Buses, games and trains
a) 
A bus has 67 passengers. It drops off 39 people.
Show how to work out how many people are still on the bus.




b) 
Pere has 127 computer games. He gives away 35 of his old games.
Show how to work out how many computer games he still has.




c) 
A train has 264 people. 92 people get off at a station.
Show how to work out how many people stay on the train.



 Using or describing countingback strategies suggests achievement (Number Framework Stage 4).
 Using the above partitioning strategies with incorrect compensation (Number Framework Stage 5). The student recognises the role of hundreds, tens and ones but cannot use them to compensate or cross the tens boundary correctly.

Place value partitioning both numbers using hundreds, tens and ones with correct compensation (Number Framework Stage 5).

Place value partitioning the smaller number into tens and ones, then subtracting in parts (Number Framework Stage 6).

Partitioning by
 complementary addition to jump through tidy numbers,
 rounding one number to a tidy number then compensation, or
 rounding and compensation to jump through tidy numbers, each with correct compensation (Year 6, Number Framework Stage 6).
 Place value partitioning using hundreds, tens and ones with correct compensation involving negative numbers. Negative numbers do not arise until curriculum level 4, but some students show readiness for them earlier.
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 "127  35 = 92" need to be questioned how they got their answer, and then compared to the table above.
Y6 (10/2010)  
a) 
28 Any 1 of:
[Using partitioning strategies with incorrect compensation.]* 
moderate easy easy 
b) 
92 Using any 1 of these strategies:
[Using partitioning strategies with incorrect compensation.]* 
easy very easy [very easy] 
c) 
172 Using any 1 of these strategies.
[Using partitioning strategies with incorrect compensation.]* 
moderate moderate [easy] 
Based on a representative sample of 201 students.
NOTES:
 (*) Many students start off with a partitioning strategy but cannot complete it correctly because of incorrect compensation, most commonly compensation in the incorrect direction, but also ignoring compensation, or otherwise misunderstand compensation. For subtraction, keeping track of the direction of compensation is an essential part of demonstrating understanding and is much harder than in addition problems. For example, in part c) it is easy to partition numbers sensibly, but the difficulty level of also compensating in the correct direction is moderate.
 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 who use the vertical algorithm and get a correct answer should be asked how and why it works to see if it is just procedural knowledge rather than displaying understanding.
 Students who used the vertical algorithm and made a conceptual error were given no credit.
Patterns within the strategies used
 Place value partitioning the smaller number into hundreds, tens and ones, then subtracting these in parts in the correct direction. This was used by the group with the second highest mean ability and was highly successful (86%), which shows that it is both sophisticated and successful. This is perhaps because it can be used with compensation always in a predictable (downwards) direction.
 On the other hand, using Partitioning by rounding one number to a tidy number then compensation or Partitioning using rounding and compensation to jump through tidy numbers was used by students with a slightly lower mean ability and had a lower success rate (72%). It is harder to control the direction of compensation correctly with this strategy. This demonstrates that although the strategy is higher on the Number Framework, it tends to break down with larger numbers.
 Partitioning using complementary addition to jump through tidy numbers was rarely used but was highly successful, as the direction of jumps (compensations) is consistently in the positive direction.
Other noteworthy points were:
 Students using the vertical algorithm had a lower mean ability and a relatively low success rate (73%).
 Students who merely stated the equation were of high mean ability for part a) [67  39], but lower for the other two parts. This may indicate that using mental strategies on two digit numbers is more effective than using them on larger numbers.
See the Analysis of student responses [pdf] for more detailed information about the strategies.
Common response  Likely misconception  
a) b) c) 
26, 27, 29 or 30 90, 91, 93 or 94 170, 171, 172 or 174 
Miscounts by 1 or 2 The student is using a counting strategy but miscounts by a small amount. It also occurs with students who use partwhole partitioning strategies, suggesting that these students were using counting strategies for the ones 
a) b) c) 
38 192 272 
Place value error  Crosses the tens or hundreds boundary without "renaming" The student does not compensate for performing 15  7 instead of 5  7. The student gets an answer ten more (or occasionally less) than the correct answer. This happened often with place value partitioning but did not occur with any students using the vertical algorithm. It was reasonably uncommon in b) and c), but did not occur in part a) 
a) b) c) 
32 112 or 88 232 or 132 
Over generalising ("won't go") error Students swap the subtrahend (number being subtracted) and the minuend (number being subtracted from) when they subtract the smaller number from a larger number. For example in the problem 67  39, students might attempt to subtract 9 from 7, but when that doesn't go, so they will subtract 7 from 9 to get 2, and then 60  30 = 30, 30 + 2 = 32. This was a common error, and occurred often when students used place value partitioning. It was far less prevalent with the vertical algorithm than with place value partitioning even though it is most often associated with the former. 
Miscounts by 1 or 2
Students who used just counting back strategies could be encouraged to start to explore the partwhole additive strategies they use when subtracting from numbers under 20. This is referred to ascomplimentary addition or reversibility. After developing this understanding students should describe how to solve a subtraction problem with "recombination" or borrowing (see Place value error), or use number lines.
Students who use partwhole partitioning strategies coupled with counting strategies for the ones need to work on their subtraction and addition basic facts. These are an important part of solving problems by removing the reliance on strategies that are at earlier stages of the Number Framework.
Place value error  Crosses the tens or hundreds boundary without "renaming"
Students who don't take into account the need to compensate when "borrowing" a ten may need to explore what happens when numbers are subtracted from each other involving recombination, i.e., indicate what a subtraction can look like with abacus, place value blocks, tens frames, etc. For example, students could use an abacus to show what happens when we subtract 39 from 67. At some stage we want to take away from 9 and will need to work out how to do this (borrowing, renaming, splitting numbers into parts, or some other way). These students could also work with problems representing them on number lines.
If the student was using place value partitioning, then they could decompose 67  39 (for example) as (50 + 10  30) + (7  9) = (50  30) + (10 + 7  9) = 20 + (17  9). This shows 60 renamed as 50 + 10 and then regrouping (commuting) the numbers.
Over generalising ("won't go") error
Students could be exposed to some problems where this strategy clearly leads to a wrong answer. Good examples would be 65  57 or 85  79. Both must be single digit answers, but these students' strategies would lead to 12 and 14 respectively, which are both too large. The ones digit is also incorrect. They could solve these examples by counting back, or solve the equivalent problems 57 + = 65 or 79 + = 85 by counting on. The students could then draw these on number lines, then incorporate the tens to do the original problems (use the keywords: subtraction AND number lines). Alternatively, they could work with the "renaming" ideas below.
Numeracy Project Resources
Book 5: Teaching Addition, Subtraction, and Place Value, 2007, Subtraction in parts (Early additive partwhole)
 Buying groceries
 Numbers at an art show
 Bank account
 Cans of fruit drink
 Temperature changes
 City populations
 How much farther?
 Addition wheel
 Subtraction wheel
 Writing word problems
 Tests and marks
 Keep fit programme
 Sharing Jelly beans
 At the canteen
 Student answers
 Number line addition and subtraction III
 Number line subtraction II
 Eating fractions of pie, pizza and cake
 Going on camp
 Fractions of cake
 Solving more maths problems
 Easy or harder subtractions
 Collecting beads
 Making a cake
 Fruit and vegetables
 Buying a phone