# Resources: Maths

# Relevant resources

Algebra | What's Possible? | |

Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make? This lesson idea is about exploring and noticing structure^{(ta)}.
The collection of NRICH activities are designed to develop students capacity to work as a mathematician. Exploring, questioning, working systematically, visualising, conjecturing, explaining, generalising, justifying, proving are all at the heart of mathematical thinking. This particular resource has been adapted from an original NRICH resource. NRICH promotes the learning of mathematics through problem solving. NRICH provides engaging problems, linked to the curriculum, with support for teachers in the classroom. Working on these problems will introduce students to key mathematical process skills. They offer students an opportunity to learn by exploring, noticing structure and discussing their insights, which in turn can lead to conjecturing, explaining, generalising, convincing and proof. The Teachers’ Notes provided focus on the pedagogical implications of teaching a curriculum that aims to provoke mathematical thinking. They assume that teachers will aim to do for students only what they cannot yet do for themselves. As a teacher, consider how this particular lesson idea can provoke mathematical thinking. How can you support students' exploration? How can you support conjecturing, explaining, generalising, convincing and proof?. | ||

Algebra | Seven Squares | |

Choose a few of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning? This lesson idea is about reasoning, justifying, convincing and proof^{(ta)}.
The collection of NRICH activities are designed to develop students capacity to work as a mathematician. Exploring, questioning, working systematically, visualising, conjecturing, explaining, generalising, justifying, proving are all at the heart of mathematical thinking. This particular resource has been adapted from an original NRICH resource. NRICH promotes the learning of mathematics through problem solving. NRICH provides engaging problems, linked to the curriculum, with support for teachers in the classroom. Working on these problems will introduce students to key mathematical process skills. They offer students an opportunity to learn by exploring, noticing structure and discussing their insights, which in turn can lead to conjecturing, explaining, generalising, convincing and proof. The Teachers’ Notes provided focus on the pedagogical implications of teaching a curriculum that aims to provoke mathematical thinking. They assume that teachers will aim to do for students only what they cannot yet do for themselves. As a teacher, consider how this particular lesson idea can provoke mathematical thinking. How can you support students' exploration? How can you support conjecturing, explaining, generalising, convincing and proof?. | ||

Algebra | Factorising with Multilink | |

Can you find out what is special about the dimensions of rectangles you can make with squares, sticks and units? This lesson idea is about visualising and explaining^{(ta)}.
The collection of NRICH activities are designed to develop students capacity to work as a mathematician. Exploring, questioning, working systematically, visualising, conjecturing, explaining, generalising, justifying, proving are all at the heart of mathematical thinking. This particular resource has been adapted from an original NRICH resource. NRICH promotes the learning of mathematics through problem solving. NRICH provides engaging problems, linked to the curriculum, with support for teachers in the classroom. Working on these problems will introduce students to key mathematical process skills. They offer students an opportunity to learn by exploring, noticing structure and discussing their insights, which in turn can lead to conjecturing, explaining, generalising, convincing and proof. The Teachers’ Notes provided focus on the pedagogical implications of teaching a curriculum that aims to provoke mathematical thinking. They assume that teachers will aim to do for students only what they cannot yet do for themselves. As a teacher, consider how this particular lesson idea can provoke mathematical thinking. How can you support students' exploration? How can you support conjecturing, explaining, generalising, convincing and proof?. | ||

Algebra | Temperature | |

Water freezes at 0°Celsius (32°Fahrenheit) and boils at 100°C (212°Fahrenheit). Is there a temperature at which Celsius and Fahrenheit readings are the same? This lesson idea is about applying and consolidating^{(ta)}. The collection of NRICH activities are designed to develop students capacity to work as a mathematician. Exploring, questioning, working systematically, visualising, conjecturing, explaining, generalising, justifying, proving are all at the heart of mathematical thinking.
The Teachers’ Notes provided focus on the pedagogical implications of teaching a curriculum that aims to provoke mathematical thinking. They assume that teachers will aim to do for students only what they cannot yet do for themselves. As a teacher, consider how this particular lesson idea can provoke mathematical thinking. How can you support students' exploration? How can you support conjecturing, explaining, generalising, convincing and proof? | ||

Algebra | Diamond Collector | |

Collect as many diamonds as you can by drawing three straight lines. This lesson idea is about thinking strategically^{(ta)}.
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Algebra | Pair Products | |

Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice? This lesson idea is about visualising and explaining^{(ta)}.
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Algebra | Charlie's Delightful Machine | |

Here is a machine with four coloured lights. Can you develop a strategy to work out the rules controlling each light? This lesson idea is about working systematically^{(ta)}.
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Algebra | Number Pyramids | |

