2.1. Draw a diagram and use it to define and explain the three types of numbers: ordinal, cardinal, and nominal. 2.2. In the decimal number system, we can describe a number in different forms. Describe \( 5 \mathbf{4 0 5}, \mathbf{3 6 2} \) in words, in place value and in terms of the power of 10 . (6) 2.3. Design your own equation on distributive property and associative property. Explain how you will assist grade 4 learners to give solutions to each property.

### **2.1. Types of Numbers: Ordinal, Cardinal, and Nominal** **Definition and Explanation:** 1. **Cardinal Numbers:** - **Definition:** Cardinal numbers represent *quantity*; they tell *how many* of something there are. - **Examples:** 1, 2, 3, 10, 100. - **Use:** Counting objects (e.g., "There are **5** apples.") 2. **Ordinal Numbers:** - **Definition:** Ordinal numbers indicate the *position* or *order* of things in a series. - **Examples:** 1st, 2nd, 3rd, 10th, 100th. - **Use:** Describing rank or sequence (e.g., "She finished in **3rd** place.") 3. **Nominal Numbers:** - **Definition:** Nominal numbers are used for *labeling* or *identifying* without indicating quantity or order. - **Examples:** Phone numbers, jersey numbers, house numbers. - **Use:** Identification purposes (e.g., "Her jersey number is **12**.") **Diagram Explanation:** Imagine a Venn diagram with three overlapping circles, each representing one type of number. However, in reality, these types are distinct and do not overlap. Here's a textual representation: ``` +-----------------+ | Cardinal | | Numbers | | (Quantity) | +-----------------+ +-----------------+ | Ordinal | | Numbers | | (Position) | +-----------------+ +-----------------+ | Nominal | | Numbers | | (Labeling) | +-----------------+ ``` **Visual Representation Suggestions:** - **Cardinal:** Show a group of objects (e.g., 5 apples) with the number "5" indicating quantity. - **Ordinal:** Show a race with runners labeled 1st, 2nd, 3rd. - **Nominal:** Display a house with a number "12" on it for identification. --- ### **2.2. Describing the Number 5,405,362** Let's break down the number \( 5,405,362 \) into different forms: **a. In Words:** - **Five million, four hundred five thousand, three hundred sixty-two.** **b. In Place Value:** Each digit in the number has a specific place value based on its position. | Digit | Place Value | Value | |-------|--------------------|---------------| | 5 | Millions | 5,000,000 | | 4 | Hundred Thousands | 400,000 | | 0 | Ten Thousands | 0 | | 5 | Thousands | 5,000 | | 3 | Hundreds | 300 | | 6 | Tens | 60 | | 2 | Ones | 2 | **c. In Terms of Powers of 10:** Each digit can be expressed as a multiple of 10 raised to a certain power based on its position. \[ 5,405,362 = 5 \times 10^6 + 4 \times 10^5 + 0 \times 10^4 + 5 \times 10^3 + 3 \times 10^2 + 6 \times 10^1 + 2 \times 10^0 \] **Explanation:** - **Millions Place (10^6):** 5 × 1,000,000 = 5,000,000 - **Hundred Thousands Place (10^5):** 4 × 100,000 = 400,000 - **Ten Thousands Place (10^4):** 0 × 10,000 = 0 - **Thousands Place (10^3):** 5 × 1,000 = 5,000 - **Hundreds Place (10^2):** 3 × 100 = 300 - **Tens Place (10^1):** 6 × 10 = 60 - **Ones Place (10^0):** 2 × 1 = 2 --- ### **2.3. Equations and Teaching the Distributive and Associative Properties** **a. Distributive Property:** **Equation Example:** \[ 3 \times (4 + 5) = (3 \times 4) + (3 \times 5) \] **Calculation:** \[ 3 \times 9 = 12 + 15 \implies 27 = 27 \] **Explanation:** The distributive property shows how multiplication interacts with addition. It allows us to multiply a number by a sum by multiplying each addend separately and then adding the products. **b. Associative Property:** **Equation Example:** \[ (2 + 3) + 4 = 2 + (3 + 4) \] **Calculation:** \[ 5 + 4 = 2 + 7 \implies 9 = 9 \] **Explanation:** The associative property shows that the way numbers are grouped in addition or multiplication does not change their sum or product. **c. Teaching Strategies for Grade 4 Learners:** 1. **Use Concrete Examples:** - **Distributive Property:** Use physical objects like blocks. For example, if you have 3 groups of (4 blocks + 5 blocks), show that it's the same as having (3 groups of 4 blocks) plus (3 groups of 5 blocks). - **Associative Property:** Use grouping with objects. Show that grouping 2 blocks and 3 blocks first, then adding 4 blocks, gives the same total as grouping 3 blocks and 4 blocks first, then adding 2 blocks. 2. **Visual Aids:** - **Distributive Property:** Draw arrays or area models to represent the multiplication and distribution. - **Associative Property:** Use number lines or grouping circles to visualize different groupings. 3. **Interactive Activities:** - **Distributive Property:** Have students create their own examples using addition and multiplication, then apply the distributive property to solve. - **Associative Property:** Engage students in rearranging groups of objects to see that the total remains the same. 4. **Relate to Real-Life Scenarios:** - **Distributive Property:** Explain scenarios like distributing candies evenly among friends. - **Associative Property:** Use examples like organizing items into different groupings without changing the total number. 5. **Practice Problems:** - Provide a variety of problems where students can apply these properties. - Encourage peer teaching where students explain their reasoning to each other. 6. **Use Stories or Games:** - Incorporate simple stories where characters need to use these properties to solve problems. - Create matching games where students match different expressions that represent the same value using these properties. **Example Activity for Distributive Property:** *Problem:* There are 3 baskets. Each basket has 4 red apples and 5 green apples. How many apples are there in total? *Solution Using Distributive Property:* \[ 3 \times (4 + 5) = (3 \times 4) + (3 \times 5) = 12 + 15 = 27 \text{ apples} \] *Teaching Point:* Show students that multiplying the total number inside the parentheses by 3 gives the same result as multiplying each part separately and then adding them together. **Example Activity for Associative Property:** *Problem:* Calculate \( (2 + 3) + 4 \) and \( 2 + (3 + 4) \). *Solution:* \[ (2 + 3) + 4 = 5 + 4 = 9 \] \[ 2 + (3 + 4) = 2 + 7 = 9 \] *Teaching Point:* Demonstrate that grouping the numbers differently does not change the total sum. --- By using these explanations, examples, and teaching strategies, Grade 4 learners can grasp the concepts of ordinal, cardinal, nominal numbers, and understand the distributive and associative properties effectively.