Distributions and Variability
Type of Unit: Project
Students should be able to:
Represent and interpret data using a line plot.
Understand other visual representations of data.
Students begin the unit by discussing what constitutes a statistical question. In order to answer statistical questions, data must be gathered in a consistent and accurate manner and then analyzed using appropriate tools.
Students learn different tools for analyzing data, including:
Measures of center: mean (average), median, mode
Measures of spread: mean absolute deviation, lower and upper extremes, lower and upper quartile, interquartile range
Visual representations: line plot, box plot, histogram
These tools are compared and contrasted to better understand the benefits and limitations of each. Analyzing different data sets using these tools will develop an understanding for which ones are the most appropriate to interpret the given data.
To demonstrate their understanding of the concepts, students will work on a project for the duration of the unit. The project will involve identifying an appropriate statistical question, collecting data, analyzing data, and presenting the results. It will serve as the final assessment.
Students calculate the mean absolute deviation (MAD) for three data sets and use it to decide which data set is best represented by the mean.The concept of mean absolute deviation (MAD) is introduced. Students understand that the sum of the deviation of the data from the mean is zero. Students calculate the MAD and understand its significance. Students find the mean and MAD of a sample set of data.Key ConceptsThe mean absolute deviation (MAD) is a measure of how much the values in a data set deviate from the mean. It is calculated by finding the distance of each value from the mean and then finding the mean of these distances.Goals and Learning ObjectivesGain a deeper understanding of mean.Understand that the mean absolute deviation (MAD) is a measure of how well the mean represents the data.Compare data sets using measures of center (mode, median, mean) and spread (range and MAD).Show that the sum of deviations from the mean is zero.
Students make a box plot for their typical-sixth-grader data from Lesson 7 and write a summary of what the plot shows.Using the line plot from Lesson 4, students construct a box plot. Students learn how to calculate the five-number summary and interquartile range (IQR). Students apply this knowledge to the data used in Lesson 7 and describe the data in terms of the box plot. Class discussion focuses on comparing the two graphs and what they show for the sets of data.Key ConceptsA box-and-whisker plot, or box plot, shows the spread of a set of data. It shows five key measures, called the five-number summary.Lower extreme: The smallest value in the data setLower quartile: The middle of the lower half of the data, and the value that 25% of the data fall belowMedian: The middle of the data setUpper quartile: The middle of the upper half of the data, and the value that 25% of the data are aboveUpper extreme: The greatest value in the data setThis diagram shows how these values relate to the parts of a box plot.The length of the box represents the interquartile range (IQR), which is the difference between the lower and upper quartile.A box plot divides the data into four equal parts. One quarter of the data is represented by the left whisker, two quarters by each half of the box, and one quarter by the right whisker. If one of these parts is long, the data in that quarter are spread out. If one of these quarters is short, the data in that quarter are clustered together.Goals and Learning ObjectivesLearn how to construct box plots, another tool to describe data.Learn about the five-number summary, interquartile range, and how they are related to box plots.Compare a line plot and box plot for the same set of data.
Students critique and improve their work on the Self Check from Lesson 13.Key ConceptsMeasures of spread (five-number summary) show characteristics of the data. It is possible to generate an appropriate data set with this information.Goals and Learning ObjectivesApply knowledge of statistics to solve problems.Identify the five-number summary, and understand measures of center and use their properties to solve problems.Track and review choice of strategy when problem solving.
Groups begin presentations for their unit project. Students provide constructive feedback on others' presentations.Key ConceptsThe unit project serves as the final assessment. Students should demonstrate their understanding of unit concepts:Measures of center (mean, median, mode) and spread (MAD, range, interquartile range)The five-number summary and its relationship to box plotsRelationship between data sets and line plots, box plots, and histogramsAdvantages and disadvantages of portraying data in line plots, box plots, and histogramsGoals and Learning ObjectivesPresent projects and demonstrate an understanding of the unit concepts.Provide feedback for others' presentations.Review the concepts from the unit.
