geometry problem solving task

Free Lesson Library

These incredibly powerful, flexible activities can be used with a variety of content and contexts.

These games are chosen for their simplicity and depth.

Problems that resist easy solutions while encouraging perseverance and deeper understanding.

• All Lessons

Selecting this shows all the free lessons in our library.

Every student deserves to have the opportunity to problem-solve and engage in genuine mathematical thinking. Rich tasks are designed to make these rich learning experiences possible. We’ve written these tasks to launch quickly, engage students, and promote the habits of mind mathematicians need: perseverance & pattern-seeking, courage & curiosity, organization & communication.

This free video PD series will help you get the most out of the tasks below.

Lessons - Rich Tasks

• Common Core Math Practices
• Counting and Cardinality
• Expressions and Equations
• Measurement and Data
• Number System
• Numbers and Operations in Base Ten
• Numbers and Operations in Fraction
• Operations and Algebraic Thinking
• Ratio and Proportion
• Exploration
• Divisibility
• Measurement
• Mental Math Games
• Multiplication
• Place Value
• Subtraction

Join Our Mailing List

Get tons of free content, like our Games to Play at Home packet, puzzles, lessons, and more!

Not a Member?

Read more about the Member Benefits of MAV and find out how to join MAV or renew your membership.

• Authentic tasks
• F - 10 Resources

Authentic tasks are designed to help students see mathematics as worthwhile and important. When students understand the purpose of a given problem in mathematics, they are more likely to persist when challenged. Authentic tasks generally have an ‘open middle’ which means that students can use different representations and solutions to communicate their knowledge and reasoning.

These curated links provide MAV members with access to nine authentic tasks from some of our primary consultants’ favourite resources. The 11 criteria provide MAV members with a research-informed context to consider each task’s potential impact on student thinking, ways of working, attitudes towards mathematics, their knowledge and understanding.

The following criteria was used to select the tasks based on their potential:

Criteria	Elaboration
Intriguing contexts capture the students’ interests and curiosities	An opportunity for students to relate learning to their own lives and communities. High student motivation/enjoyment and sense of purpose can be anticipated or observed.
Problem solving is required to overcome obstacles	Exploring non-routine questions, real life challenges, posing problems and designing investigations. There are obstacles that students have to overcome in order to succeed.
Low entry/ high ceiling, and an open-middle encourages different strategies	Caters for a range of student abilities. Open-middle allows for different possibilities, strategies, materials and products to emerge. Task may adapt depending on student progress.
Opportunities for creative thinking and or visualising	Students invent, discover and imagine new ways to solve a problem. Students make connections and see relationships by visualising the problem or representing solutions visually.
Encourages reasoning and critical thinking	Logical, rational and critical thinking. eg: estimating, hypothesising, justifying, generalising, comparing, explaining, interpreting and looking back.
Opportunity to collaborate and see others working mathematically	Students challenge each other, the computer, the teacher etc and observe how they work mathematically. Make decisions in groups to communicate findings, engage with different ideas, monitor and regulate each other's thinking.
Opportunities for students to develop fluency	Estimating, collecting and interpreting data, using mathematical language, continuing patterns, choosing appropriate unit of measurement, recalling factual knowledge and concepts readily.
Promotes feedback and metacognition.	Self and peer reflection targeted at specific aspects of the work. eg: knowledge; how your thinking is changed as a result of lesson. Affective traits eg: problem solving attitude, collaborative skills.
Extends knowledge or applies knowledge in new contexts	Provides students with access to forms of knowledge beyond what they can pick up in everyday life or via the Internet.
Promotes an understanding of the ‘why’ and ‘how’ of mathematics	Seeing patterns, connecting related ideas based on previously constructed knowledge. Represent concepts (big Ideas) in different ways eg: developing number sense, place value.
Guides future learning	Provides data on student growth to guide future direction. eg: identifies patterns or errors, level of progress towards goal. Makes suggestions for future learning.

Used with permission © Martin Holt Educational Consultant 2017

If you would like to learn more about this approach to assessing or using tasks contact [email protected]

Statistics and probability

NRICH problem solving task:	Scootle lesson sequence:
Target Level: F - 2	Target Level: 1 - 6

Measurement and geometry

Wildmaths interactive game:	Teach Engineering investigation:
Target Level: 2-4	Target Level: F-2

Number and algebra

NZMaths lesson sequence:	MAV problem solving task:	ReSolve Maths by Inquiry lesson sequence:
Target Level: 5-6	Target Level: 1-6	Target Level: 5-6

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

• Build Me Up
• Document Listing
• Undergrad and first year teachers
• Picture books
• Mathspiration
• News and research
• Curriculum and resources
• VCE Resources
• Investigations - Vinculum
• Learning Activities Years F to 9
• Town Squared-game
• Mathematics in Careers

The Common Denominator 2/24

DOWNLOAD NOW

• Join or Renew Membership
• Register for Professional Learning
• Maths Talent Quest
• Girls in STEM
• VCE Revision Lectures
• Resources in the MAV Shop

Problem Solving Activities: 7 Strategies

• Critical Thinking

Problem solving can be a daunting aspect of effective mathematics teaching, but it does not have to be! In this post, I share seven strategic ways to integrate problem solving into your everyday math program.

In the middle of our problem solving lesson, my district math coordinator stopped by for a surprise walkthrough. 

I was so excited!

We were in the middle of what I thought was the most brilliant math lesson– teaching my students how to solve problem solving tasks using specific problem solving strategies. 

It was a proud moment for me!

Each week, I presented a new problem solving strategy and the students completed problems that emphasized the strategy. 

Genius right? 

After observing my class, my district coordinator pulled me aside to chat. I was excited to talk to her about my brilliant plan, but she told me I should provide the tasks and let my students come up with ways to solve the problems. Then, as students shared their work, I could revoice the student’s strategies and give them an official name. 

What a crushing blow! Just when I thought I did something special, I find out I did it all wrong. 

I took some time to consider her advice. Once I acknowledged she was right, I was able to make BIG changes to the way I taught problem solving in the classroom. 

When I Finally Saw the Light

To give my students an opportunity to engage in more authentic problem solving which would lead them to use a larger variety of problem solving strategies, I decided to vary the activities and the way I approached problem solving with my students. 

Problem Solving Activities

Here are seven ways to strategically reinforce problem solving skills in your classroom. 

Seasonal Problem Solving

Many teachers use word problems as problem solving tasks. Instead, try engaging your students with non-routine tasks that look like word problems but require more than the use of addition, subtraction, multiplication, and division to complete. Seasonal problem solving tasks and daily challenges are a perfect way to celebrate the season and have a little fun too!

Cooperative Problem Solving Tasks

Go cooperative! If you’ve got a few extra minutes, have students work on problem solving tasks in small groups. After working through the task, students create a poster to help explain their solution process and then post their poster around the classroom. Students then complete a gallery walk of the posters in the classroom and provide feedback via sticky notes or during a math talk session.

Notice and Wonder

Before beginning a problem solving task, such as a seasonal problem solving task, conduct a Notice and Wonder session. To do this, ask students what they notice about the problem. Then, ask them what they wonder about the problem. This will give students an opportunity to highlight the unique characteristics and conditions of the problem as they try to make sense of it. 

Want a better experience? Remove the stimulus, or question, and allow students to wonder about the problem. Try it! You’ll gain some great insight into how your students think about a problem.

Math Starters

Start your math block with a math starter, critical thinking activities designed to get your students thinking about math and provide opportunities to “sneak” in grade-level content and skills in a fun and engaging way. These tasks are quick, designed to take no more than five minutes, and provide a great way to turn-on your students’ brains. Read more about math starters here ! 

Create your own puzzle box! The puzzle box is a set of puzzles and math challenges I use as fast finisher tasks for my students when they finish an assignment or need an extra challenge. The box can be a file box, file crate, or even a wall chart. It includes a variety of activities so all students can find a challenge that suits their interests and ability level.

Calculators

Use calculators! For some reason, this tool is not one many students get to use frequently; however, it’s important students have a chance to practice using it in the classroom. After all, almost everyone has access to a calculator on their cell phones. There are also some standardized tests that allow students to use them, so it’s important for us to practice using calculators in the classroom. Plus, calculators can be fun learning tools all by themselves!

Three-Act Math Tasks

Use a three-act math task to engage students with a content-focused, real-world problem! These math tasks were created with math modeling in mind– students are presented with a scenario and then given clues and hints to help them solve the problem. There are several sites where you can find these awesome math tasks, including Dan Meyer’s Three-Act Math Tasks and Graham Fletcher’s 3-Acts Lessons . 

Getting the Most from Each of the Problem Solving Activities

When students participate in problem solving activities, it is important to ask guiding, not leading, questions. This provides students with the support necessary to move forward in their thinking and it provides teachers with a more in-depth understanding of student thinking. Selecting an initial question and then analyzing a student’s response tells teachers where to go next. 

