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Designing Assignments for Learning

The rapid shift to remote teaching and learning meant that many instructors reimagined their assessment practices. Whether adapting existing assignments or creatively designing new opportunities for their students to learn, instructors focused on helping students make meaning and demonstrate their learning outside of the traditional, face-to-face classroom setting. This resource distills the elements of assignment design that are important to carry forward as we continue to seek better ways of assessing learning and build on our innovative assignment designs.

On this page:

Rethinking traditional tests, quizzes, and exams.

  • Examples from the Columbia University Classroom
  • Tips for Designing Assignments for Learning

Reflect On Your Assignment Design

Connect with the ctl.

  • Resources and References

Cite this resource: Columbia Center for Teaching and Learning (2021). Designing Assignments for Learning. Columbia University. Retrieved [today’s date] from https://ctl.columbia.edu/resources-and-technology/teaching-with-technology/teaching-online/designing-assignments/

assignment design for maths

Traditional assessments tend to reveal whether students can recognize, recall, or replicate what was learned out of context, and tend to focus on students providing correct responses (Wiggins, 1990). In contrast, authentic assignments, which are course assessments, engage students in higher order thinking, as they grapple with real or simulated challenges that help them prepare for their professional lives, and draw on the course knowledge learned and the skills acquired to create justifiable answers, performances or products (Wiggins, 1990). An authentic assessment provides opportunities for students to practice, consult resources, learn from feedback, and refine their performances and products accordingly (Wiggins 1990, 1998, 2014). 

Authentic assignments ask students to “do” the subject with an audience in mind and apply their learning in a new situation. Examples of authentic assignments include asking students to: 

  • Write for a real audience (e.g., a memo, a policy brief, letter to the editor, a grant proposal, reports, building a website) and/or publication;
  • Solve problem sets that have real world application; 
  • Design projects that address a real world problem; 
  • Engage in a community-partnered research project;
  • Create an exhibit, performance, or conference presentation ;
  • Compile and reflect on their work through a portfolio/e-portfolio.

Noteworthy elements of authentic designs are that instructors scaffold the assignment, and play an active role in preparing students for the tasks assigned, while students are intentionally asked to reflect on the process and product of their work thus building their metacognitive skills (Herrington and Oliver, 2000; Ashford-Rowe, Herrington and Brown, 2013; Frey, Schmitt, and Allen, 2012). 

It’s worth noting here that authentic assessments can initially be time consuming to design, implement, and grade. They are critiqued for being challenging to use across course contexts and for grading reliability issues (Maclellan, 2004). Despite these challenges, authentic assessments are recognized as beneficial to student learning (Svinicki, 2004) as they are learner-centered (Weimer, 2013), promote academic integrity (McLaughlin, L. and Ricevuto, 2021; Sotiriadou et al., 2019; Schroeder, 2021) and motivate students to learn (Ambrose et al., 2010). The Columbia Center for Teaching and Learning is always available to consult with faculty who are considering authentic assessment designs and to discuss challenges and affordances.   

Examples from the Columbia University Classroom 

Columbia instructors have experimented with alternative ways of assessing student learning from oral exams to technology-enhanced assignments. Below are a few examples of authentic assignments in various teaching contexts across Columbia University. 

  • E-portfolios: Statia Cook shares her experiences with an ePorfolio assignment in her co-taught Frontiers of Science course (a submission to the Voices of Hybrid and Online Teaching and Learning initiative); CUIMC use of ePortfolios ;
  • Case studies: Columbia instructors have engaged their students in authentic ways through case studies drawing on the Case Consortium at Columbia University. Read and watch a faculty spotlight to learn how Professor Mary Ann Price uses the case method to place pre-med students in real-life scenarios;
  • Simulations: students at CUIMC engage in simulations to develop their professional skills in The Mary & Michael Jaharis Simulation Center in the Vagelos College of Physicians and Surgeons and the Helene Fuld Health Trust Simulation Center in the Columbia School of Nursing; 
  • Experiential learning: instructors have drawn on New York City as a learning laboratory such as Barnard’s NYC as Lab webpage which highlights courses that engage students in NYC;
  • Design projects that address real world problems: Yevgeniy Yesilevskiy on the Engineering design projects completed using lab kits during remote learning. Watch Dr. Yesilevskiy talk about his teaching and read the Columbia News article . 
  • Writing assignments: Lia Marshall and her teaching associate Aparna Balasundaram reflect on their “non-disposable or renewable assignments” to prepare social work students for their professional lives as they write for a real audience; and Hannah Weaver spoke about a sandbox assignment used in her Core Literature Humanities course at the 2021 Celebration of Teaching and Learning Symposium . Watch Dr. Weaver share her experiences.  

​Tips for Designing Assignments for Learning

While designing an effective authentic assignment may seem like a daunting task, the following tips can be used as a starting point. See the Resources section for frameworks and tools that may be useful in this effort.  

Align the assignment with your course learning objectives 

Identify the kind of thinking that is important in your course, the knowledge students will apply, and the skills they will practice using through the assignment. What kind of thinking will students be asked to do for the assignment? What will students learn by completing this assignment? How will the assignment help students achieve the desired course learning outcomes? For more information on course learning objectives, see the CTL’s Course Design Essentials self-paced course and watch the video on Articulating Learning Objectives .  

Identify an authentic meaning-making task

For meaning-making to occur, students need to understand the relevance of the assignment to the course and beyond (Ambrose et al., 2010). To Bean (2011) a “meaning-making” or “meaning-constructing” task has two dimensions: 1) it presents students with an authentic disciplinary problem or asks students to formulate their own problems, both of which engage them in active critical thinking, and 2) the problem is placed in “a context that gives students a role or purpose, a targeted audience, and a genre.” (Bean, 2011: 97-98). 

An authentic task gives students a realistic challenge to grapple with, a role to take on that allows them to “rehearse for the complex ambiguities” of life, provides resources and supports to draw on, and requires students to justify their work and the process they used to inform their solution (Wiggins, 1990). Note that if students find an assignment interesting or relevant, they will see value in completing it. 

Consider the kind of activities in the real world that use the knowledge and skills that are the focus of your course. How is this knowledge and these skills applied to answer real-world questions to solve real-world problems? (Herrington et al., 2010: 22). What do professionals or academics in your discipline do on a regular basis? What does it mean to think like a biologist, statistician, historian, social scientist? How might your assignment ask students to draw on current events, issues, or problems that relate to the course and are of interest to them? How might your assignment tap into student motivation and engage them in the kinds of thinking they can apply to better understand the world around them? (Ambrose et al., 2010). 

Determine the evaluation criteria and create a rubric

To ensure equitable and consistent grading of assignments across students, make transparent the criteria you will use to evaluate student work. The criteria should focus on the knowledge and skills that are central to the assignment. Build on the criteria identified, create a rubric that makes explicit the expectations of deliverables and share this rubric with your students so they can use it as they work on the assignment. For more information on rubrics, see the CTL’s resource Incorporating Rubrics into Your Grading and Feedback Practices , and explore the Association of American Colleges & Universities VALUE Rubrics (Valid Assessment of Learning in Undergraduate Education). 

Build in metacognition

Ask students to reflect on what and how they learned from the assignment. Help students uncover personal relevance of the assignment, find intrinsic value in their work, and deepen their motivation by asking them to reflect on their process and their assignment deliverable. Sample prompts might include: what did you learn from this assignment? How might you draw on the knowledge and skills you used on this assignment in the future? See Ambrose et al., 2010 for more strategies that support motivation and the CTL’s resource on Metacognition ). 

