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Maneuvering the Middle
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Math Problem Solving Strategies
How many times have you been teaching a concept that students are feeling confident in, only for them to completely shut down when faced with a word problem? For me, the answer is too many to count. Word problems require problem solving strategies. And more than anything, word problems require decoding, eliminating extra information, and opportunities for students to solve for something that the question is not asking for. There are so many places for students to make errors! Let’s talk about some problem solving strategies that can help guide and encourage students!
Grab your Problem Solving Strategy Freebie Here !
Update for 2020: Scroll down to the bottom to read how we address showing your work during distance learning.
1. C.U.B.E.S.
C.U.B.E.S stands for circle the important numbers, underline the question, box the words that are keywords, eliminate extra information, and solve by showing work.
- Why I like it: Gives students a very specific ‘what to do.’
- Why I don’t like it: With all of the annotating of the problem, I’m not sure that students are actually reading the problem. None of the steps emphasize reading the problem but maybe that is a given.
2. R.U.N.S.
R.U.N.S. stands for read the problem, underline the question, name the problem type, and write a strategy sentence.
- Why I like it: Students are forced to think about what type of problem it is (factoring, division, etc) and then come up with a plan to solve it using a strategy sentence. This is a great strategy to teach when you are tackling various types of problems.
- Why I don’t like it: Though I love the opportunity for students to write in math, writing a strategy statement for every problem can eat up a lot of time.
3. U.P.S. CHECK
U.P.S. Check stands for understand, plan, solve, and check.
- Why I like it: I love that there is a check step in this problem solving strategy. Students having to defend the reasonableness of their answer is essential for students’ number sense.
- Why I don’t like it: It can be a little vague and doesn’t give concrete ‘what to dos.’ Checking that students completed the ‘understand’ step can be hard to see.
4. Maneuvering the Middle Strategy AKA K.N.O.W.S.
Here is the strategy that I adopted a few years ago. It doesn’t have a name yet nor an acronym, (so can it even be considered a strategy…?)
UPDATE: IT DOES HAVE A NAME! Thanks to our lovely readers, Wendi and Natalie!
- Know: This will help students find the important information.
- Need to Know: This will force students to reread the question and write down what they are trying to solve for.
- Organize: I think this would be a great place for teachers to emphasize drawing a model or picture.
- Work: Students show their calculations here.
- Solution: This is where students will ask themselves if the answer is reasonable and whether it answered the question.
Here is where I typically struggle with problem solving strategies: 1) modeling the strategy in my own teaching weeks after I have taught students to use the strategy and 2) enforcing students to do it. So… basically everything. This might be why I haven’t been able to stick with a strategy from year to year.
5. Digital Learning Struggle
Many teachers are facing how to have students show their work or their problem solving strategy when tasked with submitting work online. Platforms like Kami make this possible. Go Formative has a feature where students can use their mouse to “draw” their work. If your students don’t have access to a touchscreen, then ha ving them submit images of their work might be your best bet. To simplify this process, I would recommend asking students to submit an image for all of their work — not individual problems. We do not want to create additional barriers for students.
If you want to spend your energy teaching student problem solving instead of writing and finding math problems, look no further than our All Access membership . Click the button to learn more.
Students who plan succeed at a higher rate than students who do not plan. Do you have a go to problem solving strategy that you teach your students?
Editor’s Note: Maneuvering the Middle has been publishing blog posts for nearly 6 years! This post was originally published in September of 2017. It has been revamped for relevancy and accuracy.
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Reader Interactions
18 comments.
October 4, 2017 at 7:55 pm
As a reading specialist, I love your strategy. It’s flexible, “portable” for any problem, and DOES get kids to read and understand the problem by 1) summarizing what they know and 2) asking a question for what they don’t yet know — two key comprehension strategies! How about: “Make a Plan for the Problem”? That’s the core of your rationale for using it, and I bet you’re already saying this all the time in class. Kids will get it even more because it’s a statement, not an acronym to remember. This is coming to my reading class tomorrow with word problems — thank you!
October 4, 2017 at 8:59 pm
Hi Nora! I have never thought about this as a reading strategy, genius! Please let me know how it goes. I would love to hear more!
December 15, 2017 at 7:57 am
Hi! I am a middle school teacher in New York state and my district is “gung ho” on CUBES. I completely agree with you that kids are not really reading the problem when using CUBES and only circling and boxing stuff then “doing something” with it without regard for whether or not they are doing the right thing (just a shot in the dark!). I have adopted what I call a “no fear word problems” procedure because several of my students told me they are scared of word problems and I thought, “let’s take the scary out of it then by figuring out how to dissect it and attack it! Our class strategy is nearly identical to your strategy:
1. Pre-Read the problem (do so at your normal reading speed just so you basically know what it says) 2. Active Read: Make a short list of: DK (what I Definitely Know), TK (what I Think I Know and should do), and WK (what I Want to Know– what is the question?) 3. Draw and Solve 4. State the answer in a complete sentence.
This procedure keep kids for “surfacely” reading and just trying something that doesn’t make sense with the context and implications of the word problem. I adapted some of it from Harvey Silver strategies (from Strategic Teacher) and incorporated the “Read-Draw-Write” component of the Eureka Math program. One thing that Harvey Silver says is, “Unlike other problems in math, word problems combine quantitative problem solving with inferential reading, and this combination can bring out the impulsive side in students.” (The Strategic Teacher, page 90, Silver, et al.; 2007). I found that CUBES perpetuates the impulsive side of middle school students, especially when the math seems particularly difficult. Math word problems are packed full of words and every word means something to about the intent and the mathematics in the problem, especially in middle school and high school. Reading has to be done both at the literal and inferential levels to actually correctly determine what needs to be done and execute the proper mathematics. So far this method is going really well with my students and they are experiencing higher levels of confidence and greater success in solving.
October 5, 2017 at 6:27 am
Hi! Another teacher and I came up with a strategy we call RUBY a few years ago. We modeled this very closely after close reading strategies that are language arts department was using, but tailored it to math. R-Read the problem (I tell kids to do this without a pencil in hand otherwise they are tempted to start underlining and circling before they read) U-Underline key words and circle important numbers B-Box the questions (I always have student’s box their answer so we figured this was a way for them to relate the question and answer) Y-You ask yourself: Did you answer the question? Does your answer make sense (mathematically)
I have anchor charts that we have made for classrooms and interactive notebooks if you would like them let me me know….
October 5, 2017 at 9:46 am
Great idea! Thanks so much for sharing with our readers!
October 8, 2017 at 6:51 pm
LOVE this idea! Will definitely use it this year! Thank you!
December 18, 2019 at 7:48 am
I would love an anchor chart for RUBY
October 15, 2017 at 11:05 am
I will definitely use this concept in my Pre-Algebra classes this year; I especially like the graphic organizer to help students organize their thought process in solving the problems too.
April 20, 2018 at 7:36 am
I love the process you’ve come up with, and think it definitely balances the benefits of simplicity and thoroughness. At the risk of sounding nitpicky, I want to point out that the examples you provide are all ‘processes’ rather than strategies. For the most part, they are all based on the Polya’s, the Hungarian mathematician, 4-step approach to problem solving (Understand/Plan/Solve/Reflect). It’s a process because it defines the steps we take to approach any word problem without getting into the specific mathematical ‘strategy’ we will use to solve it. Step 2 of the process is where they choose the best strategy (guess and check, draw a picture, make a table, etc) for the given problem. We should start by teaching the strategies one at a time by choosing problems that fit that strategy. Eventually, once they have added multiple strategies to their toolkit, we can present them with problems and let them choose the right strategy.
