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Universal Design for Learning Guidelines

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Universal Design for Learning Guidelines

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Engagement, Representation, Action & Expression

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The UDL Guidelines are a tool used in the implementation of Universal Design for Learning, a framework to improve and optimize teaching and learning for all people based on scientific insights into how humans learn. Learn more about the Universal Design for Learning framework from CAST. The UDL Guidelines can be used by educators, curriculum developers, researchers, parents, and anyone else who wants to implement the UDL framework in a learning environment. These guidelines offer a set of concrete suggestions that can be applied to any discipline or domain to ensure that all learners can access and participate in meaningful, challenging learning opportunities.

Reenvisioning the UDL Guidelines

Because the UDL Guidelines are meant to be informed by feedback from the field as well as new research, they have been updated several times in the past. We are in the process of updating the Guidelines once again in our UDL Rising to Equity initiative.

This update will focus specifically on addressing systemic barriers that result in inequitable learning opportunities and outcomes. CAST aims to develop a transparent, inclusive, and community-driven process. If you are interested in collaborating and staying updated on our progress, we invite you to complete a brief survey.

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The UDL Guidelines are a tool used in the implementation of Universal Design for Learning, a framework to improve and optimize teaching and learning for all people based on scientific insights into how humans learn. Learn more about the Universal Design for Learning framework from CAST. The UDL Guidelines can be used by educators, curriculum developers, researchers, parents, and anyone else who wants to implement the UDL framework in a learning environment. These guidelines offer a set of concrete suggestions that can be applied to any discipline or domain to ensure that all learners can access and participate in meaningful, challenging learning opportunities.

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Open Middle Math Depth Of Knowledge Matrices - Robert Kaplinsky

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If you love Open Middle problems, then you'll love these matrices that compare problems across the grade levels at different Depth of Knowledge levels. Use the button below to download high quality, printable PDFs of them all or preview what you'll get by scrolling below. Click on a link below to jump to the...

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Hit the button - building fluency with number bonds, multiplication and division

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Hit the button - building fluency with number bonds, multiplication and division

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QR CODE

Scan this on mobiles and tablets to quickly open this web page.

Hit the Button is an interactive maths game with quick fire questions on number bonds, times tables, doubling and halving, multiples, division facts and square numbers. The games, which are against the clock, challenge and develop mental maths skills. An untimed, practise mode is available in our Hit the Button app along with lots more extra features.

The activities can be matched to appropriate mathematical ability. Regular use of Hit the Button can help students to sharpen their recall of vital number facts in a fun way and learning multiplication tables need not be boring.

AGE

6-11 Years

SUBJECTS

Maths Times Tables Division Number Bonds Doubles Halves Square Numbers

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Hit the Button - Quick fire maths practise for 6-11 year olds

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Quick fire questions on number bonds, doubles, halves, times tables, division facts and square numbers against the clock. Superb!

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Playlist of videos from my MANSW 2022 Conference presentation.....  Join my Mathigon Class with a Student Account (just use a different email address) and experience what a student does. Class code is: DNBP-CYFW

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Playlist of videos from my MANSW 2022 Conference presentation.....  Join my Mathigon Class with a Student Account (just use a different email address) and experience what a student does. Class code is: DNBP-CYFW

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MANSW 2022 Annual Conference - Polypad Across the Strands

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Each video contains a link to the associated polypad canvas. You can save and modify these to your own teacher account under Files and assign to your classes.

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Assessment Review Ideas - The Dice is Right

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Assessment Review Ideas - The Dice is Right

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plus more ideas on this website https://www.smore.com/83624-how-to-go-over-a-test?fbclid=IwAR0FoGIlmOPjblkGfjTn3xnrHMIuq8hkImOAFeHD8xdqVLRZHg0IUWC3NbU

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Something went wrong when loading this Polypad. Please try again later!
Generate Embed Code

You can embed a Polypad canvas on other websites like Wordpress: just select the features required below, and then copy the code block into the HTML of your website. We provide this service only for non-commercial, educational purposes. If you want to use Polypad commercially, please contact us.

What's New on Polypad

July

Significantly updated "File" menu: organise saved Polypads into folders, more granular sharing settings (public, students/teachers, private), examples and assignments appear as a special folder, and we moved the share and changelog panels.

Re-organised "Tiles" sidebar with improved grouping of tile types.

Buttons to change the denominator of fraction bars (expand and reduce), including support for denominators up to 1/24.

New "Number Dot Arrangement" component.

New number tables (addition, multiplication and integers) and number frame components.

Remember the weight of balance scale tiles when you save a Polypad. Fix support for algebra tiles on the balance scale: you can now change the value of "x" and even solve quadratic equations.

New algebra tiles for y, y² and xy, including support on balance scales.

New advanced options to lock the position of a tile (but allow selecting/copying/etc) and to make a tile invisible.

Improved change handling with the colour picker (create a single "undo" state when moving a slider or typing in the RGB fields).

June

Polyhedral Dice.

Stack, shuffle and draw playing cards, standard deck template in sidebar.

New "join" action to merge multiple polygons. Improved "cut" algorithm for polygons.

Double-click dice, coins and spinners to randomise, and playing cards to draw a card.

Tutorial Pages for every tile type.

Improved numberline resizing and snapping, and flip actions for ruler and protractor.

Significant performance improvements.

Improved graph plotting with implicit functions.

Hide Exploding Dots labels when the "hide number labels" option is selected.

Pride Month background.

May

Transparency support in colour picker.

Drag to select ranges of cells in tables, and copy/delete them.

New Domino tiles under Probability.

Irregular pentagons under Geometry.

Handle to partially highlight fraction bars.

Function plotting: create a coordinate system and an equation editor component. Next, drag the blue triangle on the right side of the equation editor onto the coordinate system. You can plot any expression with one variable, an equation of the form y = f(x), an implicit function containing x and y, as well as polar functions of the form r = f(θ).

Buttons to quickly change the axes scales of coordinate systems.

Decimal and percentage labels for fraction bars and fraction circles. You can switch between the different display modes in the settings panel on the right.

Direct number inputs for the RGB values in the colour picker.

New table component in the Tools section of the sidebar.

Layering and locking for tiles: enable Advanced Options in the settings panel on the right. Then you can use the new options in the toolbar to force tiles to stay in the foreground or background, or lock tiles. To revert locking, click Unlock All Tiles in the settings panel.

Increased limit of saved Polypad canvases per account from 100 to 250.

Equation editor improvements.

Keyboard Shortcuts and Accessibility

These are the keyboard shortcuts supported by Polypad for accessibility:

SHIFT: hold while you click on multiple tiles to select all of them at once

ALT: hold before clicking and dragging on a tile to move a copy

BACKSPACE: delete your current selection

C: duplicate your current selection

CTRL/CMD + X/Y: undo or redo your last change

ARROWS: move the currently selected tiles up, down, left or right (hold SHIFT to reduce the shift size, for more precise moving)

R: rotate the currently selected tile by 15° (hold SHIFT to rotate by -15°)

S: move the focus to the sidebar on the left

T: move the focus to the toolbar at the bottom

U: move the focus to the canvas area

ESC: clear the current selection and close any open popups and modals

TAB: cycle through all buttons on the page and tiles on the canvas (hold SHIFT to reverse direction)

Use CTRL/CMD + X/C/V to cut, copy or paste tiles. You can even copy and paste tiles from one canvas onto another!

Points and tiles will automatically snap to nearby shapes or gridlines as you move them. You can disable this behaviour by pressing the SHIFT key while moving an item with your mouse or trackpad.

You can learn more about our commitment to accessibility, as well as a WCAG checklist, at mathigon.org/accessibility
It is impossible to balance these weights.
Image Upload Failed

Unfortunately, we couldn't upload this image. Make sure you are signed into your Mathigon account, and that the image is a PNG or JPEG file less than 1MB.

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Something went wrong when loading this Polypad. Please try again later! You can embed a Polypad canvas on other websites like Wordpress: just select the features required below, and then copy the code block into the HTML of your website. We provide this service only for non-commercial, educational purposes.

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ITP Moving Digits

I am remaking the ITPs so that they will work on all modern browsers and tablets. They will remain freely available to all without the need for a subscription.

View full screen in your browser. This ITP allows you to demonstrate the effect of multiplying and dividing by 10 and 100. By dragging cards from the set of digit cards, different decimal numbers can be displayed. These decimal numbers can be placed in either of the two middle rows. The cards in the first row move when the multiply or divide buttons are selected. The cards will not move if the result is greater than 99 999 or less than 0.01. The cards in the second row do not move in order to highlight the shift in the digits when multiplying or dividing.
The ITP can be used to explore, and to encourage children to predict the effect of multiplying and dividing by 10 and 100. The ITP can also be used to demonstrate the effect of repeated multiplication and division by 10.
For more multiplication and division resources click here.

Scan to open this game on a mobile device. Right-click to copy and paste it onto a homework sheet.

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ITP moving digits, multiply and divide by 10 and 100

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Diagnostic Testing resources plus lots more....

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I first set up the Mr Barton Maths website back in 2007. If you find the website useful to help you teach or learn maths, please consider supporting my work by becoming a Patreon. Thanks so much!

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Base 10 blocks - decimal place value
Balancing - for equations
Algebra Tiles
and more....

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The store will not work correctly in the case when cookies are disabled.

JavaScript seems to be disabled in your browser. For the best experience on our site, be sure to turn on Javascript in your browser.

Hands-On Math Goes Digital In and Beyond the Classroom!

Explore Our Collection of FREE Virtual Manipulatives

From Number Lines to Base Ten, Pattern Blocks to Place Value, Disks our collection of free Virtual Manipulatives gives educators and families access to visual and hands-on digital resources that lead to understanding. Students can use these resources in class and at home to practice math concepts, reinforce their growing math understanding, technology they love in the process.

Free Resources & Instructions for Teachers and Families

We've developed these resources for easy access and easy use:

How-To Videos

There's a short video for each of the Virtual Manipulatives in our collection so teachers can use these digital materials easily and confidently with students.
Watch Now

Activities

Activities from the Didax books that teachers use most for easy download. Engaging challenges address core math operations, concepts and skills, from early numeracy all the way to introductory geometry and algebra.
Access Now

FAQ

Download this document of FAQ's to answer your Virtual Manipulative questions.
Download Now

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Demonstrate key math concepts in your classroom using our new virtual manipulatives!

