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math lessons solving problems

Teaching Problem Solving in Math

  • Freebies , Math , Planning

Problem solving tends to REALLY throw students for a loop when they're first introduced to it. Up until this point, math has been numbers, but now, math is numbers and words. I discuss four important steps I take in teaching problem solving, and I provide you with examples as I go. You can also check out my math workshop problem solving unit for third grade!

Every year my students can be fantastic at math…until they start to see math with words. For some reason, once math gets translated into reading, even my best readers start to panic. There is just something about word problems, or problem-solving, that causes children to think they don’t know how to complete them.

Every year in math, I start off by teaching my students problem-solving skills and strategies. Every year they moan and groan that they know them. Every year – paragraph one above. It was a vicious cycle. I needed something new.

Problem solving tends to REALLY throw students for a loop when they're first introduced to it. Up until this point, math has been numbers, but now, math is numbers and words. I discuss four important steps I take in teaching problem solving, and I provide you with examples as I go. You can also check out my math workshop problem solving unit for third grade!

I put together a problem-solving unit that would focus a bit more on strategies and steps in hopes that that would create problem-solving stars.

The Problem Solving Strategies

First, I wanted to make sure my students all learned the different strategies to solve problems, such as guess-and-check, using visuals (draw a picture, act it out, and modeling it), working backward, and organizational methods (tables, charts, and lists). In the past, I had used worksheet pages that would introduce one and provide the students with plenty of problems practicing that one strategy. I did like that because students could focus more on practicing the strategy itself, but I also wanted students to know when to use it, too, so I made sure they had both to practice.

I provided students with plenty of practice of the strategies, such as in this guess-and-check game.

Problem solving tends to REALLY throw students for a loop when they're first introduced to it. Up until this point, math has been numbers, but now, math is numbers and words. I discuss four important steps I take in teaching problem solving, and I provide you with examples as I go. You can also check out my math workshop problem solving unit for third grade!

There’s also this visuals strategy wheel practice.

Problem solving tends to REALLY throw students for a loop when they're first introduced to it. Up until this point, math has been numbers, but now, math is numbers and words. I discuss four important steps I take in teaching problem solving, and I provide you with examples as I go. You can also check out my math workshop problem solving unit for third grade!

I also provided them with paper dolls and a variety of clothing to create an organized list to determine just how many outfits their “friend” would have.

Problem solving tends to REALLY throw students for a loop when they're first introduced to it. Up until this point, math has been numbers, but now, math is numbers and words. I discuss four important steps I take in teaching problem solving, and I provide you with examples as I go. You can also check out my math workshop problem solving unit for third grade!

Then, as I said above, we practiced in a variety of ways to make sure we knew exactly when to use them. I really wanted to make sure they had this down!

Problem solving tends to REALLY throw students for a loop when they're first introduced to it. Up until this point, math has been numbers, but now, math is numbers and words. I discuss four important steps I take in teaching problem solving, and I provide you with examples as I go. You can also check out my math workshop problem solving unit for third grade!

Anyway, after I knew they had down the various strategies and when to use them, then we went into the actual problem-solving steps.

The Problem Solving Steps

I wanted students to understand that when they see a story problem, it isn’t scary. Really, it’s just the equation written out in words in a real-life situation. Then, I provided them with the “keys to success.”

S tep 1 – Understand the Problem.   To help students understand the problem, I provided them with sample problems, and together we did five important things:

  • read the problem carefully
  • restated the problem in our own words
  • crossed out unimportant information
  • circled any important information
  • stated the goal or question to be solved

We did this over and over with example problems.

Problem solving tends to REALLY throw students for a loop when they're first introduced to it. Up until this point, math has been numbers, but now, math is numbers and words. I discuss four important steps I take in teaching problem solving, and I provide you with examples as I go. You can also check out my math workshop problem solving unit for third grade!

Once I felt the students had it down, we practiced it in a game of problem-solving relay. Students raced one another to see how quickly they could get down to the nitty-gritty of the word problems. We weren’t solving the problems – yet.

Problem solving tends to REALLY throw students for a loop when they're first introduced to it. Up until this point, math has been numbers, but now, math is numbers and words. I discuss four important steps I take in teaching problem solving, and I provide you with examples as I go. You can also check out my math workshop problem solving unit for third grade!

Then, we were on to Step 2 – Make a Plan . We talked about how this was where we were going to choose which strategy we were going to use. We also discussed how this was where we were going to figure out what operation to use. I taught the students Sheila Melton’s operation concept map.

Problem solving tends to REALLY throw students for a loop when they're first introduced to it. Up until this point, math has been numbers, but now, math is numbers and words. I discuss four important steps I take in teaching problem solving, and I provide you with examples as I go. You can also check out my math workshop problem solving unit for third grade!

We talked about how if you know the total and know if it is equal or not, that will determine what operation you are doing. So, we took an example problem, such as:

Sheldon wants to make a cupcake for each of his 28 classmates. He can make 7 cupcakes with one box of cupcake mix. How many boxes will he need to buy?

We started off by asking ourselves, “Do we know the total?” We know there are a total of 28 classmates. So, yes, we are separating. Then, we ask, “Is it equal?” Yes, he wants to make a cupcake for EACH of his classmates. So, we are dividing: 28 divided by 7 = 4. He will need to buy 4 boxes. (I actually went ahead and solved it here – which is the next step, too.)

Step 3 – Solving the problem . We talked about how solving the problem involves the following:

  • taking our time
  • working the problem out
  • showing all our work
  • estimating the answer
  • using thinking strategies

We talked specifically about thinking strategies. Just like in reading, there are thinking strategies in math. I wanted students to be aware that sometimes when we are working on a problem, a particular strategy may not be working, and we may need to switch strategies. We also discussed that sometimes we may need to rethink the problem, to think of related content, or to even start over. We discussed these thinking strategies:

  • switch strategies or try a different one
  • rethink the problem
  • think of related content
  • decide if you need to make changes
  • check your work
  • but most important…don’t give up!

To make sure they were getting in practice utilizing these thinking strategies, I gave each group chart paper with a letter from a fellow “student” (not a real student), and they had to give advice on how to help them solve their problem using the thinking strategies above.

Problem solving tends to REALLY throw students for a loop when they're first introduced to it. Up until this point, math has been numbers, but now, math is numbers and words. I discuss four important steps I take in teaching problem solving, and I provide you with examples as I go. You can also check out my math workshop problem solving unit for third grade!

Finally, Step 4 – Check It.   This is the step that students often miss. I wanted to emphasize just how important it is! I went over it with them, discussing that when they check their problems, they should always look for these things:

  • compare your answer to your estimate
  • check for reasonableness
  • check your calculations
  • add the units
  • restate the question in the answer
  • explain how you solved the problem

Then, I gave students practice cards. I provided them with example cards of “students” who had completed their assignments already, and I wanted them to be the teacher. They needed to check the work and make sure it was completed correctly. If it wasn’t, then they needed to tell what they missed and correct it.

Problem solving tends to REALLY throw students for a loop when they're first introduced to it. Up until this point, math has been numbers, but now, math is numbers and words. I discuss four important steps I take in teaching problem solving, and I provide you with examples as I go. You can also check out my math workshop problem solving unit for third grade!

To demonstrate their understanding of the entire unit, we completed an adorable lap book (my first time ever putting together one or even creating one – I was surprised how well it turned out, actually). It was a great way to put everything we discussed in there.

Problem solving tends to REALLY throw students for a loop when they're first introduced to it. Up until this point, math has been numbers, but now, math is numbers and words. I discuss four important steps I take in teaching problem solving, and I provide you with examples as I go. You can also check out my math workshop problem solving unit for third grade!

Once we were all done, students were officially Problem Solving S.T.A.R.S. I just reminded students frequently of this acronym.

Stop – Don’t rush with any solution; just take your time and look everything over.

Think – Take your time to think about the problem and solution.

Act  – Act on a strategy and try it out.

Review – Look it over and see if you got all the parts.

Problem solving tends to REALLY throw students for a loop when they're first introduced to it. Up until this point, math has been numbers, but now, math is numbers and words. I discuss four important steps I take in teaching problem solving, and I provide you with examples as I go. You can also check out my math workshop problem solving unit for third grade!

Wow, you are a true trooper sticking it out in this lengthy post! To sum up the majority of what I have written here, I have some problem-solving bookmarks FREE to help you remember and to help your students!

