Part A: Numeracy
1
Number Foundations
Master whole numbers, integers, and order of operations
Learning Objectives
- Apply order of operations (BODMAS) correctly
- Work confidently with negative numbers
- Round and estimate to check answers quickly
- Understand place value in large and decimal numbers
- Solve real-world number problems in teaching contexts
Chapter Progress: 0%
1. Place Value & Number Sense
Place Value Breakdown
Understanding place value is the foundation of numeracy. Each digit in a number has a value based on its position: ones, tens, hundreds, thousands, and decimal places like tenths and hundredths.
Example 1
ACER Authentic Problem: Ordering Numbers and Place Value
❓ Question: Arrange the four student reading fluency scores in ascending order (from smallest to largest): 3.14, 3.2, 3.015, and 3.104.
📄 Source Passage / Reference Data:
Reading Fluency Scores: Student A = 3.14, Student B = 3.2, Student C = 3.015, Student D = 3.104.
📝 Step-by-Step Method:
1
Align decimal points vertically to compare place values directly.
2
Add trailing zeros so every value has three decimal places (thousandths).
3
Compare digits from left to right: all have 3 units. Comparing fractional thousandths: 015 < 104 < 140 < 200.
🔍 Evidence Extracted: "3.015 < 3.104 < 3.140 < 3.200"
🎯 Final Answer: 3.015, 3.104, 3.14, 3.2
💡 Why It Is Correct: Adding trailing zeros prevents the common error of confusing number length with value: 3.015 is smaller than 3.104, followed by 3.14, with 3.2 being the largest.
⚡ LANTITE Exam Strategy: LANTITE Strategy: When comparing decimal values, equalise decimal places with trailing zeros.
2. Operations with Whole Numbers
The Four Basic Operations
Mastering addition (+), subtraction (-), multiplication (×), and division (÷) without relying on a calculator is essential for the LANTITE exam.
Example 2
ACER Authentic Problem: Multi-Item Budgeting & Change (Sept 2023 Q7)
❓ Question: Kim pays for both dinners with a $50 note. How much change should Kim receive?
📄 Source Passage / Reference Data:
PAYING FOR DINNER
Kim and Lee meet for dinner. Kim has a burger for $15.90 and Lee has a salad for $14.90. They each have a drink costing $4.00 each.
📝 Step-by-Step Method:
1
Calculate total cost of food items.
2
Add the cost of two drinks (2 × $4.00 = $8.00).
3
Subtract the total expenditure from the $50 note.
🔍 Evidence Extracted: "Total spent = $38.80. Change from $50 = $11.20."
🎯 Final Answer: $11.20
💡 Why It Is Correct: Summing food ($30.80) and drinks ($8.00) gives $38.80. Subtracting from $50 yields $11.20 change.
⚡ LANTITE Exam Strategy: Official ACER Method: In mental subtraction, subtract dollars first ($50 - $38 = $12), then subtract cents ($12.00 - $0.80 = $11.20).
Keywords for Operations
| Addition (+) | Subtraction (-) | Multiplication (×) | Division (÷) |
|---|---|---|---|
| Sum, altogether | Difference, fewer | Product, times | Quotient, average |
| Combined, plus | Remaining, less than | Each, per, groups of | Share, split, per person |
3. Order of Operations (BODMAS)
What is BODMAS?
BODMAS stands for Brackets, Orders (powers/roots), Division, Multiplication, Addition, and Subtraction. It tells you the correct sequence to solve mathematical expressions.
The BODMAS Rule
B -> O -> D / M -> A / S
Example 3
Step-by-step BODMAS Calculation
❓ Question: Calculate the value of the numerical expression: 10 + 2 × 5 using the correct order of operations.
📄 Source Passage / Reference Data: BODMAS Operational Expression: 10 + 2 × 5.
📝 Step-by-Step Method:
1
Identify the operations involved: Addition (+) and Multiplication (×).
2
According to BODMAS, perform multiplication before addition.
3
Add the result to the initial number.
🔍 Evidence Extracted: "Multiplication 2 × 5 = 10 precedes Addition (+ 10)"
🎯 Final Answer: 20
💡 Why It Is Correct: Multiplication takes precedence over addition. Evaluate 2 × 5 = 10 first, then 10 + 10 = 20.
⚡ LANTITE Exam Strategy: Always evaluate multiplication and division before addition and subtraction unless brackets dictate otherwise.
Example 4
Nested Brackets Calculation
❓ Question: Evaluate the mathematical expression with nested brackets: 3 × (4 + (6 ÷ 2)).
