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20 Math Critical Thinking Questions to Ask in Class Tomorrow

• November 20, 2023

The level of apathy towards math is only increasing as each year passes and it’s up to us as teachers to make math class more meaningful . This list of math critical thinking questions will give you a quick starting point for getting your students to think deeper about any concept or problem. 

Since artificial intelligence has basically changed schooling as we once knew it, I’ve seen a lot of districts and teachers looking for ways to lean into AI rather than run from it.

The idea of memorizing formulas and regurgitating information for a test is becoming more obsolete. We can now teach our students how to use their resources to make educated decisions and solve more complex problems.

With that in mind, teachers have more opportunities to get their students thinking about the why rather than the how.

Table of Contents

Looking for more about critical thinking skills? Check out these blog posts:

• Why You Need to Be Teaching Writing in Math Class Today
• How to Teach Problem Solving for Mathematics
• Turn the Bloom’s Taxonomy Verbs into Engaging Math Activities

What skills do we actually want to teach our students?

As professionals, we talk a lot about transferable skills that can be valuable in multiple jobs, such as leadership, event planning, or effective communication. The same can be said for high school students. 

It’s important to think about the skills that we want them to have before they are catapulted into the adult world. 

Do you want them to be able to collaborate and communicate effectively with their peers? Maybe you would prefer that they can articulate their thoughts in a way that makes sense to someone who knows nothing about the topic.

Whatever you decide are the most essential skills your students should learn, make sure to add them into your lesson objectives.

When should I ask these math critical thinking questions?

Critical thinking doesn’t have to be complex or fill an entire lesson. There are simple ways that you can start adding these types of questions into your lessons daily!

Start small

Add specific math critical thinking questions to your warm up or exit ticket routine. This is a great way to start or end your class because your students will be able to quickly show you what they understand. 

Asking deeper questions at the beginning of your class can end up leading to really great discussions and get your students talking about math.

Add critical thinking questions to word problems

Word problems and real-life applications are the perfect place to add in critical thinking questions. Real-world applications offer a more choose-your-own-adventure style assignment where your students can expand on their thought processes. 

They also allow your students to get creative and think outside of the box. These problem-solving skills play a critical role in helping your students develop critical thinking abilities.

Keep reading for math critical thinking questions that can be applied to any subject or topic!

When you want your students to defend their answers.

• Explain the steps you took to solve this problem
• How do you know that your answer is correct?
• Draw a diagram to prove your solution.
• Is there a different way to solve this problem besides the one you used?
• How would you explain _______________ to a student in the grade below you?
• Why does this strategy work?
• Use evidence from the problem/data to defend your answer in complete sentences.

When you want your students to justify their opinions

• What do you think will happen when ______?
• Do you agree/disagree with _______?
• What are the similarities and differences between ________ and __________?
• What suggestions would you give to this student?
• What is the most efficient way to solve this problem?
• How did you decide on your first step for solving this problem?

When you want your students to think outside of the box

• How can ______________ be used in the real world?
• What might be a common error that a student could make when solving this problem?
• How is _____________ topic similar to _______________ (previous topic)?
• What examples can you think of that would not work with this problem solving method?
• What would happen if __________ changed?
• Create your own problem that would give a solution of ______________.
• What other math skills did you need to use to solve this problem?

Let’s Recap:

• Rather than running from AI, help your students use it as a tool to expand their thinking.
• Identify a few transferable skills that you want your students to learn and make a goal for how you can help them develop these skills.
• Add critical thinking questions to your daily warm ups or exit tickets.
• Ask your students to explain their thinking when solving a word problem.
• Get a free sample of my Algebra 1 critical thinking questions ↓

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• Critical Thinking

How To Encourage Critical Thinking in Math

By Mary Montero

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Critical thinking is more than just a buzzword… It’s an essential skill that helps students develop problem-solving abilities and make logical connections between different concepts. By encouraging critical thinking in math, students learn to approach problems more thoughtfully, they learn to analyze and evaluate math concepts, identify patterns and relationships, and explore different strategies for finding the solution. Critical thinking also involves a great deal of persistence. Those are critical life skills!

When you think about it, students are typically asked to solve math problems and find the answer. Showing their work is frequently stressed too, which is important, but not the end. Instead, students need to be able to look at math in different ways in order to truly grasp a complete understanding of math concepts. Mathematics requires logical reasoning, problem-solving, and abstract thinking.

What Does Critical Thinking in Math Look Like?

When I think about critical thinking in math, I focus on:

• Solving problems through logical thinking . Students learn how to break down complex problems, analyze the different parts, and understand how they fit together logically.
• Identifying patterns and making connections. Students learn how to identify patterns across different math concepts, make connections between seemingly unrelated topics, and develop a more in-depth understanding of how math works.
• Evaluating and comparing solutions. Students learn to evaluate which solution is best for a given problem and identify any flaws in their reasoning or others’ reasoning when looking at different solutions

Mathematician Posters

These FREE Marvelous Mathematician posters have been a staple in my classroom for the last 8+ years! I first started using a version from MissMathDork and adapted them for my classroom over the years. 

I print, laminate, and add magnetic stickers on the back. At the beginning of the year, I only put one or two up at a time depending on our area of focus. Now, they are all hanging on my board, and I’ll pull out different ones depending on our area of focus. They are so empowering to my mathematicians and help them stay on track!

A Marvelous Mathematician:

• knows that quicker doesn’t mean better
• looks for patterns
• knows mistakes happen and keeps going
• makes sense of the most important details
• embraces challenges and works through frustrations
• uses proper math vocabulary to explain their thinking
• shows their work and models their thinking
• discusses solutions and evaluates reasonableness
• gives context by labeling answers
• applies mathematical knowledge to similar situations
• checks for errors (computational and conceptual)

Critical Thinking Math Activities

Here are a few of my favorite critical thinking activities. 

