# 5.1E: Spring Problems I (Exercises) - Mathematics

In the following exercises assume that there’s no damping.

## Q5.1.1

1. An object stretches a spring (4) inches in equilibrium. Find and graph its displacement for (t>0) if it is initially displaced (36) inches above equilibrium and given a downward velocity of (2) ft/s.

2. An object stretches a string (1.2) inches in equilibrium. Find its displacement for (t>0) if it is initially displaced (3) inches below equilibrium and given a downward velocity of (2) ft/s.

3. A spring with natural length (.5) m has length (50.5) cm with a mass of (2) gm suspended from it. The mass is initially displaced (1.5) cm below equilibrium and released with zero velocity. Find its displacement for (t>0).

4. An object stretches a spring (6) inches in equilibrium. Find its displacement for (t>0) if it is initially displaced (3) inches above equilibrium and given a downward velocity of (6) inches/s. Find the frequency, period, amplitude and phase angle of the motion.

5. An object stretches a spring (5) cm in equilibrium. It is initially displaced (10) cm above equilibrium and given an upward velocity of (.25) m/s. Find and graph its displacement for (t>0). Find the frequency, period, amplitude, and phase angle of the motion.

6. A (10) kg mass stretches a spring (70) cm in equilibrium. Suppose a (2) kg mass is attached to the spring, initially displaced (25) cm below equilibrium, and given an upward velocity of (2) m/s. Find its displacement for (t>0). Find the frequency, period, amplitude, and phase angle of the motion.

7. A weight stretches a spring (1.5) inches in equilibrium. The weight is initially displaced (8) inches above equilibrium and given a downward velocity of (4) ft/s. Find its displacement for (t > 0).

8. A weight stretches a spring (6) inches in equilibrium. The weight is initially displaced (6) inches above equilibrium and given a downward velocity of (3) ft/s. Find its displacement for (t>0).

9. A spring–mass system has natural frequency (7sqrt{10}) rad/s. The natural length of the spring is (.7) m. What is the length of the spring when the mass is in equilibrium?

10. A (64) lb weight is attached to a spring with constant (k=8) lb/ft and subjected to an external force (F(t)=2sin t). The weight is initially displaced (6) inches above equilibrium and given an upward velocity of (2) ft/s. Find its displacement for (t>0).

11. A unit mass hangs in equilibrium from a spring with constant (k=1/16). Starting at (t=0), a force (F(t)=3sin t) is applied to the mass. Find its displacement for (t>0).

12. A (4) lb weight stretches a spring (1) ft in equilibrium. An external force (F(t)=.25sin8 t) lb is applied to the weight, which is initially displaced (4) inches above equilibrium and given a downward velocity of (1) ft/s. Find and graph its displacement for (t>0).

13. A (2) lb weight stretches a spring (6) inches in equilibrium. An external force (F(t)=sin8t) lb is applied to the weight, which is released from rest (2) inches below equilibrium. Find its displacement for (t>0).

14. A (10) gm mass suspended on a spring moves in simple harmonic motion with period (4) s. Find the period of the simple harmonic motion of a (20) gm mass suspended from the same spring.

15. A (6) lb weight stretches a spring (6) inches in equilibrium. Suppose an external force (F(t)={3over16}sinomega t+{3over8}cosomega t) lb is applied to the weight. For what value of (omega) will the displacement be unbounded? Find the displacement if (omega) has this value. Assume that the motion starts from equilibrium with zero initial velocity.

16. A (6) lb weight stretches a spring (4) inches in equilibrium. Suppose an external force (F(t)=4sinomega t-6cosomega t) lb is applied to the weight. For what value of (omega) will the displacement be unbounded? Find and graph the displacement if (omega) has this value. Assume that the motion starts from equilibrium with zero initial velocity.

17. A mass of one kg is attached to a spring with constant (k=4) N/m. An external force (F(t)=-cosomega t-2sinomega t) n is applied to the mass. Find the displacement (y) for (t>0) if (omega) equals the natural frequency of the spring–mass system. Assume that the mass is initially displaced (3) m above equilibrium and given an upward velocity of (450) cm/s.

18. An object is in simple harmonic motion with frequency (omega_0), with (y(0)=y_0) and (y'(0)=v_0). Also, find the amplitude of the oscillation and give formulas for the sine and cosine of the initial phase angle.

19. Two objects suspended from identical springs are set into motion. The period of one object is twice the period of the other. How are the weights of the two objects related?

20. The weight of one object is twice the weight of the other. How are the periods of the resulting motions related?

21. Two identical objects suspended from different springs are set into motion. The period of one motion is (3) times the period of the other. How are the two spring constants related?

## Assignments

Part I of each problem set consists of problems which are either rather routine or for which solutions are available in the back of the book, or in 18.03 Notes and Exercises by Prof. Mattuck.

Part II contains more challenging and novel problems. They will help you develop an understanding of the material.

[EP] refers to a problem in the course textbook: Edwards, C., and D. Penney. Elementary Differential Equations with Boundary Value Problems. 6th ed. Upper Saddle River, NJ: Prentice Hall, 2003. ISBN: 9780136006138

[Notes] refers to an exercise in the "18.03 Notes and Exercises" written by Prof. Mattuck.

Most problem sets also contains one problem based around one the "Mathlets."

Problem set 3: Second Half (PDF)

Problem set 5: Second half (PDF)

Problem set 8: Second half (PDF)

## Mathematics Resources

Challenge your students with creative mathematics lessons, printable worksheets, activities, quizzes, and more during Math Education Month (April)—or any time of the year! Focus on various mathematical themes, such as geometry, algebra, probability and statistics, money, measurement, and more! Incorporating other subjects—such as art, reading, and science—into your math lessons will help hold your students' interest in the subject. You'll find resources here that are appropriate for whichever grade level you teach.

