Construction of Number Sentences - More About

Introduction to logical structures of story problems

Consider the following two story problems:

Lucy has 3 marbles. Jane has 5 marbles. How many more does Jane have?
Australia has 20 million people. Singapore has 3 million people. How many more people are there in Australia than in Singapore?

These are both solved by subtraction (5 − 3 = 2 or 20 − 3 = 17). While the numbers and contexts are different (marbles and populations), both problems have the same logical structure of comparing two quantities.

Now consider the following two story problems:

Lucy has 3 marbles. Jane has 5 marbles. How many more does Jane have?
Ken had 5 marbles, but lost 3. How many does Ken now have?

In these two problems the numbers are the same and they both involve subtraction, but the logical structures are different. The first is about comparing 2 quantities and the second is about a change to one quantity. The one operation of subtraction applies in both of these real world situations.

The different types of logical structures for the four basic operations are summarized below.

Logical structures of addition and subtraction story problems

The number sentences in an addition/subtraction fact family (e.g. 3 + 5 = 8, 8 – 5 = 3, 8 – 3 = 5) can describe real world situations with three basic logical structures:

• combine quantities
• change a quantity (includes ‘take away’ subtraction)
• compare quantities (includes ‘compare’ subtraction)

Examples of three basic logical structures for addition and subtraction
Combine quantities
Lucy had 3 marbles. Jane had 5 marbles. How many did they have altogether?	3 + 5 = ?
Lucy and Jane had 8 marbles altogether. If Lucy has 3 marbles, how many does Jane have?	8 - 3 = ? or 3 + ? = 8
Australia has 20 million people. Singapore has 3 million people. How many do they have altogether?	20 + 3 = ?
Change a quantity
Ben is 7. How old will he be in 4 years time?	7 + 4 = ?
Ben is 7. In how many years will he be 11?	7 + ? = 11 or 11 - 7 = ?
In 4 years time, Ben will be 11. How old is he now?	? + 4 = 11 or 11 - 4 = ?
Matthew had 15 lollies and he ate 9. How many did he have left?	15 - 9 = ?
Matthew had 15 lollies. He dropped some and then had only 9 left. How many did he drop?	15 - ? = 9 or 15 - 9 = ?
Nicholas had 15 budgerigars. Some flew away and he had only 9 left. How many flew away?	15 - ? = 9 or 15 - 9 = ?
Nicholas had some budgerigars, but 6 flew away and he had 9 left. How many did he start with?	? - 6 = 9 or 9 + 6 = ?
Compare quantities
Carol is 15 years old. She is 9 years older than Timmy. How old is Timmy?	15 = ? + 9 or 15 - 9 = ?
Carol is 15 years old. Leila is 9 years old. How much older than Leila is Carol?	15 = 9 + ? or 15 - 9 = ?
My car took 45 litres of petrol and your car took 60 litres. How much more did your car take?	60 - 45 = ?

Logical structures of multiplication and division story problems

The number sentences in a multiplication/division fact family (e.g. 3 × 4 = 12, 4 × 3 = 12, 12 ÷ 3 = 4) can describe real world situations with several different logical structures including:

• equal groups
• arrays
• rates
• Cartesian product
• product of measures.

NOTE: Students will encounter all the different addition and subtraction situations by Level 3. However, it is expected that with multiplication and division, students only work with equal groups and arrays in the early levels, and study the other multiplicative situations in Levels 4 - 6.

Examples of logical structures for multiplication and division
Equal groups
Lucy had 3 packets of 4 pencils. How many pencils did she have altogether?	3 × 4 = ? or 4 + 4 + 4 = 12
Lucy got 12 pencils by opening 3 packets. How many pencils were in each packet? (This is partition division - the 12 pencils came from 3 equal groups)	3 × ? = 12 or 12 ÷ 3 = ?
Lucy put 12 pencils into 3 packets. How many pencils were in each packet? (This is partition division - the 12 pencils are shared into 3 equal groups)	12 ÷ 3 = ?
Lucy got 12 pencils by opening some packets each containing 4 pencils. How many packets did she open? (This is quotition divison - the pencils came in 'quotas' of 4)	? × 4 = 12 or 12 ÷ 4 = ?
Lucy had 12 pencils and put them into packets each containing 4 pencils. How many packets did she need? (This is quotition divison - each packet gets a quota of 4)	12 ÷ 4 = ?
Arrays
A flag has 10 rows of stars, each containing 5 stars. How many stars on the flag altogether?	10 × 5 = ?
I want to pack 24 tins into a box which holds 6 in a row. How many rows will there be?	? × 6 = 24 or 12 ÷ 6 = ?
Rates
I bought 2.8 kg of apples for $5.50 per kilogram. How much should that cost?	2.8 × 5.50 = ?
I spent $6 on apples at $5.50 per kg. What weight did I buy?	6 ÷ 5.50 = ?
Cartesian product
A girl has 3 t-shirts and 4 skirts. How many different outfits can she make?	3 × 4 = ?
A girl has 3 t-shirts and 4 skirts and 5 hats. How many different outfits can she make?	3 × 4 × 5 = ?
Product of measures
A rectangle measures 3 cm by 5 cm. What is its area?	3 × 5 = ?
What is the momentum of a mass of 3kg travelling at 4 metres per second?	3 × 4 = ?