(6.MP) Ratios and Proportional Relationships

(6.RP) The Number System

(6.NS) Expressions and Equations

(6.EE) Geometry

(6.G) Statistics and Probability

(6.SP)

Standard 6.MP.1

Make sense of problems and persevere in solving them. Explain the meaning of a problem and look for entry points to its solution. Analyze givens, constraints, relationships, and goals. Make conjectures about the form and meaning of the solution, plan a solution pathway, and continually monitor progress asking, "Does this make sense?" Consider analogous problems, make connections between multiple representations, identify the correspondence between different approaches, look for trends, and transform algebraic expressions to highlight meaningful mathematics. Check answers to problems using a different method.

Standard 6.MP.2

Reason abstractly and quantitatively. Make sense of the quantities and their relationships in problem situations. Translate between context and algebraic representations by contextualizing and decontextualizing quantitative relationships. This includes the ability to decontextualize a given situation, representing it algebraically and manipulating symbols fluently as well as the ability to contextualize algebraic representations to make sense of the problem.

Standard 6.MP.3

Construct viable arguments and critique the reasoning of others. Understand and use stated assumptions, definitions, and previously established results in constructing arguments. Make conjectures and build a logical progression of statements to explore the truth of their conjectures. Justify conclusions and communicate them to others. Respond to the arguments of others by listening, asking clarifying questions, and critiquing the reasoning of others.

Standard 6.MP.4

Model with mathematics. Apply mathematics to solve problems arising in everyday life, society, and the workplace. Make assumptions and approximations, identifying important quantities to construct a mathematical model. Routinely interpret mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

Standard 6.MP.5

Use appropriate tools strategically. Consider the available tools and be sufficiently familiar with them to make sound decisions about when each tool might be helpful, recognizing both the insight to be gained as well as the limitations. Identify relevant external mathematical resources and use them to pose or solve problems. Use tools to explore and deepen their understanding of concepts. „„

Standard 6.MP.6

Attend to precision. Communicate precisely to others. Use explicit definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose. Specify units of measure and label axes to clarify the correspondence with quantities in a problem. Calculate accurately and efficiently, and express numerical answers with a degree of precision appropriate for the problem context.

Standard 6.MP.7

Look for and make use of structure. Look closely at mathematical relationships to identify the underlying structure by recognizing a simple structure within a more complicated structure. See complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, see 5 - 3(x - y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.

Standard 6.MP.8

Look for and express regularity in repeated reasoning. Notice if reasoning is repeated, and look for both generalizations and shortcuts. Evaluate the reasonableness of intermediate results by maintaining oversight of the process while attending to the details.

- Grade 6 Math Module 1: Ratios and Unit Rates (EngageNY)

Students begin their sixth grade year investigating the concepts of ratio and rate. They use multiple forms of ratio language and ratio notation, and formalize understanding of equivalent ratios. Students apply reasoning when solving collections of ratio problems in real world contexts using various tools (e.g., tape diagrams, double number line diagrams, tables, equations and graphs). Students bridge their understanding of ratios to the value of a ratio, and then to rate and unit rate, discovering that a percent of a quantity is a rate per 100. The 35 day module concludes with students expressing a fraction as a percent and finding a percent of a quantity in real world concepts, supporting their reasoning with familiar representations they used previously in the module. - Ratios and Proportional Relationships (6.RP) - 6th Grade Core Guide

The Utah State Board of Education (USBE) and educators around the state of Utah developed these guides for Mathematics Grade 6 - Ratios and Proportional Relationships

- Chapter 1 - Mathematical Foundations (UMSMP)

This is Chapter 1 of the Utah Middle School Math: Grade 6 textbook. It provides a Mathematical Foundation for Ratio Relations. - Chapter 1 - Student Workbook (UMSMP)

This is Chapter 1 of the Utah Middle School Math: Grade 6 student workbook. It covers the following topics: Ratio Relations. - Chapter 2 - Mathematical Foundations (UMSMP)

This is Chapter 2 of the Utah Middle School Math: Grade 6 textbook. It provides a Mathematical Foundation for Percent, Division with Fractions, and Measurement Conversion. - Chapter 2 - Student Workbook (UMSMP)

This is Chapter 2 of the Utah Middle School Math: Grade 6 student workbook. It covers the following topics: Percent, Division with Fractions, and Measurement Conversion. Engage NY - Grade 6 Math Module 1: Ratios and Unit Rates (EngageNY)

Students begin their sixth grade year investigating the concepts of ratio and rate. They use multiple forms of ratio language and ratio notation, and formalize understanding of equivalent ratios. Students apply reasoning when solving collections of ratio problems in real world contexts using various tools (e.g., tape diagrams, double number line diagrams, tables, equations and graphs). Students bridge their understanding of ratios to the value of a ratio, and then to rate and unit rate, discovering that a percent of a quantity is a rate per 100. The 35 day module concludes with students expressing a fraction as a percent and finding a percent of a quantity in real world concepts, supporting their reasoning with familiar representations they used previously in the module. - Grade 6 Unit 2: Rate, Ratio, and Proportional Reasoning Using Equivalent Fractions (Georgia Standard

In this unit, students will gain a deeper understanding of proportional reasoning through instruction and practice, develop and use multiplicative thinking, develop a sense of proportional reasoning, develop the understanding that ratio is a comparison of two numbers or quantities, find percents using the same processes for solving rates and proportions and solve real-life problems involving measurement units that need to be converted. - Grade 6 Unit 4: One Step Equations and Inequalities (Georgia Standards)

In this unit students will: Determine if an equation or inequality is appropriate for a given situation. Solve mathematical and real-world problems with equations. Represent real-world situations as inequalities. Interpret the solutions to equations and inequalities. Represent the solutions to inequalities on a number line. Analyze the relationship between dependent and independent variables through the use of tables, equations and graphs.