Ready to jump in? Grab a free set of problem solving challenges like the ones pictured using the form below. 

Which of the problem solving activities will you try first? Respond in the comments below.

Shametria Routt Banks

• Assessment Tools
• Content and Standards
• Differentiation
• Math & Literature
• Math & Technology
• Math Routines
• Math Stations
• Virtual Learning
• Writing in Math

You may also like...

2 Responses

This is a very cool site. I hope it takes off and is well received by teachers. I work in mathematical problem solving and help prepare pre-service teachers in mathematics.

Thank you, Scott! Best wishes to you and your pre-service teachers this year!

Leave a Reply Cancel reply

Your email address will not be published. Required fields are marked *

This site uses Akismet to reduce spam. Learn how your comment data is processed .

©2024 The Routty Math Teacher.   All Rights Reserved.  Designed by Ashley Hughes.

Privacy overview, grade level.

• Mindset Rubric
• Thought Leaders
• Special Education
• Mathematical Freedom
• Number Sense
• Visual Mathematics
• Growth Mindset
• High Quality Teaching Examples
• Assessment & Grading
• Depth Not Speed
• Brain Science
• Week of Inspirational Math(s)
• Indigenous Mathematical Art
• Advice on Teaching Content
• Youcubed at Home
• Maths and Art
• Exploring Calculus
• Mathematical Mindset Algebra
• Poly Machines
• Mathematical Mindset Teaching Resources
• Parent Resources
• Student Resources
• Why It Is Important
• Online Course
• High School Data Science Course
• Data Big Ideas and Tasks
• Grade 6-10 Data Science Unit
• Films & Videos
• Teaching to Big Ideas
• Summer Camps
• Rethinking Giftedness Film
• Mindset Boosting Videos
• Videos from Teachers
• Teaching Videos
• Online Student Course
• Online Courses for Teachers
• Professional Development
• Webinar for Parents & Guardians
• Fluency Without Fear
• Mindset Mathematics Summer Camps
• Mindset Evidence
• Neuroscience & Education Article
• Special Edition Ed Sciences
• Seeing as Understanding
• Short Impact Papers
• Research Articles
• In the News
• TV, Radio and Podcasts
• Newsletters

Tasks Where do youcubed tasks come from?

Join us this fall for NEW online workshops on Teaching Mathematics as a Conceptual & Connected Subject by grade bands! More info here

Rich Problems – Part 1

Rich problems – part 1, by marvin cohen and karen rothschild.

One of the underlying beliefs that guides Math for All is that in order to learn mathematics well, students must engage with rich problems. Rich problems allow ALL students, with a variety of neurodevelopmental strengths and challenges, to engage in mathematical reasoning and become flexible and creative thinkers about mathematical ideas. In this Math for All Updates, we review what rich problems are, why they are important, and where to find some ready to use. In a later Math for All Updates we will discuss how to create your own rich problems customized for your curriculum.

What are Rich Problems?

At Math for All, we believe that all rich problems provide:

• opportunities to engage the problem solver in thinking about mathematical ideas in a variety of non-routine ways.
• an appropriate level of productive struggle.
• an opportunity for students to communicate their thinking about mathematical ideas.

Rich problems increase both the problem solver’s reasoning skills and the depth of their mathematical understanding. Rich problems are rich because they are not amenable to the application of a known algorithm, but require non-routine use of the student’s knowledge, skills, and ingenuity. They usually offer multiple entry pathways and methods of representation. This provides students with diverse abilities and challenges the opportunity to create solution strategies that leverage their particular strengths.

Rich problems usually have one or more of the following characteristics:

• Several correct answers. For example, “Find four numbers whose sum is 20.”
• A single answer but with many pathways to a solution. For example, “There are 10 animals in the barnyard, some chickens, some pigs. Altogether there are 24 legs. How many of the animals are chickens and how many are pigs?”
• A level of complexity that may require an entire class period or more to solve.
• An opportunity to look for patterns and make connections to previous problems, other students’ strategies, and other areas of mathematics. For example, see the staircase problem below.
• A “low floor and high ceiling,” meaning both that all your students will be able to engage with the mathematics of the problem in some way, and that the problem has sufficient complexity to challenge all your students. NRICH summarizes this approach as “everyone can get started, and everyone can get stuck” (2013). For example, a problem could have a variety of questions related to the following sequence, such as: How many squares are in the next staircase? How many in the 20th staircase? What is the rule for finding the number of squares in any staircase?

• An expectation that the student be able to communicate their ideas and defend their approach.
• An opportunity for students to choose from a range of tools and strategies to solve the problem based on their own neurodevelopmental strengths.
• An opportunity to learn some new mathematics (a mathematical residue) through working on the problem.
• An opportunity to practice routine skills in the service of engaging with a complex problem.
• An opportunity for a teacher to deepen their understanding of their students as learners and to build new lessons based on what students know, their developmental level, and their neurodevelopmental strengths and challenges.

Why Rich Problems?

All adults need mathematical understanding to solve problems in their daily lives. Most adults use calculators and computers to perform routine computation beyond what they can do mentally. They must, however, understand enough mathematics to know what to enter into the machines and how to evaluate what comes out. Our personal financial situations are deeply affected by our understanding of pricing schemes for the things we buy, the mortgages we hold, and fees we pay. As citizens, understanding mathematics can help us evaluate government policies, understand political polls, and make decisions. Building and designing our homes, and scaling up recipes for crowds also require math. Now especially, mathematical understanding is crucial for making sense of policies related to the pandemic. Decisions about shutdowns, medical treatments, and vaccines are all grounded in mathematics. For all these reasons, it is important students develop their capacities to reason about mathematics. Research has demonstrated that experience with rich problems improves children’s mathematical reasoning (Hattie, Fisher, & Frey, 2017).

Where to Find Rich Problems

Several types of rich problems are available online, ready to use or adapt. The sites below are some of many places where rich problems can be found:

• Which One Doesn’t Belong – These problems consist of squares divided into 4 quadrants with numbers, shapes, or graphs. In every problem there is at least one way that each of the quadrants “doesn’t belong.” Thus, any quadrant can be argued to be different from the others.
• “Open Middle” Problems – These are problems with a single answer but with many ways to reach the answer. They are organized by both topic and grade level.
• NRICH Maths – This is a multifaceted site from the University of Cambridge in Great Britain. It has both articles and ready-made problems. The site includes  problems for grades 1–5 (scroll down to the “Collections” section) and problems for younger children . We encourage you to explore NRICH more fully as well. There are many informative articles and discussions on the site.
• Rich tasks from Virginia – These are tasks published by the Virginia Department of education. They come with complete lesson plans as well as example anticipated student responses.
• Rich tasks from Georgia – This site contains a complete framework of tasks designed to address all standards at all grades. They include 3-Act Tasks , YouCubed Tasks , and many other tasks that are open ended or feature an open middle approach.

The problems can be used “as is” or adapted to the specific neurodevelopmental strengths and challenges of your students. Carefully adapted, they can engage ALL your students in thinking about mathematical ideas in a variety of ways, thereby not only increasing their skills but also their abilities to think flexibly and deeply.

Hattie, J., Fisher, D., & Frey, N. (2017). Visible learning for mathematics, grades K-12: What works best to optimize student learning. Thousand Oaks, CA: Corwin Mathematics.

NRICH Team. (2013). Low Threshold High Ceiling – an Introduction . Cambridge University, United Kingdom: NRICH Maths.

The contents of this blog post were developed under a grant from the Department of Education. However, those contents do not necessarily represent the policy of the Department of Education, and you should not assume endorsement by the Federal Government.

Math for All is a professional development program that brings general and special education teachers together to enhance their skills in planning and adapting mathematics lessons to ensure that all students achieve high-quality learning outcomes in mathematics.

Our Newsletter Provides Ideas for Making High-Quality Mathematics Instruction Accessible to All Students

Sign up for our newsletter, recent blogs.

• Social Justice in the Math Classroom May 30, 2024
• Parents and Teachers as Co-Constructors of Children’s Success as Mathematical Learners May 1, 2024
• The Problem with Word Problems March 11, 2024
• Honoring Diversity: What, why, and how? February 22, 2024
• Looking at a Student at Work January 3, 2024

Team & Partners

Testimonials

Newsletter Signup

Teachers: Resources for Middle Grades (6-8)

The North Carolina Collaborative for Mathematics Learning (NC 2 ML) aims to support NC math educators in implementing the revised mathematics content standards in ways that align with what we know from research on students’ mathematical thinking, mathematics teaching, and teacher learning. To do so, we bring together mathematics educators to co-design research-based resources and professional learning opportunities.

6-8 Resources Home

First Week Problem Solving Tasks

The Instructional Frameworks at each grade level recommend spending the first week of school doing general, high cognitive demand tasks with students in order to establish strong communication practices (SMP 3). Students can be enculturated into the discourse, listening and writing practices essential for strong mathematical reasoning while working these problems.

