Mathematical Foundations and Real World Reasoning · 3rd Year · Additive Thinking and Mental Strategies · Autumn Term

Mental Math: Bridging and Compensation

Developing flexible strategies like bridging through ten and using near doubles.

• NCCA Curriculum Specifications
• NCCA: Primary - Number
• NCCA: Primary - Operations

About This Topic

Mental math strategies, specifically bridging through ten and compensation, are fundamental to developing number fluency in third-year students. Bridging through ten involves breaking down an addition problem so that one number is added to reach the next multiple of ten, and then the remainder is added. For example, to calculate 27 + 15, a student might bridge through ten by thinking 27 + 3 = 30, and then 30 + 12 = 42. Compensation is a related strategy where a number is adjusted to a more convenient number, usually a multiple of ten, and then the adjustment is accounted for in the final answer. For 27 + 15, a student might compensate by thinking 27 + 10 = 37, and then adding the remaining 5, or by adjusting 15 to 13 to make 27 + 13 = 40, and then adding the remaining 2, resulting in 42.

These strategies are crucial because they move students beyond rote memorization of algorithms and encourage flexible thinking about numbers. Understanding how to manipulate numbers to simplify calculations builds confidence and efficiency. It also lays the groundwork for more complex algebraic thinking, where similar adjustments are made to equations. The ability to choose the most efficient mental path for a given problem is a hallmark of strong mathematical reasoning, fostering a deeper conceptual understanding of addition and number relationships. Active learning, through varied practice and peer discussion, solidifies these flexible strategies.

Key Questions

1. Explain how changing a number to a nearby multiple of ten can make mental addition easier.
2. Analyze why different people use different mental paths to reach the same sum.
3. Assess when it is faster to calculate mentally than to write it down.

Watch Out for These Misconceptions

Common Misconception

Compensation means always adding 10 to the second number.

What to Teach Instead

Compensation involves adjusting a number to a nearby 'friendly' number, which isn't always 10. Students need to understand they can adjust by any amount and then account for that adjustment. Discussion of various examples helps clarify this.

Common Misconception

Bridging through ten is only useful for numbers close to a multiple of ten.

What to Teach Instead

Bridging through ten is a versatile strategy applicable to many addition problems. Exploring how to bridge through tens for numbers like 18 + 17, by first adding 2 to reach 20, then adding the remaining 15, shows its broad utility.

Active Learning Ideas

Bridging Board Game

Students roll two dice and use bridging through ten to add the numbers. They move their game piece along a number line, explaining their bridging steps. The first to reach the end wins.

30 min·Small Groups

Compensation Challenge Cards

Provide cards with addition problems. Students must solve each problem using a compensation strategy, writing down both the adjusted calculation and the final answer. They can work individually or in pairs.

25 min·Pairs

Mental Math Relay

Divide the class into teams. Present an addition problem to the first student in each line. They solve it using bridging or compensation and run to tag the next teammate, who solves the next problem. The first team to finish wins.

20 min·Whole Class

Frequently Asked Questions

Why are bridging and compensation important for third-year students?

These strategies build number sense and mental flexibility, moving students beyond rigid algorithms. They enable students to solve addition problems more efficiently and confidently, fostering a deeper understanding of number relationships and preparing them for more advanced mathematical concepts.

How can I help students choose the best mental math strategy?

Encourage students to analyze the numbers in a problem. Discuss when bridging might be easier (e.g., 47 + 5) versus when compensation is more efficient (e.g., 38 + 21). Provide opportunities for them to explain their reasoning and compare different approaches.

What is the difference between bridging and compensation?

Bridging through ten means breaking down a number to reach the next multiple of ten. Compensation involves adjusting one of the numbers to a more convenient value (like a multiple of ten) and then correcting for that adjustment. Both aim to simplify calculations.

How does active learning benefit the teaching of mental math strategies?

Games, card challenges, and relays allow students to practice these strategies in engaging ways, making abstract concepts concrete. Collaborative problem-solving and peer teaching help students articulate their thinking, discover alternative methods, and solidify their understanding through active engagement.