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10 Summative Assessment Examples to Try This School Year

Written by Jordan Nisbet

• Teaching Strategies

• A formative and summative assessment definition
• Difference between formative and summative assessment
• Pros and cons of summative assessment
• 9 effective and engaging summative assessment examples
• Helpful summative assessment strategies

When gauging student learning, two approaches likely come to mind: a formative or summative assessment.

Fortunately, feeling pressure to choose one or the other isn’t necessary. These two types of learning assessment actually serve different and necessary purposes. 

Definitions: What’s the difference between formative and summative assessment?

Formative assessment occurs regularly throughout a unit, chapter, or term to help track not only how student learning is improving, but how your teaching can, too.

According to a WestEd article , teachers love using various formative assessments because they help meet students’ individual learning needs and foster an environment for ongoing feedback.

Take one-minute papers, for example. Giving your students a solo writing task about today’s lesson can help you see how well students understand new content.

Catching these struggles or learning gaps immediately is better than finding out during a summative assessment.

Such an assessment could include:

• In-lesson polls
• Partner quizzes
• Self-evaluations
• Ed-tech games
• One-minute papers
• Visuals (e.g., diagrams, charts or maps) to demonstrate learning
• Exit tickets

So, what is a summative assessment?

Credit: Alberto G.

It occurs at the end of a unit, chapter, or term and is most commonly associated with final projects, standardized tests, or district benchmarks.

Typically heavily weighted and graded, it evaluates what a student has learned and how much they understand.

There are various types of summative assessment. Here are some common examples of summative assessment in practice:

• End-of-unit test
• End-of-chapter test
• Achievement tests
• Standardized tests
• Final projects or portfolios

Teachers and administrators use the final result to assess student progress, and to evaluate schools and districts. For teachers, this could mean changing how you teach a certain unit or chapter. For administrators, this data could help clarify which programs (if any) require tweaking or removal.

The differences between formative and summative assessment

While we just defined the two, there are five key differences between formative and summative assessments requiring a more in-depth explanation.

Formative assessment:

• Occurs through a chapter or unit
• Improves how students learn
• Covers small content areas
• Monitors how students are learning
• Focuses on the process of student learning

Summative assessment:

• Occurs at the end of a chapter or unit
• Evaluates what students learn
• Covers complete content areas
• Assigns a grade to students' understanding
• Emphasizes the product of student learning

During vs after

Teachers use formative assessment at many points during a unit or chapter to help guide student learning.

Summative assessment comes in after completing a content area to gauge student understanding.

Improving vs evaluating

If anyone knows how much the learning process is a constant work in progress, it’s you! This is why formative assessment is so helpful — it won’t always guarantee students understand concepts, but it will improve how they learn.

Summative assessment, on the other hand, simply evaluates what they’ve learned. In her book, Balanced Assessment: From Formative to Summative, renowned educator Kay Burke writes, “The only feedback comes in the form of a letter grade, percentage grade, pass/fail grade, or label such as ‘exceeds standards’ or ‘needs improvement.’”

Little vs large

Let’s say chapter one in the math textbook has three subchapters (i.e., 1.1, 1.2 and 1.3). A teacher conducting formative assessments will assign mini tasks or assignments throughout each individual content area.

Whereas, if you’d like an idea of how your class understood the complete chapter, you’d give them a test covering a large content area including all three parts.

Monitoring vs grading

Formative assessment is extremely effective as a means to monitor individual students’ learning styles. It helps catch problems early, giving you more time to address and adapt to different problem areas.

Summative assessments are used to evaluate and grade students’ overall understanding of what you’ve taught. Think report card comments: did students achieve the learning goal(s) you set for them or not?

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Process vs product

“It’s not about the destination; it’s about the journey”? This age-old saying sums up formative and summative assessments fairly accurately.

The former focuses on the process of student learning. You’ll use it to identify areas of strength and weakness among your students — and to make necessary changes to accommodate their learning needs.

The latter emphasizes the product of student learning. To discover the product’s “value”, you can ask yourself questions, such as: At the end of an instructional unit, did the student’s grade exceed the class standard, or pass according to a district’s benchmark?

In other words, formative methods are an assessment for learning whereas summative ones are an assessment of learning .

Now that you’ve got a more thorough understanding of these evaluations, let’s dive into the love-hate relationship teachers like yourself may have with summative assessments.

Perceived disadvantages of summative assessment

The pros are plenty. However, before getting to that list, let’s outline some of its perceived cons. Summative assessment may:

1) Offer minimal room for creativity

Rigid and strict assignments or tests can lead to a regurgitation of information. Some students may be able to rewrite facts from one page to another, but others need to understand the “why” before giving an answer.

2) Not accurately reflect learning

“Teaching to the test” refers to educators who dedicate more time teaching lessons that will be emphasized on district-specific tests.

A survey conducted by Harvard’s Carnegie-Knight Task Force on the Future of Journalism asked teachers whether or not “preparing students to pass mandated standardized tests” affects their teaching.

A significant 60% said it either “dictates most of” or “substantially affects” their teaching. While this can result in higher scores, curriculum distortion can prevent students from learning other foundational subject areas.

3) Ignore (and miss) timely learning needs

Because summative assessment occurs at the end of units or terms, teachers can fail to identify and remedy students’ knowledge gaps or misconceptions as they arise.

Unfortunately, by this point, there’s often little or no time to rectify a student’s mark, which can affect them in subsequent units or grades.

4) Result in a lack of motivation

The University of London’s Evidence for Policy and Practice conducted a 19-study systematic review of the impact summative assessment and tests have on students’ motivation for learning.

Contrary to popular belief, researchers found a correlation between students who scored poorly on national curriculum tests and experienced lower self-esteem, and an unwillingness to put more effort into future test prep. Beforehand, interestingly, “there was no correlation between self-esteem and achievement.”

