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Gyosu.ai - Generative AI for Math Teachers

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Generative AI for math teachers. Create worksheets, problems, quizzes, and exams easily.

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Gyosu.ai - Generative AI for Math Teachers

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Generative AI for math teachers. Create worksheets, problems, quizzes, and exams easily.

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Symbolab Math Calculator - Step by Step calculator

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Symbolab: equation search and math solver - solves algebra, trigonometry and calculus problems step by step

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GeoGebra - the world's favorite, free math tools used by over 100 million students and teachers

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Free digital tools for class activities, graphing, geometry, collaborative whiteboard and more

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Uh oh! Wolfram|Alpha doesn't run without JavaScript.

Please enable JavaScript. If you don't know how, you can find instructions . Once you've done that, refresh this page to start using Wolfram|Alpha.

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Wolfram|Alpha: Making the world's knowledge computable

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Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. For math, science, nutrition, history, geography, engineering, mathematics, linguistics, sports, finance, music...

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Classroom Resources - National Council of Teachers of Mathematics

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MagicSchool.ai - AI for teachers - lesson planning and more!

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Join thousands of teachers saving hours of time using AI to help lesson plan, differentiate, write assessments, write IEPs, communicate clearly, and more!

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These entertaining, imaginative resources from KET include standards-based interactives and videos for grades 5-8. Think Math offers 43 KET-produced resources from Math at the Core: Middle School, including 17 Scale City videos and interactives about proportional reasoning, 14 Wild Fractions animations and games that feature visual models for fraction multiplication and division, nine Logical Leaps interactives about number lines, and three Maritime Mysteries interactives involving Cartesian graphs.

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Model your word problems with Thinking Blocks! Only at MathPlayground.com.

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Classroom Challenges are lessons that support teachers in formative assessment. There are 100 lessons in total, 20 at each grade from 6 to 8 and 40 for 'Career and College Readiness' at High School Grades 9 and above. Some lessons are focused on developing math concepts, others on solving non-routine problems.

The Brief Guide for teachers and administrators (PDF) is recommended reading before using these lessons for the first time.

Each lesson is downloadable as an all-in-one PDF to print out, and a supporting PowerPoint presentation.

Finding a Lesson

There are several ways of navigating:

1. Use the menu on the left to browse by grade and topic area, or search for a key phrase.

2. Go to the Standards tab to find lessons linked to a particular content standard or practice.

3. The complete set of lessons is listed below in alphabetic order.

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Classroom Challenges are lessons that support teachers in formative assessment. There are 100 lessons in total, 20 at each grade from 6 to 8 and 40 for 'Career and College Readiness' at High School Grades 9 and above.

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Notes: A task based on this image appears in the excellent book, Extending the Challenge in Mathematics, by Linda Jensen Sheffield, published by Corwin Press in 2003. See her book for a detailed discussion that includes examples of students' thinking. The problem is similar to the Triangle Sums activity but works especially well for younger students.

Tasks and questions: Insert a different number into each square so that the sum of the bottom row equals the sum of the left column. How many solutions are possible? Does every solution make the same sum? What sums are possible? Are there patterns in the sums? Which numbers can be in the corner? Why does the corner have to be an odd number? What would happen if there were an extra square in each row and column and I use the numbers 1 through 7? Would it still make a solution if I added 1 to each number? What would the new sum be? Would the corner still have to be an odd number? Why or why not? Would it still make a solution if I doubled each number? What would the new solution be? What would happen if I kept subtracting 1 from the numbers in one of my solutions? Can negative numbers, fractions, or decimals be use in the puzzle? What happens if you multiply the numbers instead of add them?

Keep asking new questions!

Notes: This problem relates to points on a grid (called lattice points) that lie on a polygon ( boundary points) and points that are inside the polygon ( interior points). You create the sides of the polygons by connecting lattice points with line segments.

Tasks and questions: What is special about the large black dots? What it special about the open dots? How many boundary points does the polygon have? How many interior points does the polygon have? Can I make more polygons that have same number of boundary and interior points? When a point is very close to the side of a polygon, how can I be sure whether or not it is a boundary point? What is the smallest number of boundary points that a polygon can have? What is the smallest number of interior points that a polygon can have? Can I make a polygon that has the smallest possible number of both boundary and interior points? How many polygons like this can I make? Do they have anything else in common? Is it possible to create polygons with every possible combination of boundary and interior points (greater than the minimum)? Are there patterns in creating these polygons? Are there patterns in the appearances of the polygons? What are the areas my polygons? What happens to the number of boundary points, interior points, and area when I move one vertex of a polygon one lattice point to the horizontally or vertically? Or both? Are these changes predictable or do they depend on the details of the shape? Is it possible to predict the area of a polygon using the number of boundary points and interior points?