Provide students with opportunities to practice

Design your assignment to be a learning experience and prepare students for success on the assignment. If students can reasonably expect to be successful on an assignment when they put in the required effort ,with the support and guidance of the instructor, they are more likely to engage in the behaviors necessary for learning (Ambrose et al., 2010). Ensure student success by actively teaching the knowledge and skills of the course (e.g., how to problem solve, how to write for a particular audience), modeling the desired thinking, and creating learning activities that build up to a graded assignment. Provide opportunities for students to practice using the knowledge and skills they will need for the assignment, whether through low-stakes in-class activities or homework activities that include opportunities to receive and incorporate formative feedback. For more information on providing feedback, see the CTL resource Feedback for Learning . 

Communicate about the assignment 

Share the purpose, task, audience, expectations, and criteria for the assignment. Students may have expectations about assessments and how they will be graded that is informed by their prior experiences completing high-stakes assessments, so be transparent. Tell your students why you are asking them to do this assignment, what skills they will be using, how it aligns with the course learning outcomes, and why it is relevant to their learning and their professional lives (i.e., how practitioners / professionals use the knowledge and skills in your course in real world contexts and for what purposes). Finally, verify that students understand what they need to do to complete the assignment. This can be done by asking students to respond to poll questions about different parts of the assignment, a “scavenger hunt” of the assignment instructions–giving students questions to answer about the assignment and having them work in small groups to answer the questions, or by having students share back what they think is expected of them.

Plan to iterate and to keep the focus on learning 

Draw on multiple sources of data to help make decisions about what changes are needed to the assignment, the assignment instructions, and/or rubric to ensure that it contributes to student learning. Explore assignment performance data. As Deandra Little reminds us: “a really good assignment, which is a really good assessment, also teaches you something or tells the instructor something. As much as it tells you what students are learning, it’s also telling you what they aren’t learning.” ( Teaching in Higher Ed podcast episode 337 ). Assignment bottlenecks–where students get stuck or struggle–can be good indicators that students need further support or opportunities to practice prior to completing an assignment. This awareness can inform teaching decisions. 

Triangulate the performance data by collecting student feedback, and noting your own reflections about what worked well and what did not. Revise the assignment instructions, rubric, and teaching practices accordingly. Consider how you might better align your assignment with your course objectives and/or provide more opportunities for students to practice using the knowledge and skills that they will rely on for the assignment. Additionally, keep in mind societal, disciplinary, and technological changes as you tweak your assignments for future use. 

Now is a great time to reflect on your practices and experiences with assignment design and think critically about your approach. Take a closer look at an existing assignment. Questions to consider include: What is this assignment meant to do? What purpose does it serve? Why do you ask students to do this assignment? How are they prepared to complete the assignment? Does the assignment assess the kind of learning that you really want? What would help students learn from this assignment? 

Using the tips in the previous section: How can the assignment be tweaked to be more authentic and meaningful to students? 

As you plan forward for post-pandemic teaching and reflect on your practices and reimagine your course design, you may find the following CTL resources helpful: Reflecting On Your Experiences with Remote Teaching , Transition to In-Person Teaching , and Course Design Support .

The Columbia Center for Teaching and Learning (CTL) is here to help!

For assistance with assignment design, rubric design, or any other teaching and learning need, please request a consultation by emailing [email protected]

Transparency in Learning and Teaching (TILT) framework for assignments. The TILT Examples and Resources page ( https://tilthighered.com/tiltexamplesandresources ) includes example assignments from across disciplines, as well as a transparent assignment template and a checklist for designing transparent assignments . Each emphasizes the importance of articulating to students the purpose of the assignment or activity, the what and how of the task, and specifying the criteria that will be used to assess students. 

Association of American Colleges & Universities (AAC&U) offers VALUE ADD (Assignment Design and Diagnostic) tools ( https://www.aacu.org/value-add-tools ) to help with the creation of clear and effective assignments that align with the desired learning outcomes and associated VALUE rubrics (Valid Assessment of Learning in Undergraduate Education). VALUE ADD encourages instructors to explicitly state assignment information such as the purpose of the assignment, what skills students will be using, how it aligns with course learning outcomes, the assignment type, the audience and context for the assignment, clear evaluation criteria, desired formatting, and expectations for completion whether individual or in a group.

Villarroel et al. (2017) propose a blueprint for building authentic assessments which includes four steps: 1) consider the workplace context, 2) design the authentic assessment; 3) learn and apply standards for judgement; and 4) give feedback. 

References 

Ambrose, S. A., Bridges, M. W., & DiPietro, M. (2010). Chapter 3: What Factors Motivate Students to Learn? In How Learning Works: Seven Research-Based Principles for Smart Teaching . Jossey-Bass. 

Ashford-Rowe, K., Herrington, J., and Brown, C. (2013). Establishing the critical elements that determine authentic assessment. Assessment & Evaluation in Higher Education. 39(2), 205-222, http://dx.doi.org/10.1080/02602938.2013.819566 .  

Bean, J.C. (2011). Engaging Ideas: The Professor’s Guide to Integrating Writing, Critical Thinking, and Active Learning in the Classroom . Second Edition. Jossey-Bass. 

Frey, B. B, Schmitt, V. L., and Allen, J. P. (2012). Defining Authentic Classroom Assessment. Practical Assessment, Research, and Evaluation. 17(2). DOI: https://doi.org/10.7275/sxbs-0829  

Herrington, J., Reeves, T. C., and Oliver, R. (2010). A Guide to Authentic e-Learning . Routledge. 

Herrington, J. and Oliver, R. (2000). An instructional design framework for authentic learning environments. Educational Technology Research and Development, 48(3), 23-48. 

Litchfield, B. C. and Dempsey, J. V. (2015). Authentic Assessment of Knowledge, Skills, and Attitudes. New Directions for Teaching and Learning. 142 (Summer 2015), 65-80. 

Maclellan, E. (2004). How convincing is alternative assessment for use in higher education. Assessment & Evaluation in Higher Education. 29(3), June 2004. DOI: 10.1080/0260293042000188267

McLaughlin, L. and Ricevuto, J. (2021). Assessments in a Virtual Environment: You Won’t Need that Lockdown Browser! Faculty Focus. June 2, 2021. 

Mueller, J. (2005). The Authentic Assessment Toolbox: Enhancing Student Learning through Online Faculty Development . MERLOT Journal of Online Learning and Teaching. 1(1). July 2005. Mueller’s Authentic Assessment Toolbox is available online. 

Schroeder, R. (2021). Vaccinate Against Cheating With Authentic Assessment . Inside Higher Ed. (February 26, 2021).  

Sotiriadou, P., Logan, D., Daly, A., and Guest, R. (2019). The role of authentic assessment to preserve academic integrity and promote skills development and employability. Studies in Higher Education. 45(111), 2132-2148. https://doi.org/10.1080/03075079.2019.1582015    

Stachowiak, B. (Host). (November 25, 2020). Authentic Assignments with Deandra Little. (Episode 337). In Teaching in Higher Ed . https://teachinginhighered.com/podcast/authentic-assignments/  

Svinicki, M. D. (2004). Authentic Assessment: Testing in Reality. New Directions for Teaching and Learning. 100 (Winter 2004): 23-29. 

