Powerful twenty five

Powerful twenty five

Pencil and paper
Overview
Using this Resource
Connecting to the Curriculum
Marking Student Responses
Working with Students
Further Resources
This task is about how you multiply or divide numbers by twenty five.
Answer each problem below and describe how you actually did it.  If you did it in your head, try and show your thinking. State any number facts you used.


a) Show how to solve 25 × 3.

Show your working.
 
 
 
 
 

Answer: __________ 

b) Show how to solve 25 × 8.

Show your working.
 
 
 
 
 

Answer: __________

c) Show how to solve 25 × 25.

Show your working.
 
 
 
 
 

Answer: __________
 
 
 
 
 

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

Show your working.
 
 
 
 
 

Answer: __________
Task administration: 
This task is completed with pencil and paper only.
Levels:
3, 4
Curriculum info: 
Maths, Number and Algebra, Number Strategies
Key Competencies: 
Using language, symbols, and texts
Keywords: 
multiplication, division, basic facts
Description of task: 
Students answer questions based on multiplication or division by 25, and explain how they arrived at their answer.
Curriculum Links: 
This resource can help to identify students' ability to apply additive or multiplicative strategies and basic facts to whole numbers to solve multiplication and division problems.
Key competencies:
This resource involves recording the strategies students use to solve multiplication and division problems. This relates to the Key Competency: Using language, symbols and text.
Learning Progression Frameworks
This resource can provide evidence of learning associated with
Multiplicative thinking, sets 4-5
Multiplicative thinking, sets 6-7
within the Mathematics Learning Progressions Frameworks.
Read more about the Learning Progressions Frameworks.
Answers/responses: 
  Y8 (11/2007)
a)

i)
ii)

75
Evidence of "knowing" that 3 × 25 = 75, or 2 × 25 = 50 or 25 + 25 = 50
or
Use of other acceptable computational strategies.

very easy
very easy
b) i)
ii)

200
Evidence of "knowing" that 4 × 25 = 100 or other basic facts about 25
or
Use of other acceptable computational strategies.

easy
very easy
c) i)
ii)

625
Evidence of "knowing" and using some basic facts about 25
or
Use of other acceptable computational strategies.

moderate
very easy
d) i)
ii)

18
Evidence of "knowing" and using some basic facts about 25
or
Use of other acceptable computational strategies.

 moderate
easy
Based on a representative sample of 164 students. 
Teaching and learning: 
  1. This task aims to see if the students can utilise basic facts knowledge of the 25-times table.  It can also give information about students’ multiplication and division strategies, even if they have no knowledge of basic facts about 25.
  2. Only four basic facts are needed for the 25-times 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.)
  3. The 25-times 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
Diagnostic and formative information: 
  Common error Likely calculation Likely misconception
c) 425

 

525

 (5 × 5) + (20 × 20)

 

(5 × 5) + (20 × 25)
  • Cross product errors
    Ignores the diagonal cross products [i.e. (5 × 20) and (20 × 5)] in the multiplication either in the vertical algorithm form or in the place value decomposition form: (5 × 5) + (5 × 20) + (20 × 5) + (20 × 20)
  • Ignores the diagonal cross product (5 × 20)
  • Incorrect multiplication.
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 three-quarters in questions c) and d) which were harder than questions a) and b).

2.  Traditional computation.
About 25-30% percent of students used traditional computational methods on each question.
The success rate for students using these strategies was about three-quarters for part c), but dropped to just over 50% for question d).
 
3.  Other numeracy strategies. 
Between 30-40% 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.

Next steps: 
Cross product errors
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 "twenty-five times table" then the following may help:

  1. Ask if they know the following: 2 × 25; 3 × 25; 4 × 25 (they probably will!)
  2. If not, explore this with materials. Click on the link: Multiplying by 25 (p.14),
    Book 8: Teaching number sense and algebraic thinking (pdf).
  3. If they do, you may inform them that they know their "25-times table", for this is all that they need to know. The rest can be derived from this, (especially if they know their 4-times 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.8-9), Number Book Three, L3-4.

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 part-whole 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.