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Sixth Grade Math Worksheets

Free grade 6 worksheets from k5 learning.

Our printable grade 6 math worksheets delve deeper into earlier grade math topics (4 operations, fractions, decimals, measurement, geometry) as well as introduce exponents, proportions, percents and integers .

Choose your grade 6 topic:

4 operations.

Place Value & Scientific Notation

Add & Subtract

Multiply & Divide

Decimals & Percents

Fractions to / from Decimals

Decimals - Add and Subtract

Decimals  - Multiply

Decimals - Divide

Add & Subtract Fractions

Multiply & Divide Fractions

Converting Fractions

Sample Grade 6 Math Worksheet

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6th Grade Math Test

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Prepare for your states math test with this Grade 6 practice assessment for Common Core.

• 40 multiple choice questions with detailed answers to support test preparation
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• Aligned to Common Core Standards

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Get ready for 6th grade

Unit 1: get ready for ratios, rates, and percentages, unit 2: get ready for arithmetic operations, unit 3: get ready for negative numbers, unit 4: get ready for equations, expressions, and inequalities, unit 5: get ready for geometry.

Free Printable Performance Tasks Worksheets for 6th Grade

Performance Tasks: Discover a vast collection of free printable math worksheets for Grade 6 students, designed to enhance their problem-solving skills and understanding of mathematical concepts. Dive into our resources and help your students excel in their learning journey!

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Explore printable Performance Tasks worksheets for 6th Grade

Performance Tasks worksheets for Grade 6 are essential tools for teachers to help their students excel in Math. These worksheets are designed to challenge students with a variety of Math Word Problems, which are crucial for developing problem-solving skills and critical thinking. By using these worksheets, teachers can ensure that their Grade 6 students are exposed to a wide range of mathematical concepts, from basic arithmetic to more advanced topics such as algebra, geometry, and data analysis. The worksheets are also an excellent resource for assessing students' understanding of the material, as they provide a clear and measurable way to track progress. Teachers can easily incorporate these Performance Tasks worksheets for Grade 6 into their lesson plans, ensuring that their students are well-prepared for success in Math.

Quizizz is an innovative platform that offers a variety of resources for teachers, including Performance Tasks worksheets for Grade 6 Math. In addition to these worksheets, Quizizz also provides engaging and interactive quizzes that can be used to assess students' understanding of Math Word Problems and other mathematical concepts. Teachers can easily create their own quizzes or choose from a vast library of pre-made quizzes, which can be customized to suit their specific needs. The platform also offers real-time feedback and analytics, allowing teachers to identify areas where their Grade 6 students may need additional support or practice. By incorporating Quizizz into their teaching strategies, educators can ensure that their students have access to a comprehensive and engaging learning experience, which will ultimately help them excel in Math.

Math Mammoth placement tests for grades 1-8 (free math assessment)

These free diagnostic tests help you discover your child or student's level in math, and to find out EXACTLY where they have gaps (if any). They are end-of-year (EOY) tests — in other words, meant to be taken AFTER studying the particular grade.

While the tests ARE placement tests for Math Mammoth complete curriculum , they work equally well as GENERIC math assessment tests.

Using the tests will allow you to tailor the instruction to help the child/student right where they need help the most. You can EASILY find out if your student has not learned specific topics, because the tests group the questions by topic . Just look at the test results per "sections" of similar problems (addition, multiplication, measuring, geometry, place value, problem solving).

You can always send me your child's test results if in doubt. (Just send me the point counts for each test item.)

To start, look over the tests and try to find the highest level one that you THINK your student or child will pass. Then you would naturally administer that test.

The video below explains the basics of HOW to use these tests:

This table shows the possible score ranges and how to interpret the result:

However, it's quite possible that just that one test will not give you a good overview of what the student has mastered. So don't be afraid of administering two or even three tests — or parts of tests. For example, let's say a student gets 70% on a test, and in particular, they get a low score in the geometry section of the test. In that case, it is beneficial to administer the GEOMETRY sections of the next lower level tests also, in order to find out how "large" a gap they have in geometry.

Sometimes, a student who is seriously behind can get a low score on even THREE subsequent tests. Basically, their skills and knowledge are "all over the place" so to speak. Such students can benefit from a tailored approach where the materials they will study will be exactly matched to the gaps they have.

One more important thing. To reduce the test-taking stress, don't put any strict time limit for the completion of the test. We don't want students to build any anxiety over this placement test or tests. Now, some children have already developed anxiety towards test-taking and/or towards math in general. In such cases, it might be best to avoid even these tests, or at least postpone the test-taking until the anxiety has subsided somewhat.

• End-of-Year Test, Grade 1 Answer Key
• End-of-Year Test, Grade 2 Answer Key
• End-of-Year Test, Grade 3 (2024 edition) Answer Key (2024 edition)
• End-of-Year Test, Grade 3 (2018 edition) Answer Key (2018 edition)
• End-of-Year Test, Grade 4 Answer Key
• End-of-Year Test, Grade 5 Answer Key
• End-of-Year Test, Grade 6 Answer Key
• End-of-Year Test, Grade 7 (pre-algebra) Answer Key
• End-of-Year Test, Grade 8 Answer Key

Canadian versions

These are all-metric and formatted to Letter size paper. For grades 1-2, the tests use Canadian coins and bills.

