Word problems

These are the four steps already known (search, try, check, say), made more precise by the model: searching becomes understanding the story; trying becomes drawing a model and choosing the operation; checking and saying are in the last step: we calculate, make sure the answer makes sense, then give it.

1. Understand the Story: Read the problem to identify the parts and the whole.
2. Draw a Model: Use a part-whole bar model to organize the information.
3. Choose the Operation: If the whole is unknown, you add the parts. If a part is unknown, you subtract the known part from the whole.
4. Solve and Check: Calculate the answer and make sure it makes sense in the context of the story.

The same bar model works for big numbers. Remember: the bar does not need to be drawn to scale — its job is to help choose the operation, not to measure.

• If the whole is unknown, add the parts.
• If a part is unknown, subtract the known part from the whole.
• Compute in your head when the numbers allow it, otherwise set it out in columns.

In a comparison problem, nothing is split into parts: two quantities are compared. So we draw two separate bars, one under the other, and we mark the difference.

• Draw one bar per quantity: the bigger one is longer.
• Read the model, not the words: “more than” or “less than” tells you who has more — the model then tells you whether to add or subtract.
• To find the bigger: add the difference. To find the smaller: subtract it. To find the difference: subtract the smaller from the bigger.

Some problems cannot be solved with one calculation: the question hides another number that you must find first.

1. Find the hidden number: Read the story and find the number the question needs but does not give. Calculate it first.
2. Answer the question: Use the hidden number in a second calculation to answer the question.
3. Check: Make sure the answer makes sense in the story.

The “hidden number” method still works with big numbers. To stay organized:

• one step per line: name the hidden number, then write its calculation;
• compute in your head when the numbers allow it, otherwise set it out in columns;
• a bar model for each step helps choose each operation.

In a mixed problem, one step is a multiplication, the other an addition or a subtraction.

• Equal groups in the story (“packs of…”, “boxes of…”, “rows of…”): when the number of groups and the size of a group are both known, the multiplication gives the hidden number.
• Then read the rest of the story as a part-whole problem: it gives the second step.
• One step per line — the times tables and the whole tens do the work.

In an equal share, the bar shows the whole on top and the parts underneath. Two questions can be asked:

• How many parts? The size of a part is known. Look for how many times it makes the whole: \(60\) children in teams of \(5\): \(5 \times \ldots = 60\), so \(12\) teams.
• How many in each part? The number of parts is known. Look for the number that, taken that many times, makes the whole: \(200\) pupils in \(40\) teams: \(40 \times \ldots = 200\), so \(5\) pupils per team.

When the table is not enough, count by jumps: \(5\), \(10\), \(15\)… up to \(60\), and count the jumps: \(12\). Always check: the parts put together must make the whole.

Sometimes the parts do not make the whole exactly. Find the biggest number of full parts, then say what is left, and answer the question that was asked:

• \(189\) photos, \(10\) per page: \(10 \times 18 = 180\), \(9\) photos are left. They need a page too: \(19\) pages are needed.
• \(38\) eggs, to be sold in boxes of \(4\): \(4 \times 9 = 36\), \(2\) eggs are left. They do not fill a box: \(9\) boxes can be sold.

The same leftover counts one more page, but not one more box: read the question before answering.