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Dan Finkel - create a culture where mistakes are a part of the learning culture

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Five Principles of Extraordinary Math Teaching | Dan Finkel | TEDxRainier

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Remote Learning, Collaboration tech apps

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Remote Learning, Collaboration tech apps

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Best Student-Collaboration Tools | Common Sense Education

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Best Student-Collaboration Tools is a list of 32 apps, games, and websites curated by Common Sense Education editors.

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Developing opportunities and ensuring progression in the development of reasoning skills.

The aims of the National Curriculum are to develop fluency and the ability to reason mathematically and solve problems. Reasoning is not only important in its own right but impacts on the other two aims. Reasoning about what is already known in order to work out what is unknown will improve fluency; for example if I know what 12 × 12 is, I can apply reasoning to work out 12 × 13. The ability to reason also supports the application of mathematics and an ability to solve problems set in unfamiliar contexts.

Research by Nunes (2009) identified the ability to reason mathematically as the most important factor in a pupil's success in mathematics. It is therefore crucial that opportunities to develop mathematical reasoning skills are integrated fully into the curriculum. Such skills support deep and sustainable learning and enable pupils to make connections in mathematics.

This resource is designed to highlight opportunities and strategies that develop aspects of reasoning throughout the National Curriculum programmes of study. The intention is to offer suggestions of how to enable pupils to become more proficient at reasoning throughout all of their mathematics learning rather than just at the end of a particular unit or topic.

We take the Progression Map for each of the National Curriculum topics, and augment it with a variety of reasoning activities (shaded sections) underneath the relevant programme of study statements for each year group. The overall aim is to support progression in reasoning skills. The activities also offer the opportunity for children to demonstrate depth of understanding, and you might choose to use them for assessment purposes as well as regular classroom activities.

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Developing opportunities and ensuring progression in the development of reasoning skills

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Resources & Downloads | Dr Paul Swan Maths

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This entire site has valuable professional development on different aspects of teaching mathematics. The author, Dr Catherine Attard is the NSW equivalent of the prominent Dr Jo Boaler from the UK. She presents some effective and fun maths games

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Framework for Engagement with Mathematics (Word Doc below) is a product of my research on student engagement and is a useful tool for planning mathematics learning.
FRAMEWORK FOR ENGAGEMENT WITH MATHEMATICS
Reflecting on Games in Mathematics A set of student reflection prompts based on the Framework for Engagement with Mathematics for use when games are used in mathematics lessons.

Engaging Maths Circle Geometry This is a hands-on, engaging and open-ended task that can be used to link geometry to measurement concepts while promoting the use of mathematical language and reasoning.

Unifix Grab: A hands-on task to teach averages. Suitable for Grades 4 - 8 but can be adapted to suit younger students.

Getting To Know You Mathematically activity

Final report for my research project on the use of financial literacy to improve student engagement with mathematics.

My latest research report evaluating the Matific resources and their ability to improve student engagement. The report includes units of work that incorporate the Matific episodes.
matific-research-report-final-111016
FREE PD session.

This session is a recorded version of a presentation I did at the AAMT Connect with Maths Day on the 18th March 2016, on Engaging Contemporary Learners with Mathematics
Engaging Contemporary Learners with Mathematics FREE iPad Book
A free resource with lots of teaching ideas. Over 120 apps are listed for use in mathematics teaching and learning.
Looking for Patterns
An addition grid with a range of investigations that promote the identifying and describing of patterns, an essential part of the problem solving process.
Password Possibilities
A mathematical investigation based on the context of a forgotten mobile phone password.
The Twelve Days of Christmas
A great range of activities to keep students engaged at the end of the year!
The Travel Game
Free game instructions, gameboard, and blank template so students can design their own games.
The Arrays Game
A game that promotes multiplicative thinking using arrays.
Blast off!
Two games that use pattern blocks to help to build students' understanding of place value.

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The Framework for Engagement with Mathematics (Word Doc below) is a product of my research on student engagement and is a useful tool for planning mathematics learning. FRAMEWORK FOR ENGAGEMENT WITH MATHEMATICS Reflecting on Games in Mathematics A set of student reflection prompts based on the Framework for Engagement with Mathematics for use when games are...

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Maths teaching ideas using digital resources

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Videos on Discovery Education Espresso contain great real-life context. I'm going to demonstrate how to utilise this using an example from KS2 Maths.
One of the videos is Fraction, decimal and percentage problem and is set on a space station.

Play until just past the first minute and pause on the screen below

It's unlikely that students will be thinking about fractions at this point, but may well have involved themselves in the story of space.

Pause on this screen to provide a great opportunity for students to create their own maths problems based on the information given.
Areas of maths this video could also support:
Time e.g. dividing up 24 hours of an astronaut's life

Ratio e.g. the proportion of jobs to leisure or travel

Algebra e.g. representing the tasks and time as an algebraic sentence

Measurement e.g. converting between hours and minutes

Geometry e.g. plotting the space station activity on 4 quadrants with coordinates

Statistics e.g. constructing the information as a line graph

The problems generated by students, can be shared, displayed, grouped and worked on. If your maths focus was percentages, then after looking at their generated questions you could continue with the video, pausing to solve the problems together.

Tip: If you don't want to scroll through modules, you can access resources in other ways from the subject home page:

1. By curriculum topic - a drop down at the top of the subject home page

2. By curriculum objective - a button on the right of the subject home page

3. Search - a search box at the top of the page

FURTHER IDEAS:

This printable resource has word problems to extend students, keeping with the theme of space.

You could post one on a large sheet of paper at different parts of the room and ask students to rotate round spending a few minutes at each to work out the answer, recording for others to see.

Students could create a pie chart to represent time with a set of events with different durations, labelling with fractions.

And this activity is a fun way to check knowledge of equivalent fractions and percentages against the clock.

To continue with the theme of space, you could use the Earth moon and sun science module looking at distances between the planets as well as diameter and mass. A poster of facts could be drawn up and displayed and students could be asked in groups to write word problems involving incorporating fractions, decimals and percentages in relation to planets.

NB For those children who need consolidation, use the Fractions and decimals (lower) module for resources that support earlier development of these concepts.
I'd love to hear if you have looked at this or other resources in maths on Discovery Education Espresso to support areas of your numeracy curriculum. Please comment in the box!

Find the featured resources:

Fraction, decimal and percentage problem
KS2 > Maths > Fractions, decimals and percentages (Upper) > Video > Fraction, decimal and percentage problem
Printable resource
KS2 > Maths > Fractions, decimals and percentages (Upper) > Printable resources > Fraction, decimal and percentage problems
Activity
KS2 > Maths > Fractions, decimals and percentages (Upper) > Activities > Snap: percentage and fraction images
Earth moon and sun
KS2 > Science > Eath, moon and sun

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Digital numeracy tips & tricks: Using footage of real life contexts to teach fractions and more

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Videos on Discovery Education Espresso contain great real-life context. I'm going to demonstrate how to utilise this using an example from KS2 Maths. One of the videos is Fraction, decimal and percentage problem and is set on a space station.

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How to start writing mathematics journals - good way of improving students reasoning skills

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Math journaling is a great way to get insight into your students' thoughts about math and their problem-solving strategies and a wonderful addition to

ANY math program from grades K-12. This page will help you set up and manage an easy-to-maintain math journal system in your classroom.

FAQs about math journaling

What's a math journal?

Many teachers have a regular journaling time in their classroom in which students reflect on a variety of topics. Math journals work the same way, except the prompts are about math.

Here are a few math journal entries produced by K-4 students in New York City during my demonstration lessons as a math coach. These are students' initial attempts at math journaling (their very first prompts). Most of these lessons were taught in the second semester of the school year.

Why math journaling?

There are a number of reasons why math journals are gaining popularity at all grade levels:Why math journaling?

Kids have the opportunity to reflect on their strategies and assess their own learning

Students practice putting their knowledge into words both verbally and in writing

Instructional focus is shifted from computation to problem solving and real-life application

The teacher gains insight into children's abilities, opinions, understandings, and misconceptions

It creates a documented portfolio-like record of student growth and progress

Math journaling is an open-ended and naturally differentiated assessment tool

Who should use math journals?

I recommend math journals to teachers of grades K-12. They're an incredibly versatile tool that can be used by students of all ability and age levels. Because the topics are open-ended, math journals are an easy way to incorporate differentiated instruction and meet the needs of every child in your class.

How do math journals work?

Many teachers choose to have students respond to math journal prompts in a separate math journal. Having a special math journal that's different from everyday school materials can be very motivating for children. Other teachers have students respond to math journal topics as part of a Math Notebook: one composition book holds a student's journal reflections, class notes, and practice activities. Still other teachers have students keep one journal in which they respond to a wide variety of prompts across different content areas (this could be termed as a 'Writer's Notebook'.) There's no one right way to do math journaling, so feel free to pick a format that makes sense for you.

How often do students write in them?

Once a week is a beneficial and attainable goal for teachers just starting with math journals. Personally, I like to have students reflect on a topic before we begin the unit of study so I can gauge prior knowledge, and then after a few days of instruction, I have students reflect on their strategies and use their writing to assess progress and address misconceptions. As we finish the skill, I assign a topic that will provide an assessment, such as having students explain how to solve a particular type of problem or connect what we've learned to other skills. Because of the variety of ways I like to use math journals, it usually works out to two assigned math journal topic each week. I also encourage children to write in their math journals even when a topic hasn't been assigned; it's very own personal place for reflection and extension.

What do students write about?

I recommend three different types of prompts:

-Prompts That Assess Attitudes: Students write about their personal thoughts and feelings about math. Examples: When it comes to math, I find it difficult to..., I love math because..., People who are good at math..., and When I study for a math test, I....

-Prompts That Assess Learning: Students write about what they've learned and reflect on what they know (and don't know). Examples: The most important thing I learned today is..., I could use today's skill in my real life when I..., Today I used math when..., At the end of this unit, I want to be able to..., and Some good test questions for this skill are....

-Prompts That Assess Process: Students explain how to solve problems or discuss a particular skill or strategy. Examples: Two ways to solve this problem are..., I knew my answer was right when..., Another strategy I could have used to solve this problem is..., If I missed a step in this problem, I could have..., and The most important part of solving this problem is to remember....

How long should students write for?

