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Outdoor Maths Activities KS1 -Maths Outdoor Learning

Outdoor Maths Activities KS1  -Maths Outdoor Learning

The following is a list of some of my favourite outdoor maths learning activities for KS1 (Key stage 1 – ages approximately 5-7).  Maths in outdoor and outside learning is a fun way for children to use maths in real, hands-on situations. It also promotes making connections between different areas of maths learning.

Younger children, in particular, need the experience of manipulating real-life materials and exploring mathematical thinking.   These experiences help them to develop an understanding of the value of numbers, and later, what happens during arithmetic operations. Children need to have a wide variety of opportunities to practice counting and problem solving using tangible objects.  It allows them to develop a deep understanding of numbers, number facts and the changes that take place during various operations (Anghileri, 2006).  It does take children longer to learn through exploration and hands-on methods rather than learning by rote. However, they will gain a deeper understanding, including how and why they work, rather than just the process (Carruthers & Worthington, 2004).  In the long-term, this will help them to build confidence in maths, as well as allow them to apply their knowledge in problem-solving.

Outdoor Maths Activities KS1

I have grouped theses outdoor maths activities based on different areas of learning for KS1.  They are primarily for children ages 5-7, but they can be adapted for younger and older children. You can also see my post on Outdoor Maths for KS2 or Outdoor Maths for EYFS for more ideas.

*Please note that this post on Outdoor Maths Activities for KS1- Outdoor Learning contains affiliate links to help with the running cost of this website. Thank you for your support so that we can keep writing!

Number & Place Value (including counting)

• Counting objects from nature   –  This might include doing nature hunts for a certain number of objects and even  counting objects with number frames or numicon for support. Collecting objects provides many opportunities for counting and learning maths in the outdoors.

• Number rocks  or number logs – Children can practice ordering numbers, and then may go on to practice ordering/counting by 2s, 5s (e.g. skip counting).

• Number games – Children may play number games with rocks – ex.  Swapping numbers  or  missing number games .
• Nature number line – Hang rope between trees (or along the fence if concerned about children running into the rope) and provide pegs. Children can hang up and order numbers to make a number line. Children might collect things like leaves to pin to correspond with the value. Which number comes first? Which number is bigger (has a larger value)? How do you know?

• Place Value – Place value frame (e.g. tens and ones) with sticks or rocks- Children can practice representing tens and ones using, for example, 1 stick for each 1 and a bundle of 10 sticks for each ten (or children may swap a large rock to replace a bundle of 10 sticks).  Ex. 35 can be shown by 3 bundles of sticks and 5 sticks or 3 large rocks and 5 sticks (or even pebbles). Which is greater? Which is less? Can you prove it?

• Number bonds – Practice number bonds to tens with sticks or rocks.  Children can find all the ways to add two numbers together to make 10 (and even all numbers 1-10).  For even deeper learning children can explore combining 3 or more values to add up to 10 (or numbers to 10) (see below in arithmetic). Is there a way to check you have found all the number bonds? Can you record them? Show me…
• Greater or less than – Children can practice representing greater than or less than with sticks.  They can see this visually by fitting in rocks (see example below) to see which is bigger or smaller < >. Which is greater? Which is less? How can you prove it?

• 100 square – Make a massive 100 square outdoors on the pavement with chalk.  Children can fill in the 100 square using number rocks or number log slices, or even writing numbers onto the square with chalk. Which number comes first? Do you notice anything about the hundred square? What happens when you count up /down 10?
• Counting picture – Children can work collaboratively to create a picture using 10/20/30 objects they find in nature. Children have to work together to find the objects and make sure they have the exact number of objects.
• Number hunt  – children can search for numerals, written out words for numbers, and/or dice or other value representations of numbers hidden outside.  They can then match different representations of the same number together (and even order them). How many different ways can you make 5? Children may use things such as a tens frame or numicon to help them represent the numbers.

• Number writing – Children can practice writing numbers with chalk or tracing over chalk numbers by painting with water.
• Parachute or Circle games with numbers – Children can be given a number and then children swap (or run in / out of the parachute or swap places in the circle depending on if the statement is true or falls). For example, the teacher might say odd numbers, even numbers, numbers less than 5, numbers greater than 5, numbers for counting by 2 or by 5, etc.  
• Minibeast Counting – Go on a minibeast hunt and have children count and keep track of what they find with tally marks or tally chart. You may want to discuss why tally marks work well for this (rather than writing down numbers). You could come together as a large group at the end and create a pictograph using their findings.
• Skip counting – Children can use number rocks to practice skip counting (e.g. practice counting in 2’s, 5’s and 10’s).  Children may want to pair the numbers with the corresponding numicon . What do you notice about the numbers when counting in 2’s? What about in 5’s or 10’s?
• Counting in groups – They can use number rocks and natural objects to count out objects in 2’s (or 5’s or whatever they are counting by) and then match with the numeral for each group (e.g. first group of 2 seashells with a number 2, second group of 2 seashells with a number 4, third group of 2 seashells with a number 6, etc.).

• Using leaves for multiplication – Children can practice repeating addition as a way to help them understand multiplication. They can count the blades on the leaves to help them do this. For example, maple and horse chestnut leaves have 5 blades each so children can use them to count in 5’s. Buttercup and clover leaves have 3 blades so children can use them to count in 3’s. What do you notice about counting in 2’s, 3’s, 5’s etc?

• 100 square problem solving – Children can make a large number line or 100 square using rock numbers or chalk (described above in number and place value section). Children can use this to help them solve addition and subtraction problems. As they count up or back along the number line (to add or subtract) they might even step along it (if it is big enough).   *As children become confident with adding, they can practise counting on and even counting up in 10’s when adding and subtracting double-digit numbers. What do you notice when you count up or down in 10?
• Nim – Nim is a mathematical strategy game where two players take turns removing objects from a pile. Each player must take at least one object per turn. The goal is to either get or avoid taking the last object from the pile. Children can play nim with a pile of sticks or rocks.
• Number bonds – Children can practice making all the number bonds for numbers 1-10 using sticks or rocks (see above in number and place value). Is there a way to keep track an record your number bonds?