Square Of Numbers

I love to incorporate challenge problems (use Nrich and Openmiddle to get started) because they teach my students so much more than how to solve a math problem. They learn important lessons in teamwork, persistence, resiliency, and growth mindset. We talk about strategies for tackling difficult problems and the importance of not giving up when things get hard.

This square of numbers challenge was a hit!

ALL kids need to feel and learn to embrace challenge. Oftentimes, kids I see have rarely faced an academic challenge. Things have just come easy to them, so when it doesn’t, they can lack strategies that will help them. In fact, they will often give up before they even get started.

I tell them it’s my job to make sure I’m helping them stretch and grow their brain by giving them challenges. They don’t love it at first, but they eventually do! 

This domino challenge was another one from Nrich . I’m always on the hunt for problems like this!!  How would you guide students toward an answer??

Fifteen Cards

This is a well-loved math puzzle with my students, and it’s amazing for encouraging students to consider all options when solving a math problem.

We have number cards 1-15 (one of each number) and only seven are laid out. With the given clues, students need to figure out which seven cards should be put out and in what order. My students love these, and after they’ve done a few, they enjoy creating their own, too! Use products, differences, and quotients to increase the challenge.

This is also adapted from Nrich, which is an AMAZING resource for math enrichment!

This is one of my favorite fraction lessons that I’ve done for years! Huge shout out to Meg from The Teacher Studio for this one. I give each child a slip of paper with this figure and they have to silently write their answer and justification. Then I tally up the answers and have students take a side and DEBATE with their reasoning! It’s an AMAZING conversation, and I highly recommend trying it with your students. 

Sometimes we leave it hanging overnight and work on visual models to make some proofs. 

Logic Puzzles

Logic puzzles are always a hit too! You can enrich and extend your math lessons with these ‘Math Mystery’ logic puzzles that are the perfect challenge for 4th, 5th, and 6th grades. The puzzles are skills-based, so they integrate well with almost ANY math lesson. You can use them to supplement instruction or challenge your fast-finishers and gifted students… all while encouraging critical thinking about important math skills!

Three levels are included, so they’re perfect to use for differentiation.

• Introductory logic puzzles are great for beginners (4th grade and up!)
• Advanced logic puzzles are great for students needing an extra challenge
• Extra Advanced logic puzzles are perfect for expert solvers… we dare you to figure these puzzles out! 

Do you have a group of students who are ready for more of a fraction challenge? My well-loved fraction puzzlers are absolutely perfect for fraction enrichment. They’ll motivate your students to excel at even the most challenging tasks! 

Math Projects

Math projects are another way to differentiation while building critical thinking skills. Math projects hold so much learning power with their real-world connections, differentiation options, collaborative learning opportunities, and numerous avenues for cross curricular learning too. 

If you’re new to math projects, I shared my best tips and tricks for using math projects in this blog post . They’re perfect for cumulative review, seasonal practice, centers, early finisher work, and more.

I use both concept-based math projects to focus on specific standards and seasonal math projects that integrate several skills.

Error Analysis

Finally, error analysis is always a challenging way to encourage critical thinking. When we use error analysis, we encourage students to analyze their own mistakes to prevent making the same mistakes in the future.

For my gifted students, I use error analysis tasks as an assessment when they have shown mastery of a unit during other tasks. For students in the regular classroom needing enrichment, I usually have them complete the tasks in a center or with a partner.

For students needing extra support, we complete error analysis in small groups.  We go step-by-step through the concept and they are always able to eventually identify what the error is. It is so empowering to students when they finally figure out the error AND it helps prevent them from making the same error in the future!

My FREE addition error analysis is a good place to start, no matter the grade level. I show them the process of walking through the problem and how best to complete an error analysis task.

When you’re ready for more, this bundle of error analysis tasks contains more than 240 tasks to engage and enrich your students in critical thinking practice.

If you want to dig even deeper, visit this conceptual vs computational error analysis post to learn more about using error analysis in the classroom. 

Related Critical Thinking Posts

• How to Increase Critical Thinking and Creativity in Your “Spare” Time
• More Tips to Increase Critical Thinking

Critical thinking is essential for students to develop a deeper understanding of math concepts, problem-solving skills, and a stronger ability to reason logically. When you learn how to encourage critical thinking in math, you’re setting your students up for success not only in more advanced math subjects they’ll encounter, but also in life. 

How do you integrate critical thinking in your classroom? Come share your ideas with us in our FREE Inspired In Upper Elementary Facebook group .

Mary Montero

I’m so glad you are here. I’m a current gifted and talented teacher in a small town in Colorado, and I’ve been in education since 2009. My passion (other than my family and cookies) is for making teachers’ lives easier and classrooms more engaging.

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Mary Thankyou for your inspirational activities. I have just read and loved the morning talk activities. I do have meetings with my students but usually at end of day. What time do you

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Engaging Maths

Dr catherine attard, promoting creative and critical thinking in mathematics and numeracy.

• by cattard2017
• Posted on June 25, 2017

What is critical and creative thinking, and why is it so important in mathematics and numeracy education?

Numeracy is often defined as the ability to apply mathematics in the context of day to day life. However, the term ‘critical numeracy’ implies much more. One of the most basic reasons for learning mathematics is to be able to apply mathematical skills and knowledge to solve both simple and complex problems, and, more than just allowing us to navigate our lives through a mathematical lens, being numerate allows us to make our world a better place.

The mathematics curriculum in Australia provides teachers with the perfect opportunity to teach mathematics through critical and creative thinking. In fact, it’s mandated. Consider the core processes of the curriculum. The Australian Curriculum (ACARA, 2017), requires teachers to address four proficiencies : Problem Solving, Reasoning, Fluency, and Understanding. Problem solving and reasoning require critical and creative thinking (). This requirement is emphasised more heavily in New South wales, through the graphical representation of the mathematics syllabus content , which strategically places Working Mathematically (the proficiencies in NSW) and problem solving, at its core. Alongside the mathematics curriculum, we also have the General Capabilities , one of which is Critical and Creative Thinking – there’s no excuse!

