Welcome to Strategies and Algorithms for the Four Operations!
Have you ever wondered why there are so many different ways to solve the exact same maths problem? In the Primary Years Programme (PYP), mathematics is all about making sense of the world around us. In this chapter, we explore how the four operations—addition, subtraction, multiplication, and division—work together. We will discover that an algorithm is simply a step-by-step set of instructions to solve a calculation, but it is only one of many powerful tools you can choose from!
Note: For working with place values up into the millions or mastering rapid fact recall, check out our companion chapters on "Place value to thousands, millions and beyond" and "Automatic recall of multiplication and division facts".
---The Big Mathematical Secret: The Operations Are Connected!
Don't worry if large numbers seem intimidating at first. The four operations are not completely different systems—they are close relatives that help each other out.
Addition and Subtraction: Inverse Operations
Inverse means "opposite" or "undoing". Addition puts quantities together, while subtraction takes quantities apart.
If you know that \(450 + 250 = 700\), then you automatically know two subtraction facts without doing any extra hard work:
\(700 - 250 = 450\)
\(700 - 450 = 250\)
Multiplication and Division: Equal Groups
Just like addition and subtraction, multiplication and division are inverse operations:
• Multiplication is repeated addition of equal groups: \(4 \times 25\) means adding \(25\) four times (\(25 + 25 + 25 + 25 = 100\)).
• Division is repeated subtraction or sharing into equal groups: \(100 \div 25\) tells us how many times we can subtract \(25\) from \(100\) until we reach zero (\(4\) times).
If you know that \(8 \times 60 = 480\), then you also know:
\(480 \div 8 = 60\) and \(480 \div 60 = 8\)
Key Terms to Remember:
• Multiple: The result of multiplying a whole number by another whole number (for example, \(12, 18, 24\) are multiples of \(6\)).
• Product: The answer to a multiplication problem (the product of \(4\) and \(5\) is \(20\)).
• Quotient: The answer to a division problem (in \(36 \div 4 = 9\), the quotient is \(9\)).
• Prime number: A whole number greater than \(1\) that has exactly two factors: \(1\) and itself (for example, \(2, 3, 5, 7, 11\)).
Key Takeaway: Because operations undo each other, you can always use the inverse operation to check if your solution makes sense!
---Mental Strategies: Thinking Flexible and Fast
An expert mathematician does not always reach for a pencil and paper right away. Often, using a flexible mental strategy is faster, easier, and less prone to simple slips.
1. Partitioning (Breaking Apart by Place Value)
You can break numbers into hundreds, tens, and ones to make them simpler to work with.
Example for Addition: Calculate \(340 + 255\)
• Break \(255\) into \(200 + 50 + 5\).
• Step 1: \(340 + 200 = 540\)
• Step 2: \(540 + 50 = 590\)
• Step 3: \(590 + 5 = 595\)
Example for Multiplication: Calculate \(6 \times 43\)
• Partition \(43\) into \(40\) and \(3\).
• Multiply each part: \(6 \times 40 = 240\) and \(6 \times 3 = 18\).
• Combine the products: \(240 + 18 = 258\).
2. Compensation (Adjusting for Friendly Numbers)
Sometimes a number is very close to a friendly multiple of \(10\) or \(100\). Adjust the number to make it easy, then compensate at the end.
Example for Subtraction: Calculate \(562 - 198\)
• \(198\) is very close to \(200\) (it is \(2\) less).
• Subtract the friendly number: \(562 - 200 = 362\).
• Since you subtracted \(2\) too many, add \(2\) back on: \(362 + 2 = 364\).
3. Doubling and Halving
This is a brilliant trick for multiplication. If you double one factor and halve the other, the product stays the same!
Example: Calculate \(16 \times 25\)
• Halve \(16\) to get \(8\), and double \(25\) to get \(50\). Now solve \(8 \times 50 = 400\).
• Or halve and double again: Halve \(8\) to get \(4\), and double \(50\) to get \(100\). Now solve \(4 \times 100 = 400\). Both methods give the exact same result!
Key Takeaway: Mental strategies let you manipulate numbers flexibly. Look for "friendly numbers" like multiples of \(10\) or \(100\) to simplify your work.
---Written Strategies and Algorithms: One Way to Represent an Operation
In the PYP, an algorithm is defined as one way to represent an operation—not the only way! Algorithms are step-by-step written methods that help us keep track of place value when dealing with larger numbers.
Addition Algorithm (Regrouping Ones and Tens)
When the sum of digits in any place value column equals \(10\) or more, we regroup into the next column to the left.
Example: \(1,468 + 875\)
• Ones column: \(8 + 5 = 13\) ones. Write down \(3\) ones, regroup \(1\) ten.
• Tens column: \(6 + 7 + 1\text{ (regrouped)} = 14\) tens. Write down \(4\) tens, regroup \(1\) hundred.
• Hundreds column: \(4 + 8 + 1\text{ (regrouped)} = 13\) hundreds. Write down \(3\) hundreds, regroup \(1\) thousand.
