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Lesson 18: Apologizing Can Help

In this lesson, students will learn about apologizing as a way of showing kindness and as a tool for problem-solving in a variety of scenarios. Lesson Plan (PDF) Lesson Presentation

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In this lesson, students will learn which clues tell them when others might be frustrated, and a new way to feel calm when they’re feeling frustrated themselves. Lesson Plan (PDF) Lesson Presentation

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In this lesson, students will learn how to identify when and where to work on solving a problem, and who should be included. Lesson Plan (PDF) Lesson Presentation

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Polygons - Lesson 11.1

Triangles - Lesson 11.2

Quadrilaterals - Lesson 11.3

Three Dimensional Figures - Lesson 11.5

Unit Cubes and Solid Figures - Lesson 11.6

Understanding Volume - Lesson 11.7

Estimate Volume - Lesson 11.8

Volume of a Rectangular Prism - Lesson 11.9

Apply Volume Formulas - Lesson 11.10

Finding Volume of Composite Formulas - Lesson 11.12

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Multiply Fractions and Whole Numbers - Lesson 7.2

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Multiply Fractions - Lesson 7.4

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Area and Mixed Numbers - Lesson 7.7

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Multistep Measurement Problems - Lesson 10.4

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Elapsed Time - Lesson 10.7

Division Patterns with Decimals - Lesson 5.1

Divide Decimals by Whole Numbers - Lesson 5.2

Estimate Quotients - lesson 5.3

Division of Decimals by Whole Numbers - Lesson 5.4

Decimal Division - Lesson 5.5

Divide Decimals - Lesson 5.6

Write Zeros in the Dividend - Lesson 5.7

Problem Solving - Decimal Operations - Lesson 5.8

Divide Fractions and Whole Numbers - Lesson 8.1

Problem Solving - Use Multiplication - Lesson 8.2

Connect Fractions to Division - Lesson 8.3

Fraction and Whole Number Division - Lesson 8.4

Interpret Division with Fractions - Lesson 8.5

Addition with Unlike Denominators - Lesson 6.1

Subtraction with Unlike Denominators - Lesson 6.2

Estimate Fraction Sums and Differences - Lesson 6.3

Common Denominators and Equivalent Fractions - Lesson 6.4

Add or Subtract Fractions - Lesson 6.5

Add or Subtract Mixed Numbers - Lesson 6.6

Subtraction with Renaming - Lesson 6.7

Patterns with Fractions - Lesson 6.8

Problem Solving with Addition and Subtraction - Lesson 6.9

Use Properties of Addition - Lesson 6.10

Multiplication Patterns with Decimals - Lesson 4.1

Multiply Decimals and Whole Numbers - Lesson 4.2

Multiply Decimals and Whole Numbers - Lesson 4.3

Multiply Using Expanded Form - Lesson 4.4

Problem Solving - Multiply Money - Lesson 4.5

Decimal Multiplication - Lesson 4.6

Multiply Decimals - Lesson 4.7

Thousandths - Lesson 3.1

Place Value of Decimals - Lesson 3.2

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Decimal Subtraction - Lesson 3.6

Estimate Decimal Sums and Differences - Lesson 3.7

Add Decimals - Lesson 3.8

Subtract Decimals - Lesson 3.9

Patterns with Decimals - Lesson 3.10

Problem Solving Add and Subtract Money - Lesson 3.11

Choose a Method - Lesson 3.12

Performance Task on Chapter 3

Place the First Digit - Lesson 2.1

Divide by 1-Digit Divisors - Lesson 2.2

Division with 2-Digit Divisors - Lesson 2.3

Partial Quotients - Lesson 2.4

Estimate with 2-Digit Divisors - Lesson 2.5

Divide by 2-Digit Divisors - Lesson 2.6

Interpret the Remainder - Lesson 2.7

Adjust Quotients - Lesson 2.8

Problem Solving - Division - Lesson 2.9

Performance Task on Chapter 2

Fifth Grade 

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Problem solving

Problem solving lesson plan

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Problem Solving: Lesson plan

Problem solving: Presentation slides

Demonstrating your skills quick fire activity

Problem solving in practice: Interactive worksheet

Our problem solving content focuses on one of these skills and develops understanding of the six stages of problem solving, as well as identifying different types of situations in which young people might already be using these skills. Furthermore, it encourages them to use an adaptive approach, explaining that different types of problems can be approached in different ways.

The activities on this page support your teaching of these skills through an independent activity, quick activities or a full length, curriculum-linked lesson plan. 