• Counting sets – Children can throw a set number of rocks towards a hula hoop laying on the ground /circle drawn on the ground (this is a way to create number bonds). Then they can count how many rocks land inside and outside of the hoop (as well as count the total). It will help them see that no matter how many different ways they land (e.g. number bonds), the total will stay constant. Similarly, children can explore playing around with a set number of rocks on a number frame to see that it is the same value even when it looks different. Is it still 7? How can you check?

• Skittles & bowling – Children can play games such as skittles , bowling and other target games . They can identify numbers on the games, count the number of skittles or targets that are hit, add up points, take away how many objects have been knocked over, and see how many are left. There are lots of opportunities for counting, adding and subtracting with these types of garden games.

• Garden counting – As children pick tomatoes or other fruits or vegetables from the garden, get them to count how many they picked from each plant. Then how many have they picked altogether?
• Sharing garden crops – As children pick flowers, strawberries, or other things from the garden help them practice using different types of division (e.g. sharing & grouping). For example – [Sharing] if you’ve picked 12 tomatoes how many will we each get (e.g. you & me)? If another child comes along – now how can we share them between us?  [Grouping] If we are selling baskets with 6 tomatoes in each basket, how many baskets can we make to sell? How many apples do you need to make a pie?… How many pies can you make with the number of apples you have picked? You might also get into remainders if there are some leftover.

• Drawing fractions – Children can draw a large square on the pavement (or in the sand) and then find different ways to shade in ½. This is a great way for them to see that ½ can look different, but it always must add up to the same amount. As children advance, they can see how many different ways they can make ¼ or 1/8. How do you know its ½ or ¼? Is there a way you can prove it?
• Fractions with sticks – Using sticks can be a great way to introduce children to see fractions visually. If you cut sticks so there is one that is whole, 2 that are ½ , and 4 that are ¼ they can see visually how fractions are divided up. It also makes it easy to see how 2(½) = 1 and ½ = 2(¼).

Measurement

• Ordering objects by length  – children usually find it easy to compare two objects but may need more practice when comparing 3 or more objects. I have a post –  ordering sticks by length , which reviews common misconceptions, ways to help children to learn this and questions to ask. Which is longer? Which is the longest? How can you prove it?

• Measuring with non-standard units – Children can practice measuring objects with non-standard units (e.g. how many stones long is the stick). This is the next step after comparing lengths, but before measuring with standard units such as cm or inches. How many pinecones long is your toy bus? Which is shorter? How do you know?

• Measuring height – Children can measure their height in rocks, pinecones or sticks (e.g. non-standard units) by laying down on the ground.  Children can then count to see how many pinecones, sticks, or rocks tall they are. Who is taller? How do you know? Is there another way to show this?
• Measuring natural objects – Children can measure natural objects, such as plants, with a ruler.  They can also go on a ‘meter hunt’ or ‘foot hunt’ to see if they can find things in nature that are a foot or meter.
• Meter or foot with natural objects – Children can try making a meter or foot using sticks, rocks pinecones or other natural objects. How many sticks/rocks/pinecones did it take to make a meter?
• They may measure the circumference of a tree.
• Plant measuring – They may also practice measuring the height of plants (e.g. non-standard to start – e.g. 5 sticks high, then with a ruler for standard units).
• Measuring growing – Children can use measuring to help them plant seeds or seedlings.  They may use a stick that is 12 inches to help them measure the distance between plants with nonstandard units.  Children may then move on to using a ruler to help them measure the recommended distance between seeds or seedlings. 
• Chalk clocks – Children may make clocks with sticks and chalk or with rocks, numbers and chalk to practice showing time.

• Counting 1 minute – Children can practice counting how many times you can jump, skip, or hop, or how far you can walk, etc. in one minute. How many did you do? Did you do fewer or more than last time? If you did more/less does that mean you are getting faster or slower? Can you find a way to keep track of how many hops, skips you do in a minute? What else do you think you can do in a minute?
• Timing – They may also time themselves to see how long it takes to run from one point to another, to bicycle 1 mile, to hop 20 times, etc.  How can you tell if you are getting faster or slower?
• Potions – Children can make up or follow potion recipes. They can follow instructions to measure (with standard or non-standard units) and combine ‘ingredients.’ Children might also compare relative measurements such as full, half-full, empty, etc. You can challenge children – which container will hold the most potion? How do you know? Can you figure out how to order the containers by which will hold the least to which will hold the most?

• Weighing – Children can use balance scales to compare the weights of different objects. How many horse chestnuts are equal to the weight of your rock? Can you prove which object is the heaviest?
• Measuring garden crops – There are lots of opportunities for measuring when picking fruit and vegetables from your garden. What is the volume of the containers you filled with raspberries? How much do the apples or squash weigh? Which is the longest courgette? Can you order them by length? Can you measure them with your ruler? If you sell some of your crops or are, instead, getting your vegetables at a “pick your own farm”, there are opportunities to discuss money as well. If we have 2 pounds of tomatoes, how much will that cost (ex. at £0.50 per £)?  
• Snail Races – see how far a snail can go in one minute. Children can help you think of ways to best measure the snail’s movement. This is also a way for children to help count 1 minute and get an idea of how lone one minute feels like.
• Sorting and ordering – Children can sort / order (gradient) natural objects (e.g. leaves, rocks) by a specific feature (ex. Shape, size, colour, or other features).
• Dam building / Den building  – Children use materials such as sticks and rocks to build a fort/den or to block off or dam a stream. They could also build obstacle courses and use directional language to help each other get through it. This is an excellent opportunity for children to practice estimating length and using spatial rotation to help them construct.

• Nature symmetry – Children may explore symmetry in nature. They may use a mirror to help and even make their own creations (see mandalas below).
• Symmetry transient art – They can make symmetrical pictures or  mandalas with natural objects .