- 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.

- 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. - Fraction Conversion 2 (with percents)

When completing this lesson students will understand how to convert fractions, decimals, and percentages. - Grid and Percent It

This lesson plans provides a 10 x 10 model so that students can understand how to solve percent problems. - Gross Domestic Product: Unit Rates in the Real World

Using interactive maps from KQED students will examine the economic divide in European countries by looking at GDP. In the activity data from the maps is used to compare GDP among other countries. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. - IXL Game: Ratios, proportions, and percents

This game helps sixth graders understand ratios, proportions, and percents, specifically percents of numbers and money amounts. This is just one of many online games that supports the Utah Math core. Note: The IXL site requires subscription for unlimited use. - Inverse Proportions and Shadows in Practice

in this interactive a figure's shadow is projected on to a screen. Students then observe how the shadow changes as the figure moves farther away . Students also complete a chart that shows the distance of the figure from the light source and the height of the shadow helping them understand inverse proportions. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. - Life Expectancy: Finding Ratio Relationships

This lesson using infographics examines how health factors such as obesity and hypertension have changed in the U.S. over the past decades. The classroom activity has students work with the mathematical concept of statistical analysis as they compare the life expectancy of men and women. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. - Living Wages in CA: Ratio and Rate in the Real World

The use of infographics helps us understand the costs of basic living expenses. The classroom activity has students look at real-life examples and data to calculate whether the minimum wage in their state can be a living wage. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. - Mixtures

This activity will help students understand percentages and mixture problems by working with two piles of colored chips. - One-Dimensional Scaling To Find Unknown Heights

An interactive activity helps students understand real-world application of ratios and asks them to scale a model of a T-Rex for a diorama. In the classroom activity students are asked to draw scale models. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. - Real World Ratio and Rate Problem: Bianca's Fifty Percent Solution

Viewers follow Bianca as she's drawn into a store by the discounts advertised in this video from Cyberchase. While shopping she understands that while discounts are nice they still can add up when shopping. The classroom activity asks students to calculate the savings on an item when various discounts are applied. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. - Real-World Proportional Relationships: Gender Wage Gap

Students use an infographic to understand how wages of today compare with those of 50 years ago in this lesson plan. The classroom activity helps students understand and calculate the wage gap using media salaries for men and women. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. - Scaling Up Rectangles Using Simulations

This interactive allows students to create murals in different sizes by understanding two-dimensional scaling. The activity for the classroom has students use graph paper to draw and compare squares with different proportional dimensions and record the data as they change. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. - Scaling Up Rectangles in the Real World

By looking at murals in this video students see how artists use proportion and measurement to create them. The class then does an activity where they enlarge a drawing by using scale and then create a classroom mural. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. - Similar Figures and Unknown Heights in Practice

Students will learn about proportional reasoning in this interactive and how it can help find an unknown height as well as exploring proportional relationships among similar triangles. The classroom activity uses the interactive as a catalyst into a discussion of these math concepts. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. - Similar Figures in the Real World

A visit to the Louisville Slugger Museum and Factory shows students how measurement and engineering are involved in the creation of new baseball bats. The math skills of proportional reasoning and equivalent ratios are used in the classroom activity involving an oversized bat and a player. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.

Standard 6.RP.1

Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. The following are examples of ratio language:“The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C received nearly three votes.”Standard 6.RP.2

Understand the concept of a unit ratea/bassociated with a ratioa:bwithb≠ 0, and use rate language in the context of a ratio relationship. The following are examples of rate language:"This recipe has a ratio of four cups of flour to two cups of sugar, so the rate is two cups of flour for each cup of sugar." “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger."(In sixth grade, unit rates are limited to non-complex fractions.)Standard 6.RP.3

Use ratio and rate reasoning to solve real-world (with a context) and mathematical (void of context) problems, using strategies such as reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations involving unit rate problems.

- Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.
- Solve unit rate problems including those involving unit pricing and constant speed.
For example, if it took four hours to mow eight lawns, how many lawns could be mowed in 32 hours? What is the hourly rate at which lawns were being mowed?- Find a percent of a quantity as a rate per 100. Solve problems involving finding the whole, given a part and the percent.
(For example, 30% of a quantity means 30/100 times the quantity.)- Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.

- The Number System (6.NS) - 6th Grade Core Guide

The Utah State Board of Education (USBE) and educators around the state of Utah developed these guides for Mathematics Grade 6 - The Number System.

- Grade 6 Math Module 2: Arithmetic Operations Including Division of Fractions (EngageNY)

In Module 2, students complete their understanding of the four operations as they study division of whole numbers, division by a fraction and operations on multi-digit decimals. This expanded understanding serves to complete their study of the four operations with positive rational numbers, thereby preparing students for understanding, locating, and ordering negative rational numbers (Module 3) and algebraic expressions (Module 4).

- Grade 6 Unit 1: Number System Fluency (Georgia Standards)

In this unit students will find the greatest common factor of two whole numbers less than or equal to 100. Find the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1-100 with a common factor as a multiple of a sum of two whole numbers with no common factor. Interpret and compute quotients of fractions. Solve word problems involving division of fractions by fractions using visual fraction models and equations to represent the problem. Fluently divide multi-digit numbers using the standard algorithm. Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.

- 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.