Teaching resources:

• Problem solving: Lesson plan and presentation slides – full lesson plan including icebreaker for use with a group of students in the classroom
• Demonstrating your skills: Quick-fire activity  – 10 minute activity for a group of students in the classroom, can be used as an icebreaker for the lesson plan
• Problem solving in practice: Interactive worksheet – activity for independent learning whether remote or in class

Lesson plan

(60 -75 minutes)

This lesson is designed to equip young people with an adaptable approach to solving problems, large or small. It includes a short film and scenarios that encourage development of practical problem solving skills which can be useful for learning, day to day life, and when in employment.

By the end of the lesson, students will be able to:

• Identify problems of different scales and what is needed to solve them
• Illustrate the use of an adaptable approach to solving problems
• Understand that problem solving is a core transferable skill and identify its usefulness in a work setting
• Work on a problem solving activity in a team

The lesson aims to reinforce students’ understanding of the potential future applications of this skill as they move into the world of work, particularly in an activity differentiated for an older or more able group on creating new opportunities.

Quick-fire activity

(5 - 10 minutes)

The demonstrating your skills quick-fire activity focuses on helping young people understand the key skills that are needed in the workplace, including the importance of problem solving.

Students will be asked to name the skills being demonstrated in a variety of scenarios, and identify ways they’re already using those skills in this short activity.

You might find it useful as a starter or icebreaker activity to begin a lesson, or at the end to allow students to put what they have just learnt in the Problem solving lesson into practice.

Interactive worksheet

(20 - 25 minutes)

Please note that students below the age of 14 cannot sign up for their own LifeSkills account. Any independent tasks must be printed or downloaded and provided digitally for them to complete as they are currently hosted on educator pages.

The Problem solving in practice interactive worksheet introduces some of the themes from the full lesson plan and gives students some practical strategies for problem solving, including introducing the six stages of problem solving. The worksheet can be printed or completed digitally, so can be used flexibly to give students practise putting their problem solving skills into action. You might choose to assign it:

• As homework following the Problem solving lesson
• For independent study
• For remote learning

Looking for more ways to boost self confidence with LifeSkills?

Other lessons that may prove useful for students to build on these activities include the  Adaptability  and  Innovation and idea generation  lessons. Alternatively, consider encouraging them to apply their skills through  Steps to starting a business  or the  Social action toolkit .

Why not build problem solving in as a focus in your students’ wider curriculum? Refer to our  Content guide to find out how this resources can be used as part of your teaching.

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Why not try one of these next?

Staying positive (resilience)

Staying positive and learning through experience are key to succeeding in challenging situations. Try this lesson and help your students succeed at work.

Good leadership styles and effective teamwork can help students excel in their future workplace. Read more about team leader skills in this lesson.

Social action toolkit

Build a comprehensive social action programme and support young people to access enriching experiences that build transferable skills for work.

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QuickMath will automatically answer the most common problems in algebra, equations and calculus faced by high-school and college students.

• The algebra section allows you to expand, factor or simplify virtually any expression you choose. It also has commands for splitting fractions into partial fractions, combining several fractions into one and cancelling common factors within a fraction.
• The equations section lets you solve an equation or system of equations. You can usually find the exact answer or, if necessary, a numerical answer to almost any accuracy you require.
• The inequalities section lets you solve an inequality or a system of inequalities for a single variable. You can also plot inequalities in two variables.
• The calculus section will carry out differentiation as well as definite and indefinite integration.
• The matrices section contains commands for the arithmetic manipulation of matrices.
• The graphs section contains commands for plotting equations and inequalities.
• The numbers section has a percentages command for explaining the most common types of percentage problems and a section for dealing with scientific notation.

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• Add Fractions
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Teaching problem solving.

Print Version

Tips and Techniques

Expert vs. novice problem solvers, communicate.

• Have students  identify specific problems, difficulties, or confusions . Don’t waste time working through problems that students already understand.
• If students are unable to articulate their concerns, determine where they are having trouble by  asking them to identify the specific concepts or principles associated with the problem.
• In a one-on-one tutoring session, ask the student to  work his/her problem out loud . This slows down the thinking process, making it more accurate and allowing you to access understanding.
• When working with larger groups you can ask students to provide a written “two-column solution.” Have students write up their solution to a problem by putting all their calculations in one column and all of their reasoning (in complete sentences) in the other column. This helps them to think critically about their own problem solving and helps you to more easily identify where they may be having problems. Two-Column Solution (Math) Two-Column Solution (Physics)

