Math - Sixth Grade

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Sixth Grade
Instructional Tasks

Stand alone tasks are organized to support learning of content standards. These tasks can be used as initial instruction or to support students who are struggling with a particular topic.

 

Strand: MATHEMATICAL PRACTICES (6.MP)
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Strand: RATIOS AND PROPORTIONAL RELATIONSHIPS (6.RP)
Understand ratio concepts and use ratio reasoning to solve problems (Standards 6.RP.1–3)
  • Anna in D.C.
    The purpose of this task is to give students an opportunity to solve a multi-step percentage problem that can be approached in many ways.
  • Apples to Apples
    The purpose of this task is to connect students' understanding of multiplicative relationships to their understanding of equivalent ratios.
  • Bag of Marbles
    The purpose of this task is to help students develop fluency in their understanding of the relationship between fractions and ratios. It provides an opportunity to translate from fractions to ratios and then back again to fractions.
  • Baking Bread 2
    The primary purpose of this task is to represent ratios of two or more quantities with parallel tape diagrams. Note that the solution to this task assumes that students have already studied equivalent ratios and understand that when you have a context with 8 units of one quantity and 2 units of another quantity, you can say the ratio is 4:1 because it is an equivalent ratio.
  • Constant Speed
    The purpose of this task is for students to learn to reason about whether or not ratios are equivalent using a diagram.
  • Converting Square Units
    Given the dimensions of a rectangular board, students must convert inches to feet, find the area of the board, and critique the reasoning the student in the problem uses the find the area.
  • Currency Exchange
    Given a scenario of a man traveling to another country and converting money students must determine the amount of the foreign currency he gets in exchange for his US dollars.
  • Dana's House
    In this task students are given the size of a lot on which a house is to be built. Given the square footage of the house, they must determine which percentage of the lot will be covered by the house.
  • Data Transfer
    This task asks the students to solve a real-world problem involving unit rates (data per unit time) using units that many teens and pre-teens have heard of but may not know the definition for. While the computations involved are not particularly complex, the units will be abstract for many students.
  • Equivalent Ratios and Unit Rates
    This task should come after students have done extensive work with representing equivalent ratios and understand that for any ratio a:b, the ratio sa:sb is equivalent to it for any s>0. The purpose of this task is to make explicit the fact that equivalent ratios have the same unit rate.
  • Evaluating Ratio Statements
    The goal of this task is to assess student understanding of ratios. The task offers five questions, some of which can be addressed using ony the given ratio, whereas others require knowledge of the total number of students.
  • Exam scores
    The goal of this task is to show how to apply ratio reasoning to calculate a percent. In order to do this task, students must know the meaning of percent, that is they need to know that a percent is a rate out of 100. The teacher may wish to encourage students to work with three different representations for the calculation: diagrams, ratio tables, and double number lines.
  • Examining California's Prison System: Real-World Ratio
    Using an infographic students look at such factors as age, gender and race to examine how the prison population in California compares to the general population. Students then apply an understanding of how they can find the value of a part by using a whole and a percent in order to look at how that can lead to recommendations for how to prevent crime. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
  • Fizzy Juice
    The goal of this task is to provide an engaging context for students to work with ratios.
  • Friends Meeting on Bicycles
    Given a story about two friends who ride bikes to meet each other and the rate at which they travel, students must calculate the distance between them at specific times.
  • Fruit Salad
    The purpose of this task is for students to solve a contextual problem where there is a multiplicative relationship between several quantities in the context. These relationships can either be represented in a ratio table or with a linear equation.
  • Games at Recess
    In this task, given the scenario of students playing games at recess, students are asked to compare different aspects of the games using ratios and then writing sentences to express those ratios.
  • Gianna's Job
    The purpose of this task is to apply reasoning about ratios to solve a rate problem. This problem introduces a rate whose units are dollars per hour of work. Using this information, students need to make two separate calculations, one with units of dollars and the other with units of hours.
  • Hippos Love Pumpkins
    The purpose of this task is for students to find unit rates in different situations involving unusual units. Most students are familiar with miles per hour, but students are unlikely to have encountered the idea of pumpkins per hippo or goats per pizza. By working with unusual (even silly) units, students must reason abstractly and quantitatively in order to answer the questions because they can't rely on their experience with the situation to guide them through it.
  • Hunger Games versus Divergent
    This is an engaging introductory lesson for a unit on ratio and proportional relationships.