Remaining groups present their unit projects. Students discuss teacher and peer feedback.Key ConceptsThe unit project serves as the final assessment. Students should demonstrate their understanding of unit concepts:Measures of center (mean, median, mode) and spread (MAD, range, interquartile range)The five-number summary and its relationship to box plotsRelationship between data sets and line plots, box plots, and histogramsAdvantages and disadvantages of portraying data in line plots, box plots, and histogramsGoals and Learning ObjectivesPresent projects and demonstrate an understanding of the unit concepts.Provide feedback for others' presentations.Review the concepts from the unit.Review presentation feedback and reflect.
Students collect data to answer questions about a typical sixth grade student. Students collect data about themselves, working in pairs to measure height, arm span, etc. Students discuss characteristics they would like to know about sixth grade students, adding these topics to a preset list. Data are collected and organized such that there is a class data set for each topic for future use. Students are asked to think about how this data could be represented and organized.Key ConceptsFor data to be useful, it must be collected in a consistent and accurate way. For example, for height data, students must agree on whether students should be measured with shoes on or off, and whether heights should be measured to the nearest inch, half inch, or centimeter.Goals and Learning ObjectivesGather data about sixth grade students.Consider how data are collected.
In this lesson, students draw a line plot of a set of data and then find the mean of the data. This lesson also informally introduces the concepts of the median, or middle value, and the mode, or most common value. These terms will be formally defined in Lesson 6.Using a sample set of data, students review construction of a line plot. The mean as fair share is introduced as well as the algorithm for mean. Using the sample set of data, students determine the mean and informally describe the set of data, looking at measures of center and the shape of the data. Students also determine the middle 50% of the data.Key ConceptsThe mean is a measure of center and is one of the ways to determine what is typical for a set of data.The mean is often called the average. It is found by adding all values together and then dividing by the number of values.A line plot is a visual representation of the data. It can be used to find the mean by adjusting the data points to one value, such that the sum of the data does not change.Goals and Learning ObjectivesReview construction of a line plot.Introduce the concept of the mean as a measure of center.Use the fair-share method and standard algorithm to find the mean.
Students make a histogram of their typical-student data and then write a summary of what the histogram shows.Students are introduced to histograms, using the line plot to build them. They investigate how the bin width affects the shape of a histogram. Students understand that a histogram shows the shape of the data, but that measures of center or spread cannot be found from the graph.Key ConceptsA histogram groups data values into intervals and shows the frequency (the number of data values) for each interval as the height of a bar.Histograms are similar to line plots in that they show the shape and distribution of a data set. However, unlike a line plot, which shows frequencies of individual data values, histograms show frequencies of intervals of values.We cannot read individual data values from a histogram, and we can't identify any measures of center or spread.Histograms sometimes have an interval with the most data values, referred to as the mode interval.Histograms are most useful for large data sets, where plotting each individual data point is impractical.The shape of a histogram depends on the chosen width of the interval, called the bin width. Bin widths that are too large or too small can hide important features of the data.Goals and Learning ObjectivesLearn about histograms as another tool to describe data.Show that histograms are used to show the shape of the data for a wider range of data.Compare a line plot and histogram for the same set of data.
Students use the Box Plot interactive, which allows them to create line plots and see the corresponding box plots. They use this tool to create data sets with box plots that satisfy given criteria.Students investigate how the box plot changes as the data points in the line plot are moved. Students can manipulate data points to change aspects of the box plot and to see how the line plot changes. Students create box plots that fit certain criteria.Key ConceptsThis lesson focuses on the connection between a data set and its box plot. It reinforces the idea that a box plot shows the spread of a data set, but not the individual data points.Students will observe the following similarities and differences between line plots and box plots:Line plots allow us to see and count individual values, while box plots do not.Line plots allow us to find the mean and the mode of a set of data, while box plots do not.Box plots are useful for very large data sets, while line plots are not.Box plots give us a better picture of how the values in a data set are distributed than line plots do, and they allow us to see measures of spread easily.Goals and Learning ObjectivesExperiment with different line plots to see the effect on the corresponding box plots.Create data sets with box plots that satisfy different criteria.Compare and contrast line plots and box plots.