Additional Supporting Articles

Herbel-Eisenmenn, B. & Breyfogle, M. (2005). Questioning our patterns of questioning. Mathematics Teaching in the Middle School, 10(9), 484-489.

Stephan, M. (2014). Establishing standards for mathematical practice. Mathematics Teaching in the Middle School, 19(9), 532-538.

Grade	Tasks
6th
7th
8th

• K-5 Resources
• 6-8 Resources
• 9-12 High School Resources
• Leaders/Administrators
• Document Library
• News & Events

Open Middle®

Area & Perimeter Task Cards

These free area & perimeter task cards are designed to provide geometry problem solving practice for your students..

Use these area & perimeter task cards to provide your students with problem solving practice during your geometry focus.

This is a set of 20 word problem task cards at varying levels that address the standards for calculating area and perimeter.

As always, this set has been created to be a free resource for classrooms and homes. Please copy and use for your personal use. They may not be sold or modified and sold in any way.

These cards might be used at a math center where students focus on problem solving. They might also be used in small groups where each group is given a different problem to solve.

These cards are designed to provide practice for students who have been taught the skills of finding area and perimeter. They are not designed to teach. It is important to make sure students have an understanding of the skills before asking them to complete any of the task cards.

We have broken these area & perimeter task cards into three levels below:

The first eight word problems are fairly straight forward types of story problems calling for students to think about real-world type situations in which they may need to calculate area or perimeter.

Task cards #9 – #16 require students to draw shapes meeting certain area or perimeter stipulations.  These are more open-ended tasks because the shapes students draw can all look very different.

The final four tasks are more challenging problems that will require students to use various problem solving strategies that include drawing pictures, using logical reasoning and guess and check. These would best be used with your most skilled mathematicians or for group work.

You can download this free set of area & perimeter task cards here:

Area & perimeter problem solving task cards.

What to do with your third grade when school is cancelled - The Measured Mom

Sunday 22nd of March 2020

[…] these free task cards from The Curriculum […]

Perimeter and Area of Rectangles - The Curriculum Corner 4-5-6

Wednesday 26th of October 2016

[…] Want another set of task cards to challenge your students?  Check out our other set here: Area & Perimeter Task Cards […]

Saturday 21st of May 2016

Very useful to link Maths with examples from life. Thank you

Home Educators Association of Virginia » The Homeschool Classroom – February 10, 2016

Wednesday 10th of February 2016

[…] www.thecurriculumcorner.com/thecurriculumcorner456/area-perimeter-problem-solving-task-cards […]

Gabriel M. Smithson

Monday 23rd of November 2015

Thanks for the Area & Perimeter Task Cards!

Want to create or adapt books like this? Learn more about how Pressbooks supports open publishing practices.

5 Teaching Mathematics Through Problem Solving

Janet Stramel

In his book “How to Solve It,” George Pólya (1945) said, “One of the most important tasks of the teacher is to help his students. This task is not quite easy; it demands time, practice, devotion, and sound principles. The student should acquire as much experience of independent work as possible. But if he is left alone with his problem without any help, he may make no progress at all. If the teacher helps too much, nothing is left to the student. The teacher should help, but not too much and not too little, so that the student shall have a reasonable share of the work.” (page 1)

What is a problem  in mathematics? A problem is “any task or activity for which the students have no prescribed or memorized rules or methods, nor is there a perception by students that there is a specific ‘correct’ solution method” (Hiebert, et. al., 1997). Problem solving in mathematics is one of the most important topics to teach; learning to problem solve helps students develop a sense of solving real-life problems and apply mathematics to real world situations. It is also used for a deeper understanding of mathematical concepts. Learning “math facts” is not enough; students must also learn how to use these facts to develop their thinking skills.

According to NCTM (2010), the term “problem solving” refers to mathematical tasks that have the potential to provide intellectual challenges for enhancing students’ mathematical understanding and development. When you first hear “problem solving,” what do you think about? Story problems or word problems? Story problems may be limited to and not “problematic” enough. For example, you may ask students to find the area of a rectangle, given the length and width. This type of problem is an exercise in computation and can be completed mindlessly without understanding the concept of area. Worthwhile problems  includes problems that are truly problematic and have the potential to provide contexts for students’ mathematical development.

There are three ways to solve problems: teaching for problem solving, teaching about problem solving, and teaching through problem solving.

Teaching for problem solving begins with learning a skill. For example, students are learning how to multiply a two-digit number by a one-digit number, and the story problems you select are multiplication problems. Be sure when you are teaching for problem solving, you select or develop tasks that can promote the development of mathematical understanding.

Teaching about problem solving begins with suggested strategies to solve a problem. For example, “draw a picture,” “make a table,” etc. You may see posters in teachers’ classrooms of the “Problem Solving Method” such as: 1) Read the problem, 2) Devise a plan, 3) Solve the problem, and 4) Check your work. There is little or no evidence that students’ problem-solving abilities are improved when teaching about problem solving. Students will see a word problem as a separate endeavor and focus on the steps to follow rather than the mathematics. In addition, students will tend to use trial and error instead of focusing on sense making.

Teaching through problem solving  focuses students’ attention on ideas and sense making and develops mathematical practices. Teaching through problem solving also develops a student’s confidence and builds on their strengths. It allows for collaboration among students and engages students in their own learning.

Consider the following worthwhile-problem criteria developed by Lappan and Phillips (1998):

• The problem has important, useful mathematics embedded in it.
• The problem requires high-level thinking and problem solving.
• The problem contributes to the conceptual development of students.
• The problem creates an opportunity for the teacher to assess what his or her students are learning and where they are experiencing difficulty.
• The problem can be approached by students in multiple ways using different solution strategies.
• The problem has various solutions or allows different decisions or positions to be taken and defended.
• The problem encourages student engagement and discourse.
• The problem connects to other important mathematical ideas.
• The problem promotes the skillful use of mathematics.
• The problem provides an opportunity to practice important skills.

Of course, not every problem will include all of the above. Sometimes, you will choose a problem because your students need an opportunity to practice a certain skill.

Key features of a good mathematics problem includes:

• It must begin where the students are mathematically.
• The feature of the problem must be the mathematics that students are to learn.
• It must require justifications and explanations for both answers and methods of solving.

Problem solving is not a  neat and orderly process. Think about needlework. On the front side, it is neat and perfect and pretty.

But look at the b ack.

It is messy and full of knots and loops. Problem solving in mathematics is also like this and we need to help our students be “messy” with problem solving; they need to go through those knots and loops and learn how to solve problems with the teacher’s guidance.

When you teach through problem solving , your students are focused on ideas and sense-making and they develop confidence in mathematics!

Mathematics Tasks and Activities that Promote Teaching through Problem Solving

Choosing the Right Task

Selecting activities and/or tasks is the most significant decision teachers make that will affect students’ learning. Consider the following questions:

• Teachers must do the activity first. What is problematic about the activity? What will you need to do BEFORE the activity and AFTER the activity? Additionally, think how your students would do the activity.
• What mathematical ideas will the activity develop? Are there connections to other related mathematics topics, or other content areas?
• Can the activity accomplish your learning objective/goals?

Low Floor High Ceiling Tasks

By definition, a “ low floor/high ceiling task ” is a mathematical activity where everyone in the group can begin and then work on at their own level of engagement. Low Floor High Ceiling Tasks are activities that everyone can begin and work on based on their own level, and have many possibilities for students to do more challenging mathematics. One gauge of knowing whether an activity is a Low Floor High Ceiling Task is when the work on the problems becomes more important than the answer itself, and leads to rich mathematical discourse [Hover: ways of representing, thinking, talking, agreeing, and disagreeing; the way ideas are exchanged and what the ideas entail; and as being shaped by the tasks in which students engage as well as by the nature of the learning environment].

The strengths of using Low Floor High Ceiling Tasks:

• Allows students to show what they can do, not what they can’t.
• Provides differentiation to all students.
• Promotes a positive classroom environment.
• Advances a growth mindset in students
• Aligns with the Standards for Mathematical Practice

Examples of some Low Floor High Ceiling Tasks can be found at the following sites:

• YouCubed – under grades choose Low Floor High Ceiling
• NRICH Creating a Low Threshold High Ceiling Classroom
• Inside Mathematics Problems of the Month

Math in 3-Acts

Math in 3-Acts was developed by Dan Meyer to spark an interest in and engage students in thought-provoking mathematical inquiry. Math in 3-Acts is a whole-group mathematics task consisting of three distinct parts:

Act One is about noticing and wondering. The teacher shares with students an image, video, or other situation that is engaging and perplexing. Students then generate questions about the situation.

In Act Two , the teacher offers some information for the students to use as they find the solutions to the problem.

Act Three is the “reveal.” Students share their thinking as well as their solutions.