June 22, 2018 at 12:19 pm
That’s brilliant! Thank you for sharing!
May 31, 2018 at 12:15 pm
Mrs. Brack is setting up her second Christmas tree. Her tree consists of 30% red and 70% gold ornaments. If there are 40 red ornaments, then how many ornaments are on the tree? What is the answer to this question?
June 22, 2018 at 10:46 am
Whoops! I guess the answer would not result in a whole number (133.333…) Thanks for catching that error.
July 28, 2018 at 6:53 pm
I used to teach elementary math and now I run my own learning center, and we teach a lot of middle school math. The strategy you outlined sounds a little like the strategy I use, called KFCS (like the fast-food restaurant). K stands for “What do I know,” F stands for “What do I need to Find,” C stands for “Come up with a plan” [which includes 2 parts: the operation (+, -, x, and /) and the problem-solving strategy], and lastly, the S stands for “solve the problem” (which includes all the work that is involved in solving the problem and the answer statement). I find the same struggles with being consistent with modeling clearly all of the parts of the strategy as well, but I’ve found that the more the student practices the strategy, the more intrinsic it becomes for them; of course, it takes a lot more for those students who struggle with understanding word problems. I did create a worksheet to make it easier for the students to follow the steps as well. If you’d like a copy, please let me know, and I will be glad to send it.
February 3, 2019 at 3:56 pm
This is a supportive and encouraging site. Several of the comments and post are spot on! Especially, the “What I like/don’t like” comparisons.
March 7, 2019 at 6:59 am
Have you named your unnamed strategy yet? I’ve been using this strategy for years. I think you should call it K.N.O.W.S. K – Know N – Need OW – (Organise) Plan and Work S – Solution
September 2, 2019 at 11:18 am
Going off of your idea, Natalie, how about the following?
K now N eed to find out O rganize (a plan – may involve a picture, a graphic organizer…) W ork S ee if you’re right (does it make sense, is the math done correctly…)
I love the K & N steps…so much more tangible than just “Read” or even “Understand,” as I’ve been seeing is most common in the processes I’ve been researching. I like separating the “Work” and “See” steps. I feel like just “Solve” May lead to forgetting the checking step.
March 16, 2020 at 4:44 pm
I’m doing this one. Love it. Thank you!!
September 17, 2019 at 7:14 am
Hi, I wanted to tell you how amazing and kind you are to share with all of us. I especially like your word problem graphic organizer that you created yourself! I am adopting it this week. We have a meeting with all administrators to discuss algebra. I am going to share with all the people at the meeting.
I had filled out the paperwork for the number line. Is it supposed to go to my email address? Thank you again. I am going to read everything you ahve given to us. Have a wonderful Tuesday!
10 Math Problem Solving Activities for Middle School
Published on june 26, 2017 at 7:25 am by ethan jacobs in lists , news.
Looking for some math problem-solving activities for middle school ? Good, you’re at the right page then.
Right before children enter Middle School (around the age of 11 or 12), they enter a critical developmental stage known as Piaget’s fourth and final stage of cognitive development. It’s at this stage that children demonstrate marked growth in a number of areas, ranging from making hypotheses and inferences to thinking abstractly and using advanced reasoning skills. In line with this crucial phase of a child’s development, Middle School Math curricula are designed to stretch the bounds of adolescent thinking while also helping them to establish new skills and sound mathematical habits.
One way that educators try to ensure this is through common core standards that can be applied to Middle School-aged students. These standards seek to achieve eight distinct objectives, which help foster the developmental transition addressed by Piaget. The objectives:
- Make sense of problems and persevere in solving them;
- Reason abstractly and quantitatively;
- Construct viable arguments and critique the reasoning of others;
- Model with mathematics;
- Use appropriate tools strategically;
- Attend to precision;
- Look for and make use of structure;
- Look for and express regularity in repeated reasoning
allow for a lot of leeways as well as creativity in the way that problems are both presented to and solved by students.
Marijus Auruskevicius/Shutterstock.com
The first objective, for example, emphasizes a student’s ability to not simply apply an algorithm to a problem, but more pointedly, make a decision and implement it. This process can draw out drastically different reactions in different students. For some, the prospect of being creative and innovative in thinking of ways to solve brain-bending problems is exciting, and often even addicting. On the other hand, getting past the roadblocks that come along with solving a tough problem can be frustrating and, at times, discouraging for students. It is in these moments that establishing math skills that promote perseverance are most critical.
A quick and easy way of avoiding that anticipated frustration that students might encounter in the face of challenging math problems is equipping them with an arsenal of tools and approaches through which they can tackle such problems.
If, for example, you told me that I was a bird with a short, stubby beak that had to find a way to drink water from a glass that was only half-full with only a pile of stones at hand, I might get frustrated pretty quickly upon realizing that my beak did not reach far enough down to allow me to drink. I might peck a few times in vain but would remain parched.
With the right set of dynamic problem-solving skills at my disposal, however, I might think of the problem in a different light, and realize that by dropping enough stones into the glass, I could make the water level rise enough that my beak could easily extract all the water I desired.
Applying these kinds of problem-solving skills to questions that are appropriate for Middle School students can fortify grit, the quality of not giving up easily, and help students to solve problems they may face in their own lives.
Taking all of this into consideration, there are a few basic skills and approaches that students can use to help them crack just about any age-appropriate problem that you, the teacher, throw at them. In the interest of time, we’ll introduce just four here, though plenty of others can certainly be applied where appropriate. The most common methods for solving problems that students may encounter are:
- Guess and Check;
- Draw a picture;
- Work Backward; and
- Use an Equation with a variable.
The first method on our list of math problem-solving activities for middle school is fairly self-explanatory. In a sense, it involves a bit of reverse-engineering, as the student starts with a proposed solution and works his or her way back to the beginning of the problem to see if that solution is effective.
Drawing a picture may be more effective for visual learners, as it enables students to lay eyes on the problem and conceive of a solution in ways that they may not have otherwise.
Working backward is like a more scientific version of guessing and checking. Students can use the information provided to step backward one piece at a time, like Guy Pearce in Memento, until they reach the solution that is in accordance with all of the details provided in the problem.
Finally, an equation that uses a variable can be effective when information is missing, or when an approach unlike the first three is required.
Again–these approaches are mere suggestions that students can apply to solving problems that they may encounter. Ultimately, a healthy combination of different tactics can serve a student well in handling any problem thrown their way. Skills such as these, though tough to develop at first, can go a long way toward helping US students stand up to their peers around the world in global math benchmarks , while also making day-to-day problems that they face easier to solve.
Without further ado, here are ten math problem-solving activities for middle school students that can help them develop a number of crucial skills. If you find these interesting, you may also like our article on the Best Problem Solving Activities For Middle School . Beyond just math, there are other areas where problem-solving can be extremely useful for that age group. And now, the problems.