• End-of-Year Test, Grade 1 (Canadian currency) Answer key
• End-of-Year Test, Grade 2 (Canadian currency) Answer key
• End-of-Year Test, Grade 3 Answer key

International versions (all metric)

The international versions are formatted to A4 paper size, use a space for thousands separator (35 230), and only use metric units. The tests for grades 1 and 2 include a section on counting coins, which is with Australian currency. The rest of the tests don't have questions about coins or bills, and use the "$" symbol for currency.

• End-of-Year Test, Grade 1 (International version) Answer key
• End-of-Year Test, Grade 2 (International version) Answer key
• End-of-Year Test, Grade 3 (International version) Answer key
• End-of-Year Test, Grade 4 (International version) Answer key
• End-of-Year Test, Grade 5 (International version) Answer key
• End-of-Year Test, Grade 6 (International version) Answer key
• End-of-Year Test, Grade 7 (International version) Answer key

International-UK versions with pounds & pence (all metric)

The versions below are formatted to A4 paper size and only use metric units. They use British coins and bills.

• End-of-Year Test, Grade 1 (British version) Answer key
• End-of-Year Test, Grade 2 (British version) Answer key
• End-of-Year Test, Grade 3 (British version) Answer key

International-euro versions with euro coins and bills & decimal comma

The versions below are formatted to A4 paper size, use a space for thousands separator (35 230), and only use metric units. They use euro coins and bills, and a decimal comma for money amounts.

• End-of-Year Test, Grade 1 (euro) Answer key
• End-of-Year Test, Grade 2 (euro) Answer key
• End-of-Year Test, Grade 3 (euro) Answer key

Tip: If you notice your child has problems with one or two concept areas in a particular test, but otherwise does fine, you can use Math Mammoth Blue Series books to "fill in those gaps". You can also email me with your test results if you have trouble deciding which Blue Series book you should use for such remedial work.

Math Mammoth grade-level review workbooks can also be used to pinpoint areas of weakness or to review a whole grade level of topics.

Background : the student began homeschooling the same year as the tests were taken and is technically in 7th grade. She had started doing Teaching Textbooks Algebra 1. However, the parent noticed that she didn't know some basic fraction information, decided to look for a supplement, and came across Math Mammoth.

The student took three end-of-year tests: for grade 6, grade 5, and grade 4. She did not do very well in the grade 6 test, but you can see her work for grade 4 and grade 5 tests in the links below. If you decide to send me your child's test results, it is not necessary to send me the complete test like this person did, but it makes a good example. Most people send me only the point counts for the various test items.

• Grade 5 test, complete work of the student
• Grade 5 test, the student's score
• Grade 6 test, complete work of the student
• Grade 6 test, the student's score

And here is my assessment and recommendations:

• Early Geometry , because she doesn't seem to understand the concepts of area and perimeter – which are of paramount importance. This book is for grades 1-3 so some topics in it are too easy and she can skip them.
• Geometry 1 – meant for grades 4-5. She will need to study the lessons that are in 4th grade level first (angles, quadrilaterals, triangles, etc.) Please note some of the content here overlaps with MM grade 5. She can study the overlapping lessons from either source.
• Measuring 1 – it is for grades 1-3 but she may need to review everything from the beginning when it comes to measuring units. Some lessons may be too easy and can be omitted - use your judgment on that.
• Measuring 2 – for grades 4-6. Please note some of the content here overlaps with MM grades 5 and 6. She can study the overlapping lessons from either source.
• Decimals 1 She has not grasped the basics of how fractions and decimals tie together – which, once again, is a very fundamental concept. She should study Decimals 1 book before starting the decimals chapter in grade 5.

Questions about placement

• My K student wants more challenge and to do 1st grade math. How can I know if the Light Blue grade 1 is the right choice for my child? The Light Blue grade 1 starts out with the same material as these books from the Blue Series: Addition 1 and Subtraction 1 . Look over those, see their samples and contents, and that will probably tell you if your children are ready for grade 1.
• I thought my child had been doing fine in math previously (getting A's and B's), but he/she still failed the placement test for their particular grade level. This is common for a child coming from public school or from another math curriculum. It is typically due to the facts that (i) Math Mammoth uses multi-step word problems extensively from 2nd grade onward; and (ii) the child may not have covered all the topics in the other curriculum as what are covered in MM. Geometry & measuring often end up being such areas. To catch up, please see the Tip above about using the Blue Series books. Another thing you can do is accelerate through the earlier levels of Math Mammoth, for example by studying the chapter reviews, and stopping to study any concept that the child has not understood. This kind of process can typically be done in only a few months, so don't worry! It is much more important to get a solid foundation in math than to look at the grade levels and just go on. In math, it is crucial to understand the previous concepts, because if not, the student WILL eventually fail and most also start hating math. See also this thread on a homeschool discussion forum: Unexpected experience with end-of-year test for Math Mammoth .

Student attempted the 2nd grade EOY test: Add & Subtract 2-A, Add & Subtract 2-B, and half of Add & Subtract 3.

Student attempted the 3rd grade EOY test: (possibly Add & Subtract 2-A) Add & Subtract 2-B, Add & Subtract 3, Multiplication 1, and Division 1.

Student attempted the 4th grade EOY test: Add & Subtract 3, Multiplication 1, Division 1, Multiplication 2, and Division 2.