“Math in 3 Acts” is a fun way to engage your students, there is a low entry point that gives students confidence, there are multiple paths to a solution, and it encourages students to work in groups to solve the problem. Some examples of Math in 3-Acts can be found at the following websites:

• Dan Meyer’s Three-Act Math Tasks
• Graham Fletcher3-Act Tasks ]
• Math in 3-Acts: Real World Math Problems to Make Math Contextual, Visual and Concrete

Number Talks

Number talks are brief, 5-15 minute discussions that focus on student solutions for a mental math computation problem. Students share their different mental math processes aloud while the teacher records their thinking visually on a chart or board. In addition, students learn from each other’s strategies as they question, critique, or build on the strategies that are shared.. To use a “number talk,” you would include the following steps:

• The teacher presents a problem for students to solve mentally.
• Provide adequate “ wait time .”
• The teacher calls on a students and asks, “What were you thinking?” and “Explain your thinking.”
• For each student who volunteers to share their strategy, write their thinking on the board. Make sure to accurately record their thinking; do not correct their responses.
• Invite students to question each other about their strategies, compare and contrast the strategies, and ask for clarification about strategies that are confusing.

“Number Talks” can be used as an introduction, a warm up to a lesson, or an extension. Some examples of Number Talks can be found at the following websites:

• Inside Mathematics Number Talks
• Number Talks Build Numerical Reasoning

Saying “This is Easy”

“This is easy.” Three little words that can have a big impact on students. What may be “easy” for one person, may be more “difficult” for someone else. And saying “this is easy” defeats the purpose of a growth mindset classroom, where students are comfortable making mistakes.

When the teacher says, “this is easy,” students may think,

• “Everyone else understands and I don’t. I can’t do this!”
• Students may just give up and surrender the mathematics to their classmates.
• Students may shut down.

Instead, you and your students could say the following:

• “I think I can do this.”
• “I have an idea I want to try.”
• “I’ve seen this kind of problem before.”

Tracy Zager wrote a short article, “This is easy”: The Little Phrase That Causes Big Problems” that can give you more information. Read Tracy Zager’s article here.

Using “Worksheets”

Do you want your students to memorize concepts, or do you want them to understand and apply the mathematics for different situations?

What is a “worksheet” in mathematics? It is a paper and pencil assignment when no other materials are used. A worksheet does not allow your students to use hands-on materials/manipulatives [Hover: physical objects that are used as teaching tools to engage students in the hands-on learning of mathematics]; and worksheets are many times “naked number” with no context. And a worksheet should not be used to enhance a hands-on activity.

Students need time to explore and manipulate materials in order to learn the mathematics concept. Worksheets are just a test of rote memory. Students need to develop those higher-order thinking skills, and worksheets will not allow them to do that.

One productive belief from the NCTM publication, Principles to Action (2014), states, “Students at all grade levels can benefit from the use of physical and virtual manipulative materials to provide visual models of a range of mathematical ideas.”

You may need an “activity sheet,” a “graphic organizer,” etc. as you plan your mathematics activities/lessons, but be sure to include hands-on manipulatives. Using manipulatives can

• Provide your students a bridge between the concrete and abstract
• Serve as models that support students’ thinking
• Provide another representation
• Support student engagement
• Give students ownership of their own learning.

Adapted from “ The Top 5 Reasons for Using Manipulatives in the Classroom ”.

any task or activity for which the students have no prescribed or memorized rules or methods, nor is there a perception by students that there is a specific ‘correct’ solution method

should be intriguing and contain a level of challenge that invites speculation and hard work, and directs students to investigate important mathematical ideas and ways of thinking toward the learning

involves teaching a skill so that a student can later solve a story problem

when we teach students how to problem solve

teaching mathematics content through real contexts, problems, situations, and models

a mathematical activity where everyone in the group can begin and then work on at their own level of engagement

20 seconds to 2 minutes for students to make sense of questions

Mathematics Methods for Early Childhood Copyright © 2021 by Janet Stramel is licensed under a Creative Commons Attribution 4.0 International License , except where otherwise noted.

Share This Book

Fun teaching resources & tips to help you teach math with confidence

25+ Engaging Math Tasks That Promote a Growth Mindset

In our culture, we are bombarded with messages implying that some people are good at math and some people aren’t. There’s this notion that some people have that elusive “math gene” and some people don’t. Overcoming these negative attitudes and baggage to encourage kids that they can in fact learn and enjoy math and there is actually no such thing as a “math person,” is a challenge. But more and more research is showing up that kids with a growth mindset towards math do better on standardized tests, are more engaged in class and have a better attitude about math in general. Today I want to focus on how rich math tasks can excite and engage kids and begin to develop a growth mindset .  As you go through this list of math tasks that promote a growth mindset , pick out your favorites to try with your kids!

* Please Note : This post contains affiliate links which help support the work of this site. Read our full disclosure here .*

What is a Growth Mindset?

The notion of a fixed mindset versus a growth mindset began with the work of Carol Dweck . She explains that everyone has a mindset or idea about how they learn. Those with a fixed mindset believe that you can’t change your level of intelligence. You can only learn so much or go so far.

Those with a growth mindset, on the other hand, view intelligence as something that can be achieved through hard work and perseverance .

In other words, “smarts” is not something you’re born with, it’s something that can develop over time if you’re willing to put in the effort.

Because so many in our culture have a fixed mindset towards math, it is important that we do what we can to dispel these notions as more and more brain research sheds light on how we learn new things. Because the truth is that our brains grow when we make mistakes . And our brains can grow and change and adapt even if half of the brain is removed .

If the human brain is capable of these remarkable achievements, surely our brains are capable of learning Algebra !

How to Develop a Growth Mindset Towards Math:

Changing our own mindset towards learning math can be a challenge, much less changing our kids’ mindsets . But it’s not impossible.

One thing that has to change is the way math is viewed . All too often, math is seen as a closed, fixed subject to be memorized and then forgotten.

But math is a creative, open and exciting topic that touches every single aspect of our lives! We see and use math everyday, and to help kids see that they can learn and achieve math at high levels, we have to get them excited about it.

One way to encourage kids to see math as creative and present in the world around us is to explore math in nature . These concepts are great because they are accessible for young kids but complex enough to challenge older, advanced learners .

Jo Boaler, in her book Mathematical Mindsets , explains the importance of rich and open math tasks. No matter what curriculum you use, you are the teacher and have the opportunity to present math in a way that is engaging.

In the book, Boaler discusses several ways tasks can be rich and engaging for kids . Here is a quick overview:

• Can you open the task to encourage multiple methods, pathways and representations?
• Can you make it an inquiry task?
• Can you ask the problem before teaching the method?
• Can you add a visual component?
• Can you make it low floor and high ceiling?
• Can you add the requirement to convince & reason?

Using these criteria, I want to share a growing list of math tasks and websites where you can find meaningful lessons to engage your kids in math learning and teach with a growth mindset .

And if you have any ideas to add to this list, be sure to share in the comments!

Math Tasks That Promote a Growth Mindset:

Which Number Doesn’t Belong? I love this set of challenges because there are countless answers for each set of numbers. This is sure to spark some interesting math debates and deep thinking! (all ages)

How Many Seeds in a Pumpkin? This fun, hands on lesson is perfect for fall and can be combined with a fun math read aloud as well. (grades K-2)

Addition & Subtraction Equation Search : These fun challenges are more than just addition and subtraction and there are lots of different right answers. (grades 1-2)

Geoboard Challenge Cards : This set of challenges promotes exploration and discovery and includes some basic shapes and more advanced challenges. (all ages)

Compose Shapes with Geometiles : These challenging visual puzzles are created to use with Geometiles and are a great visual challenge for kids. (all ages)

What’s for Lunch? A Real Life Decimal Lesson : This introduction to adding and subtracting decimals is a great example of math in the real world. (grades 3+)

Gumball Estimation : This printable gumball challenge has different variations for increasing challenges or different ages. (grades 3+)

Surface Area of 3D Shapes : I have two investigations for determining the surface area of 3D shapes. Surface area of prisms and cylinders | Surface area of pyramids and cones (grades 7+)

Pattern Block Fraction Games : Pattern blocks provide a fantastic visual for understanding fractions. These games provide practice for kids to help them understand difficult operations.(grades 3-5):

Subtracting Mixed Numbers Game | Adding Fractions Game

Cut & Paste Logic Puzzles : Similar to sudoku puzzles, these are a fun introduction for young kids. (grades PreK-K)

Pattern Block Logic Puzzles : These puzzles are similar to sudoku, but are perfect for young kids not yet ready for more advanced logic puzzles. (grades k-2)

Grid & Sudoku Logic Puzzles : Older kids will love the challenge of these unique math problems that we don’t often see in textbooks. (grades 4+):

Thanksgiving Puzzles | Christmas Puzzles | Valentine’s Day Puzzles | 4th of July Puzzles

Missing Number Puzzles: Addition & Subtraction : These digital puzzles challenge kids to go beyond the standard algorithm for adding & subtracting large numbers and encourages deep thinking. (grades 3-4)

Pattern Puzzles for Google Slides : This set of missing number challenges encourages kids to see patterns, while building algebraic thinking. (grades 4-6)

Order of Operations Puzzles for Google Slides : Each problem in this set includes an expression with some of the numbers missing. Students must then use their knowledge of order of operations to figure out which number is missing.