Problem solving tends to REALLY throw students for a loop when they're first introduced to it. Up until this point, math has been numbers, but now, math is numbers and words. I discuss four important steps I take in teaching problem solving, and I provide you with examples as I go. You can also check out my math workshop problem solving unit for third grade!

You can grab these problem-solving bookmarks for FREE by clicking here .

You can do any of these ideas without having to purchase anything. However, if you are looking to save some time and energy, then they are all found in my Math Workshop Problem Solving Unit . The unit is for grade three, but it  may work for other grade levels. The practice problems are all for the early third-grade level.

math lessons solving problems

  • freebie , Math Workshop , Problem Solving

math lessons solving problems

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Math Problem Worksheets Word Problems Hub Page

Welcome to the Math Salamanders' Math Problem Worksheets.

Here you will find a range of different math problem sheets to help you learn to apply your math facts to solve a range of problems.

There are a range of different word problems and fun math challenges for each grade.

We also have a selection of word problems sorted by math area including fractions and ratio.

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Math Problem Worksheets

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Look no further! The pages you need are below!

Here is our selection of different Math problem worksheets.

Included in this page are a range of math problem pages from 1st grade to 5th grade. There are also fraction problems, ratio problems as well as addition, subtraction, multiplication and division problems.

Many of the problem sheets use 'real life' data, so your child can learn some interesting facts while they are solving problems, and also hopefully see the point of all the math facts they have learnt at the same time.

Each problem solving sheet comes with a separate answer sheet.

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Here you will find a range of math problems aimed at first grade level. Each problem sheet is based on an interesting theme such as parties or the seaside.

Using these sheets will help your child to:

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Here are a range of problems solving sheets for 2nd graders. Most of the sheets contain 'real-life' problems related to animal facts.

Using the sheets will help your child to:

  • apply their addition, subtraction, and multiplication skills;
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All the 2nd grade math problem worksheets in this section support Elementary math benchmarks.

These sheets involve solving one or two more challenging longer problems.

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These sheets involve solving a range of division problems.

Using this link will open our 2nd Grade Math Salamanders website in a new browser window.

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3rd Grade Math Word Problems

Here are a range of problems solving sheets for 3rd graders. Most of the sheets contain 'real-life' problems which involve real data.

  • develop their knowledge of fractions;

All the 3rd grade sheets in this section support Elementary math benchmarks.

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These sheets involve solving 3-digit and 4-digit addition word problems.

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Here is our set of 4th grade math problems to help your child with their problem solving skills.

Each problem sheet comes complete with answers, and is available in both standard and metric units where applicable.

Many of the problems are based around 'real-life' problems and data such as the world's heaviest animals.

  • apply their addition, subtraction and problem solving skills;
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Using the problems in this section will help your child develop their problem solving and reasoning skills.

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Here you will find a range of subtraction word problems to help your child apply their subtraction facts.

The worksheets cover subtraction problems from 1st to 3rd grade.

These sheets involve solving a range of subtraction word problems up to 100.

  • Subtraction Word Problems 2nd grade

Addition & Subtraction Problems

Here you will find a range of addition and subtraction word problems to help your child apply their knowledge.

  • 2nd Grade Addition and Subtraction Word Problems

Here you will find a range of multiplication word problems to help your child apply their multiplication facts.

The worksheets cover multiplication problems from 2nd to 5th grade.

  • Multiplication Word Problems 5th Grade

Here you will find a range of division word problems to help your child apply their division facts.

The worksheets cover division problems from 3rd to 5th grade.

Inequalities Word Problems

These sheets involve changing a word problem into an inequality.

Here you will find a range of fraction word problems to help your child apply their fraction learning.

The worksheets cover a range of fraction objectives, from adding and subtracting fractions to working out fractions of numbers. The sheets support fraction learning from 2nd grade to 5th grade.

  • Fraction Riddles for kids (easier)
  • Free Printable Fraction Riddles (harder)
  • Percentage Word Problems 5th Grade

Here you will find a range of ratio word problems to help your child understand what a ratio is and how ratios work.

The sheets support ratio learning at a 5th grade level.

This is our finding all possibilities area where all the worksheets involve finding many different answers to the problem posed.

The sheets here encourage systematic working and logical thinking.

The problems are different in that, there is typically only one problem per sheet, but the problem may take quite a while to solve!

  • Finding all Possibilities problems
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This is our logic problems area where all the worksheets involve using reasoning and logical thinking skills.

The sheets here are designed to get children thinking logically and puzzling the problems out.

There are a range of different logic problems from 1st through 5th grade!

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Common Core Math Explained: 8 Common Core Math Examples To Use In The Classroom 

Samantha dock.

Navigating the world of Common Core math can be tricky for teachers who are trying to meet the individual needs of their students. It can be helpful to have a bank of Common Core math examples to hand when planning your lessons. 

Embraced by the majority of states in the U.S., the Common Core math standards help to develop students’ conceptual understanding, problem-solving skills, and real-world applications.  In this article, we explore what Common Core math is, 8 Common Core math examples and top tips for educators teaching Common Core math.

What is Common Core math?

How is common core math different from traditional math, 8 common core math examples, tips for teaching common core math, common core math and the wider world.

Common Core math is a set of educational standards for mathematics adopted by forty states in the United States. Each standard outlines the math knowledge students should know and be able to do at each grade level, from kindergarten through to high school. 

These standards aim to provide a more focused and coherent set of learning goals for students, with an emphasis on conceptual understanding, problem-solving, and critical thinking skills.  

Often, Common Core math involves multiple strategies and approaches to solving problems. In turn, this encourages students to understand the underlying concepts rather than simply memorizing algorithms.  

One of the aims of the Common Core State Standards is to move away from traditional memorization of procedures and algorithms towards a deeper understanding of connections between mathematical concepts.  

Common Core math standards are organized by grade level and cover a wide range of mathematical topics, including:  

  • Arithmetic 
  • Probability

Each standard is divided into domains, which represent broad categories of mathematical content such as: 

  • Counting and cardinality
  • Operations and algebraic thinking
  • Numbers and operations in base ten
  • Measurement and data

Common Core math standards have been controversial in some areas due to concerns about curriculum changes, standardized testing, and complexity. But some argue that these standards provide a more coherent approach to mathematics education and better prepare students for higher education than traditional math. 

Read more: Why is Math Important?

Common Core math and traditional math represent two different approaches to teaching mathematics.

Traditional math typically refers to methods of teaching mathematics that were used before the adoption of the Common Core standards. These methods often focused on rote memorization of formulas and procedures, with less emphasis on understanding the concepts or on real-life application of mathematical skills.

Here are some key differences between the two:

  • Focus on Conceptual Understanding vs. Memorization A strong emphasis is placed on developing students’ conceptual understanding of math concepts under the common core. It aims to help learners understand the “why” behind mathematical procedures rather than just memorizing algorithms.  Traditional math often focuses more on rote memorization of formulas and procedures without necessarily understanding the underlying concepts needed to approach math questions.
  • Problem-Solving and Critical Thinking vs. Rote Practice Common Core math problems encourage critical thinking skills. They promote multiple approaches to solving new math problems and require students to justify their reasoning. Often, traditional math involves repetitive practice of standard procedures with less emphasis on problem-solving and critical thinking.
  • Real-World Applications vs. Abstract Exercises Connections between mathematical concepts to real-world situations is valued under the common core. This helps students see the relevance of the math skills they are learning. Tasks and problems require the application of mathematical skills in practical contexts.  Traditional math lessons focus more on abstract exercises and textbook problems that may not always have clear real-world connections.
  • Depth of Understanding vs. Breadth of Coverage Rather than covering a wide range of topics, Common Core math aims for depth of understanding and maths mastery. Fewer topics at each grade level allow for deeper exploration and mastery of key concepts.  In contrast, traditional math tends to cover a broader range of topics in less depth.
  • Flexibility and Multiple Strategies vs. One Correct Method Students are encouraged to use multiple strategies and approaches to solve problems through the Common Core math standards. Flexibility and creativity are valued when approaching problem-solving.  Emphasis on a single “correct” method or algorithm for solving problems is the general approach in traditional math. Overall, Common Core State Standards aim to develop students’ mathematical proficiency in alignment with the demands of the modern world. This includes the need for critical thinking, problem-solving, and application of mathematical concepts to real-world situations

Overall, Common Core State Standards aim to develop students’ mathematical proficiency in alignment with the demands of the modern world. This includes the need for critical thinking, problem-solving, and application of mathematical concepts to real-world situations.  