Villarroel, V., Bloxham, S, Bruna, D., Bruna, C., and Herrera-Seda, C. (2017). Authentic assessment: creating a blueprint for course design. Assessment & Evaluation in Higher Education. 43(5), 840-854. https://doi.org/10.1080/02602938.2017.1412396    

Weimer, M. (2013). Learner-Centered Teaching: Five Key Changes to Practice . Second Edition. San Francisco: Jossey-Bass. 

Wiggins, G. (2014). Authenticity in assessment, (re-)defined and explained. Retrieved from https://grantwiggins.wordpress.com/2014/01/26/authenticity-in-assessment-re-defined-and-explained/

Wiggins, G. (1998). Teaching to the (Authentic) Test. Educational Leadership . April 1989. 41-47. 

Wiggins, Grant (1990). The Case for Authentic Assessment . Practical Assessment, Research & Evaluation , 2(2). 

Wondering how AI tools might play a role in your course assignments?

See the CTL’s resource “Considerations for AI Tools in the Classroom.”

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Teaching Commons > Teaching Guides > Assignment Design

Assignment Design

Strategies

Here's a short list of some general assignment design strategies that apply to a wide variety of disciplines.

Aligning with Learning Goals

A number of strategies for deterring plagiarism are discussed, including asking your students to write about current topics relevant to your course and staging essay assignments throughout the quarter.

Integrative Learning

​Integrative learning occurs when students make connections among ideas and experiences in order to transfer learning to new contexts.​

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  • Maths Project

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Learn Maths project with Vedantu

Mathematics is a word that most of the students in their early grade until then and now.

The early graders normally fear the word "Mathematics". But in reality, maths is a subject full of logic and proof.

This is no longer primary to teach the students in the 21st century. Including Mathematics Project in the upper primary stage aids students in picturing the fundamental concepts, principles, theorems, and the underlying procedure occupied in solving them. Mathematics is a word that is feared by the largest of the students in their early grades.

Following are some of the advantages of Maths projects in our schools:

While traditional learning is not much suggested to solve the maths problems, it is recommended to the teachers and board to grant maths education in a more rational and challenging way through project works.

It has been seen that rote learning is not effective in the long term. Maths related projects work not only to help in improving the problem-solving ability but also will be able to learn it in a better way for their lifetime.

These projects help the students to improve their planning and critical thinking ability as they employ “habit of thinking and mind skills”.

Including this concept in the curriculum will also help improve the reasoning skills of the student.

Remember when you try to learn something relating to the real world you understand it better. Similarly, when you try to learn a concept with more examples and relate it to the real world the concept gets deeper into your mind and retains forever and ever. Henceforth, it is always good to do a project on the concept you have understood.

Project 1 

Probability:.

Probability is such a native part of your life that you rarely think about it. Though, every time you use a word like “might,” “may,” “undoubtedly,” “without fail,” or “maybe,” you can see and even a probability that an event will occur.

Scientists and great mathematicians like to express probability more accurately. For example, if you toss a coin in the air, the probability (P) that it will land heads or tails.

Materials Required:

A book and a pencil.

Four coins.

Procedure: 

Using a paper and pencil, draw circles with an “H” or a “T” in the centre of the paper to illustrate the different results when you toss these three coins.

Using the circles that you drew as mentioned above, express the following:

The probability of getting three heads while tossing the coins.

The probability of getting three tails while tossing the coins.

The probability of getting one head and three tails while tossing the coins.

The probability of getting one tail and three heads while tossing the coins.

Hint: There are eight distinctly different possibilities so make sure you haven’t left any of them out.

Try tossing three coins 16 times and writing down the outcomes. Are the probabilities roughly equal as you calculated in step 2? Try tossing three coins 24 times. Are the probabilities any closer?

Project 2 

Pythagoras theorem:.

This theorem states that the square on the hypotenuse of a right-angled triangle is equal in area to the sum of the squares on the other two sides.

(Image will be Uploaded Soon)

Colour papers

Ruler and sketches

Make a right angle Triangle, of 3cm, 4cm and 5cm as shown 3cm 5cm.

Make 3 square sheets of 3 * 3 cm, 4 * 4 cm, 5 * 5 cm.

Fix these square sheets to the sides of the triangle.

Make the square sheets into 3 * 3 cm such that 9 squares of equal length

Similarly, repeat the same thing with the remaining square sets. 4cm

By this work, we can prove the above-said theorem.

In the above work, we have considered a = 3cm, b = 4cm and c = 5cm.

Henceforth, 3 square + 4 square = 5 square. 

3 * 3 + 4 * 4 = 5 * 5 

9 + 16 = 25 

Hence the theorem is proved. 

Assessment: 

Geometry map project:.

Angles, Lines, and Triangles: Your assignment is to design a map that includes several distinct kinds of lines, angles, and triangles. Your map can be of a town, your neighbourhood or a made-up place of your own. It must, however, incorporate the following: 2 sets of streets that are parallel Two sets of streets that are perpendicular One street that intersects another street to form an obtuse angle.

Restaurant Menu Project: 

Create a menu for a themed restaurant. The student will write word problems using their menu. It can be a very fundamental element, just adding prices of different items, to more complex problems, like finding the cost of a meal for you and a friend plus tax or figuring the cost of the meal for grandma with her senior discount of 10%. The criteria for each of the following items in the menu are as follows: 4 different types of appetizers, 4 different types of beverages and 4 main dishes.

Compare Two Fractions:

Two proper fractions are given to you with different denominators (show how to write these fractions as decimal numbers and as a percentage. Compare both the fractions using an appropriate mathematical symbol.) Be sure to explain what you are thinking!

Supplementary and Complementary Angles:

Label and highlight the given angles from 6 different pictures. Select six of the following angles: acute angle, obtuse angle, straight, right angle, supplementary angles, and complementary angle. Then measure the acute angle, obtuse angle, supplementary angle, and complementary angles, and write their measures inside the interior of each angle.

Design Your Dream House: 

Draw a 2D version of how you want the front of your house to look like. Must include four windows, and two doors. All lines must be drawn with the help of a ruler and must be in centimetres. Students must then determine the perimeter and area of each window and the door. You must also do the same for the front of the house such that the perimeter and the area. Then you must convert these measurements to millimetres. Each dimension must be written in a typical drafting fashion. Houses must be uniformly coloured, and students may add additional features.

Mathematics Article: 

Draft an article about mathematics, which may include a mathematician. Write 2 important things you learned from the article. At least one-page minimum. Note, write the summary in your own words!

Maths Project assists pupils in developing their critical thinking and reasoning skills. Most schools provide exhibits for classes 5, 6, 7, 8, 9, and 10, where students may display their unique ideas and project work using mathematical models, allowing them to study the topic in a creative way.

There's an ancient adage that goes, "Tell me, and I'll forget; show me, and I'll remember; involve me, and I'll comprehend." Confucius, a Chinese philosopher, said

Today's youth expects an interactive learning paradigm that will keep them engaged and help them absorb the facts quickly. Previously, the primary instrument used by schools to impart instruction was:

writing, 

testing, 

This is no longer necessary to teach pupils in the twenty-first century. This is why the CBSE board of governors has approved a project-based training and learning programme for pupils. Incorporating a Maths project into the CBSE and ICSE curricula has aided millions of students in their professional pursuits.