Explaining How to Make a Bar Graph

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Use a lesson that provides students with an opportunity to identify and apply median, mode, range, and mean.

The Concept of Measurement

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Students learn how to solve music and math problems by finding patterns. Money Word Problems I (Grade 2)

Promote real-world problem-solving skills Covers addition, subtraction, and fractions Includes answer key and…

General KWL Chart (Version 3)

This simple KWL chart can be used in all curriculum areas and all grade levels. The activity will keep your students…

Scoots are a way for students to practice math skills on the move. Question cards are placed around the room and on the “Scoot!” command students rotate through stations with an answer sheet until they have completed all questions.

## First Grade Math Worksheets and Printables You may not remember the first time you understood how and why 2 + 2 = 4, but rest assured, it was a monumental moment for your young self. It was also the first time you realized how challenging first grade math can be. Thankfully, your child has one advantage you didn’t: access to our first grade math worksheets. What makes these worksheets particularly worthwhile is the variety of concepts that are offered, not to mention the way in which they are presented.

From basic addition and subtraction activities to measurement worksheets to puzzles where solving math problems unlocks the answer to a “What’s the word?” question, there is no end to the number of ways your child can improve his math skills. What’s particularly great about our first grade math worksheets is they inject some fun into what can be an overwhelming challenge (learning math). So if your child struggles with reading a clock, print out our Telling Time with Clockwork Cat worksheet. Or access The Pet Store Tally worksheet to help your student better understand sorting and counting.

With dozens of unique activities and alluring graphics/characters, your child will inevitably discover the joy of learning first grade math — something you may have never thought possible as a kid!

## Projects

You have the option of completing a project to replace your two lowest homework grades. For the project, you need to read a research paper and complete the following tasks:

• Explain in detail the mathematical model and the techniques used to analyze it.
• Summarize the results of the paper and, if possible, attempt to reproduce them.
• Propose one or more potential extensions to the model and discuss how the results may change.

## 5.1E: Spring Problems I (Exercises) - Mathematics

STAT 453/553 STATISTICAL INFERENCE

Department of Mathematics and Statistics, UNM

Office: 314 SMLC Building

email:ghuerta AT math DOT unm DOT edu

Class Time: Tue-Thurs 14:00-15:15.

Classroom: Collaborative Teaching and Learning Building (CTLB) 210.

Office Hours: T 12:00-1:00. Th 12:30-1.45pm, TR: 15:30-16:30 (for 15.30 only if I don't have a faculty meeting/seminar). By appointment for other times or days.

INSTRUCTOR'S DESCRIPTION

The goal of this course is to give an introduction to the theory (and mathematics) of statistical inference. The first part of the course will discuss issues about random sampling, likelihood and sufficiency. The second part will discuss aspects on point estimation and hypothesis testing. The final part of the course will address confidence intervals and a brief introduction to asymptotic evaluations. The approach of this course is unified in the sense that will try to discuss concepts from the 3 main standpoints of statistical inference: frequentist, likelihood and bayesian. The course material along with Stat 461/561 are the basis for In-class portion of the Statistics qualifying exam.

Transformations of univariate and multivariate distributions to obtain the special distributions important in statistics. Concepts of estimation and hypothesis testing in both large and small samples with emphasis on the statistical properties of the more commonly used procedures, including Students t-tests, F-tests and chi-square tests. Confidence intervals. Performance of procedures under non-standard conditions (i.e., robustness).

Stat 461/561: Probability (offered in the Fall). For Stat 443/553 its important to have a good understanding of basic Probability, probability distributions, expectations, variances and moments in general. Also some good familiarity with joint distributions, conditional and marginal distributions is required.

Casella, G. and Berger, R.L. (2001) Statistical Inference. 2ed. Duxbury Press. (C&B, main text)

Young, G. A. and Smith, R.L. (2010) Essentials of Statistical Inference. Cambridge University Press. (to support or complement material in Casella and Berger)

Chapter 5: Random samples and related concepts (mostly review).

Chapter 5: Convergence and generating random samples (review).

Chapter 6: Sufficiency, ancillary and completeness. The likelihood principle.

Chapter 7: Point estimation: Methods and properties of estimators.

Chapter 8: Hypothesis Testing: Methods for finding and evaluating tests.

Chapter 9: Interval estimation: Methods for finding and evaluating interval estimators.

Chapter 10: Asymptotic Evaluations

MY CLASS NOTES (based on the Casella and Berger book)

The grading will be based on a midterm exam, a final exam and quizzes. I plan to give a quiz roughly ever two weeks with a first quiz on January 28th and then one on February 11th. Each of the exams is worth 35% of the final grade. The quizzes are worth 30% of the final grade, there will be about 7-8 quizzes of about 15-20 minutes each during class. I have a policy of no make-up tests or quizzes. Only in the case of an emergency that that can be fully documented through a medical or police report, I will consider rescheduling an exam at an alternative date. Exercises from the textbook and other sources will be assigned regularly as Homework (HW). HW will not be graded but I highly encourage you to do it, since exam or quizzes problems can be taken out from HW. I will post a list of HW exercises through this webpage or our UNM learn webpage. I highly encourage you to work the HW problems in groups. The more you work on exercises from the text assigned or not assigned or from other sources, the more you will be prepared for the exams.

Midterm: Thursday, March 10. Will cover material up to the class of March 8. review

Final exam: Tuesday May 10th, 10:00am-12:00pm in our regular classroom. Review for Final

HOMEWORKS (to be revised)

HW1: Exercises from Chapter 5 (C&B): 5.4, 5.15, 5.16, 5.24, 5.31, 5.35, 5,39, 5.40, 5.49, 5.50, 5.52, 5.56

HW2: Exercises from Chapter 6 (C&B): 6.6, 6.12, 6.15, 6.18, 6.21, 6.30, 6.34, 6.36

HW3: Exercises from Chapter 7 (C&B): 7.1, 7.8, 7.10, 7.11, 7.12, 7.19, 7.20, 7.24, 7.38,7.47,7.49,7.58

HW4: Exercises from Chapter 8 (C&B): 8.8a (only part (a) for this problem.),8.10,8.14,8.16,8.17, 8.18,8.22

HW5: Exercises from Chapter 9 (C&B): 9.10,9.13,9.17,9.23,9.26,9.39

This is a minimum list of problems you should work on. The more problems you work on, the better prepared you will be specially for the in-class quals. Solutions

to selected problems are available here. These are previous semesters so if you have questions, please let me know.