Lesson OverviewStudents complete a card sort that requires them to match sets of statistics with the corresponding line plots.Students match cards with simple line plots to the corresponding card with measures of center. Some cards include mode, mean, median, and range, and some have one or two measures missing. Students discuss how they determined which cards would match.Key ConceptsTo complete the card sort in this lesson efficiently, students must be able to relate statistical measures with line plots. If they start with the measures that are easy to see, they can narrow down the possible matches.The mode is the easiest measure to see immediately. It is simply the number with the tallest column of dots.The range can be found easily by subtracting the least value in the plot from the greatest.The median can be found fairly quickly by counting to the middle dot or by pairing dots on the ends until reaching the middle.The mean must be calculated by adding data values and dividing.Goals and Learning ObjectivesApply knowledge of measures of center and range to solve problems.Discuss and review strategy choices when problem solving.
Students will apply what they have learned in previous lessons to analyze and draw conclusions about a set of data. They will also justify their thinking based on what they know about the measures (e.g., I know the mean is a good number to use to describe what is typical because the range is narrow and so the MAD is low.).Students analyze one of the data sets about the characteristics of sixth grade students that was collected by the class in Lesson 2. Students construct line plots and calculate measures of center and spread in order to further their understanding of the characteristics of a typical sixth grade student.Key ConceptsNo new mathematical ideas are introduced in this lesson. Instead, students apply the skills they have acquired in previous lessons to analyze a data set for one attribute of a sixth grade student. Students make a line plot of the data and find the mean, median, range, MAD, and outliers. They use these results to determine a typical value for their data.Goals and Learning ObjectivesDescribe an attribute of a typical sixth grade student using line plots and measures of center (mean and median) and spread (range and MAD).Justify thinking about which measures are good descriptors of the data set.
Students form groups and identify a question to investigate for the unit project. Each group submits a proposal outlining the statistical question, the data collection method, and a prediction of results.Key ConceptsStudents will apply what they have learned from the first two lessons to begin the unit project.Goals and Learning ObjectivesChoose a statistical question to answer over the course of the unit.Determine the necessary data collection method.Predict the results.Write a proposal that outlines the project.
GalleryCreate a Data SetStudents will create data sets with a specified mean, median, range, and number of data values.Bouncing Ball Experiment How high does the class think a typical ball bounces (compared to its drop height) on its first bounce? Students will conduct an experiment to find out.Adding New Data to a Data Set Given a data set, students will explore how the mean changes as they add data values.Bowling Scores Students will create bowling score data sets that meet certain criteria with regard to measures of center.Mean Number of Fillings Ten people sit in a dentist's waiting room. The mean number of fillings they have in their teeth is 4, yet none of them actually have 4 fillings. Students will explain how this situation is possible.Forestland Students will examine and interpret box plots that show the percentage of forestland in 20 European countries.What's My Data?Students will create a data set that fits a given histogram and then adjust the data set to fit additional criteria.What's My Data 2? Students will create a data set that fits a given box plot and then adjust the data set to fit additional criteria.Compare Graphs Students will make a box plot and a histogram that are based on a given line plot and then compare the three graphs to decide which one best represents the data.Random Numbers What would a data set of randomly generated numbers look like when represented on a histogram? Students will find out!No Telephone? The U.S. Census Bureau provides state-by-state data about the number of households that do not have telephones. Students will examine two box plots that show census data from 1960 and 1990 and compare and analyze the data.Who Is Taller?Who is taller—the boys in the class or the girls in the class? Students will find out by separating the class height data gathered earlier into data for boys and data for girls.
Students explore how adjusting the bin width or adding, deleting, or moving data values affects a histogram.Students use the Histogram interactive to explore how the bin width can affect how the data are displayed and interpreted. Students also explore how adjusting the line plot affects the histogram.Key ConceptsAs students learned in the last lesson, a histogram shows data in intervals. It shows how much data is in each bin, but it does not show individual data. In this lesson, students will see that the same histogram can be made with different sets of data. Students will also see that the bin width can greatly affect how the histogram looks.Goals and Learning ObjectivesExplore what the shape of the histogram tells us about the data set and how the bin width affects the shape of the histogram.Clarify similarities and differences between histograms and line plots.Compare a line plot and histogram for the same set of data.
Students write statistical questions that can be used to find information about a typical sixth grade student. Then, the class works together to informally plan how to find the typical arm span of a student in their class.Key ConceptsStatistical thinking, in large part, must deal with variability; statistical problem solving and decision making depend on understanding, explaining, and quantifying the variability in the data.“How tall is a sixth grader?” is a statistical question because all sixth graders are not the same height—there is variability.Goals and Learning ObjectivesUnderstand what a statistical question is.Realize there is variability in data and understand why.Describe informally the range, median, and mode of a set of data.