  • Jim and Jesse's Money
    This task reads "Jim and Jesse each had the same amount of money. Jim spent $58 to fill the car up with gas for a road-trip. Jesse spent $37 buying snacks for the trip. Afterward, the ratio of Jims money to Jesse's money is 1:4. How much money did each have at first?"
  • Kendall's Vase - Tax
    For this task students are given this problem: "Kendall bought a vase that was priced at $450. In addition, she had to pay 3% sales tax. How much did she pay for the vase?"
  • Mangos for Sale
    The purpose of this task is to generate a classroom discussion about ratios and unit rates in context.
  • Many Ways to Say It
    The purpose of this task is to help students understand and use ratio language.
  • Mixing Concrete
    Given that the ratio of sand and cement of 5 : 3 is needed to make concrete, students must determine how many cubic feet of each are needed to make 160 cubic feet of concrete mix?
  • Mixtures
    This activity will help students understand percentages and mixture problems by working with two piles of colored chips.
  • Overlapping Squares
    In this task students are given a drawing showing two overlapping congruent squares. They must determine the area of the overlap.
  • Painting a Barn
    Given the dimensions of a barn, the square footage covered by a gallon of paint, and the price of the paint, students must find the cost of painting the barn and explain their work.
  • Party Planning
    The goal of this task is to provide a ratio problem which can be solved efficiently with a wide variety of techniques. While it could be used at many points in a ratio unit (with or without additional instructions on which technique to apply) one possible use of the task is as a summative assessment.
  • Pennies to Heaven
    The goal of this task is to give students a context to investigate large numbers and measurements. Students need to fluently convert units with very large numbers in order to successfully complete this task.
  • Perfect Purple Paint I
    The goal of this task is to provide a good context for engaging students in reasoning about ratios.The teacher may wish to use this task to demonstrate or introduce some of the different representations of ratios (ratio table, double number line, graphing points in the coordinate plane). The numbers are small so that the focus can be on the methods and not performing arithmetic.
  • Price per pound and pounds per dollar
    This task could be used by teachers to help students develop the concept of unit rates. Its purpose is to help students see that when you have a context that can be modeled with a ratio and associated unit rate, there is almost always another ratio with its associated unit rate (the only exception is when one of the quantities is zero), and to encourage students to flexibly choose either unit rate depending on the question at hand.
  • Representing a Context with a Ratio
    The purpose of this task is to introduce students to ratios and ratio language.
  • Riding at a Constant Speed, Assessment Variation
    Riding at a Constant Speed addresses aspects of 6.RP.2 "Understand the concept of a unit rate a/b associated with a ratio a:b" and 6.RP.3 "Use ratio and rate reasoning to solve real-world and mathematical problems." The numbers are chosen so that it would be easy to implement this task as a fill-in-the-blank item.
  • Running at a Constant Speed
    The purpose of this task is to give students experience in reasoning with equivalent ratios and unit rates from both sides of the ratio when given information about a runner and their pace.
  • Same and Different
    The purpose of this task is to analyze some very common contexts that can be represented by ratios and to motivate the idea of equivalent ratios for different kinds of contexts. It can also be used to introduce students to double number line diagrams.
  • Security Camera
    Students are given the scenario of a shop owner wants to prevent shoplifting. They are shown the shop floor plan and the rotation ability of the camera. They then must answer questions about which parts and percentages of the shop are now seen by the camera.
  • Shirt Sale
    In this task students are given the scenario of a student who buys a shirt at a percentage of the original price. They must calculate the original price and explain and show their work.
  • Simple Unit Conversion Using Ratio Reasoning
    The purpose of this instructional task is for students to use rate and ratio reasoning to solve unit conversion problems. In grade 6, unit conversion should be approached as a case of ratio reasoning, rather than a separate procedure to learn, and this task is an example of what that might look like. This task should come after students have spent time building up their understanding of equivalent ratios and are comfortable with some different representations of equivalent ratios.
  • Speed Conversions
    The goal of this task is to perform a unit conversion in the context of speed while also focusing on the precision of the conversion factor. Because the conversion rate is a decimal, this task should be used after students have gained some familiarity with ratio and rate reasoning.
  • The Escalator, Assessment Variation
    This task presents a scenario about someone riding an escalator. Students are then given a series of statements such as "He traveled 2 meters every 5 seconds" and then asked to determine which of the statements are true.