• Making Patterns – Children can make patterns with natural objects. This might mean repeating patterns, or it might mean making more complicated patterns such as (x+1) or 2x or x2 etc.
• Shapes – Children can make shapes out of sticks, rocks, etc.  They can copy over ones drawn in chalk or create their own freehand. To take this further, if children are using objects that are very similar in size/length (e.g. rocks or leaves) they can use them to do a non-standard unit measure of the perimeter – e.g. the rectangle is 1 leaf wide and 3 leaves long. It’s a great way to see the difference between squares and rectangles visually.

• String shapes – Children can use loops of string or large bands to make shapes (this can be done in partners/groups) and see how manipulating them changes the way the shape looks or may turn it into a different shape. How do you make a triangle or a square? Can you show me different ways to make a triangle?

• Shape pictures – Children can draw pictures (in chalk) using 2D shapes. What shapes did you use to make a house, car, etc?
• Shape hunt – Children may go on 2D and 3D shape hunts in nature. Which shape is it? How do you know?

Data Handling

• Pictographs – Children may organise natural objects such as leaves or flowers by features such as colour, size, type, etc. on a  pictograph.

• Venn Diagrams  – Using hula-hoops to sort objects by two different features (e.g. leaves by colour and size, etc.)

I hope you find this list of outdoor maths activities for KS1 helpful.  They can provide a great way to enhance and complement the learning that children do in class.  If you decide to try out some maths outdoor learning, let me know how you get on!

References – Outdoor Maths Activities KS1 -Maths Outdoor Learning

Carruthers, E. and Worthington, M. (2004).  ‘Young children exploring early calculation’.  Mathematics Teaching , (187), 30-34.

Anghileri, J. (2006).  Teaching number sense , (Ch. 4, pp. 49-70).  London: Continuum.

Arithmetic , Data, Patterns & Sorting , Geometry , Maths , Measurement , Natural , Number & Place Value , Preschooler , Rocks , School Age , Sticks

Dirt , flowers , hands-on learning , Learning Outdoors , Leaves , numerals , numicon , Nuts , outdoor learning , Rocks , Sand , Sticks , Water

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• Education, training and skills
• School curriculum
• Primary curriculum, key stage 1
• Tests and assessments (key stage 1)

Key stage 1: mathematics test framework

• Standards & Testing Agency

Updated 20 May 2024

© Crown copyright 2024

This publication is licensed under the terms of the Open Government Licence v3.0 except where otherwise stated. To view this licence, visit nationalarchives.gov.uk/doc/open-government-licence/version/3 or write to the Information Policy Team, The National Archives, Kew, London TW9 4DU, or email: [email protected] .

Where we have identified any third party copyright information you will need to obtain permission from the copyright holders concerned.

This publication is available at https://www.gov.uk/government/publications/key-stage-1-mathematics-test-framework/key-stage-1-mathematics-test-framework

This test framework is based on the national curriculum programme of study (2014) for mathematics, introduced for teaching in schools from September 2014 and first assessed in the summer term 2016. The framework specifies the purpose, format, content and cognitive domain of the optional key stage 1 (KS1) mathematics tests; it is not designed to be used to guide teaching and learning.

This document has been produced to aid the test development process.

1.1 Purposes of STA optional assessments

The purpose of the optional assessments is to give schools access to test papers to support in the evaluation of pupil achievement and help to understand where they need additional support as they transition into key stage 2 (KS2).

While the Government encourages schools to administer the tests, there is no requirement to do so or to report results to parents or local authorities, and they will not be used for accountability purposes.

2 What is a test framework?

The purpose of the test framework is to provide the documentation to guide the development of the tests. The framework is written primarily for those who write test materials and to guide subsequent development and test construction. It is being made available to a wider audience for reasons of openness and transparency.

The framework includes those parts of the programme of study as outlined in the national curriculum (2014) that will be covered in the test (the content domain). The cognitive processes associated with the measurement of mathematics are also detailed in the cognitive domain.

The test framework also includes a test specification from which valid, reliable and comparable tests can be constructed each year. This includes specifics about test format, question types, response types, marking and a clear test-level reporting strategy.

By providing all of this information in a single document, the test framework answers questions about what the test will cover, and how, in a clear and concise manner. The framework does not provide information on how teachers should teach the national curriculum.

The test development process used by the Standards and Testing Agency (STA) embeds within it the generation of validity and reliability evidence through expert review and trialling. Given that the optional KS1 tests will be internally marked by teachers, an additional study to consider the reliability of marking will be undertaken as part of the ‘technical pre-test’ trial in the first year. The test framework does not provide detail of the validity and reliability of individual tests; this will be provided in the test handbook, which will be published on the Department for Education’s website following the administration of the test.

The test framework should be used in conjunction with the national curriculum (2014) and the annual ‘Optional KS1 tests guidance’ document.

3 Nature of the test

The KS1 mathematics test forms part of the optional assessment arrangements for pupils at the end of KS1.

The test is based on the national curriculum statutory programme of study (2014) for mathematics at KS1.

The mathematics test will cover the aspects of the curriculum that lend themselves to paper-based testing.

The optional KS1 mathematics test will be marked by teachers.

3.1 Population to be assessed

The KS1 tests are optional. They are made available to schools to support the evaluation of pupil achievement and help to understand where they need additional help as they transition into KS2.

3.2 Test format

The optional mathematics test comprises 2 components, which are presented to pupils as 2 separate test papers. The first paper is an arithmetic paper. The second paper presents a range of mathematical reasoning and problem-solving questions. The test is administered on paper.

The tests are designed to enable pupils to demonstrate their attainment and as a result are not strictly timed, since the ability to work at pace is not part of the assessment. However, elements within the curriculum state that pupils should be able to use quick recall of mathematical facts and the arithmetic paper is designed to assess some of these elements. Guidance will be provided to schools to ensure that pupils are given sufficient time to demonstrate what they understand, know and can do without prolonging the test inappropriately. Table 1 provides an indication of suggested timings for each component. The total testing time is approximately 55 minutes. If teachers or administrators change the time significantly, the test outcomes will be less reliable.