Keep asking new questions!

Notes: This problem comes from the book Advanced Common Core Math Explorations: Numbers and Operations, Prufrock Press, 2015 and is used by permission. It is about an imaginary planet called Torr whose inhabitants group by fours instead of tens for their place value, which changes the way they write their numerals.

Tasks and questions: The inhabitants of the planet Torr have two fingers on each hand. Figure out how the Torrans write their numerals. What is the value of each place in the Torran place value system? What are the patterns in these values and what causes them? Can I discover fast ways to convert one of our numerals into a Torran numeral? Can I discover fast ways to convert a Torran numeral into one of our numerals? Why is the number nineteen written as a three-digit numeral in the Torran system? What would Torran place value blocks look like? (Think of the "base 10" blocks that our students use to learn about place value.) Do the Torran numerals ever have a digit greater than 3? Why or why not? Are there patterns in the ways that the Torrans count? What would the Torran algorithms for addition, subtraction, multiplication, and division look like? Would these Torran calculations be easier or harder to learn than ours? Why? Would Torrans have anything like our "decimals"? How would these work? Would our repeating decimals always be repeating for them? Would their prime numbers be the same as our prime numbers? Would they look the same when I write them down? What if there were another culture on Torr that grouped by eights instead of fours? How would I answer all of these questions for them? Would there be a quick way to translate between the numeration systems of these two cultures (without using our system at all)? How long is a Torran day? A Torran year? How might Torrans measure these types of things?

Keep asking new questions!

Notes: This list of fractions comes from the book Advanced Common Core Math Explorations: Ratios, Proportions, and Similarity. Prufrock Press, 2016 and is used by permission. The list can be extended forever, but in ways that you might not expect! It is the basis of an excellent activity for students who are learning about comparing fractions, but there is much more to it than meets the eye. Professional mathematicians are still discovering new things about these ideas!

Tasks and questions: There seem to be manyaddition or subtraction equations with the numerators and denominators of these fractions. Why does this happen? Why are the smallest numerators and denominators in the left and right fractions? Why are the largest numerators and denominators in the second and fourth fractions? What pattern was used to create the list? Can I use the pattern to extend the list to the left and/or right? Can I use the patterns to extend the list in other ways? How long can I make the list? Why are all of the fractions in simplest form? Are all of the fractions still in simplest form when I extend the list? Why or why not? What happens if I arrange the fractions in order of size? I wonder why the fractions always increase from left to right when I extend the list? Do the fractions increase at a constant rate? Do they increase according to some other pattern? What happens to the list if I start with new fractions on the left and right? What happens to the list if I begin with the greater fraction on the left? What happens to the list if I begin with fractions that are not in simplest form? What happens to the list if I begin with two equivalent fractions? Is it possible to find two numbers to start with so that every possible fraction eventually appears in the list when I extend it? Or does this already happen with the list we began with? What would happen if I used multiplication (or another operation) on the numerators and denominators to create the fractions in between?

Keep asking new questions!

Notes: This image comes from Advanced Common Core Math Explorations: Factors and Multiples,Prufrock Press, 2014 and is used by permission. It is a personal favorite of mine. I like to use it at the beginning of the school year with upper-elementary / middle school students. After we figure out what the grid is about, we use it all year long. By the way, this is a great problem for adults, too! See Math Building Blocks to learn more.

Tasks and questions: What patterns do you see? Do you see anything related to rows, columns, or diagonals? What happens in the cells that have only white blocks? What about the ones with only red blocks? How can I predict the next cell that contains a red block? How can I predict the next cell that contains at least two white blocks? How can I predict the next appearance of other combinations of colors? Is it possible to continue the grid downward? How long will the grid continue? Why is the upper left cell blank? What happens if I number each cell (left to right, top to bottom as you would read a book)? If I do this, should I start with 1 or 0? Why? is there a pattern to how the blocks are arranged within each cell? Is this pattern important from a mathematical standpoint? If I continue the grid, will every new cell have blocks in it? Will every cell have its own unique collection of blocks? Will every possible combination of blocks occur in some cell? What patterns can I find in the number of white blocks in each cell? The number of red blocks in each cell? Other colors? How are all of these patterns related to each other?