For some students, summative assessment can sometimes be seen as 'high stakes' testing due to the pressure on them to perform well. That said, 'low-stakes' assessments can also be used in the form of quizzes or practice tests.

Repeated practice tests reinforce the low self-image of the lower-achieving students… When test scores are a source or pride and the community, pressure is brought to bear on the school for high scores.

Similarly, parents bring pressure on their children when the result has consequences for attendance at high social status schools. For many students, this increases their anxiety, even though they recognize their parents as being supportive.

5) Be inauthentic

Summative assessment has received criticism for its perceived inaccuracy in providing a full and balanced measure of student learning.

Consider this, for example: Your student, who’s a hands-on, auditory learner, has a math test today. It comes in a traditional paper format as well as a computer program format, which reads the questions aloud for students.

Chances are the student will opt for the latter test format. What’s more, this student’s test results will likely be higher and more accurate.

The reality is that curricula — let alone standardized tests — typically don’t allow for this kind of accommodation. This is the exact reason educators and advocates such as Chuck Hitchcock, Anne Meyer, David Rose, and Richard Jackson believe:

Curriculum matters and ‘fixing’ the one-size-fits-all, inflexible curriculum will occupy both special and general educators well into the future… Students with diverse learning needs are not ‘the problem’; barriers in the curriculum itself are the root of the difficulty.

6) Be biased

Depending on a school district’s demographic, summative assessment — including standardized tests — can present biases if a group of students is unfairly graded based on their race, ethnicity, religion, gender, or social class.

In his presentation at Kansas State University, emeritus professor in the UCLA Graduate School of Education and Information Studies, Dr. W. James Popham, explained summative assessment bias:

This doesn’t necessarily mean that if minority students are outperformed on a summative test by majority students that the test is biased against that minority. It may instead indicate that the minority students have not been provided with the appropriate instruction…

An example of content bias against girls would be one in which students are asked to compare the weights of several objects, including a football. Since girls are less likely to have handled a football, they might find the item more difficult than boys, even though they have mastered the concept measured by the item.

Importance and benefits of summative assessment

Overall, these are valid points raised against summative assessment. However, it does offer fantastic benefits for teachers and students alike!

Summative assessment can:

1) Motivate students to study and pay closer attention

Although we mentioned lack of motivation above, this isn’t true for every student. In fact, you’ve probably encountered numerous students for whom summative assessments are an incredible source of motivation to put more effort into their studies.

For example, final exams are a common type of summative assessment that students may encounter at the end of a semester or school year. This pivotal moment gives students a milestone to achieve and a chance to demonstrate their knowledge.

In May 2017, the College Board released a statement about whether coaching truly boosts test scores:

Data shows studying for the SAT for 20 hours on free Official SAT Practice on Khan Academy is associated with an average score gain of 115 points, nearly double the average score gain compared to students who don’t use Khan Academy. Out of nearly 250,000 test-takers studied, more than 16,000 gained 200 points or more between the PSAT/NMSQT and SAT…

In addition to the 115-point average score increase associated with 20 hours of practice, shorter practice periods also correlate with meaningful score gains. For example, 6 to 8 hours of practice on Official SAT Practice is associated with an average 90-point increase.

2) Allow students to apply what they’ve learned

It’s one thing to memorize multiplication tables (which is a good skill), but another to apply those skills in math word problems or real-world examples.

Summative assessments — excluding, for example, multiple choice tests — help you see which students can retain and apply what they’ve learned.

3) Help identify gaps in student learning

Before moving on to a new unit, it’s vital to make sure students are keeping up. Naturally, some will be ahead while others will lag behind. In either case, giving them a summative assessment will provide you with a general overview of where your class stands as a whole.

Let’s say your class just wrote a test on multiplication and division. If all students scored high on multiplication but one quarter of students scored low on division, you’ll know to focus more on teaching division to those students moving forward.

4) Help identify possible teaching gaps

Credit: woodleywonderworks

In addition to identifying student learning gaps , summative assessment can help target where your teaching style or lesson plans may have missed the mark.

Have you ever been grading tests before, to your horror, realizing almost none of your students hit the benchmark you hoped for? When this happens, the low grades are not necessarily related to study time.

For example, you may need to adjust your teaching methods by:

• Including/excluding word problems
• Incorporating more visual components
• Innovative summative assessments (we list some below!)

5) Give teachers valuable insights

Credit: Kevin Jarrett

Summative assessments can highlight what worked and what didn’t throughout the school year. Once you pinpoint how, where and what lessons need tweaking, making informed adjustments for next year becomes easier.

In this world nothing can be said to be certain, except death and taxes… and, for teachers, new students year after year. So although old students may miss out on changes you’ve made to your lessons, new ones get to reap the benefits.

This not only improves your skills as an educator, but will ensure a more enriching educational experience for generations of students to come.

6) Contribute positively to learning outcomes

Certain summative assessments also provide valuable data at district, national, and global levels. Depending on average test scores, this can determine whether or not certain schools receive funding, programs stay or go, curriculum changes occur, and more. Burke writes:

Summative assessments also provide the public and policymakers with a sense of the results of their investment in education and give educators a forum for proving whether instruction works – or does not work.

The seven aims of summative assessment

Dr. Nancy P. Gallavan, a professor of teacher education at the University of Central Arkansas, believes teachers can use performance-based summative assessments at any grade level.

However, in an article for Corwin , she suggests crafting yours with seven aims in mind:

• Accompanied  with appropriate time and task management
• Achievable  as in-class activities and out-of-class assignments
• Active  involvement in planning, preparation, and performance
• Applicable  to academic standards and expectations
• Appropriate  to your students’ learning styles, needs, and interests
• Attractive  to your students on an individual and group level
• Authentic  to curricular content and context

Ideally, the assessment method should also measure a student’s performance accurately against the learning objectives set at the beginning of the course.

Keeping these goals in mind, here’s a list of innovative ways to conduct summative assessments in your classroom!