10 Best Problem Solving Activities For Middle School 10 Math Problem Solving Activities for Middle School Castaways and Coconuts Math Problem solving activity for middle school students Common Core Educational Standards for Middle School Students Cookie Decorations Math Problem solving activity for middle school students Domino Effect Math Problem solving activity for middle school students Drawing a picture math problem solution method Family Matters Math Problem solving activity for middle school students Full HD Math Problem solving activity for middle school students Guess and Check Math Problem Solution Method List XFinance math problem-solving activities for middle school Missing Mangoes math problem solving activity Mystery Weight Math Problem solving activity for middle school students Piaget's Stages of Cognitive Development Show Me the Money Math Problem solving activity for middle school students Slideshow Supersize Me Math Problem solving activity for middle school students Use an Equation with a variable math problem solution method Wild Dog Math Problem solving activity for middle school students Work Backward Math Problem Solution Method Show more... Show less
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Can Your Middle Schoolers Solve These Math Puzzles?
May 1, 2017 by Anthony Persico
Are you looking for some fun, printable math games and puzzles for your middle school students?
These 5 math challenges will engage your students and get them thinking creatively and visually about math topics including fractions, areas models, the order of operations, and even algebra!
So, can your middle schoolers solve these fun math puzzles?
Pre-Algebra
Solution: Pizza = 3, Hamburger = 2, Taco = 8
Multiplication Table
Solution: Volcano = 1, Statue of Liberty = 3, Rocket = 6, Race Car = 2, Ferris Wheel = 18
Logic Puzzle
Solution: Multiple solutions exist using each value only once.
Solution: Pretzel = 8, Mustard = 3, Chocolate-Covered = 50, Salt = 40, Sticks = 24
Fraction Model
Solution: Pink = 16, Chocolate = 8, Purple = 2, Ice Cream Cup = 24, Ice Cream Cone = 3
Are you looking for more daily math challenges and puzzles to share with your kids?
My best-selling workbook 101 Math Challenges for Engaging Your Students in Grades 3-8 is now available as a PDF download. You can get yours today by clicking here.
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By Anthony Persico
Anthony is the content crafter and head educator for YouTube's MashUp Math and an advisor to Amazon Education's ' With Math I Can ' Campaign. You can often find me happily developing animated math lessons to share on my YouTube channel . Or spending way too much time at the gym or playing on my phone.
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20 Problem-Solving Activities for Middle School Students
June 28, 2022 // by Lesa M.K. Bullins, EdS
Problem-solving skills are important to the building of critical thinking, which in turn strengthens student executive function. Good problem solvers can build stronger cognitive flexibility, a critical component of executive functioning.
The teenage years are a crucial time for neuroplasticity, so it is a prime time for learning and developing important cognitive skills along with critical information. Bring problem-solving to life in your middle school classroom with these 20 activities.
1. Feelings Expression Scenarios
A huge part of problem-solving is properly expressing your own feelings. Students often struggle to state how they feel without combative, aggressive, or accusatory language; therefore opportunities to practice with realistic situations is a key problem-solving component. You can create scenario task cards to support students in realistic applications for relatable situations, or use pre-made cards.
Learn More: Pinterest
2. Empathy Empowered Discussions
In addition to being able to calmly and kindly express one's feelings, empathy is a key problem-solving component. Teens can often struggle to express empathy as they have difficulty recognizing and interpreting due to the teenage brain functioning.
Teenage brains are still developing, so different areas of the brain are controlling different functions than we see in adult brains; furthermore, since teens are still figuring out what they think and feel about a variety of things, it can be difficult for them to recognize and consider the feelings and thoughts of others. You can instigate empathy discussions through relatable content like this short video.
Learn More: Austin Wideman
3. Model, Model, Model...and then Model Some More!
Students learn more from what they see you do than what they hear you say! This means you have to be an active and purposeful model of what you expect. So make sure you are aware of your actions and words in front of your students!
Learn More: Education World
4. Get Out of the Way
We need to allow students the time and space to solve problems. We cannot intervene every time they struggle to find the answer right away. Constant intervention hinders critical thinking and decision-making skills.
Make sure to leave some room for students to figure out solutions. Keep safe proximity so students have the comfort to know you are there if they cannot find a solution, but resist the urge to jump in as soon as you see them struggling.
Learn More: Brookings
5. Plan a Road Trip
Engage problem-solving skills within context while reinforcing math, research, geography, and communication skills, too! Students can plan a road trip from start to finish in small groups. As an added bonus, you can let students travel virtually to the places they planned for their trip using Google Earth.
If time allows, they can even take screenshots and stage selfies for a presentation to share their trip with the class! This is a really great cross-curricular activity for the digital classroom, too!
6. Escape the Room
Escape rooms were made for problem-solving, so what better way to build these skills for students in an exciting way! Create different challenge activities surrounding a variety of subjects and skills to reinforce while lettings students put problem-solving to use finding practical solutions to escape the room!
Divide kids into teams and get on this engaging problem-solving activity!
7. Teach Explicit Strategies for Reflection
Students can build analytical skills by reflecting on their problem-solving process. Teach explicit skills to help students recognize and reflect on how they solve problems to reinforce future use and strengthen overall critical thinking abilities. Check out how Ellie from Cognitive Cardio made it work even in the time constraints of middle school schedules!
Learn More: Cognitive Cardio With MSMM
8. Daily Practice
Give students short, interesting, and challenging problems to solve during the morning and afternoon transition times. Daily practice solving challenges is important for cognitive development and reinforces academic skills! You can find tons of daily challenges online or create your own.
9. Build Something
Let students work together in teams to build something from simple building materials. Increase the challenge by limiting resources or requiring students to pick their own resources for building blocks from a variety of random items. You can check out the marshmallow toothpick tower-building activity!
Learn more: Wow Sci
10. Blind Drawing Partners
Students can work in partner pairs or small groups to develop a vast array of abilities through this problem-solving activity. Blind team-building activities are excellent, low-prep ways to engage students' critical thinking and communication!
There are different ways you can implement this, but check out this video for an example of one application of the blind drawing game.
Learn more: Philip Barry
11. Laser Maze
Create a laser maze for students to get active in problem-solving. Create and implement different time durations to increase the challenge. Do not have lasers? No budget for lasers? Don't worry, red painter's tape will do the job!
Learn More: That Phillips Family
12. Shared Story Puzzles
Creating story puzzles that force students to work in groups together to put together, add on, and create a cohesive story that is meaningful is another challenging task to engage in collaborative problem-solving.
Learn More: Secondary English Coffee Shop
13. Yarn Webs
This social-skill-building collaborative problem-solving activity is fun for any age. Organize students into teams then let them choose a color of yarn, build a team web, and see who can navigate. There are so many ways this activity can be adapted, but you can watch a video of one interpretation here .
Learn More: KEYSAmeriCorps
14. Scavenger Hunt
Create a series of clues that students must solve to progress through the game. Working in groups can help build conflict resolution and social skills as well. Check out how to create scavenger hunts for the classroom in this video by Learning Life.
Learn more: Learning Life
15. Boom! Math!
An excellent way to build advanced problem-solving skills, as well as mathematical analysis, is to create math Boom Cards with word problems like these from Math in the Middle. Boom cards are a great activity for students to practice and build skills!