UNM POLICY ON DISABILITIES

Qualified students with disabilities needing appropriate academic adjustments should contact me to ensure your needs are met in a timely manner.

## Common Core Standards For Mathematics

How can the Common Core Math be implemented in the Classroom?
How can I teach the Common Core Math at home?
How can I get homework help for the Common Core Math?

What is the Common Core Standards? The Common Core State Standards Initiative is a U.S. education initiative that seeks to bring diverse state curricula into alignment with each other by following the principles of standards-based education reform. The initiative is sponsored by the National Governors Association (NGA) and the Council of Chief State School Officers (CCSSO).

The Mathematics Standards include Standards for Mathematical Practice and Standards for Mathematical Content.

### Mathematical Practice

The eight principles of mathematical practice are:

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

### Mathematical Content

Mathematical content is organized in a number of domains.

### Common Core Standards By Grades

Lessons, Worksheets and Games that are suitable for the Common Core Math sorted by Grades. We have Common Core Math Lessons, Common Core Math Worksheets, Common Core Math Games and Common Core Math Lesson Plans.

Common Core for Kindergarten Counting & Cardinality
Know number names and the count sequence.
Count to tell the number of objects.
Compare numbers.
Operations & Algebraic Thinking
Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.
Number & Operations in Base Ten
Work with numbers 11-19 to gain foundations for place value.
Measurement and Data
Describe and compare measurable attributes.
Classify objects and count the number of objects in each category.
Geometry
Identify and describe shapes.
Analyze, compare, create, and compose shapes.

Common Core for Grade 1 Operations & Algebraic Thinking
Represent & solve problems involving addition & subtraction.
Understand & apply properties of operations and the relationship between addition and subtraction.
Work with addition and subtraction equations.
Number & Operations in Base Ten
Extend the counting sequence.
Understand place value.
Use place value understanding and properties of operations to add and subtract.
Measurement and Data
Measure lengths indirectly and by iterating length units.
Tell and write time.
Represent and interpret data.
Geometry
Reason with shapes and their attributes.

Common Core for Grade 2 Operations & Algebraic Thinking
Represent and solve problems involving addition and subtraction.
Work with equal groups of objects to gain foundations for multiplication.
Number & Operations in Base Ten
Understand place value.
Use place value understanding and properties of operations to add and subtract.
Measurement and Data
Measure & estimate lengths in standard units.
Relate addition and subtraction to length.
Work with time and money.
Represent and interpret data.
Geometry
Reason with shapes and their attributes.

Common Core for Grade 3 Operations & Algebraic Thinking
Represent and solve problems involving multiplication and division.
Understand properties of multiplication and the relationship between multiplication and division.
Multiply and divide within 100.
Solve problems involving the four operations, and identify and explain patterns in arithmetic.
Number & Operations in Base Ten
Use place value understanding and properties of operations to perform multi-digit arithmetic.
Number & Operations & Fractions
Develop understanding of fractions as numbers.
Measurement and Data
Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects.
Represent and interpret data.
Geometric measurement: understand concepts of area and relate area to multiplication and to addition.
Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures.
Geometry
Reason with shapes and their attributes.

Common Core for Grade 4 Operations & Algebraic Thinking
Use the four operations with whole numbers to solve problems.
Gain familiarity with factors and multiples.
Generate and analyze patterns.
Number & Operations in Base Ten
Generalize place value understanding for multi-digit whole numbers.
Use place value understanding and properties of operations to perform multi-digit arithmetic.
Number & Operations & Fractions
Extend understanding of fraction equivalence and ordering.
Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.
Understand decimal notation for fractions, and compare decimal fractions.
Measurement and Data
Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.
Represent and interpret data.
Geometric measurement: understand concepts of angle and measure angles.
Geometry
Draw and identify lines and angles, and classify shapes by properties of their lines and angles.

Common Core for Grade 5 Operations & Algebraic Thinking
Write and interpret numerical expressions.
Analyze patterns and relationships.
Number & Operations in Base Ten
Understand the place value system.
Perform operations with multi-digit whole numbers and with decimals to hundredths.
Number & Operations &Fractions
Use equivalent fractions as a strategy to add and subtract fractions.
Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
Measurement and Data
Convert like measurement units within a given measurement system.
Represent and interpret data.
Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.
Geometry
Graph points on the coordinate plane to solve real-world and mathematical problems.
Classify two-dimensional figures into categories based on their properties.

Common Core for Grade 6 Ratios & Proportional Relationships
Understand ratio concepts and use ratio reasoning to solve problems.
The Number System
Apply and extend previous understandings of multiplication and division to divide fractions by fractions.
Multiply and divide multi-digit numbers and find common factors and multiples.
Apply and extend previous understandings of numbers to the system of rational numbers.
Expressions & Equations
Apply and extend previous understandings of arithmetic to algebraic expressions.
Reason about and solve one-variable equations and inequalities.
Represent and analyze quantitative relationships between dependent and independent variables.
Geometry
Solve real-world and mathematical problems involving area, surface area, and volume.
Statistics & Probability
Develop understanding of statistical variability.
Summarize and describe distributions.