- Keep, Change, Flip

Students are taught the "Keep, Change, Flip" rule for dividing fractions by viewing this clever Flocabulary rap song. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. - Modeling Fraction Division Using Comparison, Group Number Unknown

In this lesson students will learn how to solve a word problem involving the division of fractions by viewing an animation about a hedgehog's hibernation. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. - Modeling Fraction Division, Equal Groups, Group Size Unknown

The skill of dividing two fractions by groups of unknown size is the focus of this video. Students will learn how to solve a word problem using this process. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. - Modeling Fraction Division, Equal Groups, Number of Groups Unknown

This animated video shows students a model they can use to solve word problems involving the division of fractions. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. - 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.

Standard 6.NS.1

Interpret and compute quotients of fractions.

- Compute quotients of fractions by fractions,
for example, by applying strategies such as visual fraction models, equations, and the relationship between multiplication and division, to represent problems.- Solve real-world problems involving division of fractions by fractions.
For example, how much chocolate will each person get if three people share 1/2 pound of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mile and area 1/2 square mile?- Explain the meaning of quotients in fraction division problems.
For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.)

- The Number System (6.NS) - 6th Grade Core Guide

The Utah State Board of Education (USBE) and educators around the state of Utah developed these guides for Mathematics Grade 6 - The Number System.

- Chapter 0 - Mathematical Foundations (UMSMP)

This is Chapter 0 of the Utah Middle School Math: Grade 6 textbook. It provides a Mathematical Foundation for Fluency. - Chapter 0 - Student Workbook (UMSMP)

This is Chapter 0 of the Utah Middle School Math: Grade 6 student workbook. It covers the following topics: Fluency. - Chapter 6 - Mathematical Foundations (UMSMP)

This is Chapter 6 of the Utah Middle School Math: Grade 6 textbook. It provides a Mathematical Foundation for Expressions and Equations. - Chapter 6 - Student Workbook (UMSMP)

This is Chapter 6 of the Utah Middle School Math: Grade 6 student workbook. It covers the following topics: Expressions and Equations. Engage NY - Grade 6 Math Module 2: Arithmetic Operations Including Division of Fractions (EngageNY)

In Module 2, students complete their understanding of the four operations as they study division of whole numbers, division by a fraction and operations on multi-digit decimals. This expanded understanding serves to complete their study of the four operations with positive rational numbers, thereby preparing students for understanding, locating, and ordering negative rational numbers (Module 3) and algebraic expressions (Module 4). - Grade 6 Unit 1: Number System Fluency (Georgia Standards)

In this unit students will find the greatest common factor of two whole numbers less than or equal to 100. Find the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1-100 with a common factor as a multiple of a sum of two whole numbers with no common factor. Interpret and compute quotients of fractions. Solve word problems involving division of fractions by fractions using visual fraction models and equations to represent the problem. Fluently divide multi-digit numbers using the standard algorithm. Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation. - Grade 6 Unit 3: Expressions (Georgia Standards)

In this unit students will represent repeated multiplication with exponents. Evaluate expressions containing exponents to solve mathematical and real world problems. Translate verbal phrases and situations into algebraic expressions. Identify the parts of a given expression. Use the properties to identify equivalent expressions. Use the properties and mathematical models to generate equivalent expressions.

- 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.

- Distributive Property with Variables

Algebra tiles are used to generate equivalent expressions using the distributive property in this instructional video. The classroom activity asks student to further explore the distributive property. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. - Equivalent Expressions with the Distributive Property

This animated Math Shorts video explains how the distributive property can help students model and create equivalent expressions. In the accompanying classroom activity, students play a quick game where they identify common factors within an expression and work on a series of problems that expand their understanding of how to apply the distributive property. While the problems begin with whole number expressions, students soon work toward algebraic notation and eventually develop the idea that ax + bx can be rewritten as x(a + b). NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. - Factors

This lesson is designed to help students understand factors of whole numbers. - Finding Factors

This lesson plan's activities give students practice in finding the factors of whole numbers. - Greatest Common Factor

This video from Math Shorts shows students how to find the greatest common factor of 2 numbers. The classroom activity shows them how Venn diagrams, multiplication and prime factors can also be used to find the greatest common factor. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. - Manipulating Graphs

This video demonstrates how to use the slope-intercept of a line to the graph of that line. The classroom activity has them demonstrate their understanding by finding equations for a set of lines through the origin. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. - Patterns of Factors

This interactive activity asks students to sort numbers based on the number of factors or prime factors. They are asked to also identify one real-life example of the usefulness of divisibility. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. - Sets and the Venn Diagram

This lesson is designed to help students understand the ideas surrounding sets and Venn diagrams.

Standard 6.NS.2

Fluently divide multi-digit numbers using the standard algorithm.Standard 6.NS.3

Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.

- Fluently divide multi-digit decimals using the standard algorithm, limited to a whole number dividend with a decimal divisor or a decimal dividend with a whole number divisor.
- Solve division problems in which both the dividend and the divisor are multi-digit decimals; develop the standard algorithm by using models, the meaning of division, and place value understanding.
Standard 6.NS.4

Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers with no common factor.For example, express 36 + 8 as 4 (9 + 2).

- The Number System (6.NS) - 6th Grade Core Guide

The Utah State Board of Education (USBE) and educators around the state of Utah developed these guides for Mathematics Grade 6 - The Number System.

- Chapter 3 - Student Workbook (UMSMP)

This is Chapter 3 of the Utah Middle School Math: Grade 6 student workbook. It covers the following topics: Extending the Number System. - Chapter 3 - Mathematical Foundations (UMSMP)

This is Chapter 3 of the Utah Middle School Math: Grade 6 textbook. It provides a Mathematical Extending the Number System. Engage NY - Grade 6 Math Module 3: Rational Numbers (EngageNY)

Students are familiar with the number line and determining the location of positive fractions, decimals, and whole numbers from previous grades. Students extend the number line (both horizontally and vertically) in Module 3 to include the opposites of whole numbers. The number line serves as a model to relate integers and other rational numbers to statements of order in real-world contexts. In this module's final topic, the number line model is extended to two-dimensions, as students use the coordinate plane to model and solve real-world problems involving rational numbers. - Grade 6 Unit 7: Rational Explorations: Numbers and their Opposites (Georgia Standards)

In this unit students will understand that positive and negative numbers are used together to describe quantities having opposite directions or values, understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates. Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line. Recognize that the opposite of the opposite of a number is the number itself.