Learn more: Boom Learning
16. Wheel of Solutions
Give students practice in exercising a number of different kinds of problem-solving skills by spinning and communicating a solution using the skills on which they land. You can make one in the classroom with a posterboard or create a digital wheel. Such a fun interactive resource! Use this great pre-made digital activity from Resource Haven on Boom Learning or create your own!
Learn More: Boom Learning
17. Collaborative Math
Another activity for team building that supports mathematical concept reinforcement is students working together to collaboratively solve math problems. Check out how Runde's Room made sure everyone is engaged in working on solving parts of the problem through the sticky-note collaborative math activity.
18. Get Mysterious
Math Mysteries are a fun activity that builds out-of-the-box thinking and creates an inquisitive environment. Problem-solving develops through the process of inquisition! You can create your own or use Lee and Miller's 40 Fabulous Math Mysteries Kid's Can't Resist Scholastic book found here.
19. Logic Puzzles and Games
In addition to logic-building games like Chess, you can provide logic puzzles for morning and afternoon transitions, during downtime, or for early finishers. Logic puzzles help students think critically. You can make your own or get some prefabricated resources like the ones found in this book by Chris King .
Learn More: Brainzilla
20. Lead Number Talks
Number talks are important to building problem-solving. Number talks allow students to build on one another in a collaborative way, discuss how they have solved problems before, consider how those solutions may be applicable to new skills they are about to learn, and build depth in math concepts.
So instead of getting quiet, get them talking!
Learn More: North Dakota Teaching Kayla Durkin
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20 Grade-School Math Questions So Hard You'll Wonder How You Graduated
Seriously, who can do these?!
Unless you grew up to be an engineer, a banker, or an accountant, odds are that elementary and middle school math were the bane of your existence. You would study relentlessly for weeks for those silly standardized tests—and yet, come exam day, you'd still somehow have no idea what any of the equations or hard math problems were asking for. Trust us, we get it.
While logic might lead you to believe that your math skills have naturally gotten better as you've aged, the unfortunate reality is that, unless you've been solving algebra and geometry problems on a daily basis, the opposite is more likely the case.
Don't believe us? Then put your number crunching wisdom to the test with these tricky math questions taken straight from grade school tests and homework assignments and see for yourself.
1. Question: What is the number of the parking space covered by the car?
This tricky math problem went viral a few years back after it appeared on an entrance exam in Hong Kong… for six-year-olds. Supposedly the students had just 20 seconds to solve the problem!
Answer: 87.
Believe it or not, this "math" question actually requires no math whatsoever. If you flip the image upside down, you'll see that what you're dealing with is a simple number sequence.
2. Question: Replace the question mark in the above problem with the appropriate number.
This problem shouldn't be too difficult to solve if you play a lot of sudoku.
All of the numbers in every row and column add up to 15! (Also, 6 is the only number not represented out of numbers 1 through 9.)
3. Question: Find the equivalent number.
This problem comes straight from a standardized test given in New York in 2014.
You're forgiven if you don't remember exactly how exponents work. In order to solve this problem, you simply need to subtract the exponents (4-2) and solve for 3 2 , which expands into 3 x 3 and equals 9.
4. Question: How many small dogs are signed up to compete in the dog show?
This question comes directly from a second grader's math homework. Yikes.
Answer: 42.5 dogs.
In order to figure out how many small dogs are competing, you have to subtract 36 from 49 and then divide that answer, 13, by 2, to get 6.5 dogs, or the number of big dogs competing. But you're not done yet! You then have to add 6.5 to 36 to get the number of small dogs competing, which is 42.5. Of course, it's not actually possible for half a dog to compete in a dog show, but for the sake of this math problem let's assume that it is.
5. Question: Find the area of the red triangle.
This question was used in China to identify gifted 5th graders. Supposedly, some of the smart students were able to solve this in less than one minute.
In order to solve this problem, you need to understand how the area of a parallelogram works. If you already know how the area of a parallelogram and the area of a triangle are related, then adding 79 and 10 and subsequently subtracting 72 and 8 to get 9 should make sense—but if you're still confused, then check out this YouTube video for a more in-depth explanation.
6. Question: How tall is the table?
YouTuber MindYourDecisions adapted this mind-boggling math question from a similar one found on an elementary school student's homework in China.
Answer: 150 cm.
Since one measurement includes the cat's height and subtracts the turtle's and the other does the opposite, you can essentially just act like the two animals aren't there. Therefore, all you have to do is add the two measurements—170 cm and 130 cm—together and divided them by 2 to get the table's height, 150 cm.
7. Question: If the cost of a bat and a baseball combined is $1.10 and the bat costs $1.00 more than the ball, how much is the ball?
This problem, mathematically speaking, is very similar to one of the other ones on this list.
Answer: $0.05.
Think back to that problem about the dogs at the dog show and use the same logic to solve this problem. All you have to do is subtract $1.00 from $1.10 and then divide that answer, $0.10 by 2, to get your final answer, $0.05.
8. Question: When is Cheryl's birthday?
If you're having trouble reading that, see here:
"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 not 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?"
It's unclear why Cheryl couldn't just tell both Albert and Bernard the month and day she was born, but that's irrelevant to solving this problem.
Answer: July 16.
Confused about how one could possibly find any answer to this question? Don't worry, so was most of the world when this question, taken from a Singapore and Asian Schools Math Olympiad competition, went viral a few years ago. Thankfully, though, the New York Times explains step-by-step how to get to July 16, and you can read their detailed deduction here.
9. Question: Find the missing letter.
This one comes from a first grader's homework.
Answer: The missing letter is J.
When you add together the values given for S, B, and G, the sum comes out to 40, and making the missing letter J (which has a value of 14) makes the other diagonal's sum the same.
10. Question: Solve the equation.
This problem might look easy, but a surprising number of adults are unable to solve it correctly.
Start by solving the division part of the equation. In order to do that, in case you forgot, you have to flip the fraction and switch from division to multiplication, thus getting 3 x 3 = 9. Now you have 9 – 9 + 1, and from there you can simply work from left to right and get your final answer: 1.
11. Question: Where should a line be drawn to make the below equation accurate?
5 + 5 + 5 + 5 = 555.
Answer: A line should be drawn on a "+" sign.
When you draw a slanted line in the upper left quadrant of a "+," it becomes the number 4 and the equation thusly becomes 5 + 545 + 5 = 555.
12. Question: Solve the unfinished equation.
Try to figure out what all of the equations have in common.
Answer: 4 = 256.
The formula used in each equation is 4 x = Y. So, 4 1 = 4, 4 2 = 16, 4 3 = 64, and 4 4 = 256.
13. Question: How many triangles are in the image above?
When Best Life first wrote about this deceiving question, we had to ask a mathematician to explain the answer!
Answer: 18.
Some people get stumped by the triangles hiding inside of the triangles and others forget to include the giant triangle housing all of the others. Either way, very few individuals—even math teachers—have been able to find the correct answer to this problem. And for more questions that will put your former education to the test, check out these 30 Questions You'd Need to Ace to Pass 6th Grade Geography.
14. Question: Add 8.563 and 4.8292.
Adding two decimals together is easier than it looks.
Answer: 13.3922.
Don't let the fact that 8.563 has fewer numberrs than 4.8292 trip you up. All you have to do is add a 0 to the end of 8.563 and then add like you normally would.