Try entering different sets of numbers in the number pyramids. How does the total at the top change? This lesson idea is about posing questions and making conjectures^{(ta)}.
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Area | Circles, frustums and cylinders revision | |

Measure the volumes of objects This resource offer students the opportunity to engage in active learning^{(ta)} - measuring and calculating using large size cylinders and frustums. This lesson brings great opportunity for small group "dialogic teaching^{(ta)}". Open-ended and closed questioning^{(ta)} of students can be used to draw on their existing knowledge and extend their understanding. The teacher provides a practical commentary below.
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Assessment | Using Assessment to Raise Achievement in Maths | |

Learning goals; self & peer assessment; effecting questioning; marking and case studies This resource explores approaches to assessment^{(ta)} in maths, including the sharing of learning objectives^{(ta)}, group work^{(ta)}, whole class^{(ta)} assessment, questioning^{(ta)} and more. Four case studies serve as useful discussion prompts to share practice^{(ta)}. This .doc version of the QCA's 'Using assessment^{(ta)} to Raise Achievement in Maths' allows schools to select parts of the document that are most relevant to them.
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Assessment | Diagnostic Questions in Maths Teaching | |

Using questions to probe what pupils do, and do not, understand These questions provide a useful starting point from which to think about the use of diagnostic questions^{(ta)} for assessment^{(ta)} for learning and whole class^{(ta)} dialogic teaching^{(ta)}. They may be useful for teachers in their own right as sample questions, or to think about the best way to deliver feedback, use ICT tools effectively, and support learners through assessment. In this context the questions should be considered with a critical eye. Teachers might like to think about:
- The interface, and the way the questions are presented (could the questions be labelled better, appear better, what is the functionality like, etc.?)
- The style of questions asked
- The sort of feedback given, both on individual questions, and overall on the completion of the quizzes
Teachers might take this as an opportunity to engage in sharing practice | ||

Assessment | Changing KS3 Questions for Engaging Assessment | |

A large set of questions grouped by topic, paper, and national curriculum level Test questions are often seen as uninteresting and useful only to assess pupils summatively. This resource however allows questioning^{(ta)} to be used to support pupils’ revision, creativity and higher order^{(ta)} problem-solving in class. The tasks could be conducted via whole class^{(ta)} discussion^{(ta)} or assessment^{(ta)}, perhaps using mini-whiteboards^{(tool)}, or in small group work^{(ta)} situations.
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Capacity | Smoothie Capacity Challenge | |

Investigating capacity and getting wet wet wet... This is a practical session to be used with a whole class in small groups^{(ta)} of 2 or 3 –perhaps with the added supervision of a teaching assistant if the class is excitable or particularly young. There is scope within this activity for the following different methods of learning:
- Whole class
^{(ta)}dialogue - Discussion of each part of the activity - Open-ended questions
^{(ta)}– How can we find out if the containers hold the same amount? What can we tell just by looking at them? - Project work – as part of a wider maths topic on measuring volume and capacity.
- Enquiry-based learning
^{(ta)}– children are discovering the answer to questions that they are encouraged to pose themselves. - Arguing and reasoning
^{(ta)}– persuading each other about their ideas. - Exploring ideas – developing practical, physical understanding of key mathematical principles.
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Consecutive Sums | Using Prime and Square Numbers - How Old Am I? | |

Last year I was square, but this year I am in my prime. How old am I? This short activity offers opportunity for pupils to engage in mathematical thinking^{(ta)} and higher order^{(ta)} problem solving/reasoning^{(ta)}. They should be able to make links between different areas of mathematics and explore their ideas in whole class^{(ta)} discussion^{(ta)} and questioning^{(ta)}.
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Geometry | Marbles in a Box | |

In a three-dimensional version of noughts and crosses, how many winning lines can you make? This lesson idea is about visualising and explaining^{(ta)}.
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Geometry | Attractive Tablecloths | |

Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs? This lesson idea is about applying and consolidating^{(ta)}.
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Geometry | Painted Cube | |

Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces? This lesson idea is about exploring and noticing structure^{(ta)}.
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Geometry | Tilted Squares | |

It's easy to work out the areas of most squares that we meet, but what if they were tilted? This lesson idea is about posing questions and making conjectures^{(ta)}.
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Geometry | Can They Be Equal? | |

Can you find rectangles where the value of the area is the same as the value of the perimeter? This lesson idea is about working systematically^{(ta)}.
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Geometry | Kite in a Square | |

Can you make sense of the three methods to work out the area of the kite in the square? This lesson idea is about reasoning, justifying, convincing and proof^{(ta)}.
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Geometry | Warmsnug Double Glazing | |

