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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 ↓

8 thoughts on “20 Math Critical Thinking Questions to Ask in Class Tomorrow”

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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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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.

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Top 5 Trickiest Mathematics Questions From Around the World

Math(s) can be a tricky subject for many students. But some questions are trickier than others.

Put on your thinking caps because we searched the internet for the top 5 trickiest mathematics questions from all around the world.

Solutions are provided at the end of all the questions (but no peeking).

If you’re up for an extra challenge, we’ve even got a bonus question at the end.

But before that… a quick announcement. World Maths Day – the world’s largest mathematics competition is back!

World Maths Day, happening on 8 March 2023, is a global celebration of mathematics where millions of students aged 5 to 18 across the world compete in Live Mathletics challenges. It’s all-inclusive, free, and open to schools as well as students learning from home. Learn more about it here .

Now, let’s jump in!

1. People on a Train 🚂

Country of origin: England

In a since-deleted tweet, a mum from England tweeted this word problem in a test meant for kids aged 6 to 7 in 2016. It went viral and even some adults were having trouble figuring out the answer.

The Question:

There were some people on a train.

19 people get off the train at the first stop.

17 people get on the train.

Now there are 63 people on the train. How many people were on the train to begin with?

2. You’ll Never Forget Cheryl’s Birthday 📅

Country of origin: Singapore

Problems that test logical reasoning are common in Math(s) Olympiads. But this question from the 2015 Singapore and Asian Schools Math Olympiad contest for students 14 to 15 years old got the whole world stumped.

Albert and Bernard just became friends with Cheryl, and they want to know when her birthday is.

Cheryl gives them a list of 10 possible dates:

• May 15, May 16, May 19
• June 17, June 18
• July 14, July 16
• August 14, August 15, August 17

Cheryl then tells Albert and Bernard separately the month and the day of her birthday respectively.

Albert: I don’t know when Cheryl’s birthday is, but I know that Bernard doesn’t know too.

Bernard: At first I don’t know when Cheryl’s birthday is, but I know now.

Albert: Then I also know when Cheryl’s birthday is.

So when is Cheryl’s birthday?

3. Taming the Snake 🐍

Country of origin: Vietnam 

This question is not only tricky but might also take a while. According to VNEXPRESS, this puzzle is meant for third graders/year 3 students (8 year olds) in Vietnam!

The Puzzle:

Image source: VN Express

All you have to do is use the digit 1 to 9 once to fill in the boxes to make the entire equation equal to 66. The expression should be read from left to right.

Sounds easy? Not quite.

In case you’re wondering, the boxes containing colon represents division.

4. Remember Where You Parked Your Car 🚗

Country of origin: Hong Kong

This problem has been around for a while but resurfaced on an elementary/primary school entrance exam in Hong Kong.

Apparently, six-year-olds were expected to know the answer in 20 seconds or less.

What is the car’s parking spot number?

5. The Red Triangle 🔺

Country of origin: China

This question came from China and was used to identify gifted fifth grade/year 5 students (10 to 11 years old). It’s said that some of them were able to solve this question in less than one minute!

ABCD is a parallelogram. In the diagram, the areas of yellow regions are 8, 10, 72 and 79.

Find the area of the red triangle. The diagram is not to scale.

Image source: Mind Your Decisions

BONUS Tricky Math(s) Question

If you still have head space for one more, try this.

6. A Mass of Money: Helen and Ivan’s coins 💰

In 2021, a Primary School Leaving Exam mathematics question left some 12-year-old students in tears. Supposedly, this question was meant to be solved in a matter of minutes, as it is only allocated 4 marks in total.

Note: This two-part question could have been recalled from memory and rewritten by an adult, which could explain the grammatical errors.

Helen and Ivan had the same number of coins.

Helen had a number of 50-cent coins, and 64 20-cent coins. These coins had a mass of 1.134kg.

Ivan had a number of 50-cent coins and 104 20-cent coins.

(a) Who has more money in coins and by how much?

(b) given that each 50-cent coin is 2.7g heavier than a 20-cent coin, what is the mass of Ivan’s coins in kilograms?

Could You Solve These Tricky Mathematics Questions?

Or were you confused and stumped? Well, you’re not alone.

We had a really tough time understanding and solving them too.

If you’re a teacher and looking for problem and reasoning questions , consider a mathematics resource to sharpen your student’s logical thinking skills.

Now let’s get to the answers…

Question 1 answer.

19 people getting off the train can be represented by -19, and 17 people getting on the train as +17.