• I'm a Math-U-See user and my child has not studied some of the materials you cover earlier than MUS. How do I switch over to Math Mammoth? I'd simply recommend you first have her take a placement test to know exactly what she knows and what she doesn't about those topics that MUS does not cover as much, such as measuring, money, clock, geometry and problem solving. Then once you have had the child take the test, you can choose books from the Blue Series to address specific topic areas (see the chart above). Remember also, I'm ready to guide you with these decisions. Just let me know the child's test results by sections.

By Maria Miller

© 2006-2024 MathMammoth.com

6th Grade Logic and Problem Solving Worksheets

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• 6th Grade Math
• Solving word problems

Grade 6 Math Word Problems: Tips, Tricks, and Answers

Do you want to stimulate your 6 th grader’s creative thinking skills? Then, enjoy this brilliant math article. In this comprehensive guide, we will provide you with a treasure trove of fun solving strategies, tips, tricks, and answers to tackle those tricky grade 6 math word problems that have been confusing your students for some time now.

In this page, you will discover why math word problems are important for 6th-graders and simple methods of breaking down complex word problems into manageable steps.

Nevertheless, we will introduce you to Mathskills4kids.com , an outstanding website with thousands of common types of grade 6 math word problems and a step-by-step approach to solving them. Interestingly, we will illustrate how to use diagrams and models to solve math word problems efficiently.

Learn to love Grade 6 Math word problems with these worksheets and answers

Hello and welcome to Grade 6 Math word problems worksheets and answers , where your 6 th Grade students will learn to love and solve math problems and activities at all times.

We understand that word problems can often frustrate students, as they require a solid understanding of mathematical concepts and the ability to interpret and apply them to real-life situations. That's why we have compiled a collection of proven strategies and techniques to empower your students to approach word problems confidently and accurately.

From understanding problem-solving strategies to breaking down complex questions into manageable steps, this guide will equip you with the knowledge and resources to make math word problems a breeze. So, let's dive in and unlock the secrets to conquering grade 6 math word problems together!

BROWSE THE WEBSITE

Download free worksheets, grade 6 math topics.

• Whole numbers

Multiplication

• Exponents and square roots
• Number theory
• Add & subtract decimals
• Multiply & divide decimals
• Fractions & mixed numbers
• Add & subtract fractions
• Multiply fractions
• Divide fractions
• Operations with integers
• Mixed operations
• Rational numbers
• Problems solving
• Ratio & proportions
• Percentages
• Measuring units
• Consumer math
• Telling time
• Coordinate graph
• Algebraic expressions
• One step equations
• Solve & graph inequalities
• Two-step equations
• 2D Geometry
• Symmetry & transformation
• Geometry measurement
• Data and Graphs
• Probability

Start practice on Sixth Grade here

Why are math word problems important for 6th-graders.

Math word problems are about more than just doing calculations. They are also about applying your math knowledge to real-world situations. Math word problems are important for 6 th graders because they help them to:

• Learn how to use different math concepts and skills in various contexts
• Develop their logical thinking and reasoning abilities
• Enhance their communication and literacy skills
• Prepare them for standardized tests and future math courses

Math word problems also make math more exciting and relevant. They show us how math can solve everyday problems and challenges. They also expose us to different topics and scenarios we may not encounter in our regular math lessons.

Strategies for solving Grade 6 math word problems

Solving Grade 6 math word problems can be intimidating, especially involving multiple steps or operations. But don't worry. Some general strategies will help your students confidently approach any word problem. Here are some of them:

• Please encourage them to read the problem carefully and identify the given information, the unknowns, and the question.
• They should rewrite the problem in their own words or summarize it in a sentence.
• Let them choose a suitable method or strategy to solve the problem. Some standard methods are guessing and checking, making a table or chart, drawing a picture or diagram, using a formula or equation, working backward, or using logical reasoning.
• They must show their work and explain each step clearly. Use appropriate units, labels, symbols, and terms.
• Lastly, tell them to check their answer by plugging it back into the problem or using a different method. Ensure their answer makes sense and answers the question.

Breaking down complex word problems into manageable steps

Some word problems may seem too complex or confusing at first glance. They may have too much information, too many steps, or too many operations. In such cases, breaking down the problem into smaller and simpler parts is helpful. Here are some tips on how to do that:

Words like "difference," "subtract," "take away," or "minus" indicate s ubtraction .

Words like "product," "multiply," "times," or "of" indicate multiplication .

Words like "quotient," "divide," "per," or "out of" indicate division .

Words like "ratio," "fraction," "percent," or "part" indicate fractions or decimals .

Words like "equal," "same as," or "is" indicate equations .

Words like "more than,” “less than," "greater than," or "smaller than" indicate inequalities .

Words like "average," "mean," or "median" indicate statistics .

Words like "area," "perimeter," "volume," or "surface area" indicate geometry , etc.

• Use parentheses, brackets, or other symbols to group the parts of the problem that belong together . For example, if the problem says:

You can rewrite it as:

• (John has 12 apples) + (Mary has 8 apples) = (total number of apples) / (4 people) = (number of apples per person)

This way, you can see the structure of the problem more clearly and focus on one part at a time.

• Solve each part of the problem separately and write down the intermediate results . For example, using the previous problem:
• (John has 12 apples) + (Mary has 8 apples) = (total number of apples)
• 12 + 8 = 20
• (total number of apples) / (4 people) = (number of apples per person)

This way, you can keep track of your work and avoid making mistakes.