  • Ticket Booth
    The goal of this task is to compare unit rates in a real world context. In addition to solving the problem by finding unit rates, students could also make a ratio table.
  • Unit Conversions
    The goal of this task is to study conversion between some volume and weight units. The focus of this task is understanding the relationship between multiplication, linear measurements, area, and volume.
  • Voting for Three, Variation 1
    In this first problem of three, students define the simple ratios that exist among three candidates in an election. It opens an opportunity to introduce unit rates.
  • Voting for Three, Variation 2
    In this problem, the total number of votes in the election and the number of votes for individual candidates is not provided. It provides the ratio of John's votes to Will's votes and enough information to compute the number of votes for Marie.
  • Voting for Three, Variation 3
    This is the last problem of seven in a series about ratios set in the context of a classroom election. Since the number of voters is not known, the problem is quite abstract and requires a deep understanding of ratios and their relationship to fractions.
  • Walk-a-thon 1
    In this task, students are given information about a context where there is a proportional relationship between two quantities in a table that has missing values. Students need to fill in the missing values, plot the corresponding points in the coordinate plane, and find the two unit rates that are associated with this proportional relationship.
  • Which detergent is a better buy?
    This purpose of this task is to provide a context for comparing ratios by using the example of laundry detergents, their costs, and how many loads they can do.
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Strand: THE NUMBER SYSTEM (6.NS)
Apply and extend previous understandings of multiplication and division of whole numbers to divide fractions by fractions (Standard 6.NS.1)
  • Baking Cookies
    This task requires students to complete a series of steps in order to find a solution, and because they need to analyze constraints, it addresses some aspects of mathematical modeling. Students must first add fractions with familiar but unlike denominators, which is a skill developed in the 5th grade. Then students need to divide fractions by fractions.
  • Cup of Rice
    Students are given a word problem "One serving of rice is 23 of a cup. I ate 1 cup of rice. How many servings of rice did I eat?" They must choose between 2 possible solutions and explain their reasoning.
  • Dan's Division Strategy
    The purpose of this task is to help students explore the meaning of fraction division and to connect it to what they know about whole-number division.
  • Drinking Juice, Variation 2
    This task builds on a fifth grade fraction multiplication task, 5.NF Drinking Juice. This task uses the identical context, but asks the corresponding Number of Groups Unknown division problem. See Drinking Juice, Variation 3 for the Group Size Unknown version.
  • Drinking Juice, Variation 3
    This task builds on a fifth grade fraction multiplication task, 5.NF Drinking Juice. This task uses the identical context, but asks the corresponding Group Size Unknown division problem. See Drinking Juice, Variation 2 for the Number of Groups Unknown version.
  • How Many Batches/What Fraction of a Batch?
    The purpose of this task is to help students extend their understanding of multiplication and division of whole numbers to multiplication and division of fractions. The task does not ask students to find the product or quotient since the task is more about learning how to represent the situation, but teachers might choose to ask students to find or estimate the answers, if desired.
  • How Many Containers in One Cup / Cups in One Container?
    These two fraction division tasks use the same context and ask "How much in one group?" but require students to divide the fractions in the opposite order.
  • How Much in One Batch?
    The purpose of this task is to help students extend their understanding of multiplication and division of whole numbers to multiplication and division of fractions.
  • How many _______ are in. . . ?
    This task provides a list of problems. They require that the students model each problem with some type of fractions manipulatives or drawings. The problems are meant to be a progression which require more sophisticated understandings of the meaning of fractions as students progress through them. If the task is used to help students see the connections to the invert-and-multiply rule for fraction division (as described in the solution) then they should already be familiar with and comfortable solving Number of Groups Unknown (a.k.a. "How many groups?") division problems with visual models.
  • Making Hot Cocoa, Variation 1
    This is the first of two fraction division tasks that use similar contexts to highlight the difference between the "Number of Groups Unknown" a.k.a. "How many groups?" when the quotient is a fraction (or mixed number) greater than 1 (Variation 1) and when the quotient is a fraction that is less than 1 (Variation 2).
  • Making Hot Cocoa, Variation 2
    This is the second of two fraction division tasks that use similar contexts to highlight the difference between the "Number of Groups Unknown" a.k.a. "How many groups?" when the quotient is a fraction (or mixed number) greater than 1 (Variation 1) and when the quotient is a fraction that is less than 1 (Variation 2).