Keep asking new questions!

Notes: Sometimes, even the most familiar pictures can lead to can lead to great problems that never end-simply by paying close attention to the details as you play with the ideas. This problem comes from Advanced Common Core Math Explorations: Measurement and Polygons, Prufrock Press, 2014 and is used by permission.

Tasks and questions: How can I find the measures of the angles at the tips of the stars by using what I know about the angles in the pentagon? How many different strategies can I find for doing this? Notice how the star is formed by extending the sides of a regular pentagon. Create new stars by extending the sides of other regular polygons. What are their star-tips angles? Are there patterns in the sizes of the star-tip angles of stars formed from different regular polygons? Can I write an algebraic formula or expression relating the number of sides in the polygons to the star-tip angles? Is there more than one formula? How does each formula relate to a different strategy or a different way of thinking about the star-tip angles? Can I use algebra to prove that these formulas are equivalent? What happens if I try to use the formula for a triangle or quadrilateral? How does this relate to what happens when I try to create a star from these polygons? What happens to the star-tip angles as the number of sides keeps increasing? How can I understand this using a formula? A table? The pictures of the stars? What are the sums of the star-tip angles in a star? What patterns are there in these sums? How do these patterns relate to the sums of the interior angles in the original polygons? Why do some polygons create multiple layers of stars when I extend their sides? How can I quickly predict the number of layers? What are the patterns in the angles for the star-tips of the different layers? Can I create a single formula that calculates the star-tip angle for any layer of any star? Is it possible to create stars from polygons that are not regular? Is there a quick way to predict which polygons will create complete stars? Are there formulas these star-tip angles or their sums?

Keep asking new questions!

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Notes: A task based on this image appears in the excellent book, Extending the Challenge in Mathematics , by Linda Jensen Sheffield, published by Corwin Press in 2003. See her book for a detailed discussion that includes examples of students' thinking. The problem is similar to the Triangle Sums activity but works especially well for younger students.

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We cover 100% of the Next Generation Science Standards (NGSS) and about 95% of educational standards in other states. Lookup your state here!

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This applet allows you to examine various two-dimensional figures to determine which ones can be folded into a cube.

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Fitted
If you are a teacher click here for a version of the problem suitable for classroom use, together with supporting materials. Otherwise, read on ...
Nine squares with side lengths $1, 4, 7, 8, 9, 10, 14, 15$ and $18$ cm can be fitted together with no gaps and no overlaps, to form a rectangle.

What are the dimensions of the rectangle?

Once you've had a chance to think about it, click below to see how three different pupils began working on the task.

This is how Anna started:

Here is what Brendan tried:

Here is Chandra's initial approach to the problem:

Can you take each of these starting ideas and develop them into a solution?

This problem is a challenging context in which to explore area. It can be solved in many different ways and reflection on different methods is encouraged through inclusion of the three sample approaches. You may well need more than one lesson for this activity.

Pose the problem orally, or project the text onto a screen, without mentioning the examples of how some children started. Give the class a few minutes to consider, individually, how they might go about tackling the problem, then pair them up and suggest that they talk to their partner about their ideas so far. Try to stand back and observe, and resist the temptation to make helpful suggestions!

Allow pairs to work on the task so that you feel they have made some progress, but do not worry if they have not completed it or if they report being stuck. The aim at this stage is for everyone to 'get into' the problem and work hard on trying to solve it, but not necessarily to achieve a final solution. Make sure that the children have easy access to any resources that they are likely to need but don't put anything out on tables already otherwise this may lead them down particular routes.

At a suitable time, hand out this sheet to pairs. Suggest to the class that when they've finished or can't make any further progress, they should look at the sheet showing three approaches used by children working on this task. Pose the question, "What might each do next? Can you take each of their starting ideas and develop them into a solution?". You may like pairs to record their work on large sheets of paper, which might be more easily shared with the rest of the class in the plenary.

