mathematical practices practice and problem solving exercises

Problem Solving Activities: 7 Strategies

• Critical Thinking

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

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

I was so excited!

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

It was a proud moment for me!

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

Genius right? 

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

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

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

When I Finally Saw the Light

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

Problem Solving Activities

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

Seasonal Problem Solving

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

Cooperative Problem Solving Tasks

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

Notice and Wonder

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

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

Math Starters

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

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

Calculators

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

Three-Act Math Tasks

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

Getting the Most from Each of the Problem Solving Activities

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

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

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

Shametria Routt Banks

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2 Responses

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

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

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Algebra Practice Problems – Master Your Skills with Exercises and Solutions!

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Understanding Algebra Fundamentals

Working with equations, practicing algebra with worksheets, mastering fractions and percentages, utilizing algebra in geometry, solving real-world problems.

To master algebra , practicing problems and going through their solutions is crucial. It’s a bit like learning to play an instrument – practice is key to improving.

For many students, the leap from arithmetic to algebra can be challenging. It requires a shift from working with concrete numbers to thinking abstractly about numbers and variables. But I’ve found that with a set repertoire of practice problems, from basic equations to more complex ones involving exponents and polynomials, it becomes much clearer.

What’s more, each problem has a set of steps to reach the solution, revealing patterns and strategies that can be applied to future challenges. Having a diverse array of practice problems is beneficial not only for students but also for educators and parents seeking to support their learner’s educational journey.

Hands-on experience with algebraic expressions and equations solidifies understanding and boosts confidence. So let’s get our hands on some algebra problems and start tackling them – who knows what new connections we’ll uncover in the intricacies of algebraic thinking.

Algebra is like a puzzle where I use mathematical symbols and letters (known as variables) to find unknown quantities. Variables represent values that can change; for example, in the equation $x + 2 = 5$, the variable $x$ takes the value $3$.

I need to understand that variables allow me to describe general math truths or perform operations without knowing the exact values.

Here’s what I often start with when exploring fundamental algebraic concepts :

Expressions and Equations : An expression is a combination of numbers and variables, like $3x + 4$. When an expression equals a number, like $3x + 4 = 19$, it becomes an equation. Equations set the stage for finding the value of the variable.

Operations with Variables : I treat variables the same way as numbers in operations. Addition, subtraction, multiplication, and division acts on variables just like on numbers, except that I don’t combine unlike terms (e.g., $x + y$ remains as it is unless I know the relationship between $x$ and $y$).

Below is a simple table detailing how basic arithmetic operations work with variables:

• Solving the Equations : To solve an equation, I perform operations that isolate the variable on one side. For example, if $x – 4 = 10$, I add $4$ to both sides to get $x = 14$.

Remembering mathematical order of operations is vital. In algebra, I follow this hierarchy: operations inside parentheses first, exponents second, then multiplication and division, and finally, addition and subtraction. This rule ensures I simplify expressions and solve equations correctly.

When I solve equations, my goal is to isolate the variable I’m interested in. This often involves moving terms from one side of the equation to the other.

For example, in a simple linear equation like ( ax + b = c ), I would solve for ( x ) by subtracting ( b ) from both sides and then dividing by ( a ).

When dealing with absolute values , like ( |x + a| = b ), I remember that the solution considers both the positive and negative counterparts because the absolute value represents the distance from zero without considering direction.

I always check my solutions by substituting them back into the original equation to ensure they satisfy the equation. This verification step is crucial, especially when initial equations involve absolute values or higher degree terms.

For practice problems, Khan Academy and Paul’s Online Math Notes are great resources that offer a variety of algebra problems, including linear equations and absolute value equations. I appreciate how Khan Academy’s platform provides instant feedback.

Working through these problems helps reinforce my understanding. Solving equations can be like a puzzle, and I find it satisfying when I find the correct solution. Each equation brings a new challenge, which keeps my algebra skills sharp and ready.

When I dive into algebra, I find that worksheets are a vital tool for mastering concepts. They offer a structured approach, enabling me to tackle one problem at a time.

While practice problems may vary in complexity, they often share a common focus on fundamental algebraic structures, such as linear equations and variables. 

For linear equations, I often start with simple formats like ( y = mx + b ) and gradually work up to more complex problems. This incremental approach helps me understand the relationship between variables and coefficients.

Sample Linear Equation Worksheet

Italicized tip: Always isolate the variable you’re solving for by performing inverse operations.

I make sure worksheets are varied, combining problems that involve solving for one variable with those requiring multiple steps. It’s the repetition and gradual escalation in difficulty that really solidify my understanding.

