Picture this.... visualising maths problems
Post date: Mar 28, 2015 11:05:17 PM
Problems and problem-solving continues to be an area, with which many children can struggle. Even children who manage well with written calculations etc., can often be stumped when facing maths problems, particularly unfamiliar ones. As I've often heard teachers exclaim "they just don't see it!". And that is exactly the point, they can't visualise the problem, its components or the connections between those components and the unknown variable.
More mathematically able children can visualise all of this, which is most likely what makes them more able; their ability to visualise and manipulate elements of the problems and rearrange them, often mentally, in such a way as they can recognise what calculation and operation(s) is required in order to find that missing part or parts. However, visualising problems, is a skill that can be learnt. While more able children will have it naturally, this skill can be taught to your classes, so that more children will be able to "see" the problem and a way to solve it. In this post, I'm going to take you through three of my most used strategies for enabling children to visualise and ultimately solve problems.
Bar Model Drawing
This is simply a drawing that looks like bar, like that which would be more commonly seen in a bar graph, and is used to illustrate number relationships. Also known as strip diagram, tape diagram, fraction strip, or linear/length model, it is a core element of Singapore Maths, where, the process of using such drawings to represent and solve problems, is also referred to as model drawing.
Models tend to be of two main types, part-whole models or comparison models. Part-whole models can be used to represent a whole, that is subdivided into parts, for example in the image below, the number 5 (the whole) is broken into a part worth 2 and a part worth 3. Comparison models are used when comparing two or more quantities, eg the amount of money two different children have, the population of two different places etc. Below are some examples of the main types of bar models:
In the earlier classes, the bar models tend to be simpler, usually only involving two parts/quantities and only one step. As the problems develop in difficulty, the accompanying bar model drawings may involve more than one part/quantity and involve multiple steps to calculate the final answer.
In Singapore, bar models are introduced in Primary 1 where the focus is on using real objects initially to represent problems, then progressing to including cubes and/or counters as well as drawings of the real objects and concrete materials. In Irish schools, we could use this approach in the infants classes. The example opposite shows how the following problem from NZ Maths Problem Solving could be represented in this way:
Cars in Garages: I own 5 cars and a very large garage.If I can see 2 cars parked outside the garage, how many are inside?How many different ways can I park my cars inside and outside the garage?This site has a whole suite of very suitable problems in Level 1, that could be represented in such a way, and would suit junior and senior infants and even first class, once the numbers involved were adjusted accordingly to suit the different class levels, number limits and abilities.
In Primary 2 & 3 in Singapore, model drawing is extended to include larger numbers. Having bigger numbers also means that it is now inefficient to have exact drawings, and instead an entire bar represents a bigger number eg instead of having a bar subdivided into 7 individual sections to represent the number 7, we use just one bar for each number (see image opposite). More importantly, where possible, the bars should be in proportion ie if you have a bar representing 7 and a bar representing 5 then the 5-bar should be shorter than the 7-bar. This helps the children develop their number sense and their ability to estimate, all of which rely heavily on the child's ability to visualise numbers. The image above comes from the excellent Thinking Blocks website, and specifically their latest addition, Thinking Blocks Junior, which allows you to select a number limit of 10 or 20, making it ideal for 1st & 2nd classes and even an able senior infants class. The site also includes interactive examples of model drawings to demonstrate addition & subtraction (with number limits of 50 or 300), multiplication & division (would suit 3rd class up), fractions (would suit 5th class up) and ratios (6th class). To read more about how I have used this site in the past check out this post: Thinking Blocks
If you have never used bar models with a class before I would recommend the following:
Check out the Thinking Blocks website, try out some of the models* and decide what would be most suitable for your class situation.
Do a whole class demonstration on the IWB, ask children to come up and place the labels/numbers etc in the correct place. Or alternatively, let the children work through a given section, individually or in pairs, during computer/tablets time (Thinking Blocks are also available as free iPad apps).
Look up word problems in your own text books and workbooks. Could you draw bar models to show and represent the problem? With practice, I find that the children get quite good as recognising when bar models could be used and how to draw them.
*There is a "Change Model" in Addition and Subtraction on the site that I don't tend to use, as I find it inefficient as a way to model that particular type of problem. Instead an empty number line (below) is more suitable.
Singapore Math Bar Model Strategy
Step by Step Model Drawing: digital book by Char Forsten
Empty number line
Also known as a blank or open number line, this is a horizontal line, initially with no numbers or markings, that helps develop a child's number sense, their ability to visualise numbers and to compute mentally. Empty number lines can be used to show elapsed time, operations, skip counting, fractions, decimals, measures, money (making change) and much more. Here are some examples below, of how they can be used. The problems used here are taken from the most recent TIMSS study in 2011
A T-chart is a type of table, that helps children recognise patterns and connections within problems. The are so called after their T shape. Below are some examples of how they can be used. Again, the problems used here are taken from the most recent TIMSS study in 2011:
Of course, these are not the only ways that problems can be represented pictorially; however there are very few problems that children would encounter in primary maths that cannot be represented by one or more than one of these strategies.
Check out also my Pinterest Board on Bar Models