  • Modeling Fraction and Mixed Number Division Using Arrays
    Students will learn how to solve word problems that involve dividing fractions and mixed numbers by using a visual model. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
  • Reciprocity
    The purpose of this task is to help students understand why dividing by a fraction gives the same result as multiplying by its reciprocal. This is accomplished by writing the division equation along with related multiplication equations and diagrams showing the situation for several different contexts.
  • Running to School, Variation 2
    This task builds on a fifth grade fraction multiplication task, "5.NF Running to School, Variation 1." This task uses the identical context, but asks the corresponding "Number of Groups Unknown" division problem. See "6.NS Running to School, Variation 3" for the "Group Size Unknown" version.
  • Running to School, Variation 3
    This task builds on a fifth grade fraction multiplication task, "5.NF Running to School, Variation 1." "6.NS Running to School, Variation 3" uses the identical context, but asks the corresponding "Group Size Unknown" division problem. See "6.NS Running to School, Variation 2" for the "Number of Groups Unknown" version.
  • Standing in Line
    The purpose of this task is for students to solve a problem in context that can be solved in different ways, but in particular by dividing a whole number by a unit fraction.
  • Traffic Jam
    This task posits this word problem to students: "You are stuck in a big traffic jam on the freeway and you are wondering how long it will take to get to the next exit, which is 1 1/2 miles away. You are timing your progress and find that you can travel 2/3 of a mile in one hour. If you continue to make progress at this rate, how long will it be until you reach the exit? Solve the problem with a diagram and explain your answer."
  • Video Game Credits
    "It requires 1/4 of a credit to play a video game for one minute." Given this information, students are asked to answer questions about how long a student can play given a specific number of credits.
Compute (add, subtract, multiply and divide) fluently with multi-digit numbers and decimals and find common factors and multiples (Standards 6.NS.2–4)
  • 12 Rectangular Units
    The purpose of this task is for students to notice how the decimal point behaves when numbers in different place-value places (for example, 0.04 and 0.3) are multiplied.
  • 2 Units Wide and 3 Units Long
    The purpose of this task is for students to notice how the decimal point behaves when numbers in the same place (both in the hundreds, both in the thousandths, etc) are multiplied.
  • Adding Base Ten Numbers, Part 1
    The goal of this task is to demonstrate that since digits in the same place represent the same-sized units, we can always add digits in the same place. This is one of three tasks relating to this.
  • Adding Base Ten Numbers, Part 2
    This is the second in a set of three tasks generalizing an addition algorithm whole numbers to all base-ten numbers.
  • Adding Base Ten Numbers, Part 3
    This is the third in a set of three tasks generalizing an addition algorithm from whole numbers to all base-ten numbers.
  • Adding Multiples
    This task is appropriate for assessing students understanding of repeated reasoning and generalizing that understanding to prepare them for deeper algebraic thinking needed in the expressions and equations domain.
  • Bake Sale
    This problem requires students to apply the concepts of factors and common factors in a context.
  • Batting Average
    The goal of this task is to perform and analyze division with whole numbers in a sports context.
  • Buying Gas
    There are two aspects to fluency with division of multi-digit numbers: knowing when it should be applied, and knowing how to compute it. While this task is very straightforward, it represents the kind of problem that sixth graders should be able to recognize and solve relatively quickly.
  • Changing Currency
    The purpose of this task is for students to notice that if the dividend and divisor both increase by a factor of 10, the quotient remains the same. This sets them up to understand the rules for moving decimal points when performing long division.
  • Factors and Common Factors
    This task requires students to apply the concepts of factors and common factors in a context.
  • Gifts from Grandma, Variation 3
    The purpose of this task is to show three problems that are set in the same kind of context, but the first is a straightforward multiplication problem while the other two are the corresponding "How many groups?" and "How many in each group?" division problems.
  • How Many Staples?
    The goal of this task is to perform long division with remainder in a context. Students are shown a box of staples and asked to find inconsistencies in the information on it.
  • Interpreting a Division Computation
    In this task, students are shown a division problem and then asked to find the products of a group of numbers related to that problem.
  • Jayden's Snacks
    Building on their fifth grade experiences with operations on decimal numbers, sixth grade students should find the task to be relatively easy. The emphasis in this task is on whether students are actually fluent with the computations, so teachers could use this as a formative assessment task if they monitor how students solve the problem.
  • Movie Tickets
    The purpose of this task is for students to solve problems involving decimals in a context involving a concept that supports financial literacy, namely inflation.