Encourage Independence

• Model the problem solving process rather than just giving students the answer. As you work through the problem, consider how a novice might struggle with the concepts and make your thinking clear
• Have students work through problems on their own. Ask directing questions or give helpful suggestions, but  provide only minimal assistance and only when needed to overcome obstacles.
• Don’t fear  group work ! Students can frequently help each other, and talking about a problem helps them think more critically about the steps needed to solve the problem. Additionally, group work helps students realize that problems often have multiple solution strategies, some that might be more effective than others

Be sensitive

• Frequently, when working problems, students are unsure of themselves. This lack of confidence may hamper their learning. It is important to recognize this when students come to us for help, and to give each student some feeling of mastery. Do this by providing  positive reinforcement to let students know when they have mastered a new concept or skill.

Encourage Thoroughness and Patience

• Try to communicate that  the process is more important than the answer so that the student learns that it is OK to not have an instant solution. This is learned through your acceptance of his/her pace of doing things, through your refusal to let anxiety pressure you into giving the right answer, and through your example of problem solving through a step-by step process.

Experts (teachers) in a particular field are often so fluent in solving problems from that field that they can find it difficult to articulate the problem solving principles and strategies they use to novices (students) in their field because these principles and strategies are second nature to the expert. To teach students problem solving skills,  a teacher should be aware of principles and strategies of good problem solving in his or her discipline .

The mathematician George Polya captured the problem solving principles and strategies he used in his discipline in the book  How to Solve It: A New Aspect of Mathematical Method (Princeton University Press, 1957). The book includes  a summary of Polya’s problem solving heuristic as well as advice on the teaching of problem solving.

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A little bit of problem-solving in every lesson: 10 tips to make it happen

Ideas to embedd proble-solving consistently in your curriculum.

This newsletter is made possible because of Eedi. Check out our brand-new set of diagnostic quizzes, videos, and practice questions for every single maths topic, ready to use in the classroom, and all for free, here .

Welcome to a series of posts where I will share 10 ideas to supercharge some of my favourite student practice activities. We will cover:

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SSDD Problems

Intelligent Practice

A little bit of problem-solving in each lesson (this post)

If you find these posts useful, the best way to support this newsletter is to share it with your colleagues. Thanks so much!

A little bit of problem-solving in every lesson

I wrote recently about the work I have been doing with a maths department to ensure that problem-solving is embedded in their curriculum so that all students get regular opportunities to try unfamiliar problems. I recommend you read that post first as it describes:

What I mean by problem-solving

Why embedding problem-solving is a challenge for many departments

Approaches that don’t work

The approach we developed

What happened when teachers tried it

Then come back to this post where I will share 10 practical tips to help you do the same.

Problem-solving: 10 top tips

Find good sources of problems. As maths teachers, we are so lucky with the quantity of high-quality, free resources available. But, of course, there is a downside: the time it takes to sift through them all to find what we are after. So, for the type of problem-solving I am talking about here - short, sharp problems that can be done in around 10 minutes - I limit myself to three main sources:

Eedi: We have four problems for every maths topic on one giant spreadsheet.

Open Middle: Fill in the gap challenges amidst various constraints

NRICH short problems: questions adapted from the UK maths challenge

Do the problems yourself as a mathematician. Don’t be tempted to just give the problem a cursory glance. It is only when we sit down and do a problem that we notice fun twists and turns, can predict where our students might struggle, or get inspired to think of variations and extensions. If doing this in a departmental meeting, I would recommend 5 minutes working on a problem on your own, and then 5 minutes sharing where you have got to with a colleague.

Then put your teacher hat on. It is important not to get too engrossed in the problem. I have arrived at many lessons excited to share a problem with my students, only to find that my students struggled, finished early, or simply did not enjoy it. The mistake I made was not pausing to think about how to help my students get the most out of the problem. So, after you have wrestled with a problem, I find these are two good questions to consider:

How would you support a student who is struggling?

How would you challenge a student who has finished?

Set an alarm when there are 10 minutes of the lesson left. Time files in lessons, and if we are not careful, we will not leave ourselves sufficient time to get the most out of the problem. Setting an alarm on my phone helps remind me to get cracking with the problem.

Plan how you will pitch the idea to students. Students find unfamiliar problems challenging, and thus may be reluctant to engage with them for fear of failure. So, how we introduce the problem is important. The pitch will be different from class to class, but here are some things to consider:

Keep the stakes low

Emphasise the importance of effort

Share your belief that they can do it

No talking or writing for 30 seconds. I am sold on this approach after trying it a few times. No speaking stops the inevitable I don’t get it . No writing stops students from diving straight in, forcing them to pause and think. Students are likely to be reluctant at first, so sharing the rationale is important. But eventually, you will find they appreciate the headspace.