Christmas Equations Algebra Challenge : These engaging problems help kids work on their algebraic reasoning and problem solving as more of a visual puzzle than a math problem. (grades 4+)

Exploring the Angles in Triangles : This hands on geometry challenge will show kids important facts about the relationships between angles in a triangle. (grades 6+)

What Makes a Triangle a Triangle? Use this lesson along with the book, “The Greedy Triangle” or on it’s on to discover what really makes a triangle a triangle. (all ages)

Would You Rather…? Tasks for Google Slides : This unique set of challenges covers ratios & percents by giving students a scenario and then asking a “would you rather…?” question. This forces them to think about the situation and the implications of the math involved.

Which Cup Holds the Most Hot Chocolate? This challenge teaches kids to think about volume as they compare different size cups. (grades 7+)

Estimating the Area of a Circle : This fun geometry challenge will help kids estimate and think about how to find the area inside a circle. (grades 7+)

Jumping Maze : This fun challenge can be done with sidewalk chalk or with pencil and paper.

Decorate a Christmas Tree : This open ended challenge is great for all ages! It can be simple enough for Kindergarten or challenging enough for high school. (all ages)

Look for Patterns in Pascal’s Triangle : This set includes several different patterns to color and observe in Pascal’s Triangle. (grades 2+)

Analyze Math Mistakes : Help kids learn from their mistakes and see them as opportunities for brain growth with these templates & classroom posters.

>> Buy Error Analysis Here!

Order of Operations Error Analysis : Deepen an understanding of order of operations with this set of order of operations error analysis tasks.

Math Websites with More Meaningful Math Tasks:

YouCubed : This site is run by Jo Boaler and her team at Stanford and includes a variety of tasks for kids of all ages.

You can also find tasks that are part of the “Week of Inspirational Math” here .

NCTM Illuminations : This has great math tasks that you can search by grade level or math standard.

GeoGebra : Find math activities and resources for all ages using their online tools for exploration.

NRich : This site offers lots of support for teachers as well as fun problems open for solution. You can search for problems to try based on age, and then you can even submit your solutions to them! The best solutions get shared on their site.

Estimation 180 : Find fun and engaging estimation challenge to get your kids thinking and problem solving.

Visual Patterns : This site offers a huge assortment of visual patterns for kids to explore and extend. This was one of my favorite ways to teach new concepts when teaching Algebra 1. This site would have made it so much easier for me!

Number Strings : Number strings are sets of related math problems, designed to support students to construct big ideas about mathematics and build their own strategies. These sets of problems can be done in small groups or as a whole class.

Yummy Math : This is a great resource for finding real world math problems that you can present to students before they’ve actually learned a method.

Mindset Mathematics Curriculum Series:

Finally, I want to share some curriculum supplement resources from Jo Boaler and the YouCubed team. Here are the books available so far, with more coming out soon:

• Mindset Mathematics | Grade 3
• Mindset Mathematics | Grade 4
• Mindset Mathematics | Grade 5
• Mindset Mathematics | Grade 6
• Mindset Mathematics | Grade 7

So that is my huge list of rich math tasks that promote a growth mindset!

I know it might be overwhelming at first to go through all these resources, so try to pick just one or two tasks to try for now , and then continue to weave them into your math routine as you get comfortable.

A lot of these open ended types of explorations are very different from how math has traditionally been taught, so it can be an adjustment to shift how you teach and think about math.

So just take it one day and one task at a time and have fun learning alongside your kids!

One Comment

• Pingback: 100+ Growth Mindset Activities to Enjoy with Your Kids - Rock Your Homeschool

Comments are closed.

Similar Posts

{FREE} Least Common Multiple Practice for Grades 4-6

Deepening Fraction Sense with Fractions Area Models

{FREE} Skittles Math: Place Value Mats

Math Curse Extension Activities! {FREE!}

Have You Checked Out Noodle Education?

Number Detectives: Addition and Subtraction Challenge Problems

Find more resources to help make math engaging, join 165k+ parents & teachers.

Who learn new tips and strategies, as well as receive engaging resources to make math fun!

• Privacy Policy

Math Time Doesn't Have to End in Tears

Join 165,000+ parents and teachers who learn new tips and strategies, as well as receive engaging resources to make math fun. Plus, receive my guide, "5 Games You Can Play Today to Make Math Fun," as my free gift to get you started!

--> -->

• Visit Two Rivers Public Charter School to see the school that inspired the Two Rivers Learning Institute.
• Course Login

Problem-Based Tasks in Math

Want to learn more about our work?

Dive deeper into our Instructional Practices and methodology by taking one of our online courses.

Providing students with opportunities to grapple with math has led to amazing things happening in my class. Students are totally excited and are driven to figure out not just how to solve a problem but why it works.

– Jessica Proffitt, Fifth-Grade Teacher at Two Rivers

Watch two rivers’s teachers and students at work on problem-based tasks in math.

Problem-Based Tasks Require Students to Apply Their Knowledge in New Contexts

Problem-based tasks are math lessons built around a single, compelling problem. The problems are truly “problematic” for students — that is, they do not offer an immediate solution.

The problems provide an opportunity for students to build conceptual understanding. Problem-based tasks require students to apply their current understanding and skills to new contexts that highlight core math concepts. For example, when students solve a problem that could be solved with multiplication before they have formally been taught what multiplication is and how it works, they build an understanding that multiplication is repeated addition.

Well-designed problem-based tasks provide multiple entry points for students to engage in problem solving, ensuring that all students have access to the same concepts. When students solve the problems in different ways—including drawing pictures, acting out the problem, writing algorithms, and using manipulatives—they make connections between the variety of models that all accurately illustrate the underlying mathematics.

Problem-Based Tasks in Math Resources

Engage your students with effective distance learning resources. ACCESS RESOURCES>>

Content standards.

Use the navigational tools below to find illustrative tasks and other resources for each standard.

Select A Grade

Explore the standards by grade level.

Select A Domain

Explore the standards by domain.

• Open Middle Problems
• Real World Problems
• Online Workshops

Problem-Based Lesson Search Engine

This search engine searches all of the sites below to quickly help you find a problem-based lesson (also called 3-Act Task, mathematical modeling, or application problem):

The links below are the pages that are being searched by the search engine:

• 101 Questions
• Andrew Gael
• Andrew Stadel
• Catherine Castillo
• Dane Ehlert
• Emergent Math’s Problem Based Curriculum Maps
• Geoff Krall
• Graham Fletcher
• Kendra Lomax
• Kristen M. Acosta
• Kyle Pearce
• Jennifer Barker
• John Scammell
• Matt Vaudrey
• Michael Van Etten
• Mike Wiernicki
• Robert Kaplinsky
• Timon Piccini

There must be many great sources of lessons that I am missing.  Please leave me a comment to let me know which websites I need to add to the search engine.

71 Comments

Wow! This is amazing! Thank you so much for doing this. One other great site I’ve found is HungryTeacher.com It has some very cool project ideas, similar to Mathalicious but not as detailed. Keep up the great work!

@EdCamposJR

Thank you Ed. I actually added HungryTeacher.com to the search engine but forgot to add it to the list of sites on this page so thanks for the heads up.

This is going be amazing for me.

This is so cool and helpful. Thank you so much for this. http://www.gogeometry.com has some challenging geometry problems.

Hi Teresa. Glad it is useful. I searched around gogeometry.com but couldn’t find anything like a problem-based lesson. Could you give me a specific link to a lesson to check out?

Thank you for this search engine. It is very helpful.

What about adding this site? http://ell.stanford.edu/teaching_resources/math

There are only a few tasks here but they focus in on our ELs.

Hi Jennifer. There are some cool lessons there but I don’t know if they would fall under the category of problem based lessons I am aspiring to capture here. Thanks for your idea!

• Pingback: Favorited Tweets #1 | Growing Exponentially

nice work, could you share how you set this up? I like the idea of embedding a search to only look through certain sites.

Also I suggest Dan Meyer’s Graphing Stories, Fawn Nguyen’s blog and the math lab: http://www.themathlab.com/Algebra/linear%20functions%20regressions%20slope/regression%20lessons/barbie%20bungee/barbbungee.htm

Hi Shaun. Check out Google’s Custom Search at https://www.google.com/cse/ . That is where you can create your own search engine. It is rather robust but not perfect. For example, it is very challenging to include a site but exclude the blog posts.