• Combine the intermediate results to get the final answer. For example, using the previous problem :

This way, you can answer the question and check your answer.

Common types of grade 6 math word problems

There are many types of word problems that you may encounter in grade 6 math . Some of the most common ones found on Mathskills4kids.com are:

• Ratio and proportion problems : These problems involve finding the relationship between two quantities with the same unit or measure. For example, if 12 pencils cost $3, how much do 20 pencils cost?
• Percent problems : These problems involve finding the part, whole, or percent of a quantity. For example, if 30% of a class is boys, and there are 24 students, how many boys are there?
• Fraction problems : These problems involve adding, subtracting, multiplying, or dividing fractions. For example, how much pizza do you have left if you have 2/3 of a pizza and eat 1/4?
• Decimal problems : These problems involve adding, subtracting, multiplying, or dividing decimals. For example, if you buy a shirt for $12.50 and a pair of jeans for $24.75, how much change do you get from $50?
• Measurement problems : These problems involve converting between different units of measurement or finding the perimeter, area, or volume of shapes. For example, if a rectangle has a length of 15 cm and a width of 10 cm, what is its area in square meters?
• Algebra problems : These problems involve finding the value of an unknown variable or expression. For example, if x + 5 = 13, what is the value of x?

A step-by-step approach to solving different types of word problems

No matter what type of word problem your 6 th graders are faced with, they can use the following steps to solve it:

• They should read the problem carefully and identify the given information and the question.
• They must choose a variable to represent the unknown quantity and write an equation or expression that relates the given information and the question.
• They can now solve the equation or expression and find the variable's value.
• They can check their answer by plugging it back into the equation or expression to see if it makes sense.
• They can now write their answer in complete sentences and include the appropriate units.

Using diagrams and models to solve math word problems

Sometimes, it can be helpful to use diagrams and models to visualize the problem and make it easier to solve. Some examples of diagrams and models for solving math word problems are:

• Tape diagrams : These are horizontal or vertical bars showing the relationship between two quantities. For example, you can use a tape diagram to show how much money each person gets when $60 is shared equally among 4 people.
• Number lines : These horizontal lines show numbers and their relative positions. For example, you can use a number line to show how to add or subtract fractions with different denominators.
• Tables : These are grids that show data in rows and columns. For example, you can use a table to show how to find equivalent ratios or fractions.
• Charts : These are graphical representations of data using bars, circles, lines, or other shapes. For example, you can use a chart to show how to find the percent of a quantity or compare different quantities.
• Drawings : These are sketches or illustrations that show shapes or objects. For example, you can use a drawing to show how to find a shape's perimeter, area, or volume.

Providing answers and explanations to sample Mathskills4kids’ Grade 6 math word problems

Here are some sample Grade 6 math word problems with answers and explanations available at Mathskills4kids.com :

Answer : 6 cups of flour

Explanation : This is a ratio problem. We can use a tape diagram to show the relationship between flour and sugar.

Flour →|<---2 cups--->|<---2 cups--->|<---2 cups---> = 6

Sugar→|<---3 cups--->|<---3 cups--->|<---3 cups---> = 9

We can see that for every 3 cups of sugar, we need 2 cups of flour. So, for 9 cups of sugar, we need 6 cups of flour.

Answer : $15

Explanation : This is a percent problem. We can use a formula to find the sale price of the shirt.

Sale price = Original price - Discount

Discount = Percent off x Original price

We know the percent off is 40%, and the original price is $25. So, we can plug these values into the formula and solve for the sale price.

Discount = 40% x $25

Discount = 0.4 x $25

Discount = $10

Sale price = $25 - $10

Sale price = $15

Tips for improving problem-solving skills in Grade 6 math

Here are some tips that can help students improve their problem-solving skills in grade 6 math :

• Please encourage them to practice regularly and try different types of word problems.
• They should review the concepts and skills they have learned and apply them to new situations.
• They can use different strategies and methods to solve word problems and compare their results.
• Let them ask for help from the teacher, parents, or peers if they get stuck or confused.
• They should learn from their mistakes and try to avoid them in the future.

Bonus: additional resources to reinforce Grade 6 math problem skills

If you want to enhance your student's Grade 6 math word problem skills , or if you need some extra help, here are some additional and useful web links that you can check out:

• Math Playground : This website has a lot of fun and interactive games that let 6 th graders practice different types of math word problems, such as fractions, decimals, ratios, proportions, and more. They can also watch videos explaining how to solve some problems. https://www.mathplayground.com/wordproblems.html .
• Khan Academy : This website has many videos and exercises covering various topics in Grade 6 math, including word problems. Students can learn at their own pace and track their progress. https://www.khanacademy.org/math/cc-sixth-grade-math .
• IXL : This website has many practice questions aligned with the Common Core standards for Grade 6 math. Students can choose from different categories of word problems, such as expressions and equations, geometry, statistics, and more. They can also get instant feedback and explanations for their answers. https://www.ixl.com/math/grade-6 .
• Math Goodies : This website has a lot of worksheets and lessons that teach 6 th graders how to solve different types of word problems, such as percent, ratio, proportion, and more. They can also find tips and tricks for solving word problems faster and easier. https://www.mathgoodies.com/math-mammoth/worksheets/pdf/grade_6_word_problems.pdf .

Thank you for sharing the links of MathSkills4Kids.com with your loved ones. Your choice is greatly appreciated.