Students analyze the data they have collected to answer their question for the unit project. They will also complete a short Self Check.Students are given class time to work on their projects. Students should use the time to analyze their data, finding the different measures and/or graphing their data. If necessary, students may choose to use the time to collect data. Students also complete a short pre-assessment (Self Check problem).Key ConceptsStudents will look at all of the tools that they have to analyze data. These include:Graphic representations: line plots, box plots, and histogramsMeasures of center and spread: mean, median, mode, range, and the five-number summaryStudents will use these tools to work on their project and to complete an assessment exercise.Goals and Learning ObjectivesComplete the project, or progress far enough to complete it outside of class.Review measures of center and spread and the three types of graphs explored in the unit.Check knowledge of box plots and measures of center and spread.
In this lesson, students are given criteria about measures of center, and they must create line plots for data that meet the criteria. Students also explore the effect on the median and the mean when values are added to a data set.Students use a tool that shows a line plot where measures of center are shown. Students manipulate the graph and observe how the measures are affected. Students explore how well each measure describes the data and discover that the mean is affected more by extreme values than the mode or median. The mathematical definitions for measures of center and spread are formalized.Key ConceptsStudents use the Line Plot with Stats interactive to develop a greater understanding of the measures of center. Here are a few of the things students may discover:The mean and the median do not have to be data points.The mean is affected by extreme values, while the median is not.Adding values above the mean increases the mean. Adding values below the mean decreases the mean.You can add values above and below the mean without changing the mean, as long as those points are “balanced.”Adding values above the median may or may not increase the median. Adding values below the median may or may not decrease the median.Adding equal numbers of points above and below the median does not change the median.The measures of center can be related in any number of ways. For example, the mean can be greater than the median, the median can be greater than the mean, and the mode can be greater than or less than either of these measures.Note: In other courses, students will learn that a set of data may have more than one mode. That will not be the case in this lesson.Goals and Learning ObjectivesExplore how changing the data in a line plot affects the measures of center (mean, median).Understand that the mean is affected by outliers more than the median is.Create line plots that fit criteria for given measures of center.
Equations and Inequalities
Type of Unit: Concept
Students should be able to:
Add, subtract, multiply, and divide with whole numbers, fractions, and decimals.
Use the symbols <, >, and =.
Evaluate expressions for specific values of their variables.
Identify when two expressions are equivalent.
Simplify expressions using the distributive property and by combining like terms.
Use ratio and rate reasoning to solve real-world problems.
Order rational numbers.
Represent rational numbers on a number line.
In the exploratory lesson, students use a balance scale to find a counterfeit coin that weighs less than the genuine coins. Then continuing with a balance scale, students write mathematical equations and inequalities, identify numbers that are, or are not, solutions to an equation or an inequality, and learn how to use the addition and multiplication properties of equality to solve equations. Students then learn how to use equations to solve word problems, including word problems that can be solved by writing a proportion. Finally, students connect inequalities and their graphs to real-world situations.
Lesson OverviewStudents apply the addition property of equality to solve equations. They are introduced to this property using a balance scale.Key ConceptsUp until this lesson, students have been solving equations informally. They used guess and check and reasoned about the quantities on either side of the equation in order to solve the equation.In this lesson, students are introduced to the addition property of equality. As equations become more and more complicated, students will need to rely on formal methods for solving them. This property states that the same quantity can be added to both sides of an equation and the new equation will be equivalent to the original equation. That means the new equation will have the same solutions as the original equation.To solve an equation such as x + 6 = 15, –6 can be added to both sides to get the resulting equation x = 9. However, since adding a negative number has not been introduced yet, students will consider both adding and subtracting a number (which is the equivalent of adding a negative number) from both sides to be an application of the addition property of equality.Students will apply the addition property of equality to an equation with the goal of getting the variable alone on one side of the equation and a number on the other.Goals and Learning ObjectivesUse the addition property of equality to keep a scale balanced.Use the addition property of equality to solve equations of the form x + p = q for cases in which p, q, and x are all non-negative rational numbers.