- 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.

- Absolute Value

This Math Shorts video uses a number line and a real-life example to explain the absolute value of a number. The classroom activity then has the student play a game where they move a penny in both positive and negative directions on a number line. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. - Adding Rational Numbers on the Number Line

In this interactive students must solve riddles about a wallaby jumping contest. But they must find equivalent fractions and common denominators to complete the riddle. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. - Addition and Subtraction of Integers

A card game in which positive and negative numbers are added together is the subject of this video teaching students how to add and subtract integers. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. - Fractions and Decimals from 0 to 1 on the Vertical Number Line

In this interactive students must solve riddles about jumping fleas by placing fractions and decimals on a 0 to 1 number line. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. - Fractions, Mixed Numbers, and Decimals on the Number Line

Using the device of a frog-jumping contest, students learn about values on a number line by placing the frogs' jump distances on the line. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. - Harry Makes a Big Splash with Positive and Negative Numbers

A video about a swim relay team opens this lesson on combining positive and negative numbers. The classroom activity involves using a number line and a game board and die. Students will write equations that represent the addition of positive and negative numbers. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. - Horizontal and Vertical Distances on the Cartesian Graph

In this activity students place marine animals on a Cartesian graph and then determine the horizontal and vertical distance between them. The classroom activity builds on the student's understanding of distances between points on a Cartesian graph. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. - IXL Game: Coordinate graphing

This game will help sixth graders learn to graph points on a coordinate plane. This is just one of many online games that supports the Utah Math core. Note: The IXL site requires subscription for unlimited use. - Locating Points on the Cartesian Graph

In this activity students use logic and clues to plot the location of marine animals on a Cartesian graph. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. - Manipulating Graphs

This video demonstrates how to use the slope-intercept of a line to the graph of that line. The classroom activity has them demonstrate their understanding by finding equations for a set of lines through the origin. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. - Rational Numbers on the Number Line

Riddles about a wallaby jumping contest must be solved in this lesson. Students must place fractions, decimals or mixed numbers representing the lengths of jumps, on a number line from -5 to +5. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. - Sample Assessment Task: Cake Weighing

This four-part sample assessment task provides a novel situation in an authentic context that all students can understand, even though it is not likely to be in their everyday experience. Use the navigation at the upper right of this page to access the task. - Using the Pythagorean Theorem on the Cartesian Graph

Students place animals on a Cartesian graph in this interactive activity. They then use the Pythagorean Theorem to determine the distance between the animals. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.

Standard 6.NS.5

Understand that positive and negative numbers are used together to describe quantities having opposite directions or values(for example, temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge);use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of zero in each situation.Standard 6.NS.6

Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.

- Recognize opposite signs of numbers as indicating locations on opposite sides of zero on the number line; recognize that the opposite of the opposite of a number is the number itself.
For example, -(-3) = 3, and zero is its own opposite.- Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.
- Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.
Standard 6.NS.7

Understand ordering and absolute value of rational numbers.

- Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram.
For example, interpret –3 > –7 as a statement that –3 is located to the right of –7 on a number line oriented from left to right.- Write, interpret, and explain statements of order for rational numbers in real-world contexts.
For example, write –3 °C > –7 °C to express the fact that –3 °C is warmer than –7 °C.- Understand the absolute value of a rational number as its distance from zero on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world context.
For example, for an account balance of –30 dollars, write |–30| = 30 to describe the size of the debt in dollars.- Distinguish comparisons of absolute value from statements about order.
For example, recognize that an account balance less than –30 dollars represents a debt greater than 30 dollars.Standard 6.NS.8

Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the samex-coordinate or the samey-coordinate.

- Expressions and Equations (6.EE) - 6th Grade Core Guide

The Utah State Board of Education (USBE) and educators around the state of Utah developed these guides for Mathematics Grade 6 - Expressions and Equations.

- Chapter 6 - Mathematical Foundations (UMSMP)

This is Chapter 6 of the Utah Middle School Math: Grade 6 textbook. It provides a Mathematical Foundation for Expressions and Equations. - Chapter 6 - Student Workbook (UMSMP)

This is Chapter 6 of the Utah Middle School Math: Grade 6 student workbook. It covers the following topics: Expressions and Equations. Engage NY - Grade 6 Math Module 4: Expressions and Equations (EngageNY)

In Module 4, Expressions and Equations, students extend their arithmetic work to include using letters to represent numbers in order to understand that letters are simply "stand-ins" for numbers and that arithmetic is carried out exactly as it is with numbers. Students explore operations in terms of verbal expressions and determine that arithmetic properties hold true with expressions because nothing has changedthey are still doing arithmetic with numbers. Students determine that letters are used to represent specific but unknown numbers and are used to make statements or identities that are true for all numbers or a range of numbers. - Grade 6 Unit 3: Expressions (Georgia Standards)

In this unit students will represent repeated multiplication with exponents. Evaluate expressions containing exponents to solve mathematical and real world problems. Translate verbal phrases and situations into algebraic expressions. Identify the parts of a given expression. Use the properties to identify equivalent expressions. Use the properties and mathematical models to generate equivalent expressions.

- 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.