Common Core for Grade 7 Ratios & Proportional Relationships
Analyze proportional relationships and use them to solve real-world and mathematical problems.
The Number System
Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
Expressions & Equations
Use properties of operations to generate equivalent expressions.
Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
Geometry
Draw, construct and describe geometrical figures and describe the relationships between them.
Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.
Statistics & Probability
Use random sampling to draw inferences about a population.
Draw informal comparative inferences about two populations.
Investigate chance processes and develop, use, and evaluate probability models.

Common Core for Grade 8 The Number System
Know that there are numbers that are not rational, and approximate them by rational numbers.
Expressions and Equations
Work with radicals and integer exponents.
Understand the connections between proportional relationships, lines, and linear equations.
Analyze and solve linear equations and pairs of simultaneous linear equations.
Functions
Define, evaluate, and compare functions.
Use functions to model relationships between quantities.
Geometry
Understand congruence and similarity using physical models, transparencies, or geometry software.
Understand and apply the Pythagorean Theorem.
Solve real-world and mathematical problems involving volume of cylinders, cones and spheres.
Statistics and Probability
Investigate patterns of association in bivariate data.

High School: Number & Quantity The Real Number System
Extend the properties of exponents to rational exponents
Classify numbers as rational or irrational
Quantities
Reason quantitatively and use units to solve problems
The Complex Number System
Perform arithmetic operations with complex numbers
Represent complex numbers and their operations on the complex plane
Use complex numbers in polynomial identities and equations
Vector and Matrix Quantities
Represent and model with vector quantities.
Perform operations on vectors.
Perform operations on matrices and use matrices in applications.

High School Algebra Seeing Structure in Expressions
Interpret the structure of expressions
Write expressions in equivalent forms to solve problems
Arithmetic with Polynomials and Rational Functions
Perform arithmetic operations on polynomials
Understand the relationship between zeros and factors of polynomials
Use polynomial identities to solve problems
Rewrite rational functions
Creating Equations
Create equations that describe numbers or relationships
Reasoning with Equations and Inequalities
Understand solving equations as a process of reasoning and explain the reasoning
Solve equations and inequalities in one variable
Solve systems of equations
Represent and solve equations and inequalities graphically

High School: Functions Interpreting Functions
Understand the concept of a function and use function notation
Interpret functions that arise in applications in terms of the context
Analyze functions using different representations
Building Functions
Build a function that models a relationship between two quantities
Build new functions from existing functions
Construct and compare linear and exponential models and solve problems
Interpret expressions for functions in terms of the situation they model
Trigonometric Functions
Extend the domain of trigonometric functions using the unit circle
Model periodic phenomena with trigonometric functions
Prove and apply trigonometric identities

High School: Geometry Congruence
Experiment with transformations in the plane
Understand congruence in terms of rigid motions
Prove geometric theorems
Make geometric constructions
Similarity, Right Triangles, and Trigonometry
Understand similarity in terms of similarity transformations
Prove theorems involving similarity
Define trigonometric ratios and solve problems involving right triangles
Apply trigonometry to general triangles
Circles
Understand and apply theorems about circles
Find arc lengths and areas of sectors of circles
Expressing Geometric Properties with Equations
Translate between the geometric description and the equation for a conic section
Use coordinates to prove simple geometric theorems algebraically
Geometric Measurement and Dimension
Explain volume formulas and use them to solve problems
Visualize relationships between two-dimensional and three-dimensional objects
Modeling with Geometry
Apply geometric concepts in modeling situations

High School: Statistics and Probability Interpreting Categorical and Quantitative Data
Summarize, represent, and interpret data on a single count or measurement variable
Summarize, represent, and interpret data on two categorical and quantitative variables
Interpret linear models
Making Inferences and Justifying Conclusions
Understand and evaluate random processes underlying
statistical experiments
Make inferences and justify conclusions from sample surveys, experiments and observational studies
Conditional Probability and the Rules of Probability
Understand independence and conditional probability and use them to interpret data
Use the rules of probability to compute probabilities of compound events in a uniform probability model
Using Probability to Make Decisions
Calculate expected values and use them to solve problems
Use probability to evaluate outcomes of decisions

### Common Core Standards By Domains

Click on the following links to find lessons suitable for the Common Core Math sorted by domains.

Counting & Cardinality Kindergarten
Know number names and the count sequence.
Count to tell the number of objects.
Compare numbers.

Operations & Algebraic Thinking Kindergarten
Represent and solve problems involving addition and subtraction.
Understand and apply properties of operations and the relationship between addition and subtraction.
Work with addition and subtraction equations.
Represent and solve problems involving addition and subtraction.
Work with equal groups of objects to gain foundations for multiplication.
Represent and solve problems involving multiplication and division.
Understand properties of multiplication and the relationship between multiplication and division.
Multiply and divide within 100.
Solve problems involving the four operations, and identify and explain patterns in arithmetic.
Use the four operations with whole numbers to solve problems.
Gain familiarity with factors and multiples.
Generate and analyze patterns.
Write and interpret numerical expressions.
Analyze patterns and relationships

Number & Operations in Base Ten Kindergarten
Work with numbers 11-19 to gain foundations for place value.
Extend the counting sequence.
Understand place value.
Use place value understanding and properties of operations to add and subtract.
Understand place value.
Use place value understanding and properties of operations to add and subtract.
Use place value understanding and properties of operations to perform multi-digit arithmetic.
Generalize place value understanding for multi-digit whole numbers.
Use place value understanding and properties of operations to perform multi-digit arithmetic.
Understand the place value system.
Perform operations with multi-digit whole numbers and with decimals to hundredths.

Number and Operations and Fractions Grade 3
Develop understanding of fractions as numbers.
Extend understanding of fraction equivalence and ordering.
Build fractions from unit fractions.
Understand decimal notation for fractions, and compare decimal fractions.

Use equivalent fractions as a strategy to add and subtract fractions.
Apply and extend previous understandings of multiplication and division.