Regarding your suggestions, those are all great sites but Graphing Stories and Fawn’s awesome blog are not really within the problem-based learning domain so that is why I haven’t included them. That being said, both of them have been links on the right side of my blog for quite some time so I definitely value them both.

I am having trouble getting the Barbie Bungee jump to load but I will try again another time to check it out.

This is an old post, but I am trying to develop my own 3Acts and have a resource page of them as well.

Hi Bryan. All of your lessons are included since they are posted on 101qs.com. Thanks!

Amazing resource!!! Thank You for creating this search engine!!! 🙂

• Pingback: Where to find the activities? | judylarsen
• Pingback: Résolution de problème | Pearltrees

Have you checked out the website: http://davidwees.com/

He has some pretty useful articles, especially on strategies for formative assessment.

Full disclosure: He’s a colleague of mine so I am biased, but I do get a lot out of what he shares and apply to my own work.

Thanks Blue. David is great but I don’t believe he is creating any problem-based lessons. If he is, please list a link to where I can find them.

http://reasonandwonder.com

Thanks Kevin. Can’t believe I forgot Michael. Problem fixed.

Rob, have you come across Tuva yet? Focused on data-based problems for math, science, and many other subjects. You can check it out at tuvalabs.com.

Hi, thanks for the resource!

Would you consider these to be examples of problem-based learning?

https://aofradkin.wordpress.com/2015/03/18/facing-the-impossible/ https://aofradkin.wordpress.com/2014/11/11/stick-figures/ https://aofradkin.wordpress.com/2014/10/10/fun-with-tetraminos/

Hello. To me the answer is both yes and no. Clearly these are problems that you can base a lesson around. However they are not of the style that we are including in this search engine. If you are interested in examples of elementary tasks of this style, check out Graham Fletcher’s problems here: http://gfletchy.com/3-act-lessons/ .

I was so excited to find this resource that I immediately emailed all of the teachers in my department a link to this post. Then, I realized that I had a double block of Algebra I approaching in 15 minutes and I could use this search tool to find a resource that might raise the engagement level for my students. Sure enough…I typed “graphing quadratic equations” into the search box, and numerous resources appeared. I selected the second hit on “Angry Birds”,and it fit seamlessly into my lesson plan. The students asked such great questions and were able to clearly answer questions about the vertex, x-intercepts, and orientation of the quadratic equations that modeled the paths of the angry bird. Of course, then I had to send out another email to my department ranting and raving about my math rush.

Isn’t it cool when it works out! Glad to hear it was useful for you in a tight situation.

• Pingback: Problem-Based Learning Frequently Asked Questions | Robert Kaplinsky - Glenrock Consulting, LLC
• Pingback: GAFEnideas | Pearltrees

What about http://www.openmiddle.com ? Thanks so much for this! You are a life saver.

Hi Alex. Obviously I love Open Middle, but it doesn’t seem to be a good fit here. This search engine is more for math modeling problems also known as 3-Act Tasks or problem-based lessons. What do you think?

Will. Change. My. Life. So excited to share this!!!!

Hi Robert, I will be sharing this with my colleagues at school that focus on the Math contents… Thanks for a great resource on behalf of all those teacher who need a little more help in the topics area.

That’s great Sonia. I hope they find it useful.

Thanks for this very useful resource.

Great search engine. Didn’t know that you could have a custom search engine.

Have you looked at nzmaths?

https://nzmaths.co.nz/problem-solving

Hi Ben. I haven’t heard of this but it seems like there are some useful problems there. I’ll add it to the list. Much appreciated!

I LOVE THIS! I’m so excited to have a search engine that will only search for mathematical tasks! I’m sharing it with everyone I know:). Thank you so much, Robert!!

Glad you like it Karen. Definitely a time saver.

Hi Robert. I love that you did this. It makes it easier to get teachers to try these out. I also have another site – tapintoteenminds.com. He has developed some tasks using the 3-Act format.

Hi Nadirah. I actually have that site on the list under Kyle Pearce, who created that site and those lessons.

Of course you did Robert! Silly me.

Thanks! This is amazing and such a great idea! Is there a way to embed this link into our own sites?

Can you explain more about what you mean to “embed this link”? You can put a link to this blog post on your own site, that’s no problem.

• Pingback: What makes PD stick? – Making Sense and Persevering

Hey! I think this is a great list of resources and so thankful for the list. It has saved me in the last a lot. I am in my grad school class and researching on PBL. I am a little confused how this qualifies has PBL and not just projects or really good math lessons. My research suggests that with PBL the learning is happening through the project and the project is not a activity we tack on at the end. I also read, “While the learning context is common to all groups, the paths may differ considerably—all leading to distinct learning. In project-based learning all students engage in a common project with unclear processes but clearly identified expected outcomes.” Can you speak a little bit about your thoughts on PBL. I am trying to find some math teachers I look up to and follow to help me learn about PBL in a secondary classroom. Please respond or email me at [email protected]

Hi Virginia. Thanks for this question. I should be clear that this is a PROBLEM-based search engine and not a PROJECT-based search engine. While the two have similarities, it sounds like you are talking about project based lessons and not what I am focusing on.

I don’t do a lot with project based learning so unfortunately I’m not a good resource for you there. Sorry! If you want to learn more about problem based learning, you can read more here: http://robertkaplinsky.com/tag/problem-based-learning-2/ .

• Pingback: 5 ways to include yourself in the online math community - without being on twitter. #MTBoS - Sara VanDerWerf

Youcubed certainly has many options as well!

Thanks Merryl. I’m not aware of any problems that Youcubed has that would be considered problem-based lessons like 3-act tasks and the ones on my site. If you can point me in the right direction, I’d love to check them out.

This is awesome! Is there any way to take out Mathalicious since it is now requires a subscription?

Hi Danielle. I understand where you’re coming from, but that isn’t a problem for me. You often get what you pay for, and as much as it feels nice to get lessons for free, the Mathalicious team’s full time job is creating high quality lessons for educators. They deserve to be compensated.

Personally, I prefer to be aware of what’s out there so that I can choose what’s best from me. Sometimes it will be worth paying for.

Have you looked at illustrative mathematics or exemplars? Both have good problem based questions.

Thanks Sarah. I think this issue is stems from the word “problem” being so broad. Yes, I have looked at those problems, but they aren’t like the 3-act tasks and other problem-based lessons I am looking for with this search engine. I want problems that are more like these: https://robertkaplinsky.com/lessons/

Is it possible to add the Phillips Exeter sets? Could it search through their PDF files?

I’m not sure Dave. Do you have a link to an example?

What about Ilustrative Mathematics? https://www.illustrativemathematics.org/

Hi Brian. I think this issue is stems from the word “problem” being so broad. Yes, I have looked at those problems, but they aren’t like the 3-act tasks and other problem-based lessons I am looking for with this search engine. I want problems that are more like these: https://robertkaplinsky.com/lessons/

• Pingback: Finding Resources * June 14 PD - Sara VanDerWerf
• Pingback: Taking Risks by @marvelousmcc - Teacher Tech
• Pingback: Performance Tasks and performance tasks | Matt Vaudrey

So many comments – did anyone else suggested the Illustrative Mathematics site

Hi Angela. Yes, a few others have suggested it. It’s not a good fit for the kinds of problems I’m referring to (which are more like you’d find on my lessons page or 3-act tasks). Thanks!

Hi. Thanks for creating that.

I’m not sure if either of these qualify, but one you may want to consider is https://mathsolutions.com/free-resources/ because it has stuff from Marilyn Burns.

I also love Steve Wyborney’s site: https://stevewyborney.com/2018/11/esti-mysteries-estimation-meets-math-mysteries/

My students really enjoy the esti-mysteries.

Thanks Jeff. Those are all great resources. The kinds of problems we’re trying to include here are more real-world or three-act task style problems, so this is not a great fit for this particular search engine.

This is AWESOME! Thank you. Wide Open School ( https://wideopenschool.org/#grades-6-8/ ) is a great resource also.

Thanks Jenny. There’s lots of great stuff there. Unfortunately, the kinds of problems we’re trying to include here are more real-world or three-act task style problems, so this is not a great fit for this particular search engine.

What a wonderful one-stop shop for problem solving tasks. I would love to link them all to my site called CURIOSEDY. I wonder if you think Curiosedy could be added to the list as it has open-ended rich tasks as well as multi-step problem solving. Would love your thoughts. Thanks again for this.

Thanks for sharing this, Francois. I can see lots of potential in what you’re doing. The kinds of tasks in this search engine (more like 3-act tasks) are a bit different from your approach, but both have value. So it’s not a great fit for this search engine but keep it going as it’ll help other students.

Hi Robert, Mark Chubb has some great stuff, especially around spatial reasoning. He deserves a spot! Thanks.