Only a few minutes is needed for math journaling: 5-10 minutes is enough for most classes, though you may want to allow for 15 minutes for more complex prompts. Allow an additional 2 minutes at the end for sharing.

Do students need to share their writing?

It's important to allow about 2 minutes at the end of the journaling period for students to share what they wrote because the sharing time:

keeps students accountable

provides an incentive to write something meaningful and cohesive

acclimates kids to expressing their thoughts and strategies verbally

allows them to hear and explore other children's strategies and techniques

Math journal sharing can be done whole class in the beginning of the year (walk around as students write and take note of any exceptional or thought-provoking responses, and ask those students to share with the group afterward). This is a great opportunity for you to encourage reflective thinking and reinforce your journaling expectations. Later, students can pair-share on their own. I would not force an unwilling student to read his or her writing, but that's my personal preference.

What if students only write one sentence?

Your students will follow your lead in math journaling: if you model lengthy and detailed responses, eventually students will follow suit. Writing about math is difficult and strange to many children in the beginning, so be very positive and encouraging with their efforts. As the weeks pass, you will see marked improvement in most students.

Also, be careful not to fuss at kids who are staring into space and not writing-math journals require a lot of thinking time, and even if you suspect the child is daydreaming, give him or her the benefit of the doubt during journaling. The time spent thinking is equally important as the time spent writing.

I recommend using math talk activities to help students get comfortable with writing about math ideas. You can use discussion starters and question stems such as these to help you:

What do students do when they're done?

Set a clear expectation on this. Either students should write until time is up, or they should begin another activity as soon as they are finished. If you want students to write for a set time, make sure it's short (like 5 minutes). You don't want kids to start writing nonsense just because they know you're watching. An alternative is to let kids draw about what they wrote when they're done-this is actually very useful and most children will stay focused on creating pictures related to the prompt.

How do I introduce math journaling to my class?

MODEL, MODEL, MODEL. In fact, the first few times you do math journaling, students may just watch you write, or copy down your example. With very young students (gr. K-1), you might choose to model every time you introduce a new prompt to the class for the first few months. The same is true if your students are older and reluctant to write: compose several journal prompts together to get students comfortable with the idea and acutely aware of your expectations.

I've written demonstration lesson plans for the teachers I work with as a math coach. These lesson plans include scripts that tell you exactly what you can say when introducing math journals for the first time. These lessons are my approach to the concept, but there are many other ways to do it. Read over the lesson plans and get a feel for how you'd like to adapt it for your class and teaching style:

How do I collect and assess math journals?

Collecting and reading math journals is critical; assessment is far less important. In fact, some teachers don't grade math journals at all, using them only as a record of students' progress and thinking. If students know you are regularly reading their math journals and you care about what they wrote, you won't have to assign a grade to get them to give their full effort.

Here are a number of assessment methods that I like-you can choose one or mix them up throughout the year:

-Walk around while students are writing and verbally confer with individuals. This is my favorite assessment method because it's the easiest on the teacher and probably the most meaningful for students. I stop to talk to kids who are stuck, and comment on what's been written so far to encourage the reluctant writers along. When I see a student drawing about the prompt (our assigned task for then they're done), I ask to read the whole response and then give verbal feedback or ask further questions to help the student improve his or her response.

-Collect only a handful of journals to read after each prompt. For one prompt, you might collect only your lowest students' journals in order to see if they've understood a concept you know the rest of the group mastered. Another time, you might select journals randomly to get a pulse for how the class is doing as a whole and keep everybody on their toes. Or, you could create a schedule in which you read half or a quarter of the class' journals this week and another group of kids' work next week. If you're going to only read a handful of journals, you can either collect them and write written comments during your planning period, or call a few students over to your desk while the others are working and verbally confer.

-Collect the entire class' journals. You can have students pass their journals in (bookmarked or left open to the current page) and write a short comment at the top of the page. Or, have students bring their journals up to you (be sure to have another activity for students to do when they're done so the class isn't playing around-this can be time consuming so make sure you've got your procedures in place). You might have children come to your desk whenever they're done writing or call over one team or group at time and give verbal feedback. I like collecting the entire class' journals every few weeks in the beginning of the year to make sure students are using the correct heading, copying the prompt (or not, depending on my directions), writing to the topic, etc.

-Give a grade on particular prompts and/or a cumulative grade once every few weeks or once a quarter. If you do feel the need to grade journals (and I like doing so every few weeks cumulatively), I encourage you to base the grade on how thorough a student's responses have been. I don't recommend taking off for spelling and grammatical errors if you want students to stay focused on the content and their problem-solving skills. You can give a checklist or rubric to students to make the scoring less subjective, but I've found it's more valuable to have students journal about what grade THEY think their responses have earned, and then simply write a comment and my grade on the top of that page. Keep in mind that the cumulative grade should be IN ADDITION to regular collection of students' journals: if you're only reading them every other month, you won't be able to use students' responses to inform instruction.

Don't feel guilty about not reading every student's journal after every single prompt. Play around with a few different approaches and find something that works well for you. Remember, the main purpose of math journaling is for students to reflect on their thinking, so as long as they're doing that and you're using their writing to help you teach, you're on the right track.

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Truth For Teachers - Everything You Need to Know About Math Journals

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Math journaling is a great way to get insight into your students' thoughts about math and their problem-solving strategies and a wonderful addition to ANY math program from grades K-12. This page will help you set up and manage an easy-to-maintain math journal system in your classroom.

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Supporting students struggling with mathematics

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http://wayback.archive-it.org/org-121/20190122152926/http://www.cehd.umn.edu/ci/rationalnumberproject/rnp1-09.html

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Cramer, K., Behr, M., Post T., Lesh, R., (2009) Rational Number Project: Initial Fraction Ideas

Originally published as
Cramer, K., Behr, M., Post T., Lesh, R., (1997) Rational Number Project: Fraction Lessons for the Middle Grades - Level 1, Kendall/Hunt Publishing Co., Dubuque Iowa.

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Cramer, K., Behr, M., Post T., Lesh, R., (2009) Rational Number Project: Initial Fraction Ideas

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Maths Books/Games/Apps/Wesbites Recommended at Different by Design MANSW 2021 - to investigate further

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Maths Books/Games/Apps/Wesbites Recommended at Different by Design MANSW 2021 - to investigate further

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Solve me mobiles
Mathsisfun Money Masters
CODAP
Times Table Brochure (adapted from Math300)
3,2,1 Maths (similar to scissors, papers, rock but with addition and multiplication)
ESSI Money game
Blooket (refer to Cassie)
MCTP Activity Bank Books by Charles Lovitt & Doug Clark
How I'd wish I taught Maths by Craig Barton
Cognitive Load Theory in Action (Eddie Woo recommendation)
Red Dragonfly
The Learning Power Approach
Tap into Teen minds
Problemo
Aiming high teacher network - Aimsetc
Maths venn
Open middle
inquiry maths
Starting point maths
Dave Stuart Jnr about motivating kids
studymaths.co.uk
Citizen Maths

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Q & A October 20, 2014 with Nalini Joshi

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Reframing feedback to improve teaching and learning

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Mathematics Pathways - University of Tasmania, Australia

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We encounter ideas related to fractions, decimals, ratios and percents on a daily basis. Confident and flexible understanding of these ideas are key to everyday estimation and mental calculation in contexts as diverse as shopping and budgeting, diluting mixtures, understanding scales on maps, interpreting probabilities and odds, and converting among metric units.

Not only is understanding these ideas essential to being numerate they are also foundational to more sophisticated mathematical ideas. If you understand fraction, decimals, ratios and percents you have the basis for understanding topics such as similarity, trigonometry, coordinate geometry, and fancy algebra

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Relationships between Fractions, Decimals, Ratios and Percentages

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We encounter ideas related to fractions, decimals, ratios and percents on a daily basis. Confident and flexible understanding of these ideas are key to everyday estimation and mental calculation in contexts as diverse as shopping and budgeting, diluting mixtures, understanding scales on maps, interpreting probabilities and odds, and converting among metric units.

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Classroom Cognitive and Meta-Cognitive Strategies for Teachers

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Classroom Cognitive and Meta-Cognitive Strategies for Teachers

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Great ideas from Maria Quigley

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Wagga Wagga Conference Geometry 2019 with extras (2)

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Find over 1 million community created, free activities, simulations, exercises, lessons, and games for math & science!

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Follow Making Math Moments Matter on FB for weekly live videos, awesome ideas, and awesome resources!

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Great critical thinking/problem solving/reasoning questions.

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Click on the images to engage in more math.

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Site contains some accessible resources without membership plus a great list of even more useful weblinks.

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This page connects MAV members to some inspirational ideas developed by some of the 'movers and shakers' that the MAV are proud to be in partnership with.

It is place where you can explore curriculum planning and implementation and teaching and assessment resources.

If you have great resources to add to this page let us know at office@mav.vic.edu.au.

MAV Teacher Tip Sheets

Teaching Ideas and Resources
Problemo Math 4 Love
Curriculum and Assessment Resources

Here is some great curriculum planning and assessment support.

Find out more here.

These support pages were produced using Strategic Partnership Program funding from the Department of Education and Training.

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common denominator; early years; family maths

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math association of victoria; mathematical; mathematical association of victoria; mathematical books; mathematical education; mathematical journals; maths revision; maths revision books; maths revision lectures; maths talent quest; maths teacher; mav; mavshop; professional development; professional learning; education consulting; mtq; primary; prime number; publications; quest; professional resources; classroom resources; resources; revision; revision books; revision lectures; school; schools; problem solving; student activities; student lectures; student; teacher; teachers; trial exams; vce; vce examinations; vce exams; vce solutions; vce trial exams; victoria; vinculum; mavcon

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primary conference; annual mathematics education conference; sac; sacs; membership; mav membership; webinar;  common denominator; early years; family maths

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math association of victoria; mathematical; mathematical association of victoria; mathematical books; mathematical education; mathematical journals; maths revision; maths revision books; maths revision lectures; maths talent quest; maths teacher; mav; mavshop; professional development; professional learning; education consulting; mtq; primary; prime number; publications; quest; professional resources; classroom resources; resources; revision; revision books; revision lectures; school; schools; problem solving; student activities; student lectures; student; teacher; teachers; trial exams; vce; vce examinations; vce exams; vce solutions; vce trial exams; victoria; vinculum; mavcon

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primary conference; annual mathematics education conference; sac; sacs; membership; mav membership; webinar; stem

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parent; maths at home; vce revision; parent help; parent tip sheets; primary mathematics resources; enhance maths teaching; authentic maths tasks; maths enrichment; maths picture books; primary maths news; primary maths research; primary mathematics curriculum; rich tasks; teacher support; primary professional learning;  early learning; early years;

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The Mathematical Association of Victoria represents mathematics teachers and educators in primary schools, secondary schools, early learning and at home; We offer professional development and professional learning in and out of schools; MAV sells professional and classroom resources through MAVshop; MAV provides student activities including provide VCE revision lectures; maths camp; games days and maths talent quest; Partners with industry to support advancement of maths education; MAV has an annual mathematics education conference and a primary mathematics education conference;

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Booklet from SA.