15. Question: There is a patch of lily pads on a lake. Every day, the patch doubles in size…
… If it takes 48 days for the patch to cover the entire lake, how long would it take for the patch to cover half of the lake?
Answer: 47 days.
Most people automatically assume that half of the lake would be covered in half the time, but this assumption is wrong. Since the patch of pads doubles in size every day, the lake would be half covered just one day before it was covered entirely.
16. Question: How many feet are in a mile?
This elementary school-level problem is a little less problem solving and a little more memorization.
Answer: 5,280.
This was one of the questions featured on the popular show Are You Smarter Than a 5th Grader?
17. Question: What value of "x" makes the equation below true?
Answer: -3.
You'd be forgiven for thinking that the answer was 3. However, since the number alongside x is negative, we need x to be negative as well in order to get to 0. Therefore, x has to be -3.
18. Question: What is 1.92 divided by 3?
You might need to ask your kids for help on this one.
Answer: 0.64.
In order to solve this seemingly simple problem, you need to remove the decimal from 1.92 and act like it isn't there. Once you've divided 192 by 3 to get 64, you can put the decimal place back where it belongs and get your final answer of 0.64.
19. Question: Solve the math equation above.
Don't forget about PEMDAS!
Using PEMDAS (an acronym laying out the order in which you solve it: "parenthesis, exponents, multiplication, division, addition, subtraction"), you would first solve the addition inside of the parentheses (1 + 2 = 3), and from there finish the equation as it's written from left to right.
20. Question: How many zombies are there?
Finding the answer to this final question will require using fractions.
Answer: 34.
Since we know that there are two zombies for every three humans and that 2 + 3 = 5, we can divide 85 by 5 to figure out that in total, there are 17 groups of humans and zombies. From there, we can then multiply 17 by 2 and 3 and learn that there are 34 zombies and 51 humans respectively. Not too bad, right?
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20 Best Math Puzzles to Engage and Challenge Your Students
Solve the hardest puzzle
Use Prodigy Math to boost engagement, offer differentiated instruction and help students enjoy math.
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It’s time for math class, and your students are bored.
It might sound harsh, but it’s true -- only about half of students report being engaged at school, and engagement levels only drop as students get older.
Math puzzles are one of the best -- and oldest -- ways to encourage student engagement. Brain teasers, logic puzzles and math riddles give students challenges that encourage problem-solving and logical thinking. They can be used in classroom gamification , and to inspire students to tackle problems they might have previously seen as too difficult.
If you want to get your students excited about math class, this post is for you. You’ll find:
- A list of 20 interesting (and sometimes amazing!) puzzles, inspired by both classic math thinking and modern classroom techniques
- Research on why math puzzles are a great idea for your classroom
- Tips on how to effectively use them in the classroom
Math Puzzles for Kids:
1. math crossword puzzles.
Puzzles to Print
Take a crossword, and make it math: that’s the basic concept behind this highly adaptable math challenge. Instead of words, students use numbers to complete the vertical and horizontal strips. Math crossword puzzles can be adapted to teach concepts like money, addition, or rounding numbers. Solutions can be the products of equations or numbers given by clues.
2. Math problem search
Have students practice their addition, subtraction, multiplication and division skills by searching for hidden math equations in a word search-style puzzle . It can be adapted to any skill you want students to practice, and promotes a solid understanding of basic math facts.
3. Math riddles
My PreCalc students love riddles... can you figure out where the other dollar went?? #MathRiddles pic.twitter.com/BclqW9nq98 — Rachel Frasier (@MsFrasierMHS) January 8, 2019
Do your students love word problems ? Try giving them some math riddles that combine critical thinking with basic math skills. Put one up on the board for students to think about before class begins, or hand them out as extra practice after they’ve finished their work.
Prodigy is an engaging, game-based platform that turns math into an adventure! While it’s not a math puzzle in the traditional sense, Prodigy uses many of the same principles to develop critical thinking skills and mathematical fluency.
Students complete curriculum-aligned math questions to earn coins, collect pets and go on quests. Teachers can deliver differentiated math content to each student, prep for standardized tests and easily analyze student achievement data with a free account.
KenKenKenKen
is a “grid-based numerical puzzle” that looks like a combined number cross and sudoku grid. Invented in 2004 by a famous Japanese math instructor named Tetsuya Miyamoto, it is featured daily in The New York Times and other newspapers. It challenges students to practice their basic math skills while they apply logic and critical thinking skills to the problem.
6. Pre-algebraic puzzles
Pre-algebraic puzzles use fun substitutions to get students ready to perform basic functions and encourage them to build problem-solving skills. They promote abstract reasoning and challenge students to think critically about the problems in front of them.As an added bonus, students who suffer from math anxiety might find the lack of complicated equations reassuring, and be more willing to attempt a solution.
7. Domino puzzle board
Games 4 Gains
There are hundreds of ways to use dominoes in your math classroom, but this puzzle gives students a chance to practice addition and multiplication in a fun, hands-on way. You can have students work alone or in pairs to complete the puzzle.
This online game and app challenges players to slide numbered tiles around a grid until they reach 2048. It’s highly addictive and not as easy as it sounds, so consider sending it home with students or assigning it after the rest of the lesson is over. It encourages students to think strategically about their next move, and it’s a great tool for learning about exponents.
Math in English
Kakuro , also called “Cross Sums,” is another mathematical crossword puzzle. Players must use the numbers one through nine to reach “clues” on the outside of the row. Decrease the size of the grid to make it easier for younger players, or keep it as is for students who need a challenge. Students can combine addition and critical thinking and develop multiple skills with one fun challenge.
10. Magic square
Magic square s have been around for thousands of years, and were introduced to Western civilization by translated Arabic texts during the Renaissance. While magic squares can be a variety of sizes, the three by three grid is the smallest possible version and is the most accessible for young students.
This is also a great math puzzle to try if your students are tactile learners. Using recycled bottle caps, label each with a number from one to nine. Have your students arrange them in a three by three square so that the sum of any three caps in a line (horizontally, vertically and diagonally) equals 15.
11. Perimeter magic triangle
This activity uses the same materials and concept as the magic square, but asks students to arrange the numbers one to six in a triangle where all three sides equal the same number. There are a few different solutions to this puzzle, so encourage students to see how many they can find.
Sudoku is an excellent after-lesson activity that encourages logical thinking and problem solving. You’ve probably already played this classic puzzle, and it’s a great choice for your students. Sudoku puzzles appear in newspapers around the world every day, and there are hundreds of online resources that generate puzzles based on difficulty.
13. Flexagon
There’s a pretty good chance that by now, fidget spinners have infiltrated your classroom. If you want to counter that invasion, consider challenging your students to create flexagons. Flexagons are paper-folded objects that can be transformed into different shapes through pinching and folding, and will keep wandering fingers busy and focused on the wonders of geometry.
14. Turn the fish
TransumThis puzzle
seems simple, but it just might stump your students. After setting up sticks in the required order, challenge them to make the fish swim in the other direction -- by moving just three matchsticks.
15. Join the dots
Cool Math 4 Kids
This puzzle challenges students to connect all the dots in a three by three grid using only four straight lines. While it may sound easy, chances are that it will take your class a while to come up with the solution. (Hint: it requires some “out of the box” thinking.)