📄 Source Passage / Reference Data: Nested Brackets Operational Expression: 3 × (4 + (6 ÷ 2)).
📝 Step-by-Step Method:
1
Start with the innermost set of brackets: (6 ÷ 2).
2
Substitute back and resolve the outer set of brackets: (4 + 3).
3
Perform the final multiplication outside the brackets.
🔍 Evidence Extracted: "Innermost (6 ÷ 2 = 3) → Outer (4 + 3 = 7) → Final (3 × 7 = 21)"
🎯 Final Answer: 21
💡 Why It Is Correct: Innermost brackets (6 ÷ 2 = 3) are solved first, then outer brackets (4 + 3 = 7), and finally 3 × 7 = 21.
⚡ LANTITE Exam Strategy: Work from the inside out when dealing with nested brackets.
4. Negative Numbers & Integers
Understanding Integers
Integers include all whole positive and negative numbers, including zero. Think of them visually on a number line to help with addition and subtraction.
Example 5
Adding and Subtracting Negative Numbers
❓ Question: Evaluate the following two integer expressions: (a) -5 + 8 and (b) -4 - 6.
📄 Source Passage / Reference Data: Canberra Weather Station Temperature Log: Overnight base = -5°C with +8°C rise; Cold front drop = -4°C with 6°C further drop.
📝 Step-by-Step Method:
1
For (a) -5 + 8: Start at -5 on the number line and move 8 units to the right.
2
For (b) -4 - 6: Start at -4 on the number line and move 6 units further to the left.
🔍 Evidence Extracted: "-5 + 8 = 3 and -4 - 6 = -10"
🎯 Final Answer: (a) 3, (b) -10
💡 Why It Is Correct: Adding positive 8 to -5 moves right past zero to +3. Subtracting 6 from -4 moves further left to -10.
⚡ LANTITE Exam Strategy: Adding a positive number moves right on the number line; subtracting a positive number moves left.
5. Rounding & Estimation
Rounding Rules
Look at the digit immediately to the right of your target place value. If it is 5 or above, round up. If it is 4 or below, keep the target digit the same (round down).
Example 5
ACER Authentic Problem: Car Speedometer Scale (July 2017 Q1)
❓ Question: The speed limit is 110 km/h. What is the difference between the speed limit and the speed shown on the dial?
📄 Source Passage / Reference Data:
CAR SPEEDO
This speedometer shows the speed at which a car is travelling.
CAR SPEEDOMETER
Posted Speed Limit: 110 km/h
📝 Step-by-Step Method:
1
Determine the scale increment on the speedometer: between 120 and 140 there are 10 tick subdivisions, so each minor tick = 2 km/h.
2
Read the needle position: 1 division past 120 km/h.
3
Calculate the difference from the posted speed limit of 110 km/h.
🔍 Evidence Extracted: "Needle points to 122 km/h. Speed limit is 110 km/h."
🎯 Final Answer: 12 km/h
💡 Why It Is Correct: The speedometer shows 122 km/h. Subtracting the speed limit of 110 km/h gives a difference of 12 km/h.
⚡ LANTITE Exam Strategy: LANTITE Dial Reading: Always establish the value of a single division before attempting to read intermediate markers.
6. Solving Word Problems
Word Problem Strategy
Tackle word problems step-by-step: Read carefully, Identify the key information, Plan your approach (choose operation), Solve, and Check if the answer makes sense in context.
Example 6
ACER Authentic Problem: School Day Duration (July 2017 Q2)
❓ Question: How long is the school day?
📄 Source Passage / Reference Data:
CLASS TIME
Start of school day
End of school day
📝 Step-by-Step Method:
1
Read the start clock: hour hand at 8, minute hand at 12.
2
Read the end clock: hour hand between 2 and 3, minute hand at 8 (40 min).
3
Calculate elapsed time from 8:00 am to 2:40 pm.
🔍 Evidence Extracted: "Start = 8:00 am; End = 2:40 pm; Elapsed = 6h 40m"
🎯 Final Answer: 6 hours and 40 minutes
💡 Why It Is Correct: From 8:00 am to 2:00 pm is 6 hours. Adding the additional 40 minutes yields 6 hours and 40 minutes.
⚡ LANTITE Exam Strategy: Clock Traps: Watch the hour hand carefully. At 8:00 am it points directly at 8, not 9.
Question 1 of 10
ACER 2017 Q1: The speedometer needle indicates 122 km/h. If the speed limit is 110 km/h, what is the difference between the speed limit and the speed shown?