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Developed by
Vince Wright, Australian Catholic University
Ann Downton, Australian Catholic University

Developed by based on
Judy Mousley, Deakin University
In Teacheris' Hands, a project of The Mathematical Association of Victoria

Fractions, Mental computation, Patterns, Reasoning, and Statistics were funded by the Australian Government Department of Education, Employment and Workplace Relations. Geometric reasoning was developed by The Australian Association of Mathematics Teachers (AAMT) Inc.

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Developed by Vince Wright, Australian Catholic University Ann Downton, Australian Catholic University Developed by based on Judy Mousley, Deakin University In Teacheris' Hands, a project of The Mathematical Association of Victoria Fractions, Mental computation, Patterns, Reasoning, and Statistics were funded by the Australian Government Department of Education, Employment and Workplace Relations.

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Online manipulatives bring all of the learning and fun without the mess. Our math virtual manipulatives help students grasp abstract concepts through modeling. Literacy virtual manipulatives create a fun space for practicing spelling, rhyming, and more!

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Virtual Manipulatives | Mathematical Modelling | Toy Theater

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Original teaching tools aimed at supporting teachers in the elementary school classroom. Two color counters, 3D dice, probability spinners, graph builders, fraction bars, base ten blocks, and more!

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My favourite formative assessment technique

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Don (UK) died in 2020 of COVID and his family are keeping his amazing website of resources alive for all of us mathematics teachers. His collection of resources are amazing!

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the rules for combining directed numbers with addition and subtraction are probably too burdensome for memory so some awareness needs to be cultivated, over time

some good graphics have been prepared by Shana McKay here
and dropbox versions here
many thanks to her

once some facility with negative number additions and subtractions has been attained, it is desirable to move towards automating (i.e. no need to think too much) these skills

this is an important step in the mathematics curriculum
many difficulties in algebra can sometimes seem more related to a lack of understanding of directed number techniques

my view is that sophisticated directed number calculators recognise types of sum and can apply a technique for that general type

presenting students with a sheet of sums, they can work together on grouping the sums by type, explaining that the result (answer) and numbers involved are not important - it's the signs that need to be focused on (the 'form'):

it may be better that students choose how to group these sums themselves, with plenty of discussion,
but another sheet has them already grouped by types:

students can then be given or find some results
and then consider and create general rules for getting results for each type of sum, for themselves

e.g. for type (c) "you subtract the smaller from the larger and give it the sign of the larger (because this 'wins')"

e.g. for type (e) "two minuses don't make a plus; one story could be: if I take £7 from you and then take £12 from you, I've taken £19 from you altogether"

they can also appreciate, after a while, that types can be 'collapsed' to other types
e.g. (e) to (b) and (g) to (a)

and here is a test!

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the rules for combining directed numbers with addition and subtraction are probably too burdensome for memory so some awareness needs to be cultivated, over time some good graphics have been prepared by Shana McKay here and dropbox versions here many thanks to her once some facility with negative number additions and subtractions has been attained, it is desirable to move towards automating (i.e.

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Online number line app

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Number Line helps students visualize number sequences and demonstrate strategies for counting, comparing, adding, subtracting, multiplying, and dividing.

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Great online resource with virtual manipulatives

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National Library of Virtual Manipulatives

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A digital library containing Java applets and activities for K-12 mathematics

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Good quality primary and middle school mathematics resources

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Welcome to Anita's awesome FREE resources! There are two ways to find what you are looking for.

1. Filter by one or more subcategories within each category.

2. In the search box, enter an Australian Curriculum Content Description (eg ACMNA057) or NSW Syllabus Outcome (eg MA2-6NA).

Wait a few moments for the list of resources below to update ... ta-da!
Click on any image in the results list to either open the downloadable PDF, or open a larger photo (for equipment pics).

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Welcome to Anita's awesome FREE resources! There are two ways to find what you are looking for. Filter by one or more subcategories within each category. In the search box, enter an Australian Curriculum Content Description (eg ACMNA057) or NSW Syllabus Outcome (eg MA2-6NA).

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Algebra Tiles - virtual or concrete manipulatives

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Algebra Tiles - virtual or concrete manipulatives

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How to use algebra tiles to teach factorisation.

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I started a series of new warm ups for my MPM2D class today. My goal is to sneak in factoring as warmups throughout the semester. By the time we need to learn it (like when we need to factor to sol...

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Jo Boaler - great place to find professional reading and rich tasks.

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At youcubed, we inspire, educate and empower teachers of mathematics, transforming the latest research on mathematics learning into accessible and practical forms.

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by Anita Chin

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Tick the boxes below to search by category, or type a curriculum code into the field underneath. Then, to download the resource, simply click on the image!

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Virtual manipulatives for Geometry

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A site has many rich tasks, games, puzzles and problem solving activities, both online and printed resources.  Site also contains professional development reading for mathematics teachers.

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search

Enriching mathematics for all learners

A free-to-use collection of mathematics activities, lessons and problems designed to nurture curious, resourceful and confident learners of mathematics. Produced by The University of Cambridge.

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The Nrich Maths Project Cambridge,England. Mathematics resources for children,parents and teachers to enrich learning. Problems,children's solutions,interactivities,games,articles.

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Usecubes is the fun app that allows you to build 3d pixel arts(voxel arts) on your iPhone, iPad and computer. Simply select cubes, then drag, draw and paint them to build a 3d model of anything you want.

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像素屋，充满乐趣的3D方块建模网站

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Just drag and draw, start modeling and create your 3d pixel arts / voxel arts online. And then use blocks to build the creative out.

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Excellent lesson suggestions, including multiple lessons, activity suggestions linked to the Australian curriculum.  Site also contains professional development reading for mathematics teachers.

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Resources for teaching STEM to students.

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Which one doesn’t belong? This promotes Working Mathematically, specifically communication and reasoning.

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This is Which One Doesn't Belong?, a website dedicated to providing thought-provoking puzzles for math teachers and students alike. There are no answers provided as there are many different, correct ways of choosing which one doesn't belong. Enjoy!

Which One Doesn't Belong?

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This is Which One Doesn't Belong?, a website dedicated to providing thought-provoking puzzles for math teachers and students alike. There are no answers provided as there are many different, correct ways of choosing which one doesn't belong. Enjoy!

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There are some activities on here for free. My favourite is the Percentage Bar Model one. There are lots of other ones.

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Converting Fractions, Decimals & Percentages

Percentages of an Amount

Percentage Change
6. Interactive Student Tasks / IWB Material*
Reverse Percentages (Flipped)
10. Interactive Student Tasks / IWB Materials* Calculating Percentages 7. Interactive Student Tasks / IWB Materials* FDP

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6. Interactive Student Tasks / IWB Material* Reverse Percentages (Flipped) 10. Interactive Student Tasks / IWB Materials* Calculating Percentages 7. Interactive Student Tasks / IWB Materials* FDP

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Also makes Flashcards

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Fraction Worksheets | Free - Distance Learning, worksheets and more: CommonCoreSheets

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The best source for free fraction worksheets. Easier to grade, more in-depth and best of all... 100% FREE! Kindergarten, 1st Grade, 2nd Grade, 3rd Grade, 4th Grade, 5th Grade and more!

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Get to know your students in an interesting way. Article title – Enhancing the development of statistical literacy through the robot bioglyph. Article discusses numeracy benefit of this activity.

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Rates, ratios and proportional reasoning plus students encounter various different units of measurement and need to use estimation (no exact answer) – this site has some interesting questions using real life data that can be answered by individuals or groups.

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A garden snail's average speed is about ​0.029 mi/h. The North American X-15's top speed is about 4,520 mi/h. How much time would it take for each to travel across the United States, a...

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A collection of sites for maths teachers made by a maths teacher.

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Grid paper (various sizes), isometric paper, graph paper, circular paper……handy link to print off

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Interactivate: Equivalent Fractions Finder

Equivalent Fractions Finder: Visually represent two unique fractions that are equivalent to a given fraction. The fractional value is shown on a number line after you check to see if your fraction is correct. Equivalent Fraction Finder is one of the Interactivate assessment explorers.

On a mission to transform learning through computational thinking, Shodor is dedicated to the reform and improvement of mathematics and science education through student enrichment, faculty enhancement, and interactive curriculum development at all levels. Student development of numerical models and simulations integrated with core curriculum provides an opportunity to gain practical experience in computational science. Additionally, the National Computational Science Institute (NCSI) provides nation-wide workshops portraying resources and instructional ideas to middle school, high school, and undergraduate instructors for use in the classroom. Resources and materials offered to these instructors are available free of charge from Shodor's website and are largely developed by Shodor student interns. Shodor's academic program efficiently guides participants from excitement to experience to expertise through computational explorations, research opportunities, and service.

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Interactivate: Equivalent Fractions Finder On a mission to transform learning through computational thinking, Shodor is dedicated to the reform and improvement of mathematics and science education through student enrichment, faculty enhancement, and interactive curriculum development at all levels. Student development of numerical models and simulations integrated with core curriculum provides an opportunity to gain practical experience in computational science.

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Australian Professional Standards for Teachers

The Australian Professional Standards for Teachers help you understand and develop your teaching practice and expertise across 4 career stages. By demonstrating the Standards you can have maximum impact on all learners.

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australian professional standards for teachers

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Australian Professional Standards for Teachers

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Mr Mathematics website with lots of lesson ideas 

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Mr Mathematics website with lots of lesson ideas 

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Requires subscription. UK curriculum outcomes. Lesson ideas/slides including background information on misconceptions and learning intentions.

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Percentages for Foundation GCSE mathematics scheme of work.. How to teach calculating a simple and compound percentage changes using calculators.

