Making number sentences II
Y6 (06/2006)  
a)  7 × 5 = 35  very easy 
b) 
For any 1 of:

moderate 
c) 
For any 1 of:

difficult 
d) 
Accept any scenario that incorporates multiplying 3 and 5 to get 15, e.g., "Hayden bought 3 packets of lollies that had 5 in every pack. How many lollies did Hayden buy?" 
moderate 
e) 
Accept any scenario that incorporates dividing 24 by 4 to get 6, e.g., "At Harry's party he had a cake with 24 slices. How many slices did each person get if only 4 people went to the party?" 
moderate 
Based on a representative sample of 177 Y6 students.
NOTE:
For questions d) and e) the correct student responses were either:
 a question involving the appropriate operation, e.g., "Me and 3 friends [sic] all shared 24 lollies. How many did we get each?"; or
 a statement with the correct operation and answer, e.g., "Lin was training for a running race in 3 days time. She trained for 5 hours a day til [sic] then. 15 hours"
Common error  Likely misconception  
b)  7 ÷ 28 = 4 
Students recognise that sharing is a division operation but write the numbers as they might be written in long or short division format. (15% of students wrote the number sentence in this way). 
b) c) 
4 21 
Solution only Students write the solution only, not the number sentence to describe the problem. This could indicate an understanding that maths is only about finding answers. 
c)  2 + 5 
Incorrect operator Students add the hours spent swimming, but do not take into account that these were rates (hours per day). Students may also think that because both expressions are × 3, that the 3's cancel each other out. 
c)  2 × 3 + 5 
Incorrect use of the distributive law Students do not multiply the 5 hours by 3. This could indicate a misuse of the distributive law, i.e. a × (b + c) = a × b + a × c. It is interpreted as a × (b + c) = a × b + c. This assumes that because the 2 has been multiplied by 3, the 5 does not also have to be multiplied by 3. 
c)  3 × 2 = 6 = 3 × 5 = 15 = 6 + 15 = 21  Students incorrectly use the equals sign in a string. 
c)  2 × 5  Students may be confusing the addition and multiplication relationship in the problem. 
NOTE: they may also be getting confused with the short and long division layouts.
Students who solved the problem without writing a number sentence may have developed an understanding that maths is only about "getting the answer". Discuss the importance of being able to "formulate" the story problem into a suitable number sentence and encourage students to write down their thinking.
For students who use an incorrect operator or misapply the distributive law, get them to model or draw the story problem and break it into parts that can be reconnected later, e.g., "If she swam for 2 hours each day for 3 days, what does this look like?" 2 and 2 and 2. If students do not make this connection try asking how many hours on each day and then ask how many altogether. "What could be a number sentence for that?" 2 + 2 + 2 "How many twos?" → 3 × 2. Then "If for the next 3 days she swam for 5 hours a day, what does this look like? What would be a number sentence for that?" 5 + 5 + 5 → 3 × 5. Combining these two expressions back together could involve learning about brackets or the order of operations. Connecting the modelling and the expression should help students recognise the predominance of multiplication over addition in the number sentence.
Students who exhibited an incorrect use of the equals sign could be encouraged to use a symbol or sign to show their processes, e.g.,
3 x 2 = 6
3 x 5 = 15
→ 6 + 15 = 21
A misuse of the equals sign could indicate a misunderstanding of the meaning of the equals sign. Students need to understand that the expression on the lefthand side of the equals sign should represent the same quantity as the expression on the righthand side of the equals sign, or that "=" means "the same as". To explore this concept further, click on the link or use the keyword equality.
17% of students wrote the equation for part c) in "pieces" e.g., 3 x 2 = 6 3 x 5 = 15 6 + 15 = 21 While not strictly incorrect, these students should be encouraged to show how to join all the expressions together into one equation, e.g., 3 x 2 + 3 x 5 = 21.
Students who have used repeated addition or subtraction to express a story problem,
e.g., 28 – 7 – 7 – 7 – 7, can be encouraged to explore the relationship between repeated addition and multiplication or repeated subtraction and division. They can then find a number sentence that better describes the relationship in the story problem, e.g., 28 ÷ 7 = 4.