Critical and creative thinking need to be embedded in every mathematics lesson . Why? When we embed critical and creative thinking, we transform learning from disjointed, memorisation of facts, to sense-making mathematics. Learning becomes more meaningful and purposeful for students.

How and when do we embed critical and creative thinking?

There are many tools and many methods of promoting thinking. Using a range of problem solving activities is a good place to start, but you might want to also use some shorter activities and some extended activities. Open-ended tasks are easy to implement, allow all learners the opportunity to achieve success, and allow for critical thinking and creativity. Tools such as Bloom’s Taxonomy and Thinkers Keys  are also very worthwhile tasks. For good mathematical problems go to the nrich website . For more extended mathematical investigations and a wonderful array of rich tasks, my favourite resource is Maths300   (this is subscription based, but well worth the money). All of the above activities can be used in class and/or for homework, as lesson starters or within the body of a lesson.

Will critical and creative thinking take time away from teaching basic concepts?

No, we need to teach mathematics in a way that has meaning and relevance, rather than through isolated topics. Therefore, teaching through problem-solving rather than for problem-solving. A classroom that promotes and critical and creative thinking provides opportunities for:

• higher-level thinking within authentic and meaningful contexts;
• complex problem solving;
• open-ended responses; and
• substantive dialogue and interaction.

Who should be engaging in critical and creative thinking?

Is it just for students? No! There are lots of reasons that teachers should be engaged with critical and creative thinking. First, it’s important that we model this type of thinking for our students. Often students see mathematics as black or white, right or wrong. They need to learn to question, to be critical, and to be creative. They need to feel they have permission to engage in exploration and investigation. They need to move from consumers to producers of mathematics.

Secondly, teachers need to think critically and creatively about their practice as teachers of mathematics. We need to be reflective practitioners who constantly evaluate our work, questioning curriculum and practice, including assessment, student grouping, the use of technology, and our beliefs of how children best learn mathematics.

Critical and creative thinking is something we cannot ignore if we want our students to be prepared for a workforce and world that is constantly changing. Not only does it equip then for the future, it promotes higher levels of student engagement, and makes mathematics more relevant and meaningful.

How will you and your students engage in critical and creative thinking?

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Encyclopedia of Mathematics Education pp 159–163 Cite as

Critical Thinking in Mathematics Education

• Eva Jablonka 2  
• Reference work entry
• First Online: 01 January 2020

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Mainstream educational psychologists view critical thinking (CT) as the strategic use of a set of reasoning skills for developing a form of reflective thinking that ultimately optimizes itself, including a commitment to using its outcomes as a basis for decision-making and problem solving. In such descriptions, CT is established as a general methodological standard for making judgments and decisions. Accordingly, some authors also include a sense for fairness and the assessment of practical consequences of decisions as characteristics (e.g., Paul and Elder 2001 ). This conception assumes rational, autonomous subjects who share a common frame of reference for representation of facts and ideas, for their communication, as well as for appropriate (morally “good”) action. Important is the difference as to what extent a critical examination of the criteria for CT is included in the definition: If education for CT is conceptualized as instilling a belief in a more or less fixed...

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Jablonka, E. (2020). Critical Thinking in Mathematics Education. In: Lerman, S. (eds) Encyclopedia of Mathematics Education. Springer, Cham. https://doi.org/10.1007/978-3-030-15789-0_35

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Getting smart collective, impact update, talking math: 100 questions that help promote mathematical discourse.

Update 2022: For a recent article on how to talk about math, click here. 

Think about the questions that you ask in your math classroom. Can they be answered with a simple “yes” or “no,” or do they open a door for students to really share their knowledge in a way that highlights their true understanding and uncovers their misunderstandings? Asking better questions can open new doors for students, helping to promote mathematical thinking and encouraging classroom discourse. Such questions help students:

• Work together to make sense of mathematics.
• Rely more on themselves to determine whether something is mathematically correct.
• Learn to reason mathematically.
• Evaluate their own processes and engage in productive peer interaction.
• Discover and seek help with problems in their comprehension.
• Learn to conjecture, invent and solve problems.
• Learn to connect mathematics, its ideas and its applications.
• Focus on the mathematical skills embedded within activities.

Dr. Gladis Kersaint

Help students work together to make sense of mathematics

• What strategy did you use?
• Do you agree?
• Do you disagree?
• Would you ask the rest of the class that question?
• Could you share your method with the class?
• What part of what he said do you understand?
• Would someone like to share ___?
• Can you convince the rest of us that that makes sense?
• What do others think about what [student] said?
• Can someone retell or restate [student]’s explanation?
• Did you work together? In what way?
• Would anyone like to add to this?
• Have you discussed this with your group? With others?
• Did anyone get a different answer?
• Where would you go for help?
• Did everybody get a fair chance to talk, to use the manipulatives, or to be recorded?
• How could you help another student without telling the answer?
• How would you explain ___ to someone who missed class today?

Refer questions raised by students back to the class.

Help students rely more on themselves to determine whether something is mathematically correct

• Is this a reasonable answer?
• Does that make sense?
• Why do you think that? Why is that true?
• Can you draw a picture or make a model to show that?
• How did you reach that conclusion?
• Does anyone want to revise his or her answer?
• How were you sure your answer was right?

Help students learn to reason mathematically

• How did you begin to think about this problem?
• What is another way you could solve this problem?
• How could you prove that?
• Can you explain how your answer is different from or the same as [student]’s?
• Let’s see if we can break it down. What would the parts be?
• Can you explain this part more specifically?
• Does that always work?
• Is that true for all cases?
• How did you organize your information? Your thinking?