Third Space Learning provides one-on-one math instruction for students who need it most. Personalized one-on-one math lessons are designed by math experts and aligned to your state’s math standards — including the Common Core State Standards. 

math lessons solving problems

3rd to 6th Grade Common Core Math test

Help your students prepare for their state math test with these 3rd - 6th grade Common Core practice math tests.

Common Core math example 1: Make sense of problems and persevere in solving them

Students should not only be able to understand problems and make sense of them, but persevere in finding solutions. 

Finding solutions may involve math skills such as: 

  • Analyzing problems
  • Making conjectures
  • Planning approaches to solving math problems 

Example: A student is faced with a word problem about finding the area of a garden. They must take the time to carefully read and understand the problem before attempting to solve it. 

This problem may require several approaches to answer the math question. Small group work and discussion can encourage students to persevere through the challenge and try different strategies until they find a solution.

Common Core math example 2: Reason abstractly and quantitatively

In order to reason abstractly, students need to be able to make sense of quantities and their relationships in mathematical situations. 

This will be easier for students if they can take abstract information from context and quantify information. Being able to decontextualize and contextualize mathematical ideas will benefit students.

Example: A graph shows the relationship between the number of hours worked and the amount earned. Students can analyze the graph to determine patterns and make predictions about future earnings based on proportional relationships between hours worked and money earned. 

For example, if John worked for 13 hours, how much money would he earn?

math lessons solving problems

Common Core math example 3: Construct viable arguments and critique the reasoning of others

Introducing math vocabulary in the classroom helps students construct viable arguments and critique the mathematical reasoning of others. Exposure to mathematics language and sentence stems will help students to reason mathematically, construct arguments, and justify their thinking, without creating cognitive overload. 

Example: During a class discussion about strategies for solving a particular math problem, you might ask students to present their solutions — justifying and explaining their reasoning. 

They can also be encouraged to critique each other’s approaches, identify strengths and weaknesses in their arguments and offer alternative methods. 

Common Core math example 4: Model mathematics

Math lessons should prepare students to use math to solve real-world problems. It may help students to do this if you represent mathematical concepts with visual models and math manipulatives . 

Example: Subtraction of fractions is a skill that many students struggle with. Using a visual model to describe and analyze the word problem can release cognitive load for students. 

For example, Paul had 11 ⅔ yards of twine. He used 6 ½ yards to make macrame wall hangings, how many yards of twine does Paul have left?

math lessons solving problems

Common Core math example 5: Use appropriate tools strategically

To solve math problems effectively and efficiently, students must be able to select and use appropriate tools. This includes recognizing when and how to use tools, as well as evaluating effectiveness and efficiency.

Example: When solving a complex geometry problem, students should recognise the effectiveness of using a protractor and ruler to accurately measure angles and lengths. 

For example, Given an angle ABC where point B is the vertex of the angle, construct an angle bisector of angle ABC using a ruler and a protractor. Then, using the angle bisector you have constructed, draw a line segment from point B to the bisected angle’s line that is exactly 5 cm long. Measure and report the angle sizes of the two new angles created by the angle bisector.

Common Core math example 6: Attend to precision

Calculations need to be carried out precisely. To do this, students need to be aware of key mathematical terminology for the Common Core Standards they are studying. This involves using appropriate units and labels and stating mathematical results clearly.

Example: A student ensures that their work is clear and organized. They pay attention to detail, avoiding errors and inaccuracies in their calculations. Below is a worked example of a student showing how to solve a word problem involving multiple percentages.

math lessons solving problems

Common Core math example 7: Look for and make use of structure

Solving math problems accurately means students need to recognize and use mathematical patterns and structure. They should be able to identify relationships between mathematical ideas and make connections between different mathematical representations.

Example: When solving a multiplication of decimals problem, a student recognizes that breaking down the whole numbers and decimal parts into their factors makes the problem easier to solve. They identify the underlying structure of the problem and use it to their advantage.

math lessons solving problems

Common Core math example 8: Look for and express regularity in repeated reasoning

Identifying and generalizing patterns and regularities in mathematical situations is key to proficiency in problem soving and reasoning . Students should be able to notice repeated reasoning and use it to solve math problems efficiently. 

Example: A student identifies similarities between a problem they’re working on and a previous math problem. They utilize the patterns in the prior example to complete the new problem. This also helps them to solve similar problems in the future.

math lessons solving problems

Teachers need to understand Common Core math standards to recognize the appropriate instructional strategies and promote a growth mindset in the classroom .  

Here are 8 tips for maximising student progress when teaching the Common Core State Standards:

1. Understand the Common Core State Standards

Familiarize yourself with the math Common Core State Standards for your specific grade level. Take the time to understand the mathematical practices, domains and teaching strategies required for your grade.

2. Focus on conceptual understanding

Prioritize conceptual understanding over rote memorization. You can achieve this by helping students understand the “why” behind math concepts and skills. Always encourage them to explain and justify their reasoning.

3. Promote multiple approaches

Offer your students a range of math strategies and approaches to problem-solving. The more methods in their math bank, the better equipped they are to find a solution. Asking students to share their thinking process helps those who are not grasping the content from the math instruction.

4. Real-world connections

Connecting mathematical concepts to real-world situations makes learning more meaningful and relevant. You can do this is by implementing math problems where students work collaboratively to solve complex, open-ended word problems with real-world relevance. 

5. Use visual representations

Diagrams, models, and manipulatives support students’ understanding of mathematical concepts by making abstract concepts more concrete and accessible. For example, you could use algebra tiles when students are first learning how to solve algebraic equations and inequalities to help them contexutalize the abstract nature of algebra.

6. Encourage discourse and collaboration

Promote a classroom environment where students feel comfortable sharing their ideas, asking questions and engaging in mathematical discourse. 

Encourage discourse by using techniques such as turn and talk, or the 3 reads method for word problems.

7. Assess progress

Use formative and summative assessments to monitor students’ progress and understanding of mathematical concepts and adjust instruction accordingly based on assessment data.  

Some examples of a formative assessment are: 

  • Exit tickets
  • Rating scales
  • Thumbs up or thumbs down 

Summative assessments include: 

  • Check for understanding quizzes
  • End-of-topic quizzes

Assessment resources:

  • Practice state assessments  

8. Professional development

Continuously seek professional development opportunities to deepen your understanding of Common Core math and improve your teaching practices. Collaborate with colleagues and participate in workshops, conferences, and online courses.

Embracing Common Core principles can help equip students for future challenges

Educators’ commitment to teaching Common Core math goes beyond math instruction. It’s about nurturing critical thinking and problem solving, ensuring students are prepared for the wider world. 

Math lessons are no longer simply giving students math worksheets and grading them on the correct answer. The American education system has developed a math curriculum that anchors mathematical concepts in real-world relevance, promotes diverse problem-solving strategies, and encourages a collaborative learning environment.  

Educators have a responsibility to ensure students have the tools and mathematical literacy they need to succeed.

Common Core math examples FAQ

1. Make sense of problems and persevere in solving them 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others 4. Model with mathematics 5. Use appropriate tools strategically 6. Attend to precision 7. Look for and make use of structure 8. Look for and express regularity in repeated reasoning

1. Focus: Emphasizes focusing deeply on a smaller number of key topics at each grade level. This is done to ensure students develop a deep understanding of foundational mathematical ideas. 2. Coherence: Emphasizes the importance of coherence in mathematical instruction. This is done to support students in making meaningful connections between different mathematical ideas, helping them see how concepts are related and reinforcing their understanding over time 3. Rigor: Focuses on increasing the rigor of mathematical instruction by demanding that students engage in conceptual understanding, procedural fluency, and application of mathematical concepts in real-world contexts. In this context, rigor means ensuring that students develop a deep understanding of mathematical concepts, are able to apply their knowledge in various contexts, and can solve complex problems through reasoning and critical thinking.

Forty states have fully adopted Common Core math, while Minnesota partially embraces it. South Carolina, Oklahoma, Indiana, Florida, and Arizona initially adopted but later repealed Common Core. Alaska, Nebraska, Texas, and Virginia never adopted it.

Do you have students who need extra support in math? Give your students more opportunities to consolidate learning and practice skills through personalized math tutoring with their own dedicated online math tutor. Each student receives differentiated instruction designed to close their individual learning gaps, and scaffolded learning ensures every student learns at the right pace. Lessons are aligned with your state’s standards and assessments, plus you’ll receive regular reports every step of the way. Personalized one-on-one math tutoring programs are available for: – 2nd grade tutoring – 3rd grade tutoring – 4th grade tutoring – 5th grade tutoring – 6th grade tutoring – 7th grade tutoring – 8th grade tutoring Why not learn more about how it works ?