Variables Practice

• Identify variables and constants
• Evaluate expressions like ( 3$x^2$ – 2x + 7 ) for given values of ( x )
• Translate word problems into algebraic equations

Algebra worksheets offer distinct advantages, as they can be tailored to focus on just the areas I’m looking to improve, providing immediate feedback when answer keys are included. Through disciplined practice , I find myself not just solving problems but also gaining a deeper appreciation of algebra’s expressive power.

I find that a solid grasp of fractions and percentages is crucial in algebra. These concepts are not only foundational in mathematics but also widely applicable in real-world scenarios. Let’s take a closer look at how we can tackle problems involving these topics.

The Fraction Fundamentals For me, working with fractions always starts with understanding the roles of the numerator (top number) and denominator (bottom number). When performing addition or subtraction with fractions, I remember that the denominator must be the same.

For example, to add $$ \frac{3}{4} $$ and $$ \frac{5}{8} $$, I first find a common denominator. Multiplying both the numerator and denominator of $$ \frac{3}{4} $$ by 2 gives me $$ \frac{6}{8} $$. Now, I can easily add the two fractions:

$$ \frac{6}{8} + \frac{5}{8} = \frac{11}{8} $$.

Multiplication and Division Multiplication of fractions is more straightforward. I simply multiply the numerators together and the denominators together. For instance:

$$ \frac{3}{5} \times \frac{2}{3} = \frac{6}{15} $$.

But when it comes to division , I flip the second fraction (find the reciprocal) and then multiply.

$$ \frac{4}{7} \div \frac{2}{5} = \frac{4}{7} \times \frac{5}{2} = \frac{20}{14} $$.

Converting to Percentages When I need to express a fraction as a percent , I multiply the fraction by 100%. For example:

$$ \frac{3}{4} = \frac{3}{4} \times 100% = 75% $$.

To solve for a percentage of a number , I convert the percent to a decimal and then multiply. If I have to find 60% of 50, I do:

$$ 60% = 0.60; \quad 0.60 \times 50 = 30 $$.

By keeping these strategies in mind, I can confidently solve a variety of problems involving fractions and percentages. Experimenting with different problems is an enjoyable way to get better at these concepts!

In exploring the relationship between algebra and geometry, I find it essential to recognize how algebra serves as a robust tool for solving geometric problems. Take the circle, for example; its properties can be unraveled using algebraic methods.

The equation of a circle in a coordinate plane, which is ( $(x – h)^2$ + $(y – k)^2$ = $r^2$ ), allows me to calculate the radius, or find points on its circumference by substituting values for ( x ) and ( y ).

When it comes to finding areas or lengths, algebra is my go-to. For a rectangle or a square, if I know one side and the area, I can easily find the missing side by setting up an algebraic equation. Let’s say the area (( A )) of a rectangle is given and also one side (( l )). The other side (( w )) can be found using ( A = l \times w ).

In geometry, I also use systems of equations, which is an algebraic approach, to find the intersection between lines, a crucial aspect in defining points in polygons or linear graphs. Here’s a scenario: I have two lines with equations ( y = 2x + 3 ) and ( y = -x + 5 ). Solving this system will give me the exact point where both lines cross.

My journey through geometric challenges often involves these algebraic techniques, showing that algebra isn’t just numbers and letters, it’s a gateway to unlocking the mysteries laid out in shapes and spaces.

When I tackle real-world problems using algebra, I begin by defining the variables that represent the unknowns I’m trying to find. Let’s say I’m trying to figure out how many apples and oranges I can buy with a certain amount of money. I’d let ( x ) represent the number of apples and ( y ) the number of oranges.

I then translate the situation into algebraic equations, usually involving addition, subtraction, multiplication, or division. For instance, if apples cost $2 each and oranges $1.50, and I have $10, the equations would look like this:

[ 2x + 1.5y = 10 ]

Next, I would use function notation to express relationships. If I want to determine the relationship between the cost of fruit and the number I buy, I might write a function:

[ f(x) = 2x ] [ g(y) = 1.5y ]

These equations and functions help me visualize the problem and perform calculations to solve for ( x ) and ( y ).

• Define variables : Represent unknown quantities with variables.
• Translate into equations : Use the context to form equations.
• Function notation : Express relationships with functions.
• Solve the equations : Find the value of the variables.

Algebra acts as a bridge between basic arithmetic and the more complex real-world scenarios. It allows me to take practical questions and find concrete answers, such as how many items I can purchase or how long a trip might take given a constant speed.

As I practice, I become adept at forming and solving these algebraic equations, making me better equipped to handle similar problems in the future.

In my experience with algebra practice problems and solutions, I’ve observed that consistency is key. Complete a variety of problems from different domains regularly to sharpen skills. It’s essential to tackle problems that challenge various difficulty levels. I’ve appreciated websites like Paul’s Online Math Notes, which provide a range of problems and detailed solutions.