Help students evaluate their own processes and engage in productive peer interaction

• What do you need to do next?
• What have you accomplished?
• What are your strengths and weaknesses?
• Was your group participation appropriate and helpful?

Help students with problem comprehension

• What is this problem about? What can you tell me about it?
• Do you need to define or set limits for the problem?
• How would you interpret that?
• Would you please reword that in simpler terms?
• Is there something that can be eliminated or that is missing?
• Would you please explain that in your own words?
• What assumptions do you have to make?
• What do you know about this part?
• Which words were most important? Why?

Help students learn to conjecture, invent and solve problems

• What would happen if ___? What if not?
• Do you see a pattern?
• What are some possibilities here?
• Where could you find the information you need?
• How would you check your steps or your answer?
• What did not work?
• How is your solution method the same as or different from [student]’s?
• Other than retracing your steps, how can you determine if your answers are appropriate?
• What decision do you think he or she should make?
• How did you organize the information? Do you have a record?
• How could you solve this using (tables, trees, lists, diagrams, etc.)?
• What have you tried? What steps did you take?
• How would it look if you used these materials?
• How would you draw a diagram or make a sketch to solve the problem?
• Is there another possible answer? If so, explain.
• How would you research that?
• Is there anything you’ve overlooked?
• How did you think about the problem?
• What was your estimate or prediction?
• How confident are you in your answer?
• What else would you like to know?
• What do you think comes next?
• Is the solution reasonable, considering the context?
• Did you have a system? Explain it.
• Did you have a strategy? Explain it.
• Did you have a design? Explain it.

Help students learn to connect mathematics, its ideas and its application

• What is the relationship of this to that?
• Have we ever solved a problem like this before?
• What uses of mathematics did you find in the newspaper last night?
• What is the same?
• What is different?
• Did you use skills or build on concepts that were not necessarily mathematical?
• Which skills or concepts did you use?
• What ideas have we explored before that were useful in solving this problem?
• Is there a pattern?
• Where else would this strategy be useful?
• How does this relate to ___?
• Is there a general rule?
• Is there a real-life situation where this could be used?
• How would your method work with other problems?
• What other problem does this seem to lead to?

Help students persevere

• Have you tried making a guess?
• What else have you tried?
• Would another recording method work as well or better?
• Is there another way to (draw, explain, say) that?
• Give me another related problem. Is there an easier problem?
• How would you explain what you know right now?

Help students focus on the mathematics from activities

• What was one thing you learned (or two, or more)?
• Where would this problem fit on our mathematics chart?
• How many kinds of mathematics were used in this investigation?
• What were the mathematical ideas in this problem?
• What is the mathematically different about these two situations?
• What are the variables in this problem? What stays constant?

Facilitating student engagement in mathematical discourse begins with the decisions teachers make when they plan classroom instruction. In the next and final blog in this series, we will dive into the specific strategies that teachers can use to foster meaningful conversations about what students are thinking, doing and learning.

This blog is part of a three-post series on the importance of mathematical discourse from Curriculum Associates   and Dr. Gladis Kersaint, the author of the recently published whitepaper Orchestrating Mathematical Discourse to Enhance Student Learning . Download your free copy here . For more on mathematical discourse and Curriculum Associates, check out:

• Talking Math: How to Engage Students in Mathematical Discourse
• Talking Math: 6 Strategies for Getting Students to Engage in Mathematical Discourse
• Curriculum Associates: Leveraging For-profit Power With a Nonprofit Purpose

Dr. Gladis Kersaint is a Professor of Mathematics Education at the University of Connecticut.

Stay in-the-know with all things EdTech and innovations in learning by signing up to receive the weekly Smart Update .  This post includes mentions of a Getting Smart partner. For a full list of partners, affiliate organizations and all other disclosures please see our Partner page .

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I love how you have the questions categorized by outcome goal. The infographic is one that I will be printing and using very often next year in my middle school classroom.

Joan Arumemi

It's an amazing application and approach in addressing math issues.Shall use them in my class .

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10 Brilliant Math Brain Teasers

Tap into rigorous problem-solving and critical thinking with these playful math brain teasers for middle and high school students.  

To break the ice during the first few awkward moments of class in the new school year, high school math teacher Lorenzo Robinson uses an unusual strategy: He reads his students’ minds.

Here’s how the mystical feat unfolds: Each student picks a number between 1 and 100. Next, they use calculators to add, subtract, multiply, and divide their way through a set of predetermined numbers. At the end, everyone ends up with the same answer: 427. Robinson isn’t psychic, of course, but because the mechanics of the teaser are always the same, regardless of which initial numbers his students select, he’s able to correctly “guess” their final result—much to their amazement. 

Starting off the lesson with a math brain teaser sets a playful tone and lowers the stakes for kids, “generating a buzz around my class,” Robinson says. “It makes students feel as though this class is not going to be scary, it’s going to be interesting. ‘We’re going to be learning, but we’re also going to have some fun.’” 

Robinson thinks of math brain teasers as a variation on brain breaks —a brief respite from dense curricular material that gives kids time to pause and process. They can also provide an opportunity to build relationships and community as kids put their heads together to find solutions. Good brain teasers can be sneaky: They get kids developing problem-solving and critical thinking skills. 

Meanwhile, before introducing a new one, Robinson works through the problem himself, identifying questions that students might have along the way and making sure his class has the background knowledge to understand how the teaser works. It’s important, he says, to provide a few minutes for kids to examine and discuss the teaser. Ask them to observe, highlight, and share things that stand out. 

“The most powerful reaction is when a kid doesn’t get the correct answer,” Robinson says, and they ask to try the problem again. “They want to feel what the other kids are feeling, that educational euphoria. They want to do it again because they want to be right.” That organic intellectual curiosity is hugely helpful in high school math, Robinson says, because it can be “parlayed into the other stuff that we do.”