Mark definitely has some great resources. Unfortunately they’re not the kinds of problems for this search engine. I’m thinking more like problems with real world context.

I believe the hyperlink for Beth Brandenburg is not working.

Thanks for catching that. I guess her site is no longer active. I just went through all of them and updated it accordingly.

Hi Robert, Thanks for sharing. Check out https://www.resolve.edu.au/teaching-resources There are many lessons on this that have problem solving and inquiry tasks. My favourites are the circumference and area of circles. Hope this fits the style of tasks you are looking for.

Thanks Lorna. This is a bit different style of problem than what I’m going for. Ideally, I’m looking for problems that are more contextual without necessarily being a word problem. These are definitely great problems, but a different kind than I’m using with this search engine. Thanks again.

Leave a Reply Cancel reply

Your email address will not be published. Required fields are marked *

Save my name, email, and website in this browser for the next time I comment.

Sign me up to receive updates including new blog posts, lessons, and events.

Don't subscribe All new comments Replies to my comments Notify me of followup comments via e-mail. You can also subscribe without commenting.

Post comment

• Skip to main content
• Skip to primary sidebar
• Skip to footer

Additional menu

Khan Academy Blog

Free Math Worksheets — Over 100k free practice problems on Khan Academy

Looking for free math worksheets.

You’ve found something even better!

That’s because Khan Academy has over 100,000 free practice questions. And they’re even better than traditional math worksheets – more instantaneous, more interactive, and more fun!

Just choose your grade level or topic to get access to 100% free practice questions:

Kindergarten, basic geometry, pre-algebra, algebra basics, high school geometry.

• Trigonometry

Statistics and probability

High school statistics, ap®︎/college statistics, precalculus, differential calculus, integral calculus, ap®︎/college calculus ab, ap®︎/college calculus bc, multivariable calculus, differential equations, linear algebra.

• Addition and subtraction
• Place value (tens and hundreds)
• Addition and subtraction within 20
• Addition and subtraction within 100
• Addition and subtraction within 1000
• Measurement and data
• Counting and place value
• Measurement and geometry
• Place value
• Measurement, data, and geometry
• Add and subtract within 20
• Add and subtract within 100
• Add and subtract within 1,000
• Money and time
• Measurement
• Intro to multiplication
• 1-digit multiplication
• Addition, subtraction, and estimation
• Intro to division
• Understand fractions
• Equivalent fractions and comparing fractions
• More with multiplication and division
• Arithmetic patterns and problem solving
• Quadrilaterals
• Represent and interpret data
• Multiply by 1-digit numbers
• Multiply by 2-digit numbers
• Factors, multiples and patterns
• Add and subtract fractions
• Multiply fractions
• Understand decimals
• Plane figures
• Measuring angles
• Area and perimeter
• Units of measurement
• Decimal place value
• Add decimals
• Subtract decimals
• Multi-digit multiplication and division
• Divide fractions
• Multiply decimals
• Divide decimals
• Powers of ten
• Coordinate plane
• Algebraic thinking
• Converting units of measure
• Properties of shapes
• Ratios, rates, & percentages
• Arithmetic operations
• Negative numbers
• Properties of numbers
• Variables & expressions
• Equations & inequalities introduction
• Data and statistics
• Negative numbers: addition and subtraction
• Negative numbers: multiplication and division
• Fractions, decimals, & percentages
• Rates & proportional relationships
• Expressions, equations, & inequalities
• Numbers and operations
• Solving equations with one unknown
• Linear equations and functions
• Systems of equations
• Geometric transformations
• Data and modeling
• Volume and surface area
• Pythagorean theorem
• Transformations, congruence, and similarity
• Arithmetic properties
• Factors and multiples
• Reading and interpreting data
• Negative numbers and coordinate plane
• Ratios, rates, proportions
• Equations, expressions, and inequalities
• Exponents, radicals, and scientific notation
• Foundations
• Algebraic expressions
• Linear equations and inequalities
• Graphing lines and slope
• Expressions with exponents
• Quadratics and polynomials
• Equations and geometry
• Algebra foundations
• Solving equations & inequalities
• Working with units
• Linear equations & graphs
• Forms of linear equations
• Inequalities (systems & graphs)
• Absolute value & piecewise functions
• Exponents & radicals
• Exponential growth & decay
• Quadratics: Multiplying & factoring
• Quadratic functions & equations
• Irrational numbers
• Performing transformations
• Transformation properties and proofs
• Right triangles & trigonometry
• Non-right triangles & trigonometry (Advanced)
• Analytic geometry
• Conic sections
• Solid geometry
• Polynomial arithmetic
• Complex numbers
• Polynomial factorization
• Polynomial division
• Polynomial graphs
• Rational exponents and radicals
• Exponential models
• Transformations of functions
• Rational functions
• Trigonometric functions
• Non-right triangles & trigonometry
• Trigonometric equations and identities
• Analyzing categorical data
• Displaying and comparing quantitative data
• Summarizing quantitative data
• Modeling data distributions
• Exploring bivariate numerical data
• Study design
• Probability
• Counting, permutations, and combinations
• Random variables
• Sampling distributions
• Confidence intervals
• Significance tests (hypothesis testing)
• Two-sample inference for the difference between groups
• Inference for categorical data (chi-square tests)
• Advanced regression (inference and transforming)
• Analysis of variance (ANOVA)
• Scatterplots
• Data distributions
• Two-way tables
• Binomial probability
• Normal distributions
• Displaying and describing quantitative data
• Inference comparing two groups or populations
• Chi-square tests for categorical data
• More on regression
• Prepare for the 2020 AP®︎ Statistics Exam
• AP®︎ Statistics Standards mappings
• Polynomials
• Composite functions
• Probability and combinatorics
• Limits and continuity
• Derivatives: definition and basic rules
• Derivatives: chain rule and other advanced topics
• Applications of derivatives
• Analyzing functions
• Parametric equations, polar coordinates, and vector-valued functions
• Applications of integrals
• Differentiation: definition and basic derivative rules
• Differentiation: composite, implicit, and inverse functions
• Contextual applications of differentiation
• Applying derivatives to analyze functions
• Integration and accumulation of change
• Applications of integration
• AP Calculus AB solved free response questions from past exams
• AP®︎ Calculus AB Standards mappings
• Infinite sequences and series
• AP Calculus BC solved exams
• AP®︎ Calculus BC Standards mappings
• Integrals review
• Integration techniques
• Thinking about multivariable functions
• Derivatives of multivariable functions
• Applications of multivariable derivatives
• Integrating multivariable functions
• Green’s, Stokes’, and the divergence theorems
• First order differential equations
• Second order linear equations
• Laplace transform
• Vectors and spaces
• Matrix transformations
• Alternate coordinate systems (bases)

Frequently Asked Questions about Khan Academy and Math Worksheets

Why is khan academy even better than traditional math worksheets.

Khan Academy’s 100,000+ free practice questions give instant feedback, don’t need to be graded, and don’t require a printer.

Math Worksheets	Khan Academy
Math worksheets take forever to hunt down across the internet	Khan Academy is your one-stop-shop for practice from arithmetic to calculus
Math worksheets can vary in quality from site to site	Every Khan Academy question was written by a math expert with a strong education background
Math worksheets can have ads or cost money	Khan Academy is a nonprofit whose resources are always free to teachers and learners – no ads, no subscriptions
Printing math worksheets use up a significant amount of paper and are hard to distribute during virtual learning	Khan Academy practice requires no paper and can be distributed whether your students are in-person or online
Math worksheets can lead to cheating or a lack of differentiation since every student works on the same questions	Khan Academy has a full question bank to draw from, ensuring that each student works on different questions – and at their perfect skill level
Math worksheets can slow down student learning since they need to wait for feedback	Khan Academy gives instant feedback after every answer – including hints and video support if students are stuck
Math worksheets take up time to collect and take up valuable planning time to grade	Khan Academy questions are graded instantly and automatically for you

What do Khan Academy’s interactive math worksheets look like?

Here’s an example:

What are teachers saying about Khan Academy’s interactive math worksheets?

“My students love Khan Academy because they can immediately learn from their mistakes, unlike traditional worksheets.”

Is Khan Academy free?

Khan Academy’s practice questions are 100% free—with no ads or subscriptions.

What do Khan Academy’s interactive math worksheets cover?

Our 100,000+ practice questions cover every math topic from arithmetic to calculus, as well as ELA, Science, Social Studies, and more.

Is Khan Academy a company?

Khan Academy is a nonprofit with a mission to provide a free, world-class education to anyone, anywhere.

Want to get even more out of Khan Academy?

Then be sure to check out our teacher tools . They’ll help you assign the perfect practice for each student from our full math curriculum and track your students’ progress across the year. Plus, they’re also 100% free — with no subscriptions and no ads.

Get Khanmigo

The best way to learn and teach with AI is here. Ace the school year with our AI-powered guide, Khanmigo. 

