The Problem with Numerical Reasoning - Worded Problems

November 14, 2019

 

I often think half the problem with worded problems, is the name.  Being called 'problems' is so negative and leads children to believe the questions will be hard before they've even started!  At least the term Numerical Reasoning is more positive.  The second half of the 'problem' is that they are genuinely hard for a lot of children at this age, when they are doing sums and enjoy the feeling of a straight forward answer. It  is also very different from being in a classroom where one topic, say fractions or shapes, is being taught largely in isolation from other mathematical topics.  For these questions it is a matter of deciphering exactly what is being asked and then combining concepts that they have been learnt to find a single answer. Even children who usually love maths can come 'unstuck' when faced with worded questions expressed in new and unusual ways; often requiring a number of steps before completion.


Before beginning these questions, the child needs to be comfortable with all the component parts.  The list below shows key topics that need to be individually understood before embarking upon multi-part questions in which the knowledge is likely to be needed.  Alongside this your child should be aware of quick ways to add, subtract, multiply and divide and how to do these operations with decimals.

 

Essential Mathematical Knowledge


Be sure your child is confident with these themes. Some of them may be new.

 

Angles
The size of a right angle?
4 types of angles?
Angles along a straight line?
Alternate (Z) angles?
Vertically opposite (X) angles?
Angles of all triangles?
Angles of isosceles triangles?
Angles of a four straight sided shape?
Angles of all straight sided shapes?
What does rotated about the origin mean?
How is a bearing measured?

 

All Shapes
What are vertices?
What is reflectional symmetry?
What is rotational symmetry?
Perimeter?  
Area of a rectangle?
Area of a triangle?
Area of a parallelogram?
How to find areas of unusual straight sided shapes?  
Does a parallelogram have reflectional symmetry?  Rotational symmetry?
Volume of  regular solid shapes such as cubes, cuboids. triangular prisms?
What is the radius of a circle?  The diameter?
Perimeter of a circle - What is it called?

 

Fractions/Decimals/Percentages
Adding and subtracting different fractions i.e.  3/4 + 2/3
Multiplying fractions?  i.e. 1/3 x 6
Where do you place the decimal place when multiplying with decimals?
How do you divide by 10, 100, 100?
What should you always do when you have finished a multiplication with decimals?
To make a fraction into a percentage multiply by 100 and do the division.
i.e.  3/7ths = 3/7 x100 = 300/7 = 42.9 (to 1 dec.place)
How do you approach a percentage question?  i.e.  35% of 230 apples?
When you have completed working out the percentage part, what must you remember to do on some questions?

 

Number Terms and Algebra
Factor?
Lowest common factor?
Highest common multiple?
BIDMAS/BODMAS - the order of operations
Input /Output functions
3b?  What does it mean?
2a + 5?  What does this mean?
2(e + 6) = 16?  What does this mean?

 

Data Presentation

Carroll diagram, Venn diagram, Pictogram, Block graph, Bar chart, Pie chart, Bar line graph, Frequency diagram


Averages

Mean, Mode, Median, Range


Ratio and Proportion


Time

including days in the months of the year, analogue, digital and twenty-four-hour time, find differences between various time zones.

 

Approaching Multi-Part Worded Questions
It is vital to pick out only the relevant information which is quite often surrounded by a lot of superfluous 'fluffy' words. Then, before beginning, the reader should check thoroughly their understanding of exactly what to do for all the parts. Perhaps jotting down notes above the different parts such as +. -, %, and underline key terms may help.  There will be no point doing parts of the questions if they don't know how to approach the whole.  As a multiple-choice test, there will be no consideration given to workings out, although working out will be necessary. 


Fortunately, there is one important tip that can lead to big improvements in scores – if used consistently and effectively – the power of estimation.  From a young age your child will have been taught that estimation is important in mathematics; in a multiple-choice test, which is most likely for part of Numerical Reasoning in the Eleven Plus, the use of estimation can be even more valuable.


Example:
Carl bought a poster for £24.56 and a lamp for £55.44.  He paid for them with 5 twenty-pound notes.  When he received his change, he put coins into a charity jar for the Royal Life Boat Association and took the notes home.  How much did he take home?
Irrelevant information: what is bought, and the name of the charity
Vital Information:  the prices, operations to be used, the vocabulary - denomination
Now let’s estimate:  Costs are approximately £80, cash available £100  Therfore the answer will be approximately £20
Answer choices:
a.  £2.04     b.  £5.90    c.  £20.04     d.  £200.40    e.  £59.00
           
If your child is competent in their basic mathematical knowledge, it can be helpful to simply go through a variety of worded questions without always doing the arithmetic in order to focus on the reasoning involved.
Here are some questions you might ask:
1.  What information is irrelevant?
2.  What might you underline - important information?
3.  What are the key operations you would use and in what and in what order?
5.  Would you work these parts out mentally, with jottings, or by writing complete calculations?
6.  Would you feel confident about completing this whole question?

 

A great free resource for these types of questions are from papers available at http://www.primarymathschallenge.org.uk/

 

Finally, many children find these questions very time consuming, which is difficult in a fast-paced CEM style test.  If your child finds them particularly challenging, they might want to pick out ones they find easier first, and then approach those they find trickier.
 

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