I often use web filters to ensure that my search results come from educational resources, especially those from trusted domains like kastatic.org and kasandbox.org , which are known for their reliable content. Interactive platforms, rather than static worksheets, offer immediate feedback and keep practice engaging.

Khan Academy, for example, offers an interactive approach that I find particularly effective. The immediate feedback helps me adjust my learning path in real-time.

When approaching algebraic word problems , I make sure to define my variables clearly, as this can often simplify complex equations. It’s also crucial to understand concepts like Venn diagrams, which visually represent logical relationships, as they frequently appear in algebra problems.

Consistency, a variety of sources, and clear problem-solving strategies have greatly contributed to my understanding of algebra . Remember, practice doesn’t just make perfect—it makes permanent. So find resources that work for you and stick to them!

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Pythagorean Theorem Exercises

Pythagorean theorem practice problems with answers.

There are eight (8) problems here about the Pythagorean Theorem for you to work on. When you do something a lot, you get better at it. Let’s get started!

Here’s the Pythagorean Theorem formula for your quick reference.

Note: drawings not to scale

Problem 1: Find the value of [latex]x[/latex] in the right triangle.

[latex]10[/latex]

Problem 2: Find the value of [latex]x[/latex] in the right triangle.

[latex]4\sqrt 7 [/latex]

Problem 3: Find the value of [latex]x[/latex] in the right triangle.

[latex]3[/latex]

Problem 4: The legs of a right triangle are [latex]5[/latex] and [latex]12[/latex]. What is the length of the hypotenuse?

[latex]13[/latex]

Problem 5: The leg of a right triangle is [latex]8[/latex] and its hypotenuse is [latex]17[/latex]. What is the measure of its other leg?

[latex]15[/latex]

Problem 6: Suppose the shorter leg of a right triangle is [latex]\sqrt 2[/latex]. The longer leg is twice the shorter leg. Find the hypotenuse.

[latex]\sqrt {10} [/latex]

Problem 7: The hypotenuse of a right triangle is [latex]4\sqrt 2[/latex]. If the longest leg is half the hypotenuse, what is the length of the shortest leg?

[latex]2\sqrt 6[/latex]

Problem 8: Find the value of [latex]x[/latex] of the right isosceles triangle.

[latex]\sqrt 6[/latex]

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Common Core State Standards Initiative

• Standards for Mathematical Practice

The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation, and connections. The second are the strands of mathematical proficiency specified in the National Research Council’s report Adding It Up : adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy).

Standards in this domain:

Ccss.math.practice.mp1 make sense of problems and persevere in solving them..

Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, "Does this make sense?" They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

CCSS.Math.Practice.MP2 Reason abstractly and quantitatively.

Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize —to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize , to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.

CCSS.Math.Practice.MP3 Construct viable arguments and critique the reasoning of others.

Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

CCSS.Math.Practice.MP4 Model with mathematics.

Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

CCSS.Math.Practice.MP5 Use appropriate tools strategically.

Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.

CCSS.Math.Practice.MP6 Attend to precision.

Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions.

CCSS.Math.Practice.MP7 Look for and make use of structure.

Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x 2 + 9 x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 - 3( x - y ) 2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y .

CCSS.Math.Practice.MP8 Look for and express regularity in repeated reasoning.

Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation ( y - 2)/( x - 1) = 3. Noticing the regularity in the way terms cancel when expanding ( x - 1)( x + 1), ( x - 1)( x 2 + x + 1), and ( x - 1)( x 3 + x 2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.

Connecting the Standards for Mathematical Practice to the Standards for Mathematical Content

The Standards for Mathematical Practice describe ways in which developing student practitioners of the discipline of mathematics increasingly ought to engage with the subject matter as they grow in mathematical maturity and expertise throughout the elementary, middle and high school years. Designers of curricula, assessments, and professional development should all attend to the need to connect the mathematical practices to mathematical content in mathematics instruction.

The Standards for Mathematical Content are a balanced combination of procedure and understanding. Expectations that begin with the word "understand" are often especially good opportunities to connect the practices to the content. Students who lack understanding of a topic may rely on procedures too heavily. Without a flexible base from which to work, they may be less likely to consider analogous problems, represent problems coherently, justify conclusions, apply the mathematics to practical situations, use technology mindfully to work with the mathematics, explain the mathematics accurately to other students, step back for an overview, or deviate from a known procedure to find a shortcut. In short, a lack of understanding effectively prevents a student from engaging in the mathematical practices.

In this respect, those content standards which set an expectation of understanding are potential "points of intersection" between the Standards for Mathematical Content and the Standards for Mathematical Practice. These points of intersection are intended to be weighted toward central and generative concepts in the school mathematics curriculum that most merit the time, resources, innovative energies, and focus necessary to qualitatively improve the curriculum, instruction, assessment, professional development, and student achievement in mathematics.

