Solving maths problems

Solving maths problems

Pencil and paper
Overview
Using this Resource
Connecting to the Curriculum
Marking Student Responses
Working with Students
Further Resources
This task is about solving maths problems.

illustration: two cows
a)
There are 21 people at a beach, and 13 more people arrive. 
Show how to work out how many people are at the beach altogether.
 
 
 
 
 
  
 
Answer: __________  

 
b)
Two classes go on camp together. One class has 37 students and the other has 28. 
Show how to work out how many students go on camp.
 
 
 
   
 
 
  
 
Answer: __________  

 
c)
A farmer has 63 cows.  He buys 59 more cows. 
Show how to work out how many cows he has altogether.
 
 
 
 
 
    
 
 
Answer: __________  
Task administration: 
This task is completed with pencil and paper only.
Levels:
2, 3
Curriculum info: 
Maths, Number and Algebra, Number Strategies
Key Competencies: 
Using language, symbols, and texts
Keywords: 
addition, work samples
Description of task: 
Students show how to solve some addition problems.
Curriculum Links: 

This resource can help to identify students' understanding of addition. Students' strategies give a better indication of curriculum level and progress than their answer. A possible progression for understanding could involve:

  • Using counting all (Number Framework Stage 3 – Counting from One).
  • Using counting-on (Number Framework Stage 4 – Advanced Counting).
  • Place value partitioning expressing tens as ones (Number Framework Stage 5 – Early Additive).
  • Place value partitioning using tens and ones (Number Framework Stage 5 – Early Additive).
  • Partitioning using rounding and compensation strategies (Number Framework Stage 6 - Advanced Additive).

Incomplete strategy use that results in a correct answer may indicate the student is operating at an early point of the relevant Number Framework stage. Students who just give an answer, or who make statements such as "21 + 13 = 34" need to be questioned how they got their answer, and then levelled according to the table above.

Click on the link Examples of students working for examples of the diffferent strategies [pdf].

Key competencies

This resource involves recording the strategies students used to solve addition problems. This relates to the Key Competency: Using language, symbols and text.

Learning Progression Frameworks
This resource can provide evidence of learning associated with
Additive thinking, set 5
Additive thinking, sets 6-7
within the Mathematics Learning Progressions Frameworks.
Read more about the Learning Progressions Frameworks.
Answers/responses: 
  Y5 (05/2010)
a) 34
Any 1 of:

  • Partitioning using rounding and compensation to cross boundaries
    e.g., 21 + 13 = (21 + 9) + 4
  • Partitioning by rounding to a tidy number then compensation
    e.g., 21 + 13 = (21 + 10) + 3
  • Place value partitioning in tens and ones
    e.g., 21 + 13 = (20 + 10) + (1 + 3) = 30 + 4
  • Place value partitioning expressing tens amounts in ones
    e.g., 21 + 13 = (2 + 1) lots of 10 + (1 + 3)
  • Visually displaying place value or other strategy
  • Counting strategy (counting all or counting on)
  • Vertical algorithm
  • Other correct strategies
 easy
easy
b) 65
Any 1 of the above strategies
Students using place value partitioning or the vertical algorithm need to take account that the sum of the ones digits is greater than ten.		 easy
easy

 
c) 122
Any 1 of the above strategies (partitioning, place value strategies, visual display, counting, vertical algorithm etc.)
Students using place value partitioning or the vertical algorithm need to take account that the sum of the ones digits is greater than ten.		 moderate
easy

 
Based on a representative sample of 159 Y5 students.

NOTE:
Students who use a strategy correctly but do not get the correct answer due to a minor error still gain credit for their working. Students using the vertical algorithm should be asked how and why it works to see if it is just procedural knowledge rather than displaying understanding.  This is just one way of solving addition, and should only be explored after students have explored their own strategies for solving such problems.