Lesson OverviewStudents apply the multiplication property of equality to solve equations.Key ConceptsIn the previous lesson, students solved equations of the form x + p = q using the addition property of equality. In this lesson, they will solve equations of the form px = q using the multiplication property of equality. They will multiply or divide both sides of an equation by the same number to obtain an equivalent equation.Since multiplication by a is equivalent to division by 1a, students will see that they may also divide both sides of the equation by the same number to get an equivalent equation. Students will also apply this property to solving a particular kind of equation, a proportion.Goals and Learning ObjectivesUse the multiplication property of equality to keep an equation balanced.Use the multiplication property of equality to solve equations of the form px = q for cases in which p, q, and x are all non-negative rational numbers.Use the multiplication property of equality to solve proportions.
Lesson OverviewStudents solve a classic puzzle about finding a counterfeit coin. The puzzle introduces students to the idea of a scale being balanced when the weight of the objects on both sides is the same and the scale being unbalanced when the objects on one side do not weigh the same as the objects on the other side.Key ConceptsThe concept of an inequality statement can be modeled using an unbalanced scale. The context—weighing a set of coins in order to identify the one coin that weighs less than the others—allows students to manipulate the weight on either side of the scale. In doing so, they are focused on the relationship between two weights—two quantities—and whether or not they are equal.Goals and Learning ObjectivesExplore a balance scale as a model for an equation or an inequality.Introduce formal meanings of equality and inequality.
Gallery OverviewAllow students who have a clear understanding of the content thus far in the unit to work on Gallery problems of their choosing. You can then use this time to provide additional help to students who need review of the unit's concepts or to assist students who may have fallen behind on work.Gallery DescriptionsKeep It BalancedStudents will use reasoning to complete some equations to make them true.Equation SortStudents will sort equations into three groups: equations with one solution, equations with many solutions, and equations with no solutionsOn the Number LineStudents will use a number line to identify numbers that make an equation or inequality true.How Many Colors?Students will write and solve an equation to find the number of different colored blocks in a box.Value of sStudents will use a property of equality to solve an equation with large numbers.Marbles in a CupStudents are given information about the weight of a cup with two different amounts of marbles in it. They use this information to find the weight of the cup.When Is It True?Students will use what they know about 0 and 1 to decide when a certain equation is true.
Lesson OverviewStudents represent real-world situations using inequality statements that include a variable.Key ConceptsInequality statements tell you whether values in a situation are greater than or less than a given number and also tell you whether values in the situation can be equal to that number or not.The symbols < and > tell you that the unknown value(s) in a situation cannot be equal to a given number: the unknown value(s) are strictly greater than or less than the number. The inequality x < y means x must be less than y. The inequality x > y means x must be greater than y.The symbols ≤ and ≥ tell you that the unknown value(s) in a situation can also be equal to a given number: the unknown value(s) are less than or equal to, or greater than or equal to, the number. The inequality x ≤ y means x is less than or equal to y. The inequality x ≥ y means x is greater than or equal to y.Goals and Learning ObjectivesUnderstand the inequality symbols <, >, ≤, and ≥.Write inequality statements for real-world situations.ELL: When writing the summary, provide ELLs access to a dictionary and give them time to discuss their summary with a partner before writing, to help them organize their thoughts. Allow ELLs who share the same primary language to discuss in their native language if they wish.
Lesson OverviewStudents practice solving equations using either the addition or the multiplication property of equality.Key ConceptsStudents will solve equations of the form x + p = q using the addition property of equality.They will solve equations of the form px = q using the multiplication property of equality.They will need to look at the variable and decide what operation must be performed on both sides of the equation in order to isolate the variable on one side of the equation.If a number has been added to the variable, they will subtract that number from both sides of the equation. If a number has been subtracted from the variable, they will add that number to both sides of the equation. If the variable has been multiplied by a number, students will either divide both sides of the equation by that number or multiply by the reciprocal of that number. If the variable has been divided by a number, students will multiply by that number. Students will see how this can be applied to solving a proportion such as xc=ab.Goals and Learning ObjectivesPractice solving equations using either the addition or the multiplication property of equality.Distinguish between equations that can be solved using the addition property of equality from equations that can be solved using the multiplication property of equality.Solve a proportion by solving an equation.