We combed through dozens of math brain teasers to find 10 good ones—including several of Robinson’s tried-and-true favorites.  

Number Magic: I’ll Bet Your Number is… 427

• Start by having students pick any number between 1 and 100. 
• Add 28. 
• Multiply that number by 6. 
• Subtract 3. 
• Divide that number by 3. 
• Subtract 3 more than your original number. 
• Add 8. 
• Subtract 1 less than your original number. 
• Multiply that number by 7. 

And voilà, you’ll correctly identify each student’s final result as 427. Courtesy of: Lorenzo Robinson  

Can Your Shoe Size Tell Your Age? 

• Start with your shoe size. If you are a half size—for example, size 8.5—round up to 9.
• Multiply your shoe size by 5. 
• Add 50. 
• Multiply that number by 20. 
• Subtract the year you were born—for example, 1991. Add 1 if you already had your birthday this year. 

The first digit(s) are your shoe size, and the last two digits are your age.  Courtesy of: Lorenzo Robinson.  

Cutting Across a Cross

Ask students to draw a cross on a sheet of paper. Drawing one on the board as a point of reference is helpful. Next, ask students to draw two straight lines that will segment or cut the cross into pieces. The goal is to produce the most pieces.

The solution can be found here .  Sourced from: MathisFun.com .

Number Magic: I’ll Bet Your Final Number is… 5  

• Start with a positive number. Students shouldn’t say the number out loud.
• Square that number. 
• Add 10x the original number to what you have now. 
• Add 25 to the result of the previous step. 
• Now take the square root of that number, rounding to the nearest whole number. 
• Subtract your original number.
• Before students share their final figure, reveal that you guess their collective result is 5. 

Courtesy of: Lorenzo Robinson.

Birthday Math 

Have students work in pairs and share the following instructions with their partner:

• Start with the number 7.
• Multiply that by the month of your birth. For example, if you were born in September, you’d use the number 9 to represent your birth month. 
• Subtract 1. 
• Multiply that number by 13. 
• Add the day of your birth. 
• Add 3. 
• Multiply that number by 11. 
• Subtract the month of your birth. 
• Subtract the day of your birth. 
• Divide by 10.
• Add 11 to that number. 
• Divide by 100. 

The result on the calculator screen should be their partner’s birthday.  Sourced from: Dr. Mike’s Math Games for Kids .

Coin Conundrum

Ask students to imagine that they have two coins that total 30 cents in value. Have them try to figure out what the two coins are, only providing them with a singular piece of information: One of the coins is not a nickel. The answer: A quarter and a nickel. (Only one of the coins is not a nickel.)

Sourced from: WeAreTeachers .

The Phone Number Trick  

• Ignoring your area code, type the first three digits of your phone number into a calculator.
• Multiply that number by 80. 
• Add 1. 
• Multiply that number by 250. 
• Add the last four digits of your phone number. 
• Add the last four digits of your phone number again. 
• Subtract 250. 
• Divide this number by 2. Do you recognize your phone number?

Courtesy of: Lorenzo Robinson. 

A Number Challenge

For a slightly more independent teaser, challenge students to produce a math equation that works using these four numbers—2, 3, 4, and 5—as well as a plus (+) and equal sign (=). Students can work individually, in pairs, or in small groups as they try to create a valid equation. The answer: 2 + 5 = 3 + 4.

Sourced from: WeAreTeachers .  

Math Mind Reader 

Students can work in pairs with this teaser. One person will start off by holding the calculator so their partner cannot see it; the other person can read the steps aloud to the partner with the calculator. 

• The student with the calculator starts by choosing a whole number from 1 to 20 and writing it down on a piece of paper without letting their partner see it. 
• Next, the student with the calculator enters their secret number into the calculator. 
• Multiply that number by 3. 
• Add the secret number, then subtract 5. 
• Multiply by 3, then multiply by 3 again. 
• Add the secret number, then subtract the number of their favorite month (you don’t have to know what month it is). For example, 9 represents the ninth month of the year, September. 
• Multiply by 3, then multiply by 3 again, and then again a third time. 
• Add the secret number, then subtract their favorite day of the month (again, you don’t have to know what it is on your end). 
• Ask them to show the non-calculator partner the result. At this stage, the non-calculator partner can guess the original secret number, even though what appears on the screen may be a very large number. 

If the result is negative, their secret number is 1.

If the result has only three digits, their secret number is 2.

In all other cases, ignore the last three digits, and then add 2 to get the secret number! 

Sourced from: Dr. Mike’s Math Games for Kids .

What’s Unique About This Number?

After writing the number 8,549,176,320 on the board, ask students to observe the number and tell you everything they think is unique about the number.

Answer: It is the digits 0 to 9 in alphabetical order (eight, five, four, nine, one, seven, six, three, two, zero), but it’s surprising and fun to see what students come up with. This number can also be evenly divided by the digits 1 through 9 except for the number 7, for example. 

Sourced from: MathisFun.com . 

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100+ Critical Thinking Questions for Students To Ask About Anything

Critical thinkers question everything.

In an age of “fake news” claims and constant argument about pretty much any issue, critical thinking skills are key. Teach your students that it’s vital to ask questions about everything, but that it’s also important to ask the right sorts of questions. Students can use these critical thinking questions with fiction or nonfiction texts. They’re also useful when discussing important issues or trying to understand others’ motivations in general.

“Who” Critical Thinking Questions

Questions like these help students ponder who’s involved in a story and how the actions affect them. They’ll also consider who’s telling the tale and how reliable that narrator might be.

• Is the protagonist?
• Is the antagonist?
• Caused harm?
• Is harmed as a result?
• Was the most important character?

• Is responsible?
• Is most directly affected?
• Should have won?
• Will benefit?
• Would be affected by this?