Measurement and Data Kindergarten
Describe and compare measurable attributes.
Classify objects and count the number of objects in each category.
Measure lengths indirectly and by iterating length units.
Tell and write time.
Represent and interpret data.
Measure and estimate lengths in standard units.
Relate addition and subtraction to length.
Work with time and money.
Represent and interpret data.
Solve problems involving measurement and estimation.
Represent and interpret data.
Geometric measurement: understand concepts of area and relate area to multiplication and to addition.
Geometric measurement: recognize perimeter.
Solve problems involving measurement and conversion of measurements.
Represent and interpret data.
Geometric measurement: understand concepts of angle and measure angles.
Convert like measurement units within a given measurement system.
Represent and interpret data.
Geometric measurement: understand concepts of volume.

Ratios and Proportional Relationships Grade 6
Understand ratio concepts and use ratio reasoning to solve problems.
Analyze proportional relationships and use them to solve real-world and mathematical problems.

Apply and extend previous understandings of multiplication and division to divide fractions by fractions.
Compute fluently with multi-digit numbers and find common factors and multiples.
Apply and extend previous understandings of numbers to the system of rational numbers.
Apply and extend previous understandings of operations with fractions.
Know that there are numbers that are not rational, and approximate them by rational numbers.

Apply and extend previous understandings of arithmetic to algebraic expressions.
Reason about and solve one-variable equations and inequalities.
Represent and analyze quantitative relationships between dependent and independent variables.
Use properties of operations to generate equivalent expressions.
Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
Expressions and Equations Work with radicals and integer exponents.
Understand the connections between proportional relationships, lines, and linear equations.
Analyze and solve linear equations and pairs of simultaneous linear equations.

Define, evaluate, and compare functions.
Use functions to model relationships between quantities.

Geometry Kindergarten
Identify and describe shapes.
Analyze, compare, create, and compose shapes.
Reason with shapes and their attributes.
Reason with shapes and their attributes.
Reason with shapes and their attributes.
Draw and identify lines and angles, and classify shapes by properties of their lines and angles.
Graph points on the coordinate plane to solve real-world and mathematical problems.
Classify two-dimensional figures into categories based on their properties.
Solve real-world and mathematical problems involving area, surface area, and volume.
Draw construct, and describe geometrical figures and describe the relationships between them.
Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.
Understand congruence and similarity using physical models, transparencies, or geometry software.
Understand and apply the Pythagorean Theorem.
Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.

Develop understanding of statistical variability.
Summarize and describe distributions.
Use random sampling to draw inferences about a population.
Draw informal comparative inferences about two populations.
Investigate chance processes and develop, use, and evaluate probability models.
Investigate patterns of association in bivariate data. Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

## List of Area Worksheets

The children in the 2nd grade and 3rd grade enhance practice with this interesting collection of pdf worksheets on finding the area by counting unit squares. Included here area exercises to count the squares in the irregular figures and rectangular shapes.

Give learning a head start with these finding the area of a square worksheets. Figure out the area of squares using the formula, determine the side lengths, find the length of the diagonals and calculate the perimeter using the area as well.

Strengthen skills in finding the area of a rectangle with these pdf worksheets featuring topics such as determine the area of rectangles, area of rectilinear shapes, rectangular paths and solve word problems. Recommended for grade 3, grade 4, grade 5 and up.

Augment your practice on finding the area with our area of rectilinear figures worksheets! With two or more non-overlapping rectangles composing them, these rectilinear shapes require adding the areas of those non-overlapping parts to arrive at their area.

Focusing on finding the area of triangles, this set of worksheets features triangles whose dimensions are given as integers, decimals and fractions involving conversion to specified units as well. Approx. grade levels: 5th grade, 6th grade and 7th grade.

The area of a parallelogram worksheets comprise adequate skills to find the area of a parallelogram, compute the value of the missing dimensions - base or height, practice finding the area by converting to specific units and more. The exercises are presented as geometric illustrations and also in word format.

This collection of area worksheets encompasses a variety of PDFs to find the area of trapezoids whose dimensions are given as integers, fractions and decimals. Determine the missing parameters by substituting the values in the formulas, solve exercises involving unit conversions too.

Emphasizing on how to find the area of a rhombus the worksheets here contain myriad PDFs to practice the same with dimensions presented as integers, decimals and fractions. Find the diagonal lengths, missing parameters, compute the area, learn to convert to a specified unit and much more.

Improve efficiency in finding the area of kites with these printable worksheets comprising illustrations and exercises in word format. Calculate the area of the kite, find the missing diagonal lengths using the area and much more!

Calculate the area of quadrilaterals whose dimensions are presented as whole numbers and fractions. The worksheets on area of a quadrilateral consist of exercises on rectangles, trapezoids, kites in the form of illustrations, on grids and in word format. Practice conversion to a specified unit in the process.

Reaffirm the concept of finding the area of a circle by using these practice worksheets. Learn to find the area or circumference using the given radius or diameter, calculate the area and circumference, compute the radius and diameter from the area or circumference given and a lot more.

An Annulus is the region enclosed between two concentric circles. How about calculating their area? Plug into this compact set of printable worksheets and practice calculating the area of those circular rings!

The children of grade 5, grade 6 and grade 7 can reinforce their skills in finding the area of mixed shapes by practicing this set of printable worksheets.

Incorporate these area of polygons worksheets comprising examples and adequate exercises to find the area of regular polygons like triangles, quadrilaterals and irregular polygons using the given side lengths, circumradius and apothem. Free worksheets are available for practice.

The area of compound shapes worksheets consist of a combination of two or more geometric shapes, find the area of the shaded parts by adding or subtracting the indicated areas, calculate the area of rectilinear shapes (irregular figures) and rectangular paths as well. This practice set is ideal for 4th grade through 7th grade.

Develop practice in finding the area of a segment of a circle with these practice pdfs. Adequate exercises in finding the area of the triangle and the area of the sector using one of the parameters given are sure to help students master calculating the area of the segment in no time.