Math word problems can be challenging for many 6th-graders, but they are also important for developing their mathematical thinking and reasoning skills.

Using the strategies and steps we discussed in this article, your student's ability to solve different types of word problems in Grade 6 math can be improved.

You can also use the diagrams and models we have shown you to help your students visualize the problem and find the solution. Encourage them to practice as much as possible, and they must not be afraid to ask for help if needed.

You can also use the web links that we have provided to reinforce your student’s learning and have fun with math. We hope this article has helped your 6th-grade students feel more confident and prepared for tackling Grade 6 math word problems.

Happy problem-solving!

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Mathematics Assessment Project

Classroom challenges, formative assessment lessons for grade 6, index of classroom challenges.

Classroom Challenges are lessons that support teachers in formative assessment. There are 100 lessons in total, 20 at each grade from 6 to 8 and 40 for ‘Career and College Readiness’ at High School Grades 9 and above. Some lessons are focused on developing math concepts, others on solving non-routine problems.

The Brief Guide for teachers and administrators (PDF) is recommended reading before using these lessons for the first time.

Finding a Lesson

There are several ways of navigating:

• Use the menu on the left to browse by grade and topic area, or search for a key phrase.
• The complete set of lessons is listed below in alphabetic order.

Adding and Subtracting Directed Numbers

Creating a Measure of Slope

Designing 3D Products: Candy Cartons

Evaluating Statements About Number Operations

Evaluating Statements: Consecutive Sums

Finding Factors and Multiples

Interpreting Equations

Interpreting Multiplication and Division

Maximizing Profit: Selling Soup

Modeling Relationships: Car Skid Marks

Optimizing Coverage: Security Cameras

Representing Data With Grouped Frequency Graphs and Box Plots

Representing the Laws of Arithmetic

Representing Variability with Mean, Median, Mode, and Range

Sharing Costs Equitably: Traveling to School

Translating between Fractions, Decimals and Percents

Using Coordinates to Interpret and Represent Data

Using Proportional Reasoning

Using Space Efficiently: Packing a Truck

Using Standard Algorithms for Number Operations

Analyzing Games of Chance

Applying Angle Theorems

Classifying Proportion and Non-Proportion Situations

Comparing Data Using Statistical Measures

Comparing Strategies for Proportion Problems

Describing and Defining Quadrilaterals

Describing and Defining Triangles

Designing a 3D Product in 2D: A Sports Bag

Designing: A Game of Chance

Drawing to Scale: A Garden

Estimating Volume: The Money Munchers

Evaluating Statements About Probability

Finding Areas of Circles

Increasing and Decreasing Quantities by a Percent

Maximizing Area: Gold Rush

Modeling: Hot and Cold

Representing: Road Race

Sampling and Estimating: Counting Trees

Solving Linear Equations

Using Positive and Negative Numbers in Context

Applying Properties of Exponents

Building and Solving Linear Equations

Classifying Solutions to Systems of Equations

Comparing Fuel Consumption: Buying Cars

Comparing Lines and Linear Equations

Comparing Value for Money: Baseball Jerseys

Defining Lines by Points, Slopes and Equations

Discovering the Pythagorean Theorem

Estimating Length Using Scientific Notation

Finding the Shortest Route: A Schoolyard Problem

Generalizing Patterns: The Difference of Two Squares

Identifying Similar Triangles

Interpreting and Using Data: Setting Taxi Fares

Interpreting Distance–Time Graphs

Matching Situations, Graphs and Linear Equations

Modeling: Making Matchsticks

Representing and Combining Transformations

Sampling and Estimating: How Many Jellybeans

Solving Linear Equations in One Variable

Translating Between Repeating Decimals and Fractions

Using Data: Testing a New Product

High School

Building and Solving Complex Equations

Calculating Arcs and Areas of Sectors of Circles

Calculating Volumes of Compound Objects

Classifying Equations of Parallel and Perpendicular Lines

Classifying Rational and Irrational Numbers

Deducting Relationships: Floodlight Shadows

Devising a Measure: Correlation

Evaluating Conditions for Congruency

Evaluating Statements About Enlargements

Evaluating Statements About Length and Area

Evaluating Statements about Radicals

Evaluating Statements about Rational and Irrational Numbers

Generalizing Patterns: Table Tiles

Generating Polynomials from Patterns

Inscribing and Circumscribing Right Triangles

Interpreting Algebraic Expressions

Interpreting Data: Muddying the Waters

Maximizing Profits: Selling Boomerangs

Modeling Motion: Rolling Cups

Modeling Population Growth: Having Kittens

Proving the Pythagorean Theorem

Representing 3D Objects in 2D

Representing Conditional Probabilities 1

Representing Conditional Probabilities 2

Representing Data with Box Plots

Representing Data with Frequency Graphs

Representing Functions of Everyday Situations

Representing Inequalities Graphically

Representing Linear and Exponential Growth

Representing Polynomials Graphically

Representing Probabilities: Medical Testing

Representing Quadratic Functions Graphically

Representing Trigonometric Functions

Solving Linear Equations in Two Variables

Solving Problems with Circles and Triangles

Solving Quadratic Equations

Sorting Equations and Identities

Sorting Equations of Circles 1

Sorting Equations of Circles 2

Transforming 2D Figures

State, district and CCSSI standards appear courtesy of their respective authors. All other material Copyright © 2007-2015 Mathematics Assessment Resource Service, University of Nottingham.