Summative assessment examples: 9 ways to make test time fun

If you want to switch things up this summative assessment season, keep reading. While you can’t change what’s on standardized tests, you can create activities to ensure your students are exhibiting and applying their understanding and skills to end-of-chapter or -unit assessments. In a refreshing way.

Why not give them the opportunity to express their understanding in ways that apply to different learning styles?

Note : As a general guideline, students should incorporate recognition and recall, logic and reasoning, as well as skills and application that cover major concepts and practices (including content areas you emphasized in your lessons).

1) One, two, three… action!

Write a script and create a short play, movie, or song about a concept or strategy of your choosing.

This video from Science Rap Academy is a great — and advanced — example of students who created a song about how blue-eyed children can come from two brown-eyed parents:

Using a tool such as iPhone Fake Text Generator , have students craft a mock text message conversation conveying a complex concept from the unit, or each chapter of that unit.

Students could create a back-and-forth conversation between two historical figures about a world event, or two friends helping each other with complex math concepts.

Have your students create a five to 10-minute podcast episode about core concepts from each unit. This is an exciting option because it can become an ongoing project.

Individually or in groups, specific students can be in charge of each end-of-chapter or -unit podcast. If your students have a cumulative test towards the end of the year or term, the podcast can even function as a study tool they created together.

Credit : Brad Flickinger

You can use online tools such as Record MP3 Online or Vocaroo to get your class started!

4) Infographic

Creating a detailed infographic for a final project is an effective way for students to reinforce what they’ve learned. They can cover definitions, key facts, statistics, research, how-to info, graphics, etc.

You can even put up the most impressive infographics in your classroom. Over time, you’ll have an arsenal of in-depth, visually-appealing infographics students can use when studying for chapter or unit tests.

5) Compare and contrast

Venn diagrams are an old — yet effective — tool perfect for visualizing just about anything! Whether you teach history or social studies, English or math, or something in between, Venn diagrams can help certain learners visualize the relationship between different things.

For example, they can compare book characters, locations around the world, scientific concepts, and more just like the examples below:

6) Living museum

This creative summative assessment is similar to one, two, three… action! Individuals will plan and prepare an exhibit (concept) in the Living Museum (classroom). Let’s say the unit your class just completed covered five core concepts.

Five students will set up around the classroom while the teacher walks from exhibit to exhibit. Upon reaching the first student, the teacher will push an imaginary button, bringing the exhibit “to life.” The student will do a two to three-minute presentation; afterwards, the teacher will move on to the next one.

7) Ed-Tech games

Now more than ever, students are growing up saturated with smartphones, tablets, and video games. That’s why educators should show students how to use technology in the classroom effectively and productively.

More and more educators are bringing digital tools into the learning process. Pew Research Center surveyed 2,462 teachers and reported that digital technologies have helped in teaching their middle and high school students.

Some of the findings were quite eye-opening:

• 80% report using the internet at least weekly to help them create lesson plans
• 84% report using the internet at least weekly to find content that will engage students
• 69% say the internet has a “major impact on their ability to share ideas with other teachers
• 80% report getting email alerts or updates at least weekly that allow them to follow developments in their field
• 92% say the internet has a “major impact” on their ability to access content, resources, and materials for their teaching
• 67% say the internet has a “major impact” on their ability to interact with parents and 57% say it has had such an impact on enabling their interaction with students

To make the most of EdTech, find a tool that actually engages your students in learning and gives you the insightful data and reports you need to adjust your instruction

Tip: Teaching math from 1st to 8th grade? Use Prodigy!

With Prodigy Math, you can:

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8) Shark Tank/Dragon’s Den

Yes, just like the reality TV show! You can show an episode or two to your class or get them to watch the show at home. Next, have students pitch a product or invention that can help change the world outside of school for the better.

This innovative summative assessment is one that’ll definitely require some more thought and creativity. But it’s important that, as educators, we help students realize they can have a huge positive impact on the world in which they live.

9) Free choice

If a student chooses to come up with their own summative assessment, you’ll need to vet it first. It’ll likely take some collaboration to arrive at something sufficient.

However, giving students the freedom to explore content areas that interest them most could surprise you. Sometimes, it’s during those projects they form a newfound passion and are wildly successful in completing the task.

We’re sure there are countless other innovative summative assessment ideas out there, but we hope this list gets your creative juices flowing.

With the exclusion of standardized state and national tests, one of the greatest misconceptions about summative assessments is that they’re all about paper and pencil. Our hope in creating this list was to help you see how fun and engaging summative assessments can truly be.

10) Group projects

Group projects aren't just a fun way to break the monotony, but a dynamic and interactive form of summative assessment. Here's why:

• Collaborative learning: Group projects encourage students to work as a team, fostering their communication and collaboration skills. They learn to listen, negotiate, and empathize, which are crucial skills in and beyond the classroom.
• Promotes critical thinking: When students interact with each other, they get to explore different perspectives. They challenge each other's understanding, leading to stimulating debates and problem-solving sessions that boost critical thinking.
• In-depth assessment: Group projects offer teachers a unique lens to evaluate both individual performances and group dynamics. It's like getting a sneak peek into their world - you get to see how they perform under different circumstances and how they interact with each other.
• Catering to different learning styles: Given the interactive nature of group projects, they can cater to different learning styles - auditory, visual, and kinesthetic. Every student gets a chance to shine!

However, it's important to set clear instructions and criteria to ensure fairness. Remember, it's not just about the final product - it's about the process too.

Some interesting examples of group projects include:

• Create a Mini Documentary: Students could work together to research a historical event and create a mini documentary presenting their findings.
• Plan a Community Service Project: This could involve identifying a problem in the local community and creating a detailed plan to address it.
• Design a Mobile App: For a more tech-focused project, students could identify a problem and design an app that solves it.