Table 1: Format of the test

3.3 resource list.

The resource list for the test is:

• Paper 1: arithmetic – a pencil; ruler; rubber (optional)
• Paper 2: mathematical reasoning – a pencil; a sharp, dark pencil for mathematical drawing; ruler (showing centimetres and millimetres); mirror; rubber (optional)

Pupils will not be permitted to use a calculator, tracing paper, number apparatus or other supporting equipment in either of the components.

4 Content domain

The content domain sets out the relevant elements from the national curriculum programme of study (2014) for mathematics at KS1 that are assessed in the optional mathematics test. The tests will, over time, sample from each area of the content domain.

The content domain also identifies elements of the programme of study that cannot be assessed in the KS1 tests (section 4.3). Attainment in these elements will be monitored through teacher assessment.

Tables 2 and 3 detail content from the national curriculum (2014). Elements from the curriculum are ordered to show progression across the years. The curriculum has been grouped into subdomains and these are detailed in the ‘strand’ column.

The numbering in Table 2 is not sequential because content that relates to KS2 has been removed from it.

4.1 Content domain referencing system

A referencing system is used in the content domain to indicate the year, the strand and the substrand - for example, ‘1N1’ equates to:

• strand – Number and place value
• substrand – 1

Table 2 shows the references for the strands and substrand, and Table 3 shows the progression across the years.

Table 2: Content domain strands and substrands

4.2 content domain for key stage 1 mathematics, table 3: content domain, 4.3 elements of the national curriculum that cannot be assessed fully.

The table below identifies areas that are difficult to fully assess in a paper-based format. Some of the points below may be partially assessed.

Table 4: Elements of the curriculum that cannot be assessed fully

5 cognitive domain.

The cognitive domain seeks to make the thinking skills and intellectual processes required for the optional KS1 mathematics test explicit. Each question will be rated against the 4 strands of the cognitive domain listed in sections 5.1 to 5.4 below to provide an indication of the cognitive demand.

The cognitive domain will be used during test development to ensure comparability of demand as well as difficulty for tests in successive years. The national curriculum (2014) aims of solving mathematical problems, fluency and mathematical reasoning are reflected within the cognitive domain.

5.1 Depth of understanding

This strand is used to assess the demand associated with recalling facts and using procedures to solve problems.

Questions requiring less depth of understanding require simple procedural knowledge, such as the quick and accurate recall of mathematical facts or the application of a single procedure to solve a problem.

At intermediate levels of demand, a question may require the interpretation of a problem or the application of facts or procedures. However, the component parts of these questions are simple and the links between the parts and processes are clear.

At a high level of demand, a greater depth of understanding is expected. Questions may require that facts and procedures will need to be used flexibly and creatively to find a solution to the problem.

Table 5: Depth of understanding - rating scale

5.2 computational complexity.

This strand is used to assess the computational demand of questions.

In questions with low complexity, there will be no numeric operation.

At an intermediate level of complexity, more than one numeric step or computation will be needed to solve the problem.

At a high level of complexity, questions will involve more than 2 processes or numeric operations.

Table 6: Computational complexity - rating scale

5.3 spatial reasoning and data interpretation.

This strand is used to assess the demand associated with the representation of geometrical problems involving 2-dimensional and 3-dimensional shapes, position and movement. This strand is also used to assess the demand associated with interpreting data.

There is a low level of demand when all the resources or information required to answer the question are presented within the problem (for example, counting the number of sides of a given 2-D shape).

At intermediate levels of demand, spatial reasoning will be needed to manipulate the information presented in the question to solve the problem (for example, find a line of symmetry on a simple shape or interpret a 2-D representation of a 3-D shape). Pupils may need to select the appropriate information in order to complete the problem (for example, from a table, chart or graph).

At the highest level of demand, there may be the need to use complex manipulation or interpretation of the information as part of the problem.

Table 7: Spatial reasoning and data interpretation - rating scale

5.4 response strategy.

This strand describes the demand associated with constructing a response to a question.

At a low level of demand, the strategy for solving a problem is given as part of the presentation of the problem.

At a lower intermediate level of demand, the strategy for solving a problem is clear. Very little construction is required to complete the task.

At an upper intermediate level of demand, there may be simple procedures to follow that will lead to completion of the problem.

At a high level of demand, the question will require that a simple strategy is developed (and perhaps monitored) to complete the task. The answer may need to be constructed, organised and reasoned.

Table 8: Response strategy - rating scale

6 test specification.

This section provides details of each test component.

6.1 Summary

The test comprises 2 components, which will be presented to pupils as 2 separate papers.

Table 9: Format of the test

6.2 breadth and emphasis.

The content and cognitive domains for the optional mathematics tests are specified in sections 4 and 5. The test will sample from the content domain in any given year. Although every element may not be included within each test, the full range of assessable content detailed in this document will be assessed over time. The questions in each test will be placed in an approximate order of difficulty.

The following sections show the proportion of marks attributed to each of the areas of the content and cognitive domains in a test.

6.2.1 Profile of content domain

Each of the 7 strands listed in Table 10 will be tested on a yearly basis and these will be present in the tests in the proportions shown.

Table 10 shows the distribution of marks across the content domain.

Table 11 shows the distribution of marks across the components of the test and by national curriculum element.

Table 10: Profile of content domain

Table 11: profile of marks by paper and curriculum element.

The total number of marks for both papers is 60.

6.2.2 Profile of cognitive domain

The cognitive domain is specified in section 5. The allocation of marks across each strand and demand rating is detailed in Table 12.

Table 12: Profile of marks by cognitive domain strand

6.3 format of questions and responses, 6.3.1 paper 1.

Paper 1 (arithmetic) will be comprised of constructed response questions, presented as context-free calculations. The arithmetic questions will each be worth one mark.