  • Multiples and Common Multiples
    This task requires students to apply the concepts of multiples and common multiples in a context.
  • Reasoning about Multiplication and Division and Place Value, Part 1
    The three tasks in this set are not examples of tasks asking students to compute using the standard algorithms for multiplication and division because most people know what those kinds of problems look like. Instead, these tasks show what kinds of reasoning and estimation strategies students need to develop in order to support their algorithmic computations.
  • Reasoning about Multiplication and Division and Place Value, Part 2
    The three tasks (including part 1 and part 3) in this set are not examples of tasks asking students to compute using the standard algorithms for multiplication and division because most people know what those kinds of problems look like. Instead, these tasks show what kinds of reasoning and estimation strategies students need to develop in order to support their algorithmic computations.
  • Setting Goals
    The purpose of this task is for students to solve problems involving division of decimals in the real-world context of setting financial goals. The focus of the task is on modeling and understanding the concept of setting financial goals, so fluency with the computations will allow them to focus on other aspects of the task.
  • Tenths of Tenths and Hundredths of Hundredths
    The purpose of this task is to relate what students know about multiplication, area, and fractions to multiplying decimals for powers of ten that are less than 1. The questions are carefully sequenced so that students are lead to construct an argument for why, from a geometric perspective, 0.10.1 is 0.01 and 0.010.01 is 0.0001. Eventually, students should generalize their understanding and know that the number of decimal places in a product is the same as the total number of decimal places in the factors. This task gives a geometric basis for understanding why that is true.
  • The Florist Shop
    Students are given the scenario of a florist ordering roses and asked to find the smallest number of bunches she could order and explain their reasoning.
  • What is the Best Way to Divide?
    The purpose of this task is to have students think strategically about their method for solving a division problem. This task shows an example of focusing on the choice of strategy as opposed to applying an algorithm without first considering options.
Apply and extend previous understandings of numbers to the system of rational numbers (Standards 6.NS.5–8)
  • Above and Below Sea Level
    The purpose of this task is to help students interpret signed numbers in a context as a magnitude and a direction and to make sense of the absolute value of a signed number as its magnitude.
  • Comparing Temperatures
    The purpose of the task is for students to compare signed numbers in a real-world context. It could be used for either assessment or instruction if the teacher were to use it to generate classroom discussion.
  • Distances Between Points
    The purpose of this task is for students to solve a mathematical problem using points in the coordinate plane.
  • Extending the Number Line
    The purpose of this task is to understand that there are natural mathematical questions to ask for which there are no answers if we restrict ourselves to the positive numbers. The idea is to motivate the need for negative numbers and to see that there is a natural representation of them on the number line.
  • Fractions on the Number Line
    In this task students are given a number line and they must label a number of fractions on the line. When given a selection of statements about inequality they must state which are true.
  • Integers on the Number Line 1
    Given a number line, students are asked to find and label two numbers. Then given several inequalities, they must decide whether the inequality is true or false.
  • Integers on the Number Line 2
    The goal of this task is to study, with a number line, why it makes sense for a whole number a that -(-a)=a.
  • It's Warmer in Miami
    The purpose of this task is for students to apply their knowledge of integers in a real-world context.
  • Jumping Flea
    This purpose of this task is to help students understand the absolute value of a number as its distance from 0 on the number line. The context is not realistic, nor is meant to be; it is a thought experiment to help students focus on the relative position of numbers on the number line.
  • Locations in the Coordinate Plane
    The goal of this task is to introduce students to the relationships between the locations and coordinates of points graphed in all four quadrants of the coordinate plane. When describing the things they notice about the point locations and coordinates, the teacher should encourage students to use terms such as quadrant, distance, origin, sign, axis, and coordinate.
  • Mile High
    The first two parts of this task ask students to interpret the meaning of signed numbers and reason based on that meaning in a context where the meaning of zero is already given by convention.
  • Nome, Alaska
    The purpose of this task is for students to solve a real-world problem by interpreting and comparing points in the coordinate plane. This task focuses students' attention on the y-values of the points, asking for the greatest y-value and the least y-value, as well as the greatest difference between y-values when the x-values are the same.
  • Plotting Points in the Coordinate Plane
    The goal of this task is to provide experience labeling coordinate axes appropriately to plot a given set of points, which will mean choosing an appropriate scale.