Summative assessment strategies for keeping tests clear and fair

In addition to using the summative assessment examples above to accommodate your students’ learning styles, these tips and strategies should also help:

• Use a rubric  — Rubrics help set a standard for how your class should perform on a test or assignment. They outline test length, how in-depth it will be, and what you require of them to achieve the highest possible grades.
• Design clear, effective questions  — When designing tests, do your best to use language, phrases, and examples similar to those used during lessons. This’ll help keep your tests aligned with the material you’ve covered.
• Try blind grading  — Most teachers prefer knowing whose tests they’re grading. But if you want to provide wholly unbiased grades and feedback, try blind grading. You can request your students write their names on the bottom of the last test page or the back.
• Assess comprehensiveness  — Make sure the broad, overarching connections you’re hoping students can make are reasonable and fluid. For example, if the test covers measurement, geometry and spatial sense, you should avoid including questions about patterning and algebra.
• Create a final test after, not before, teaching the lessons  — Don’t put the horse before the carriage. Plans can change and student learning can demand different emphases from year to year. If you have a test outline, perfect! But expect to embrace and make some changes from time to time.
• Make it real-world relevant  — How many times have you heard students ask, “When am I going to use this in real life?” Far too often students assume math, for example, is irrelevant to their lives and write it off as a subject they don’t need. When crafting test questions, use  culturally-relevant word problems  to illustrate a subject’s true relevance.

Enter the Balanced Assessment Model

Throughout your teaching career, you’ll spend a lot of time with formative and summative assessments. While some teachers emphasize one over the other, it’s vital to recognize the extent to which they’re interconnected.

In the book Classroom Assessment for Student Learning , Richard Stiggins, one of the first educators to advocate for the concept of assessment for learning, proposes something called “a balanced assessment system that takes advantage of assessment of learning and assessment for learning.”

If you use both effectively, they inform one another and “assessment becomes more than just an index of school success. It also serves as the cause of that success.”

In fact, Stiggins argues teachers should view these two types of assessment as “in sync.”

They can even be the  exact same thing — only the purpose and the timing of the assessment determine its label. Formative assessments provide the training wheels that allow students to practice and gain confidence while riding their bikes around the enclosed school parking lot.

Once the training wheels come off, the students face their summative assessment as they ride off into the sunset on only two wheels, prepared to navigate the twists and turns of the road to arrive safely at their final destination.

Conclusion: Going beyond the test

Implementing these innovative summative assessment examples should engage your students in new and exciting ways.

What’s more, they’ll have the opportunity to express and apply what they’ve learned in creative ways that solidify student learning.

So, what do you think — are you ready to try out these summative assessment ideas? Prodigy is a game-based learning platform teachers use to keep their students engaged.

Sign up for a free teacher account  and set an  Assessment  today!

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Turn math assessments into enjoyable experiences with Prodigy's game-based approach. Get ready for eager learners!

Free Math Worksheets by Math-Drills

Math-Drills.com includes over 70,000 free math worksheets that may be used to help students learn math. Our math worksheets are available on a broad range of topics including number sense, arithmetic, pre-algebra, geometry, measurement, money concepts and much more. There are two interactive math features: the math flash cards and dots math game.

Math-Drills.com was launched in 2005 with around 400 math worksheets. Since then, tens of thousands more math worksheets have been added. The website and content continues to be improved based on feedback and suggestions from our users and our own knowledge of effective math practices.

Most Math-Drills users are classroom teachers or parents. Classroom teachers use our math worksheets to assess student mastery of basic math facts, to give students extra math practice, to teach new math strategies, and to save precious planning time. Parents use our math worksheets to give their children extra math practice over school breaks and to enhance their math education. Home schools use our math worksheets in their programs to develop and strengthen math skills in their children.

Most Popular Free Math Worksheets this Week

Supporting Student Learning with Free Math Worksheets by Math-Drills

Math-Drills believes that education should be accessible to all children despite their socioeconomic situation or any other factors. Since it began in 2005, all the math worksheets on Math-Drills have been free-to-use with students learning math. The Math-Drills website works well on any device and worksheets can be printed or used on a screen.

Math-Drills worksheets are also used in special education, adult education, tutoring, colleges, high schools, prisons and a variety of other situations. Special education teachers especially like that we break down math skills and provide large-print options. Adult learners appreciate the simple uncluttered format that our worksheets offer. Tutors use our math worksheets to reduce their costs and focus on student learning. High school, college and university educators sometimes need remedial resources for students to allow them to continue with more advanced topics.

Students who practice their math skills with our math worksheets over school breaks keep their math skills sharp for upcoming school terms. Because we provide answer keys, students are able to self-assess and use the immediate feedback provided by an answer key to analyze and correct errors in their work. Our interactive (fillable) math worksheets allow them to fill in their answers on the screen and save or print the results.

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Top-Rated Tips From Experts: How to Write a Maths Assignment

Mathematics Mastery: Trust Our Experts for Impeccable Assignment Writing, Ensuring Precision and Academic Excellence.

Writing a mathematics assignment is rewarding yet challenging, requiring precision, clarity, and a deep understanding of mathematical concepts. Navigating this academic challenge successfully can be exhaustive for students. Under this circumstance, consider the insights and tips our expert professionals share. These tips enhance your approach to creating high-quality and accurate math assignments.

From effective problem-solving strategies to the importance of clear communication, these expert-recommended guidelines offer a comprehensive framework for achieving excellence in math homework. Whether you're a student looking to improve your assignment skills or an educator guiding others through the process, incorporating these tips can elevate the quality and impact of your math assignments.

Mastering Math Assignments: Strategies for Success

Embracing the world of mathematics can be enjoyable and intellectually stimulating when approached with the right mindset. Here are some valuable tips to enhance your ability to solve math assignments efficiently.