- Algebra Four

This lesson contains a game activity designed to help students practice solving algebraic equations. - 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. - Distributive Property with Variables

Algebra tiles are used to generate equivalent expressions using the distributive property in this instructional video. The classroom activity asks student to further explore the distributive property. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. - Equivalent Expressions with the Distributive Property

This animated Math Shorts video explains how the distributive property can help students model and create equivalent expressions. In the accompanying classroom activity, students play a quick game where they identify common factors within an expression and work on a series of problems that expand their understanding of how to apply the distributive property. While the problems begin with whole number expressions, students soon work toward algebraic notation and eventually develop the idea that ax + bx can be rewritten as x(a + b). NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. - IXL Game: Algebra: Evaluate expressions

This game is designed to help sixth graders understand how to evaluate expressions involving integers. This is just one of many online games that supports the Utah Math core. Note: The IXL site requires subscription for unlimited use. - Linear Function Machine

By putting different values into the linear function machine students will explore simple linear functions. - Order of Operations: PEMDAS

A Flocabulary rap song instructs students on the order of operations and then they apply that knowledge to the classroom activity. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.

Standard 6.EE.1

Write and evaluate numerical expressions involving whole-number exponents.Standard 6.EE.2

Write, read, and evaluate expressions in which letters stand for numbers.

- Write expressions that record operations with numbers and with letters representing numbers.
For example, express the calculation "Subtract y from 5" as 5 - y and express "Jane had $105.00 in her bank account. One year later, she had x dollars more. Write an expression that shows her new balance" as $105.00 + x.- Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity.
For example, describe the expression 2(8 + 7) as a product of two factors; view (8 + 7) as both a single entity and a sum of two terms.- Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, applying the Order of Operations when there are no parentheses to specify a particular order.
For example, use the formulas V = s^{3}and A = 6s^{2}to find the volume and surface area of a cube with sides of length s = 1/2.Standard 6.EE.3

Apply the properties of operations to generate equivalent expressions.For example, apply the distributive property to the expression 3(2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y.Standard 6.EE.4

Identify when two expressions are equivalent.For example, the expressions y + y + y and 3y are equivalent because they name the same number, regardless of which number y represents.

- Expressions and Equations (6.EE) - 6th Grade Core Guide

The Utah State Board of Education (USBE) and educators around the state of Utah developed these guides for Mathematics Grade 6 - Expressions and Equations.

- Chapter 6 - Mathematical Foundations (UMSMP)

This is Chapter 6 of the Utah Middle School Math: Grade 6 textbook. It provides a Mathematical Foundation for Expressions and Equations. - Chapter 6 - Student Workbook (UMSMP)

This is Chapter 6 of the Utah Middle School Math: Grade 6 student workbook. It covers the following topics: Expressions and Equations.

- Grade 6 Unit 4: One Step Equations and Inequalities (Georgia Standards)

In this unit students will: Determine if an equation or inequality is appropriate for a given situation. Solve mathematical and real-world problems with equations. Represent real-world situations as inequalities. Interpret the solutions to equations and inequalities. Represent the solutions to inequalities on a number line. Analyze the relationship between dependent and independent variables through the use of tables, equations and graphs.

- 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.

- Algebra Four

This lesson contains a game activity designed to help students practice solving algebraic equations. - Balancing Scales To Solve Equations

The focus of this lesson and interactive is balancing algebraic equations. Students engage with the Annenberg interactive to solve 3 balance problems. The classroom activity then asks students to change balance problems into algebraic equations where variables represent unknown amounts. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. - Distributive Property with Variables

Algebra tiles are used to generate equivalent expressions using the distributive property in this instructional video. The classroom activity asks student to further explore the distributive property. 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 Patterns to Make Predictions

This activity asks students to identify and contemplate mathematical patterns that we see around us. They are asked to represent them in a table and predict the pattern to the 7th, 9th, and nth terms. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. - Graphing Inequalities: Fractions and Decimals from 0 to 1

In this interactive, use logic to solve three riddles involving high-jump performers in a flea circus. Then, using knowledge of inequalities, place the fleas in the appropriate range on a vertical number line. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. - Graphing Inequalities: Fractions, Mixed Numbers, and Decimals

In this interactive, use logic to solve three riddles involving a jumping frog competition. Then, using knowledge of inequalities and rational numbers, place the frogs in the correct range from 0 to 5 on a number line. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. - Graphing Inequalities: Rational Numbers

In this interactive, use logic to solve three riddles involving a jumping wallaby competition. Then, using knowledge of inequalities and rational numbers, place the wallabies on the correct range from 5 to 5 on the number line. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.

Standard 6.EE.5

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 6.EE.6

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 6.EE.7

Solve real-world and mathematical problems by writing and solving equations of the formx + a = bandax = bfor cases in whicha, bandxare all non-negative rational numbers.Standard 6.EE.8

Write an inequality of the formx > corx < cto represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the formx > corx < chave infinitely many solutions; represent solutions of such inequalities on number line diagrams.

- Expressions and Equations (6.EE) - 6th Grade Core Guide

The Utah State Board of Education (USBE) and educators around the state of Utah developed these guides for Mathematics Grade 6 - Expressions and Equations.

- Grade 6 Unit 4: One Step Equations and Inequalities (Georgia Standards)

In this unit students will: Determine if an equation or inequality is appropriate for a given situation. Solve mathematical and real-world problems with equations. Represent real-world situations as inequalities. Interpret the solutions to equations and inequalities. Represent the solutions to inequalities on a number line. Analyze the relationship between dependent and independent variables through the use of tables, equations and graphs.

- 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.