Give students a mini-whiteboard. Mini-whiteboards are the ideal vehicle to tackle unfamiliar problems. Students have less fear about writing their ideas down as they can be rubbed away if needed. And as I wrote about here , the simple act of putting two mini-whiteboards together provides the catalyst for focussed, positive student collaboration.

Use the 4-2 approach. You want students to work independently on a problem, but also to reap the benefits of collaboration. I find the 4-2 approach works well here, with students spending 4 minutes working on their own to see how far they can get, and then having 2 minutes to collaborate with their partner to compare progress and approaches.

Ask questions to find the best pairs to call upon. After the 2-minute paired discussion is a good time to start collecting ideas about how to solve the problem. Here are three good prompts to help choose which pairs to hear from:

Put your hand up if you disagree with the answer of your partner

Put your hand up if you changed your mind during your discussion

Put your hand up if your partner said something that you found interesting

Showcase students’ work. It is always a good idea to share students’ work with the rest of the class, especially when tackling unfamiliar problems. Students can benefit from thinking hard about the approach of others. There are loads of different ways to share students’ work, including under a visualiser, recreating it on the board, or simply holding it up. My favourite is to use the infinite canvas on Jake Gordon’s Maths Universe website . It is completely free, and it is amazing!

Three of my favourite problems:

Challenge your students with some of these… or enjoy trying them yourself. And remember, you can find all the Eedi problem-solving resources here :

Angles in parallel lines

Equations of straight lines

Mean from a list of numbers

What is your experience with problem-solving?

Do you like any of these ideas?

Do you have any extra tips to share?

Let me know in the comments below!

🏃🏻‍♂️ Before you go, have you…🏃🏻‍♂️

… checked out our incredible, brand-new, free resources from Eedi?

… read my latest Tips for Teachers newsletter about hiding your tell when calling upon students to explain their thinking?

… listened to my most recent podcast about feedback cycles, lesson observations and Exit Tickets?

… considered booking some CPD, coaching, or maths departmental support?

… read my Tips for Teachers book ?

Thanks so much for reading and have a great week!

Ready for more?

10 Best Problem-Solving Therapy Worksheets & Activities

Cognitive science tells us that we regularly face not only well-defined problems but, importantly, many that are ill defined (Eysenck & Keane, 2015).

Sometimes, we find ourselves unable to overcome our daily problems or the inevitable (though hopefully infrequent) life traumas we face.

Problem-Solving Therapy aims to reduce the incidence and impact of mental health disorders and improve wellbeing by helping clients face life’s difficulties (Dobson, 2011).

This article introduces Problem-Solving Therapy and offers techniques, activities, and worksheets that mental health professionals can use with clients.

Before you continue, we thought you might like to download our three Positive Psychology Exercises for free . These science-based exercises explore fundamental aspects of positive psychology, including strengths, values, and self-compassion, and will give you the tools to enhance the wellbeing of your clients, students, or employees.

This Article Contains:

What is problem-solving therapy, 14 steps for problem-solving therapy, 3 best interventions and techniques, 7 activities and worksheets for your session, fascinating books on the topic, resources from positivepsychology.com, a take-home message.

Problem-Solving Therapy assumes that mental disorders arise in response to ineffective or maladaptive coping. By adopting a more realistic and optimistic view of coping, individuals can understand the role of emotions and develop actions to reduce distress and maintain mental wellbeing (Nezu & Nezu, 2009).

“Problem-solving therapy (PST) is a psychosocial intervention, generally considered to be under a cognitive-behavioral umbrella” (Nezu, Nezu, & D’Zurilla, 2013, p. ix). It aims to encourage the client to cope better with day-to-day problems and traumatic events and reduce their impact on mental and physical wellbeing.

Clinical research, counseling, and health psychology have shown PST to be highly effective in clients of all ages, ranging from children to the elderly, across multiple clinical settings, including schizophrenia, stress, and anxiety disorders (Dobson, 2011).

Can it help with depression?

PST appears particularly helpful in treating clients with depression. A recent analysis of 30 studies found that PST was an effective treatment with a similar degree of success as other successful therapies targeting depression (Cuijpers, Wit, Kleiboer, Karyotaki, & Ebert, 2020).

Other studies confirm the value of PST and its effectiveness at treating depression in multiple age groups and its capacity to combine with other therapies, including drug treatments (Dobson, 2011).

