Describe the six key steps in mathematical thinking for problem-solving and illustrate their relevance with an example of calculating the cost of a group lunch.

Here are the six key steps in mathematical thinking for problem-solving, illustrated with an example of calculating the cost of a group lunch:

1. Understand the Problem: Clearly identify the goal, the given information, and any implicit assumptions or constraints.
2. Plan the Approach: Determine the necessary steps, formulas, or logical sequence to move from the given information to the solution.
3. Execute the Plan: Carry out the calculations or logical operations systematically.
4. Check for Accuracy and Reasonableness: Verify calculations, units, and assess if the answer makes sense in the real-world context.
5. State the Solution Clearly: Present the final answer in a concise and understandable manner, addressing the original question.
6. Reflect and Generalize: Consider alternative methods, potential pitfalls, or how the learned approach could be applied to similar problems.

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Example: Calculating the Cost of a Group Lunch

Scenario: A group of 5 friends went out for lunch. Three friends ordered a burger combo each at $12.50 per combo, and two friends ordered a salad each at $9.75 per salad. They also shared an appetizer for $8.00. A 15% tip is to be added to the total bill. If they split the final cost (including tip) equally, what is each person's share?

1. Understand the Problem: * Goal: Find the individual share of the total lunch cost, including tip. * Given Information: * Number of friends: 5 * Burger combo cost: $12.50 (3 ordered) * Salad cost: $9.75 (2 ordered) * Appetizer cost: $8.00 * Tip percentage: 15% * Implicit: Need to calculate the subtotal, then the tip, then the total bill, and finally divide by the number of friends.

2. Plan the Approach: * Step 1: Calculate the total cost of the burger combos. * Step 2: Calculate the total cost of the salads. * Step 3: Sum the costs of burgers, salads, and the appetizer to find the subtotal. * Step 4: Calculate the tip amount by multiplying the subtotal by the tip percentage. * Step 5: Add the tip amount to the subtotal to get the total bill. * Step 6: Divide the total bill by the number of friends to find each person's share.

3. Execute the Plan: * Step 1: Cost of burgers: 3 \times \12.50 = $37.50$* **Step 2:** Cost of salads:$2 \times $9.75 = $19.50$* **Step 3:** Subtotal:$$37.50 + $19.50 + $8.00 = $65.00$* **Step 4:** Tip amount:$0.15 \times $65.00 = $9.75$* **Step 5:** Total bill:$$65.00 + $9.75 = $74.75$* **Step 6:** Each person's share:$$74.75 \div 5 = $14.95$

4. Check for Accuracy and Reasonableness: * All calculations were performed correctly. * The units (dollars) are consistent throughout. * A cost of $14.95 per person for a lunch with an appetizer and tip seems reasonable. * To verify: 5 \times \14.95 = $74.75$, which matches the total bill.

5. State the Solution Clearly: * Each person's share of the group lunch, including the 15% tip, is $14.95.

6. Reflect and Generalize: * This systematic approach can be applied to any group dining scenario, regardless of the number of items, prices, tip percentages, or people. * For more complex situations (e.g., different tax rates, individual items not split equally), the planning phase would involve breaking down the problem into more granular steps. * This method helps ensure all costs are accounted for and distributed fairly.