Ultimate Guide to Problem Solving Techniques [FREE]

Are you trying to build problem solving and reasoning skills in the classroom?

Here are 9 ready-to-go printable problem solving techniques that all your students should know, including challenges, short explanations and questioning prompts.

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Featured series.

A series of random questions answered by Harvard experts.

Explore the Gazette

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One way is the wrong way to do math. Here’s the right way.

Harvard Staff Writer

Research by Ed School psychologist reinforces case for stressing multiple problem-solving paths over memorization

There’s never just one way to solve a math problem, says Jon R. Star , a psychologist and professor of education at the Harvard Graduate School of Education. With researchers from Vanderbilt University, Star found that teaching students multiple ways to solve math problems instead of using a single method improves teaching and learning. In an interview with the Gazette, Star, a former math teacher, outlined the research and explained how anyone, with the right instruction, can develop a knack for numbers.

Jon R. Star

GAZETTE: What is the most common misconception about math learning?

STAR: That you’re either a math person or you’re not a math person — that some people are just born with math smarts, and they can do math, and other people are just not, and there’s not much you can do about it.

GAZETTE: What does science say about the process of learning math?

STAR: One thing we know from psychology about the learning process is that the act of reaching into your brain, grabbing some knowledge, pulling it out, chewing on it, talking about it, and putting it back helps you learn. Psychologists call this elaborative encoding. The more times you can do that process — putting knowledge in, getting it out, elaborating on it, putting it back in — the more you will have learned, remembered, and understood the material. We’re trying to get math teachers to help students engage in that process of elaborative encoding.

GAZETTE: How did you learn math yourself?

STAR: Learning math should involve some sense-making. It’s necessary that we listen to what our teacher tells us about the math and try to make sense of it in our minds. Math learning is not about pouring the words directly from the teacher’s mouth into the students’ ears and brains. That’s not the way it works. I think that’s how I learned math. But that’s not how I hope students learn math and that’s not how I hope teachers think about the teaching of math. Teachers should teach math in a way that encourages students to engage in sense-making and not merely to memorize or internalize exactly what the teacher says or does.

GAZETTE: Tell us about the teaching method described in the research.

STAR: One of the strategies that some teachers may use when teaching math is to show students how to solve problems and expect that the student is going to end up using the same method that the teacher showed. But there are many ways to solve math problems; there’s never just one way.

The strategy we developed asks that teachers compare two ways for solving a problem, side by side, and that they follow an instructional routine to lead a discussion to help students understand the difference between the two methods. That discussion is really the heart of this routine because it is fundamentally about sharing reasoning: Teachers ask students to explain why a strategy works, and students must dig into their heads and try to say what they understand. And listening to other people’s reasoning reinforces the process of learning.

GAZETTE: Why is this strategy an improvement over just learning a single method?

STAR: We think that learning multiple strategies for solving problems deepens students’ understanding of the content. There is a direct benefit to learning through comparing multiple methods, but there are also other types of benefits to students’ motivation. In this process, students come to see math a little differently — not just as a set of problems, each of which has exactly one way to solve it that you must memorize, but rather, as a terrain where there are always decisions to be made and multiple strategies that one might need to justify or debate. Because that is what math is.

For teachers, this can also be empowering because they are interested in increasing their students’ understanding, and we’ve given them a set of tools that can help them do that and potentially make the class more interesting as well. It’s important to note, too, that this approach is not something that we invented. In this case, what we’re asking teachers to do is something that they do a little bit of already. Every high school math teacher, for certain topics, is teaching students multiple strategies. It’s built into the curriculum. All that we’re saying is, first, you should do it more because it’s a good thing, and second, when you do it, this is a certain way that we found to be especially effective, both in terms of the visual materials and the pedagogy. It’s not a big stretch for most teachers. Conversations around ways to teach math for the past 30 or 40 years, and perhaps longer, have been emphasizing the use of multiple strategies.

GAZETTE: What are the potential challenges for math teachers to put this in practice?

STAR: If we want teachers to introduce students to multiple ways to solve problems, we must recognize that that is a lot of information for students and teachers. There is a concern that there could be information overload, and that’s very legitimate. Also, a well-intentioned teacher might take our strategy too far. A teacher might say something like, “Well, if comparing two strategies is good, then why don’t I compare three or four or five?” Not that that’s impossible to do well. But the visual materials you would have to design to help students manage that information overload are quite challenging. We don’t recommend that.

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Math Problem Solving Strategies

In these lessons, we will learn some math problem solving strategies for example, Verbal Model (or Logical Reasoning), Algebraic Model, Block Model (or Singapore Math), Guess & Check Model and Find a Pattern Model.

Related Pages Solving Word Problems Using Block Models Heuristic Approach to Problem-Solving Algebra Lessons

Problem Solving Strategies

The strategies used in solving word problems:

  • What do you know?
  • What do you need to know?
  • Draw a diagram/picture

Solution Strategies Label Variables Verbal Model or Logical Reasoning Algebraic Model - Translate Verbal Model to Algebraic Model Solve and Check.

Solving Word Problems

Step 1: Identify (What is being asked?) Step 2: Strategize Step 3: Write the equation(s) Step 4: Answer the question Step 5: Check

Problem Solving Strategy: Guess And Check

Using the guess and check problem solving strategy to help solve math word problems.

Example: Jamie spent $40 for an outfit. She paid for the items using $10, $5 and $1 bills. If she gave the clerk 10 bills in all, how many of each bill did she use?

Problem Solving : Make A Table And Look For A Pattern

  • Identify - What is the question?
  • Plan - What strategy will I use to solve the problem?
  • Solve - Carry out your plan.
  • Verify - Does my answer make sense?

Example: Marcus ran a lemonade stand for 5 days. On the first day, he made $5. Every day after that he made $2 more than the previous day. How much money did Marcus made in all after 5 days?

Find A Pattern Model (Intermediate)

In this lesson, we will look at some intermediate examples of Find a Pattern method of problem-solving strategy.

Example: The figure shows a series of rectangles where each rectangle is bounded by 10 dots. a) How many dots are required for 7 rectangles? b) If the figure has 73 dots, how many rectangles would there be?

a) The number of dots required for 7 rectangles is 52.

b) If the figure has 73 dots, there would be 10 rectangles.

Example: Each triangle in the figure below has 3 dots. Study the pattern and find the number of dots for 7 layers of triangles.

The number of dots for 7 layers of triangles is 36.

Example: The table below shows numbers placed into groups I, II, III, IV, V and VI. In which groups would the following numbers belong? a) 25 b) 46 c) 269

Solution: The pattern is: The remainder when the number is divided by 6 determines the group. a) 25 ÷ 6 = 4 remainder 1 (Group I) b) 46 ÷ 6 = 7 remainder 4 (Group IV) c) 269 ÷ 6 = 44 remainder 5 (Group V)

Example: The following figures were formed using matchsticks.

a) Based on the above series of figures, complete the table below.

b) How many triangles are there if the figure in the series has 9 squares?

c) How many matchsticks would be used in the figure in the series with 11 squares?

b) The pattern is +2 for each additional square.   18 + 2 = 20   If the figure in the series has 9 squares, there would be 20 triangles.

c) The pattern is + 7 for each additional square   61 + (3 x 7) = 82   If the figure in the series has 11 squares, there would be 82 matchsticks.

Example: Seven ex-schoolmates had a gathering. Each one of them shook hands with all others once. How many handshakes were there?

Total = 6 + 5 + 4 + 3 + 2 + 1 = 21 handshakes.

The following video shows more examples of using problem solving strategies and models. Question 1: Approximate your average speed given some information Question 2: The table shows the number of seats in each of the first four rows in an auditorium. The remaining ten rows follow the same pattern. Find the number of seats in the last row. Question 3: You are hanging three pictures in the wall of your home that is 16 feet wide. The width of your pictures are 2, 3 and 4 feet. You want space between your pictures to be the same and the space to the left and right to be 6 inches more than between the pictures. How would you place the pictures?

The following are some other examples of problem solving strategies.

Explore it/Act it/Try it (EAT) Method (Basic) Explore it/Act it/Try it (EAT) Method (Intermediate) Explore it/Act it/Try it (EAT) Method (Advanced)

Finding A Pattern (Basic) Finding A Pattern (Intermediate) Finding A Pattern (Advanced)

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Mathematical Thinking: An Essential Skill For Everyday Living

Created: February 21, 2024

Last updated: February 21, 2024

Every day, kids and adults alike encounter challenges that require mathematics. You need a basic knowledge of mathematics, whether you’re going to the grocery store or just want to count the coins you gave in your piggy bank. These are classic examples of locations and situations where you regularly engage your mathematical thinking capacity.