Free Mathematics Tutorials

Math word problems with answers for grade 6.

Grade 6 maths word problems with answers are presented. Some of these problems are challenging and need more time to solve. Also detailed solutions and full explanations are included.

• Two numbers N and 16 have LCM = 48 and GCF = 8. Find N.
• If the area of a circle is 81pi square feet, find its circumference.
• Find the greatest common factor of 24, 40 and 60.
• In a given school, there are 240 boys and 260 girls. a) What is the ratio of the number of girls to the number of boys? b) What is the ratio of the number of boys to the total number of pupils in the school?
• If Tim had lunch at $50.50 and he gave 20% tip, how much did he spend?
• Find k if 64 ÷ k = 4.
• Little John had $8.50. He spent $1.25 on sweets and gave to his two friends $1.20 each. How much money was left?
• What is x if x + 2y = 10 and y = 3?
• A telephone company charges initially $0.50 and then $0.11 for every minute. Write an expression that gives the cost of a call that lasts N minutes.
• A car gets 40 kilometers per gallon of gasoline. How many gallons of gasoline would the car need to travel 180 kilometers?
• A machine fills 150 bottles of water every 8 minutes. How many minutes it takes this machine to fill 675 bottles?
• A car travels at a speed of 65 miles per hour. How far will it travel in 5 hours?
• A small square of side 2x is cut from the corner of a rectangle with a width of 10 centimeters and length of 20 centimeters. Write an expression in terms of x for the area of the remaining shape.
• A rectangle A with length 10 centimeters and width 5 centimeters is similar to another rectangle B whose length is 30 centimeters. Find the area of rectangle B.
• A school has 10 classes with the same number of students in each class. One day, the weather was bad and many students were absent. 5 classes were half full, 3 classes were 3/4 full and 2 classes were 1/8 empty. A total of 70 students were absent. How many students are in this school when no students are absent?
• The perimeter of square A is 3 times the perimeter of square B. What is the ratio of the area of square A to the area of square B.
• John gave half of his stamps to Jim. Jim gave gave half of his stamps to Carla. Carla gave 1/4 of the stamps given to her to Thomas and kept the remaining 12. How many stamps did John start with?
• Two balls A and B rotate along a circular track. Ball A makes 4 full rotations in 120 seconds. Ball B makes 3 full rotation in 60 seconds. If they start rotating now from the same point, when will they be at the same starting point again?
• A segment is 3 units long. It is divided into 9 parts. What fraction of a unit are 2 parts of the segment?
• A car is traveling 75 kilometers per hour. How many meters does the car travel in one minute?
• Carla is 5 years old and Jim is 13 years younger than Peter. One year ago, Peter's age was twice the sum of Carla's & Jim's age. Find the present age of each one of them.
• Linda spent 3/4 of her savings on furniture. She then spent 1/2 of her remaining savings on a fridge. If the fridge cost her $150, what were her original savings?
• The distance bewteen Harry and Kate is 2500 meters. Kate and Harry start walking toward one another and Kate' dog start running back and forth between Harry and Kate at a speed of 120 meters per minute. Harry walks at the speed of 40 meters per minute while Kate walks at the speed of 60 meters per minute. What distance will the dog have travelled when Harry and Kate meet each other?

Answers to the Above Questions

• a) 13:12 b)12:25
• 0.50 + N * 0.11
• 4.5 gallons
• 450 centimeters squared
• 108 cubic centimeters
• 1250 meters/minute
• Carla:5 years, Jim: 6 years, Peter: 19 years.
• 3000 meters

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Education Standards

Wyoming standards for mathematics.

Learning Domain: Ratios and Proportional Relationships

Standard: Use ratio and rate reasoning to solve real-world and mathematical problems.

Maryland College and Career Ready Math Standards

Learning Domain: Expressions and Equations

Standard: Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.

Standard: Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.

Standard: Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

Common Core State Standards Math

Cluster: Understand ratio concepts and use ratio reasoning to solve problems

Cluster: Reason about and solve one-variable equations and inequalities

Which Property Do I Use?

Problem solving exercise.

Lesson Overview

Students practice solving equations using either the addition or the multiplication property of equality.

Key Concepts

Students will solve equations of the form x + p = q using the addition property of equality.

They will solve equations of the form px = q using the multiplication property of equality.

They will need to look at the variable and decide what operation must be performed on both sides of the equation in order to isolate the variable on one side of the equation.

If a number has been added to the variable, they will subtract that number from both sides of the equation. If a number has been subtracted from the variable, they will add that number to both sides of the equation. If the variable has been multiplied by a number, students will either divide both sides of the equation by that number or multiply by the reciprocal of that number. If the variable has been divided by a number, students will multiply by that number. Students will see how this can be applied to solving a proportion such as x c = a b .

Goals and Learning Objectives

• Practice solving equations using either the addition or the multiplication property of equality.
• Distinguish between equations that can be solved using the addition property of equality from equations that can be solved using the multiplication property of equality.
• Solve a proportion by solving an equation.

Lesson Guide

Have students work in pairs to identify the property they can use to solve each equation.

ELL: When monitoring students, pay special attention to ELLs to ascertain that they are on task and clear about what needs to be done. Provide enough time for students to discuss which properties they used and why.

