3 exercise books cost €6: additive instead of proportional thinking
Why children solve tasks like “3 exercise books cost €6” additively instead of proportionally, and how teachers can build the functional relationship.
In short: Asked “3 exercise books cost €6. How much do 6 exercise books cost?”, many children calculate €6 + €3 = €9 because three more books are added. They transfer the difference in quantity to the price, although doubling the quantity also doubles the price. Helpful approaches are tables with arrows, asking for the price of one item, doubling and halving, and counterexamples in which the relationship is not proportional. Mazio Math organises its content by the five content areas, including patterns and structures, and flags it to the teacher when a common misconception shows up in the class.
This article is for teachers of grades 3 and 4. It shows typical errors in tasks involving quantity and price, explains their cause and describes how you can build up the relationship with tables and calculation methods. How children learn to understand word problems in general is covered in the article on steps to understanding word problems.
Typical errors with quantity and price
| Error | Example | What lies behind it |
|---|---|---|
| Difference instead of multiple | 3 exercise books cost €6. For 6 books the child calculates €6 + €3 = €9. The correct answer is €12. | The increase in quantity is added to the price. |
| Price for several taken as price for one | 2 pencils cost €2. For 5 pencils the child keeps counting in twos: €2, €4, €6, €8, €10. The correct answer is €5. | The price for two items is added for every further item. |
| Always doubling | 4 kg cost €8. For 6 kg the child doubles to €16. The correct answer is €12. | “More” is only associated with doubling. |
| Numbers combined without meaning | 4 kg cost €12. For 1 kg the child calculates 12 × 4 = €48. The correct answer is €3. | The operation is guessed and the size of the result is not checked. |
| Proportional where it does not fit | A child is 110 cm tall at age 5, so 220 cm tall at age 10. | Every relationship is treated as “twice as much, twice as much”. |
The page Maths in grades 3 and 4 (in German) also lists the first example as a typical misconception in the content area patterns and structures.
Why children think additively
In the first years of school, children mainly meet additive relationships: something is added, something is taken away. Language reinforces this, because “three more exercise books” sounds like plus. Proportional relationships require a focus on multiples: if the quantity is doubled, the price doubles.
The German project Mathe sicher können describes this in its support module on proportional relationships with the idea that “for each portion, the same amount is always added”. It follows that a multiple of the quantity goes with the same multiple of the price. The module distinguishes four calculation methods: adding step by step, scaling up, working back to one portion, which is also known as the unitary method, and calculating with the constant factor. Among the typical errors it lists taking the price for two items as the price for one portion and then adding it step by step, and associating “proportional” only with doubling. The module is written for lower secondary school, but the foundations are laid in grades 3 and 4.
Building the relationship
Tables with arrows. Quantity on the left, price on the right. The children note how they get from one row to the next: “times 2” on the left means “times 2” on the right. Small tables with only the rows that are needed are enough.
| Exercise books | Price |
|---|---|
| 3 | €6 |
| 1 | €2 |
| 6 | €12 |
| 9 | €18 |
The price of one. The question “How much does one exercise book cost?” turns many tasks into a simple one. €6 for 3 books is €2 per book. From there, any quantity can be calculated.
Doubling and halving. Double the quantity, double the price. Half the quantity, half the price. Recipes are good for practising this: pancake batter for 4 people is doubled for 8 people and halved for 2.
Strip diagram. Each exercise book is a strip of two boxes for €2. Six books are six strips, so twelve boxes. This shows that each book adds the same amount.
Counterexamples. Not every relationship is proportional. Height does not grow in proportion to age, and large packs often cost less than the quantity would suggest. The PIKAS material on money values points out that in everyday life goods and prices rarely behave proportionally because of bulk discounts. Children should therefore ask whether “twice as much” really applies here.
Asking diagnostic questions
- “How much does a single exercise book cost?”
- “How many more exercise books are there now? And how many times as many?”
- “Show me your method in the table.”
- “Can 6 exercise books cost €9 if one book costs €2?”
- “Does this apply here too: twice as much, twice the price?”
If a child works out the price of one item correctly but does not use it, they need the table as a tool. If they cannot work out the price of one item, the hurdle is more likely to be division and understanding “per item”.
How Mazio Math supports functional thinking
Mazio Math is a learning platform for maths lessons in primary school. The content is organised by the five content areas of the German national education standards, including patterns and structures, and is designed for the two-year bands of grades 1/2 and 3/4. For grades 3 and 4 there are units in all five content areas. In “Checking calculation methods”, the children decide which method fits a question in a word problem.
Finding content. The “Curriculum” page in the dashboard shows whether there is matching material for a focus in the curricula of the German federal states. With the “Grade 3/4” filter and a focus expanded, you see the matching units, Basiscamp exercises and Knobelbox tasks under “Matching content in Mazio Math”. On the shelf, you can use the search to filter by title, topic or competence.
Securing the basics. Children who cannot work out the price of one item often need the multiplication tables and division. The Basiscamp practises these in the “Multiply and Divide” world and has worlds on patterns, balance and puzzles. It remembers for each child which task types they can already do confidently. With the plus on a card on the shelf, you put a whole sub-world on the class board.
A lesson sequence for grade 4. The lesson begins with the exercise book task and the answers €9 and €12. The class votes on them and then decides with a table. Next, you use the plus to put a suitable piece of content on the board, with a note such as “Ask yourselves first: how much does one cost?”. The row shows you how many children have finished. If a child makes a mistake, the learning assistant Mazio helps directly in the task with a question, with markings in the picture and, if needed, with a given partial step. He does not give the solution.
Misconceptions in the class. Once a week, Mazio Math checks whether a common misconception shows up in the class’s wrong answers. The support card explains the error in thinking with an example, suggests steps for teaching and proposes suitable content for the board. You decide what goes on the board.
Next step. Register for free to create an account and look at the shelf and the curriculum view. The article How Mazio Math evaluates the class’s errors in thinking describes how Mazio Math makes errors in thinking visible at class level. Using Mazio Math with a class, and trying it out beforehand, is currently available to schools in Germany. If your school is in another country, registration records your interest.
Frequently asked questions
Do children learn the unitary method in primary school?
As a formal procedure, usually later. The idea behind it, first finding the price of one item and then scaling up, is something children in grades 3 and 4 can already use with tables. When the topic is covered depends on the curriculum where you teach.
How can I tell whether a child is thinking proportionally?
The child expects twice the price for twice the quantity and can explain why, for example with the price of one item. If the child simply adds the increase in quantity to the price, they are still thinking additively.
Is every relationship between quantity and price proportional?
No. With large packs or special offers, the price per item often falls. In lessons you first work with situations in which every item costs the same and then deliberately discuss counterexamples.