• Makes the decisions?

“What” Critical Thinking Questions

Ask questions that explore issues more deeply, including those that might not be directly answered in the text.

• Background information do I know or need to know?
• Is the main message?
• Are the defining characteristics?

• Questions or concerns do I have?
• Don’t I understand?
• Evidence supports the author’s conclusion?
• Would it be like if … ?
• Could happen if … ?
• Other outcomes might have happened?
• Questions would you have asked?
• Would you ask the author about … ?
• Was the point of … ?
• Should have happened instead?
• Is that character’s motive?
• Else could have changed the whole story?

• Can you conclude?
• Would your position have been in that situation?
• Would happen if … ?
• Makes your position stronger?
• Was the turning point?
• Is the point of the question?
• Did it mean when … ?
• Is the other side of this argument?
• Was the purpose of … ?
• Does ______ mean?
• Is the problem you are trying to solve?
• Does the evidence say?
• Assumptions are you making?
• Is a better alternative?
• Are the strengths of the argument?

• Are the weaknesses of the argument?
• Is the difference between _______ and _______?

“Where” Critical Thinking Questions

Think about where the story is set and how it affects the actions. Plus, consider where and how you can learn more.

• Would this issue be a major problem?
• Are areas for improvement?
• Did the story change?
• Would you most often find this problem?

• Are there similar situations?
• Would you go to get answers to this problem?
• Can this be improved?
• Can you get more information?
• Will this idea take us?

“When” Critical Thinking Questions

Think about timing and the effect it has on the characters or people involved.

• Is this acceptable?
• Is this unacceptable?

• Does this become a problem?
• Is the best time to take action?
• Will we be able to tell if it worked?
• Is it time to reassess?
• Should we ask for help?
• Is the best time to start?
• Is it time to stop?
• Would this benefit society?

• Has this happened before?

“Why” Critical Thinking Questions

Asking “why” might be one of the most important parts of critical thinking. Exploring and understanding motivation helps develop empathy and make sense of difficult situations.

• Is _________ happening?
• Have we allowed this to happen?
• Should people care about this issue?

• Is this a problem?
• Did the character say … ?
• Did the character do … ?
• Is this relevant?
• Did the author write this?
• Did the author decide to … ?
• Is this important?

• Did that happen?
• Is it necessary?
• Do you think I (he, she, they) asked that question?
• Is that answer the best one?
• Do we need this today?

“How” Critical Thinking Questions

Use these questions to consider how things happen and whether change is possible.

• Do we know this is true?
• Does the language used affect the story?
• Would you solve … ?
• Is this different from other situations?

• Is this similar to … ?
• Would you use … ?
• Does the location affect the story?
• Could the story have ended differently?
• Does this work?
• Could this be harmful?
• Does this connect with what I already know?
• Else could this have been handled?
• Should they have responded?

• Would you feel about … ?
• Does this change the outcome?
• Did you make that decision?
• Does this benefit you/others?
• Does this hurt you/others?
• Could this problem be avoided?

More Critical Thinking Questions

Here are more questions to help probe further and deepen understanding.

• Can you give me an example?

• Do you agree with … ?
• Can you compare this with … ?
• Can you defend the actions of … ?
• Could this be interpreted differently?
• Is the narrator reliable?
• Does it seem too good to be true?

• Is ______ a fact or an opinion?

What are your favorite critical thinking questions? Come exchange ideas on the WeAreTeachers HELPLINE group on Facebook .

Plus, check out 10 tips for teaching kids to be awesome critical thinkers ., you might also like.

5 Critical Thinking Skills Every Kid Needs To Learn (And How To Teach Them)

Teach them to thoughtfully question the world around them. Continue Reading

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Mathematical Reasoning™

Bridging the gap between computation and math reasoning.

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Forget boring math lessons and dreaded drill sheets.  These fun, colorful books use engaging lessons with easy-to-follow explanations, examples, and charts to make mathematical concepts easy to understand.  They can be used as textbooks or comprehensive workbooks with your textbooks to teach the math skills and concepts that students are expected to know in each grade—and several concepts normally taught in the next grade. Every lesson is followed with a variety of fun, colorful activities to ensure concept mastery.  The lessons and activities spiral slowly, allowing students to become comfortable with concepts, but also challenging them to continue building their problem-solving skills.  These books teach more than mathematical concepts; they teach mathematical reasoning, so students learn to devise different strategies to solve a wide variety of math problems.  All books are written to the standards of the National Council of Teachers of Mathematics.

Beginning 1 , Beginning 2 , and Level A Contents

Understanding Pre-Algebra This book teaches and develops the math concepts and critical thinking skills necessary for success in Algebra I and future mathematics courses at the high school level.  It was written with the premise that students cannot problem solve or take leaps of reasoning without understanding the concepts and elements that lead to discovery.  The author—with 35 years of experience teaching mathematics—is a firm believer that understanding leads to confidence and confidence gives students the resolve to succeed in higher level mathematics rather than fear it.   It is standards-based, but what makes it different from other pre-algebra books is that it organizes concepts in a logical fashion, stressing practice and critical thinking. It avoids the mistakes—found in many other math books—of trying to teach new concepts before students receive the prerequisite skills and practice necessary for success. The concepts are presented clearly and in connection to other concepts. Math vocabulary is very important to success in higher mathematics, so this book includes easy-to-follow explanations and a user-friendly glossary.

Free Detailed Solutions are available!

Understanding Pre-Algebra Contents

Understanding Geometry The successful completion of this colorful 272-page book will prepare middle schoolers for high school geometry. It covers more than 50% of the concepts taught in high school geometry using a step-by-step approach and teaches the reasoning behind the properties taught in geometry–instead of merely asking them to memorize them. Students are also taught the basics of geometric proofs and coordinate geometry in a way middle school students can understand. Students who struggle with high school geometry usually have lower standardized test scores because it is a fundamental subject in high school standardized testing. A glossary of terms that every student should master is included. This book can be used as a classroom textbook in Grades 7, 8, or 9 (usually over a two-year period) or as a reference for high school students. This book covers more than the National Math Standards for middle school mathematics.

