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In this feature we return to the popular Low Threshold High Ceiling (LTHC) tasks, which have become so well associated with NRICH (although we are not sure if we invented the phrase*). At NRICH our aim is to offer rich tasks which develop deep understanding of mathematical concepts. There are many NRICH tasks which can be used in this way, but the ones in this article are particularly strong
in offering different levels of thinking, choice and challenge. It is the range of possibilities that allow children to think, reason and make decisions, in particular, that makes a task stand out as a LTHC task.

In addition to the activities listed below have a look at the Primary Curriculum Mapping documents to see how NRICH tasks link to the aims and curriculum content of the Primary Mathematics National Curriculum.

The last time we looked exclusively at LTHC tasks was back in August 2012 and so we thought it would be a good idea to return to this area and explore it further. Some of the tasks are the same but we have also added in some different ones too.

**What does LTHC mean?**

Many of you will be familiar with the concept and purpose of LTHC tasks. However, here is a reminder.

Imagine a room. You can get into the room easily, without any problem. Once inside the room there are lots of possibilities to do some activities. Many of these activities are unproblematic, while others are more challenging. The only limit placed on your choice of activitiy is the space of the room and the height to which you can rise.

So, a LTHC task is a mathematical activity where everyone in the group can begin and then work at their own level, yet the task also offers lots of possibilities for learners to do much more challenging mathematics too.

**Why do we like LTHC tasks?**

One of the main reasons we like LTHC tasks is because they provide opportunities for all children to work like mathematicians. While the content within these tasks can be quite simple, the level of thinking required is sophisticated. In this way LTHC tasks fit very well with the aims of the primary mathematics National Curriculum (DfE, 2013) of fluency, reasoning and problem solving. Developing
problem solving skills alongside the conceptual understanding that LTHC tasks offer provides valuable opportunities for children to think mathematically.

LTHC tasks also promote a positive classroom culture because the whole class are working on the same activity. We know that for many learners maths can be a lonely subject that they often believe they can't do.

Sometimes classrooms are organised into two types of organisation. The first is where the class is split into groups (often by attainment), each of which is engaged in a different activity. Sometimes all the groups are working on different parts of the same mathematical topic, but not always. It can be difficult to feel part of a classroom community when you have no idea what the others are
doing. The second type of organisation might be where everyone does the same, often closed, acitivity. This time everyone knows what they are supposed to be doing, but those learners at either end of the attainment scale either finish first and get bored, or get left behind and feel excluded.

LTHC tasks offer activities which are suitable for pretty much everyone, allowing the less confident learners to stay close to the original task and consolidate their understanding while offering the more confident a chance to explore and challenge themselves. Everyone has a sense of what is going on and plenaries are much more meaningful as learners hear about how their peers have worked on the
activity.

So, another reason we like using LTHC tasks is that they offer an opportunity to those learners who may not otherwise choose to explore the mathematics in different ways. They allow learners to see what they can focus on and not what they can't. As teachers it can be easy to predict how well our learners will cope with a particular piece of mathematics and sometimes that can be a self-fulfilling
prophecy. Yet, when the ceiling is raised it can be surprising what heights learners can achieve.

A third reason we like LTHC tasks is that they offer many possibilities for learners to focus on sophisticated process skills rather than just more content knowledge. It's often mistakenly thought that the only way to challenge learners is to offer them content at a higher level. However, learners need to be challenged by the choices they can make within mathematical problems. So, often the
content in LTHC tasks remains quite simple but the level of challenge can become quite sophisticated.

At NRICH we always write activities with an eye on the high ceiling, but the tasks below are ones that fit particularly well with LTHC.

**Magic Vs** is one of our favourite tasks. The threshold is counting to 15 as children try to make both sides of the V add up to the same number. Others will start to recognise patterns and begin to make generalisations about them. More
confident children will make up their own questions such as What if ...? and test their conjectures.

**Sort the Street** is a lovely task where children are asked to sort the houses in the street in different ways. It is particularly suited to children working in pairs or small groups as children can reason and talk about why they have (or have not) placed houses into different groupings.

The **Poly Plug Rectangles** task provides opportunities for children to make hypotheses, explain their reasoning and test out their ideas, all through the medium of rectangles.

Another favourite is a task called **Noah**. Children try to decide how many animals entered the ark (based on 12 legs, or more if you change the number!). One of the strengths of this activity is the range of different answers available. The task encourages children to reason about their thinking as well as employing a systematic approach to record their
answers.

**Number Differences** is a more challenging version of **Ring a Ring of Numbers** or **More Numbers in the Ring**. Children need to think where to
position the numbers 1 to 9 to meet certain criteria. Lots of explanation is encouraged in this task as well as supporting children towards making general statements about odd and even numbers.

In our experience children love the **Worms** task, probably because of its title to begin with but then because of the way the activity unfolds. This task relies on children finding good ways of recording their results and for those more confident children, asking questions such as 'I wonder what would happen if ...?'.

** In the context of learning technologies, the phrase LTHC or 'low threshold high ceiling' (also sometimes referred to as 'low threshold no ceiling) originates in Seymour Papert's description of the central design principle of the Logo programming language (Papert, 1980). Another example of a widely used LTHC platform is Net Logo, a modelling environment. In the context of programming
languages 'low threshold' means that new users, including those who never programmed before, should find it easy to get started, whereas 'no ceiling' (or high ceiling) means the language shouldn't be limiting for advanced users.*