SWD: This lesson is a review and application of skills learned so far in this unit. Consider the prerequisite skills for this lesson. Students with disabilities may need review, direct instruction, and guided practice with the skills needed to complete the tasks that follow. Make sure all students understand the terms included in this lesson. Students should understand these domain-specific terms:

• addition property of equality
• multiplication property of equality

Re-teach/review these terms to students with disabilities.

Mathematics

After pairs have made their matches, have students describe how they decided which property to use. Students should explain that they looked to see what operation was performed already on the variable. If an amount has been added to or subtracted from the variable, they use the addition property of equality. If the variable has been multiplied or divided by some number, they use the multiplication property of equality.

• Equations you can solve using the addition property of equality
• Equations you can solve using the multiplication property of equality

HANDOUT: Which Property Do I Use?

Math Mission

Discuss the Math Mission. Students will solve equations using either the addition property of equality or the multiplication property of equality.

Solve equations using either the addition property of equality or the multiplication property of equality.

An Equation with Decimals

Have students work in pairs on all problems in Tasks 3, 4, 5, and 6.

SWD: Students with disabilities may use an incorrect property. Ask students to read the equation to you to see if they correct the misconception. Pay special attention to how they read their equation.

ELL: When discussing this example, make a point of writing the questions on the board, along with students' responses. This will assist ELLs by giving them written and oral access to the questions. Make sure students record important information in their Notebook.

Mathematical Practices

Mathematical Practice 1: Make sense of problems and persevere in solving them.

Look for students who check their work using substitution and revise accordingly.

Mathematical Practice 3: Construct viable arguments and critique the reasoning of others.

Look for students who make errors so that these can be discussed during Ways of Thinking.

Mathematical Practice 6: Attend to precision.

Look for students who correctly record the steps for solving the equations and correctly identify the properties they are using.

Mathematical Practice 8: Look for and express regularity in repeated reasoning.

Look for students who see that the overall goal in solving each equation is the same—isolate the variable—even when the operation used to do so differs.

Interventions

Student uses the wrong operation when trying to isolate the variable.

• What operation has been applied to the variable now?
• What is the opposite of that operation?
• How do you undo addition (multiplication)?

Student is unable to identify the sides of the equation because the variable is on the right.

• What is the equal sign?
• Now tell me what is on the left side of the equal sign.
• Now tell me what is on the right side of the equal sign.

Student gets the wrong answer.

• Write the equation you are trying to solve.
• Substitute your solution for the variable. Do you get a true equation?
• Try solving the equation again.
• 8.05 = x + 4.9 8.05 – 4.9 = x + 4.9 – 4.9      Addition property of equality x = 3.15 Check: 8.05 = 3.15 + 4.9 80.5 = 8.05

An Equation With Decimals

• Use either the addition or multiplication property of equality to solve this equation: 8.05 = x + 4.9
• Write the name of the property, and show each step you used to find the solution.
• Check your solution by substituting it for x in the equation. Does your solution make the equation true?
• Try to get x by itself on one side of the equation.
• If substituting your solution does not result in a true equation, go back and check the steps you used to find your solution.

An Equation With Whole Numbers

Mathematical Practice 7: Look for and make use of structure.

Look for students who use division and for students who multiply by the reciprocal.

Student does not understand why you can use multiplication or division.

• Dividing by 7 gives the same result as multiplying by ….
• 7 y = 329   7 y 7 = 329 7                Multiplcation property of equality y = 47 Check: 7 • 47 = 329 329 = 329
• Use either the addition or multiplication property of equality to solve this equation: 7 y = 329
• Check your solution by substituting it for y in the equation. Does your solution make the equation true?
• Think about what operation you will use, and whether you will use a whole number or a fraction with that operation.

An Equation With a Fraction

• Dividing by 2 3 gives the same result as multiplying by ….
• 2 3 m = 12 3 2  • 2 3 m = 3 2  • 12           Multiplication property of equality m = 18 Check: 2 3  • 18 = 12 12 = 12
• Use either the addition or multiplication property of equality to solve this equation: 2 3 m = 12
• Check your solution by substituting it for m in the equation. Does your solution make the equation true?

A Proportional Equation

Mathematical Practice 2: Reason abstractly and quantitatively.

Look for students who check their answer to p 7 = 12 4 by solving the proportion in a different way.

• p 7 = 12 4 p 7   •  7 =  12 4  • 7        Multiplication property of equality p =  84 4 p = 21 Check: 21 7  =  12 4   3 = 3
• Use either the addition or multiplication property of equality to solve this equation: p 7 = 12 4
• Check your solution by substituting it for p in the equation. Does your solution make the equation true?

Which property would you use to get p by itself on one side of the equation?

Prepare a Presentation

Preparing for ways of thinking.

Look for students who have different solution methods to share in the Ways of Thinking discussion.

Challenge Problem

• 5 x + 2 = 2 x  + 9 5 x – 2 x + 2 = 2 x – 2 x + 9        Addition property of equality 3 x + 2 = 9 3 x + 2 – 2 = 9 – 2                   Addition property of equality 3 x  = 7 3 x 3 =  7 3                                     Multiplicaton property of equality x = 2 1 3

Choose one of the equations you solved to present.

• Show the steps you took to solve the equation.
• Explain how you used one of the properties of equality to solve.
• How could you use the addition and multiplication properties of equality to solve the equation 5 x + 2 = 2 x  + 9?