Benefits of Maths Project Work:

Incorporating a Maths Project into the upper elementary stage assists children with conceptualising the fundamental ideas, theorems, and principles, as well as the underlying method required in solving them. Mathematics is a word that most pupils in their early grades despise. Some pupils utilise arithmetic techniques to solve problems, while others study hard. Maths is a topic that is full of reasoning and proof. The following are some of the benefits of maths projects in schools:

While rote learning is not suggested for solving arithmetic problems, instructors and the board are encouraged to deliver maths education in a more sensible and demanding manner through project work.

Role-learning or memorization is ineffective for long-term memory. Maths-related project work not only improves problem-solving abilities but also allows students to learn in a rewarding method that will last a lifetime.

Maths projects assist pupils to enhance their planning and critical thinking abilities by utilising "habit of thinking and cognitive skills." Incorporating maths projects into the curriculum will also assist students to enhance their thinking abilities.

Students are required to showcase what they have made for their projects in front of the class. This, in turn, assists students in improving their communication and presenting abilities, which will serve them well in their professional careers.

Keeping all of these benefits in mind for the Maths project will aid in producing a memorable learning experience for the pupils.

Maths Project Ideas

Maths Project enables students to put their ideas into practice and get real-world experience. It will assist students in developing interpersonal skills and cognitive ability, as well as increasing their degree of confidence in the topic. Let's look at several maths models here;

Real-world Mathematical Number System

Area and perimeter of various sorts of forms and shapes

Angle Types

Probability and statistics 

Algebraic Equations and Quadratic 

Pythagoras' Theorem

Conic Sections

These are the few general themes for which students can develop a model.

Class 10 Maths Model

For pupils in class 10, we've included some project ideas for maths that are related to their curriculum and can be simply reproduced.

Surface area and volume of a Cube and a Cuboid: 

Students in Grade 10 must be able to compute the surface areas and volumes of specified 3D objects such as a cube, a cuboid, and so on. Try to recognise the 3D forms around you and investigate how we can calculate their surface area and volume.

Areas of two comparable triangles are compared:

The notion of similar triangles is simple to memorise; however, understanding the link between the areas of two similar triangles using their sides is essential.

Finding the angles using trigonometric ratios: 

As we all know, trigonometry can be used to locate the missing sides or angles of a right triangle. More similar situations will be practised with the assistance of a maths project model on trigonometry. The use of trigonometric ratios will aid in determining the needed parameters.

The following are the mean values of the provided data:

In general, we deal with many numbers in our daily lives, and it is often necessary to know the average of these figures, such as the average time to complete a given activity. In this scenario, the idea of mean values is the most efficient way to get the desired outcome.

Probability of random experiments:  

This probability maths project aids in understanding different types of random experiments and determining the probabilities of occurrences related to them. For example, in the experiment of tossing a coin three times, receiving exactly two heads.

Model for 9th Grade Maths Project

Students in Class 9 can utilise these project ideas to construct maths models that are aligned with their curriculum. Try out these 9th grade Math project ideas and have fun while learning.

Numbers are represented in a number line.

In Coordinate Geometry, the Cartesian Plane is used.

Geometry Shapes Types of Triangles Euclid's Geometry Model.

Maths Project Ideas for Class 8 

Secondary students in Class 8 can create some of the greatest functional models based on the following topics:

Creating various types of quadrilaterals

Number line representation of rational numbers

Data grouping, organisation, and display using charts and graphs.

Profit and loss for commodities, as well as the discovery of simple interest

Having fun with numbers

Linear graphs (use matchsticks to represent)

3D Object Visualisation

7th Grade Maths Project

With the aid of practical models, students in Class 7 may easily grasp Mathematics and related principles. They can acquire several project ideas from here to make such models. These models will assist students in visualising topics and developing their confidence in any given area.

Here are the subjects on which students will base their projects.

Integer data types (positive and negative)

Fractional Forms (Proper and Improper fractions)

In two-dimensional space, what are lines and angles?

Triangle Shapes (Scalene, Isosceles and Equilateral)

Quantity Comparison

Imagining Solid Shapes

Maths Project for 6th Grade

When pupils advance from class 5th to class 6th, their educational level rises. They will be introduced to numerous new topics that they did not study in elementary school.

As a result, doing maths projects based on various topics and properly understanding them will be quite fascinating for children.

Knowing and comparing various figures

Whole-number patterns

Defining a Point, a Line, and an Angle (Basic geometry)

Line models of parallel and perpendicular lines

What exactly are decimals and fractions?

Matchstick Patterns in Algebra

Formal symmetry

Mathematical Working Models for Exhibits

Several mathematical projects may be made in school displays, such as:

Calculator: The calculator is constructed of cardboard and has four holes, with the first, third, and fourth holes containing moveable numbers and the second hole containing all the symbols depending on the operations performed: addition, subtraction, division, and multiplication. In this manner, we might construct a man-made calculator and assess the abilities of the exhibition's attendees.

Here are some more nice ideas for developing workable models based on mathematical concepts:

Construct a school project in which each building is represented by a distinct form. For example, the school's roof will be shaped like a triangle.

Create a model using LED lights that are based on trigonometric ratios (Sine, Cosine, and Tangent).

Make a "height and distance" model out of cardboard, paper, pulleys, threads, and other materials.

Using LED lights, match the following model for square and cube numbers.

A model that represents the centroid of several sorts of triangles (Acute, Obtuse and right triangle)

Make a model that depicts the many sections of a circle (radius, diameter, the centre of the circle, chord, sector, arc, etc.)

Geometrical forms in three dimensions

CBSE has included maths projects in its formative and summative classes, and the percentage of marks assigned to the projects ranges from 10% to 15%. The project might take the shape of high-level thinking skills questions or case studies. The class teacher may also reach the pupils through an open book assessment test at times.

The benefits underlie the strategy of accessing their children through maths projects and out-of-school learning. This assists students in making real-life decisions and acquiring mathematical information in a more holistic manner, which will benefit them in their career and professional lives.

We approach making students learn through interactive sessions, and films allow students to study in a dynamic manner.

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FAQs on Maths Project

1. How to get the list of Maths project models and examples?

Students can find a list of Maths projects based on different chapters on various online platforms. Vedantu is one such platform that provides a curated list of projects on different chapters for students. These projects help students to improve their critical thinking and planning skills and imbibe the habit of thinking. With the help of these projects, students can enrich their knowledge regarding the chapters and also get a real-time learning experience.

2. What are some of the Maths topics on which students can find projects on Vedantu’s site?

Following are some of the topics on which Maths projects are available on Vedantu:

Probability

Pythagoras Theorem

Geometry Map Project

Restaurant Menu Project

Compare Two Fractions

Design Your Dream House

Supplementary and Complementary Angles Project

Maths Article

Students can find such self-assessment based project topics on Vedantu’s site. These projects play a vital role in increasing their enthusiasm for learning and also guide them to explore the subject.

3. What is the restaurant menu project? How will this project help them?

In the Restaurant Menu Project, students will be asked to create a menu for a themed restaurant. This is an interesting project that will help students to solve complex mathematical problems. For example, students will be able to calculate the cost of a meal for two including tax or a meal plan for senior citizens with a 10% discount. This is a fun way to learn percentage problems and other arithmetic numerical-based problems with the menu card that they have created.

4. What value will Maths Project add to a student's academic growth?

Students must be asked to take up Maths projects and assignments based on the practical application of the subject. Such projects will improve their subject knowledge and will also hone their problem-solving and research skills. Mathematics is not just about solving textbook problems but applying those concepts to real-life situations. Such projects on topics like probability, profit-loss, percentage, etc. will make students aware of the subject matter in a more comprehensive manner. This will not only improve the subject knowledge of students but will also encourage them to pursue Mathematics during higher studies. Students can avail various fun Maths project topics on Vedantu.