For learners     For teachers     For parents

K-5 Math Centers

K-5 math ideas, 3rd grade math, need help organizing your k-5 math block, 5 ways to include math problem solving activities in your classroom.

Are you looking for math problem solving activities that are not too easy and not too hard, but juuust right? I’ve got something just for you and your students.

Solve and Explain Problem Solving Tasks are open-ended math tasks that provide just the right amount of challenge for your kids. Here’s a little more about them.

Open-ended math problem solving tasks:

• promote multiple solution paths and/or multiple solutions
• boost critical thinking and math reasoning skills
• increase opportunities for developing perseverance
• provide opportunities to justify answer choices
• strengthen kids written and oral communication skills

What Makes These So Great?

• All Common Core Standards are covered for your grade level
• 180+ Quality questions that are rigorous yet engaging
• They are SUPER easy to assemble
• Provide opportunities for meaningful math discussions
• Perfect for developing a growth mindset
• Easily identify student misconceptions so you can provide assistance
• Very versatile (check out the different ways to use them below)

You can find out more details for your grade level by clicking on the buttons below.

I’m sure you really want to know how can you use these with your kids. Check out the top 5 ideas on how to use Solve and Explain Problem Solving Tasks in your classroom.

How and When Can I Use Them?

Solve and Explain Tasks Cards are very versatile. You can use them for:

• Math Centers  – This is my favorite way to use these! Depending on your grade level, there are at least two (Kinder – 2nd) or three (3rd-5th) tasks types per Common Core standard. And each task type has 6 different questions. Print out each of the different tasks types on different color paper. Then, let students choose which one question from each task type they want to solve.

• Problem of the Day  – Use them as a daily math journal prompt. Print out the recording sheet and project one of the problems on your white board or wall.  Students solve the problem and then glue it in their spiral or composition notebooks.

• Early Finisher Activities  -No more wondering what to do next!Create an early finishers notebook where students can grab a task and a recording sheet. Place the cards in sheet protectors and make copies of the Early Finisher Activity Check-Off card for your kids to fill out BEFORE they pull a card out to work on. We want to make sure kids are not rushing through there first assignment before moving on to an early finisher activity.

• Weekly Math Challenges  – Kids LOVE challenges! Give students copies of one of the problems for homework. Then give them a week to complete it. Since many of the questions have multiple solutions and students have to explain how they got their answers, you can have a rich whole group discussion at the end of the week (even with your kindergarten and 1st grade students).

Shop Recommended Resources

• Formative Assessments  – Give your students a problem to solve. Then use the Teacher Scoring Rubric to see how your kids are doing with each standard. Since they have to explain their thinking, this is a great way to catch any misconceptions and give feedback to individual students.

So this wraps up the top 5 ways that you can use problem solving tasks in your classroom.  Click your grade level below to get Solve and Explain problem solving tasks for your classroom.

• Read more about: K-5 Math Ideas

You might also like...

Reflect and Reset: Tips for Becoming a Better Math Teacher

Student Math Reflection Activities That Deepen Understanding

5 Math Mini-Lesson Ideas that Keep Students Engaged

A Rigorous Elementary Math Curriculum for Busy Teachers

What We Offer:

Follow us here:.

10 Math Tasks for the Beginning of the Year

I have a rule about the first day of school: always do some math. No, that doesn't mean you have to start Lesson 1.1 as soon as students walk through the door, but it does mean that you should give your students a preview of the kind of thinking, reasoning, puzzling, and sense-making that they'll be doing in your class this year. Ideally, students will be so highly engaged that they barely even recognize they're doing math -- and certainly not the kind of math they're used to in school.

I also have an inordinate appreciation for what I call "interesting problems". These are tasks that use mathematical thinking and strategy, but don't require specific content knowledge like the formula for the equation of a circle or knowing what a composite function does. They are highly accessible, highly engaging, and have multiple solution strategies. The task itself can be explained in a few sentences and students can work on them for 20 minutes or 2 hours, depending on how far they want to take it. I scour the internet for tasks like these and have been collecting them for YEARS on my computer. I decided this was the year to bring them to the light and share them with the Math Medic community.

These tasks don't require formal content knowledge, but they do help students engage in the mathematical practices and develop mathematical habits of mind, such as:

Looking for and making use of structure

Representing one's thinking

Working systematically

Visualizing

Developing a convincing argument

Conjecturing and generalizing

While I've curated this list with high school students in mind, many of these tasks could be done with middle schoolers or even with adults. The inspiration for these questions came from all over this great big internet, but have been adapted and reformatted for classroom use. So, without further ado, here are my (current) top 10 "interesting problems" to do on the first day of school.

10 Interesting Problems

The Shopping Cart Task

A linear context in a LOT of disguise. Many solution strategies and great opportunities for representing one's thinking with a model or visual.

The Locker Problem

This one is set up with multiple parts providing lots of natural extensions. Thinking about a number's properties is key to this task! Make sure to print the 100s chart that is on page two on a separate sheet of paper. You can offer it to everyone or as an optional support.

A Leg to Stand On

Loads of solution strategies on this one as well. Your teacher brain might scream system of equations with 4 variables, but you'll be surprised at the intuitive solutions your students find to solve this problem.

To Run or Not to Run?

Perfect after a summer of olympics. Students deal with rates in this problem, which is an important concept for any age group and relevant for any math course.

Four 4's

This one's been famous for a long time but I'm sharing it anyway because students do great with it!

Page Turner

This one and the next two all encourage students to think systematically. There's a brute force solution but making use of structure will illuminate an easier way.

How Many 7's?

This is a good intro to thinking systematically and has a nice extension. I would use this in an Algebra 1 or Geometry course.

Oddball Numbers

This one is very difficult, so we recommend saving it for your Precalc or above courses.

This one is the most recent in my collection and I'm still thinking about the extension part!

Toothpick Challenge

I've often done this one with Geometry students because of the shapes and visual reasoning components.

Editable versions of these tasks can be found in this Google Drive folder .

How to use these in your classroom:

Pick ONE task for students to work on. We don't recommend giving multiple tasks back-to-back because it can start to feel like a worksheet, rather than a puzzle.

Solve the problem yourself first! We are purposefully not giving solutions here , so make sure you've wrestled with the problem yourself before handing it out to students.

Have students work on these in groups of 2, 3, or 4. Make sure they have enough materials available to hash out their ideas and represent their strategies. These are great to do on vertical non-permanent surfaces or poster paper.

Decide how long you will let students work. If doing this on the first day of school, we recommend about 20 minutes. If students don't have a solution by then, that is totally fine. A surprising number of them will keep thinking about the problem throughout the day or even at home.

Be ready with some extensions for groups who finish early, but make sure they understand what "done" means. Have they clearly communicated their strategy? Have they convinced themselves and others that their strategy will hold up? If giving an extension, make sure it's related to the given task, not just a different task. It's important that students are challenged in the depth of their reasoning, not in the quantity of problems.

If you're looking for more tasks like these, I highly recommend the NRICH site from the University of Cambridge.

• Math Medic Core

Recent Posts

What Does It Mean to "Make Sense of Problems and Persevere in Solving Them?"

What Does It Mean to "Construct Viable Arguments and Critique the Reasoning of Others"?

5 Myths About Math

Or search by topic

Number and algebra.

• Place value and the number system
• Fractions, decimals, percentages, ratio and proportion
• Calculations and numerical methods
• Algebraic expressions, equations and formulae
• Coordinates, functions and graphs
• Patterns, sequences and structure
• Properties of numbers

Geometry and measure

• 3D geometry, shape and space
• Transformations and constructions
• Vectors and matrices
• Measuring and calculating with units
• Pythagoras and trigonometry
• Angles, polygons, and geometrical proof

Probability and statistics

• Handling, processing and representing data
• Probability (spec_group)

Working mathematically

• Thinking mathematically
• Mathematical mindsets

Advanced mathematics

• Decision mathematics and combinatorics
• Advanced probability and statistics

For younger learners

• Early years foundation stage

Problem Solving

Problem solving and the new curriculum

Developing a classroom culture that supports a problem-solving approach to mathematics

Developing excellence in problem solving with young learners

Using NRICH Tasks to Develop Key Problem-solving Skills

Trial and Improvement at KS1

Trial and Improvement at KS2

Working Systematically - Primary teachers

Number Patterns

Working Backwards at KS1

Working Backwards at KS2

Visualising at KS1 - Primary teachers

Visualising at KS2 - Primary teachers

Conjecturing and Generalising at KS1 - Primary teachers

Conjecturing and Generalising at KS2 - Primary teachers

Mathematical Problem Solving in the Early Years

Low threshold high ceiling - an introduction.

What's all the talking about?

Group-worthy tasks and their potential to support children to develop independent problem-solving skills

Developing the classroom culture: using the Dotty Six Activity as a springboard for investigation