The major concepts

Effective coping varies depending on the situation, and treatment typically focuses on improving the environment and reducing emotional distress (Dobson, 2011).

PST is based on two overlapping models:

Social problem-solving model

This model focuses on solving the problem “as it occurs in the natural social environment,” combined with a general coping strategy and a method of self-control (Dobson, 2011, p. 198).

The model includes three central concepts:

• Social problem-solving
• The problem
• The solution

The model is a “self-directed cognitive-behavioral process by which an individual, couple, or group attempts to identify or discover effective solutions for specific problems encountered in everyday living” (Dobson, 2011, p. 199).

Relational problem-solving model

The theory of PST is underpinned by a relational problem-solving model, whereby stress is viewed in terms of the relationships between three factors:

• Stressful life events
• Emotional distress and wellbeing
• Problem-solving coping

Therefore, when a significant adverse life event occurs, it may require “sweeping readjustments in a person’s life” (Dobson, 2011, p. 202).

• Enhance positive problem orientation
• Decrease negative orientation
• Foster ability to apply rational problem-solving skills
• Reduce the tendency to avoid problem-solving
• Minimize the tendency to be careless and impulsive

D’Zurilla’s and Nezu’s model includes (modified from Dobson, 2011):

• Initial structuring Establish a positive therapeutic relationship that encourages optimism and explains the PST approach.
• Assessment Formally and informally assess areas of stress in the client’s life and their problem-solving strengths and weaknesses.
• Obstacles to effective problem-solving Explore typically human challenges to problem-solving, such as multitasking and the negative impact of stress. Introduce tools that can help, such as making lists, visualization, and breaking complex problems down.
• Problem orientation – fostering self-efficacy Introduce the importance of a positive problem orientation, adopting tools, such as visualization, to promote self-efficacy.
• Problem orientation – recognizing problems Help clients recognize issues as they occur and use problem checklists to ‘normalize’ the experience.
• Problem orientation – seeing problems as challenges Encourage clients to break free of harmful and restricted ways of thinking while learning how to argue from another point of view.
• Problem orientation – use and control emotions Help clients understand the role of emotions in problem-solving, including using feelings to inform the process and managing disruptive emotions (such as cognitive reframing and relaxation exercises).
• Problem orientation – stop and think Teach clients how to reduce impulsive and avoidance tendencies (visualizing a stop sign or traffic light).
• Problem definition and formulation Encourage an understanding of the nature of problems and set realistic goals and objectives.
• Generation of alternatives Work with clients to help them recognize the wide range of potential solutions to each problem (for example, brainstorming).
• Decision-making Encourage better decision-making through an improved understanding of the consequences of decisions and the value and likelihood of different outcomes.
• Solution implementation and verification Foster the client’s ability to carry out a solution plan, monitor its outcome, evaluate its effectiveness, and use self-reinforcement to increase the chance of success.
• Guided practice Encourage the application of problem-solving skills across multiple domains and future stressful problems.
• Rapid problem-solving Teach clients how to apply problem-solving questions and guidelines quickly in any given situation.

Success in PST depends on the effectiveness of its implementation; using the right approach is crucial (Dobson, 2011).

Problem-solving therapy – Baycrest

The following interventions and techniques are helpful when implementing more effective problem-solving approaches in client’s lives.

First, it is essential to consider if PST is the best approach for the client, based on the problems they present.

Is PPT appropriate?

It is vital to consider whether PST is appropriate for the client’s situation. Therapists new to the approach may require additional guidance (Nezu et al., 2013).

Therapists should consider the following questions before beginning PST with a client (modified from Nezu et al., 2013):

• Has PST proven effective in the past for the problem? For example, research has shown success with depression, generalized anxiety, back pain, Alzheimer’s disease, cancer, and supporting caregivers (Nezu et al., 2013).
• Is PST acceptable to the client?
• Is the individual experiencing a significant mental or physical health problem?

All affirmative answers suggest that PST would be a helpful technique to apply in this instance.

Five problem-solving steps

The following five steps are valuable when working with clients to help them cope with and manage their environment (modified from Dobson, 2011).

Ask the client to consider the following points (forming the acronym ADAPT) when confronted by a problem:

• Attitude Aim to adopt a positive, optimistic attitude to the problem and problem-solving process.
• Define Obtain all required facts and details of potential obstacles to define the problem.
• Alternatives Identify various alternative solutions and actions to overcome the obstacle and achieve the problem-solving goal.
• Predict Predict each alternative’s positive and negative outcomes and choose the one most likely to achieve the goal and maximize the benefits.
• Try out Once selected, try out the solution and monitor its effectiveness while engaging in self-reinforcement.