At its core, mathematical thinking consists of working methodically and visualizing math. You don’t just cram math concepts, but go ahead and memorize and know them so they can be applied to daily activities.

So, what exactly is mathematical thinking, and what are some steps that can be taken to think mathematically? Keep reading to know more.

What is Mathematical Thinking?

Mathematical thinking is dissecting issues, identifying their constituent parts, and then developing rational plans to solve them. It is a mindset that uses mathematical concepts to address practical problems.

When kids think mathematically, they can conveniently use mathematical reasoning to think creatively in real life and handle their problems quickly. They will enjoy classroom lessons more and be able to do mental math , which will serve them well now and in the future. This use of math to glide past mental hurdles further incentivizes kids to use innovative problem-solving techniques to break down math issues into manageable chunks.

Examples of Mathematical Thinking in Everyday Life

Everyone uses mathematical thinking in everyday life. For instance, estimating how long it would take you to get to the park is already mathematical thinking. Even counting how much time you have to do a particular task is mathematical thinking.

For more detailed examples, here are three common mathematical thinking examples in everyday life:

Grocery Shopping

Nobody can escape from doing a little math when grocery shopping. First, you need to do money math by budgeting how much you will spend on groceries. Then, you estimate how much time you’ll need to shop.

After buying all the items you need, you’ll need to figure out the total cost by adding the cost prices of all the goods and deducting their corresponding discounts. Finally, you pay the cashier and use subtraction to determine the amount of change owed.

Mathematical thinking is also used in the kitchen, especially when using recipes in baking snacks like cookies. Here, the precise measurements of each ingredient are determined by applying proportions and ratios.

For example, a baker must determine whether half a teaspoon or one teaspoon of sugar will be used for the cookies. Similarly, chefs need to know their measurements to avoid falling short of customers’ demands.

Keeping Track of Cash

Mathematical thinking will help kids learn skills that will be useful in managing their finances. With time, understanding simple and compound interest, for example, will come in handy.

Mathematical thinking will also help young high school or college students select an ideal first bank account and credit card plan. And when they start building their credit scores, it will ensure they get familiar with interest rates much faster. This could lead them to save and invest better, becoming more astute at managing their finances.

Steps to Approach Problems Mathematically

According to Polya, the steps to approach problems mathematically are understanding the problem, devising a plan, carrying out the plan, and looking back. It’s more focused on identifying the issue, formulating a strategy, putting it into action, and assessing the results.

Here is a math problem to solve for fun:

Let’s pretend that the issue at hand is that you are organizing a party and would want to provide each guest with a gift bag containing five toys. So, how many toys will you purchase for the gift bags if fifteen kids attend?

Understanding the problem

Before solving any problem, students should take a moment to understand the issue. Kids need to actively seek ways on how to solve math problems. Some tactics they can use are looking for the unknown and similarities between the problems and math concepts.

In understanding the problem, you first need to clarify the information on the number of guests and how many toys will be in each gift bag.

Devising a plan

The process of developing a solution to an issue is called devising a plan (translation). In this stage, students will try to establish relationships between the data and the unknown. You may set up an equation, create a diagram, or make a chart to solve your issue. Pick a suitable approach to handle the problem.

Simply multiply the number of guests by the product of the number of toys per bag to get the total number of toys. The formula should be something like this:

P = toys per bag (t) × number of attendees (n), where toys per bag should be 5.

Carrying out the plan

The next phase is for the student to put the plan into action. The students should double-check the solution strategy at every stage. 

Carry out the measures specified in the strategy:

  • Choose n = 15 and t = 5 toys per bag.
  • Enter the new value into the formula: P = 5 multiplied by 15.
  • Multiply the numbers: P = 75 toys.

Looking back

Kids should reflect on the answers and verify that the answer is correct and addresses the initial issue. Assuming each gift bag contains five toys, a total of 75 toys would be needed. The answer is straightforward and makes sense in light of the problem description; it solves the mathematical issue.

Polya’s approach is beneficial for students as they work to solve mathematical issues and develop mathematical thinking.

Games and Activities to Enhance Mathematical Thinking

One excellent technique to help kids improve their mathematical thinking and creative reasoning abilities is to play mathematics games with them. Playing the games lets them participate in interactive and practical math classes. 

Here are three games and activities to enhance mathematical thinking in the classroom.

Math Facts Race

Split the class in half and have them stand in two lines. Put out two sets of addition or multiplication facts grids at the front of the classroom. The first person on each team has to fill in one of the numbers on their grid. For instance, 3 x 4 = 12. 

The next step is to return to the beginning of the queue and transfer the marker to the first person standing there. The winner is the team whose grid is completed first.

Please, Sign My Worksheet

Playing this math game in class requires students to solve problems in a non-traditional manner. Give each student about ten math problems to solve.

Then, instead of completing their worksheets , pupils are tasked with tackling the worksheets of others. After the children solve any math problem from someone else’s worksheet, they are to sign.

Addition and Subtraction Bingo

To play, just make a deck of bingo cards and fill them out with the solutions to some basic math questions. Instead of numbers, read out math problems like “8 + 6” or “9 – 4.” The next step for students to win the bingo game is to get the correct answers.

Applying Mathematical Thinking to Solve a Sample Problem

Imagine that you’re preparing for a bake sale. According to your recipe, one cup of flour and two eggs could give you 18 cookies. Based on your estimation, you must prepare enough cookies to feed 100 individuals, providing three cookies for each person. How much of flour is required?

To solve this problem, do the following:

  • Find out how many cookies are required in total: 100 people x 3 cookies per person = 300 cookies.
  • Find out how many batches are required: 300 cookies ÷ 18 cookies per batch = 16.6667 batches.
  • Round up to the nearest whole batch: You can’t bake a partial batch, so round 16.6667 batches up to 17 batches.
  • Calculate the total flour needed: 17 batches x 1 cup/batch = 17 cups of flour.

Encouraging Mathematical Thinking at Home and School with Brighterly

Brighterly is a math educational website with tutors encouraging kids to understand math concepts instead of cramming formulas. The vetted professional tutors at Brighterly tailor lessons to children’s specific requirements instead of following the traditional school curriculum. With the individualized lesson plan from the platform, kids can learn with tutors who teach the kids real-life applications.

The instructors also make kids study math online with study-related games and videos that put them in charge of their learning process. So, the lesson plans are simple. Instead, students can conveniently learn from the comfort of their homes and apply the concepts they learned to things around them.

One notable part about using the Brighterly platform is that you can first have a trial class with them before signing up. So, it’s a foolproof registration.

Mathematical thinking is essential as people go about their daily activities. This article has discussed examples of its use case in the real world, proving your kids can definitely benefit from an introduction to mathematical thinking before they come of age.

Brighterly is a fun math learning website with professional tutors acknowledging that kids should do more than cram math concepts. They believe that students should grasp these concepts in their entirety to develop their mathematical critical thinking skills.

So, knowing what you know now, do you want to give your child an excellent opportunity to learn the necessary mathematical thinking skills? Register Now on Brighterly!

Jessica is a a seasoned math tutor with over a decade of experience in the field. With a BSc and Master’s degree in Mathematics, she enjoys nurturing math geniuses, regardless of their age, grade, and skills. Apart from tutoring, Jessica blogs at Brighterly. She also has experience in child psychology, homeschooling and curriculum consultation for schools and EdTech websites.

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Math Word Problems

Welcome to the math word problems worksheets page at Math-Drills.com! On this page, you will find Math word and story problems worksheets with single- and multi-step solutions on a variety of math topics including addition, multiplication, subtraction, division and other math topics. It is usually a good idea to ensure students already have a strategy or two in place to complete the math operations involved in a particular question. For example, students may need a way to figure out what 7 × 8 is or have previously memorized the answer before you give them a word problem that involves finding the answer to 7 × 8.

