Powerful twenty five
a) Show how to solve 25 × 3.


b) Show how to solve 25 × 8.


c) Show how to solve 25 × 25.


d) Show how to work out how many classes of exactly 25 students could be made with 450 students.

Y8 (11/2007)  
a) 
i) 
75 
very easy 
b) 
i) ii) 
200 
easy 
c) 
i) ii) 
625 
moderate 
d) 
i) ii) 
18 
moderate easy 
 This task aims to see if the students can utilise basic facts knowledge of the 25times table. It can also give information about students’ multiplication and division strategies, even if they have no knowledge of basic facts about 25.

Only four basic facts are needed for the 25times table
1 × 25 = 25 (so 5 × 25 = 125; 9 × 25 = 225)
2 × 25 = 50 (so 6 × 25 = 150 10 × 25 = 250)
3 × 25 = 75 (so 7 × 25 = 175 11 × 25 = 275)
4 × 25 = 100 (so 8 × 25 = 200 12 × 25 = 300, etc.)  The 25times table is useful for:
 doing whole number computations involving 25

linking with percentages: 25% is a quarter of 100% because 4 × 25 = 100.
Also 25/100 is equivalent to 1/4; and 
relating to decimals. 0.25 is a quarter of 1 because 4 × 0.25 = 1.00
so 1/4 ≡ 0.25 because 1 ÷ 4 = 0.25
Common error  Likely calculation  Likely misconception  
c) 
425
525 
(5 × 5) + (20 × 20)
(5 × 5) + (20 × 25) 

b) c) 
56 175 
(8 × 5) + (8 × 2) (5 × 25) + (2 × 25) 
Place value errors in multiplication Uses (8 × 2) instead of (8 × 20) Uses (2 × 25) instead of (20 × 25) 
Strategies
1. Basic Facts about 25
Close to a quarter of students utilised basic facts about 25 on each part. However nearly a half of the students used some basic facts in at least one part of the question.
The success rate for students using basic facts knowledge was consistently the highest, remaining at about threequarters in questions c) and d) which were harder than questions a) and b).
2. Traditional computation.
About 2530% percent of students used traditional computational methods on each question.
The success rate for students using these strategies was about threequarters for part c), but dropped to just over 50% for question d).
3. Other numeracy strategies.
Between 3040% percent of students used numeracy strategies for questions a) – c), but only about 15% used them on question d).
The success rate for students using these strategies was relatively low especially for question c). Place value partitioning was quite unsuccessful for this, with just over 10% of students who used it obtaining a correct answer.
Some students incorporated other basic facts into these strategies.
Example: There is nine fifties in 450 (because 9 × 5 = 45)
Half the 50s makes 25
Count how many 25s there are: 18 classes
4. Other strategies or no strategy given.
The percentage of students giving other strategies (which largely showed little or no understanding) rose from about 6% for question a) through to about 15% for question d).
The success rate for these students using strategies was relatively low, dropping from about a half for question a) to about a third for question d).
For details of the percentages of responses and success rates, click on the link Appendix of student strategies for NM0163.
The student needs to be at the advanced additive or advanced multiplicative stages to understand correct multiplication strategies of this sort.
 For a correct way to decompose the multiplication, if the student was working "horizontally" click on the link to the following numeracy book”: Cross products (p.37), Book 6: Teaching multiplication and division (pdf), or go to Next steps on resource NM1246.
 If the student was using a vertical algorithm and made errors of this sort, click on the link: Paper power (p.34), Book 6: Teaching multiplication and division (pdf).
Place value errors
The student has not taken into account that the number is in the tens position rather than in the ones position. The two links above may help.
Another idea is to get the student doing multiplications such as 20 × 25 first and seeing that it is 20 lots of 25, not 2 lots of 25.
Basic facts of twenty five
If the student has not utilised any basic facts of the "twentyfive times table" then the following may help:
 Ask if they know the following: 2 × 25; 3 × 25; 4 × 25 (they probably will!)

If not, explore this with materials. Click on the link: Multiplying by 25 (p.14),
Book 8: Teaching number sense and algebraic thinking (pdf). 
If they do, you may inform them that they know their "25times table", for this is all that they need to know. The rest can be derived from this, (especially if they know their 4times table).
7 × 25 = (4 × 25) + (3 × 25)
Other resources
Connected Books
The article The finishing touch in Connected 2, 2007 explores multiplication by 25.
Figure it Out:
Number patterns (pp.89), Number Book Three, L34.
Appendix of student strategies for NM0163.
Strategy name 
a)
% use %correct 
b)
% use % correct 
c)
% use % correct 
d)
% use % correct 
Basic facts about 25  
Using basic fact knowledge of 25  7% 100%  17% 96%  18% 73%  20% 76% 
Using basic fact that 2 × 25 = 50  16% 93%  7% 92%  2% 50%  6% 70% 
Computational – Traditional  
Vertical algorithm (× or ÷)  23% 97%  23% 89%  30% 71%  24% 58% 
Addition using vertical algorithm  4% 100%  2% 100%  2% 25%  1% 100% 
Other numeracy strategies  
Place value partitioning  32% 91%  24% 79%  23% 11%  1% 0% 
Other partwhole strategies  3% 100%  6% 90%  2% 33%  1% 0% 
Successive addition or skip counting  7% 92%  6% 80%  3% 40%  8% 62% 
Doubling and halving  0%   2% 100%  1% 0%  2% 75% 
Deducing from previous parts of the question     1% 100%  2% 33%  4% 67% 
Other  
Other Strategy given  4% 57%  7% 18%  9% 0%  12% 16% 
No strategy given  2% 50%  5% 38%  7% 0%  21% 12% 
Results are based on a sample of 164 Year 8 students.
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