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Includes differentiated teaching activities for students identified from pre-test. It also contains a teachers manual.. Pre-test identifies specific misconceptions and levels. https://www.pearson.com.au/insights-and-news/supporting-educators/breakthrough-in-secondary-mathematics-education-for-graduate-and-out-of-field-teachers/

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How to use with ICT

In the simplest form, the teacher poses a question and students respond by displaying their device scree with their chosen card from one of the templates below.

As an extension of this, there are plugins that add live polling to both Microsoft PowerPoint and Google Slides presentations. All of these will require students to have ready access to a mobile device or laptop with internet access. In Google G Suite, you can use a form to collect data, use a spreadsheet to collate the data, and then make a live chart that can be inserted into a presentation. The chart will update as students respond to the question.

Single computer

If using presentation software in class, include an ABCD question slide. Plickers requires the teacher to have a mobile device with a camera and students need a set of printed cards, however, it collates answers to prearranged questions quickly. Websites such as Kahoot and Mentimeter template are engaging ways to pose questions and record live results.

External resources

Disclaimer
Links to third-party websites:
If you use the links provided on this website to access a third party's website, you acknowledge that the terms of use, including licence terms set out on the third party's website apply to the use which may be made of the materials on that third party's website or where permitted by the Copyright Act 1968 (Cth).

The department accepts no responsibility for content on third-party websites.

Personalised learning

Disability, Learning and Support

When planning to use technology in the classroom it is important to consider the diversity of your learners. Universal Design for Learning (UDL) is a framework to guide the design of learning environments that are accessible and effective for all. For UDL guidelines, information and additional materials, visit the CAST website.

Many students require technology as an adjustment to support their access to learning. Adjustments (NESA) are actions taken that enable a student with disability and additional learning needs to access syllabus outcomes and content on the same basis as their peers. Enrol in the Personalised learning with technology online course to help you make more informed decisions regarding technology.

For a range of simple, how-to videos visit the Assistive Technology page on the Disability, Learning and Support website. Resources are organised into four sections; Literacy and Learning, Vision, Hearing, Physical and Motor Skills.

High potential and gifted learning and support

When planning to use technology in the classroom it is important to consider the full range of abilities of all learners. High potential and gifted learners may require additional adjustments and deliberate talent development. These strategies include differentiation, grouping, enrichment and advanced learning pathways so students can be engaged, grow and achieve their personal best.

Assessing and identifying high potential and gifted learners will help teachers decide which students may benefit from extension and additional challenge. Effective strategies and contributors to achievement for high potential and gifted learners helps teachers to identify and target areas for growth and improvement. School leaders can access the Evaluation and Planning Tool to support strategic improvement planning.

Recognising the diversity of high potential and gifted students represented in classrooms across 4 domains of potential can be explored further by accessing illustrations of practice.

For further support and advice about how to tailor learning for high potential and gifted students from all backgrounds, visit the High Potential and Gifted Education web section, High Potential and Gifted Education Policy or attend one of the professional learning courses on offer.

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State of New South Wales, Department of Education, Australia

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ABCD cards can be established as a way for students to give quick, silent, individual responses to a multiple-choice question posed during class.

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Create quizzes for diagnostic assessments

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Don't spend hours marking books

Diagnostic Questions is a world leading assessment tool that provides detailed insights into student understanding in a fraction of the time, simultaneously reducing your workload and improving student results and confidence.

25,000 Questions & Quizzes

Create your own or choose from the thousands created by teachers, awarding bodies and publishers.

Learn more

Actionable Insights

Use our powerful Data Analytics, Feedback and Planning tools to understand your students better than ever.

Learn more

Free forever

We're in the business of helping teachers. We know how tight departmental budgets are, we know the pain of change and we know how frustrating it is when a product you've come to depend upon starts charging. Well, we're giving you, the teacher in the classroom, a promise that Diagnostic Questions will always remain free. It's a small gesture, but we hope it helps.

Supported Subjects

Expected in November 2016, the ultimate GCSE Computing companion, with Scheme of Work and 1000s of questions from OCR and teachers.

750 questions from the Royal Society of Chemistry plus 1000s of others from teachers across the UK covering, Biology, Chemistry and Physics.

We want to support the most popular subjects that you teach. Help us focus on subjects that will make a difference by voting for our next subject.

Resolving misconceptions improves performance

A continuous system of misconception identification provides teachers with the foresight they need to intervene effectively in students' learning.

"Good diagnostic questions encapsulate our best information about what students find difficult, and are an indispensable part of any teacher's repertoire."
Dr. Dylan Wiliam, Emeritus Professor of Educational Assessment @UCL

"...the single greatest step forward in AFL in the maths classroom for years. I think it would be hard to find any better AFL strategy out there."
William Emeny, greatmathsteachingideas.com

"...good questions help us know more about our students' learning and help us to address misconceptions. An outstanding new resource to help us do just that."
Colleen Young, colleenyoung.wordpress.com

"Discover key misconceptions before you teach, track progress, review students explanations and quickly give personalised feedback."
Mohammed Ladak, mathedup.co.uk

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Better results, more confidence, less time

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Accessible to all students. Every option has a reason why they don't belong.

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This is Which One Doesn't Belong?, a website dedicated to providing thought-provoking puzzles for math teachers and students alike. There are no answers provided as there are many different, correct ways of choosing which one doesn't belong. Enjoy!

Which One Doesn't Belong?

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This is Which One Doesn't Belong?, a website dedicated to providing thought-provoking puzzles for math teachers and students alike. There are no answers provided as there are many different, correct ways of choosing which one doesn't belong. Enjoy!

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Differentiation Adjustment Strategies by DoE

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Differentiation Adjustment Strategies by DoE

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Use differentiation adjustments, strategies and examples to cater for the needs of high potential and gifted students.

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Equations activity - Fun and engaging for kids. 

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Equations activity - Fun and engaging for kids. 

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Students work up levels, developing their ability to solve equations using the balancing method.

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Complete the lesson as noted, or use it as stimulus for your own. Lots of mathematical riddles, just search riddles. Great rich, open ended task inspiration from this site.

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TED-Ed lessons on all subjects. TED-Ed celebrates the ideas of teachers and students around the world. Discover hundreds of animated lessons, create customized lessons, and share your big ideas.

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The Mathematics Assessment Project is part of the Math Design Collaborative initiated by the Bill & Melinda Gates Foundation. The project set out to design and develop well-engineered tools for formative and summative assessment that expose students' mathematical knowledge and reasoning, helping teachers guide them towards improvement and monitor progress. The tools are relevant to any curriculum that seeks to deepen students' understanding of mathematical concepts and develop their ability to apply that knowledge to non-routine problems.

More about the Math Assessment Project

100 lessons for formative assessment, some focused on developing math concepts, others on solving non-routine problems. A Brief Guide for teachers and administrators (PDF) is recommended reading before using these lessons for the first time.

A set of 94 exemplar summative assessment tasks to illustrate the range of performance goals required by CCSSM. The tasks come with scoring rubrics and examples of scored student work.

Complete summative test forms and rubrics designed to help teachers and students monitor their progress using a range of task types similar to the 'Tasks' section.

5 Prototype modules that encourage groups of teachers to explore the practical and pedagogical concepts behind the materials, such as formative assessment, collaborative learning and the use of unstructured problems.

The Teaching for Robust Understanding of Mathematics (TRU Math) suite is a set of tools with applications in Professional Development and research based around a framework for characterizing powerful learning environments.

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The Mathematics Assessment Project is part of the Math Design Collaborative initiated by the Bill & Melinda Gates Foundation. The project set out to design and develop well-engineered tools for formative and summative assessment that expose students' mathematical knowledge and reasoning, helping teachers guide them towards improvement and monitor progress.

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LaTeX - resources. Make exam writing easier

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LaTeX - resources. Make exam writing easier

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https://www.overleaf.com/project
http://mathquill.com/
https://www.dropbox.com/s/cqo5x805yjdhbg7/LaTeX_for_Teaching.pdf?dl=0

For Windows:
https://pctex.com ( not free )
https://miktex.org (free)
For mac:
https://www.tug.org/mactex/ (free)

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An online LaTeX editor that's easy to use. No installation, real-time collaboration, version control, hundreds of LaTeX templates, and more.

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Welcome to 52 weeks of math! The list of activities is below, or there is a blog view here. Enjoy! Week 1 - Embroider Curves with Lines Week 2 - Spiral of Theodorus Week 3 - Sonobe Origami Octahedron Week 4 - Polyhedral Study Week 5 - Tell a Tale (or Comic) Week 6 -...

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State of New South Wales, Department of Education, Australia

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The Digital learning selector helps you make fast decisions about a technology you can use for learning with your class. Find support to get started, contribute new tools and share them with your colleagues.

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Choose a topic that is of interest to students. It might be about sport, travel, or music. [Image © Shutterstock/Neale Cousland]

In Mathematical concepts - context is key I argued that context can provide the challenge, the motivation and the purpose for understanding and using mathematics, and that one approach was to use a problem solving, investigative, open-ended approach-one which uses real materials and situations, and identifies the connections between maths and the real world.

However, teaching mathematics using a problem solving approach, especially one that is based around making connections with the real world, can be demanding-certainly harder than teaching out of a textbook. It requires more planning and preparation in advance if it is going to work successfully.

So, how can you plan what to do and how to link maths with the real world? And what can you do in the classroom and out?

The following flow chart maps out one possible process to use for negotiating and planning what to teach that supports this approach. This process is often best done in collaboration with other teachers - it makes it more creative and you can share the development time.

The first and crucial task is to decide what the theme or context is that will be the topic for students to investigate and work on.

Importantly, it should be of interest to the students. It could be a current popular topic, or an issue of concern. It might be about sports, food, travel, cars, holidays, fashion or music. However, it is important to remember that one topic may not be of interest to all students in the group, so a few different options can help.

It can also help if students are involved in the decision making process, so that they can feel ownership in some way. There are several ways to involve students: discussing ideas with them, suggesting topics, brainstorming with them, working in small groups to develop ideas, getting them to write down ideas on cards, and much more.
Image © Shutterstock/Monkey Business Images
Talking with other teachers who teach the same group of students will give you an idea of what the students might be working on in their other classes. If topics are chosen that are related to what students are learning in their other subjects, this provides a cross-curricular approach to teaching and the curriculum.