Understanding Geometry Contents

NOTE:  It is our recommendation that students complete Understanding Pre-Algebra (see description above) before attempting Understanding Geometry .

Understanding Algebra I This is a one-year Algebra I course for Grades 7-9. Students who have a solid algebra background will have no trouble with the algebra problems from SAT and even the GRE.  This 384-page book highlights vocabulary and notation, and has examples from the history of math. What makes this book unique and different from other algebra textbooks is that it is built from the experiences of an award-winning algebra teacher with more than 30 years of teaching experience. Many textbooks are written by a committee of authors, and many of those authors have little experience teaching beginning algebra students in middle school or high school. Understanding Algebra I presents the most essential concepts and skills needed to fully understand and gain confidence in algebra in a step-by-step fashion, teaching students that algebra is generalized arithmetic. It helps students see the connection between mathematics that they already know and algebra, so that learning algebra becomes easier and less abstract. This book provides students with real strategies to succeed in solving word problems by using charts and translating strategies that guarantee success.

Understanding Algebra I Contents

Essential Algebra for Advanced High School and SAT

Discover Essential Algebra for Advanced High School and SAT , a 241-page math book in the esteemed Mathematical Reasoning series written by award-winning author and teacher with 30 years of expertise in secondary mathematics. This powerful resource teaches the ‘essential’ connection of arithmetic and geometric concepts with algebraic concepts. Without this understanding, students tend to memorize Algebra I problem-solving steps—which is sufficient to pass Algebra I—but leaves them unprepared for math courses beyond Algebra I and the SATs. Algebra, the essential language of all advanced mathematics, lies at the core of this book's teachings. By delving into the generalized arithmetic that underpins algebra, students develop a solid foundation in the rules governing number and fraction operations, including factors and multiples. This vital knowledge empowers students to move beyond mere memorization of Algebra I problem-solving steps and confidently tackle the complexities of math courses beyond Algebra I. Without the knowledge and skills taught in this book, students often struggle or even fail in advanced mathematics courses and on the SATs. Imagine a good high school student who sees a problem like 3•x•y•4 and hesitates to write 12xy due to uncertainty about the rules governing multiplication. Or not understanding how to add 2x to 1/4y to combine it into a single fraction. Or why –6 2 is different than (–6) 2 . It is easy to see that not having a strong understanding of the foundational rules of algebra can stop even the smartest students from succeeding in advanced high school math courses. Essential Algebra for Advanced High School and SAT serves as a companion to an Algebra I course or aids in post-Algebra I readiness. To ensure students’ long-term success in advanced math beyond Algebra I, this book teaches the following 'essential' mathematics skills and concepts:

• Understanding Terms and Order of Operations
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• Working with Terms and Polynomials
• Polynomial Division, Factoring, and Rational Expressions
• Solving Equations and Inequalities
• Ratio, Proportion, and Percent
• How Algebra is Used in Geometry
• Understanding Functions
• Working With Quadratic Equations and Functions

Mathematical Reasoning™ Supplements These supplemental books reinforce grade math concepts and skills by asking students to apply these skills and concepts to non-routine problems. Applying mathematical knowledge to new problems is the ultimate test of concept mastery and mathematical reasoning. These user-friendly, engaging books are made up of 50 theme-based collections of problems, conveniently grouped in self-contained, double-sided activity sheets that provide space for student work. Each collection contains relevant math facts at the end of the worksheet in case students need hints to solve the problems. Calculators are allowed on activity sets that have a calculator icon at the top of the front side of the set. Each activity set is accompanied by a single-sided answer sheet containing strategy tips and detailed solutions. Teachers and parents will appreciate the easy-to-understand, comprehensive solutions. These books are a wonderful enrichment tool, but also can be used to assess how well students have learned their grade level's math concepts.

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Critical thinking definition

Critical thinking, as described by Oxford Languages, is the objective analysis and evaluation of an issue in order to form a judgement.

Active and skillful approach, evaluation, assessment, synthesis, and/or evaluation of information obtained from, or made by, observation, knowledge, reflection, acumen or conversation, as a guide to belief and action, requires the critical thinking process, which is why it's often used in education and academics.

Some even may view it as a backbone of modern thought.

However, it's a skill, and skills must be trained and encouraged to be used at its full potential.

People turn up to various approaches in improving their critical thinking, like:

• Developing technical and problem-solving skills
• Engaging in more active listening
• Actively questioning their assumptions and beliefs
• Seeking out more diversity of thought
• Opening up their curiosity in an intellectual way etc.

Is critical thinking useful in writing?

Critical thinking can help in planning your paper and making it more concise, but it's not obvious at first. We carefully pinpointed some the questions you should ask yourself when boosting critical thinking in writing:

• What information should be included?
• Which information resources should the author look to?
• What degree of technical knowledge should the report assume its audience has?
• What is the most effective way to show information?
• How should the report be organized?
• How should it be designed?
• What tone and level of language difficulty should the document have?

Usage of critical thinking comes down not only to the outline of your paper, it also begs the question: How can we use critical thinking solving problems in our writing's topic?

Let's say, you have a Powerpoint on how critical thinking can reduce poverty in the United States. You'll primarily have to define critical thinking for the viewers, as well as use a lot of critical thinking questions and synonyms to get them to be familiar with your methods and start the thinking process behind it.

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A Crash Course in Critical Thinking

What you need to know—and read—about one of the essential skills needed today..

