Some maths problems
| a) |
At a beach volley ball game there are 154 people. A bus brings 38 more people.
Show how to work out how many people are at the game altogether.
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| b) |
At a kapa haka competition there are 357 children and 162 adults.
Show how to work out how many people are at the competition altogether.
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| c) |
At a table tennis competition there are 326 players and 279 people in the crowd.
Show how to work out how many people are at the competition altogether.
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| d) |
A farmer has 736 sheep. He buys 589 more sheep.
Show how to work out how many sheep he has altogether.
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- Place value partitioning both numbers using hundreds, tens and ones with correct compensation. (Number Framework Stage 5).
- Place value partitioning one number into hundreds, tens and ones then adding it on to the other in parts (Number Framework Stage 6).
- Partitioning using rounding and compensation to jump through tidy numbers or Partitioning by rounding one number to a tidy number then compensation (Number Framework Stage 6).
- Students who uss the vertical algorithm need to be questioned why it works. Are the using it procedurally without understanding?
- An incomplete strategy 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 "154 + 38 = 192" need to be questioned how they got their answer, and then levelled according to the table above.
Key competencies
| Y6 (10/2010) | ||
| a) |
192 Any 1 of:
- rounding and compensation to jump through tidy numbers
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easy very easy |
| b) |
519 Any 1 of the above strategies |
moderate very easy |
| c) |
605 Any 1 of the above strategies |
easy very easy |
| d) |
1325 Any 1 of the above strategies |
moderate very easy |
Based on a representative sample of 201 students.
NOTES
- Students who used a strategy correctly but did not get the correct answer due to a minor error still gained credit for their working. This included the very small number of students who compensated in the wrong direction.
- Students who did some of the two-digit calculations mentally were given full credit. e.g., 357 + 162 = (350 + 160) + (7 + 2) or (300 + 100) + (57 + 62). It may be worth asking them how they did these two-digit calculations mentally.
- 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.
| Common response | Likely misconception | |
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a) b) c) d) |
189 - 191, 193 - 195 516 - 518, 520 - 522 602 - 604, 606 - 608 1322 -1324, 1326 - 1328 |
Answer within 3 of correct answer - some counting involved Students are likely to be using counting strategies on the ones digits, and was often coupled with partitioning strategies. These students probably do some of the compensations or rounding by counting on (or back) rather than by using automated basic facts. There was no direct evidence of students using only counting strategies, due largely to the magnitude of the numbers making it inefficient and unmanageable even in part a). |
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a) b) c) d) |
182 419 505 or 595 1215, 1225 or 1315 |
Crosses the tens or hundreds boundary incorrectly (not "carrying") The student gets an answer that is ten, a hundred (or 110) less than the correct answer. This happens both with place value partitioning as well as failing to "carry" in the vertical algorithm. It was most prevalent in parts b) and d), and only happened occasionally in part a). Several students got answers close to these, suggesting they were also using counting strategies, but probably just on the final compensation of the ones. |
Student Strategies
Students who used any of the first four partitioning strategies had approximately equal mean abilities. Students who used the vertical algorithm met with roughly equal success rates, but were of a markedly lower mean ability than those using partitioning.
Strategy
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Success rates and student ability
Notably less successful than other partitioning strategies (including the vertical algorithm). More successful in parts a) and c). Most common for Part a) where it was very successful and students had high mean ability. Less successful in part b)-d) and students had a lower mean ability. By far the most common strategy. Highest success rate. May be good for larger numbers, and Links to the vertical algorithm. Students had a lower mean ability |
Many students who used strategies 2 - 4 (above), showed only part of their working, particularly in parts b) - d). They were clearly doing some two-digit calculations mentally. For example:
(b) 357 + 162 = (350 + 160) + (7 + 2) or (300 +100) + (57 + 62)
(c) 326 + 270 = 596; 596 + 9 = 605
(d) 740 + 590 = 1330; 1330 - 4 - 1 = 1325
These students had approximately equal mean abilities as those who did not use two-digit mental arithmetic, but had somewhat lower success rates due to errors in their mental arithmetic.
Answer within 3 of correct answer - some counting involved
Students who use partitioning strategies coupled with counting strategies for the ones need to work on their addition (and subtraction) 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. Students could be using other addition resources involving smaller numbers such as Solving maths problems.
Crosses the tens or hundreds boundary incorrectly (not "carrying")
These students need to take account of when the ones digits, the tens digits, or the hundreds 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 "154 + 38 = 182" get them to do the sum 154 + 35. They will most likely come up with 189. They can then see that the first sum is 3 more than 189. Alternatively get them to calculate 150 + 30 = 180 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. Students could also explore Number line addition & subtraction and Number line addition.
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
- Addition wheel
- Writing word problems
- Estimate these II
- Tests and marks
- Keep fit programme
- Sharing Jelly beans
- Knock the can
- At the canteen
- Number line addition II
- Estimating with addition III
- Estimating addition IV
- Student answers
- Number line addition and subtraction III
- Estimating with addition
- Estimation or not?
- Eating fractions of pie, pizza and cake
- Saving for a pet
- Fractions of cake
- Solving maths problems
- Collecting beads
- Making a cake
- Fruit and vegetables