As teachers, we need to remember that negotiation is a two way process-not just one way from the students' perspective. The teacher is there as an educational leader and has a vital role to play with their knowledge and expertise about what topics are appropriate, what investigations are achievable and manageable, and what mathematics needs to be covered to meet curriculum guidelines.
Stay tuned: In following articles I will illustrate the different stages of the planning process in more detail with examples. References
Tout, D. and Motteram, G. (2006). Foundation numeracy in context. Melbourne: Australian Council for Educational Research (ACER)

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Planning for context-based teaching - Teacher Magazine

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In Mathematical concepts - context is key I argued that context can provide the challenge, the motivation and the purpose for understanding and using mathematics, and that one approach was to use a problem solving, investigative, open-ended approach-one which uses real materials and situations, and identifies the connections between maths and the real world.

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Geoboard, fractions, number lines....

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Sharing in an app allows educators to pose problems and investigations for students by using an 8-character code or link. Students can share their thinking with their teachers and classmates. Share codes remain available for continued use for 18 months from the date they are last accessed.
Watch the video to learn how sharing works.

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Sharing in an app allows educators to pose problems and investigations for students by using an 8-character code or link. Students can share their thinking with their teachers and classmates. Share codes remain available for continued use for 18 months from the date they are last accessed.

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Mathematics Pathways - University of Tasmania, Australia

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All two-dimensional (2D) and three-dimensional (3D) objects can be classified according to their specific properties. Understanding these shapes and their properties provides the tools to help us to represent and visualise our world.

Reasoning with geometric ideas is deductive in nature. There is a logical connection among the ideas.

Geometry can be used to model the world around us. Shapes and objects can be manipulated and transformed using geometric rules and tools.

Please fill out the Research Survey after completing this module, provide your feedback to help improve the pathway.

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All two-dimensional (2D) and three-dimensional (3D) objects can be classified according to their specific properties. Understanding these shapes and their properties provides the tools to help us to represent and visualise our world. Reasoning with geometric ideas is deductive in nature. There is a logical connection among the ideas.

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Learning intentions and success criteria

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Learning intentions and success criteria

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This website will help in learning how to write learning intentions and success criteria in mathematics.

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The Toolkits

Our Teaching and Learning Toolkit (the Toolkit) and the Early Childhood Education Toolkit (ECE Toolkit) are accessible summaries of education research. The two Toolkits are designed to support teachers, school leaders and early childhood education educators in making evidence-informed decisions about how to improve learning outcomes, particularly for children and young people from disadvantaged settings.

The Toolkits do not make definitive claims as to what will work to improve outcomes in a given setting. Rather they provide high quality information about what is likely to be beneficial based on existing evidence.

The Toolkits aim to:

support evidence-informed decision making in Australian schools and early childhood education settings;

provide guidance for school leaders and teachers, and early childhood education educators on how to use their resources to improve educational outcomes for their students, particularly those surrounded by disadvantage;

act as an introduction to educational research.

The Toolkits synthesise international and Australian research, and presents a wide range of educational interventions, summarised in terms of:

1. the average months' worth of learning progress;

2. the strength of the evidence;

3. the cost.

The Toolkits are live resources updated regularly as new studies are published in Australia and internationally.

Watch this video on how the Toolkits are designed to support educators who are making decisions about approaches they're considering implementing in their schools to improve learning outcomes, particularly for students from settings surrounded by disadvantage. (11:43).

Read more about the Teaching & Learning Toolkit and how you can use it (1 MB, PDF).

Read more about the Early Childhood Education Tookit (726 KB, PDF)

' In my role, I spend a great deal of time collecting, reading and unpacking research around teaching practice and approaches with our teachers. The [Teaching & Learning] Toolkit is neat, concise and easy to use. It will save me countless hours in the way it lights the path directly to the most relevant and reliable research.' - Frances Roberts,Head of Curriculum, Bounty Boulevard State School after attending a workshop led by Evidence for Learning's Dr Tanya Vaughan.

Months' impact

Months' impact is estimated in terms of the additional months' progress you can likely expect children and young people to make as a result of implementing an approach, compared to similar children and young people who did not receive the approach. The months' impact takes the average progress over a year as a benchmark.

For example, the ' feedback ' strand in the Teaching & Learning Toolkit shows that improving the quality of feedback provided to students has an average impact of eight months. This means that students in a class where high quality feedback is provided can be expected to make, on average, eight months more progress over the course of a year compared to another class of students who are performing at the same level at the start of the year.
Effect sizes
These impact estimations are based on 'effect sizes' reported in comparative data (see table below). Effect sizes are quantitative measures of the impact of different approaches on learning. The Toolkits prioritise effect sizes derived from systematic reviews of research and quantitative syntheses of data such as meta-analyses of experimental studies. To be included in the analysis an approach needs to have some quantifiable evidence base for comparison.

Effect sizes describe the size of the difference between two groups in a standard and comparable way. However, it can be difficult to understand what a given effect size actually means for the progress of children and young people. That is why the Toolkits translates effect sizes into the months' progress measure, as shown in the table below.

Cost

Cost estimations in the Toolkits are based on the approximate cost of implementing an approach in a class of 25 students. Cost estimates commonly include the cost of additional resources, the cost of training or professional development or the cost of activities for students.

Evidence security

Evidence security 'padlock' ratings provide an overall estimate of the robustness of the evidence. It is based on:

the quantity of evidence available (i.e. the number of systematic reviews or meta-analyses and the number of primary studies which they synthesise);

the outcomes measured in those studies

the methodological quality of the available evidence;

the consistency of impact estimates across the reviews and meta-analyses that have been synthesised.

No evidence reviews available, only individual research studies.

At least one evidence review. Reviews include studies with relevant outcomes, and studies with methods which enable researchers to draw weak conclusions about impact.

At least two evidence reviews. Reviews include studies with relevant outcomes, and studies with methods and analysis which enable researchers to draw moderate conclusions about impact.

At least three evidence reviews. Reviews include studies with highly relevant outcomes, and studies with methods and analysis which enable researchers to draw strong conclusions about impact. Impact estimates are broadly consistent across studies.

At least five evidence reviews. Reviews are recent, and include studies with highly relevant outcomes, and studies with methods and analysis which enable researchers to draw strong conclusions about impact. Impact estimates are consistent across studies.

Who wrote the Toolkits

The Teaching & Learning Toolkit was developed by two charities based in the UK, the Education Endowment Foundation (EEF) and the Sutton Trust, in collaboration with academics at Durham University. In 2015, the Australian Teaching & Learning Toolkit was launched, with support from the Department of Education and Training in Victoria, Social Ventures Australia (SVA) and a number of other organisations in Australia. The Australasian Research Summaries for approaches in the Teaching & Learning Toolkit were produced in collaboration with Melbourne Graduate School of Education and launched in 2016.

The Early Childhood Education Toolkit was produced by the EEF in collaboration with a team of academics at Durham University, led by Professor Steve Higgins. The Australasian Research Summaries for the 12 approaches in the Early Childhood Education Toolkit were produced in collaboration with Telethon Kids. In 2019, the Early Childhood Education Toolkit was launched in Australia with support from The Bryan Foundation. In 2019, the Early Childhood Education Toolkit was launched with support from The Bryan Family Foundation.
The full references for the Toolkits are:
The Teaching & Learning Toolkit

Education Endowment Foundation. (2019). Evidence for Learning Teaching & Learning Toolkit: Education Endowment Foundation. Retrieved from https://evidenceforlearning.org.au/the-toolkits/the-teaching-and-learning-toolkit/

Evidence for Learning in collaboration with Melbourne Graduate School of Education. (2019). Australasian Research Summaries. Retrieved from https://evidenceforlearning.org.au/the-toolkits/the-teaching-and-learning-toolkit/australasian-research-summaries/

The Early Childhood Education Toolkit

Education Endowment Foundation. (2019). Evidence for Learning Early Childhood Education Toolkit: Education Endowment Foundation. Retrieved from https://evidenceforlearning.org.au/the-toolkits/early-childhood-education-toolkit/

Evidence for Learning in collaboration with Telethon Kids. (2019). Australasian Research Summaries. Retrieved from https://evidenceforlearning.org.au/the-toolkits/early-childhood-education-toolkit/australasian-research-summaries/

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Our Teaching and Learning Toolkit (the Toolkit) and the Early Childhood Education Toolkit (ECE Toolkit) are accessible summaries of education research. The two Toolkits are designed to support teachers, school leaders and early childhood education educators in making evidence-informed decisions about how to improve learning outcomes, particularly for children and young people from disadvantaged settings.

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Start with learning area content that aligns with students' chronological age to:

plan age-appropriate teaching and learning programs

develop rigorous programs that challenge and engage all students

ensure that all students progress.

CASE: Abilities

Adjust the teaching and learning programs and classroom instruction to:

draw on students' interests and abilities to personalise age-equivalent learning area content

use the general capabilities and/or cross-curriculum priorities to adjust the learning focus

align age-equivalent learning area content with individual learning goals.

CASE: Standards

Assess student progress in relation to achievement standards and individual learning goals.

Note: Requirements for assessment and reporting may differ across states and territories.

CASE: Evaluation

Evaluate actions taken to personalise student learning. Teachers review these actions in consultation with the student, their family/carers and other professionals, to evaluate and respond to the impact of personalised learning strategies. Evaluation of these strategies acknowledges successes and determines whether alternate personalised learning strategies are required to ensure that the student progresses.

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Start with learning area content that aligns with students' chronological age to: plan age-appropriate teaching and learning programs develop rigorous programs that challenge and engage all students ensure that all students progress.

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Reflective practice questions to ask yourself.

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Reflective practice resource for all NSW DoE teachers with useful and targeted questions when reflecting on teaching practice.

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A lot of college level teacher training programs talk about the importance of developing a "reflective practitioner" but what exactly does this mean for the classroom teacher? If you haven't yet accepted responsibility for student learning than it doesn't mean much at all. You'll have some success as a teacher and some failures which you'll most certainly blame on any number of factors. If however you believe that all students can learn at a high level and that your performance as a teacher has a direct impact on student learning, than reflection should be an integral part of what you do.

Below you will find a list of 30 Questions Teachers should be regularly asking themselves to ensure they're classroom is as focused on learning as they would like it to be. Feel free to beg, borrow, steal and share however you see fit.