6.3.2 Paper 2

For Paper 2, mathematical reasoning problems are presented in a wide range of formats to ensure pupils can fully demonstrate mathematical fluency, mathematical problem solving and mathematical reasoning. There will be 6 aural questions at the start: one practice question and 5 test questions. These questions will help the pupils settle into the test; they will be placed in approximate order of difficulty. All questions may be read aloud, so that reading ability does not impair a pupil’s ability to demonstrate his or her mathematical attainment.

Paper 2 will include both selected response and constructed response questions.

Selected response questions, where pupils are required to select which option satisfies the constraint given in the question, will include question types such as:

• multiple choice, where pupils are required to select their response from the options given
• matching, where pupils are expected to indicate which options match correctly
• true / false or yes / no questions, where pupils are expected to choose one response for each statement

Constructed response questions, where pupils are required to construct an answer rather than simply select one or more options, will include the following:

• constrained questions, where pupils are required to provide a single or best answer; these might involve giving the answer to a calculation, completing a chart or table, or drawing a shape (for questions worth more than one mark, partial credit will be available)
• less constrained questions, where pupils are required to communicate their approach to solving a problem

Questions in Paper 2 will comprise items presented in context and out of context.

6.4 Marking and mark schemes

The end of KS1 tests are optional and will be marked internally by teachers.

The mark schemes will give specific guidance for the marking of each question, together with general principles to ensure consistency of marking.

The mark schemes will provide the total number of marks available for each question and the criteria by which teachers should award the marks to pupils’ responses. Where multiple correct answers are possible, examples of different types of correct answer will be given in the mark schemes. Where applicable, additional guidance will indicate minimally acceptable and unacceptable responses. The mark schemes will provide a content domain reference, so it is possible to determine what is assessed in each question.

For all questions, the mark schemes will be developed during the test development process and will combine the expectations of experts with examples of pupils’ responses obtained during trialling.

For two-mark questions, where the correct answer has not been obtained, the mark scheme will indicate how marks can be awarded for correctly following a process or processes through the problem.

Within the mark schemes, examples of responses will be developed for ‘method’ questions. This is because the questions are open, leading to pupils giving a wide range of responses that are very close to the border between creditworthy or non-creditworthy. The additional examples help to improve marking reliability by providing examples of responses that fall just either side of the border of what is creditworthy or non-creditworthy.

6.5 Reporting

The raw score on the test (the total marks achieved out of the 60 marks available) will be converted into a scaled score using a conversion table. Scaled scores retain the same meaning from one year to the next. Therefore, a particular scaled score reflects the same standard of attainment in one year as in the previous year, having been adjusted for any differences in difficulty of the test.

Additionally, each pupil will receive an overall result indicating whether or not he or she has achieved the required standard on the test. A standard-setting exercise will be conducted on the first live test in 2016 to determine the scaled score needed for a pupil to be considered to have met the standard. This process will be facilitated by the performance descriptor in section 6.7, which defines the performance level required to meet the standard. In subsequent years, the standard will be maintained using appropriate statistical methods to translate raw scores on a new test into scaled scores with an additional judgemental exercise at the expected standard. The scaled score required to achieve the expected standard on the test will remain the same.

6.6 Desired psychometric properties

While the focus of the outcome of the test will be whether a pupil has achieved the expected standard, the test must measure pupils’ ability across the spectrum of attainment. As a result, the test must aim to minimise the standard error of measurement at every point on the reporting scale, particularly around the expected standard threshold.

The provision of a scaled score will aid in the interpretation of pupils’ performance over time, as the scaled score that represents the expected standard will be the same year-on-year. However, at the extremes of the scaled score distribution, as is standard practice, the scores will be truncated such that above or below a certain point all pupils will be awarded the same scaled score to minimise the effect for pupils at the ends of the distribution, where the test is not measuring optimally.

6.7 Performance descriptor

This performance descriptor describes the typical characteristics of pupils whose performance in the optional KS1 test is at the threshold of the expected standard. Pupils who achieve the expected standard in the tests have demonstrated sufficient knowledge to be well-placed to succeed in the next phase of their education, having studied the full KS1 programme of study in mathematics. This performance descriptor will be used by panels of teachers to set the standards on the new tests following their first administration in May 2016. It is not intended to be used to support teacher assessment, since it reflects only the elements of the programme of study that can be assessed in a written test (see content domain in section 4).

6.7.1 Overview

Pupils working at the expected standard will be able to engage with all questions within the test. However, they will not always achieve full marks on each question, particularly if working at the threshold of the expected standard.

Questions will range from those requiring recall of facts or application of learned procedures to those requiring understanding of how to use facts and procedures creatively to decide how to solve more complex and unfamiliar problems. There will be a variety of question formats including selected response, short answer and more complex calculations involving a small number of steps.

Question difficulty will be affected by the strands of the cognitive domain such as computational complexity and spatial reasoning and data interpretation. This should be borne in mind when considering the remainder of this performance descriptor, since pupils working at the threshold of the expected standard may not give correct responses to all questions. In cases where there are multiple interrelated computational steps and/or a need to infer new information or to visualise or represent a more abstract problem, some pupils may find the question difficult to understand in a test setting. This will be true even when the performance descriptor determines that a skill should be within the pupil’s capacity if working at the expected standard.

The following sections describe the typical characteristics of pupils in year 2 working at the threshold of the expected standard. It is recognised that different pupils will exhibit different strengths, so this is intended as a general guide rather than a prescriptive list. References in [square brackets] refer to aspects of the content domain specified in section 4.