- Algebra Four

This lesson contains a game activity designed to help students practice solving algebraic equations. - Finding Patterns to Make Predictions

This activity asks students to identify and contemplate mathematical patterns that we see around us. They are asked to represent them in a table and predict the pattern to the 7th, 9th, and nth terms. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. - Linear Function Machine

By putting different values into the linear function machine students will explore simple linear functions. - Linear Inequalities

This online tutorial is designed to help the student to understand the vocabulary of inequalities and then use addition, subtraction, multiplication and division to solve linear inequalities.

Standard 6.EE.9

Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation.For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.

- Geometry (6.G) - 6th Grade Core Guide

The Utah State Board of Education (USBE) and educators around the state of Utah developed these guides for Mathematics Grade 6 - Geometry.

- Chapter 5 - Mathematical Foundations (UMSMP)

This is Chapter 5 of the Utah Middle School Math: Grade 6 textbook. It provides a Mathematical Foundation for Geometry. - Chapter 5 - Student Workbook (UMSMP)

This is Chapter 5 of the Utah Middle School Math: Grade 6 student workbook. It covers the following topics: Geometry. Engage NY - Grade 6 Math Module 5: Area, Surface Area, and Volume Problems (EngageNY)

In this module, students utilize their previous experiences in order to understand and develop formulas for area, volume, and surface area. Students use composition and decomposition to determine the area of triangles, quadrilaterals, and other polygons. Extending skills from Module 3 where they used coordinates and absolute value to find distances between points on a coordinate plane, students determine distance, perimeter, and area on the coordinate plane in real-world contexts - Grade 6 Unit 5: Area and Volume (Georgia Standards)

In this unit students will: Find areas of right, equilateral, isosceles, and scalene triangles, and special quadrilaterals. Find areas of composite figures and polygons by composing into rectangles and decomposing into triangles and other shapes. Solve real-world and mathematical problems involving area. Decipher and draw views of rectangular and triangular prisms from a variety of perspectives. Recognize and construct nets for rectangular and triangular prisms. Find the surface area of rectangular and triangular prisms by using manipulatives and by constructing nets. - Grade 6 Unit 7: Rational Explorations: Numbers and their Opposites (Georgia Standards)

In this unit students will understand that positive and negative numbers are used together to describe quantities having opposite directions or values, understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates. Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line. Recognize that the opposite of the opposite of a number is the number itself.

- 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).

- 2D Nets and 3D Decorative Boxes

Calculating the surface area of cardboard boxes is the focus of this interactive activity. The classroom activity takes this knowledge and asks the students for figure out how many square inches of wrapping paper is needed to wrap a gift. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. - A Place in Space

In this lesson the student is asked to describe a point in space using coordinates. - Area

This lesson is designed to help students be able to calculate the area of a random shape on a grid, explain the correlation between the size of the perimeter and the number of different possible areas that can be contained within that perimeter. - Area Explorations

In this lesson, students will explore the area of irregular shapes to find multiple different methods for calculating area - Areas of Irregular Shapes: Building Sailboats

Learn how wooden boat builders use a variety of mathematical concepts when custom designing their vessels. This video focuses on how area, volume, and measurements of irregular shapes are used in the engineering process, taking math out of the classroom and into real world problem solving. - Boxed In and Wrapped Up

This lesson asks students to find the volume and surface area of a rectangular box and then convert it into a cubical box with the same volume. - Cartesian Coordinate System

This lesson is designed to help students understand the Cartesian plane, specifically how to plot points, read coordinates and find the ratio of the rise over run for slope. - 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. - Great Modeling Tasks in Three Acts - File Cabinet

This surface area activity has students answer the question: How many stickies cover the cabinet? - Horizontal and Vertical Distances on the Cartesian Graph

In this activity students place marine animals on a Cartesian graph and then determine the horizontal and vertical distance between them. The classroom activity builds on the student's understanding of distances between points on a Cartesian graph. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. - Pentagon Puzzles

By deconstructing pentagons into triangles, students in this activity learn how to calculate the area of pentagons. - Scale Models and Three-Dimensional Scaling in Practice

Students can use this interactive to explore how an object changes when enlarged by a factor of 10. They put this understanding to use in the activity when they compare two cubes of different sizes by volume and surface area. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. - Surface Area , Area and Volume: The Big Sleep

In this video, Bianca is planning a sleepover for friends. She has to figure out how many people she can invite because the floor will only hold so many sleeping bags. She must calculate both the surface area of the floor and the surface area of a sleeping bag. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. - Surface Area and Volume

An online activity is the focus of this lesson plan to help students understand the concepts of surface area and volume. - Surface Area of Prisms

In this lesson students will understand surface area and how solve for the surface area of triangular prisms. - Surface Area of a Rectangular Prisms

This lesson will help students understand surface area and solve problems using the surface area of a rectangular prism. - Table for 22: A Real-World Geometry Project

This Teaching Channel video has students apply knowledge of area and perimeter to solve real-world problems. This site provides a lesson plan and student handouts. (13 minutes) - The Largest Container: Problems Using Volume and Shape

By using a single sheet of paper this interactive leads students to construct shapes, calculate volume, and think about the relationships between different shapes. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. - Triangle Area

This interactive lesson plan will help students understand how to find the area of a right triangle. - Triangle Explorer

The applet in this lesson allows students to draw triangles and calculate their area. - Volume of Prisms

This is a lesson designed to help students understand how to solve problems for the volume of triangular prisms. - Volume of Rectangular Prisms

This lesson is designed to help students understand how to solve for the volume of rectangular prisms. - What's Fun About Surface Area?