-19 + 17 = 2, meaning that there was a net loss of two people.

Originally, the train had 2 more people.

So if there are 63 people on the train now, that means there were 65 people to begin with.

Question 2 Answer

You can solve this by the process of elimination, based on what each person says.

Let’s go through the information line by line.

[Line 7] Cheryl then tells Albert and Bernard separately the month and the day of her birthday respectively.

This is an important piece of information because it tells us that Albert knows the month , and Bernard knows the day .

So Albert knows it’s either May, June, July or August, and Bernard knows that it’s either 14, 15, 16, 17, 18 or 19.

[Line 8] Albert: I don’t know when Cheryl’s birthday is, but I know that Bernard doesn’t know too .

The second part is the clue. The fact that Albert claims that Bernard doesn’t know means it can’t be 18 or 19. Why?

If it were 19, then Bernard would know the exact birthday, as May is the only date with 19.

If Bernard was told the date was 18, he would also know that the birthday must be June 18, as that’s the only date with 18.

So you can rule out May 19 and June 18.

But how is Albert sure that Bernard didn’t hear 18 or 19?

It must be because Albert knows the birthday is not in May or June.

If Albert was told the month was May, he couldn’t be sure that Bernard wasn’t thinking of the number 19. Therefore, you can cross out May.

And if Albert was told the month of June, he couldn’t’ be sure if Bernard wasn’t thinking of the number 17. So June is also out.

In other words, Albert was told either July or August .

Based on the above information, you can eliminate these five dates – May 15, May 16, May 19, June 17 and June 18.

Dates left: July 14, July 16, August 14, August 15 and August 17.

[Line 9] Bernard: At first I don’t know when Cheryl’s birthday is, but now I know.

Upon hearing Albert’s statement, Bernard now figures this out.

If Bernard was told the date was 14, it would still be ambiguous whether the month was July or August. So you can rule out he was not told 14.

You are now left with three dates – July 16, August 15 and August 17.

[Line 10] Albert: Then I also know when Cheryl’s birthday is.

Albert couldn’t have been told it was August, as there are two dates in August. So you can deduce that he must have been told it’s July.

Therefore, the answer is July 16 .

Question 3 Answer

Let’s start by breaking the puzzle into bite-size pieces, one step at a time.

First, write the expression in the normal way you usually write mathematical expressions. This makes it easier to put in the numbers.

__ + 13 × __ ÷ __ + __ + 12 × __ – __ – 11 + __ × __ ÷ __ – 10 = 66

Next, let’s look at how many ways are there to put the numbers 1 to 9 in these 9 different boxes.

You can put 9 different numbers in the first box.

So that’s 9 possibilities in the first box, 8 possibilities in the second box, followed by 7 boxes in the third box and so forth.

Applying this logic, you will have one less possibility for each box, until we get to the last box.

In total, there are 9 factorial (9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 9!) or 362,880 possibilities .

Now that’s a lot of possibilities to try and work by purely guessing and checking.

So let’s try working out the solution logically.

Remember the BEDMAS/BIDMAS/PEDMAS/PEMDAS rule you learnt in school?

To respect the order of operations, add parentheses or brackets to the equation. This means that multiplication or division comes before addition or subtraction.

__ + ( 13 × __ ÷ __ ) + __ + ( 12 × __ ) – __ – 11 + ( __ × __ ÷ __ ) – 10 = 66

Now it’s time to fill in some numbers to guess and check our assumptions.

What if you first used the numbers 1 to 9, from left to right ?

1 + (13 × 2 ÷ 3 ) + 4 + (12 × 5 ) – 6 – 11 + ( 7 × 8 ÷ 9 ) – 10 = 52.88…

Hey, that’s pretty close to 66!

What if you wrote the numbers in descending order , from 9 to 1?

9 + (13 × 8 ÷ 7 ) + 6 + (12 × 5 ) – 4 – 11 + ( 3 × 2 ÷ 1 ) – 10 = 70.85…

That also gets you pretty close to the answer.

So how can you modify this expression to get to 66? The key is to look at the numbers and their positions.

In the next few steps, we used trial and error – testing and moving the numbers around until we got to 66.

Here’s one solution we got:

9 + (13 × 4 ÷ 8 ) + 5 + (12 × 6 ) – 7 – 11 + ( 1 × 3 ÷ 2 ) – 10 = 66

Now for the keen observers out there, you’d notice that you can switch the numbers that are being added, to generate another solution.