Make Connections

• As you discuss the Work Time problems, be sure to include students who may have misconceptions so that you can engage the class in active discussion.
• For 7 y = 329 and 2 3 m = 12, try to include both students who used multiplication and students who used division.
• Look for students who use precise vocabulary in describing the steps they used to solve an equation and in naming the properties used.
• If possible, allow a student with an incorrect solution to show what happens when the incorrect solution is substituted into the equation. The resulting equation will be false.
• If students have incorrect solutions due to their not understanding fraction or decimal operations, include a discussion of those incorrect solutions as well. There may be other students who could benefit from a review of these operations.
• For the proportion p 7 = 12 4 , take this opportunity to allow students to demonstrate alternative ways to solve the proportion. For example, a student might use the ratio 12 4 to determine the unit rate: 3 to 1. The student can then reason that 21 7 would represent that same unit rate, and so 21 must be the solution to the proportion.

If any students attempted the Challenge Problem, have them share their solutions. If the solutions are incorrect, have the class work together to find the correct solution. If no one attempted the problem, provide the first step: Subtract 2 x from both sides. Then allow students to attempt the rest of the solution.

SWD: On a chart, write the key points that students make during their presentations of problems from Tasks 3, 4, 5, 6.

Performance Task

Ways of thinking: make connections.

• Take notes about the properties and steps your classmates used to solve the equations.

As your classmates present, ask questions such as:

• How did you know which property to use?
• Why do you get the same result whether you multiply by 1 7 or divide by 7?
• You divided by 2 3 . Could you multiply by 3 2 instead?

Properties of Equality

• Have pairs quietly discuss the information about the properties of equality. Have them give examples of how to use the properties to solve equations.
• As student pairs work together, listen for students who may still have misconceptions so that you can address them in the class discussion.
• If a number has been added to (subtracted from) a variable in an equation, that number will need to be subtracted from (added to) both sides of the equation to solve it. This process makes use of the addition property of equality. It can be used to solve equations such as x + 4 = 10 and x − 2 = 12.
• If a variable has been multiplied by a number, both sides of the equation will need to be divided by that number to solve it. This process makes use of the multiplication property of equality. It can be used to solve equations such as 4 x = 12.
• If a variable has been divided by a number, both sides of the equation will need to be multiplied by that number to solve it. This process makes use of the multiplication property of equality. It can be used to solve equations such as x 2  = 12 and can also be used to solve proportions such as x 2 = 10 12 .
• Dividing by a number gives the same result as multiplying by the reciprocal of that number. So, when using the multiplication property of equality, you can use either operation. To solve 7 x = 56, you can divide both sides of the equation by 7 or you can multiply both sides by 1 7 . The result is the same. Similarly, you can solve 2 3 x = 6 by either multiplying by 3 2 or dividing by 2 3 .

Formative Assessment

Summary of the math: properties of equality.

Read and Discuss

• Addition property of equality : If a = b , then a + c = b + c .
• Multiplication property of equality : If a  = b , then ac  = bc .
• Add the same value to both sides or subtract the same value from both sides and the equation will still be true.
• Multiply or divide both sides by the same value and the equation will still be true.
• Provide examples of equations that can be solved using the addition property of equality, and explain the steps for using the property to solve each equation?
• Provide examples of equations that can be solved using the multiplication property of equality, and explain the steps for using the property to solve each equation?

Reflect On Your Work

Have each student write a quick reflection before the end of the class. Review the reflections to see whether students understand how to use properties to solve equations.

Write a reflection about the ideas discussed in class today. Use the sentence starters below if you find them to be helpful.

I can use the addition property of equality when …

I can use the multiplication property of equality when …

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Ontario Junior Math Resources

Superb lesson ideas and supports for grades 4 -6 teachers and parents.

• Number Sense and Numeration
• Measurement
• Geometry and Spatial Sense
• Patterning and Algebra
• Data Management and Probability
• Math Centers
• Three-part Lessons and Bansho
• Anchor Charts

Problem-Solving Tasks

• Sites for Online Practice
• Graphic Organizers
• Interactive Whiteboard Tools
• Guides to Effective Instruction
• Assessment Resources
• Research into Practice Series
• Capacity Building Series
• Paying Attention to Mathematics Education
• OAME Resources
• Parent Resources
• Special Ed Support
• ESL Support
• The Language of Math
• Create a Classroom Economy

2. Classroom Challenges (Mathematics Assessment Resource Services, Nottingham) – Formative assessment lessons for Grade 6 – The resources on this website are amazing! You will find many wonderful problem solving tasks with teacher guides describing what to do before, during and after the lesson. For example, preview this resource entitled Solving Real-Life Problems: Selling Soup (University of Nottingham & UC Berkeley, 2014)

2. Problems of the Month, from Inside Mathematics ( http://www.insidemathematics.org/index.php ). These tasks are developed and owned by the  Shell Centre for Mathematical Education , University of Nottingham, England.

– 4th Grade                               – 5th Grade                                 – 6th Grade

3. Robert Kaplinsky’s tasks  are beautifully designed and bring problem solving to life!

4. Stuffed with Pizza – A fractions performance task, New York City Department of Education.

5. Houghton Mifflin Math – Problem Solving  application of strategies

6. Houghton Mifflin Math – Investigations

7. Parallel and Open Task Problem-Solving Math Bank (Ontario Teachers’ Federation)

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