6.7.2 Number

Pupils working at the expected standard are able to:

• count in multiples of 2, 5 and 10, to 100, forwards and backwards [N1]
• count forwards in multiples of 3 to 30 [N1]
• count in steps of 10, to 100, forward and backward (for example, 97, 87, 77, 67, …) [N1]
• read and write numbers to at least 100 in numerals, and make recognisable attempts to write numbers to 100 in words [N2]
• use place value in whole numbers up to 100 to compare and order numbers, using less than (<), equals (=) and greater than (>) signs correctly [N2]
• identify, represent and estimate numbers within a structured environment (for example, estimate 33 on a number line labelled in multiples of ten) [N4]
• use place value and number facts to solve problems (for example, 60 – ▢ = 20) [N6]
• use addition and subtraction facts [C1]
• a two-digit number and ones (for example, 65 + 8, 79 – 6)
• a two-digit number and tens (for example, 62 + 30, 74 – 20)
• 2 two-digit numbers (for example, 36 + 41, 56 – 22)
• 3 one-digit numbers (for example, 9 + 6 + 8) [C2]
• use inverse operations to solve missing number problems for addition and subtraction (for example, given 9 + 5 = 14, complete 14 – ▢ = 9 and ▢ – 9 = ▢) [C3]
• solve simple 2-step problems with addition and subtraction (for example, Ben has 5 red marbles and 6 blue marbles. He gives 7 of his marbles to a friend. How many marbles does he have left?) [C4]
• recall and use multiplication and division facts for the 10 multiplication table using the appropriate signs (×, ÷ and =) (for example, 80 ÷ 10 = ▢) [C6, C7]
• recall and use multiplication facts for the 2 and 5 multiplication tables and begin to recall and use division facts for the 2 and 5 multiplication tables using appropriate signs (×, ÷ and =) (for example, 2 × ▢ = 16, 5 × 6 = ▢) [C6, C7]
• recognise odd and even numbers [C6]
• solve problems involving multiplication and division (for example, Ben shares 15 grapes between 3 friends; how many grapes do they each receive?) [C8]
• know that addition and multiplication of 2 small numbers can be done in any order (commutative) and subtraction of one number from another cannot (for example, 5 × 6 = 6 × 5, but 19 – 12 is not equal to 12 – 19) [C9]
• recognise and find half of a set of objects or a quantity (for example, find 12 of 18 pencils) and begin to find 1/3 (one-third) or 1/4 (one-quarter) or 3/4 (three-quarters) of a small set of objects (for example, find 1/3 (one-third) of nine pencils) [F1]
• recognise, find and name fractions 1/2 (one-half), 1/3 (one-third), 1/4 (one-quarter), 2/4 (two-quarters) and 3/4 (three-quarters) of a shape (for example, shade 1/4 (one-quarter) or 3/4 (three-quarters) of a square split into 4 equal rectangles, or shade 1/2 (one-half) of a symmetrical shape split into 8 equal parts [F1]
• recognise the equivalence of 2 quarters and one half in practical contexts [F2]

6.7.3 Measurement

Pupils working at the expected standards are able to:

• compare and order lengths, mass, volume / capacity (for example, 30 cm is longer than 20 cm [M1]
• choose and use appropriate standard units to measure length / height in any direction (m / cm); mass (kg / g); temperature (°C); capacity (litres / ml) to the nearest appropriate unit (for example, the bucket contains 4 litres of water, scale marked every litre and labelled at 5 litres) using rulers, scales, thermometers and measuring vessels and begin to make good estimates (for example, the book is about 20 cm long) [M2]
• recognise and use symbols for pounds (£) and pence (p); combine amounts to make a particular value and find different combinations of coins to equal the same amounts of money (for example, find two different ways to make 48p) [M3]
• recognise, tell and write the times: o’clock, half past and quarter past and quarter to the hour; draw hands on a clock face to show half past and o’clock times [M4]
• begin to tell and write the time to 5 minutes, including quarter past / to the hour and draw hands on a clock face to show these times [M4]
• solve problems in a practical context involving addition and subtraction of money of the same unit, including giving change (for example, Mrs Smith buys a cake for 12p and a biscuit for 5p; how much change does she get from 20p?) [M9]

6.7.4 Geometry

• compare and sort common 2-D shapes (for example, semi-circle, rectangle and regular polygons such as pentagon, hexagon and octagon) and everyday objects, identifying and describing their properties (for example, the number of sides or vertices, and recognise symmetry in a vertical line) [G1, G2]
• compare and sort common 3-D shapes (for example, cone, cylinder, triangular prism, pyramid) and everyday objects, identifying and describing their properties (for example, flat / curved surfaces, and beginning to count number of faces and vertices correctly) [G1, G2]
• identify 2-D shapes on the surface of 3-D shapes and images of them (for example, a circle on a cylinder and a triangle on a pyramid) [G3]
• order and arrange combinations of mathematical objects in patterns (for example, continue a repeating pattern such as: circle, circle, star, triangle, circle, circle, star, triangle, circle, ▢) [P1]
• use mathematical vocabulary to describe position, direction (for example, left and right) and movement, including movement in a straight line, and distinguish between rotation as a turn, and in terms of right angles for quarter and half turns [P2]

6.7.5 Statistics

• interpret simple pictograms (where the symbols show one-to-one correspondence), tally charts, block diagrams (where the scale is divided into ones, even if only labelled in multiples of 2) and simple tables [S1]
• answer questions by counting the number of objects in each category and sorting the categories by quantity [S2]
• answer questions about totalling and begin to compare simple categorical data (for example, when the pictures or blocks are adjacent) [S2]

6.7.6 Solve problems and reason mathematically

• use place value and number facts to solve problems (for example, 40 + ▢ = 70) [N6,C1]
• use inverse operations to solve missing number problems for addition and subtraction (for example, There were some people on a bus, six got off leaving seventeen people on the bus. How many were on the bus to start with?) [C3]
• solve simple 2-step problems with addition and subtraction, which require some retrieval (for example, There are 12 kittens in a basket, 6 jump out and only 2 jump back in; how many are in the basket now?) [C4]
• solve simple problems involving multiplication and division (for example, Ahmed buys 3 packs of apples. There are 4 apples in each pack. How many apples does he buy?) [C8]
• solve problems with one or 2 computational steps using addition, subtraction, multiplication and division and a combination of these (for example, Joe has 2 packs of 6 stickers; Mina gives him 2 more stickers; how many stickers does he have altogether?) [C4, C8]
• solve simple problems in a practical context involving addition and subtraction of money of the same unit, including giving change (for example, Identify three coins with a total value of 24p or find the two items which cost exactly £1 altogether from a list such as: 70p, 40p, 50p and 30p) [M3, M9]

7 Diversity and inclusion

The Equality Act 2010 sets out the principles by which national curriculum assessments and associated development activities are conducted. During the development of the tests, STA’s test development division will make provision to overcome barriers to fair assessment for individuals and groups wherever possible.