If the client is not satisfied with their solution, they can return to step ‘A’ and find a more appropriate solution.

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Positive self-statements

When dealing with clients facing negative self-beliefs, it can be helpful for them to use positive self-statements.

Use the following (or add new) self-statements to replace harmful, negative thinking (modified from Dobson, 2011):

• I can solve this problem; I’ve tackled similar ones before.
• I can cope with this.
• I just need to take a breath and relax.
• Once I start, it will be easier.
• It’s okay to look out for myself.
• I can get help if needed.
• Other people feel the same way I do.
• I’ll take one piece of the problem at a time.
• I can keep my fears in check.
• I don’t need to please everyone.

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PST practitioners have many different techniques available to support clients as they learn to tackle day-to-day or one-off trauma.

5 Worksheets and workbooks

Problem-solving self-monitoring form.

Ask the client to complete the following:

• Describe the problem you are facing.
• What is your goal?
• What have you tried so far to solve the problem?
• What was the outcome?

Reactions to Stress

It can be helpful for the client to recognize their own experiences of stress. Do they react angrily, withdraw, or give up (Dobson, 2011)?

The Reactions to Stress worksheet can be given to the client as homework to capture stressful events and their reactions. By recording how they felt, behaved, and thought, they can recognize repeating patterns.

What Are Your Unique Triggers?

Helping clients capture triggers for their stressful reactions can encourage emotional regulation.

When clients can identify triggers that may lead to a negative response, they can stop the experience or slow down their emotional reaction (Dobson, 2011).

The What Are Your Unique Triggers ? worksheet helps the client identify their triggers (e.g., conflict, relationships, physical environment, etc.).

Problem-Solving worksheet

Imagining an existing or potential problem and working through how to resolve it can be a powerful exercise for the client.

Use the Problem-Solving worksheet to state a problem and goal and consider the obstacles in the way. Then explore options for achieving the goal, along with their pros and cons, to assess the best action plan.

Getting the Facts

Clients can become better equipped to tackle problems and choose the right course of action by recognizing facts versus assumptions and gathering all the necessary information (Dobson, 2011).

Use the Getting the Facts worksheet to answer the following questions clearly and unambiguously:

• Who is involved?
• What did or did not happen, and how did it bother you?
• Where did it happen?
• When did it happen?
• Why did it happen?
• How did you respond?

2 Helpful Group Activities

While therapists can use the worksheets above in group situations, the following two interventions work particularly well with more than one person.

Generating Alternative Solutions and Better Decision-Making

A group setting can provide an ideal opportunity to share a problem and identify potential solutions arising from multiple perspectives.

Use the Generating Alternative Solutions and Better Decision-Making worksheet and ask the client to explain the situation or problem to the group and the obstacles in the way.

Once the approaches are captured and reviewed, the individual can share their decision-making process with the group if they want further feedback.

Visualization

Visualization can be performed with individuals or in a group setting to help clients solve problems in multiple ways, including (Dobson, 2011):

• Clarifying the problem by looking at it from multiple perspectives
• Rehearsing a solution in the mind to improve and get more practice
• Visualizing a ‘safe place’ for relaxation, slowing down, and stress management

Guided imagery is particularly valuable for encouraging the group to take a ‘mental vacation’ and let go of stress.

Ask the group to begin with slow, deep breathing that fills the entire diaphragm. Then ask them to visualize a favorite scene (real or imagined) that makes them feel relaxed, perhaps beside a gently flowing river, a summer meadow, or at the beach.

The more the senses are engaged, the more real the experience. Ask the group to think about what they can hear, see, touch, smell, and even taste.

Encourage them to experience the situation as fully as possible, immersing themselves and enjoying their place of safety.

Such feelings of relaxation may be able to help clients fall asleep, relieve stress, and become more ready to solve problems.

We have included three of our favorite books on the subject of Problem-Solving Therapy below.

1. Problem-Solving Therapy: A Treatment Manual – Arthur Nezu, Christine Maguth Nezu, and Thomas D’Zurilla

This is an incredibly valuable book for anyone wishing to understand the principles and practice behind PST.

Written by the co-developers of PST, the manual provides powerful toolkits to overcome cognitive overload, emotional dysregulation, and the barriers to practical problem-solving.

Find the book on Amazon .