  • Reflecting Points Over Coordinate Axes
    The goal of this task is to give students practice plotting points and their reflections.
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Strand: EXPRESSIONS AND EQUATIONS (6.EE)
Apply and extend previous understandings of arithmetic to algebraic expressions involving exponents and variables (Standards 6.EE.1–4)
  • Anna in D.C.
    The purpose of this task is to give students an opportunity to solve a multi-step percentage problem that can be approached in many ways.
  • Commutative and Associative Equations
    This lesson focuses on how to rearrange and combine parts of algebraic expressions by using the commutative and associative properties of addition. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
  • Determining Surface Area with Unit Blocks, Rulers, and Nets
    In this video students are shown how to calculate the surface area of a prism. The classroom activity in the lesson requires that students apply this knowledge and measure the surface areas of real 3-Dl objects. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
  • Distance to School
    This task asks students to find equivalent expressions by visualizing a familiar activity involving distance.
  • Equivalent Expressions (6th grade)
    In this problem students have to transform expressions using the distributive, commutative and associative properties to decide which expressions are equivalent.
  • Exponent Experimentation 1
    The purpose of this task is to give students experience working with exponential expressions and to promote making use of structure to compare exponential expressions.
  • Exponent Experimentation 2
    The purpose of this task is to give students experience experimenting with equivalent numerical expressions. This work supports fluency because students practice working with operations, decomposing numbers, and recognizing perfect squares and perfect cubes.
  • Exponent Experimentation 3
    The purpose of this task is to give students experience working with exponential expressions and with what is meant by a solution to an equation.
  • Families of Triangles
    The purpose of this task is to introduce students to the idea of a relationship between two quantities by using a familiar geometic context. In order to benefit from this task, students should have already developed and become comfortable with a formula for the area of a triangle. The focus of this task should be on noticing the relationship between height and area and creating a graphical and algebraic representation of this relationship, not on understanding the meaning behind the geometric terms.
  • Reciprocity
    The purpose of this task is to help students understand why dividing by a fraction gives the same result as multiplying by its reciprocal. This is accomplished by writing the division equation along with related multiplication equations and diagrams showing the situation for several different contexts.
  • Rectangle Perimeter 1
    This tasks gives a verbal description for computing the perimeter of a rectangle and asks the students to find an expression for this perimeter. They then have to use the expression to evaluate the perimeter for specific values of the two variables.
  • Rectangle Perimeter 2
    This task is a natural follow up for task Rectangle Perimeter 1. After thinking about and using one specific expression for the perimeter of a rectangle, students now extend their thinking to equivalent expressions for the same quantity.
  • Seven to the What?!?
    Students are asked to find the last digit and the last two digits of 7 to the 2011th power. So the purpose of this task is to give students an opportunity to practice working with positive integer exponents.
  • Sierpinski's Carpet
    The purpose of this task is to help motivate the usefulness of exponential notation in a geometric context and to give students an opportunity to see that sometimes it is easier to write a number as a numeric expression rather than evaluating the expression.
  • The Djinnis Offer
    The purpose of this task is to introduce the idea of exponential growth and then connect that growth to expressions involving exponents. It illustrates well how fast exponential expressions grow.
They reason about and solve one-variable equations and inequalities (Standards 6.EE.5–8)
  • Anna in D.C.
    The purpose of this task is to give students an opportunity to solve a multi-step percentage problem that can be approached in many ways.
  • Firefighter Allocation
    In this task students are asked to write an equation to solve a real-world problem. There are two natural approaches to this task. In the first approach, students have to notice that even though there is one variable, namely the number of firefighters, it is used in two different places. In the other approach, students can find the total cost per firefighter and then write the equation
  • Fishing Adventures 1
    This particular task could be used for instruction or assessment. The context lends itself to the use of inequalities, so it could also be used to introduce inequalities.
  • Fruit Salad
    The purpose of this task is for students to solve a contextual problem where there is a multiplicative relationship between several quantities in the context. These relationships can either be represented in a ratio table or with a linear equation.
  • Height Requirements
    The goal of this task is to express constraints from a real world context using one or more inequalities. In addition to writing inequalities, students also display the numbers (heights) satisfying the inequalities on a number line.
  • Log Ride
    In this instructional task students are given two inequalities, one as a formula and one in words, and a set of possible solutions. They have to decide which of the given numbers actually solve the inequalitie
  • Make Use of Structure
    The purpose of this task is to help students reason about the meaning of equations and the solution of an equation, and to give them an opportunity to make connections with operations with fractions and decimals.