5. What are the benefits of a Maths project?

Maths Project assists students with visualising fundamental concepts, theorems, and principles, as well as the underlying method required in solving them. Maths-related project work not only improves problem-solving abilities but also allows students to learn in a rewarding method that will last a lifetime. Maths projects assist pupils to enhance their planning and critical thinking abilities by utilising "habit of thinking and cognitive skills." Maths projects assist pupils to enhance their planning and critical thinking abilities by utilising "habit of thinking and cognitive skills." Incorporating maths projects into the curriculum will also assist students to enhance their thinking abilities.

NCERT Study Material

MS Word Cover Page Templates

Download, personalize & print, math project cover pages.

Posted By: admin 25/07/2020

Educational institutes require students to submit assignments and projects to get awarded with grades or marks, which reflect in the student’s report card. The projects assigned by the instructors need to be submitted timely and in a presentable way. Few instructors require projects in the form of a presentation while others require in a report form.

The cover page is also known as the title or representative page of the project, which provides the key information of the project’s title, and the information of the person submitting the project and to whom the project is being submitted.

Mathematics projects are the assigned tasks to solve any mathematical problem, which could be an equation, algebraic expression, geometric exercise, or any other question of a similar kind. The projects are assigned by the course instructor and are graded upon submission.

The project needs to have a cover page, to provide information on what the project is about, the entire content of the project needs to come after the cover page. If the project covers all the essentials but is presented with a dull or unattractive cover page, then the project fails to gain attention and is often graded low. Therefore, it is important to design the cover page in a way that serves its purpose.

The cover page of the Mathematics project should include the following essentials:

  • Project title- A project is incomplete without a title, and that title needs to be written on the cover page. There could be multiple projects of the same course, but giving/adding title helps in finding the relevant project or assignment later when it needs to be revised or reviewed before the examination.
  • Course/Subject name- Adding subject name could help in sorting and categorizing the projects.
  • Submitted by- the name of the person submitting the project should be mentioned on the cover page, as it helps the instructor to know who the student is, and the project can be marked for the same student.
  • Submitted to- The faculty room or office often has a bundle of projects, adding the name of the instructor to whom the project is being submitted helps in identifying the relevant instructor and minimizes the probability of the project being misplaced.
  • Submission date- the date is important because the projects need to be submitted timely or else either the marks are deducted, or the project is not accepted. Therefore, to ensure that the project is duly submitted on time, one must mention the date.

Before designing or inserting a cover page before the content of the project, one must go through the project submission guidelines (if any). This would help in finding out the requirements of the cover page.

Few instructors require the students to submit the project using a simple cover page, with added information and with no images. While few instructors provide their own template.

Moreover, the formatting guidelines could also be defined by the instructor, and he/she may ask the students to prepare and submit the project as per the guidelines defined by American Psychological Association (APA), Modern Language Association (MLA), Harvard, Chicago or any other format.

In case no guidelines are provided then designing a project cover page requires time, and there is a need to carefully design the page to make the project presentable. This could be done by inserting a blank page or cover page in MS Office using the Menu bar.

There are a variety of templates already available in the database of MS Office, which can be customized and edited as per the requirement. Using online resources, which offer freely available templates that can be downloaded and customized as per the requirement and can be then inserted in the project’s documentation.

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Math Poster Templates

Customize math poster templates.

Symbols Poster

If you're assigning this to your students, copy the poster to your account and save. When creating an assignment, just select it as a template!

Math Symbol Poster Example

Why Educational Math Posters?

In the realm of education, visual aids play a crucial role in facilitating effective learning experiences. Having posters in class can make learning more visual for kids and serve as reminders for important symbols, equations, and vocabulary.

Imagine stepping into a classroom adorned with vibrant educational posters that showcase intricate designs and creative border designs. These posters not only add an aesthetically pleasing element to the environment but also serve as visual aids that help kids grasp complex mathematical concepts more easily. These resources are instrumental in presenting clear and concise explanations of formulas, equations, and diagrams that might otherwise be challenging to convey solely through words during a lesson or presentation.

Designing Effective Math Posters

Creating an effective poster involves careful consideration of elements such as color schemes, typography, and layout. An eye-catching template can be customized to match the theme of the lesson, making the learning experience more engaging and memorable. With the availability of free math templates, teachers can experiment with various designs and layouts to find the one that best resonates with their class.

Interactive Math Posters

Complementing posters with printable worksheets adds an interactive dimension to the learning process. These posters and free math PowerPoint templates serve as opportunities for kids to apply the concepts they've learned from the posters. Math worksheet creators and math quiz generators are readily available online, allowing teachers to tailor exercises to their class's needs. From problem-solving activities to critical thinking challenges, interactive worksheets offer a platform for students to strengthen their mathematical skills.

The integration of posters and worksheets extends beyond the traditional classroom setting. Homeschoolers, tutors, and online educators can harness the potential of these resources to create comprehensive lesson plans or an engaging presentation. By providing students with visually appealing (not boring!) ideas, images, and interactive exercises, you will foster a deeper understanding of mathematical concepts and encourage self-directed learning.

Benefits and Outcomes

The synergy between posters and worksheets yields a multitude of benefits. Learners experience improved comprehension and retention, increased engagement, and enhanced problem-solving capabilities. As they interact with fun math Google slides themes and designs, and solve problems on printable math worksheets, their confidence in tackling mathematical challenges grows.

Educators worldwide have embraced this innovative approach to teaching mathematics. In classrooms and online platforms, educators have shared their examples and success stories about how combining visual presentations with interactive exercises has transformed learning experiences.

Resources for Creating Math Posters and Worksheets

A variety of online tools and resources are available for educators to create designs, a single template, and worksheets. Free math templates and math worksheet makers streamline the process, allowing educators to focus on the content and educational value. Here are some tips for making a poster:

  • Select a Relevant Math Concept: Choose a specific topic or concept that you want to teach or reinforce through your poster. Whether it's algebra, geometry, calculus, or statistics, make sure the concept aligns with your educational goals.
  • Gather Visual Resources: Collect relevant visual resources such as diagrams, equations, graphs, and illustrations that represent the chosen concept. These visuals will be the core of your poster component.
  • Design the Poster Component: Utilize a template or create your own design that suits your teaching theme. Incorporate designs and creative math border designs to make the poster visually appealing. Arrange the visuals logically, ensuring they effectively convey the mathematical information.
  • Integrate Interactive Exercises: Access a math worksheet maker or math quiz creator tool to design interactive exercises that complement the poster content. Craft relevant problems, puzzles, or questions that require children to apply the concepts presented on the poster.
  • Use a Math Lesson Plan Template: Incorporate your poster and worksheet components into a comprehensive lesson plan template. This template will serve as the structure for your lesson.
  • Ensure Clarity and Accessibility: Check that the visuals, equations, and text on both the poster and the worksheet are clear and easy to understand.
  • Incorporate Math Illustrations: Enhance the visual appeal of your poster by incorporating engaging and relevant images alongside the content.
  • Evaluate and Adjust: After the lesson, assess students' understanding by reviewing their performance on the interactive exercises. Use the insights gained to adjust your teaching approach and refine future work.

By following these steps, you'll create a comprehensive poster that combines visual appeal, interactive engagement, and effective educational content, fostering a deeper understanding of mathematical concepts.