Eliminate Distractions: Dedicate focused time to complete your math assignments, as they often involve multi-step problem-solving. Minimize distractions by putting away your phone and notifying others of your study plans. Seeking assistance from math assignment help services can provide expert guidance and support you require for math assignments.

Utilize Mathematical Tricks: Learn and apply quick mathematical tricks from your professors for calculations. You can also opt for assistance from a math assignment helper to ace different techniques. Familiarity with these tricks can significantly expedite the problem-solving of math writing assignment.

Thoroughly Read and Follow Instructions: Carefully read the instructions provided by your instructor before diving into the assignment. Use a notepad to jot down essential information and create an outline for a better understanding. Clearly articulate your problem-solving process, ensuring you address all aspects mentioned in the instructions.

Start with Challenging Problems: Prioritize your math problems based on difficulty and importance. Tackle the most challenging problems first and gradually work your way down. This approach ensures efficient utilization of your energy and time.

Take Regular, Short Breaks: Maintain concentration by taking short breaks to alleviate stress and prevent burnout. Breaks of 15 to 20 minutes allow you to recharge without losing focus on your assignments.

Study with a Friend: Collaborating with a study companion can provide motivation and a fresh perspective. Explaining concepts to each other enhances understanding and promotes effective learning.

Seek Expert Assistance: Don't hesitate to ask for help from experts at any stage of your assignment. Technology facilitates instant access to professionals who can assist with complex problems, ensuring academic success.

Practice Regularly: Building a strong foundation in mathematics requires consistent practice. Regular practice eases complex computations and explores various methods. It also helps you to analyze shortcuts for rapid problem-solving.

Common Worries About Math Assignments:

Many students experience common worries about math assignments. They often feel anxious about understanding complex concepts and solving difficult problems. Time management is another concern, as math assignments can be time-consuming and require focused attention. The fear of making mistakes and the pressure to achieve high grades can also contribute to stress. Additionally, some students worry about not having adequate resources or support to complete their assignments successfully. These concerns can impact their confidence and performance, but with the right strategies and help, such as tutoring and practice, students can overcome these challenges and excel in their math assignments.

Understanding why students may struggle with math assignments is essential. Common reasons include:

Problems in Math Concept Understanding

Difficulty in understanding math arises from a combination of factors.

Some learners find the abstract nature of mathematical concepts challenging. Others face problems due to confusion from previously perceived knowledge. Most students acknowledged they often find it hard to visualize and relate math problems to real-world scenarios. Foundational knowledge gaps hinder comprehension. Effective math understanding requires a supportive learning environment, including several techniques that we have explained in the visual. These steps emphasize building a solid foundation for understanding and solving mathematical problems.

Challenges Due to Teaching

Students face diverse study challenges due to teaching methodologies. One of them is the lack of effective communication between educators and students. If teachers fail to articulate concepts clearly, students may struggle to grasp subject materials. Inadequate teacher-student interaction can hinder the resolution of doubts or concept clarification. Outdated teaching methods fail to engage students, resulting in a lack of interest and motivation to study. Effective teaching involves adapting to the diverse needs of students. It fosters a positive and interactive study environment for a better learning experience.

Inconsistency in Practice:

Inconsistency in practice poses a significant challenge in various contexts, ranging from skill development to routine tasks. Irregular practice can impede progress. Inconsistency in practicing new skills or habits can hinder the cultivation of expertise or the formation of positive routines. The key to overcoming this challenge is establishing and adhering to a consistent, structured practice regimen. By doing so, individuals can enhance their learning, skill acquisition, and personal growth, fostering a more sustainable and effective path toward mastery and success.

Quick Tips for Improved Math Skills

Improve your math skills with these quick tips: practice regularly, review and understand basic concepts, seek help when needed, use online resources, and solve various problem types. Stay organized, work on time management, and maintain a positive attitude towards learning. Consistency and persistence are key to mastering math.

Enhance your arithmetic abilities and speed up problem-solving with these additional tips:

Master the Basics:

Proficiency in fundamental operations is crucial for tackling more advanced topics. It ensures a solid foundation by mastering the core principles of arithmetic.

Understand Concepts, Don't Memorize:

Focus on understanding the underlying concepts rather than memorizing procedures. It enables flexible problem-solving and a deeper grasp of mathematical principles.

Use Online Resources:

Explore online platforms offering interactive math exercises and tutorials. Math assignment help websites provide valuable resources for offering academic assistance.

Break Down Complex Problems:

Break down complex problems into smaller, manageable steps. This approach makes problem-solving more approachable and helps prevent feeling overwhelmed.

Teach Others:

Teaching a concept to someone else is an effective way to solidify your understanding. This process forces you to articulate ideas and reinforces your knowledge.

Seek Help When Needed:

Don't hesitate to ask for help when you encounter difficulties. Teachers, classmates, or online communities can provide valuable insights and clarification.

Review and Reflect:

Regularly review previously learned material and reflect on your progress. Identify areas of weakness and focus on reinforcing those concepts.

Utilize Educational Apps and Games:

Explore educational apps and games that make learning math enjoyable. These approaches can add an element of fun to your practice sessions.

Assignments are integral to academic growth, and mastering math is a journey that requires dedication and effective time management. Seek assistance when needed, and you'll develop strong math skills with consistent practice.

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Math Worksheets

Test your math skills! Ace that test! See how far you can get! You can view them on-screen, and then print them, with or without answers.

Every worksheet has thousands of variations, so you need never run out of practice material.

Choose your Subject !

* Note: the worksheet variation number is not printed with the worksheet on purpose so others cannot simply look up the answers. If you want the answers, either bookmark the worksheet or print the answers straight away.

Also! Our forum members have put together a collection of Math Exercises .

Free Math Printable Worksheets with Answer Keys and Activities

Other free resources.

Feel free to download and enjoy these free worksheets on functions and relations. Each one has model problems worked out step by step, practice problems, as well as challenge questions at the sheets end. Plus each one comes with an answer key.