  • Morning Walk
    This task presents a straight forward question that can be solved using an equation in one variable. The numbers are complicated enough so that it is natural to set up an equation rather than solve the problem in one's head.
Represent and analyze quantitative relationships between dependent and independent variables in a real-world context (Standard 6.EE.9)
  • Chocolate Bar Sales
    In this task students use different representations to analyze the relationship between two quantities and to solve a real world problem. The situation presented provides a good opportunity to make connections between the information provided by tables, graphs and equations.
  • Families of Triangles
    The purpose of this task is to introduce students to the idea of a relationship between two quantities by using a familiar geometic context. In order to benefit from this task, students should have already developed and become comfortable with a formula for the area of a triangle. The focus of this task should be on noticing the relationship between height and area and creating a graphical and algebraic representation of this relationship, not on understanding the meaning behind the geometric terms.
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Strand: GEOMETRY (6.G)
Solve real-world and mathematical problems involving area, surface area, and volume (Standards 6.G.1–4)
  • 24 Unit Squares
    The purpose of this activity is to help students think a little more flexibly about the concept of area before studying, generally, the areas of triangles and special quadrilaterals.
  • Areas of Right Triangles
    This task is intended to help build understanding as students work toward deriving a general formula for the area of any triangle. The purpose of this task is for students to use what they know about area and express regularity in repeated reasoning to generate a formula for area of a right triangle.
  • Areas of Special Quadrilaterals
    The purpose of this task is for students to use what they know about area to find the areas of special quadrilaterals. Depending on previous instruction, methods may include decomposing the figures into right triangles and rectangles, or drawing a rectangle to encircle the figure and subtracting areas of right triangles that are not part of the original figure.
  • Banana Bread
    The purpose of this task is two-fold. One is to provide students with a multi-step problem involving volume. The other is to give them a chance to discuss the difference between exact calculations and their meaning in a context.
  • Base and Height
    In this scenario a teacher has given students a task to label the base and height of a triangle and shows 3 students' solutions. Students must then identify which, if any, of the solutions are correct and explain why.
  • Computing Volume Progression 1
    This is the first in a series of four tasks that gradually build in complexity. The purpose of this series of tasks is to build in a natural way from accessible, concrete problems involving volume to a more abstract understanding of volume. The purpose of this first task is to see the relationship between the side-lengths of a cube and its volume.
  • Computing Volume Progression 2
    This is the second in a series of four tasks that gradually build in complexity. The purpose of this series of tasks is to build in a natural way from accessible, concrete problems involving volume to a more abstract understanding of volume. In this iteration, we do away with the lines that delineate individual unit cubes (which makes it more abstract) and generalize from cubes to rectangular prisms. However, the calculations are the same as in 6.G Computing Volume Progression 1.
  • Computing Volume Progression 3
    This is the third in a series of four tasks that gradually build in complexity. The purpose of this series of tasks is to build in a natural way from accessible, concrete problems involving volume to a more abstract understanding of volume. Here, we are given the volume and are asked to find the height.
  • Computing Volume Progression 4
    This is the last in a series of four tasks that gradually build in complexity. The purpose of this series of tasks is to build in a natural way from accessible, concrete problems involving volume to a more abstract understanding of volume. This problem is based on Archimedes Principle that the volume of an immersed object is equivalent to the volume of the displaced water.
  • Determining Surface Area with Unit Blocks, Rulers, and Nets
    In this video students are shown how to calculate the surface area of a prism. The classroom activity in the lesson requires that students apply this knowledge and measure the surface areas of real 3-Dl objects. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
  • Finding Areas of Polygons (6th grade)
    This task asks students to find the area of polygons that are best suited for increasingly abstract methods.
  • Nets for Pyramids and Prisms
    The goal of this task is to work with nets for three-dimensional shapes and use them to calculate surface area.
  • Polygons in the Coordinate Plane
    The purpose of this task is for students to practice plotting points in the coordinate plane and finding the areas of polygons. This task assumes that students already understand how to find areas of polygons by decomposing them into rectangles and triangles.
  • Same Base and Height, Variation 1
    This is the first version of a task asking students to find the areas of triangles that have the same base and height, and is the most concrete.
  • Same Base and Height, Variation 2
    This is the second version of a task asking students to find the areas of triangles that have the same base and height. This presentation is more abstract as students are not using physical models.