## Math Worksheets Worksheets with basic, single-digit addition facts (sums up to 18).

Practice adding 3-digit numbers together with these printable worksheets, task cards, and games.

Challenge your students to solve addition problems with 4 and 5-digit numbers.

Column addition problems with 3 or more addends of 1, 2, 3, and 4-digit numbers.

The worksheets, flashcards, and number lines on this page have a mix of basic, single-digit addition and subtraction problems.

Add fractions and mixed numbers with like denominators, as well as unlike denominators.

Build logical thinking skills with these addition square brainteaser worksheets.

Basic fact family triangles, number bonds, and fact squares worksheets for addition and subtraction.

On these printable game boards, students color pairs of numbers that equal a given sum.

Write algebraic expressions, learn to identify independent/dependent variables, solve for variables in equations, work with inequalities, and more.

Using protractors to find measurements of right, acute, and obtuse angles.

Download worksheets to practice geometry concepts related to vertical, complementary, and supplementary angles.

Use the formula A = 1/2 x (b x h) to calculate the area of triangles

Calculate the area (in square units) of the shapes on these worksheets.

Use these worksheets to help students learn about statistics and creating box-and-whisker plots. There are also worksheets on calculating Q1, Q3, median, minimum, and maximum values.

Calendars to help students with years, months, weeks and days on a calendar.

Volume or capacity worksheets (gallons, quarts, pints, & cups).

Calculate the diameter and radius of circles also includes circumference and area worksheets.

Color the mystery pictures according to the number key at the bottom.

Use these worksheets to practice counting Australian money.

Count toonies, loonies, quarters, nickels, and dimes with these Canadian currency worksheets.

Learn to count pounds and pence, coins used in the United Kingdom.

Practice counting American money (pennies, nickels, dimes, and quarters).

Learn to count and write numbers up to 30.

Learn how to accurately count two, three, and four-digit numbers. Complete the number lines, tell what number comes before, and skip counting.

This area has math daily word problems for grades 1 through 5. There are hundreds of graphical word problems for students to solve, with plenty of space to show their work.

Add and subtract decimal numbers with tenths, hundredths, and thousandths place values.

Practice long division with decimal numbers.

Practice multiplication problems that have decimal factors and products.

Naming and working with decimal numbers.

Basic division facts worksheets, games, and activities.

Long division worksheets with 2, 3, and 4-digit dividends.

Learn to balance simple algebraic equations and find the value of variables.

Identifying odd and even numbers

Find the exponents of single-digit numbers on these printable worksheets and task cards.

Complete factor trees, find greatest common factors, and least common multiples.

Reducing fractions, ordering fractions, equivalent fractions, and comparing fractions.

Identifying basic fractions, fraction strips, fraction manipulatives.

Find sums of fractions and mixed numbers. Includes worksheets with like-denominator fractions, as well as unlike-denominator fractions.

Divide fractions and mixed numbers. Many of these worksheets include illustrated problems, graphical model problems, as well as word problems.

Practice basic mixed number skills.

Multiply the fractions and mixed numbers. Many worksheets include models and diagrams, as well as word problems.

Print these worksheets to help students learn about reciprocal fractions.

Practice subtracting fractions and mixed numbers. Includes like and unlike denominators.

Use frames and arrows activities to build up logical thinking skills.

Interpret the line graphs on the worksheets and answer questions.

Line plots are a special type of number line that represents frequency of data.

Study the pictographs on the worksheets and answer questions.

Analyze the pie graphs on the worksheets and answer questions.

Compare numbers greater than, less than, and equal to.

Use these helpful place value charts, hundreds charts, ninety-nines charts.

Complete these printable input and output boxes, or rule boxes.

Solve and graph the inequalities. Includes single-variable, one-step, and two-step inequalities.

Compare, order, add, and subtract the positive and negative numbers.

Determine the least common multiple of each set numbers

Learn about lines, line segments, rays, parallel lines, and perpendicular lines.

Solve the math problems and use the answers to complete the crossword puzzles.

Solve the math problems to decode the answer to funny riddles. Includes a wide variety of math skills, including addition, subtraction, multiplication, division, place value, rounding, and more.

This page has a set of whole-page reading passages. Students use information from the passages to solve math problems. These are much longer than "regular" word problems.

Calculate the mean, or average, of numbers.

Calculate the mode, median, mean, and range of the given numbers.

Select the types of measurement worksheets you're looking for, including linear measurement, capacity, and temperature.

Practice metric linear measurement: centimeters, millimeters, and meters.

Learn American linear measurement inches, feet, and yards.

Measure the weight and convert to and from grams and kilograms.

Estimate and convert capacity measurements in liters and milliliters.

Measure the weight and convert to and from ounces and pounds.

This index page will link you to dozens of middle school math topics on this site. Topics include inequalities, absolute value, algebra, and more.

Drill-and-practice sheets for basic adding, subtracting, multiplying, and dividing.

These worksheets will help students learn to find and identify multiples of numbers.

Learn about the associative, distributive, commutative, and identity properties of multiplication.

Print these multiplication charts and tables for students to use as reference.

Solve two and three-digit multiplication problems.

Learn basic multiplication facts with these worksheets, mystery pictures, and games.

Basic number bonds, fact family worksheets and triangles for division and multiplication.

Multiply by 2, 3, or 4 digit numbers with a lattice grid.

Plot the ordered pairs to reveal mystery pictures.

Solve the addition, subtraction, multiplication, and division facts to reveal a mystery picture.

Students will used the clues posted each day to figure out the weekly secret number.

Printable number line worksheets for teaching counting, addition, subtraction, number patterns, fractions, and decimals.

Practice order of operations: parenthesis and exponents, then multiply and divide, then add and subtract.

Coordinate planes and ordered pair worksheets.