Lesson OverviewStudents use reasoning to identify solutions to equations. They initially do this using the balance scale. They also learn that some equations may have all numbers as solutions and some equations may have no solutions.Key ConceptsBefore beginning the formal process of solving equations, students need opportunities to use reasoning to find solutions. Students study examples where reasoning pays off. For example, in the equation 4b + 15 = 3b + 6b, students can reason that 4b + 15 = 3b + 6b, so 5b must be equal to 15, an equation which they can solve by understanding multiplication.Students also discover that there are equations that can have every number as a solution or no number as a solution. They may recognize some equations with all numbers as solutions by recognizing that they show a property of operations, such as the commutative property of addition.SWD: Students with disabilities may struggle to determine salient information in lessons. Preview the goals with students to support saliency determination as they move through the instruction and tasks.Students with disabilities may struggle to self-monitor their progress through the lesson. Provide students with a copy of the lesson goals to use as a checklist as they move through the different tasks. Have students indicate when they have reached each goal for the lesson. This will also promote engagement, independence, and self-management of learning.Goals and Learning ObjectivesUse reasoning to identify the solution to an equation.Recognize equations that have any number as a solution and equations that have no solutions.
Lesson OverviewStudents represent inequalities on a number line, find at least one value that makes the inequality true, and write the inequality using words.SWD:When calling on students, be sure to call on ELLs and to encourage them to actively participate. Understand that their pace might be slower or they might be shy or more reluctant to volunteer due to their weaker command of the language.SWD:Thinking aloud is one strategy for making learning visible. When teachers think aloud, they are externalizing their internal thought processes. Doing so may provide students with insights into mathematical thinking and ways of tackling problems. It also helps to model accurate mathematical language.Key ConceptsInequalities, like equations, have solutions. An arrow on the number line—pointing to the right for greater values and to the left for lesser values—can be used to show that there are infinitely many solutions to an inequality.The solutions to x < a are represented on the number line by an arrow pointing to the left from an open circle at a.Example: x < 2The solutions to x > a are represented on the number line with an arrow pointing to the right from an open circle at a.Example: x > 2The solutions to x ≤ a are represented on the number line with an arrow pointing to the left from a closed circle at a.Example: x ≤ 2The solutions to x ≥ a are represented on the number line with an arrow pointing to the right from a closed circle at a.Example: x ≥ 2Goals and Learning ObjectivesRepresent an inequality on a number line and using words.Understand that inequalities have infinitely many solutions.
Students work in pairs to critique and improve their work on the Self Check from the previous lesson.Key ConceptsTo critique and improve the task from the Self Check and to complete a similar task with a partner, students use what they know about solving equations and relating the equations to real-world situations.Goals and Learning ObjectivesSolve equations using the addition or multiplication property of equality.Write word problems that match algebraic equations.Write equations to represent a mathematical situation.
Lesson OverviewStudents solve problems using equations of the form x + p = q and px = q, as well as problems involving proportions.Key ConceptsStudents will extend what they know about writing expressions to writing equations. An equation is a statement that two expressions are equivalent. Students will write two equivalent expressions that represent the same quantity. One expression will be numerical and the other expression will contain a variable.It is important that when students write the equation, they define the variable precisely. For example, n represents the number of minutes Aiko ran, or x represents the number of boxes on the shelf.Students will then solve the equations and thereby solve the problems.Students will solve proportion problems by solving equations. This makes sense because a proportion such as xa=bc is really just an equation of the form xp = q where p=1a and q=bc.Students will also compare their algebraic solutions to an arithmetic solution for the problem. They will see, for example, that a problem that might be solved arithmetically by subtracting 5 from 78 can also be solved algebraically by solving x + 5 = 78, where 5 is subtracted from both sides—a parallel solution to subtracting 5 from 78.Goals and Learning ObjectivesUse equations of the form x + p = q and xp = q to solve problems.Solve proportion problems using equations.ELL: ELLs may have difficulty verbalizing their reasoning, particularly because word problems are highly language dependent. Accommodate ELLs by providing extra time for them to process the information. Note that this problem is a good opportunity for ELLs to develop their literacy skills since it incorporates reading, writing, listening, and speaking skills. Encourage students to challenge each others' ideas and justify their thinking using academic and specialized mathematical language.