16. Brain teasers
While they don’t always deal directly with math skills, brain teasers can be important tools in the development of a child’s critical thinking skills. Incorporate brain teasers into a classroom discussion, or use them as math journal prompts and challenge students to explain their thinking.
Bonus: For a discussion on probability introduce an older class to the Monty Hall Problem, one of the most controversial math logic problems of all time.
17. Tower of Hanoi
This interactive logic puzzle was invented by a French mathematician named Edouard Lucas in 1883. It even comes with an origin story: According to legend, there is a temple with three posts and 64 golden disks.
Priests move these disks in accordance with the rules of the game, in order to fulfill a prophecy that claims the world will end with the last move of the puzzle. But not to worry -- it’s going to take the priests about 585 billion years to finish, so you’ll be able to fit in the rest of your math class.
Starting with three disks stacked on top of each other, students must move all of the disks from the first to the third pole without stacking a larger disk on top of a smaller one. Older students can even learn about the functions behind the solution: the minimum number of moves can be expressed by the equation 2n-1, where n is the number of disks.
18. Tangram
Tangram puzzles -- which originated in China and were brought to Europe during the early 19th century through trade routes -- use seven flat, geometric shapes to make silhouettes. While Tangrams are usually made out of wood, you can make sets for your class out of colored construction paper or felt.
Tangrams are an excellent tool for learners who enjoy being able to manipulate their work, and there are thousands of published problems to keep your students busy.
Similar to Sudoku, Str8ts challenges players to use their logic skills to place numbers in blank squares. The numbers might be consecutive, but can appear in any order. For example, a row could be filled with 5, 7, 4, 6 and 8 . This puzzle is better suited to older students, and can be used as a before-class or after-lesson activity to reinforce essential logic skills.
20. Mobius band
Is it magic? Is it geometry? Your students will be so amazed they might have a hard time figuring it out. Have them model the problem with strips of paper and see for themselves how it works in real life. With older students, use mobius bands to talk about geometry and surface area.
Why use math puzzles to teach?
Math puzzles encourage critical thinking.
Critical thinking and logic skills are important for all careers, not just STEM-related ones. Puzzles challenge students to understand structure and apply logical thinking skills to new problems.
A study from the Eurasia Journal of Mathematics, Science and Technology Education found that puzzles “develop logical thinking, combinatorial abilities, strengthen the capacity of abstract thinking and operating with spatial images, instill critical thinking and develop mathematical memory.”
All these skills allow young students to build a foundation of skills they’ll draw on for the rest of their lives, no matter what kind of post-secondary route they pursue.
They help build math fluency
Math games can help students build a basic understanding of essential math concepts, and as another study shows, can also help them retain concepts longer .
In the study, early elementary students gradually moved from using the “counting” part of their brains to complete math problems to the “remembering” part that adults use, suggesting math puzzles and repeated problems can help build the essential skill of math fluency .
Many of the math puzzles above allow students to practice essential addition, subtraction, multiplication and division skills, while advanced or modified problems can be used to introduce pre-algebraic concepts and advanced logic skills.
Math puzzles connect to existing curricula
No matter what curriculum you’re using, there’s a good chance it emphasizes problem-solving, critique and abstract thinking. This is especially true of Common Core math and similar curricula.
How Math Skills Impact Student Development
Math puzzles allow students to develop foundational skills in a number of key areas, and can influence how students approach math practically and abstractly. You can also tie them into strategies like active learning and differentiated instruction.
Instead of just teaching facts and formulas, math puzzles allow you to connect directly with core standards in the curriculum. You can also use them to provide a valuable starting point for measuring how well students are developing their critical thinking and abstract reasoning skills.
Tips for using math puzzles in the classroom
View this post on Instagram A post shared by Sarah Werstuik (@teach.plan.love)
Now that you’ve got some great math puzzles, it might be tricky to figure out how to best incorporate them into your classroom. Here are some suggestions for making the most of your lesson time:
Make sure the puzzles are the right level for your class
If the problems are too easy, students will get bored and disengage from the lesson. However, if the problems are too difficult to solve, there’s a good chance they’ll get frustrated and give up early.
There’s a time and a place
While math puzzles are a great way to engage your students in developing critical thinking skills, they’re not a tool for teaching important math concepts. Instead, use them to reinforce the concepts they’ve already learned.
Kitty Rutherford , a Mathematics Consultant in North Carolina, emphasizes that math puzzles and games shouldn’t be based solely on mental math skills , but on “conceptual understanding” that builds fluency over time. Math puzzles help build the essential balance between thinking and remembering.
Give them space to figure it out
Rachel Keen , from the Department of Psychology at the University of Virginia, conducted a study about problem-solving skills in preschoolers. She found that “playful, exploratory learning leads to more creative and flexible use of materials than does explicit training from an adult.”
Give your students space to struggle with a problem and apply their own solutions before jumping in to help them. If the problem is grade-appropriate and solvable, students will learn more from applying their own reasoning to it than just watching you solve it for them.
Model puzzles for your students
Use problems like the mobius strip to awe and amaze your students before drawing them into a larger discussion about the mathematical concept that it represents. If possible, make math puzzles physical using recycled craft supplies or modular tools.
Afterward, have a class discussion or put up math journal prompts. What methods did your students try? What tools did they use? What worked and what didn’t? Having students explicitly state how they got to their solution (or even where they got stuck) challenges them to examine their process and draw conclusions from their experience.
Final thoughts on math puzzles
Be aware that it might take a while to get all your students on board -- they could be hesitant about approaching unfamiliar problems or stuck in the unenthusiasm that math class often brings. Consider creating a weekly leaderboard in your classroom for the students that complete the most puzzles, or work through a few as a class before sending students off on their own.
Instead of yawns and bored stares , get ready to see eager participants and thoughtful concentration. Whether you choose to use them as an after-class bonus, a first day of school activity or as part of a targeted lesson plan, math puzzles will delight your students while also allowing them to develop critical skills that they’ll use for the rest of their lives.
What are you waiting for? Get puzzling!
>> Create or log in to your teacher account on Prodigy – an engaging, game-based learning platform for math that’s easy to use for educators and students alike. Aligned with curricula across the English-speaking world, it’s used by more than a million teachers and 50 million students.
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Math Activities for Middle School Enrichment: Critical Thinking at a Critical Age
Gifted Free Activities for Middle School Students
May 1, 2018, by The Critical Thinking Co. Staff
Good Mathematical Habits for Young Adolescents
Mathematics content is best learned in a way that fosters good habits of mathematical thinking. The Common Core State Standards in Mathematics ( www.corestandards.org ) supplement their K-12 standards for content with eight standards for mathematical practice:
- Make sense of problems and persevere in solving them.
- Reason abstractly and quantitatively.
- Construct viable arguments and critique the reasoning of others.
- Model with mathematics.
- Use appropriate tools strategically.
- Attend to precision.
- Look for and make use of structure.
- Look for and express regularity in repeated reasoning. These mathematical practices, and how they relate to content, mean very different things depending upon student age.
Middle School as a Critical Transition Period
The middle school years mark a critical transition in a child’s cognitive development – how a child thinks and learns. Generally at age 11 or 12 children enter the fourth and final stage in Piaget’s four stages of cognitive development , called the formal operational stage. During this time children show significant growth in their ability to think abstractly, use advanced reasoning skills, make hypotheses and inferences, and draw logical conclusions. Ideally, the middle school years provide educators with new opportunities to foster good thinking habits and mathematical practices.
