Mathematics · Year 3 · Additive Thinking and Mental Strategies · Term 2

Mental Subtraction Strategies

Applying mental strategies like counting back, compensation, and bridging to subtract multi-digit numbers.

• ACARA Content Descriptions
• AC9M3N03
• AC9M3N04

About This Topic

Flexible addition strategies move students away from rigid algorithms and towards mental fluency. In Year 3, students learn to use jump, split, and compensation strategies to solve problems efficiently. This flexibility allows them to choose the best tool for the specific numbers involved, such as using compensation for 39 + 45 (changing it to 40 + 44). This topic aligns with AC9M3N03 and AC9M3N04, focusing on efficient mental and written strategies.

Developing these strategies is vital for everyday life, such as calculating costs or measuring ingredients. It also builds the confidence needed for more complex mathematics in later years. This topic comes alive when students can compare their different methods through structured discussion, seeing that there are many valid paths to the correct answer.

Key Questions

1. Design a mental strategy to subtract 99 from a three-digit number efficiently.
2. Compare the effectiveness of counting back versus using compensation for different subtraction problems.
3. Evaluate when it is more appropriate to use a mental strategy versus a written algorithm for subtraction.

Learning Objectives

• Design a mental strategy to subtract 99 from a three-digit number efficiently.
• Compare the effectiveness of counting back versus using compensation for different subtraction problems.
• Evaluate when it is more appropriate to use a mental strategy versus a written algorithm for subtraction.
• Calculate the difference between two three-digit numbers using at least two different mental strategies.
• Explain the steps involved in bridging to subtract a multiple of ten.

Before You Start

Counting and Number Sequences

Why: Students need a solid understanding of number order and how to count forwards and backwards to apply strategies like counting back.

Place Value to Hundreds

Why: Understanding hundreds, tens, and ones is crucial for strategies like bridging to ten and compensation involving multiples of ten.

Key Vocabulary

Counting Back	A mental strategy where you start with the larger number and subtract in steps, often by tens or ones.
Compensation	A mental strategy where you adjust one or both numbers in a subtraction problem to make it easier to solve, then adjust the answer.
Bridging to Ten	A mental strategy that involves subtracting to reach the nearest multiple of ten, then subtracting the remainder.
Mental Algorithm	A step-by-step mental process used to solve a calculation without writing it down.

Watch Out for These Misconceptions

Common Misconception

Students may try to use the 'split' strategy for every problem, even when it is inefficient (e.g., for 99 + 45).

What to Teach Instead

Present problems where one strategy is clearly faster. Through peer discussion, help students see that 'compensation' (making 99 into 100) is much simpler than splitting 99 into 90 and 9.

Common Misconception

In the 'jump' strategy, students sometimes lose track of the original number after the first jump.

What to Teach Instead

Use an empty number line to visually record each jump. Having students 'teach' their jump steps to a partner helps them maintain the sequence of the calculation.

Active Learning Ideas

Think-Pair-Share: Strategy Showdown

The teacher presents a problem like 56 + 29. Students solve it mentally, then share their specific method with a partner. They must decide together which strategy (split, jump, or compensation) was the most efficient for those specific numbers.

15 min·Pairs

Gallery Walk: Strategy Posters

Small groups are assigned one strategy (e.g., 'The Jump Method'). They create a poster showing how to use it for three different problems. The class then walks around, leaving 'sticky note' questions or praise for each method.

40 min·Small Groups

Inquiry Circle: The Budget Challenge

Students are given a small 'budget' for a school canteen and must add up various items using mental strategies. They work in pairs to check each other's work by using a different strategy than their partner used.

30 min·Pairs

Real-World Connections

• A baker calculating the remaining ingredients after using some for a recipe. For example, if a recipe needs 250g of flour and they started with 500g, they might mentally subtract 250g to find they have 250g left.
• A shopper at a supermarket mentally calculating the change they should receive after a purchase. If an item costs $19 and they pay with a $50 note, they might mentally subtract $20 from $50 to get $30, then add $1 back to get $31.

Assessment Ideas

Quick Check

Present students with the problem: 'Subtract 47 from 132'. Ask them to write down the strategy they used and their answer. Observe which students are using counting back, compensation, or bridging.

Discussion Prompt

Pose the question: 'When is it easier to subtract 19 mentally compared to subtracting 10 and then 9?'. Facilitate a discussion where students compare these methods and explain their reasoning.

Exit Ticket

Give each student a card with a subtraction problem, e.g., '156 - 38'. Ask them to write the answer and briefly describe the mental strategy they used to solve it.

Frequently Asked Questions

What is the 'compensation' strategy in Year 3?

Compensation involves adjusting one number to make it a 'friendly' number (like a multiple of 10), solving the problem, and then adjusting the answer to compensate for the original change. For example, for 48 + 25, you might do 50 + 25 = 75, then subtract 2 to get 73.

How can active learning help students learn addition strategies?

Active learning, particularly through 'strategy sharing', exposes students to different ways of thinking. When a student explains their 'jump' strategy to a peer who used 'split', both students gain a deeper understanding of number relationships. This collaborative environment encourages experimentation and reduces the fear of making a mistake, as the focus is on the process rather than just the final answer.

When should students move from mental to written strategies?

In Year 3, the focus is on developing strong mental models. Written strategies should be used to record and support mental thinking, especially as numbers get larger or more complex. The goal is for students to choose the most efficient method, whether it's done entirely in their head or supported by a quick sketch.

How do I help a student who is stuck on counting by ones?

Provide visual supports like ten-frames or number lines to encourage 'chunking'. Use peer modeling where a more confident student demonstrates how they 'jump' by tens. Small, frequent 'number talks' can also help build the habit of looking for groups of ten.