More Storyboard That Resources and Free Printables

  • Math Infographic
  • Math Vocabulary Worksheet
  • Test and Quiz
  • Math Activities

How to Make a Math Template

Choose one of the premade templates.

We have lots of amazing templates to choose from. Take a look at our colorful example for inspiration!

Click on “Copy Template”

Once you do this, you will be directed to the storyboard creator.

Give Your Poster a Name!

Be sure to call it something related to the topic so that you can easily find it in the future.

Edit Your Poster

Our templates are fully customizable, so this is where you will include details, text, images, and make any aesthetic changes that you would like. The options are endless for math enthusiasts!

Click "Save and Exit"

When you are finished with your poster, click this button in the lower right hand corner to exit your storyboard.

From here you can print, download as a PDF, use as a math PowerPoint template, turn into a presentation on Google slides, attach it to an assignment and use it digitally, and more!

Happy Creating!

Frequently Asked Questions About Math Posters

How can a math poster benefit students.

Maths posters offers a comprehensive learning experience by providing visual explanations of concepts, and allowing students to apply those concepts through interactive exercises. This combination enhances understanding, engagement, and retention of the material.

What types of math concepts work well with posters?

They are useful in teaching various topics such as arithmetic, algebra, geometry, calculus, and statistics. For example, when teaching geometry, a maths poster can visually display shapes and properties, supported by interactive exercises on a worksheet to calculate areas and angles. In algebra, a poster can illustrate solving equations step-by-step, while the accompanying worksheet offers practice problems to reinforce the process. The combination of visual aids, interactive tasks, and informative posters accommodates diverse learning needs, promoting comprehensive understanding and engagement across numerous disciplines.

What types of interactive exercises can I include in a math poster?

You can include a variety of interactive exercises such as multiple-choice questions, fill-in-the-blank problems, matching exercises, true/false questions, and even open-ended problem-solving challenges. The choice of exercises depends on the learning objectives and the concepts being taught.

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15 Best & Fun Math Projects for Students

A child doing math problems

Super Easy and Super Fun Math Project Ideas for Grade 1 Students

Exploratory ideas for math projects for grade 2 students, project-based learning math ideas for grade 3 students, math project-based learning ideas for grade 4 students, advanced math projects for students in grade 5, frequently asked questions (faqs).

Math projects for students are a great way to get kids interested in math . They can be used to teach new concepts, review old ones, or just provide some fun and engaging math practice. There are tons of great math projects out there, but we’ve compiled a list of fifteen easy and engaging math projects for elementary school students—the best of the best!

Abacus with beads of different colors

1. Scoop and Cone Matching Game

What you need:

  • Cones and scoops made from felt or cardstock
  • Marker or sketch pens

Description:

Write a number on the cone. Write different combinations of addition and subtraction equations to represent the number on the scoop.

Students have to solve the equations and match the correct scoop to the cone.

Skills Learned:

Addition, subtraction, and the concept of equations

2. More or Less Dot Games

  • Ten frame cards
  • A set of dots (or colorful buttons or plastic corks to use as dots)
  • A deck of cards

Give a student a card and add some dots to it. Ask them, “How many dots are there on the card?” Once students master this, you may ask them, “What number is one more/one less?” You can also give them two cards and ask which one has more or less dots.

For two or more students, card games are a gold mine! Take a deck of cards. Snip off their corners with numerals written on them. Place the cards with their face downward. Ask each student to turn up a card. Ask them to tell whose card is “more” or “less.” Each correct answer wins them a point!

Visualizing numbers, understanding the concept of more or less, comparing numbers, addition, and subtraction

3. Shape Graphs

  • Different geometric shapes in different colors and sizes
  • Graph papers with large rows and columns (with rows mentioning shape names and columns mentioning numbers)
  • Some crayons

Distribute some graph paper among the children. Spread out some shapes in front of them. They have to find out how many shapes of each type there are and color that many boxes of relevant columns.

Recognition of geometric shapes by their names, and understanding and representing data in pictorial form

Drawing line on a sheet of paper with a ruler

4. Elementary Architects

  • Instructions and photos of room designing projects
  • 2-page student project sheet to promote reading in math
  • Note-taking forms
  • Sample blueprints for reference
  • Brainstorming sheet
  • Grid paper templates

Ask the students to design their rooms, calculate areas, and estimate flooring needs by reading the instructions, looking at the photos, and taking notes.

Students love to play architects. Allow them sufficient room for being creative to promote their spatial awareness.

Reading comprehension, estimation, area, and perimeter calculation

5. M&M’s Math Game

  • A box of colorful m&m’s
  • Graph papers for kids

Let your students dig into the box of m&m’s and take a few each. They have to count how many m&m’s of each color they got. If they count m&m’s of each color correctly, they can eat them! Otherwise, they have to return the m&m’s to the box and try again!

As they master their skills, you can take this math game to the next level. They can make a graph using graph paper and crayons! You may have to help them label the graph and the graphing part itself.

Counting, addition, making graphs

6. Hit a Home Run for Math Fact Fluency

  • DIY baseball game board with math facts
  • Number cards
  • Counters to use as baseball players—9 for each team

Write the numbers 1 to 9 in one row and 0 in the next row to make a baseball diamond.

Help your students write math facts such as doubles (2 + 2, 3 + 3, etc.), near doubles (9 + 8), addition/subtraction of 10 (8 + 2, 5 + 5), and related subtraction facts (7 – 3, 9 – 6) on the number cards.

To play, have each student roll two dice. They get to move one of their baseball players the number of spaces corresponding to the first die and then answer the math fact that corresponds to the number they landed on. If they answer correctly, they get to roll again. The first player to get three of their baseball players “home” wins!

Math facts fluency, addition, subtraction

A tamarin monkey on a tree branch

7. Place Value in the Wild Math Project

  • Digital and printable version of a student guide with detailed instructions and visuals
  • Student printables or digital recording sheets guiding students on how to select a habitat, research animals of that habitat, note sizes and lifespans of these animals, etc.

As third graders research animals as expedition scouts for Wildlife Explorers International, they learn about place values through various activities, such as representing numbers in different ways, comparing numbers, and estimating lengths, heights, and lifespans of animals.

You can ask students to use standard numbers, expanded forms, and word forms of numbers. They may also be introduced to decimals through this project.

Place value, estimation, decimals

8. The Time of Your Life

  • A printable or digital student guide with detailed instructions, visuals, and student printables
  • Analog and digital clocks (one per student pair)

In this project, students learn to read the time on both analog and digital clocks. They also practice setting the time on these clocks.

As they work in pairs, they take turns being the “teacher” and the “student.” The teacher explains to the student how to read the time on a clock. Then, the student sets the time on the clock according to the teacher’s instructions.

Or they tell how many seconds, minutes, or hours have elapsed in doing an activity.

It’s a great activity for third graders, where students can win prizes for being the best timekeepers!

Telling time, elapsed time

9. What’s Your Angle, Pythagoras?

  • A scorecard
  • Child-safe compass (optional)

Pythagorean principles are put to the test in this game! Players use a protractor and ruler (or child-safe compass) to draw angles and then measure the length of the sides of right triangles. The goal is to have the longest hypotenuse at the end of the game.

You can call out “Right-Angled Triangle” randomly, and the students have to arrange themselves in the shape in a flash. Those who do it correctly win!

You may also call out “Right Angle”, “Acute Angle”, or “Obtuse Angle” where students have to pair up instantly. If some fail to do it, they are out.