For example:

9 + (13 × 4 ÷ 8) + 5 + (12 × 6) – 7 – 11 + (1 × 3 ÷ 2) – 10 = 66 OR (switch 5 and 9) 5 + (13 × 4 ÷ 8) + 9 + (12 × 6) – 7 – 11 + (1 × 3 ÷ 2) – 10 = 66

Similarly, you can switch the numbers that are multiplied, and it won’t affect the final answer.

9 + (13 × 4 ÷ 8) + 5 + (12 × 6) – 7 – 11 + ( 1 × 3 ÷ 2) – 10 = 66 OR (switch 1 and 3) 9 + (13 × 4 ÷ 8) + 5 + (12 × 6) – 7 – 11 + ( 3 × 1 ÷ 2) – 10 = 66

This means anytime you come up with one way to solve it, you can generate a total of four ways – because multipclation and addition are commutative (it doesn’t what the order of the numbers are, the answer is the same).

In fact, there are multiple answers to this puzzle. 136 to be exact. How do we know?

Now, that’s a problem to solve for another time. 😉

Question 4 Answer

The ‘trick’ to this question is that it requires no math(s) at all!

All you have to do is to look at it from a different perspective – literally.

Turn the question upside down, and you’ll see that it’s a simple number sequence, with the answer being 87.

Question 5 Answer

Even though it looks complicated, this question can actually be solved with a simple calculation: 79 + 10 – 72 – 8 = 9

Wait, what? But how?

To get there, you need to understand basic arithmetic and know that the area of a parallelogram and the area of a triangle are related.

The ‘secret’ is to identify triangles with areas that are half of the parallelogram.

The area of a triangle is (base × height) ÷ 2, and the area of a parallelogram is base × height.

A triangle whose base equals one side of the parallelogram, and whose height reaches the opposite side of the parallelogram, has exactly half the area of a parallelogram.

This is true for a pair of triangles as well – if the pair of triangles span one side and if their heights reach the opposite side.

To make solving this easier, you can start by labelling the unknown areas with letters a to f . And let the area of the red triangle be x .

Presh Talwalkar from Mind You Decisions, breaks down the solution in his video here .

Question 6 Answer (Part a)

The key is to remember that Helen and Ivan have the same number of coins.

Let’s look and compare the total number of coins for each type.

Ivan has 40 more 20-cent coins than Helen. For them to have the same number of coins, you have to ‘balance’ this out in terms of the 50-cent coins.

This means Helen must have 40 more of the 50-cent coins than Ivan.

Let’s now compare the amount of money of each coin type that Helen has, minus that of Ivan.

Since Helen has 40 fewer (104 – 64) of the 20-cent coins, so Helen will have:

– 40 × 0.2 = – 8

This means she has $8 less than Ivan (in 20-cent coins).

On the other hand, Helen has 40 more of the 50-cent coins than Ivan. So she will have:

+ 40 × 0.5 = 20

This means she has $20 more than Ivan (in 50-cent coins).

Now, you can add this together to find out how much more or less money Helen has.

– 8 + 20 = 12

Therefore, Helen has $12 more than Ivan.

Question 6 Answer (Part b):

The total mass of Helen’s coin is 1.134kg. And you know that a 50-cent coin is 2.7g heavier than a 20-cent coin.

From the first part of the question, you can see that if you had Helen’s coins, you can ‘exchange’ 40 of the 50-cent coins for 40 of the 20-cent coins, that will be the total coins Ivan has. And you can get the weight difference from that.

Let’s compare the weight of Helen’s coins to Ivan’s coins.

In terms of the 20-cent coins, subtract 40 of the 20-cent coins, multiplied by the weight of the coins.

– 40 × 0.2 weight

In terms of the 50-cent coins, add 40 of the 50-cent coins, multiplied by the weight.

+40 × 0.5 weight

So the net impact of this, Helen compared to Ivan, has 40 more of the heavier coins – 40 more of the 50-cent coins, compared to the 20-cent coins than Ivan.

+ 40 × 0.5 weight / 40 × (0.5 – 0.2 weight)

You know the difference in weight between 50-cent and 20-cent coins is 2.7 grams. Therefore, you can substitute that in the equation.

+ 40 × 0.5 weight / 40 × (2.7 g) –> 40 × (2.7 g) = 108g

So Helen’s weight of coins is 108 g more than Ivan.

To get Ivan’s weight, we take Helen’s coins and subtract by 108g.

1134g – 108g = 1026g

Convert that to kilograms to get the answer, 1.026 kg .