• Thousands column: \(1 + 1\text{ (regrouped)} = 2\) thousands.
• Total: \(2,343\)
Subtraction Algorithm (Decomposing / Regrouping)
When you cannot subtract a larger digit from a smaller digit in a place value column, trade from the column to the left.
Example: \(3,042 - 1,265\)
• Ones column: You cannot do \(2 - 5\) in whole numbers. Regroup \(1\) ten from the tens column (leaving \(3\) tens), making the ones into \(12\). Now, \(12 - 5 = 7\).
• Tens column: You now have \(3\) tens and need to subtract \(6\) tens. You cannot trade from \(0\) hundreds, so go to the thousands column!
• Regroup \(1\) thousand into \(10\) hundreds, then trade \(1\) of those hundreds into the tens column (leaving \(9\) hundreds and giving you \(13\) tens).
• Now solve: \(13 - 6 = 7\) tens.
• Hundreds column: \(9 - 2 = 7\) hundreds.
• Thousands column: \(2 - 1 = 1\) thousand.
• Total: \(1,777\)
Multiplication: From Grid/Area Models to Written Steps
Before using a compact algorithm, visualising multiplication with an area model (or grid method) ensures you understand what the numbers represent.
Example: Multiply \(24 \times 35\)
Break both numbers apart: \(24 = 20 + 4\) and \(35 = 30 + 5\). Multiply each section:
• \(20 \times 30 = 600\)
• \(20 \times 5 = 100\)
• \(4 \times 30 = 120\)
• \(4 \times 5 = 20\)
Add the partial products together: \(600 + 100 + 120 + 20 = 840\). A formal vertical algorithm follows these exact same steps, keeping track of the place values row by row!
Division: Sharing and Chunking
Division is all about finding out how many equal groups can be made from a total.
Example: \(156 \div 6\)
Instead of guessing all at once, you can subtract convenient "chunks" of \(6\):
• We know that \(20 \times 6 = 120\). Subtract \(120\) from \(156\): \(156 - 120 = 36\).
• We know that \(6 \times 6 = 36\). Subtract \(36\) from \(36\): \(36 - 36 = 0\).
• Add the chunks together: \(20 + 6 = 26\). Therefore, \(156 \div 6 = 26\).
Key Takeaway: Standard algorithms are reliable tools, but they rely entirely on place value. Keep your columns neatly aligned so your numbers don't wander!
---Extending Strategies to Decimals and Fractions (Phase 4)
Once you understand how operations work with whole numbers, you can apply the exact same logic to decimal fractions and fractions!
Adding and Subtracting Decimals
Because our number system is base 10, the rules for regrouping do not change when moving past the decimal point into tenths and hundredths. Simply align the place values (tenths with tenths, hundredths with hundredths).
Example: \(4.25 + 3.8\)
• Think of \(3.8\) as \(3.80\) so both numbers show hundredths.
• Add the hundredths: \(5 + 0 = 5\) hundredths (\(0.05\)).
• Add the tenths: \(2 + 8 = 10\) tenths. Regroup \(10\) tenths into \(1\) whole unit, leaving \(0\) tenths.
• Add the whole units: \(4 + 3 + 1\text{ (regrouped)} = 8\).
• Total: \(8.05\)
Computation Ideas Applied to Fractions
Whole-number reasoning also guides how we work with fractions:
• Addition and Subtraction: Just as you can only directly add things that share the same unit (like tens with tens), you can only add or subtract fractions that share the same unit or denominator (e.g., \(\frac{2}{7} + \frac{3}{7} = \frac{5}{7}\)).
• Multiplication: Finding a fraction of a quantity works like repeated grouping. For example, \(\frac{1}{3}\) of \(12\) means dividing \(12\) into \(3\) equal parts, giving \(4\).
Note: For more on converting and comparing fractions, decimals, and percentages, refer to the chapter "Comparing, ordering and converting fractions, decimals and percentages".
Key Takeaway: Decimal and fraction calculations follow the exact same place-value and grouping principles as whole numbers!
---Common Mistakes to Avoid
• Forgetting Place Value in Subtraction: When working with numbers containing zeros (like \(504 - 168\)), take your time regrouping across columns step by step from the left.
• Misaligning Decimals: Do not line up decimal numbers by their last digit. Always line up the decimal point so that tenths match with tenths, and units match with units.
• Thinking There is Only One "Right" Method: An algorithm is just one option! If a problem like \(1,000 - 997\) can be solved mentally in two seconds by counting up (\(3\)), you don't need a long vertical algorithm.
Quick Review: Choosing Your Strategy
When faced with any calculation in your inquiry units or daily life, ask yourself these three questions:
1. Can I do this mentally? Look for friendly numbers, doubles, or partition by place value.
2. Would a quick diagram or model help? Use an open number line, grid model, or bar model to visualise what is happening.
3. Do I need a written algorithm? For larger numbers with multiple regrouping steps, write out your place-value columns clearly and work step by step.
4. Does my answer make sense? Use an inverse operation or round the numbers to check if your answer is reasonable!