Angles, Pythagorean theorem

Wooden desk calendar

10. Calendar Math in the Classroom

  • A printable or digital calendar template

A perfect math review technique for fifth graders, calendar math is a great way to engage them in the concepts of days, weeks, months, and years. You have to display a calendar in the classroom and point out various aspects of it to the students. For example, you can ask them how many days there are in February, or how many months have 31 days, etc.

You can also use the calendar to teach place value. For instance, you can ask students to name the day on which their birthdays fall this year and write it down. Then, they can find out the day on which their birthdays will fall next year and so on.

This activity can be done with a physical calendar or a digital one. Students can use real-world objects like coins or candy to help them understand the concepts of place value, addition, and subtraction.

Days, weeks, months, years, place value, addition, subtraction

11. Run a Pizza Place

  • Pizza boxes or paper plates
  • Colorful cardboard pizzas

Bring fraction to life with this fun activity! Students run their own pizza place, where they take orders, make pizzas, and serve them to customers.

They can use play money to buy pizza toppings and then charge customers for their pizzas. They can also use fraction strips or circle fractions to create pizzas of different sizes.

Such math projects for students teach them concepts like halves, thirds, fourths, eighths, and more. And children will have a blast doing it!

Fractions, equivalent fractions, comparing fractions, adding and subtracting fractions

12. Hot Cocoa Project!

  • Hot cocoa stall
  • Marshmallows
  • Whipped cream (optional)
  • Chocolate shavings (optional)
  • Recipe book
  • Play money or real money

An excellent activity for young entrepreneurs (under adult supervision), this hot cocoa project simulates a hot cocoa stand. Students can make and sell hot cocoa to their classmates, using real or play money.

They can follow a recipe to make the hot cocoa mix, and then use it to make individual cups of hot cocoa. They can also add marshmallows, whipped cream, and chocolate shavings to their hot cocoa, and charge extra for these toppings.

This activity is a great way to teach children about money, measurement, and fractions. And they’ll love getting creative with the hot cocoa mix!

Money, measurement, fractions, addition, subtraction

Children working on math project in class

13. Performance Math Art

  • Props or costumes (optional)
  • A video recording device (such as a smartphone)

Divide students into groups of 2 to 4 and ask them to prepare a performance art (dramatic poetry, song, or a skit) to explain the Order of Operations (or any other mathematical concepts, such as area and perimeter, exponents and roots, or geometry).

After they have practiced, film their final performance. Students can watch the videos to revise the concept later.

Students may also review each other’s performance in terms of delivery, clarity, and creativity to give constructive feedback.

Order of operations, area and perimeter, exponents and roots, geometry

14. Probably Probability

  • DIY probability tables

An inspirational idea for kinesthetic learners, this activity gets students up and about as they experiment with probability.

Provide each student with a die (or multiple dice) and a coin. Ask them to roll the die (or dice), flip the coin, and record their results in a table. They can create their probability tables.

Once they have collected enough data, they can look for patterns and predict the probability of certain events.

Probability, independent and dependent events, expected values

15. The Theme Park Project

  • Theme Park templates (for guidance)
  • Construction paper
  • Glue or tape
  • Markers or crayons
  • Small toys (optional)

This project is perfect for a math class that is learning about geometry and measurement. Students will use their knowledge of shapes, angles, and measurements to create a mini theme park.

They can start by choosing a template (or creating their own) and then cutting out the shapes from construction paper. Once they have all the pieces, they can assemble their theme park and add details with markers or crayons.

They can also add small toys to their theme park if they wish. Finally, they can measure the area and perimeter of their creation.

Children can dream up new rides, give them outlandish names, create menus for concession stands, and research healthy and junk foods!

A lot of math happens in everyday life if we just look for it.

Geometry, measurement, area, perimeter

By working on these fun projects, students can learn and practice various math skills, from basic counting and graphing to more advanced concepts such as fractions and decimals. These math projects for students can be used to supplement your regular math curriculum or as a standalone activity. Either way, your students are sure to enjoy them!

How can I make sure my students are engaged in the project?

Make sure to give your students a chance to be creative and have fun with the projects. For example, with the “Theme Park Project,” encourage them to develop their own designs and be as creative as possible with the details. With the “Probably Probability” project, let them experiment with different ways of collecting data and see what patterns they can find.

Do I need to prepare anything in advance?

It largely depends on the project you choose. For some projects, you may want to prepare templates in advance. For others, such as the “Probably Probability” project, you only need dice and coins.

How long should the projects take?

Again, it depends on the project. Some math projects for students require several days to complete. Others can be done in one class period or a few minutes.

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STAT151A Homework 1: Due Jan 26th

Your name here

1 Simple regression in matrix form

Consider the simple linear model \(y_n = \beta_0 + \beta_1 z_n + \varepsilon_n\) .

Let \(\bar{y}:= \frac{1}{N} \sum_{n=1}^Ny_n\) and \(\bar{z}:= \frac{1}{N} \sum_{n=1}^Nz_n\) . Recall that the ordinary least squares estimates are given by \[ \begin{aligned} \hat{\beta}_1 = \frac{\frac{1}{N} \sum_{n=1}^N(z_n - \bar{z}) (y- \bar{y})}{\frac{1}{N} \sum_{n=1}^N(z_n - \bar{z})^2} \quad\textrm{and}\quad \hat{\beta}_0 = \bar{y}- \hat{\beta}_1 \bar{z}. \end{aligned} \]

Write the set of equations

\[ y_n = \beta_0 + \beta_1 z_n + \varepsilon_n \]

for \(n \in \{1, \ldots, N\}\) in matrix form. That is, let \(\boldsymbol{X}\) denote an \(N \times 2\) matrix, \(\boldsymbol{Y}\) and \(\boldsymbol{\varepsilon}\) denote \(N \times 1\) matrices, \(\boldsymbol{\beta}= (\beta_0, \beta_1)^\intercal\) , and express the matrices \(\boldsymbol{Y}\) , \(\boldsymbol{X}\) , and \(\boldsymbol{\varepsilon}\) in terms of the scalars \(y_n\) , \(z_m\) , and \(\varepsilon_n\) so that \(\boldsymbol{Y}= \boldsymbol{X}\boldsymbol{\beta}+ \boldsymbol{\varepsilon}\) is equivalent to the set of regression equations.

Define \[ \begin{aligned} \overline{zz} := \frac{1}{N} \sum_{n=1}^Nz_n^2 \quad\textrm{and}\quad \overline{zy} := \frac{1}{N} \sum_{n=1}^Nz_n y_n \end{aligned} \]

Write an explict expressions for \(\boldsymbol{X}^\intercal\boldsymbol{X}\) , \(\boldsymbol{X}^\intercal\boldsymbol{Y}\) , and \(\left(\boldsymbol{X}^\intercal\boldsymbol{X}\right)^{-1}\) , all in terms of \(\bar{y}\) , \(\bar{z}\) , \(\overline{zz}\) , \(\overline{zy}\) , and \(N\) . Verify that the inverse is correct by direct multiplication.

Compute \((\boldsymbol{X}^\intercal\boldsymbol{X})^{-1} \boldsymbol{X}^\intercal\boldsymbol{Y}\) . Show that the first row is equal to \(\hat{\beta}_0\) and the second row is equal to \(\hat{\beta}_1\) as given by the ordinary least squares formula in the problem statement above.