There are a number of strategies used in solving math word problems; if you don't have a favorite, try the Math-Drills.com problem-solving strategy:

  • Question : Understand what the question is asking. What operation or operations do you need to use to solve this question? Ask for help to understand the question if you can't do it on your own.
  • Estimate : Use an estimation strategy, so you can check your answer for reasonableness in the evaluate step. Try underestimating and overestimating, so you know what range the answer is supposed to be in. Be flexible in rounding numbers if it will make your estimate easier.
  • Strategize : Choose a strategy to solve the problem. Will you use mental math, manipulatives, or pencil and paper? Use a strategy that works for you. Save the calculator until the evaluate stage.
  • Calculate : Use your strategy to solve the problem.
  • Evaluate : Compare your answer to your estimate. If you under and overestimated, is the answer in the correct range. If you rounded up or down, does the answer make sense (e.g. is it a little less or a little more than the estimate). Also check with a calculator.

Most Popular Math Word Problems this Week

Easy Multi-Step Word Problems

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  • Addition Word Problems One-Step Addition Word Problems Using Single-Digit Numbers One-Step Addition Word Problems Using Two-Digit Numbers
  • Subtraction Word Problems Subtraction Facts Word Problems With Differences from 5 to 12
  • Multiplication Word Problems One-Step Multiplication Word Problems up to 10 × 10
  • Division Word Problems Division Facts Word Problems with Quotients from 5 to 12
  • Multi-Step Word Problems Easy Multi-Step Word Problems

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Hello! In this article, I want to talk about MATH WORD PROBLEMS. They indeed give many children "problems", but that doesn't have to be!

math lessons solving problems

Here's the outline of the article:

The problem

The solution.

  • Problems to ponder/solve

Student's Misguide to Problem Solving

Have you ever noticed this kind of "recipe" for math lessons in many math books?

In other words, the word problems are usually in the END of the lesson, and just a few. But worse... if the lesson is about topic X, then the word problems are usually about the topic X too!

Children might be learning about multi-digit multiplication, or subtraction, or dividing decimals. After the calculation exercises come some word problems, which oddly enough are solved by using the exact operation just practiced !

Worse YET... typically the word problems in elementary grades only have TWO numbers in them. So, even if you didn't understand a word in the problem, you might be able to do it. (The misguide to problem solving at the end of this article refers to this also.)

Just try: the following made-up problem is in FINNISH... and let's say it is found within a long division lesson. I assume now that you do NOT know Finnish — but can you solve it?

Drag your mouse over the white space below to see the translation (highlight it).

My thought is that over the years, when children are exposed to such lessons over and over again, they figure out that it's mentally less demanding to NOT even READ the problem too carefully ! Why bother? Just take the two numbers, divide (or multiply, or add, or subtract) them, and that's it. They "learn" (intelligently) this unspoken rule:

It is no wonder that our students are not able to solve anything that's different from that routine!

How can you avoid this terrible situation?

Children simply need to be given good problems to solve. That way, they will get used to THINKING.

These good word problems should advance in difficulty over the grades, and foster children's logical thinking. ONE-step problems are adequate for 1st and 2nd grades, and then here and there mixed in with other types of problems.

In early elementary grades, one needs to mix up the (often one-step) word problems so that not all of them are solved by the operation just studied. I have done that often in my Math Mammoth books .

But children need to start solving MULTI-step problems as soon as they can, including even in 1st grade. (And yes, Math Mammoth books do include some two-step problems starting in 1st grade materials.)

book cover

However, children already following the Math Mammoth curriculum probably won't need that book, because they will be learning to solve word problems all along in the curriculum. The book is meant for remediation.

For a variety of word problems resources, check the list of problem solving resources at the end of this email.

Problems to ponder

:)

A similar problem is included in the 5th grade Singapore textbook:

Now, these are not anything spectacular. You can solve these types of problems for example by using a bar model (image from Math Mammoth grade 5 Light Blue series):

math lessons solving problems

Or, you can solve them with arithmetic: simply take away the given difference from the total, then divide the remaining amount evenly, and lastly add the given difference to one person's part.

Yet in the U.S., these kind of problems are generally introduced in Algebra 1 — in ninth grade (!), AND they are only solved using algebraic means.

Here is another example, of which I remember feeling aghast, found in a modern U.S. algebra textbook:

Third-grade children should know multiplication well enough to quickly find that 6 and 7 fit the problem! Why use a "backhoe" (algebra) for a problem you can solve using a "small spade" (simple multiplication)!

Another example, a 3rd grade problem from Russia:

You can draw a boy and a girl, draw two pockets for the boy, and one pocket for the girl. This visual representation easily solves the problem .

Here is an example of a Russian problem for grades 6-8:

(You can solve this in many different ways — and that is a often the case withe good word problems.)

So, once again, the only true way of learning to solve word problems is .... TO SOLVE LOTS OF THEM . They don't have to be real-life or involve awkward numbers (such as occur in real life). Realistic, complex problems might be good for a "spice", but not for the "main course". "Fantastic" (unreal) problems are fine.

One of my girls used to cherish the word problems in my Math Mammoth books MORE than the other kinds of problems. And the general consensus of the feedback I hear from Math Mammoth users is that once children get used to solving a variety of word problems, it's no longer a problem!

Some children, if not used to solving word problems, may need to start from the beginning — with a collection of simple one-step word problems (mixed up). But that's okay. They will learn! Our Foundational Word Problems book can get them started (at about 3rd grade level).

Lastly, a joke by Lynn Nordstrom:

Rule 1: If at all possible, avoid reading the problem. Reading the problem only consumes time and causes confusion.

Rule 2: Extract the numbers from the problem in the order they appear. Be on the watch for numbers written in words.

Rule 3: If rule 2 yields three or more numbers, the best bet is adding them together.

Rule 4: If there are only 2 numbers which are approximately the same size, then subtraction should give the best results.

Rule 5: If there are only two numbers and one is much smaller than the other, then divide if it goes evenly -- otherwise multiply.

Rule 6: If the problem seems like it calls for a formula, pick a formula that has enough letters to use all the numbers given in the problem.

Rule 7: If the rules 1-6 don't seem to work, make one last desperate attempt. Take the set of numbers found by rule 2 and perform about two pages of random operations using these numbers. You should circle about five or six answers on each page just in case one of them happens to be the answer. You might get some partial credit for trying hard.

I hope your students do not fit the above joke.

Sources and further resources

Math Stars Free problem-solving newsletters that include a nice mix of good problems that inspire curiosity. There is a set for each of the grades 1-8, with commentaries for teachers. I used them with all my children, and highly recommend them! The best is, they're free!

Problem Solving Decks A free collection of problems for grades 1-8, originally published by the North Carolina State Board of Education.

Math Puzzles and Brainteasers cover

Ray's Arithmetic Ray's Arithmetic is an old math book, full of word problems, starting from the simplest. It basically uses word problems to teach arithmetic. It is available FREE online.

3rd Grade Math Word Problems Worksheets Look especially for the MIXED word problem worksheets in this collection. The site offers word problem worksheets for all of the grades 1-5 (look in the menu).

Challenge Math cover

Challenge Math For the Elementary and Middle School Student For children in grades 4-7. Contains 20 chapters with instruction and problems at three levels of difficulty.

Mixed operations word problem worksheets These are all one-step problems but useful for grades 3-4 students who need such practice.

National Math Festival — collection of math newsletters that contain interesting & fun problems by themes such as "Math and Cats", "Math and Time", and "Math and Desserts".

By Maria Miller

math lessons solving problems

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Teaching and Learning

Elevating math education through problem-based learning, by lisa matthews     feb 14, 2024.

Elevating Math Education Through Problem-Based Learning

Image Credit: rudall30 / Shutterstock

Imagine you are a mountaineer. Nothing excites you more than testing your skill, strength and resilience against some of the most extreme environments on the planet, and now you've decided to take on the greatest challenge of all: Everest, the tallest mountain in the world. You’ll be training for at least a year, slowly building up your endurance. Climbing Everest involves hiking for many hours per day, every day, for several weeks. How do you prepare for that?

The answer, as in many situations, lies in math. Climbers maximize their training by measuring their heart rate. When they train, they aim for a heart rate between 60 and 80 percent of their maximum. More than that, and they risk burning out. A heart rate below 60 percent means the training is too easy — they’ve got to push themselves harder. By combining this strategy with other types of training, overall fitness will increase over time, and eventually, climbers will be ready, in theory, for Everest.

math lessons solving problems

Knowledge Through Experience

The influence of constructivist theories has been instrumental in shaping PBL, from Jean Piaget's theory of cognitive development, which argues that knowledge is constructed through experiences and interactions , to Leslie P. Steffe’s work on the importance of students constructing their own mathematical understanding rather than passively receiving information .

