Journal

1. Student task presentation

The class is about to hold a party and needs your help to purchase snacks. The specific rules are as follows

• Total budget: 100 dollars
• Purchase two kinds of snacks: chocolate ($5 each) and potato chips ($3 per pack)
• There are 30 students in the class, so the total number of snacks (the number of chocolate blocks + the number of potato chip packs) should be exactly 30 (1 per person, neither more nor less).
• Both kinds of snacks must be purchased
• The number of chocolates should not exceed 25 pieces, and the number of potato chips should not exceed 28 packs.

Your task:

• Find all the effective combinations of “chocolate quantity + potato chip quantity” that conform to the above rules;
• Choose any combination, calculate its total cost, and verify whether it is within the budget of 100 dollars.
• Explain the methods you used to find these combinations (for example: whether to analyze from the quantity of chocolate);
• If the total cost of a certain combination exceeds 100 dollars, analyze the reasons and propose adjustment methods.

2. Learning objectives

• Students should be able to:
  • Establish a linear equation based on the task constraints.
  • Calculate the cost of one of the snack combinations and then verify whether it meets the requirement of 100 dollars.
  • Find at least two methods for finding effective combinations.

• Students should understand
  • When the total quantity is fixed, adjusting the quantity of high-priced items has a greater impact on the total expenditure.
• The difficulties that students may encounter
  • An error occurred when calculating the total cost
  • Forget the requirement that “both kinds of snacks must be purchased”
  • When calculating the combination, ignore the quantity limit

Solution Methods	Assessing Questions	Advancing Questions
Strategy One: The method of setting Unknowns for derivationLet the number of chocolates be x pieces, then the number of potato chips is 30 – x packs. Considering the quantity limit: x≤25 and 30 – x≤28, it is deduced that x≥2. Therefore, the range of x is 2 to 25.Considering the budget constraint: 5x+3(30 – x)≤100, it is simplified to 2x+90≤100, and further derivation leads to x≤5.The final valid x values are 2, 3, 4, and 5, corresponding to four combinations. For example: when x=3, 3 pieces of chocolate +27 packs of potato chips, the total cost is 3×5+27×3=96 (within the budget).	Why is the number of potato chips expressed as 30−x?Why does 5x+3(30−x) become 2x+90 when simplifying the budget equation?	If the budget is raised to 110 yuan and the budget equation is redrawn, how will the effective range of x change at this time?
Strategy Two: Snacks with higher amounts will be given priority in calculationGive priority to calculating the quantity of chocolates (as the unit price of chocolates is higher and has a greater impact on the budget), x=1,2,3,… 25,	Why choose to calculate the number of chocolates first rather than the number of potato chips?When calculating x=1, the total cost is relatively low. Why is it still judged as invalid?	If you were to calculate the number of potato chips first, from which package would you start the calculation? Can the same number of effective combinations be obtained in the end?
Strategy Three: Enumeration method List all possible combinationsList all combinations of “chocolate quantity + potato chip quantity = 30”, such as (1,29), (2,28)… (29,1); Then eliminate the combinations that do not meet the conditions.	Why are (30,0) or (0,30) not included when listing combinations? How do you quickly determine which combinations violate the quantity limit? Why was the combination (6,24) still excluded when it met the quantity limit?	If the rule of “at least 5 pieces of chocolate” is added, which combinations need to be excluded additionally? How many valid combinations are there remaining?