Modeling Reflection - Questions to Ask With Students

1.Was this activity successful....why or why not?

2.If we do this again, what can I do differently to help you learn more?

3.Did this activity help you learn more than others we've done? Why?

Classroom Culture - Questions to Ask About Your Rules & Relationships

4.Are the relationships that I have with my students helping or hindering their ability to learn?

5.Could the problems I have in my classroom be solved by pre-teaching my expectations or developing rules/procedures to deal with these issues?

6.Was my demeanor and attitude towards my class today effective for student learning?

7.Am I excited to go to work today?

8.Are my students excited to come to my class today? (How much does #6 impact #7?)

9.What choices have I given my students lately?

10.Can I explain at least SOMETHING about each of my student's personal lives?

Curriculum and Instruction - Reflection on Assessment and Grading Practices

11.Does my gradebook accurately reflect student learning?

12.Do my assessments really reflect learning, or merely task completion or memorization skills?

13.Why did I REALLY choose this particular lesson to cover this objective?

14.What evidence do I have my students are learning?

15.What new strategies have I tried lately that might benefit a student I am struggling with?

16.In what ways am I challenging students who are clearly being successful in my classroom?

17.What do I do when students aren't learning in my classroom?

18.Which students benefited from this activity?

19.Which students did not benefit from this activity?

Collaboration - Questions to Ask Ourselves About Our Place in a Professional Learning Community

20.In what areas can I still improve professionally?

21.What's stopping me from improving in these areas?

22.In what ways can I support my colleagues in their student's learning?

23.Do my actions as a teacher show my belief that all students can learn at a high level?

24.Do my actions as a teacher show that I take pride in my work?

25.Are the relationships I have with my colleagues conducive to creating a collaborative culture focused on learning?

26.Are the relationships I have with my student's parents conducive to improving learning?

Mental Health - Questions to Help Teachers Maintain a Healthy Outlook

27.What new ideas have I tried in my classroom lately to keep myself energized about teaching?

28.What have I done lately to relieve stress and focus on my own mental health, to ensure I remain an effective teacher?

29.What things am I currently doing that I could realistically make less of a priority in my profession?

30.How much time have I spent with my friends and family in the last two weeks?
All through with this piece? Consider checking out...35 Questions for Student Reflection The Importance of Reflection in Education Using Reflection to Help Students Learn One Way to Handle Defiant Students

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30 Questions For Teacher Reflection - Edunators

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Edunators is a website dedicated to helping teachers overcome the obstacles that prevent them from focusing on learning in the classroom. We take pride in oversimplifying the complex and discussing some of Education's biggest ideas in a more conversational tone.

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Useful to help unpack the AITSL Teaching Standards

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Pythagoras, slope,....

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One day, while watching a student fiddling with a miniature Tech Deck skateboard before class, Middle School math teacher Clothilde Labrousse had a moment of inspiration. Her class was studying slope and the Pythagorean theorem, and she thought that a skate ramp could be a perfect representation of both concepts in the real world.

She transformed this seed of an idea into an engaging project where students designed and built their own miniature skate parks. The project allowed students to utilize three concepts they had been studying - slope, proportion, and the Pythagorean theorem - and to see how the concepts played out in the physical world.

The project was also a great example of the conceptual learning approach that INTL* emphasizes in Middle School mathematics. Instead of simply memorizing formulas and applying them in practice problems, students are expected to think critically and internalize the deeper meaning of a mathematical concept, as well as understand how it relates to other concepts they have already studied.

For the skate park project, students had full creative license to come up with a blueprint design for their park's layout. They were required to base the slope of each ramp off of the slope of a real street in San Francisco, to help them better understand what slope looks like and how it functions in the real world. They used linear functions to graph the slopes, and they used the Pythagorean theorem to determine measurements for each ramp. Next, they used proportions to figure out how to transfer their blueprint on paper into measurements for the foam core board they used as their building material. The final step was construction, which tested students' precision and spatial skills. By seeing for themselves how quickly a skateboard would accelerate down ramps of different steepness in their finished parks, students could better internalize the concepts of rise, run, and slope.

This conceptual approach is part of the MYP (Middle Years Program)* that we are adopting in the Middle School, which encourages students to think critically, inquire, and ask questions in order to understand on a deeper level. At INTL, middle school students currently have 6 hours dedicated to math each week, while most of their peers have only 4. This extra time allows students to both master the skills and go more in-depth with the concepts.

Miss Labrousse explains, "Conceptual learning takes longer, but is a much deeper and more effective way of learning. It creates much stronger thinkers and mathematicians, and builds a very strong foundation as students continue to advance in high school."

By the time students graduate from the Middle School, they will have already effectively completed Pre-Algebra, Algebra, and Geometry. INTL graduates will have also thought critically and examined mathematical concepts in a much deeper way. Hopefully, they will have developed a passion for learning math - spurred-on, no doubt, by engaging and creative projects like building their own miniature skate park.

*In 2020, the International School of the Peninsula (ISTP) formally changed its name to Silicon Valley International School (INTL) to better reflect its bilingual programs, location, and international values.

*Only schools authorized by the IB Organization can offer any of its four academic programmes: the Primary Years Programme (PYP), the Middle Years Programme (MYP), the Diploma Programme or the Career-related Programme (CP). Candidate status gives no guarantee that authorization will be granted. For further information about the IB and its programmes, visit http://www.ibo.org

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Skate Park Geometry - Learning Math Conceptually in the International Middle School

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One day, while watching a student fiddling with a miniature Tech Deck skateboard before class, Middle School math teacher Clothilde Labrousse had a moment of inspiration. Her class was studying slope and the Pythagorean theorem, and she thought that a skate ramp could be a perfect representation of both concepts in the real world.

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Love Maths offers a number of services to schools- classroom modelling, professional development days, planning effectively for maths and after-school professional development sessions. For information about how to get in touch follow the contact link.

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Flatback turtle population lesson very good for number and algebra strand, as well as statistics and reading and interpreting graphs, plus other great lessons for inspiration. Used successfully in Numeracy CEC in year 11. 

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Flatback turtle population lesson very good for number and algebra strand, as well as statistics and reading and interpreting graphs, plus other great lessons for inspiration. Used successfully in Numeracy CEC in year 11. 

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page lessons used gender specifically girls written focus stem lesson

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In this series of lessons, students are set a design challenge to create a program to automatically switch on an air conditioner. For this project, students are introduced to the Arduino microcontroller and Arduino integrated development environment for programming. However, other similar microcontrollers could be used instead following a similar process.

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Youtube video explaining the ratio's used in Minecraft when Crafting.

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Math Is Visual: Let's Teach It That Way | Free Visual Math Talk Prompts

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Access free Visual Math Talk Prompts to build fluency and flexibility with number sense and operations for your K-12 Math Class or at Home.

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A website to inspire creativity in maths

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Practice Fractions with Line and Circle Models

Many of the worksheets will let learners shade number lines and circles to illustrate fraction concepts. The circle and number line images on the following worksheets were made with the Fraction Designer pages that can be found on this web site.

These worksheets are pdf documents that may be reproduced for your own use and for your student's use.

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Practice Fractions with Line and Circle Models Many of the worksheets will let learners shade number lines and circles to illustrate fraction concepts. The circle and number line images on the following worksheets were made with the Fraction Designer pages that can be found on this web site.

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Benjamin Odgers Youtube Video Playlists - includes accompanying worksheets.

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Benjamin Odgers Youtube Video Playlists - includes accompanying worksheets.

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mathematics maths online "online maths" "online mathematics" "maths tutor" courses worksheets "work booklets" "new south wales" "maths teacher" nsw syllabus ...

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Hi, I am a maths teacher in NSW, Australia. If you check out my playlists you will find courses based on the Australian Curriculum. I also make work booklets with questions and examples linked to my YouTube videos. Read the description below each video to find links to my work booklets.

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Great scaffolded worksheets with faded examples, see Fill in the Blanks. Also code breakers.

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Maths Teaching Resources | Dr Austin Maths

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Dr Austin Maths is a free-to-use Maths teaching resources website, focussing on quality resources for KS4 IGCSE and GCSE Maths.

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Area Mazes - Why did the rectangle go to the gym?

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Area Mazes - Why did the rectangle go to the gym?

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Presented by Miriam Lees - MANSW2022 conference. Engaging and fun.

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(Skip to the resources by clicking here)

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Simultaneous Equations.....and lots more math, using the Colour Cube. 

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Simultaneous Equations.....and lots more math, using the Colour Cube. 

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The Color Math Concept
by Ken Davies

The human eye is able to decipher patterns of light according to the primary colors of the additive system: red, green and blue (RGB). However, it is the subtractive system's primaries: cyan, magenta and yellow (CMY) that best lend themselves to understanding the COLORCUBE and the concept of Color Math. This chapter unveils the inner workings of the COLORCUBE Model and the tools that are required for navigation in and about it.

Within any given subtractive color space, as defined by the three primaries CMY, the use of Color Math will allow us to map the relationships of all the colors encompassed by that cubic area. Bridging the additive and subtractive systems of color, mixing colors, selecting color complements and converting color media equivalents all become matters of mathematics rather than that of guess work or compromise.

Following a brief introduction to basic logic and scientific fundamentals, as they are relevant to color, we will expand upon the Color Math concept. By the conclusion of this chapter, we will be able to apply Color Math to a variety of tasks such as dissecting the COLORCUBE, dispelling critical aspects of traditional color theory, mixing colors, determining color complements and charting print-to-paint media conversions.

Principles of Color Math

The similarity to commonly applied basics in mathematics and algebra will make these initial color principles appear somewhat elementary. However, their value will become apparent as the problems that we will encounter become increasingly complex.

The diagram below illustrates the principle of symmetry that states that the order in which colors are added to one another does not alter the outcome.

Figure 1. Color A + Color B = Color B + Color A

The addition (or subtraction) of two or more colors will likely cause a visible change in hue but we must also pay particular attention to the cumulative volume of the operation. For example, mixing one measurable unit of color with another yields twice the volume of the resulting color. The following diagram highlights this change in quantity using simple, like colors.

Figure 2(a). 1 part Color A + 1 part Color A = 2 parts Color A

Also, this principle of cumulative volume applies to those operations involving varying colors as well.