Ordinal numbers define position in a series. (examples: first, second, third, fourth, etc.)

Students must list the numbers that come next in these patterns.

Students determine which pictures come next in the patterns.

Convert from decimal numbers and fractions to percents.

Add to find the perimeters of polygons on these worksheets.

Finding the value of the underlined digit value standard and expanded form rounding.

Learn the differences between prime and composite numbers. Also, learn to identify and find prime numbers by factoring, or by using the Sieve of Eratosthenes.

Identifying and working with polygon shapes.

Determine the probability of certain outcomes.

Printable puzzle pieces that you can cut out for students to match up. Great for learning centers, small group activities and independent practice.

Find the lengths of the sides of a right triangle with the Pythagorean theorem formula.

Students use an iPad or smartphone to scan QR codes to complete or check problems on each of these math worksheets.

Practice comparing pairs of quantities using ratios.

Identify the reflected, rotated, and translated shapes.

Learn to read and write Roman numerals with these printable worksheets and activities.

Rounding numbers to the nearest tens and hundreds.

Learn to write numbers in scientific notation.

On these worksheets, students will use the key to decode the secret numbers in each addition, subtraction, multiplication, or division problem.

Identify similar and congruent shapes.

Use these worksheets to teach students about skip counting by hundreds.

Review counting by tens with these printables.

Count elevens, multiply by elevens, and fill in missing numbers.

Fill in empty boxes and word problems by counting by twelves.

Skip counting by twenty-fives is important if you're teaching your students to count money.

Use these printables to teach students to skip count by intervals of two.

The set of worksheets on this page will help students learn to count by threes.

If you're teaching students to count by fours, try these worksheets.

When you teach students to count nickels or count by fives, these worksheets may be helpful to you.

Tell whether the shapes were flipped, slid, or turned.

On these worksheets, students will use graphs, ordered pairs, and tables to calculate the slopes of straight lines.

Name the geometric solid shapes: rectangular prisms, cubes, spheres, and cylinders.

These worksheets cover a variety of place value concepts, including even/odd, finding the value of digits, and writing numbers in expanded form.

Single and multi-digit subtractions. Includes 1, 2, 3, and 4-digit numbers. There are links to decimal and money subtraction too.

Practice basic, single-digit subtraction facts concept and drill worksheets.

Practice subtracting fractions and mixed numbers with these printable worksheets and task cards.

Calculate the surface area of rectangular prisms and other three-dimensional shapes.

Find the lines of symmetry, identify symmetrical figures, and complete the symmetrical shapes.

Learn to count objects with tally marks.

Read the Celsius and Fahrenheit thermometers and state the temperature.

Printable ten-frame activities for teaching counting, basic addition, and simple subtraction.

Teach students to recognize geometric tessellation of two-dimensional shapes.

Learn about telling time to the nearest minute, hour, and quarter hour.

Determine the amount of time that has passed.

These worksheets feature pictures of graduated cylinders. Students must write the correct volume in milliliters.

Calculate the volume of solid shapes. Includes volume "counting cubes," rectangular prisms, cones, cylinders, and spheres.

Practice word problems for addition, subtraction, multiplication, and division.

Review mixed word problem skills at different grade levels.

These word problems have multiple steps, and require students to use critical thinking skills.

Index of our basic and advanced-level fraction worksheets.

Learn area, perimeter, symmetry, polygons, solid shapes, and more.

Choose from pie graphs, bar graphs, and line graphs.

Count by 2s, 3s, 4s, 5s, 10s, 25s, or 100s.

Here's an index page that links to all of the subtraction sections of our website.

This page contains only a partial index of the math skills worksheets on S.T.W.

View the complete index of all Math, ELA, Spelling, Phonics, Grammar, Science, and Social Studies worksheets found on this website.

## Related topics Sample Word Problems Worksheet

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7th grade math introduces kids to many new concepts that build heavily on what was taught in the earlier grades. The math worksheets for Grade 7 available online are an effective way to get kids to practice math and sharpen their math skills. ### Calculate the Surface Area

Use this basic 7th grade math worksheet to learn to calculate surface area. See more ### Percent Problems

Use this basic 7th grade math worksheet to learn to calculate percentage. See more ### Scale Drawings

Use this basic 7th grade math worksheet to learn how to draw to scale. See more ### Constructing Triangles

Use this basic 7th grade math worksheet to learn how to construct a triangle. See more ### Multiplication and Division of Rational Numbers

Use this basic 7th grade math worksheet to learn to multiply and divide integers. See more ### Addition and Subtraction of Integers

Use this basic 7th grade math worksheet to learn to add and subtract integers. See more ### Volume of a Cube

Use this basic 7th grade math worksheet to understand how to calculate the volume of a cube. See more ### Value of Algebraic Expression

Use this basic 7th grade math worksheet to understand algebra. See more ### Ratio Word Problems

Use this basic 7th grade math worksheet to understand ratio. See more ### Subtracting Fractions

Use this basic 7th grade math worksheet to practice subtracting fractions. See more ### Round the Numbers

Use this basic 7th grade math worksheet to master the art of simplifying expressions. Download it now! See more ### Simplify the Expressions

Use this basic 7th grade math worksheet to master the art of simplifying expressions. Download it now! See more ### Find the Place Value – 1

Use this free printable 7th grade place value worksheet to sharpen your middle schooler’s math skills. See more ### Find the Place Value – 2

Use this free printable 7th grade place value worksheet to sharpen your middle schooler’s math skills. See more ### Find the Average – 1

Here’s a printable seventh grade math worksheet to test your math smarts can you find the average of these sets of numbers? See more ### Find the Average – 2

Finding the average of a set of numbers is easy! Try your math smarts with this simple math worksheet for middle school. See more Integer operations is a crucial topic for 7th grade. Here’s a printable math worksheet to help your child understand the rules. See more  ### Find the GCF – 1