  • Sierpinski's Carpet
    The purpose of this task is to help motivate the usefulness of exponential notation in a geometric context and to give students an opportunity to see that sometimes it is easier to write a number as a numeric expression rather than evaluating the expression.
  • Volumes with Fractional Edge Lengths
    The purpose of this task is to introduce students to fractional units for volume.
  • Walking the Block
    The purpose of this task is for students to apply the calculation of distances on a coordinate plane to a real life context. Though explicit coordinates are not given in the problem, the reasoning behind finding the side lengths of the rectangles in the plane is present and this activity could prepare for formalizing of this with the Cartesian coordinate plane later on.
  • Wallpaper Decomposition
    The purpose of this task is for students to experiment with composition and decomposition of polygons to examine shapes in a real world context. To find the area of the wall, students will decompose a pentagon into simpler shapes (for example, a rectangle and a triangle).
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Strand: STATISTICS AND PROBABILITY (6.SP)
Develop understanding of statistical variability of data (Standards 6.SP.1–3)
  • Average Number of Siblings
    The goal of this task is to compare the mean and median in a context where the data is slightly skewed to the right.
  • Buttons: statistical questions
    The purpose of this task is to provide questions related to a particular context (a jar of buttons) so that students can identify which are statistical questions. The task also provides students with an opportunity to write a statistical question that pertains to the context.
  • Describing Distributions
    In this task, students are asked to describe data distributions in terms of center, spread and overall shape and to also compare data distributions in terms of center and spread by selecting which of two distributions has a greater center and which has a greater spread.
  • Electoral College
    This task is intended to demonstrate that a graph can summarize a distribution as well as provide useful information about specific observations. With the table provided, the graph and values have context. The purpose of this task is to help students understand that a distribution can be described in terms of shape and center, and also to provide practice in selecting and calculating measures of center.
  • Examining California's Prison System: Real-World Ratio
    Using an infographic students look at such factors as age, gender and race to examine how the prison population in California compares to the general population. Students then apply an understanding of how they can find the value of a part by using a whole and a percent in order to look at how that can lead to recommendations for how to prevent crime. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
  • Identifying Statistical Questions
    he purpose of this task is to help students learn to distinguish between statistical questions and questions that are not statistical.
  • Is It Center or Is It Variability?
    The purpose of this task is to challenge students to think about whether they should be most interested in the center of the data distribution or in the spread of a data distribution in order to answer a given statistical question.
  • Puppy Weights
    Given the weights of puppies, the student is asked to draw a graph summarizing the varying weights, describe the distribution of the weights, and determine the typical weight of a puppy born in the location.
  • Statistical Questions
    The goal of this task is to promote a discussion of what makes a statistical question.
Summarize and describe distributions (Standards 6.SP.4–5)
  • Average Number of Siblings
    The goal of this task is to compare the mean and median in a context where the data is slightly skewed to the right.
  • Comparing Test Scores
    The goal of this task is to critically compare the center and spread of two data sets.
  • Describing Distributions
    In this task, students are asked to describe data distributions in terms of center, spread and overall shape and to also compare data distributions in terms of center and spread by selecting which of two distributions has a greater center and which has a greater spread.
  • Electoral College
    This task is intended to demonstrate that a graph can summarize a distribution as well as provide useful information about specific observations. With the table provided, the graph and values have context. The purpose of this task is to help students understand that a distribution can be described in terms of shape and center, and also to provide practice in selecting and calculating measures of center.
  • Examining California's Prison System: Real-World Ratio
    Using an infographic students look at such factors as age, gender and race to examine how the prison population in California compares to the general population. Students then apply an understanding of how they can find the value of a part by using a whole and a percent in order to look at how that can lead to recommendations for how to prevent crime. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.
  • Math Homework Problems
    The goal of this task is to calculate and interpret the Mean Absolute Deviation in a context. It is intended to be an introductory task but can readily be adapted for a more in depth study.
  • Mean or Median?
    The goal of this task is to examine advantages and disadvantages of the mean and median for summarizing a given data set.
  • Puppy Weights
    Given the weights of puppies, the student is asked to draw a graph summarizing the varying weights, describe the distribution of the weights, and determine the typical weight of a puppy born in the location.
  • Puzzle Times
    This is a simple task designed to assess students ability to construct a dot plot and to calculate and compare measures of center.
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