National curriculum assessments will also meet Ofqual’s core regulatory criteria. One of the criteria refers to the need for assessment procedures to minimise bias: ‘The assessment should minimise bias, differentiating only on the basis of each learner’s ability to meet national curriculum requirements’ (Regulatory framework for national assessment, published by Ofqual 2011).

The optional end of KS1 mathematics test should:

• use appropriate means to allow all pupils to demonstrate their mathematical fluency, solving problems and reasoning
• provide a suitable challenge for all pupils and give every pupil the opportunity to achieve as high a standard as possible
• provide opportunities for all pupils to achieve, irrespective of gender, disability or special educational need, social, linguistic or cultural backgrounds
• use materials that are familiar to pupils and for which they are adequately prepared
• not be detrimental to pupils’ self-esteem or confidence
• be free from stereotyping and discrimination in any form

The test development process uses the principles of universal design, as described in the ‘Guidance on the principles of language accessibility in national curriculum assessments’ (New language accessibility guidance, published by Ofqual 2012).

In order to improve general accessibility for all pupils, where possible, questions will be placed in order of difficulty. As with all national curriculum tests, attempts have been made to make the question rubric as accessible as possible for all pupils, including those who experience reading and processing difficulties and those for whom English is an additional language, while maintaining an appropriate level of demand to adequately assess the content. This includes applying the principles of plain English and universal design wherever possible, conducting interviews with pupils and taking into account feedback from expert reviewers.

For each test in development, expert opinions on specific questions are gathered - for example, at inclusion panel meetings, which are attended by experts and practitioners from across the fields of disabilities and special educational needs. This provides an opportunity for some questions to be amended or removed in response to concerns raised.

Issues likely to be encountered by pupils with specific learning difficulties have been considered in detail. Where possible, features of questions that lead to construct irrelevant variance (for example, question formats and presentational features) have been considered and questions have been presented in line with best practice for dyslexia and other specific learning difficulties.

7.1 Access arrangements

The full range of access arrangements applicable to optional KS1 assessments as set out in the ‘Optional KS1 tests guidance’ will be available to eligible pupils as required.

Teachers are able to vary the administration arrangements for pupils according to their need. Where arrangements are varied, it should follow normal classroom practice for assessments of this type.

Appendix: Glossary of terminology used in the test framework

Independent review of key stage 2 testing, assessment and accountability (2011), Lord Bew.

Hughes S., Pollit A., & Ahmed A. (1998). ‘The development of a tool for gauging demands of GCSE and A-Level exam questions’. Paper presented at the BERA conference The Queens University Belfast.

Webb L. N. (1997). ‘Criteria for alignment of expectations and assessments in mathematics and science education’. Research Monograph No. 8. Council of Chief School Officers.

Smith M.S., Stein M.K. (1998). ‘Selecting and creating mathematical tasks: from research to practice’. Mathematics teaching in middle school 3 pp344–350.

About this publication

Who is it for.

This document is aimed primarily at those responsible for developing the optional KS1 national curriculum test in mathematics. It may also be of interest to schools with pupils in KS1 and other education professionals.

What does it cover?

The framework provides detailed information to ensure an appropriate test is developed, including the:

• content domain
• cognitive domain
• test specification
• test performance descriptors

Related information

Visit the  Standards and Testing Agency homepage  for all related information.

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Millennials have a fresh take on the FIRE movement, and it's less about taking it easy in retirement

At age 36, Jace Mattinson is already over retirement . Four years ago, he sold his lumber company for seven figures, and he had enough saved that he never needed to work again.

He said that was an enticing idea after five "extremely tough" years of owning a business. During that time, he was away from his home in Austin a few nights a week and hustling to run the 135-year-old company he'd acquired. After selling the company, he needed a long break from anything laborious.

"I was golfing three, four times a week. I was going to the lake. I was doing all my hobbies that I really cared about and enjoyed, ones that for the greater part of a decade I didn't have as much time to do," Mattinson told Business Insider.

But after eight months, he decided retirement was not nearly as fulfilling as he'd imagined. He returned to a job in lumber distribution and revived his financial podcast. He said he wanted to continue to model a good work ethic for his kids.

Mattinson has all the trappings of someone in the FIRE movement . The acronym, which stands for financial independence, retire early, was coined in the 1990s in the book "Your Money or Your Life" and popularized on blogs like Mr. Money Mustache and the investment site Motley Fool. The idea was to work hard, ideally with multiple income streams, live a life of austerity, invest prudently, and build a big enough nest egg to walk away from work well before the average retirement age of 64.

But millennials, including Mattinson, who finds himself happiest when he has a balance of work and leisure, said they're not as interested in early retirement — and are creating their own versions of life after work.

Millennials often want the FI without the RE

Devotees of the FIRE movement often save or invest the majority of their income . Some take on extra jobs or delay major life milestones like marriage or having kids.

It's an exclusive club, and many hungry millennials are eager to join it. ChooseFI's Facebook group has over 108,000 members, while the r/financialindependence subreddit has 2.2 million members. But for some FIRE wannabes, the "FI" part of the equation is the biggest focus, and the "RE" half seems to be less of a foregone conclusion.

A popular rule of thumb among this group is the "4% rule," which says you should aim to save 25 times your annual expenses so you can withdraw 4% of your funds each year after you quit working. Some FIRE participants told BI that their target savings goal is between $1.5 million and $2.5 million, though many are working toward more for even greater security.