The Balance Between Mathematical Content and Practice
Students begin middle school exposed to mathematics as a very broad subject covering a wide array of topics: 2D geometry, probability, percentages, number theory, logic, patterns, statistics, graphing, number operations, proportions, elementary algebra, 3D geometry, and so on. They finish middle school and begin high school usually embarking on year-long studies of content-intensive mathematical subject areas: a year of Algebra 1, then a year of Geometry, then a year of Algebra 2, and so on. Though young adolescents begin middle school ready to think with more power, creativity, and independence, the accompanying increase in content expectations means that a balance between mathematical content and practice can be difficult to achieve. Developing good thinking and learning habits requires investment of time and patience, and well-intended educators can be drawn away from quality mathematical practices when the drive to learn content becomes too formidable.
Committing to Critical Thinking at the Middle School Level
Content can be learned in ways that ask young adolescents to harness and develop their new cognitive abilities. For example, a traditional 2D geometry question might ask:
Calculate the perimeter and area of a rectangle with a 15-inch length and a 9-inch width.
This question can be answered by performing a routine calculation using formulas for the perimeter and area of a rectangle. Similar content can be studied with a question that asks for critical thinking:
For what whole number values of length and width will the rectangle have an area of 60 square yards and a perimeter of 38 yards?
This second question (from Mathematical Reasoning™ Middle School Supplement ) requires students to develop a strategy to construct a solution. Indeed, a common approach involves making a mental or physical list of pairs of whole numbers that multiply to 60 and then searching for the pair of numbers that add up to 19 (since a rectangle’s perimeter is twice the sum of the length and width). The correct answer is a length of 15 inches and a width of 4 inches (assigning the larger number to length). Note the depth and value of a critical thinking opportunity: the solution strategy connects 2D geometry with the number theory technique of factoring and is a precursor to a more sophisticated factoring procedure used in Algebra 1. The second question requires greater time investment than the first question, but is worth the extra time if one is committed to young adolescents learning content in a way that fully engages their reasoning skills.
Fostering Perseverance
The first Common Core mathematical practice standard emphasizes the need to have students make sense of problems and persevere in solving them. The most important ingredient in Polya’s classic four-step problem solving strategy is the act of making decisions, as opposed to simply applying an algorithm that has been instructed. Young adolescent reaction to problem solving and decision making can be decidedly mixed. On the one hand, playing an active role in the solution process – figuring something out and being creative – can be fun, exciting, sometimes even addicting for young minds that are ready to be engaged. However, overcoming obstacles and persevering with a task that requires multiple steps and authentic reasoning can also sometimes be discouraging for early adolescent brains just learning how to tap into their emerging powers. The frustration level can depend on the difficulty level of the problem-solving situation, and a common, safe path is to keep decision making and creative expectations down to a minimum. However, if mathematics education in the United States is to reach a higher standard against a worldwide benchmark , children must be encouraged to persevere with critical thinking and decision making, to embrace both the excitement and occasional frustration of authentic reasoning and creativity.
Enrichment Activities to Stimulate Critical Thinking
The Critical Thinking Co.™ specializes in activities that stimulate use of reasoning skills and creativity when learning content. These enrichment activities challenge students to make decisions and construct solutions – to play an active role when learning content. Variety is favored over repetition, although care is taken to have common themes emphasized and connections reinforced. Presentation is often graphic intensive, resulting in visual appeal to young eyes. Real-world applications are easily identifiable. Problem-solving is supported with clear, comprehensive solutions and explanations. An example is provided with the activity sets Dimension Detective and Linear Patterns and accompanying solution pages from Mathematical Reasoning™ Middle School Supplement . In Dimension Detective students deduce missing dimensions for a variety of geometric shapes by using proportional reasoning, number theory ideas, and connections between 2D and 3D shapes. In Linear Patterns students determine number patterns and geometric patterns, and then deduce algebraic expressions to describe these patterns (a precursor to creating algebraic equations to describe linear graphs). Each activity set is accompanied by needed math facts, strategy tips, and comprehensive solutions that teachers and parents can use to help support student investigations. These sorts of enrichment activities provide middle school students with an opportunity to explore mathematical content, create or reinforce ideas, make connections, and use abstract reasoning. Young adolescents have emerging cognitive powers to accompany their rapid physical growth, and math enrichment can provide middle school students with appealing opportunities to use their maturing reasoning skills.
Free Printable Math Worksheets for Grades 6-9
- Dimension Detective (Number Theory)
- Linear Patterns
- Geometry & Introduction to Trigonometry
- Uncovering All the Angles
- Algebraic Cryptograms
- The Finest Pyramid
Visit our product pages to find hundreds more free printable activities.
Award-Winning Math Books for Middle School Students
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" Mathematical Reasoning presents math in a way that makes it not only simple to learn, but also practical to apply to everyday circumstance (obviously making math more meaningful to him)." - Katherine, Mississippi
"My son hated math. I saw your curriculum and gave it a try. Wow, what a great fit for my son! Your Mathematical Reasoning curriculum gave him a way to learn math that made sense to him. Thank you!" - Amy, Ontario, Canada
"Critical Thinking has uniquely captured the method of teaching students not only how to think logically, but to apply logic to the process of learning in every subject area. [M]any companies have begun to jump on the bandwagon and produce materials that use the critical thinking label, but the contents are the same old drill and practice, and our students know the difference right away. - Peggy, educator, Tucson, AZ
"For 12 years, I've recommended your company to my teacher-training students. Your materials are exceptionally valuable to teachers. Critical Thinking not only produces quality materials, but your fine company is unusually responsive to needs and questions of the individual teacher as well as of the big school districts." - Dr. Leonard Martin, Teacher Education Program, University of Denver
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Problem solving questions for middle school
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Middle School Math Worksheets
How to solve middle school math problems
With a simple review of your work, you can find ways to improve and understand How to solve middle school math problems!
Mathematical Problem
Students then use the structure of the problem to see if their initial guess works to solve the problem correctly. If their initial guess fails to work, they
Solving Mathematical Problems In More Than One Way
Solve equation
To solve an equation, one must find the value of the variable that makes the equation true.
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14 effective ways to help your students conquer math word.
by N Klang 2021 Cited by 13 - Mathematical problem-solving constitutes an important area of mathematics geometry in education that occurred in the mid-20th century,
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Rich Problems
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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. https://nrich.maths.org/10345
Math for All is a professional development program that brings general and special education teachers together to enhance their skills in planning and adapting mathematics lessons to ensure that all students achieve high-quality learning outcomes in mathematics.
Our Newsletter Provides Ideas for Making High-Quality Mathematics Instruction Accessible to All Students
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Is there a csc button on a calculator
You have to do 1/tan , 1/sin, 1/cos. Remember that sin = 1/csc and it works the other way too csc = 1/sin.
How to put csc sec cot in calculator
Why don't graphing calculators have secant and cosecant
So, since the cotangent is the reciprocal, then the formula for determining the cotangent is the adjacent side divided by the opposite side of
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How do you get (cot) (csc) (sec) on my calculator?
Cosecant calculator for degrees or radians.
