Solving maths problems


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.




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.




c) 
A farmer has 63 cows. He buys 59 more cows.
Show how to work out how many cows he has altogether.



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 countingon (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].
Y5 (05/2010)  
a) 
34 Any 1 of:

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

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.
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 partwhole 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 
Miscounts by 1 or 2
These students need to move from the counting stages to partwhole 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.
 Buying groceries
 Buying books II
 School swimming sports
 Numbers at an art show
 Bank account
 Temperature changes
 City populations
 Sharing fruit and vegetables
 Different dollars
 Addition wheel
 A few coins
 Adding and subtracting
 Adding sweets
 Change from $10
 Lemons, library and sports
 Adding coins
 Different coins
 Notes and coins
 Spending pocket money
 The same total
 How much change?
 Writing word problems
 Estimate these II
 Tests and marks
 Keep fit programme
 Sharing Jelly beans
 Knock the can
 Fractions
 Working with numbers II
 Money computations
 Adding and subtracting fractions III
 At the canteen
 Jack's cows
 Number line addition II
 Estimating with addition III
 Number line addition
 Estimating addition IV
 Student answers
 Number line addition and subtraction III
 Cover up
 Seeds and sweets
 Estimating with addition
 Estimation or not?
 Number line addition and subtraction
 Eating fractions of pie, pizza and cake
 Cover up II
 Saving for a pet
 Fractions of cake
 Some maths problems
 Collecting beads
 Party balloons
 Adding and subtracting more numbers
 Going on a picnic
 Spending at the shop
 Buying some things
 The right money
 Counting money
 Making a cake
 Fruit and vegetables