Diagnostic and formative information: 

Student strategies

  • The strategy that led to the correct answer most often was Partitioning using rounding and compensation to cross boundaries (partitions across boundaries). This was also used by students with the highest mean ability.
  • Partitioning by rounding to a tidy number then compensation (Partitions with nice numbers) and Place value partitioning using tens and ones (PV partitions with 10s and 1s) attracted students with the next highest mean abilities.
  • Students using the Vertical algorithm were less able that those using all the above strategies, but were about as likely to get a correct answer as the as the strategies in the second bullet point.
  • Students who used Place value partitioning expressing tens as ones, Visually displaying partitioningor Other PV strategies were of lower mean ability and had a success rate close to 50%.
  • Students who restate the problem or give the addition equation "21 + 13 = 34" were of similar ability as those in the last bullet point, but were more likely to obtain a correct answer (about 70% of them).
  • Students using counting strategies were of slightly lower mean ability, but still had a success rate of about 50%.
  • Students with incomplete strategies had a low mean ability, and had low success rates.
  • Students who made other statements, or who did not answer the question had the lowest mean abilities, and had very little success on any of the questions in the test set.

Click on Analysis of student responses for NM1328 [pdf] for supporting evidence for these.

  Common response Likely misconception
a)
b)
c)		 32, 33, 35 or 36
63, 64, 66 or 67
120, 121, 123 or 124		 Miscounts by 1 or 2
The student is using a counting strategy, either counting all or counting on. It also occurs with students who use a mixture of part-whole partitioning strategies and counting. In particular, it suggests suggesting that these students were using counting strategies for the ones.
b)
c)		 55
112		 Crosses the tens boundary incorrectly
The student gets an answer ten less the correct answer. This happens both with place value partitioning as well as failing to "carry" in the vertical algorithm. It was far more prevalent in part c), and did not occur in part a). Several students got answers close to these, suggesting they were also using counting strategies.
b)
c)		 20 (= 3 + 7 + 2 + 8)
23 (= 6 + 3 + 5 + 9)		 Counts the individual digits
Does not appreciate the role of digits in the tens place. Many students have added the tens digits and ones digits separately (e.g., for 63 + 59 student go 6 + 5=11 and 3 + 9=12), but then failed to see that 11 is equivalent to 11 tens = 110.
a)
b)
c)		 21 + 13 (= 34)
37 + 28 (= 67)
63 + 59 (=122)		 Restates the question or writes as an addition number sentence
Next steps: 

Miscounts by 1 or 2
These students need to move from the counting stages to part-whole thinking. The Numeracy Development Project: Book 5 gives strategies for this on pages 26­­–28.
Students who are already using early additive (Stage 5) ideas of partitioning need to move from counting the ones (or the tens), and rely on using additive basic facts instead.

Crosses the tens boundary incorrectly
These students need to take account of when the ones digits sum to more than 10. One way to do this is get them to perform a slightly easier but related addition. For example, if they say "63 + 59 = 112" get them to do the sum 63 + 56. They will most likely come up with 119! They can then see that the first sum is 3 more than 119. Alternatively get them to calculate 60 + 50 = 110 and then see that the answer is 12 bigger than this (not 2 bigger). The Numeracy Development Project: Book 5 gives other strategies for crossing the tens boundary on page 28.

Counts the individual digits
If the student has done 6+3+5+9, they probably do not have the concept of place value and need to work with materials then imaging to develop the idea.
Students who add the ones and the tens separately (e.g., 6 + 5=11 and 3 + 9=12) need to see that there are 11 lots of 10 (=110). Get them to do the sum 60 + 50 (= 6 tens plus 5 tens) to consolidate this idea.

Restates the question or gives the addition equation
These students need to develop their skills in showing how they arrived at an answer. They may initially do this orally and then attempt to write it down. Exemplars of other students written strategies may assist this (see Examples of students' working [pdf]).
Some students may have used scrap paper and may need to show this or be encouraged to use the box to write down their working. Students may have used a calculator. Ask them to show how they would perform the sums without a calculator by repeating the question, perhaps with slightly altered numbers.

Student strategies
Generally the further up the Numbers Framework students were, the higher their mean ability, and the more likely they were to give a correct answer.
For more detailed information click on the link Analysis of student responses for NM1328 [pdf].

Many Figure It Out and Numeracy resources explore addition, in particular Numeracy Development Project: Book 5: Teaching Addition, Subtraction, and Place Value.