You don't become a skilled mountain climber by just reading or watching others climb. You become proficient by hitting the mountains, climbing, facing challenges and getting right back up when you stumble. And that's how people learn math.

math lessons solving problems

So what makes PBL different? The key to making it work is introducing the right level of problem. Remember Vygotsky’s Zone of Proximal Development? It is essentially the space where learning and development occur most effectively – where the task is not so easy that it is boring but not so hard that it is discouraging. As with a mountaineer in training, that zone where the level of challenge is just right is where engagement really happens.

I’ve seen PBL build the confidence of students who thought they weren’t math people. It makes them feel capable and that their insights are valuable. They develop the most creative strategies; kids have said things that just blow my mind. All of a sudden, they are math people.

math lessons solving problems

Skills and Understanding

Despite the challenges, the trend toward PBL in math education has been growing , driven by evidence of its benefits in developing critical thinking, problem-solving skills and a deeper understanding of mathematical concepts, as well as building more positive math identities. The incorporation of PBL aligns well with the contemporary broader shift toward more student-centered, interactive and meaningful learning experiences. It has become an increasingly important component of effective math education, equipping students with the skills and understanding necessary for success in the 21st century.

At the heart of Imagine IM lies a commitment to providing students with opportunities for deep, active mathematics practice through problem-based learning. Imagine IM builds upon the problem-based pedagogy and instructional design of the renowned Illustrative Mathematics curriculum, adding a number of exclusive videos, digital interactives, design-enhanced print and hands-on tools.

The value of imagine im's enhancements is evident in the beautifully produced inspire math videos, from which the mountaineer scenario stems. inspire math videos showcase the math for each imagine im unit in a relevant and often unexpected real-world context to help spark curiosity. the videos use contexts from all around the world to make cross-curricular connections and increase engagement..

This article was sponsored by Imagine Learning and produced by the Solutions Studio team.

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10 Helpful Worksheet Ideas for Primary School Math Lessons

M athematics is a fundamental subject that shapes the way children think and analyze the world. At the primary school level, laying a strong foundation is crucial. While hands-on activities, digital tools, and interactive discussions play significant roles in learning, worksheets remain an essential tool for reinforcing concepts, practicing skills, and assessing understanding. Here’s a look at some helpful worksheets for primary school math lessons.

Comparison Chart Worksheets

Comparison charts provide a visual means for primary school students to grasp relationships between numbers or concepts. They are easy to make at www.storyboardthat.com/create/comparison-chart-template , and here is how they can be used:

  • Quantity Comparison: Charts might display two sets, like apples vs. bananas, prompting students to determine which set is larger.
  • Attribute Comparison: These compare attributes, such as different shapes detailing their number of sides and characteristics.
  • Number Line Comparisons: These help students understand number magnitude by placing numbers on a line to visualize their relative sizes.
  • Venn Diagrams: Introduced in later primary grades, these diagrams help students compare and contrast two sets of items or concepts.
  • Weather Charts: By comparing weather on different days, students can learn about temperature fluctuations and patterns.

Number Recognition and Counting Worksheets

For young learners, recognizing numbers and counting is the first step into the world of mathematics. Worksheets can offer:

  • Number Tracing: Allows students to familiarize themselves with how each number is formed.
  • Count and Circle: Images are presented, and students have to count and circle the correct number.
  • Missing Numbers: Sequences with missing numbers that students must fill in to practice counting forward and backward.

Basic Arithmetic Worksheets

Once students are familiar with numbers, they can start simple arithmetic. 

  • Addition and Subtraction within 10 or 20: Using visual aids like number lines, counters, or pictures can be beneficial.
  • Word Problems: Simple real-life scenarios can help students relate math to their daily lives.
  • Skip Counting: Worksheets focused on counting by 2s, 5s, or 10s.

Geometry and Shape Worksheets

Geometry offers a wonderful opportunity to relate math to the tangible world.

  • Shape Identification: Recognizing and naming basic shapes such as squares, circles, triangles, etc.
  • Comparing Shapes: Worksheets that help students identify differences and similarities between shapes.
  • Pattern Recognition: Repeating shapes in patterns and asking students to determine the next shape in the sequence.

Measurement Worksheets

Measurement is another area where real-life application and math converge.

  • Length and Height: Comparing two or more objects and determining which is longer or shorter.
  • Weight: Lighter vs. heavier worksheets using balancing scales as visuals.
  • Time: Reading clocks, days of the week, and understanding the calendar.

Data Handling Worksheets

Even at a primary level, students can start to understand basic data representation.

  • Tally Marks: Using tally marks to represent data and counting them.
  • Simple Bar Graphs: Interpreting and drawing bar graphs based on given data.
  • Pictographs: Using pictures to represent data, which can be both fun and informative.

Place Value Worksheets

Understanding the value of each digit in a number is fundamental in primary math.

  • Identifying Place Values: Recognizing units, tens, hundreds, etc., in a given number.
  • Expanding Numbers: Breaking down numbers into their place value components, such as understanding 243 as 200 + 40 + 3.
  • Comparing Numbers: Using greater than, less than, or equal to symbols to compare two numbers based on their place values.

Fraction Worksheets

Simple fraction concepts can be introduced at the primary level.

  • Identifying Fractions: Recognizing half, quarter, third, etc., of shapes or sets.
  • Comparing Fractions: Using visual aids like pie charts or shaded drawings to compare fractions.
  • Simple Fraction Addition: Adding fractions with the same denominator using visual aids.

Money and Real-Life Application Worksheets

Understanding money is both practical and a great way to apply arithmetic.

  • Identifying Coins and Notes: Recognizing different denominations.
  • Simple Transactions: Calculating change, adding up costs, or determining if there’s enough money to buy certain items.
  • Word Problems with Money: Real-life scenarios involving buying, selling, and saving.

Logic and Problem-Solving Worksheets

Even young students can hone their problem-solving skills with appropriate challenges.

  • Sequences and Patterns: Predicting the next item in a sequence or recognizing a pattern.
  • Logical Reasoning: Simple puzzles or riddles that require students to think critically.
  • Story Problems: Reading a short story and solving a math-related problem based on the context.

Worksheets allow students to practice at their own pace, offer teachers a tool for assessment, and provide parents with a glimpse into their child’s learning progression. While digital tools and interactive activities are gaining prominence in education, the significance of worksheets remains undiminished. They are versatile and accessible and, when designed creatively, can make math engaging and fun for young learners.

The post 10 Helpful Worksheet Ideas for Primary School Math Lessons appeared first on Mom and More .

Mathematics is a fundamental subject that shapes the way children think and analyze the world. At the primary school level, laying a strong foundation is crucial. While hands-on activities, digital tools, and interactive discussions play significant roles in learning, worksheets remain an essential tool for reinforcing concepts, practicing skills, and assessing understanding. Here’s a look […]

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></center></p><h2>Leap Years And Math Olympiad Problems Free Worksheets & Solutions</h2><p>Counting down for our once-in-four-years Leap Day for this Thursday! In our previous article , we explored the science and math principles behind leap years. Now, it’s time to solve some math problems related to leap years.</p><p>Whether your child is just starting to learn mathematical concepts or is already a seasoned Math Olympiad problem solver, these free Spark Math festive worksheets are sure to spark curiosity and excitement.</p><h2>Understanding Leap Years</h2><p>Before we leap into the math problems, let’s quickly recap what leap years are. A leap year occurs every four years to keep our calendar in sync with the solar year, which is slightly longer than our typical 365-day calendar year. During a leap year, an extra day is added to February, making the month 29 days instead of the usual 28.</p><h2>Free Worksheets for Leap Year Fun</h2><p>Understanding leap years shows how math impacts our world. Our Spark Math teachers have created free worksheets tailored to children of different math skill levels. These worksheets feature a range of leap year math problems aimed at improving your child’s mathematical reasoning, critical thinking, and problem-solving skills.</p><h2>For Kindergarten (Leap Year Skip Counting)</h2><p>This worksheet is designed to help kindergarten students practice counting by four, enhancing their number sense, and pattern recognition skills.</p><p><center><img style=

For Primary (Basic Leap Year Math Problems)

Problems 1-5 are tailored for primary school students, focusing on counting, calendar skills, reading comprehension, and real-life applications.

math lessons solving problems

For Primary (Advanced Math Problems)

Get ready for an extra dose of challenge! Problems 6-9 in the Advanced section draw inspiration from the mock papers of the International Junior Math Olympiad . These questions are perfect for both primary school students who are eager to put their critical thinking and problem-solving skills to the test.