Allow at least fifteen minutes for a final discussion. Invite some pairs to explain how the three different methods might be continued. You may find that some members of the class used completely different approaches when they worked on the task to begin with, so ask them to share their methods too. You can then facilitate a discussion about the advantages and disadvantages of each. Which way would they choose to use if they were presented with a similar task in the future? Why?

Tell me about this approach. What do you think he/she was doing?
How do you think this will help to solve the problem?
What do you think he/she would have done next?

Tiles on a Patio is an investigation based on similar ideas which would make a good follow-up activity.

It might help some pupils to have sets of squares cut out ready for use. However, only give them out if children explicitly decide to approach the task in this way at the start, or if they are continuing Brendan's solution, and you think that they would spend too long accurately drawing and cutting the squares themselves.

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Nine squares with side lengths 1, 4, 7, 8, 9, 10, 14, 15, and 18 cm can be fitted together to form a rectangle. What are the dimensions of the rectangle?

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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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A Three-Act Task is a whole-group mathematics task consisting of three distinct parts: an engaging and perplexing Act One, an information and solution seeking Act Two, and a solution discussion and solution revealing Act Three.

Why would I use this strategy?

To provide an engaging context for the use of mathematics and the development of mathematical understanding.

To reduce the literacy demand.

To add engagement. Students wonder what will happen next.

To create low barriers to entry, allowing the teacher to scaffold as necessary.

To provide an opportunity for estimation and reasonableness.

To provide opportunities to talk about mathematics.

To provide opportunities for reflective thought.

To build new knowledge from prior knowledge.

To encourage multiple approaches.

To honor diversity.

To create situations which require students to engage in mathematical modeling.

To build relational understandings among mathematics concepts.

To shift student ideas about justification of thinking and answers.​

How do I use this strategy?
​
ACT 1: Engage and Perplex

In Act 1, the teacher sharing with students an image, video, or other situation that is engaging and perplexing.

Students discuss what they notice and wonder. They generate question to ask about the situation.

Students decide on a question to answer and make estimates about the likely solution.

ACT 2: Seek Information and Solutions

In Act 2, students work on finding solutions to their problems.

They use information they have and ask for more information as needed.

The teacher supplies more information as requested by students.

They may adjust their question as they work.

​ACT 3: Reveal, Discuss, Extend!

In Act 3, students share their work, their thinking, and their solutions.

There is a reveal by the teacher of a solution, and the discussion that ensues may take many directions:

Students might compare their solutions to each others and to the reveal.

Students might compare their solutions to their estimates and discuss the comparison.

Students might discuss the assumptions that were made in the work.

Students might think of other questions they could pursue next

The teacher helps students connect their work to the core math of the task.

The design of a 3-Act Task is meant to help students see that the math they learn is power rather than punishment. Each of the 3 steps is critical:

1. Invite students to try a task that is intuitive, but inefficient or inaccurate.

2. Help them understand some math.

3. Invite them to re-try the task and see that with math it's more efficient and accurate.

More Resources for 3-Act Tasks:

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A Three-Act Task is a whole-group mathematics task consisting of three distinct parts: an engaging and perplexing Act One, an information and solution seeking Act Two, and a solution discussion and...

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51 Esti-Mysteries - Steve Wyborney's Blog: I'm on a Learning Mission.

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Printing the PDF files from Acrobat Reader reproduces the Challenges much better than simply printing from within your browser.

Caution: If you are printing on a color printer, you may want to set your printer to black & white or gray as the Challenges use significant amounts of color. (Or just download the black & white versions below.)

Note : It may be necessary to scale your page setup to 85% or select "Fit to Page" to print entire page on a single sheet of paper.

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Figure This! Math Challenges for Families - Download and Print the Challenges

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Printing the PDF files from Acrobat Reader reproduces the Challenges much better than simply printing from within your browser. Caution: If you are printing on a color printer, you may want to set your printer to black & white or gray as the Challenges use significant amounts of color.

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We have a variety of prompts to help you lesson plan and do administrative tasks with GenAI chatbots like ChatGPT, Claude, Bard, and Perplexity.

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Curipod - Spark curiosity with interactive lessons

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Free ready to play interactive lessons that sparks your students curiosity in the classroom

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By converting our sims to HTML5, we make them seamlessly available across platforms and devices. Whether you have laptops, iPads, chromebooks, or BYOD, your favorite PhET sims are always right at your fingertips.Become part of our mission today, and transform the learning experiences of students everywhere!