Using the Cosecant Calculator To use the tool to find the cosecant, enter the measurement of the angle in either degrees or radians, choose the units, and
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Calculating Secant, Cosecant, and Cotangent Using the TI
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Competition Math for Middle School
Written for the gifted math student, the new math coach, the teacher in search of problems and materials to challenge exceptional students, or anyone else interested in advanced mathematical problems. Competition Math contains over 700 examples and problems in the areas of Algebra, Counting, Probability, Number Theory, and Geometry. Examples and full solutions present clear concepts and provide helpful tips and tricks.
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Below you can link to hundreds of middle school-level math worksheets throughout our website. These worksheets and printable activities will help students to learn about the absolute value of numbers. Includes identifying values, opposite numbers, ordering, and comparing. Work with basic algebraic expressions, equations, and inequalities.
We've categorized some of the best resources for several middle school math topics: Arithmetic Skills. Introduction to Counting. Basic Number Sense. Exploring Equations. Properties of Right Triangles. Multiple Approaches to Problems. Squares. Using Similar Figures.
Middle School Maths (Grades 6, 7, 8, 9) - Free Questions and Problems With Answers Grade 6, 7, 8 and 9 Maths questions and problems to test deep understanding of Maths concepts and computational procedures of students are presented. Answers to the questions are provided and located at the end of each page.
I have used "20 Questions" many times over the years. The questions posed are fun and challenging. The students enjoy working on them individually or in small groups. I usually give them about 20 to 23.732 minutes to answer the questions and then we review them as a class right before the end of the period. Sometimes I offer a small treat ...
Math Problem Solving Strategies 1. C.U.B.E.S. C.U.B.E.S stands for circle the important numbers, underline the question, box the words that are keywords, eliminate extra information, and solve by showing work. Why I like it: Gives students a very specific 'what to do.'
The most common methods for solving problems that students may encounter are: Guess and Check; Draw a picture; Work Backward; and Use an Equation with a variable. The first method on our list...
Module 6: Problem solving with the coordinate plane: 5th grade (Eureka Math/EngageNY) ... Bridge (8th): Middle school math (India) Our mission is to provide a free, world-class education to anyone, anywhere. Khan Academy is a 501(c)(3) nonprofit organization. Donate or volunteer today!
120 Math word problems, categorized by skill Addition word problems Best for: 1st grade, 2nd grade 1. Adding to 10: Ariel was playing basketball. 1 of her shots went in the hoop. 2 of her shots did not go in the hoop. How many shots were there in total? 2. Adding to 20: Adrianna has 10 pieces of gum to share with her friends.
These 5 math challenges will engage your students and get them thinking creatively and visually about math topics including fractions, areas models, the order of operations, and even algebra! So, can your middle schoolers solve these fun math puzzles? Pre-Algebra Solution: Pizza = 3, Hamburger = 2, Taco = 8 Multiplication Table
Math! An excellent way to build advanced problem-solving skills, as well as mathematical analysis, is to create math Boom Cards with word problems like these from Math in the Middle. Boom cards are a great activity for students to practice and build skills! Learn more: Boom Learning 16. Wheel of Solutions
Start by solving the division part of the equation. In order to do that, in case you forgot, you have to flip the fraction and switch from division to multiplication, thus getting 3 x 3 = 9. Now you have 9 - 9 + 1, and from there you can simply work from left to right and get your final answer: 1. 11.
Math Puzzles for Kids: 1. Math crossword puzzles Puzzles to Print Take a crossword, and make it math: that's the basic concept behind this highly adaptable math challenge. Instead of words, students use numbers to complete the vertical and horizontal strips.
Problem-solving is supported with clear, comprehensive solutions and explanations. An example is provided with the activity sets Dimension Detective and Linear Patterns and accompanying solution pages from Mathematical Reasoning™ Middle School Supplement. In Dimension Detective students deduce missing dimensions for a variety of geometric ...
Open Middle - Challenging math problems worth solving Home CHALLENGING MATH PROBLEMS WORTH SOLVING DOWNLOAD OUR FAVORITE PROBLEMS FROM EVERY GRADE LEVEL Get Our Favorite Problems Take The Online Workshop WANT GOOGLE SLIDE VERSIONS OF ALL PROBLEMS? HERE'S OUR GROWING COLLECTION Get Google Slide Versions WANT TO SHARE OPEN MIDDLE WITH OTHERS?
A national middle school mathematics competition that builds problem-solving skills and fosters achievement through four levels of fun, in-person "bee" style contests. ... art and technology and challenges students to produce a video solving a math problem in a real-world setting.
Make Sense of Math. 4.8. (30) $4.00. PDF. After a long break, wake-up your student's problem-solving skills with these middle school math activities that are just right for back to school. These are tasks that require problem-solving skills to complete. These are approachable for various skill levels.
Problem solving questions for middle school - Problem solving scenarios may be easy to find online, but this selection is geared specifically for speech ... Solve math problem 6. The Problem Solving Premium Packet allows students to apply and practice problem solving skills. used with older (junior high-high school) students. ...
Text and Solutions $ 49.00. Text: 256 pages. Solutions: 120 pages. A thorough introduction for students in grades 7-10 to counting and probability topics such as permutations, combinations, Pascal's triangle, geometric probability, basic combinatorial identities, the Binomial Theorem, and more.
Much of math at the middle school level therefore takes place in the context of problem solving. As you move closer to algebra, geometry, and even trigonometry, your students will need to work ...
Middle School Math Worksheets Here you can find math worksheets for middle school students. Print Study.com worksheets to use in the classroom or hand them out as homework. Our middle...
Spanish - Middle School Math Course 1: Next week's word problems Spanish - Middle School Math: Course 2 and Pre-Algebra Spanish - Middle School Math Course 2 and Pre-Algebra: Last week's word problems ... Write an equation for each problem. Then solve the equation. Functions Complete the function table (rule and easier inputs given)
Problem solving problems for middle school - All Aboard. is a problem-solving activity that is suited well to a class-size group. Amoeba Ball Toss Balloon. ... To determine what the math problem is, you will need to look at the given information and figure out what is being asked. Once you know what the problem is, you can solve it using the ...
Resources for Middle Grades (6-8) Mathematics Teaching in the Middle School, 10(9), 484-489. Problem Solving Tasks Solutions (Word) Determine math; Instant Expert Tutoring; Improve your math performance
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.
Here, we will be discussing about Impossible to solve math problems for middle school. Solve My Task. Get support from expert professors Get calculation support online Do homework x. 15 tricky children's test questions that stumped the internet. If you thought elementary school was hard, you won't believe the tricky math questions and hard math ...
Middle School/Lower Secondary High School/Upper Secondary College/University ... The simplest math problem no one can solve. lesson duration 22:09 33,302,871 views. 05:07. Thinking & Learning Can you solve the giant spider riddle? lesson duration 05:07 ...
Competition Math for Middle School. Jason Batterson. Paperback. 390 pages. Contains over 700 examples and problems in the areas of Algebra, Counting, Probability, Number Theory, and Geometry. Examples and full solutions present clear concepts and provide helpful tips and tricks.
10 Best Problem Solving Activities For Middle School. Schools Details: WebIf you're a teacher or a parent looking to engage young students in a more interesting way, you might like this list of the best problem solving activities for middle … middle school problem solving games › Verified 5 days ago