2. Emotion-Centered Problem-Solving Therapy: Treatment Guidelines – Arthur Nezu and Christine Maguth Nezu

Another, more recent, book from the creators of PST, this text includes important advances in neuroscience underpinning the role of emotion in behavioral treatment.

Along with clinical examples, the book also includes crucial toolkits that form part of a stepped model for the application of PST.

3. Handbook of Cognitive-Behavioral Therapies – Keith Dobson and David Dozois

This is the fourth edition of a hugely popular guide to Cognitive-Behavioral Therapies and includes a valuable and insightful section on Problem-Solving Therapy.

This is an important book for students and more experienced therapists wishing to form a high-level and in-depth understanding of the tools and techniques available to Cognitive-Behavioral Therapists.

For even more tools to help strengthen your clients’ problem-solving skills, check out the following free worksheets from our blog.

• Case Formulation Worksheet This worksheet presents a four-step framework to help therapists and their clients come to a shared understanding of the client’s presenting problem.
• Understanding Your Default Problem-Solving Approach This worksheet poses a series of questions helping clients reflect on their typical cognitive, emotional, and behavioral responses to problems.
• Social Problem Solving: Step by Step This worksheet presents a streamlined template to help clients define a problem, generate possible courses of action, and evaluate the effectiveness of an implemented solution.

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While we are born problem-solvers, facing an incredibly diverse set of challenges daily, we sometimes need support.

Problem-Solving Therapy aims to reduce stress and associated mental health disorders and improve wellbeing by improving our ability to cope. PST is valuable in diverse clinical settings, ranging from depression to schizophrenia, with research suggesting it as a highly effective treatment for teaching coping strategies and reducing emotional distress.

Many PST techniques are available to help improve clients’ positive outlook on obstacles while reducing avoidance of problem situations and the tendency to be careless and impulsive.

The PST model typically assesses the client’s strengths, weaknesses, and coping strategies when facing problems before encouraging a healthy experience of and relationship with problem-solving.

Why not use this article to explore the theory behind PST and try out some of our powerful tools and interventions with your clients to help them with their decision-making, coping, and problem-solving?

We hope you enjoyed reading this article. Don’t forget to download our three Positive Psychology Exercises for free .

• Cuijpers, P., Wit, L., Kleiboer, A., Karyotaki, E., & Ebert, D. (2020). Problem-solving therapy for adult depression: An updated meta-analysis. European P sychiatry ,  48 (1), 27–37.
• Dobson, K. S. (2011). Handbook of cognitive-behavioral therapies (3rd ed.). Guilford Press.
• Dobson, K. S., & Dozois, D. J. A. (2021). Handbook of cognitive-behavioral therapies  (4th ed.). Guilford Press.
• Eysenck, M. W., & Keane, M. T. (2015). Cognitive psychology: A student’s handbook . Psychology Press.
• Nezu, A. M., & Nezu, C. M. (2009). Problem-solving therapy DVD . Retrieved September 13, 2021, from https://www.apa.org/pubs/videos/4310852
• Nezu, A. M., & Nezu, C. M. (2018). Emotion-centered problem-solving therapy: Treatment guidelines. Springer.
• Nezu, A. M., Nezu, C. M., & D’Zurilla, T. J. (2013). Problem-solving therapy: A treatment manual . Springer.

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Lesson Plan

Problem-solving steps, view aligned standards, learning objectives.

Students will be able to identify a problem and engage in problem-solving steps to come up with a solution to strengthen their responsible decision-making competency.

Introduction

• Bring students together in a circle, either seated or standing.
• Bring blocks with you to the circle.
• Show the student the blocks and ask them to watch you build a tall castle.
• After you build it, bring out two figurines that you would like to play with in the castle.
• Say out loud, "Hmm....there seems to be a problem. This castle is too small for my toys. How can I make this bigger?"
• Take a few big breaths into your belly and ask the class to breathe with you.
• Explain that deep breathing can help you stay calm and focused when solving a problem.
• Ask the class, "How can I make this castle bigger?"
• Ask, "Would someone like to come up and think of a new way to solve this problem?"
• Invite one student to come up to solve the problem (i.e., make a larger castle).
• Ask your student volunteer, "What would you do here? How would you test a solution?"
• Give the student time to come up with a solution to make a larger castle.
• Identify the problem.
• Look at what solutions have already been tried.
• Think of new ways to solve the problem.
• Try it out!
• Thank your student volunteer for showing you how to solve the problem and send them back to their seats.
• Say, "All of us sometimes come across problems we have to solve, and sometimes friends can help us, too!"

