Making number sentences II

Making number sentences II

Pencil and paper
Overview
Using this Resource
Connecting to the Curriculum
Marking Student Responses
Working with Students
Further Resources
This task is about writing number sentences and story problems.
illustration: selection of number sentences
A number sentence uses numbers and symbols instead of words.
For example, Five friends have 6 marbles each.  How many do they have altogether? (story problem)
could be written as 5 × 6 = ? (number sentence).

Question 1Change answer

a) Liline went shopping for Easter eggs. She bought 7 packets of 5 eggs.
Write a number sentence to show how many Easter eggs Liline bought altogether.

Question 1Change answer

b) Hamish gave away all his old Pokemon cards. He had 28 cards and shared them equally between his 7 friends.
Write a number sentence to show many cards they get each.

Question 1Change answer

c) Maraea was training for a swimming race. For the first 3 days she swam for 2 hours a day.
    For the next 3 days she swam for 5 hours a day.
Write a number sentence to show how many hours she swam over the 6 days.

Question 1Change answer

d) Write a story problem for the number sentence 3 x 5 = ?

Question 1Change answer

e) Write a story problem for the number sentence 24 ÷ 4 = ?
Task administration: 
This task can be completed with pencil and paper or online.
Level:
3
Curriculum info: 
Maths, Number and Algebra, Equations and expressions
Key Competencies: 
Using language, symbols, and texts
Keywords: 
number sentences, story problems, multiplication, division
Description of task: 
Students write number sentences for story problems, and create story problems from number sentences.
Curriculum Links: 
Key competencies
This resource involves writing word problems that corresponds to  given number sentences. This relates to the Key Competency: Using language, symbols and text.
Learning Progression Frameworks
This resource can provide evidence of learning associated with
Using symbols and expressions to think mathematically, set 4
within the Mathematics Learning Progressions Frameworks.
Read more about the Learning Progressions Frameworks.
Answers/responses: 
    Y6 (06/2006)
a) 7 × 5 = 35 very easy
b) For any 1 of:

  • 28 ÷ 7 = 4
  • 4 × 7 = 28
  • 28 – 7 – 7 – 7 – 7  (repeated subtraction)
 moderate
c) For any 1 of:

  • (3 × 2) + (3 × 5) = 21
  • 3 × 2 + 3 × 5 = 21
  • 3 × (2 + 5)
  • 3 × 7 = 21
  • 2 × 3 = 6, 3 × 5 = 15, 6 + 15 = 21
  • 2 + 2 + 2 + 5 + 5 + 5
  • 6 + 15 = 21 [Accept this as correct but ask students to explain/write down how they got the 6 and the 15].
 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"
Teaching and learning: 
The focus of this resource is on the ability of students to translate between number sentences and story problems rather than correct calculation.  Accordingly, the importance is on the creation of the equation not the answer.  Only a very small number of students who wrote a suitable equation got the answer incorrect.
Diagnostic and formative information: 
  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.
Next steps: 
For students who write the division in the incorrect order (i.e., 7 ÷ 28 = 4) ask them to read back the number sentence using the word "divided by" for division (as opposed to "goes into").
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 left-hand side of the equals sign should represent the same quantity as the expression on the right-hand 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.