How have "Warmsnug" arrived at the prices shown on their windows? Which window has been given an incorrect price? This lesson idea is about applying and consolidating^{(ta)}.
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Handling Data | Olympic Measures | |

These Olympic quantities have been jumbled up! Can you put them back together again? This lesson idea is about applying and consolidating^{(ta)}.
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Handling Data | Olympic Records | |

Can you deduce which Olympic athletics events are represented by the graphs? This lesson idea is about reasoning, justifying, convincing and proof^{(ta)}.
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Handling Data | Odds and Evens | |

Is this a fair game? How many ways are there of creating a fair game by adding odd and even numbers? This lesson idea is about posing questions and making conjectures^{(ta)}.
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Handling Data | Which List is Which? | |

Six samples were taken from two distributions but they got muddled up. Can you work out which list is which? This lesson idea is about exploring and noticing structure^{(ta)}.
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Handling Data | Which Spinners? | |

Can you work out which spinners were used to generate the frequency charts? This lesson idea is about exploring and noticing structure^{(ta)}.
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Handling Data | Non-transitive Dice | |

Alison and Charlie are playing a game. Charlie wants to go first so Alison lets him. Was that such a good idea? This lesson idea is about thinking strategically^{(ta)}.
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Handling Data | Place Your Orders | |

Can you rank these sets of quantities in order, from smallest to largest? Can you provide convincing evidence for your rankings? This lesson idea is about visualising and explaining^{(ta)}.
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Handling Data | M, M and M | |

If you are given the mean, median and mode of five positive whole numbers, can you find the numbers? This lesson idea is about working systematically^{(ta)}.
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ICT | T-MEDIA video case studies of ICT use | |

Multimedia tools for professional development The materials aim to stimulate debate and reflection rather than present models of ‘best practice’. They include hyperlinks to video clips from the emerging pedagogical themes, built-in prompts for thinking and discussion, suggested alternative approaches, and analytic commentary from teachers and researchers involved. They illustrate and question how the technologies can be exploited to enhance learning in real classrooms. The resources can be used by individuals or groups of colleagues / student teachers.
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Investigation | Consecutive Sums | |

Can all numbers be made in this way? For example 9=2+3+4, 11=5+6, 12=3+4+5, 20=2+3+4+5+6 By definition, a problem is something that you do not immediately know how to solve, so learning how to solve something unfamiliar is not straightforward. Tackling an extended problem is difficult.
This lesson gives pupils an opportunity to engage in mathematical thinking The plan suggests several visualisation | ||

Language | Exploring shape and its mathematical language through sorting activities | |

Using mathematical language to discuss shapes of objects either printed or hidden in 'feely bags'. Can you feel the forks? The Investigation^{(ta)} of shapes and geometry can be very rewarding. A practical approach using objects from the pupils’ environment can increase their motivation and interest. In this unit, you will be using everyday objects to help pupils develop geometrical skills, such as recognising, visualisation^{(ta)}, describing, sorting, naming, classifying and comparing.
Through games | ||

Measuring | Thinking about measurement | |

How big is it? Thinking about measurement This module introduces the uses and limitations of measuring size, as well as giving a taste of using distance learning^{(i)} materials for higher education. This resource can serve as a useful revision of various measures, while providing helpful insight into active learning techniques, use of real life examples, the importance of technical language, and the importance of study skills^{(topic)} (such as use of dictionaries, indexes, representation of information, etc.).
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Number | Factors and Multiples Game | |

A game in which players take it in turns to choose a number. Can you block your opponent? This lesson idea is about thinking strategically^{(ta)}.
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Number | What's it Worth? | |

There are lots of different methods to find out what the shapes are worth - how many can you find? This lesson idea is about reasoning, justifying, convincing and proof^{(ta)}.
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Number | Mixing Lemonade | |

Can you work out which drink has the stronger flavour? This lesson idea is about applying and consolidating^{(ta)}.
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Number | What Numbers Can We Make? | |

Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make? This lesson idea is about visualising and explaining^{(ta)}.
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Number | Summing Consecutive Numbers | |

Many numbers can be expressed as the sum of two or more consecutive integers. Can you say which numbers can be expressed in this way? This lesson idea is about exploring and noticing structure^{(ta)}.
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Number | GOT IT | |

Can you develop a strategy for winning this game with any target? This lesson idea is about working systematically^{(ta)}.
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Patterns | Exploring Pattern | |

Exploring patterns in mathematics Each chapter of this tutorial highlights the study skills^{(topic)} required to work through the real world examples and activities given. There are problems to be solved, some of which involve higher order^{(ta)} thinking skills (for example, being asked to correct a set of instructions), and all of which encourage the use of mathematical language^{(ta)} and mathematical thinking^{(ta)}. The resource could also be used in class, or as a useful homework^{(ta)} pack.
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Polygons | Exploring properties of rectangles: Perimeter and area. | |

