MIP1502 Assignment 1 (DETAILED ANSWERS) 2025 - DISTINCTION GUARANTEED

MIP1502 Assignment 1 (DETAILED ANSWERS) 2025 - DISTINCTION GUARANTEED - DISTINCTION GUARANTEED - DISTINCTION GUARANTEED Answers, guidelines, workings and references ... An engineer is designing a steel truss for a bridge using metal beams arranged in triangles. Figure 1: 1 triangle is formed using 3 beams. Figure 2: 2 triangles are formed in a row pointing up and down using 5 beams. Figure 3: 3 triangles are formed in a row using 7 beams. Draw or visualise figure 4. How many beams would be needed? Determine the general rule Tn for the number of beams in terms of n, the number of triangles. Your formula will likely include multiplication by 2. Explain what this 2 represents physically in the construction of the bridge, showing how many beams are added for each new triangle and how many are shared. Calculate which figure number n would require exactly 87 beams. A student plots this pattern on a Cartesian plane and connects the points with a solid straight line. Critique this representation and state whether the data is continuous or discrete. Explain why drawing a solid line is not correct in this context, for example what a value such as n equals 1.5 would mean. If the engineer adds 2 extra vertical support beams, one at the start and one at the end, explain how the rule Tn would change. 1.1 Algebra is often introduced in primary school through patterns, number sentences, and symbolic reasoning. Critically evaluate the rationale for introducing algebraic thinking in the Foundation and Intermediate Phases. In your response: 1.1.1 Discuss at least two pedagogical benefits of early algebra exposure. (4) 1.1.2 Identify one common misconception learners may develop and explain how it can be addressed. (3) 1.1.3 Justify how early algebra supports progression into formal algebra in later grades. (3) 1.2 Many learners struggle with the concept of multiplying negative numbers. Design a mini-lesson (not just explanations) that includes: 1.2.1 A real-world context (2) 1.2.2 A visual model (2) 1.2.3 A pattern-based reasoning approach (2) 1.2.4 Explain how each method supports conceptual understanding. (4) [20] Question 2 2.1 Translate the following real-world scenarios into algebraic expressions or equations. Then solve them. 2.1.1 A machine depreciates in value by 15% annually. If it was worth R120,000 initially, what is its value after 3 years? (4) 2.1.2 A recipe calls for 2 parts flour, 3 parts sugar, and 5 parts water. If you have 1.2 kg of sugar, how much flour and water are needed to maintain the ratio?. (4) 2.1.3 A plumber charges a call-out fee and an hourly rate. A 3-hour job costs R870, and a 5-hour job costs R1,250. Determine the call-out fee and hourly rate. (6) 2.2 Create a real-world context for the equation: 0.75 This question requires you to move beyond simply solving mathematical problems. The study guide distinguishes between an unknown in a number sentence and a variable in a functional relationship. Consider the following two mathematical statements: Statement A: 5k − 10 = 25 Statement B: y = 5k − 10 Explain the conceptual difference in the role of the symbol k in statement A compared to statement B. Which statement represents a specific value, and which statement represents a domain of input values? You are observing a Grade 6 class. A learner, Sam, writes the equation P + S = 20 to represent the sentence that there are 20 people in the room consisting of professors and students. However, when asked to represent the sentence that there are six times as many students as professors, Sam writes 6S = P. Explain why Sam’s logic is flawed. Consider the inequality 5 − 3x ≥ −4. Solve for x, ensuring you explicitly show the inverse operations. Represent the solution using correct set-builder notation. A learner looks at the solution and asks if x is less than or equal to 3, then the biggest number is 3 and whether the next biggest number is 2. Refute this claim using the property of density in real numbers and give an example of a number between 2 and 3 that satisfies the condition. Explain why the expression 5 divided by x minus 3 is undefined if x equals 3 and relate this to the inverse relationship between multiplication and division. Algebra evolved from describing problems in sentences to using symbols. Translate the rhetorical problem into a single symbolic equation: I am thinking of a number. If I square it and subtract five times the number, the result is negative six. Solve the equation algebraically. Explain why it is necessary to set the equation equal to zero to solve it effectively and explain the zero product principle. A learner, Lerato, attempts to simplify the expression 5(2a · 3). Her work is shown. Identify the specific property she applied incorrectly and explain why it is not applicable. Solve the expression correctly and name the property that allows regrouping of numbers. Provide a simple arithmetic counterexample to show that her method changes the value. Using the distributive property, prove that x · 0 = 0. A student claims that division is associative like multiplication. Disprove this claim by calculating (24 ÷ 6) ÷ 2 and comparing it to 24 ÷ (6 ÷ 2). Mr Jola is organising transport for a Grade 7 field trip using identical minibuses. If he places 12 learners in each minibus, 5 learners are left. If he places 13 learners in each minibus, there are 2 empty seats in the last minibus. Define your variables clearly. Construct two linear equations representing the total number of learners for both scenarios. Solve the system to determine the number of minibuses and total learners. Explain how you would verify your answer without using a memo. The Student Representative Council is organising a concert and needs to generate R6000 from ticket sales. Let x be the price of a ticket and y the number of people attending. The situation is represented by xy = 6000. If the ticket price is lowered by R10, 20 more people will attend, and revenue remains R6000. Set up a system of two equations. Substitute one variable into the other to show it results in a quadratic equation of the form x² − 10x − 3000 = 0. Solve for the original ticket price and explain why the negative solution is not valid in context. An engineer is designing a steel truss using triangles. Figure 1 uses 3 beams, figure 2 uses 5 beams, and figure 3 uses 7 beams. Draw or visualise figure 4 and determine how many beams are needed. Determine the general rule for the number of beams in terms of the number of triangles. Explain what the multiplication by 2 represents physically in the structure. Calculate which figure number would require exactly 87 beams. A student plots the pattern on a Cartesian plane and connects the points with a straight line. Critique this representation and state whether the data is continuous or discrete. Explain why drawing a solid line is incorrect in this context. If 2 extra vertical support beams are added at the ends of the bridge, explain how the algebraic rule changes. You will analyse functional relationships, diagnose a common learner misconception based on the principles in your learning material, and design a learning activity that aligns with the CAPS curriculum for the Intermediate Phase. 1.1 Consider the two real-world situations below. The first represents a linear function, and the second represents a non-linear function. • Situation 1 (Linear): A rental van costs R230 per day plus R4,30 per kilometre driven. • Situation 2 (Non-linear): The area of a circle depends on the length of its radius. For each situation, you must: 1.1.1 Identify the independent variable (input) and the dependent variable (output). (2) 1.1.2 Write a function rule using function notation (e.g.,