Posted April 8, 2024 | Reviewed by Michelle Quirk

• In research for "A More Beautiful Question," I did a deep dive into the current crisis in critical thinking.
• Many people may think of themselves as critical thinkers, but they actually are not.
• Here is a series of questions you can ask yourself to try to ensure that you are thinking critically.

Conspiracy theories. Inability to distinguish facts from falsehoods. Widespread confusion about who and what to believe.

These are some of the hallmarks of the current crisis in critical thinking—which just might be the issue of our times. Because if people aren’t willing or able to think critically as they choose potential leaders, they’re apt to choose bad ones. And if they can’t judge whether the information they’re receiving is sound, they may follow faulty advice while ignoring recommendations that are science-based and solid (and perhaps life-saving).

Moreover, as a society, if we can’t think critically about the many serious challenges we face, it becomes more difficult to agree on what those challenges are—much less solve them.

On a personal level, critical thinking can enable you to make better everyday decisions. It can help you make sense of an increasingly complex and confusing world.

In the new expanded edition of my book A More Beautiful Question ( AMBQ ), I took a deep dive into critical thinking. Here are a few key things I learned.

First off, before you can get better at critical thinking, you should understand what it is. It’s not just about being a skeptic. When thinking critically, we are thoughtfully reasoning, evaluating, and making decisions based on evidence and logic. And—perhaps most important—while doing this, a critical thinker always strives to be open-minded and fair-minded . That’s not easy: It demands that you constantly question your assumptions and biases and that you always remain open to considering opposing views.

In today’s polarized environment, many people think of themselves as critical thinkers simply because they ask skeptical questions—often directed at, say, certain government policies or ideas espoused by those on the “other side” of the political divide. The problem is, they may not be asking these questions with an open mind or a willingness to fairly consider opposing views.

When people do this, they’re engaging in “weak-sense critical thinking”—a term popularized by the late Richard Paul, a co-founder of The Foundation for Critical Thinking . “Weak-sense critical thinking” means applying the tools and practices of critical thinking—questioning, investigating, evaluating—but with the sole purpose of confirming one’s own bias or serving an agenda.

In AMBQ , I lay out a series of questions you can ask yourself to try to ensure that you’re thinking critically. Here are some of the questions to consider:

• Why do I believe what I believe?
• Are my views based on evidence?
• Have I fairly and thoughtfully considered differing viewpoints?
• Am I truly open to changing my mind?

Of course, becoming a better critical thinker is not as simple as just asking yourself a few questions. Critical thinking is a habit of mind that must be developed and strengthened over time. In effect, you must train yourself to think in a manner that is more effortful, aware, grounded, and balanced.

For those interested in giving themselves a crash course in critical thinking—something I did myself, as I was working on my book—I thought it might be helpful to share a list of some of the books that have shaped my own thinking on this subject. As a self-interested author, I naturally would suggest that you start with the new 10th-anniversary edition of A More Beautiful Question , but beyond that, here are the top eight critical-thinking books I’d recommend.

The Demon-Haunted World: Science as a Candle in the Dark , by Carl Sagan

This book simply must top the list, because the late scientist and author Carl Sagan continues to be such a bright shining light in the critical thinking universe. Chapter 12 includes the details on Sagan’s famous “baloney detection kit,” a collection of lessons and tips on how to deal with bogus arguments and logical fallacies.

Clear Thinking: Turning Ordinary Moments Into Extraordinary Results , by Shane Parrish

The creator of the Farnham Street website and host of the “Knowledge Project” podcast explains how to contend with biases and unconscious reactions so you can make better everyday decisions. It contains insights from many of the brilliant thinkers Shane has studied.

Good Thinking: Why Flawed Logic Puts Us All at Risk and How Critical Thinking Can Save the World , by David Robert Grimes

A brilliant, comprehensive 2021 book on critical thinking that, to my mind, hasn’t received nearly enough attention . The scientist Grimes dissects bad thinking, shows why it persists, and offers the tools to defeat it.

Think Again: The Power of Knowing What You Don't Know , by Adam Grant

Intellectual humility—being willing to admit that you might be wrong—is what this book is primarily about. But Adam, the renowned Wharton psychology professor and bestselling author, takes the reader on a mind-opening journey with colorful stories and characters.

Think Like a Detective: A Kid's Guide to Critical Thinking , by David Pakman

The popular YouTuber and podcast host Pakman—normally known for talking politics —has written a terrific primer on critical thinking for children. The illustrated book presents critical thinking as a “superpower” that enables kids to unlock mysteries and dig for truth. (I also recommend Pakman’s second kids’ book called Think Like a Scientist .)

Rationality: What It Is, Why It Seems Scarce, Why It Matters , by Steven Pinker

The Harvard psychology professor Pinker tackles conspiracy theories head-on but also explores concepts involving risk/reward, probability and randomness, and correlation/causation. And if that strikes you as daunting, be assured that Pinker makes it lively and accessible.

How Minds Change: The Surprising Science of Belief, Opinion and Persuasion , by David McRaney

David is a science writer who hosts the popular podcast “You Are Not So Smart” (and his ideas are featured in A More Beautiful Question ). His well-written book looks at ways you can actually get through to people who see the world very differently than you (hint: bludgeoning them with facts definitely won’t work).

A Healthy Democracy's Best Hope: Building the Critical Thinking Habit , by M Neil Browne and Chelsea Kulhanek

Neil Browne, author of the seminal Asking the Right Questions: A Guide to Critical Thinking, has been a pioneer in presenting critical thinking as a question-based approach to making sense of the world around us. His newest book, co-authored with Chelsea Kulhanek, breaks down critical thinking into “11 explosive questions”—including the “priors question” (which challenges us to question assumptions), the “evidence question” (focusing on how to evaluate and weigh evidence), and the “humility question” (which reminds us that a critical thinker must be humble enough to consider the possibility of being wrong).

Warren Berger is a longtime journalist and author of A More Beautiful Question .

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