How did you fare? Share this with your students or friends who love a great math(s) challenge!

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5 Ways to Stop Thinking for Your Students

Too often math students lean on teachers to think for them, but there are some simple ways to guide them to think for themselves.

Who is doing the thinking in your classroom? If you asked me that question a few years ago, I would have replied, “My kids are doing the thinking, of course!” But I was wrong. As I reflect back to my teaching style before I read Building Thinking Classrooms by Peter Liljedahl (an era in my career I like to call “pre-thinking classroom”), I now see that I was encouraging my students to mimic rather than think .

My lessons followed a formula that I knew from my own school experience as a student and what I had learned in college as a pre-service teacher. It looked like this: Students faced me stationed at the board; I demonstrated a few problems while students copied what I wrote in their notes. I would throw out a few questions to the class to assess understanding. If a few kids answered correctly, I felt confident that the lesson had gone well. Some educators might call this “ I do, we do, you do .”

What’s wrong with this formula? When it was time for them to work independently, which usually meant a homework assignment because I used most of class time for direct instruction, the students would come back to class and say, “The homework was so hard. I don’t get it. Can you go over questions 1–20?” Exhausted and frustrated, I would wonder, “But I taught it—why didn’t they get it?”

Now in the “peri-thinking classroom” era of my career, my students are often working at the whiteboards in random groups as outlined in Liljedahl’s book. The pendulum has shifted from the teacher doing the thinking to the students doing the thinking. Do they still say, “I don’t get it!”? Yes, of course! But I use the following strategies to put the thinking back onto them.

5 Ways to Get Your Students to Think

1. Answer questions with a refocus on the students’ point of view. Liljedahl found in his research that students ask three types of questions: “(1) proximity questions—asked when the teacher is close; (2) stop thinking questions—most often of the form ‘is this right’ or ‘will this be on the test’; and (3) keep thinking questions—questions that students ask so they can get back to work.” He suggests that teachers acknowledge “proximity” and “stop thinking questions” but not answer them.

Try these responses to questions that students ask to keep working:

• “What have you done so far?” 
• “Where did you get that number?” 
• “What information is given in the problem?” 
• “Does that number seem reasonable in this situation?”  

2. Don’t carry a pencil or marker. This is a hard rule to follow; however, if you hold the writing utensil, you’ll be tempted to write for them . Use verbal nudges and hints, but avoid writing out an explanation. If you need to refer to a visual, find a group that has worked out the problem, and point out their steps. Hearing and viewing other students’ work is more powerful .

3. We instead of I . When I assign a handful of problems for groups to work on at the whiteboards, they are tempted to divvy up the task. “You do #30, and I’ll do #31.” This becomes an issue when they get stuck. I inevitably hear, “Can you help me with #30? I forgot how to start.”

I now require questions to use “we” instead of “I.” This works wonders. As soon as they start to ask a question with “I,” they pause and ask their group mates. Then they can legitimately say, “ We tried #30, and we are stumped.” But, in reality, once they loop in their group mates, the struggling student becomes unstuck, and everyone in the group has to engage with the problem.

4. Stall your answer. If I hear a basic computation question such as, “What is 3 divided by 5?” I act like I am busy helping another student: “Hold on, I need to help Marisela. I’ll be right back.” By the time I return to them, they are way past their question. They will ask a classmate, work it out, or look it up. If the teacher is not available to think for them, they learn to find alternative resources.

5. Set boundaries. As mentioned before, students ask “proximity” questions because I am close to them. I might reply with “Are you asking me a thinking question? I’m glad to give you a hint or nudge, but I cannot take away your opportunity to think.” This type of response acknowledges that you are there to help them but not to do their thinking for them.

When you set boundaries of what questions will be answered, the students begin to more carefully craft their questions. At this point of the year, I am starting to hear questions such as, “We have tried solving this system by substitution, but we are getting an unreasonable solution. Can you look at our steps?” Yes!

Shifting the focus to students doing the thinking not only enhances their learning but can also have the effect of less frustration and fatigue for the teacher. As the class becomes student-centered, the teacher role shifts to guide or facilitator and away from “sage on the stage.”

As another added benefit, when you serve as guide or facilitator, the students are getting differentiated instruction and assessment. Maybe only a few students need assistance with adding fractions, while a few students need assistance on an entirely different concept. At first, you might feel like your head is spinning trying to address so many different requests; however, as you carefully sift through the types of questions you hear, you will soon be comfortable only answering the “keep thinking” questions.

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Understanding Algebra I

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