• Long Division with Remainders
• Long Division with Remainders #2 (Zeros in the Quotient)
• Long Division with 2 Digit Divisors
• Whole Number by Unit Fraction
• Equation of Circle
• Simplify Imaginary Numbers
• Adding and Subtracting Complex Numbers
• Multiplying Complex Numbers
• Dividing Complex Numbers
• Dividing Complex Number (Advanced)
• End of Unit, Review Sheet
• Distance Formula
• Simplify Rational Exponents (Algebra 2)
• Solve Equations with Rational Exponents (Algebra 2)
• Solve Equations with variables in Exponents (Algebra 2)
• Exponential Growth (no answer key on this one, sorry)
• Compound Interest Worksheet #1 (No logs)
• Compound Interest Worksheet (Logarithms required)
• Factor Trinomials Worksheet
• Factor by Grouping
• Domain and Range (Algebra 1)
• Functions vs Relations (Distinguish function from relation, state domain etc..) (Algebra 2)
• Evaluating Functions (Algebra 2)
• 1 to 1 Functions (Algebra 2)
• Composition of Functions (Algebra 2)
• Inverse Functions Worksheet (Algebra 2)
• Operations with Functions (Algebra 2)
• Functions Review Worksheet (Algebra 2)
• Logarithmic Equations
• Properties of Logarithms Worksheet
• Product Rule of Logarithms
• Power Rule of Logarithms
• Quotient Rule of Logarithms
• Solve Quadratic Equations by Factoring
• Quadratic Formula Worksheets (3 different sheets)
• Quadratic Formula Worksheet (Real solutions)
• Quadratic Formula (Complex solutions)
• Quadratic Formula (Both real and complex solutions)
• Discriminant and Nature of the Roots
• Solve Quadratic Equations by Completing the Square
• Sum and Product of Roots
• Radical Equations
• Mixed Problems on Writing Equations of Lines
• Slope Intercept Form Worksheet
• Standard Form Worksheet
• Point Slope Worksheet
• Write Equation of Line from the Slope and 1 Point
• Write Equation of Line From Two Points
• Equation of Line Parallel to Another Line and Through a Point
• Equation of Line Perpendicular to Another Line and Through a Point
• Slope of a Line
• Perpendicular Bisector of Segment
• Write Equation of Line Mixed Review
• Word Problems
• Multiplying Monomials Worksheet
• Multiplying and Dividing Monomials Sheet
• Adding and Subtracting Polynomials worksheet
• Multiplying Monomials with Polynomials Worksheet
• Multiplying Binomials Worksheet
• Multiplying Polynomials
• Simplifying Polynomials
• Factoring Trinomials
• Operations with Polynomials Worksheet
• Dividing Radicals
• Simplify Radicals Worksheet
• Adding Radicals
• Multiplying Radicals Worksheet
• Radicals Review (Mixed review worksheet on radicals and square roots)
• Rationalizing the Denominator (Algebra 2)
• Radical Equations (Algebra 2)
• Solve Systems of Equations Graphically
• Solve Systems of Equations by Elimination
• Solve by Substitution
• Solve Systems of Equations (Mixed Review)
• Activity on Systems of Equations (Create an advertisement for your favorite method to Solve Systems of Equations )
• Real World Connections (Compare cell phone plans)
• Identifying Fractions

Trigonomnetry

• Law of Sines and Cosines Worksheet (This sheet is a summative worksheet that focuses on deciding when to use the law of sines or cosines as well as on using both formulas to solve for a single triangle's side or angle)
• Law of Sines
• Ambiguous Case of the Law of Sines
• Law of Cosines
• Vector Worksheet
• Sine, Cosine, Tangent, to Find Side Length
• Sine, Cosine, Tangent Chart
• Inverse Trig Functions
• Real World Applications of SOHCATOA
• Mixed Review
• Unit Circle Worksheet
• Graphing Sine and Cosine Worksheet
• Sine Cosine Graphs with Vertical Translations
• Sine, Cosine, Tangent Graphs with Phase Shifts
• Sine, Cosine, Tangent Graphs with Change in Period, Amplitude and Phase Shifts (All Translations)
• Tangent Equation, Graph Worksheet
• Graphing Sine, Cosine, Tangent with Change in Period
• Cumulative, Summative Worksheet on Periodic Trig Functions - period, amplitude, phase shift, radians, degrees,unit circle
• Ratio and Proportion
• Similar Polygons
• Area of Triangle
• Interior Angles of Polygons
• Exterior Angles of Polygons

• Identifying Fractions Worksheet
• Associated Powerpoint
• Simplify Fractions Worksheet (Regular Difficulty)
• Associated PowerPoint
• Simplify Fractions Worksheet (Challenging Difficulty level for advanced learners)
• System of Linear Equations Worksheet
• System of Linear Equations - Real World Application
• Compositions of Reflections. Reflections Over Intersecting Lines as Rotations

All of these worksheets and activities are available for free so long as they are used solely for educational, noncommercial purposes and are not distributed outside of a specific teacher's classroom.

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Foundation year

Above satisfactory

Satisfactory

Below satisfactory

The Ultimate Guide to Writing a Problem-Solving and Modelling Task (PSMT) Report for General Maths

Unfamiliar with what the problem-solving and modelling task (PSMT) for QCAA General Maths IA1 involves? Or maybe you have written up a report for a PSMT before, but you just need a refresher!

Whatever the case is, we’re here to help. You mainly do exams for General Maths, so this is pretty much the only IA that is an assignment .

Although there is no telling what the topics of choice will be, the structure of the report , as well as various hints and tricks regarding research and format, will still apply. So we’ll go over some important considerations, as well as an overall barebones structure breakdown to make sure you maximise your marks!

Let’s dive in!

What is a Maths PSMT? Formulate Solve Evaluate and Verify Conclusion Appendix

What is a data analysis report for the QCAA General Maths IA1 PSMT?