In this Teaching Channel video an educator helps students construct an understanding of surface area. (7 minutes)

Standard 6.G.1

Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.Standard 6.G.2

Find the volume of a right rectangular prism with appropriate unit fraction edge lengths by packing it with cubes of the appropriate unit fraction edge lengths(for example, 3½ x 2 x 6), and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulasV = kWhandV = bhto find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems. (Note: Model the packing using drawings and diagrams.)Standard 6.G.3

Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems.Standard 6.G.4

Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.

- Statistics and Probability (6.SP) - 6th Grade Core Guide

The Utah State Board of Education (USBE) and educators around the state of Utah developed these guides for Mathematics Grade 6 - Statistics and Probability.

- Chapter 4 - Mathematical Foundations (UMSMP)

This is Chapter 4 of the Utah Middle School Math: Grade 6 textbook. It provides a Mathematical Foundation for Fluency. - Chapter 4 - Student Workbook (UMSMP)

This is Chapter 4 of the Utah Middle School Math: Grade 6 student workbook. It covers the following topics: Statistics. Engage NY - Grade 6 Math Module 6: Statistics (EngageNY)

In this module, students move from simply representing data into analysis of data. Students begin to think and reason statistically, first by recognizing a statistical question as one that can be answered by collecting data. Students learn that the data collected to answer a statistical question has a distribution that is often summarized in terms of center, variability, and shape. - Grade 6 Unit 6: Statistics (Georgia Standards)

In this unit students will: Analyze data from many different sources such as organized lists, box-plots, bar graphs, histograms and dot plots. Understand that responses to statistical questions may vary. Understand that data can be described by a single number. Determine quantitative measures of center (median and/or mean). Determine quantitative measures of variability (interquartile range and range).

- 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.

- Box Plots

By completing this lesson students will understand the concept of median, quartiles, and how to build a box plot. - Comparative Experimental Design

This Annenberg Learner's Learning Math interactive teaches students the difference between the design of comparative observational studies and comparative experimental studies. Students then design their own study in the classroom activity. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. - Data Analysis Using Mean, Median, Mode, and Range

There are 4 videos in this lesson helping students learn about mean, median, mode and range. The classroom activity consists of a game in which students make attempts to get a pom-pom into a can. They collect the data about attempts. then calculate the mean, median, mode and range. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. - Data Organization and Representation: Weather Forecasting

In this lesson's video students see how a meteorologist organizes weather data for a presentation on-air. Students are then given sets of recent weather data which they must analyze to find the center and variability. They then predict the next day's temperature based on those statistics. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. - 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. - Gathering Data for a Line Plot

Two interactives are used in this lesson and both involve counting raisins in boxes and then plotting a graph. Students extend their understanding of mean, median, mode, and range after creating line plots in the classroom activity. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. - How Long Is a Minute?

This activity is designed to give students an understanding of arithmetic mean, median, and range of data sets. They practice using a virtual stopwatch in the interactive to estimate how long a minute is. They collect the data from repeat guesses and use that in the activity. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. - IXL Game: Statistics

This game is designed to help sixth graders understand how to calculate mean, median, mode, and range. - Life Expectancy: Finding Ratio Relationships

This lesson using infographics examines how health factors such as obesity and hypertension have changed in the U.S. over the past decades. The classroom activity has students work with the mathematical concept of statistical analysis as they compare the life expectancy of men and women. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. - Line Plot Representation of Deviation from the Mean

The concepts of means and deviations are the focus of this interactive from Annenberg. Students manipulate dots on a line graph to experiments with deviation. The classroom activity helps them understand positive and negative deviation. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. - Make a Stem and Leaf Plot

Students use data values to create stem and leaf plots with this interactive. This is further explored in the classroom activity which also asks students to interpret data to answer a question. 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 Task: The Missing Words

Math Task Overview: Students should be able to explain a reasonable strategy for determining the number of missing words. They should accurately compute the mean and range, and select an appropriate graph for displaying the data. Students will explore the concepts of variability and distribution of a data set. - Measuring Variability Through Tracking Wildfires

Data about wildfires in the U.S. is the basis of this lesson. Students examine the data about active fires as well as historical data and use that data to find changes across decades. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. - Prison Population: Real World Statistical Variability

Statistical data about changes in state and national prison populations from 1925 is the focus of this lesson. Students examine the data to make observations, find patterns and calculate the median and range. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. - Recognize and Represent Proportional Relationships Between Quantities: Ratio, Proportion, Cross Mult

Four videos are included in this lesson explaining ratio, proportion and how to use the shortcut of cross multiply and divide. Students then play the game "Pom-Pom Nose Push" to collect data and determine the ratio of time to distance. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. - Spinner

By manipulating a spinner and its pointer students will learn about probability in this activity. - The Bell Curve

This lesson and activity introduces the student to the concept of the Bell Curve and distribution. - The Hunger Games

By analyzing the lottery system used in the novel Hunger Games students will learn about probability. Students then create their own lottery system based on criteria from the novel in the accompanying classroom activity. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. - The Median: Salary.com

By looking at the data on the website Salary.com students will understand the application of quartiles and percentiles to understand and compare salaries. In the activity students use data set about salaries to create box plots and look at pay differentials. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. - Variation About the Mean: The Boston Harbor Project

The use and statistical analysis of data in the efforts to improve the water quality in Boston Harbor is the subject of this video. Students then apply an understanding of mean absolute deviation to examine data. They also use data about bacteria found in a river in an analysis of what that data set tells. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.

Standard 6.SP.1

Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers.For example, “How old am I?” is not a statistical question, but “How old are the students in my school?” is a statistical question because one anticipates variability in students’ ages.Standard 6.SP.2

Understand that a set of data collected to answer a statistical question has a distribution that can be described by its center, spread/range and overall shape.Standard 6.SP.3

Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.