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Check out our Spark Math Competition Programme to see how our Spark Math teacher guide your child on the path to becoming math champions!

What Concepts Do The Spark Math Leap Year Worksheets Teach?

Number sense and counting by four.

Counting leap years involves basic counting skills. Since leap years occur every four years, children learn to count by fours to identify leap years. This reinforces their ability to count sequentially and identify multiples of four.

Pattern Recognition

Identifying leap years involves recognising patterns in the calendar. Children learn to identify the recurring pattern of leap years and the exceptions (such as years divisible by 100 but not by 400). This skill is crucial in mathematics and lays the groundwork for more advanced topics like algebra and geometry.

Calendar Calculations

The worksheet also provides practice in calendar calculations, as students work with dates and years to identify leap years accurately. This helps them become more proficient in manipulating and interpreting calendar information.

Problem-Solving and Critical Thinking

Leap year and calendar calculation also involves problem-solving and critical thinking skills. To determine future or past leap years, children need to calculate the number of leap years within a certain time frame. This encourages them to apply their math knowledge in different contexts and make logical deductions.

Real-World Application

By engaging with leap year calculations, students see the real-world application of mathematical concepts. They understand why leap years are necessary and how they impact our calendar system.

Answer Key and Solutions

Now, check out the answer key and step-by-step solutions for the Leap Years Spark Math worksheets here.

Leap Year Skip Counting

  • Identify the starting point.
  • Identify the leap year (+4) pattern.
  • Fill in the missing leap years.
  • Double check by counting backwards.

Leap Years And Math Olympiad Problems Free Worksheets And Solutions

Primary (Basic Math Problems)

1. February 2. four 3. 366 4. 2020, 2024, and 2028 5. 2028 Method: Add 4 years to 2024.

Primary (Advanced Math Problems)

Problems 6 – 9 are inspired by the International Junior Math Olympiad competition. Check out our Spark Math Competition Programme to see how our Spark Math teacher guide your child on the path to becoming math champions!

6. To calculate whether a year is a leap year, you can use the following steps: A – 2000: – Divisible by 4? Yes – Divisible by 100? Yes – Divisible by 400? Yes So, 2000 is a leap year. B – 2048: – Divisible by 4? Yes – Divisible by 100? No So, 2048 is a leap year. C – 2100: – Divisible by 4? Yes – Divisible by 100? Yes – Divisible by 400? No So, 2100 is not a leap year. D – 2222: – Divisible by 4? No So, 2222 is not a leap year. Therefore, among the given options, only 2000 and 2048 are leap years. Hence, the correct answers are A and B .

7. Number of times = Total days in February ÷ Days in a week = 28 ÷ 7 = 4 Therefore, the number of days in February in a non-leap year is 4 times the number of days in one week. Hence, the correct answer is A . 4 .

8. First, list out all the years from 2000 to 2024 that are divisible by 4:

= 2000, 2004, 2008, 2012, 2016, 2020, and 2024. Then, double check the list against the special rule for century leap years:

Leap Years And Math Olympiad Problems Free Worksheets And Solutions

So, there are 7 leap years from 2000 to 2024. Hence, the correct answer is E. None of the above .

9. To find out the day of the week for 1st September, 2024, we can utilise the fact that there are exactly 7 days in a week. Number of days advanced in non-leap years = 365 ÷ 7 = 52 (with remainder 1) Number of days advanced in leap years = 366 ÷ 7 = 52 (with remainder 2) Therefore, every non-leap year, the day of the week advances by 1 day, and every leap year, it advances by 2 days.

From 1st September 2018, to 1st September 2024, there are 6 years in total. Among these, there are 2 leap years (2020 and 2024) and 4 non-leap years.

So, the total number of days that the day of the week advances within the given period, can be calculated as follows:

Number of days advanced in non-leap years = 4 non-leap years × 1 day = 4 days

Number of days advanced in leap years = 2 leap years × 2 days = 4 days

Total number of days advanced = 4 days (leap years) + 4 days (non-leap years) = 8 days

Since there are 7 days in a week, advancing by 8 days means we move to the next day of the week.

Therefore, September 1st, 2024, will be Sunday .

How Spark Math Teaches Math?

Spark Math by Spark Education is designed to help K2-P6 children develop strong numeracy skills, master heuristic strategies, and excel in Mathematics. It goes beyond the Singapore MOE Math Syllabus , introducing students to advanced problems found in  international competitions  and  real-life scenarios .

Through  dual-interface technology , Spark Math teachers can interact with students in real-time. They use  gamification  and  animated content  to ensure engaged, active learning while reinforcing advanced math concepts.

With Spark Math, your child will receive  both physical and digital textbooks , workbooks, and math manipulatives, creating an effective blended learning experience conveniently at home. Our parent app also provides a one-stop solution for managing the class schedules and monitoring your child’s learning progress.

Ready to see the difference Spark Math can make for your child? Try a FREE online math test and receive a complimentary report to experience firsthand how Spark Math can help your child succeed in mathematics!

Spark Passion for Learning, Ignite Lifelong Growth

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Praxis Core Math

Course: praxis core math   >   unit 1.

  • Rational number operations | Lesson
  • Rational number operations | Worked example

Ratios and proportions | Lesson

  • Ratios and proportions | Worked example
  • Percentages | Lesson
  • Percentages | Worked example
  • Rates | Lesson
  • Rates | Worked example
  • Naming and ordering numbers | Lesson
  • Naming and ordering numbers | Worked example
  • Number concepts | Lesson
  • Number concepts | Worked example
  • Counterexamples | Lesson
  • Counterexamples | Worked example
  • Pre-algebra word problems | Lesson
  • Pre-algebra word problems | Worked example
  • Unit reasoning | Lesson
  • Unit reasoning | Worked example

What are ratios and proportions?

What skills are tested.

  • Identifying and writing equivalent ratios
  • Solving word problems involving ratios
  • Solving word problems using proportions

How do we write ratios?

  • The ratio of lemon juice to sugar is a part-to-part ratio. It compares the amount of two ingredients.
  • The ratio of lemon juice to lemonade is a part-to-whole ratio. It compares the amount of one ingredient to the sum of all ingredients.
  • Determine whether the ratio is part to part or part to whole.
  • Calculate the parts and the whole if needed.
  • Plug values into the ratio.
  • Simplify the ratio if needed. Integer-to-integer ratios are preferred.

How do we use proportions?

  • Write an equation using equivalent ratios.
  • Plug in known values and use a variable to represent the unknown quantity.
  • If the numeric part of one ratio is a multiple of the corresponding part of the other ratio, we can calculate the unknown quantity by multiplying the other part of the given ratio by the same number.
  • If the relationship between the two ratios is not obvious, solve for the unknown quantity by isolating the variable representing it.
  • (Choice A)   1 : 4 ‍   A 1 : 4 ‍  
  • (Choice B)   1 : 2 ‍   B 1 : 2 ‍  
  • (Choice C)   1 : 1 ‍   C 1 : 1 ‍  
  • (Choice D)   2 : 1 ‍   D 2 : 1 ‍  
  • (Choice E)   4 : 1 ‍   E 4 : 1 ‍  
  • (Choice A)   1 6 ‍   A 1 6 ‍  
  • (Choice B)   1 3 ‍   B 1 3 ‍  
  • (Choice C)   2 5 ‍   C 2 5 ‍  
  • (Choice D)   1 2 ‍   D 1 2 ‍  
  • (Choice E)   2 3 ‍   E 2 3 ‍  
  • Your answer should be
  • an integer, like 6 ‍  
  • a simplified proper fraction, like 3 / 5 ‍  
  • a simplified improper fraction, like 7 / 4 ‍  
  • a mixed number, like 1   3 / 4 ‍  
  • an exact decimal, like 0.75 ‍  
  • a multiple of pi, like 12   pi ‍   or 2 / 3   pi ‍  

Things to remember

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IMAGES

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VIDEO

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  24. Leap Years And Math Olympiad Problems Free Worksheets & Solutions

    Understanding leap years shows how math impacts our world. Our Spark Math teachers have created free worksheets tailored to children of different math skill levels. These worksheets feature a range of leap year math problems aimed at improving your child's mathematical reasoning, critical thinking, and problem-solving skills.

  25. Ratios and proportions

    Solving word problems using proportions How do we write ratios? Two common types of ratios we'll see are part to part and part to whole. For example, when we make lemonade: The ratio of lemon juice to sugar is a part-to-part ratio. It compares the amount of two ingredients. The ratio of lemon juice to lemonade is a part-to-whole ratio.