Teaching your child how to calculate GCF is a breeze when you have this printable math worksheet for seventh grade!See more ### Find the GCF – 2

The largest number that can evenly divide two numbers (a. k. a. Greatest Common Factor) is challenging you to find it!See more ### Understanding Pythagoras

‘Understanding Pythagoras’ is a self explanatory math worksheet for kids of 7th grade who’ve just been introduced to the distance theorem . See more ### Speedy Speed

Do you find it difficult to explain the distance metrics to your child? Or, does your child find the distance metrics so complex that she prefers to . See more ### Solving Pythagoras - 2

‘Understanding Pythagoras’ is the math worksheet that needs to be downloaded and understood by students before moving on to this one. See more ### Solving Pythagoras - 1

‘Solving Pythagoras – 1’ is a cool Pythagoras worksheet for middle school kids. Students who find it difficult to grasp the concept of Pythagoras . See more ### Distance and Speed

If it’s a session of ‘Need for Speed’ that your youngster at home has just played, then sit him down with this cool distance printable.See more ### Complex Distance

Does your child find distance problems so complex that she prefers ignoring those most often? Fret not, here is a fun distance worksheet. See more ### Make Proportion

Make peace with proportion problems with this easy, fun proportion worksheet! With a given set of numbers, students have to . See more ### Missing Proportion

Knowing how to calculate proportions is an integral part of our daily lives! Its application in simple tasks like cooking, exercising, and driving . See more ### More Proportions

Just as its name suggests, ‘More Proportion’ is a simple, fun, proportion worksheet that offers plenty of practice to the kids.See more ### Plenty of Proportion

Proportions are used in our daily lives to estimate prices of things and compare a figure with another. You can apply proportions in the simplest of daily tasks, . See more ### Proportional or Not

Proportional or Not’ is a beginner-level proportion worksheet that tests students’ preliminary understanding of the concept. See more ### Proportions and Rates

How important it is to learn proportion? Is it important at all? You will have the answer once you are introduced to the concept of proportion in your class.See more ### Proportion-X

After solving elementary proportion worksheets like ‘Plenty of Proportion’ and ‘More Proportions’ one after another, . See more ### Table Top

This is a cracking tables worksheet perfect for 7th graders that need to be filled with negative multiples. The grid has a few blank cells and a few already-filed cells. See more Repeated multiplication of the same number by itself is the theory of exponents explained simply. Any number expressed by exponent 1. See more ### All Mixed Up

Mixed equations are a great way to learn the complex steps leading to advanced math problems. 7th graders are introduced to this complex yet cool. See more How easy for you to tell the answer to this one: 2x1x3x2x4x5x10? Knowing tables and counting them effortlessly gives one a grip over the cool subject of math.See more ### Mix and Solve

Advanced math problems and equations are always made easy if there’s enough expertise and skills in solving mixed equations.See more ### Mixed 'Em Up

7th grade math is not an easy job! It is the important stage where students learn to synthesize many concepts that they’ve learnt in earlier grades. See more ### Per-cent-age

Repeated multiplication of the same number by itself is the theory of exponents explained simply. Any number expressed by exponent 1. See more ### Reducing Fractions

Reducing fractions is a great way to recollect earlier learnt concepts in math like fractions and divisions. The more 7th graders practice. See more ### Solve the Expos

The theory of exponents can be explained very simply. If a number is multiplied by itself repeatedly, it is definitely a math of exponents – also called powers.See more ### Tell the Percentage

Interesting figures are expressed in percentages almost 99% of times. Some examples are 94% of life on Earth is aquatic, 70% of the planet is ocean, 50% of the. See more ### Two a Min

Repeated multiplication of the same number by itself is the theory of exponents explained simply. Any number expressed by exponent 1. See more Repeated multiplication of the same number by itself is the theory of exponents explained simply. Any number expressed by exponent 1See more ### Characteristics of a Triangle

“Characteristics of Solids” is a great worksheet that lets students explore and manipulate the basic solid shapes in their three dimensional forms before discussing their characteristics. See more ### Cat and Rat

‘Cat and Rat’ is a cool geometry worksheet that shows kids how to design a maze of their own using only a grid and some tracing paper! See more ### Challenge! Construct Angles

Drawing angles is easy-peasy when you have a protractor, but it is also possible to draw angles without one. Challenge yourself to try and construct the angles in this worksheet without a protractor. See more ### Characteristics of Solids

“Characteristics of Solids” is a great worksheet that lets students explore and manipulate the basic solid shapes in their three dimensional forms before discussing their characteristics. See more ### Angles of Polygons

"Angles of Polygons” is a simple yet comprehensive worksheet that familiarizes sixth graders with the formula used to determine the sum of the angles of a polygon. See more ### Area of Parallelograms

Do your students find geometry formulae difficult to remember? Well that’s sure to change with this geometry worksheet, “Area of Parallelograms”. See more ### Area of Trapezoids

This simple geometry worksheet will help your students understand how the formula for the area of a trapezoid is derived.See more ### Area of Triangles

One of the earliest geometry formulae learnt is that for the area of a triangle. Help your young learner arrive at the formula herself with this simple geometry worksheet for kids. See more ### Area of Circles

“Area of Circles” is an incredibly simple geometry worksheet that will have your budding mathematician deriving geometry formulae in no time!See more ### The Whole Truth

Find out with our ‘The Whole Truth’ worksheet! In this worksheet, kids need to read each of the sentences written and fill in the blanks. See more ### Name That Angle

Do your kids know all about the different angles? Find out with our ‘Name That Angle’ worksheet. See more ### True or False?

This simple ‘True or False?’ printable geometry worksheet will be able to test how much kids know about angles. See more ### Label The Diagram

‘Label the Diagram’ is a printable geometry worksheet in which students need to label the figure drawn as per the instructions provided. See more 