perform operations in mathematical expressions: Brackets / Parentheses Orders / Exponents Division and Multiplication (from left to right) Addition and Subtraction (from left to right)

Example: 2 + 3 x (-4) - 6 ÷ 2 Multiplication: 3 x (-4) = -12 Division: 6 ÷ 2 = 3 Expression now becomes: 2 + (-12) - 3 Addition: 2 + (-12) = -10 Subtraction: -10 - 3 = -13 2.6 Properties of Integers Commutative Property: The order in which you add or multiply integers does not change the result.

Addition: a + b = b + a (e.g., 3 + (-2) = (-2) + 3 = 1)

Multiplication: a x b = b x a (e.g., 4 x (-5) = (-5) x 4 = -20)

Associative Property: The way you group integers when adding or multiplying does not change the result.

Addition: (a + b) + c = a + (b + c) (e.g., (1 + 2) + (-3) = 1 + (2 + (-3)) = 0)

Multiplication: (a x b) x c = a x (b x c) (e.g., (2 x (-1)) x 3 = 2 x (-1 x 3) = -6)

Distributive Property: Multiplying a number by a sum or difference is the same as multiplying the number by each term in the sum or difference and then adding or subtracting the products. a x (b + c) = (a x b) + (a x c) (e.g., 3 x (2 + (-4)) = (3 x 2) + (3 x (-4)) = 6 + (-12) = -6) Guided Practice (With Solutions)

Question 1: Evaluate: -8 + 5 - (-2)

Solution: Rewrite subtraction as addition of the opposite: -8 + 5 + 2 Add -8 and 5: -3 Add -3 and 2: -1 Therefore, -8 + 5 - (-2) = -1 Question 2: Calculate: (-3) x 4 ÷ (-2)

Solution: Multiplication: (-3) x 4 = -12 Division: -12 ÷ (-2) = 6 Therefore, (-3) x 4 ÷ (-2) = 6 Question 3: Simplify: 10 - 2 x (3 - 5)

Solution: Brackets: 3 - 5 = -2 Multiplication: 2 x (-2) = -4 Subtraction: 10 - (-4) = 10 + 4 = 14 Therefore, 10 - 2 x (3 - 5) = 14 Question 4: A shopkeeper made a profit of R50 on Monday, lost R25 on Tuesday, made a profit of R30 on Wednesday, lost R40 on Thursday and made a profit of R15 on Friday. Calculate his total profit or loss for the week.

Solution: Represent profit as positive and loss as negative integers: +50, -25, +30, -40, +15 Add all the integers: 50 - 25 + 30 - 40 + 15 = 30 Therefore, the shopkeeper made a total profit of R30 for the week. Independent Practice (Questions Only)

Evaluate: -12 + 7 - 3 + 8 Calculate: 5 x (-6) ÷ 3 Simplify: 15 + 3 x (4 - 7)

Evaluate: -24 ÷ (-4) + 2 x (-3)

Simplify: 8 - (5 + 2 x (-1)) A thermometer reads -3°C in the morning. The temperature rises by 8°C during the day and then falls by 5°C at night. What is the final temperature? A mine shaft is 250m below the surface. A lift descends a further 120m. What is the new depth of the lift?