To be sure, early retirees are a small slice of the population. According to Business Insider's analysis of American retirees, just 2.2% are 50 or younger. Less than 1% are below age 35. Just 0.75% of all Americans over 18 and under 50 are retired. Still, many BI spoke with retire unofficially or partially retire, taking on less responsibility at a company or moving to a lower-stakes position.

BI spoke to a dozen millennials who have achieved or are on track to achieve financial independence. While some have retired and told BI they're enjoying it, most feel retirement is pointless and still want to build their careers or give back to their communities.

"The thing I have noticed shift most is the emphasis on FI and less on RE," Scott Rieckens, the executive producer of the film " Playing With FIRE ," said. "I think it's awesome to see, as it signals that financial independence is the key motive, which it is, and that work and purpose are actually really important. Retiring early to nothing is a bad idea."

Brad Barrett, the host of the "ChooseFI" podcast, said "vanishingly few" people with the wherewithal to reach financial independence are retiring early. To him, reaching financial independence allows someone to live the life they want, but retiring early signifies turning away from everything you've worked toward.

For many, financial freedom goes beyond quitting a job you don't like. Some said it's the ability to spend on travel or leisure without much stress — which has become even more important after the pandemic's peak. Others said it helps them lead a life of purpose, whether that means educating people on a podcast or leading charity efforts.

The problem with retirement seems to be that people want to add value to their communities and within their own lives — and they believe work is the way to do that. As Bill Schaninger, a speaker, author, and thought leader on the future of work, found in research he conducted with Naina Dhingra for McKinsey , 70% of people who were surveyed said they define their purpose through work.

"Many people figured out one of the things that I get a lot of validation from is being clever, solving problems, participating, and working on something bigger than me," Schaninger told BI.

COVID-19 may have amplified this, he added. "The fragility of our condition, I think, was brought home in a way that maybe many of us had taken for granted," he said. "And so now it's like, 'Well, if I'm going to do this, it has to matter.'"

The millennial version of early retirement

Mitch, 37, said he is about to quit his high-stress job and take a mini-retirement — he has a 22-stop national parks trip planned this summer.

The Minnesota resident and vice president of a building-maintenance company, who asked that only his first name be used because of an ongoing job transition, has a net worth of about $2 million but said he's only planning to take a few months off before returning to the workforce in a lower-stress position. All the sources BI spoke with provided documentation of their net worth.

Mitch said he stumbled into the online personal-finance community in his early 30s, which inspired him and his wife to increase their savings to at least 75% of their income by avoiding spending on luxury items. He said even his high savings won't affect his decision to quit working.

"I think a lot of traditional retirees lack purpose — they take a year or two of retirement and hate it because they do whatever and lose purpose," Mitch said. "The ones that volunteer, continue to coach and consult, or do whatever it is to sharpen their brain and really have a purpose tend to be some of the happiest retirees."

Brian Luebben, a financially independent millennial , described having a panic attack shortly after he hit FI and quit his sales job.

"If you have anxiety, financial freedom is not going to solve it," he said. "If you have depression, financial freedom is not going to solve it. Be careful of the mountaintop moments. When you become a millionaire, when you become financially free, when you do all this stuff, no mariachi band follows you around and performs."

He argued that achieving financial independence and hitting a specific number is "the simplest part." After all, there's a playbook for wealth-building strategies like investing in real estate or building an e-commerce business.

"The most difficult part is figuring out what you do when you have nothing to do all day," he said. "What do you choose to work on?"

Luebben, who hosts a podcast and runs the entrepreneur resource The Action Academy to help other people achieve financial freedom, said people should think through four core questions before they're even close to achieving financial independence: "What does the perfect day look like? What does the perfect week look like? Who was with you? And where?"

Going through that exercise can help ensure that your identity doesn't become wrapped up in achieving FIRE, which is something that Grant Sabatier, who took a year and a half off from work after achieving financial independence, struggled with.

"I defined myself by the pursuit of financial independence," Sabatier, the author of " Financial Freedom ," said. "Then, once I reached it, it was like, now I no longer had to do that thing, so what am I going to do? I encourage people on the path to do that inner work. Don't delay figuring out what you really want, why you're pursuing financial independence, and what you want to do after."

Balancing work and fun

Instead of a traditional retirement, many financially independent millennials are finding a balance between work and leisure that works for them.

For Sabina Horrocks, 41, becoming a millionaire was "quite boring." She and her husband worked in six-figure managerial positions, recently achieving a net worth of about $2 million, then had a daughter in 2021. They "plowed money into investments early on," kept daily expenditures low, and purchased rental properties they eventually sold.

She quit her sales operations job but has no intention of stopping work. She's a stay-at-home mom and plans to continue her blog The Moneyaires ; she'd also like to become a financial coach or planner.

Blogging and coaching were common post-FI pursuits among the would-be early retirees BI spoke to. Michelle Schroeder-Gardner, 34, runs the blog Making Sense of Cents , and over the past decade, she and her husband have lived mostly in an RV or a sailboat.

By 2017, their blog, advertising sales, and a course they created called Making Sense of Affiliate Marketing had generated nearly $1.2 million in revenue. By 2018, they had achieved financial independence. After years of 100-hour workweeks, she now spends 10 hours a week on her business, which generates $600,000 a year.

"I'm able to travel whenever I want. I can work whenever I want. Nothing's really dependent on my work hours," she said. "My plan is pretty much to continue doing this while I like it and continue to make a little bit more money and save as much as I can."

Lauren and Steven Keys, who quit their full-time jobs in their 20s , have a similar outlook.

Steven does freelance work for his former employer but spends much of his time on an online-tutoring service called CramBetter that he cofounded in 2023. Lauren has one social-media client she works with a couple of hours a month. They also run a financial-independence blog, Trip of a Lifestyle , and earn rental income from a fully paid-off investment property.

"There's this misconception about early retirement that you'll never make another penny ever again and just sit on the beach all day for the rest of your life," Steven said. "We're never going to stop making any money whatsoever."

Are you part of the FIRE movement or living by some of its principles? Reach out these reporters at [email protected] or [email protected] .

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