A data analysis report explores or challenges a claim, aiming to prove it or disprove it by analysing datasets that are different from the ones originally used to arrive at the original theory .

In other words, your task is to find out whether a claim is true or false by collecting data, that you will analyse yourself. Your mathematical conclusions will either go with, or go against, the claim you’ve been asked to investigate.

Your mandate is to take the question given on the ISMG , and answer it clearly by the end, using maths and data.

It’s important to note that you will be given a page limit to stick to, rather than a word limit: this will be of 10 pages for the actual report, as well as an appendix (a space for tables and graphs) and a reference page .

Note: This means you should only include the graphs and tables that are essential to your explanation in the main body, not any superfluous ones. These will take up space and take away from your ability to use more words to get your ideas across.

Note that even though this assessment is marked internally, the QCAA does not allow your teachers to give specific individual feedback on the assignment!

You can easily reach out to resources like Art of Smart or a tutor to get specific PSMT feedback !

A Note About Referencing for the Maths PSMT

The referencing style will be specified by your teacher, but whenever writing about datasets obtained by someone else, or when citing concepts that do not come from your own understanding, you should always reference in-text .

The bottom line is, any reference you include in your reference list should be referenced at least once in the main body of the report for the Maths PSMT.

Aim for about ten references: these may be full datasets, explanations regarding phenomena that your data can’t explain, or opinions regarding the claim — potentially anything can be used as a source if it serves to support your analysis!

Here’s a preview of our General Maths PSMT sample response:

Download our QCAA General Maths IA1 PSMT Sample Response now!

Find out what PSMT mark you need to be on track for an ATAR above 90 …

Introduction and Context

Your introduction should discuss the general topic and issue at hand, then arrive at the claim in question . Essentially, the aim is to start general and get progressively more specific until we state the claim — take a look at this example:

The effects of climate change, as a result of mankind’s actions, are at the core of today’s political question. The following report addresses only a small fraction of the data gathered in recent years regarding climate change, specifically focusing on Canada as its subject. The stimulus report issued by the Canadian government, CCCR-Canada, introduces a statement regarding Canada’s average climate warming rate, said to be twice that of the rest of the world: “…[land surface] average temperature in Canada has increased at roughly double the global mean rate, with Canada’s mean annual temperature having risen about 1.7ºC (likely range 1.1ºC –2.3ºC) over the 1948– 2016 period,” (Xuebin Zhang, 2019). The claim was subsequently paraphrased by a further stimulus, a BBC article, reporting much the same conclusion: “Canada is warming on average at a rate twice as fast as the rest of the world, a new scientific report indicates,” (BBC News, 2019). 

Next is our research aim. By the time you’ll have started writing the report, the data will have been analysed, so you’ll know whether your data demonstrates or challenges the claim.

It is time to put this into words; have a look this example for reference:

The report tentatively foresees the rate of Canada’s warming being double that of t. Thus, it would deem the statement true. It also foresees an increasing warming rate in the last 10 to 20-year period, with this being thrice or four times higher than that of the previous years. 

Investigation Parameters for the QCAA General Maths IA1 PSMT

This section sets out the parameters of your investigation, or the criteria with which you have selected your dataset. This is so that your analysis appears less superficial, and more realistic.

Let’s suppose that our task is to investigate whether it is true that islands around the world are disappearing at a mean rate of 5mm per year due to sea level rise. Here we have three parameters to consider : first, we must define the term ‘island’ , as we couldn’t possibly take into account data from every single bit of land above sea level, as the variance in Island size is huge. Second, you are not going to be able to accurately obtain and analyse data from islands in every corner of the globe ; some of these places are hard to reach and constantly beaten by storms, so you won’t find data for every island you intend to include in your data. And finally, we must decide at what point in time we collect the data. The parameters section is there for you to explain that, for all the reasons listed above, your best bet at tackling this question is to pick 2 or 3 big islands from the southern and northern hemisphere, gather data across the last 10 years in terms of sea level rise, and compare the results.

Ultimately, the research parameters will be unique to the assessment question.

But remember, you don’t have a UN-Class team of researchers, millions in budget and a lifetime at your disposal. Be realistic regarding the data you can analyse!

Doing this PSMT at the start of Year 11? Here’s why you might be feeling that Year 11 is so hard …

Observations and Assumptions

Observations .

These are the things you may have come across while researching (so you’ll have a few in-text references here). It can be effective to use diagrams and statistics . Aim for at least four. 

Examples: The density of the materials used can heavily impact the weight and durability of the pendant Pendants sizes average between 9.5-25.4mm  Fashion pendants on average cost between $10-$99 (Lovisa, 2020) All measurements must be positive values, so when graphing, pendant must entirely be in the first quadrant to avoid negative values 

Assumptions 

As the task can be somewhat vague, some assumptions will have to be made in order to make it more specific . Aim for at least four.

Ensure you justify the reasons for/implications of each assumption. 

Examples: The pendant is for the average adult consumer, and will be considered ‘fine jewellery’ and so will have dimensions, weight and cost as such  The entire area of the pendant will have the same depth (meaning that the pendant will be flat), as creating variety in depth will make the volume of the pendant difficult to calculate  The domains and ranges of the functions will correlate to the dimensions of the pendant (i.e. 1 unit = 1cm) The cost calculated is the estimated cost for the materials of the pendant, not including the construction cost

This section is also heavily dependent on the claim you’re investigating — but that doesn’t mean there is no set way to go about it. We’ve got 6 great tips you can follow to ace the analysis of your PSMT for General Maths!

Tip 1: Remove Outliers from Your Data

Before you begin the analysis, remove any outliers from your data, or alternatively, explore the effect that outliers have when removed from the dataset compared to when they are included.

Tip 2: Focus on the Bigger Picture

Next, make sure to focus on the bigger picture. Pay attention to general trends rather than isolated fluctuations in the data , and comment on their correlation.