• How to read the grade level standards
• Introduction
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• Number & Operations in Base Ten
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• Number & Operations—Fractions¹
• Number & Operations in Base Ten¹
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• The Number System
• Expressions & Equations
• Statistics & Probability
• The Real Number System
• Quantities*
• The Complex Number System
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• Seeing Structure in Expressions
• Arithmetic with Polynomials & Rational Expressions
• Creating Equations*
• Reasoning with Equations & Inequalities
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• Building Functions
• Linear, Quadratic, & Exponential Models*
• Trigonometric Functions
• High School: Modeling
• Similarity, Right Triangles, & Trigonometry
• Expressing Geometric Properties with Equations
• Geometric Measurement & Dimension
• Modeling with Geometry
• Interpreting Categorical & Quantitative Data
• Making Inferences & Justifying Conclusions
• Conditional Probability & the Rules of Probability
• Using Probability to Make Decisions
• Courses & Transitions
• Mathematics Glossary
• Mathematics Appendix A

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• Schools & Teaching

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Powerful Problem Solving: Activities for Sense Making with the Mathematical Practices 1st Edition

How can we break the cycle of frustrated students who “drop out of math” because the procedures just don’t make sense to them?  Or who memorize the procedures for the test but don’t really understand the mathematics?  Max Ray-Riek and his colleagues at the Math Forum @ Drexel University say “problem solved,” by offering their collective wisdom about how students become proficient problem solvers, through the lens of the CCSS for Mathematical Practices.  They unpack the process of problem solving in fresh new ways and turn the Practices into activities that teachers can use to foster habits of mind required by the Common Core:

• communicating ideas and listening to the reflections of others
• estimating and reasoning to see the “big picture” of a problem
• organizing information to promote problem solving
• using modeling and representations to visualize abstract concepts
• reflecting on, revising, justifying, and extending the work.

Powerful Problem Solving shows what’s possible when students become active doers rather than passive consumers of mathematics.  Max argues that the process of sense-making truly begins when we create questioning, curious classrooms full of students’ own thoughts and ideas.  By asking “What do you notice?  What do you wonder?”  we give students opportunities to see problems in big-picture ways, and discover multiple strategies for tackling a problem.  Self-confidence, reflective skills, and engagement soar, and students discover that the goal is not to be “over and done,” but to realize the many different ways to approach problems.

Read a sample chapter .

• ISBN-10 0325050902
• ISBN-13 978-0325050904
• Edition 1st
• Publisher HEINEMANN
• Publication date September 24, 2013
• Language English
• Dimensions 8.5 x 0.4 x 11 inches
• Print length 208 pages
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Editorial Reviews

About the author.

Susan O’Connell has decades of experience supporting teachers in making sense of mathematics and effectively shifting how they teach. As a former elementary teacher, reading specialist, and math coach, Sue knows what it’s like in the classroom and her background is evident throughout her work as she unpacks best practices in a clear, practical, and upbeat way.

Sue is the lead author of the new Math by the Book series , a K-5 resource connecting math and children's literature.

She is also the lead author of Math in Practice , a grade-by-grade K-5 professional learning resource. She is also coauthor of the bestselling Putting the Practices Into Action , Mastering the Basic Math Facts in Addition and Subtraction , and Mastering the Basic Math Facts in Multiplication and Division . She served as editor of Heinemann’s popular Math Process Standards series and also wrote the bestselling Now I Get It .

Sue is a nationally known speaker and education consultant who directs Quality Teacher Development , an organization committed to providing outstanding math professional development for schools and districts across the country.

Watch an introductory Math in Practice webinar , hosted by Sue.

Click here to watch Sue talk about the links between reading and math.

Connect with Sue on Twitter: @SueOConnellMath

Max Ray-Riek is a curriculum writer at Illustrative Mathematics and is the lead author of Powerful Problem Solving: Activities for Sense Making with the Math Practices . He previously worked for The Math Forum, focusing on fostering problem solving, communication, and valuing student thinking. Max is a former secondary mathematics teacher who regularly presents at regional and national conferences.

Watch Max and Suzanne Alejandre talk about giving students effective feedback .

Follow him on Twitter @maxmathforum .

Product details

• Publisher ‏ : ‎ HEINEMANN; 1st edition (September 24, 2013)
• Language ‏ : ‎ English
• Paperback ‏ : ‎ 208 pages
• ISBN-10 ‏ : ‎ 0325050902
• ISBN-13 ‏ : ‎ 978-0325050904
• Reading age ‏ : ‎ 8 - 13 years
• Grade level ‏ : ‎ 3 - 8
• Item Weight ‏ : ‎ 1.1 pounds
• Dimensions ‏ : ‎ 8.5 x 0.4 x 11 inches
• #448 in Educational Psychology (Books)
• #607 in Math Teaching Materials
• #1,433 in Educational Certification & Development

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