Do two rectangles that have the same area also have the same perimeter? A problem to inspire higher order^{(ta)} questioning^{(ta)} especially in whole class^{(ta)} dialogic teaching^{(ta)} encouraging pupils to engage in mathematical thinking^{(ta)} and language^{(ta)}. You could use Geogebra^{(tool)} in this investigation, as an example of same-task group work^{(ta)}.
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Probability | Playing with Probability - Efron's Dice | |

I have some dice that are coloured green, yellow, red and purple... Efron's dice provide a discussion^{(ta)} topic for joint reasoning^{(ta)} - whole class^{(ta)} or in group work^{(ta)}. Pupils can explore aspects of mathematical thinking^{(ta)} particularly with relation to probability.
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Shape | Getting Your Formulae in Shape | |

Solving a card sort for perimeter, volume and area formulae This resource provides an opportunity for some revision of shape formulae - perimeter, area, and volume. It encourages pupils to engage in effectivereasoning^{(ta)}, and group talk^{(ta)}, and could be used as an effective assessment^{(ta)} tool. The task could be differentiated^{(ta)}, or extended for a whole class by cutting the 'formulae' lines off the bottom of each hexagon, and asking students to match these to the shapes, prior to matching the shapes to the formulae type.
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Simultaneous Equations | Love Food, Hate Waste - Simultaneous Equations | |

Using real world data to explore simultaneous equations Using a source that was not intended by its creators as a mathematical resource, pupils are introduced to informal ways of solving simultaneous equations.
The lesson starts with an intriguing ‘hook’, pupils are able to use reasoning | ||

Standard Index Form | An Introduction to the Standard Index Form | |

Working out the rules according to which a calculator displays large numbers The Standard Index Form is a key idea for mathematicians and scientists. The notion that we choose to write numbers in this way requires some explanation. So in this activity, pupils take part in an investigation^{(ta)} on how standard index form works. This is a higher order^{(ta)} problem solving context where students are encouraged to engage in mathematical thinking^{(ta)}. They may be involved in whole class^{(ta)} or small group work^{(ta)} discussion^{(ta)}, so they have a good opportunity to practice using mathematical language^{(ta)} and questioning^{(ta)}.
This means that students do not need to be able to explain their ideas in full: they can use the calculator's feedback to discover whether their ideas are correct or not. This is also an exciting way for pupils to realise an initial idea that fits the data may need to be extended when new data arises. This resource therefore aims to develop investigative skills, as well as introduce pupils to standard index form in a memorable way. The pupils can later use their knowledge of indices in discussion | ||

Statistics | Cubic Equations and Their Roots | |

To interactiviley explore and understand complex mathematics with GeoGebra This lesson features a ‘real life’ example for students to explore using visualisation^{(ta)} via GeoGebra. The focus on ‘real life’ increases student motivation.
The activity engages pupils in group talk | ||

Statistics | Analysing the performance of Olympic runner, Usain Bolt | |

Exploring real world statistics using the GeoGebra program Using real world data to engage pupils in mathematical thinking^{(ta)}, language^{(ta)} and reasoning^{(ta)}. This resource provides opportunities for group work or whole class exercises, with engaging questioning^{(ta)}.
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Using images | Organising images for a narrative | |

Write an essay without words The lesson encourages students to think about how to portray their knowledge through narrative^{(ta)} - which may engage some students who would usually be less interested. The lesson encourages students to think about how to capture valuable information and ensure that key elements are highlighted while not 'overloading' the viewer with data. The lesson can be tailored to any age group - for younger pupils the task could be to take before and after photos and label them. More advanced pupils might explore time-lapse photography. Pupils should be encouraged to think about how this relates to the scientific method^{(ta)} The task is interactive and could be conducted as a group work^{(ta)} activity or as an element of an inquiry-based learning project. It could also lend itself to whole class^{(ta)} dialogue^{(ta)} and the use of ICT^{(i)} including 'clicker' response systems for assessment^{(ta)} and questioning^{(ta)}.
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Visualisation | Perimeter of a rectangle. | |

Interactive GeoGebra investigation that allows children (age 6-10) to explore an element of mathematics for themselves.
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Visualisation | Kepler's Third Law | |

Using 'real life' data This lesson features a ‘real life’ example for students to explore GeoGebra. The focus on ‘real life’ increases student motivation.
The activity engages pupils in group talk(i), mathematical thinking(i) and vocabulary(i). This open ended(i) task encourages higher order(i) thinking, and encourages whole class(i) discussion(i)/questioning(i) and inquiry(i) projects. |