- Statistics and Probability (6.SP) - 6th Grade Core Guide

The Utah State Board of Education (USBE) and educators around the state of Utah developed these guides for Mathematics Grade 6 - Statistics and Probability.

- Chapter 4 - Mathematical Foundations (UMSMP)

This is Chapter 4 of the Utah Middle School Math: Grade 6 textbook. It provides a Mathematical Foundation for Fluency. - Chapter 4 - Student Workbook (UMSMP)

This is Chapter 4 of the Utah Middle School Math: Grade 6 student workbook. It covers the following topics: Statistics.

- Grade 6 Unit 6: Statistics (Georgia Standards)

In this unit students will: Analyze data from many different sources such as organized lists, box-plots, bar graphs, histograms and dot plots. Understand that responses to statistical questions may vary. Understand that data can be described by a single number. Determine quantitative measures of center (median and/or mean). Determine quantitative measures of variability (interquartile range and range).

- 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.

- Box Plot and Five Number Summaries

An understanding of the terms minimum, maximum, median, and quartiles is the focus of this lesson. Students first use the interactive to compare different representations, then learn how to create a visual representation of data. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. - Box Plots

By completing this lesson students will understand the concept of median, quartiles, and how to build a box plot. - Comparing A Stem and Leaf Plot, Histogram, and Frequency Table

This lesson is all about the different ways to represent data. Students then practice creating various representations when given data. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. - Data Analysis Using Mean, Median, Mode, and Range

There are 4 videos in this lesson helping students learn about mean, median, mode and range. The classroom activity consists of a game in which students make attempts to get a pom-pom into a can. They collect the data about attempts. then calculate the mean, median, mode and range. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. - Data Organization and Representation: Weather Forecasting

In this lesson's video students see how a meteorologist organizes weather data for a presentation on-air. Students are then given sets of recent weather data which they must analyze to find the center and variability. They then predict the next day's temperature based on those statistics. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. - Describing Distributions: New Balance

Using the shoe company New Balance, this video takes a look at how the company uses statistical data to plan what to produce and sell. Students then discuss how data can be applied in real world situation. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. - 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. - Gathering Data for a Line Plot

Two interactives are used in this lesson and both involve counting raisins in boxes and then plotting a graph. Students extend their understanding of mean, median, mode, and range after creating line plots in the classroom activity. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. - How Long Is a Minute?

This activity is designed to give students an understanding of arithmetic mean, median, and range of data sets. They practice using a virtual stopwatch in the interactive to estimate how long a minute is. They collect the data from repeat guesses and use that in the activity. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. - Line Plot Representation of Deviation from the Mean

The concepts of means and deviations are the focus of this interactive from Annenberg. Students manipulate dots on a line graph to experiments with deviation. The classroom activity helps them understand positive and negative deviation. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. - Make a Stem and Leaf Plot

Students use data values to create stem and leaf plots with this interactive. This is further explored in the classroom activity which also asks students to interpret data to answer a question. 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 Task: The Missing Words

Math Task Overview: Students should be able to explain a reasonable strategy for determining the number of missing words. They should accurately compute the mean and range, and select an appropriate graph for displaying the data. Students will explore the concepts of variability and distribution of a data set. - Pie Chart

Students enter values in this applet and create pie charts in which they can vary the number or size of sections and display as fractions or percentages. - Real-World Data Sets: Low- and High-Paying Jobs

Infographics with data about the highest and lowest paying jobs in the U.S. is the basis of this lesson. Students examine the data to find the disparity between the high and low rates and create box plots to represent the data. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. - Responses to a Court Verdict: Analyzing Samples to Gain Understanding

The infographic basis of this activity pulls the Trayvon Martin case out of the headlines to examine the data collected from a poll about the case. The students need to analyze the responses to conclude whether the results fall along racial lines. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. - Student Task: Candy Bars

In this task, students analyze a survey to decide how many candy bars students typically eat in a week. - Student Task: Suzi's Company

In this task, students must help Suzi figure out the annual salary bill for her company and check some statistics about rates of pay. - Summarize Numerical Data Sets Using Venn Diagrams

The 3 videos in this lesson teach students about Venn diagrams: ones with 2 circles; those with 3 circles; and those with a circle within a circle. Students then apply their understanding to activities where they have to represent certain things, such as students' clothing colors, with a Venn diagram. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. - The Bell Curve

This lesson and activity introduces the student to the concept of the Bell Curve and distribution. - The Hunger Games

By analyzing the lottery system used in the novel Hunger Games students will learn about probability. Students then create their own lottery system based on criteria from the novel in the accompanying classroom activity. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. - The Median: Salary.com

By looking at the data on the website Salary.com students will understand the application of quartiles and percentiles to understand and compare salaries. In the activity students use data set about salaries to create box plots and look at pay differentials. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. - Using Graphs to Identify Patterns and Trends in US Immigrant Residency

Understanding where immigrants lived and how they moved is the focus of the info graphic used in this lesson. Students interpret the graph to understand the history of immigration in our country and look at patterns, trends and growth. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort. - Variation About the Mean: The Boston Harbor Project

The use and statistical analysis of data in the efforts to improve the water quality in Boston Harbor is the subject of this video. Students then apply an understanding of mean absolute deviation to examine data. They also use data about bacteria found in a river in an analysis of what that data set tells. NOTE: You have to create a Free PBS Account to view this web page, but it is easy to do and worth the effort.

Standard 6.SP.4

Display numerical data in plots on a number line, including dot plots, histograms, and box plots. Choose the most appropriate graph/plot for the data collected.Standard 6.SP.5

Summarize numerical data sets in relation to their context, such as by:

- Reporting the number of observations.
- Describing the nature of the attribute under investigation, including how it was measured and its units of measurement.
- Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered.
- Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered.

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