Curriculum  /  Math  /  10th Grade  /  Unit 7: Circles  /  Lesson 10

Lesson 10 of 14

Criteria for Success

Tips for teachers, anchor problems.

• Problem Set

Target Task

Additional practice.

Lesson Notes

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Use angle and side length relationships with chords, tangents, inscribed angles, and circumscribed angles to solve problems. 

Common Core Standards

Core standards.

The core standards covered in this lesson

G.C.A.2 — Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

The essential concepts students need to demonstrate or understand to achieve the lesson objective

• Identify and use angle and segment relationships in circle diagrams to find missing measures. 
• State and describe the conjectures and theorems used to find missing measures in circle diagrams. 

Suggestions for teachers to help them teach this lesson

This lesson summarizes key concepts in the unit; therefore, this lesson should be used as a review day. 

Unlock features to optimize your prep time, plan engaging lessons, and monitor student progress.

Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding

25-30 minutes

In circle $$A$$ below, $$\overleftrightarrow{CE}$$ and $$\overleftrightarrow{DE}$$  are tangent at points  $$C$$ and $$D$$ , respectively.

Find $$m\angle{CED}$$  and  $$m\angle{ADG}$$ .

Guiding Questions

Geometry - 8.10 AP1 by Match Fishtank is made available by GeoGebra  under the CC BY-NC-SA 3.0  license. Copyright © International GeoGebra Institute, 2013. Accessed June 13, 2017, 11:55 a.m..

In the following diagram, the radius of circle  $$D$$  is $$5$$  cm and  $$F$$  is the midpoint of  $$\overline{AE}$$ . The measures of  $$\widehat{EB}$$  and  $$\widehat{BC}$$  are given in the diagram. Find the measures of all other unmarked angles, arcs, and segments.

Module 7: Circles a Geometric Perspective from  Geometry: A Learning Cycle Approach  made available by Mathematics Vision Project  under the CC BY 4.0  license. © 2016 Mathematics Vision Project. Accessed Oct. 19, 2017, 2:58 p.m..

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

5-10 minutes

In the diagram below,  $$\overline{PA}$$ is tangent to circle $$O$$ , and  $$\overline{AB}$$ is a chord. If $$m\widehat{ACB}=300$$ , find the measure of $$\angle BAP$$ .

G.C.A.2: Chords, Secants, and Tangents 17 is made available on JMAP by Steve Sibol and Steve Watson. Copyright © 2017  JMAP, Inc. - All rights reserved. Accessed Sept. 19, 2018, 10:43 a.m..

The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

• EngageNY Mathematics Geometry > Module 5 > Topic C > Lesson 16
• Mathematics Vision Project: Geometry Module 7: Circles a Geometric Perspective — Lesson 5, "From Polygons to Circles"
• Mathematics Vision Project: Geometry Module 7: Circles a Geometric Perspective — Lesson 6, "Circular Reasoning"
• MARS Formative Assessment Lessons for High School Solving Problems with Circles and Triangles
• Mathematics Vision Project: Geometry Module 7: Circles a Geometric Perspective — Lesson 3, "Cyclic Polygons"

Topic A: Equations of Circles

Derive the equation of a circle using the Pythagorean Theorem where the center of the circle is at the origin.

G.GPE.A.1 G.GPE.B.4

Given a circle with a center translated from the origin, write the equation of the circle and describe its features. 

G.C.A.1 G.CO.A.5 G.GPE.A.1

Write an equation for a circle in standard form by completing the square. Describe the transformations of a circle. 

G.CO.A.5 G.GPE.A.1

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Topic B: Angle and Segment Relationships in Inscribed and Circumscribed Figures

Define a chord to derive the Chord Central Angles Conjecture and Thales’ Theorem. 

Describe the relationship between inscribed and central angles in terms of their intercepted arc. 

Determine the angle and length relationships between intersecting chords. 

Prove properties of angles in a quadrilateral inscribed in a circle.

Define and determine properties of tangents and secants of circles to solve problems with inscribed and circumscribed triangles. 

G.C.A.2 G.C.A.3

Construct tangent lines to a circle to define and describe the circumscribed angle. 

G.C.A.2 G.C.A.4

Topic C: Arc Length, Radians, and Sector Area

Define, describe, and calculate arc length.

Describe the proportional relationship between arc length and the radius of a circle. Convert between degrees and radians to write the arc measure in radians. 

Calculate the sector area of a circle. Identify relationships between sector area, arc angle, and radius. 

Use sector area of circles to calculate the composite area of figures.

G.C.B.5 N.Q.A.2 N.Q.A.3

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