A good way to make sure this is executed is to comment always on the correlation coefficient and the coefficient of determination . These will be the main indications as to whether two variables are associated with each other.

Tip 3: Don’t State if Something is True or False

Always keep away from stating something is true or false. You’re never going to have the full picture or the full data, or the time to meticulously analyse it, which leads us to the next point…

Refrain from using ‘absolute’ language… instead, say something like “the data suggests xyz to be correlated to abc…”

Tip 4: Pay Attention to Your Data Sources

Who are they affiliated to? Are they credible? Reputable? You’ll get to comment on all of these factors in the limitations section of your assignment.

Tip 5: Remember That Correlation Doesn’t Always Equal Causation

Correlation does not necessarily mean causation. Two variables may be correlated, but not directly affect each other.

Use this as an opportunity to discuss the ‘Three Cs’ : Coincidence, Common Response, & Confounding!

Tip 6: Compare the Data!

The most important of all of them. COMPARE. It might be worth purposefully getting data that challenges yours.

Commenting on their similarities and differences will maximise your marks. Comparison of different datasets to the claim is essential to score high on this assessment.

There is no set way to divide your analysis. We suggest dividing the analysis into smaller sections, each of which deals with a specific aspect of the dataset, or different datasets together. This structure will ensure you achieve the best possible marks. Have a look at this marking guide from QCAA :

Displaying Results

This section is a breeze to write. Just summarise what you’ve found in your analysis compared to the claim!

Again, remember to focus on general trends, and state their plausibility with regards to your hypothesis. Do they mostly support your hypothesis or challenge it?

Download our General Maths PSMT Sample Response now!

Evaluate and verify, reasonableness .

Discuss which aspects of your design/model are reasonable and which aspects are unreasonable and provide justification. Explain how the unreasonable aspects could be improved.

Include aspects that were not taken into account and may have an effect on the design/model. Also, revisit the assumptions and consider any revisions that could be made to improve your design/model. 

Stuck doing your actual IA1 PSMT at the end of Year 11 ? Here are some helpful tips for you to use… But if you haven’t done your IA1s yet, we’ve got everything you need to know in our QCE IA1 guide !

Strengths of the Model

What are the strengths of your model? Part of the evaluation of your work is to identify the strengths of the analysis . This can be anything from the accuracy of the data to the method and parameters you chose to set yourself when analysing. 

Limitations of the Model

This section deals with the possible disadvantages of the analysis you’ve made. In order to evaluate your work effectively, ask yourself in what ways your work could be criticised, and what you wish you could have had to make your analysis more accurate or attainable. 

It’s good in this section to write about the sources you used in your investigation, as well as the source from which the original claim was taken ; discuss any bias or advantages that the sources might have in  terms of strength and reliability. Here is the marking guide for more help:

Evaluation of the Claim

Think of this as a summary of all your results, written in relation to the claim . The key questions you should be answering here are:

• What part of my results supports/challenges the claim?
• What are the limitations that could have improved my accuracy?
• Is the data analysed enough to get to a valid conclusion? Why?

Group the deductions you have made in the evaluation of the claim in a few sentences. It is unlikely you will be able to completely prove or disprove your research statement. There just isn’t enough data if the claim is complex.

What you can say is that the claim is either plausible or implausible considering your research . Refer to this example and the earlier marking criteria.

It is thus evident that the results across the period of time examined (if considered in its entirety) corroborate the claim forwarded by the CCCR Canada Report. However, the claim can be challenged if examining the results that comprise the last fifteen to twenty years, where Canada’s warming rate is much lower than that of the compared subjects.

In the main body of the report you will have included the few graphs and tables that showcase the most important trends of your data. The appendix, however, is the place where you must show your mathematical process. It should include:

• Datasets (if these are not too large)
• Any images you’ve referred to
• Operations and calculations
• Excel formulas (if this is the software you’re using to analyse the data)

We get that this assessment is confusing ! Check out our maths experts in Brisbane  for more help!

When you receive your results for this task, you should plug them into our QCE Cohort Comparison Tool to see how you’ve done compared to your cohort.

There you have it!

Now that you know exactly what to include in your QCAA General Maths PSMT report, it’s time to ace the assignment! Always remember to refer back to the marking guide to ensure that you’re doing exactly what the assessment is asking of you.

If you’re looking for other resources to help you study for QCAA General Maths, check out our practice questions and guides below:

• Unit 3 General Maths IA2 Short Answer Practice Questions
• Unit 4 General Maths IA3 Short Answer Practice Questions
• The Ultimate Guide to QCAA General Maths Unit 3: Bivariate Data, Sequences and Change, and Earth Geometry
• The Ultimate Guide to QCAA General Maths Unit 4: Investing and Networking
• General Maths Multiple Choice Practice Questions for Unit 3 & 4 External Assessment
• What IA3 mark you need to get an ATAR 90+

If you’re studying Maths Methods check out our guide to writing a PSMT here!

Are you looking for some extra guidance with the QCAA General Maths IA1 PSMT?

We have an incredible team of qce tutors and mentors.

Still looking for help? One of our Brisbane-based Maths tutors can offer one-on-one tutoring sessions to help you ACE your PSMT!

We can help you master the General Maths syllabus and ace your upcoming General Maths assessments with personalised lessons conducted one-on-one in your home or online!

We’ve supported over  8,000 students over the last 11 years , and on average our students score mark improvements of over 20%!

To find out more and get started with an inspirational QCE tutor and mentor,   get in touch today  or give us a ring on  1300 267 888!

Vittorio Manessi  is an Art of Smart tutor based in Queensland studying environmental science. He was one of the first Year 12 students to study under the new ATAR system in Queensland. He enjoys Maths, Science, English and Ancient History and is keen to share his knowledge of the QCE by making awesome resources.

• Topics: 🔢 Mathematics , ✍️ Learn

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