Lesson Notes By Weeks and Term v5 - Grade 7

Lesson Video

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Performance objectives

• Compare and order integers on a number line.
• Add and subtract integers.
• Multiply and divide integers.
• Solve word problems involving integers.
• Identify and apply the properties of integers.

Lesson summary

This week, we delve deeper into the world of whole numbers and integers, building on what you've already learned. Understanding whole numbers and integers is crucial because they form the foundation for almost all mathematical concepts you will encounter. From budgeting your pocket money to calculating distances during your holiday travels, integers are essential tools. In the South African context, understanding integers can help you analyze financial data, understand temperature variations in different regions, and even interpret sports statistics. Think about calculating goal differences in soccer or understanding profit and loss in a small business.

Lesson notes

2.1 What are Whole Numbers and Integers?

Whole Numbers: Whole numbers are the set of numbers starting from zero and continuing infinitely upwards: 0, 1, 2, 3, 4, 5, and so on. They do not include fractions or decimals.

Integers: Integers are the set of whole numbers and their negative counterparts. This means they include all positive whole numbers, zero, and all negative whole numbers.

Examples: ..., -3, -2, -1, 0, 1, 2, 3,... 2.2 The Number Line: The number line is a visual representation of integers. Zero is in the middle, positive integers are to the right, and negative integers are to the left. The further a number is to the right, the larger it is. The further a number is to the left, the smaller it is.

Example: Consider -5 and -2. -2 is to the right of -5 on the number line, therefore -2 > -5 ( -2 is greater than -5). 2.3 Adding and Subtracting Integers: Adding Integers with the same sign: Add the absolute values of the numbers and keep the same sign.

Example: (-3) + (-4) = -7 (Add 3 and 4 to get 7, then keep the negative sign).

Example: 5 + 2 = 7 (Add 5 and 2 to get 7, keep the positive sign - although we usually don't write it).

Adding Integers with different signs: Subtract the smaller absolute value from the larger absolute value. Keep the sign of the number with the larger absolute value.

Example: (-7) + 3 = -4 (The absolute values are 7 and 3. 7 - 3 =

4. Since 7 has a larger absolute value and it is negative, the answer is -4).

Example: 4 + (-1) = 3 (The absolute values are 4 and 1. 4 - 1 =

3. Since 4 has a larger absolute value and it is positive, the answer is 3).

Subtracting Integers: Subtracting an integer is the same as adding its opposite. Change the subtraction sign to an addition sign and change the sign of the number being subtracted.

Example: 5 - (-2) = 5 + 2 = 7

Example: (-3) - 4 = (-3) + (-4) = -7 2.4 Multiplying and Dividing Integers: Multiplying/Dividing Integers with the same sign: The result is positive.

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

Example: 6 / 2 = 3 Multiplying/Dividing Integers with different signs: The result is negative.

Example: (-4) 2 = -8

Example: 10 / (-5) = -2 2.5 Properties of Integers Commutative Property: The order doesn't matter in addition and multiplication. a + b = b + a (e.g., 2 + 3 = 3 + 2) a b = b a (e.g., 4 5 = 5 * 4)

Associative Property: The grouping doesn't matter in addition and multiplication. (a + b) + c = a + (b + c) (e.g., (1 + 2) + 3 = 1 + (2 + 3)) (a b) c = a (b c) (e.g., (2 3) 4 = 2 (3 * 4))

Distributive Property: Multiplying a number by a sum is the same as multiplying the number by each term in the sum and then adding the results. a (b + c) = (a b) + (a c) (e.g., 2 (3 + 4) = (2 3) + (2 * 4)) Guided Practice (With Solutions)

Question 1: Represent the following integers on a number line: -4, 2, -1, 0,

3. Solution: Draw a number line. Mark zero in the middle. Mark positive integers to the right and negative integers to the left. Plot the points -4, -1, 0, 2, and 3 on the number line. This helps visualize the order and magnitude of the integers.

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

Solution: Step 1: First, we deal with the subtraction of a negative number: (-8) + 5 - (-2) = (-8) + 5 + 2 Step 2: Now, we add from left to right: (-8) + 5 = -3 Step 3: Finally, we add the last number: -3 + 2 = -1 Therefore, (-8) + 5 - (-2) = -

1. The key is to remember that subtracting a negative is the same as adding a positive.

Question 3: Calculate: (-3) * 4 / (-6)

Solution: Step 1: Multiply (-3) * 4 = -12 (Different signs result in a negative product).

Step 2: Divide -12 / (-6) = 2 (Same signs result in a positive quotient).

Therefore, (-3) * 4 / (-6) =

2. Question 4: A thermometer reads 5°C in the morning. By the afternoon, the temperature drops by 8°

C. What is the temperature in the afternoon?

Solution: Step 1: The initial temperature is 5°

C. Step 2: The temperature drops by 8°C, meaning we subtract 8 from the initial temperature: 5 -

8. Step 3: 5 - 8 = 5 + (-8) = -3 Therefore, the temperature in the afternoon is -3°

C. Independent Practice (Questions Only) Order the following integers from smallest to largest: 7, -5, 0, -2, 3, -

8. Calculate: 12 + (-5) - 3 + (-2)

Calculate: (-6) * (-3) / 2 Evaluate: 5 * (2 - 7) A bank account has a balance of R

2

0

0. A withdrawal of R350 is made. What is the new balance? The highest point in Gauteng is 1,850 meters above sea level. The lowest point is 1,400 meters below a certain reference point. What is the difference in altitude between the highest point in Gauteng and the reference point? Assume the reference point is sea level (0 meters). Then the lowest point is at -1400 meters relative to the reference point if you take that as your base.

Simplify: -2 * (4 + (-1))

Calculate: -15 ÷ (3 + 2) * -4 The temperature at 6 AM was -2°C. By noon, it had risen by 10°C. What was the temperature at noon? Then, by 6 PM, it fell by 7°C. What was the temperature at 6 PM? A mine shaft is 300m below the ground.