Figure 2(b). 1 Color A + 1 Color B = 2 Color C

As easy as these fundamentals are to grasp, the visual aids accompanying the upcoming problems shown may be counter-intuitive. The derivations of the Color Math "building blocks" presented in the next section will be the first step in making them fully comprehensible.

Primary Colors

We have stated that all of the colors captured within a color space are functions of the primaries cyan, magenta and yellow. The vertices or corner points on a COLORCUBE are these three pure hues (labeled A, B and C below), along with their various combinations. Together, these eight colors represent the key elements used in all Color Math calculations.

Although considered primary in the additive system of color, red, green and blue are the secondary colors in the subtractive system. By combining cyan with magenta, cyan with yellow and magenta with yellow, we are able to make blue, green and red respectively. As long as each primary ingredient is at full saturation and in equivalent quantities, the following operations hold true.

When all three primaries are mixed together in equal proportion, the resulting color is black. (editor's note: this color mixture is mainly theoretical as we have yet to encounter perfect subtractive primaries that make a perfect black when mixed.)

Black is also achieved when certain combinations of the primary and secondary elements are mixed. We recall from figure 4 (c), that red is made up of equal parts magenta and yellow. Therefore, we can add cyan to red in order to form black by virtue of the equation below:

And so,

Because each secondary color is merely a combination of two primaries, black is the result in each of the following mixtures. Please note the unequal quantities of the inputs and the constant amount of the sum.

The operations shown above may beg the question: what happens when the secondary colors are added together in equal quantities, as in the diagram below?

Answering this question correctly requires a break down of each of the inputs into its' constituent CMY elements in order to investigate the exact quantities being mixed. In order to simplify this task, we will take single units of red, green and blue and discover that they yield the following:

Upon analyzing the CMY proportions in the sum, we recognize the color as black! This solution may strike you as odd. After all, if equal quantities of primary colors cyan, magenta and yellow make black, how can the same amounts of secondary colors red, green and blue possibly produce an identical result? Can the following diagram really be true?

The answer is "yes" because we have learned that, by definition; black is CMY in equal amounts at full saturation. (If the saturation or purity of the primary hues is less than full, some form of gray will emerge from the mixture. The value of the gray will depend on the amount of base color present.) So, in deconstructing red, green and blue into their respective CMY elements, we discover that these three colors cumulatively do indeed carry the correct CMY proportions to create black. Color Math will be used later in the chapter to prove the validity of this conclusion.

White as "Base" Color

In the parentheses above, we alluded briefly to the concept of the base color. This is the color that is perceived when no primary hues are present. For instance, in color media that utilize the additive system of color, black is considered to be the base color. That is, the eye responds to the absence of red, green and blue (primaries in the additive system) and its perception is the color black. The opposite occurs in the subtractive system where zero values of cyan, magenta and yellow result in the perception of white.

While the use of a non-white, non-black base is possible; the result is a truncated colorspace and a somewhat skewed COLORCUBE. Such is the case when looking through tinted eyewear or offset printing onto colored paper. Thus, in order to maximize the set of visible colors within our CMY color space, white will be the assumed base color.

To this point in our discussion, all of the colors have been fully saturated containing no white or base color. We now attempt to understand the impact of the base color by infusing white base with increasing amounts of primary hues and observe the changes to the resulting color.

Our analysis begins with total white base to which we add rising increments of yellow primary. As long as the overall volume of each color sample is held constant, the progression from 100% white leads to a fully saturated yellow. As the horizontal bars in the top left diagram show, the reduction of base color, white gives way to an increase of non-base content. The changes in the "base-to-primary" ratio results in graduated versions of a mixed color (known as tints due to the presence of white) as shown in the color circles. The upper right corner of the diagram below maps the same procedure using magenta primary.

Let us now conduct a similar operation with two primary colors adding them both to the base color, white. We should recognize that the combination of equal parts yellow and magenta creates red but as illustrated above in the bottom left box, this mixture does not affect the proportion of the white base. The base color maintains its percentage value of 100, 75, 50, 25 and 0% of the total color respectively, regardless of the amounts of multiple primaries which make up the remainder.

The same effect takes place when all three primaries are added to decreasing amounts of white base. As demonstrated in the bottom right corner above, grays and black are the result of mixing cyan, magenta and yellow with a white base. It is important to remember from these exercises that the CMY primary mixture occupies only the portion of the color that is non-base.

Constructing the COLORCUBE

Returning momentarily to the progressions of magenta and yellow tints, we in fact derived two axes of the COLORCUBE. The plane of color, constructed in the figure 12, is what bridges the color space that exists between those two axes. Using a similar strategy to that exhibited earlier, we start with 100% white and incrementally reduce the amount of base color while increasing the primary content. The non-base portions of each color shown in the diagram are mixtures containing magenta and/or yellow. Horizontal bars once again capture the systematic increases in these mixtures within the fixed overall volume of color.

It should seem obvious that moving away from the white square horizontally implies a larger presence of yellow and likewise, a vertical movement downward results in the addition of magenta. What may not be so obvious is that these increases in primaries affect only the portion of color that is non-base. For instance, although magenta and yellow are maximized at a certain point in the "25% base color" diagram, they only contribute 37.5% each to the whole color. (In case this is unclear, the "certain point" that is being referred to is in the bottom left diagram, located in the middle of the color group and is the corner of the colored squares).

The next diagram depicts the unified yellow-magenta plane previously constructed. The horizontal bars of color have been attached to their corresponding color to illustrate the proportions of base and non-base content in each. These "Color Bars" also make the pattern that exists between the relative quantities of the primaries themselves visually apparent.

We will continue to attach these Color Bars to color samples in order to assist our understanding of how the COLORCUBE is built. They are graphic and mathematical representations of the ingredients present in each color and are standardized to a constant length for the purposes of measurement and comparison.

The next phase of COLORCUBE construction involves the addition of the remaining primary, in this case, cyan. The above plane was theoretically conceptualized by taking the white-yellow axis and incrementally increasing magenta content while adjusting base content to maintain its proportion. This resulted in multiple axes containing various combinations of the white and primary colors.

The incremental addition of cyan to this entire magenta-yellow plane will yield similar results in the form of multiple planes of color. This third dimension of color, as in the diagram below, is organized in terms of base color percentage (exactly as above) and each plane is separated according to cyan content. The Color Bars once again fix the overall volume of each color to a single value and allow us to see the actual color that each formula represents.

Applying Color Math

Having completed our derivation of an entire COLORCUBE, we revisit the color theory basics from the beginning of the chapter and attempt to prove these operations using Color Math. In doing so we will introduce tools and techniques that will assist our understanding of CMY addition and subtraction as well as create a visual grounding to Color Math procedures.

Color Bars

Horizontal bars of color were used extensively during the construction of the COLORCUBE to graphically depict the base and non-base content of each color they represented. They were extremely useful in allowing us to detect and map the patterns that exist within a color space. In this section these diagrams become accessories to the addition of color and precursors to actual Color Math.
Figure 15.

Remembering that each Color Bar is standardized to a single set length, we are able to add these fixed amounts together in order to determine the configuration of their sum.

For example, a given quantity of 100% white plus an equal amount of 100% primary yellow would yield a color that is half base, half non-base with a 50% yellow hue.

Figure 15 demonstrates this graphically. It shows that once the input colors are combined and the mixture is reset to the size standard of the original Color Bar(s), the resulting color is one that contains equal parts white base and yellow primary.

Figure 16.
The diagram to the left is another situation that involves base white. However, this example shows the second input to be a color containing unequal multiple primaries.

When adding colors using Color Bars, the presence of three primaries adds complexity to the equation but the procedure for solving it is the same.

Figure 17 shows the summarization of three mixed colors. Each of the ingredients are gathered and grouped according their CMYW category.
Figure 17.

Figure 18.
Let us take for our next equation the sums of the last two operations and use them as inputs. After adding the Color Bars together, we can sort and recalibrate the sum to a fixed amount with all of the constituent elements intact.

Color Boxes

A method that addresses our need for exact numbers while retaining the benefits granted us by the Color Bars must begin with the information on hand being organized in a practical manner. The tables introduced here place the CMY formula of a color in a usable format that allows any color to be added and/or subtracted. It also takes into consideration the volume of color as well.

Suppose we have three units of color gold (C20 M40 Y60) which is one of our inputs in a Color Math operation. Our first step is to determine its percentage base content by subtracting the CMY quantity that is greatest from 100. In this instance, yellow is most dominant with a value of 60. Consequently, this color contains 40% base white.

The remaining 60% consists of primary colors cyan, magenta and yellow in relative proportion to the color's CMY formula. Thus, we divide each of the primary quantities (20, 40, and 60) by the sum of those primaries (120) and multiply each by 60%. This yields adjusted weights of 10, 20 and 30% for CMY respectively. We find that this shade of gold is 10% cyan, 20% magenta, 30% yellow and 40% white. The table below summarizes this information as well as provides the volume calculation.

The next diagram displays the above information in a graphical format that actually maps the relative strength of each of the CMY element and places the overall proportion of the base color in the forefront. This Color Box format adds a visual dimension to the CMY formula and offers insight to the color itself.

This Color Math documentation also makes use of tables and diagrams that isolate the current operation. These are fairly self-explanatory in that they summarize the actual quantities and volumes of color that are being added or subtracted in columnar form. In figure 21, "A" and "B" represent the input colors and "C" symbolizes the resultant color.
Introduction Complete
At this point, the basics of Color Math have been fully explained. We pause here to welcome comments from you that either accept or reject the theories explored here. Please contact Spittin' Image Software for more information on Color Math and look for future developments on this fascinating subject on this website.

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The Color Math Concept by Ken Davies The human eye is able to decipher patterns of light according to the primary colors of the additive system: red, green and blue (RGB). However, it is the subtractive system's primaries: cyan, magenta and yellow (CMY) that best lend themselves to understanding the COLORCUBE and the concept of Color Math.

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Pick's Theorem - Formula, Area - Cool link 

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Pick's Theorem - Formula, Area - Cool link 

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Formula, Area - Cool link between strands. Could make a good investigation for year 7 - 10.

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This works for all simple polygons made of straight lines (no crossing lines or holes). #math #mathematics #maths #mathtok #STEM #science #learnontiktok #tiktoktaughtme 

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@mathletters
This works for all simple polygons made of straight lines (no crossing lines or holes). #math #mathematics #maths #mathtok #STEM #science #learnontiktok #tiktoktaughtme
♬ original sound - Math Letters