2 Mean zero residuals.

Consider the model \(y_n = \beta z_n + \varepsilon_n\) . Let \(\hat{\beta}\) denote the least squares estimator and \(\hat{\varepsilon}_n = y_n - \hat{\beta}z_n\) .

Suppose \(z_n\) is not a constant. Is it necessarily the case that \(\frac{1}{N} \sum_{n=1}^N\hat{\varepsilon}_n = 0\) ? Prove your answer.

Suppose \(z_n\) is a constant, but \(z_n \equiv 5\) for every \(n \in \{1, \ldots, N\}\) . Is it necessarily the case that \(\frac{1}{N} \sum_{n=1}^N\hat{\varepsilon}_n = 0\) ? Prove your answer.

Now the model \(y_n = \beta_1 z_{n1} + \beta_2 z_{n2} + \varepsilon_n\) . Suppose that \(z_{n1} = 1\) is \(n\) is even, and is \(0\) otherwise. Similarly, suppose that \(z_{n2} = 1\) is \(n\) is odd, and is \(0\) otherwise. Let \(N\) be even. Is it necessarily the case that \(\frac{1}{N} \sum_{n=1}^N\hat{\varepsilon}_n = 0\) ? Prove your answer.

3 Inner products and covariances

Let \(\boldsymbol{z}= (z_1, \ldots, z_N)\) and \(\boldsymbol{y}= (y_1, \ldots, y_N)\) . Let \(\boldsymbol{X}\) denote an \(N \times P\) matrix whose \(n\) –th row is the transpose of the \(P\) -vector \(\boldsymbol{x}_n^\intercal\) .

(Note: this question involves limits of random variables, and there are many distinct ways that random variables can converge to limits. If you’re familiar with these different modes of probabilisitic convergence, feel free to state what mode of convergence applies, but if you are not, don’t worry — modes of convergence will not matter much for this class, and you can state your result heuristically.)

For a set of quantities (numbers, vectors, pairs of vectors, etc), the “empirical distribution” over that set refers to drawing an element with replacement from the set with equal probability given to each entry. For example, if \(\mathcal{Z}'\) is a drawn from the empirical distribution over the set \(\{z_1, \ldots, z_N \}\) , then \(\mathbb{P}\left(\mathcal{Z}' = z_n\right) = 1/N\) for each \(n\) . Similarly, if \((\mathcal{Z}', \mathcal{Y}')\) is drawn from the empirical distribution over the pairs \(\{(z_1, y_1), \ldots, (z_N, y_N)\}\) , then \(\mathbb{P}\left((\mathcal{Z}', \mathcal{Y}') = (z_n, y_n)\right) = 1/N\) for all \(n\) .

(Hint: it may help to recall that the bootstrap uses draws from the empirical distribution, and that, in the empirical distribution, the elements of the set are fixed and not random.)

Let \((\mathcal{Z}', \mathcal{Y}')\) denote a draw from the empirical distribution over the set \(\{(y_1, z_1), \ldots, (y_N, z_N)\}\) .

Prove that \(\frac{1}{N} \boldsymbol{z}^\intercal\boldsymbol{y}= \mathbb{E}\left[\mathcal{Z}' \mathcal{Y}'\right]\) . Then prove that \(\frac{1}{N} \boldsymbol{1}^\intercal\boldsymbol{z}= \mathbb{E}\left[\mathcal{Z}'\right]\) as a special case.

Now suppose that the entries of \(\boldsymbol{z}\) are independent and identically distributed (IID) realizations of the random variable \(\mathcal{Z}\) , and that the entries of \(\boldsymbol{y}\) are similarly IID realizations of a random variable \(\mathcal{Y}\) . Assuming that \(\mathbb{E}\left[|\mathcal{Z}|\right] < \infty\) and \(\mathbb{E}\left[|\mathcal{Y}|\right] < \infty\) , prove that

\[ \frac{1}{N} \boldsymbol{z}^\intercal\boldsymbol{y}\rightarrow \mathbb{E}\left[\mathcal{Z} \mathcal{Y}\right] \textrm{ as }N \rightarrow \infty \]

(Hint: don’t prove this from scratch, appeal to a probability theorem.)

Using only inner products involving \(\boldsymbol{y}\) , \(\boldsymbol{z}\) , and \(\boldsymbol{1}\) , write an expression for \(\mathrm{Cov}\left(\mathcal{Y}', \mathcal{Z}'\right)\) . Prove that the expression converges with probability one to \(\mathrm{Cov}\left(\mathcal{Y}, \mathcal{Z}\right)\) . (Hint: again, use your previous results and a theorem from probability.)

Now, let \((\mathcal{X}', \mathcal{Y}')\) denote a draw from the empirical distribution over \(\{(x_1, y_1), \ldots, (x_N, y_N) \}\) . (Recall that the vector \(x_n\) is a length– \(P\) column vector, and \(x_n^\intercal\) is the \(n\) –th row of the matrix \(\boldsymbol{X}\) .)

\[ \begin{aligned} \frac{1}{N} \boldsymbol{X}^\intercal\boldsymbol{X}= \mathbb{E}\left[\mathcal{X}' \mathcal{X}'^\intercal\right] \quad\textrm{and}\quad \frac{1}{N} \boldsymbol{X}^\intercal y= \mathbb{E}\left[\mathcal{X}' \mathcal{Y}'\right]. \end{aligned} \]

Now, suppose that rows of \(\boldsymbol{X}\) are IID realizations of the random \(P\) –vector \(\mathcal{X}\) , and that \(\mathbb{E}\left[|\mathcal{X}_p|\right] < \infty\) for each \(p \in \{ 1, \ldots, P \}\) . Assume, as above, that \(\mathbb{E}\left[|\mathcal{Y}|\right] < \infty\) .

Prove that, as \(N \rightarrow \infty\) ,

\[ \frac{1}{N} \boldsymbol{X}^\intercal\boldsymbol{X}\rightarrow \mathbb{E}\left[\mathcal{X} \mathcal{X}^\intercal\right] \quad\textrm{and}\quad \frac{1}{N} \boldsymbol{X}^\intercal\boldsymbol{Y}\rightarrow \mathbb{E}\left[\mathcal{X} \mathcal{Y}\right], \]

where both limits are with probability one.

Now assume that, for each \(p \in \{1, \ldots, P\}\) and \(q \in \{1, \ldots, P\}\) , \(\mathbb{E}\left[\left|\mathcal{X}'_p\right| \left|\mathcal{X}'_q\right| \mathcal{Y}^2\right] < \infty\) . Prove that, as \(N \rightarrow \infty\) ,

\[ \frac{1}{\sqrt{N}} \left( \boldsymbol{X}^\intercal\boldsymbol{Y}- \mathbb{E}\left[\boldsymbol{X}^\intercal\boldsymbol{Y}\right] \right) \rightarrow \mathcal{Z}, \]

where \(\mathcal{Z}\) is a multivariate normal random variable. What is the covariance of \(\mathcal{Z}\) ? (Hint: again, appeal to a probability theorem.)

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  23. STAT151A Homework 1: Due Jan 26th

    Write the set of equations. y n = β 0 + β 1 z n + ε n. for n ∈ { 1, …, N } in matrix form. That is, let X denote an N × 2 matrix, Y and ε denote N × 1 matrices, β = ( β 0, β 1) ⊺, and express the matrices Y, X, and ε in terms of the scalars y n, z m, and ε n so that Y = X β + ε is equivalent to the set of regression equations.

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