Lesson OverviewUsing a balance scale, students decide whether a certain value of a variable makes a given equation or inequality true. Then students extend what they learned using the balance scale to substituting a given value for a variable into an equation or inequality to decide if that value makes the equation or inequality true or false.Key ConceptsStudents will extend what they know about substituting a value for a variable into an expression to evaluate that expression.Equations and inequalities may contain variables. These equations or inequalities are neither true nor false. When a value is substituted for a variable, the equation or inequality then becomes true or false. If the equation or inequality is true for that value of the variable, that value is considered a solution to the equation or inequality.Goals and Learning ObjectivesUnderstand what solving an equation or inequality means.Use substitution to determine whether a given number makes an equation or inequality true.
Lesson OverviewStudents use weights to represent equal and unequal situations on a balance scale and represent them symbolically.Key ConceptsAn equation is a statement that shows that two expressions are equivalent. An equal sign (=) is used between the two expressions to indicate that they are equivalent. You can think of the two expressions as being “balanced.”An inequality is a statement that shows that two expressions are unequal. The symbols for “greater than” (>) and “less than” (<) are used to indicate which expression has the greater or lesser value. In an inequality, you can think of the two expressions as being “unbalanced.”Goals and Learning ObjectivesExplore a balance scale as a model for equations and inequalities.Understand that an equation states that two expressions are equivalent using an equal sign (=).Understand that an inequality states that one expression is greater than (>) or is less than (<) another expression.Use the equal sign (=) and the greater than (>) and less than (<) symbols with rational numbers.
Type of Unit: Concept
Students should be able to:
Write and evaluate simple expressions that record calculations with numbers.
Use parentheses, brackets, or braces in numerical expressions and evaluate expressions with these symbols.
Interpret numerical expressions without evaluating them.
Students learn to write and evaluate numerical expressions involving the four basic arithmetic operations and whole-number exponents. In specific contexts, they create and interpret numerical expressions and evaluate them. Then students move on to algebraic expressions, in which letters stand for numbers. In specific contexts, students simplify algebraic expressions and evaluate them for given values of the variables. Students learn about and use the vocabulary of algebraic expressions. Then they identify equivalent expressions and apply properties of operations, such as the distributive property, to generate equivalent expressions. Finally, students use geometric models to explore greatest common factors and least common multiples.
Lesson OverviewStudents use a geometric model to investigate common multiples and the least common multiple of two numbers.Key ConceptsA geometric model can be used to investigate common multiples. When congruent rectangular cards with whole-number lengths are arranged to form a square, the length of the square is a common multiple of the side lengths of the cards. The least common multiple is the smallest square that can be formed with those cards.For example, using six 4 × 6 rectangles, a 12 × 12 square can be formed. So, 12 is a common multiple of both 4 and 6. Since the 12 × 12 square is the smallest square that can be formed, 12 is the least common multiple of 4 and 6.Common multiples are multiples that are shared by two or more numbers. The least common multiple (LCM) is the smallest multiple shared by two or more numbers.Goals and Learning ObjectivesUse a geometric model to understand least common multiples.Find the least common multiple of two whole numbers equal to or less than 12.
Students use a rectangular area model to understand the distributive property. They watch a video to find how to express the area of a rectangle in two different ways. Then they find the area of rectangular garden plots in two ways.Key ConceptsThe distributive property can be used to rewrite an expression as an equivalent expression that is easier to work with. The distributive property states that multiplication distributes over addition.Applying multiplication to quantities that have been combined by addition: a(b + c)Applying multiplication to each quantity individually, and then adding the products together: ab + acThe distributive property can be represented with a geometric model. The area of this rectangle can be found in two ways: a(b + c) or ab + ac. The equality of these two expressions, a(b + c) = ab + ac, is the distributive property.Goals and Learning ObjectivesUse a geometric model to understand the distributive property.Write equivalent expressions using the distributive property.
Students analyze how two different calculators get different values for the same numerical expression. In the process, students recognize the need for following the same conventions when evaluating expressions.Key ConceptsMathematical expressions express calculations with numbers (numerical expressions) or sometimes with letters representing numbers (algebraic expressions).When evaluating expressions that have more than one operation, there are conventions—called the order of operations—that must be followed:Complete all operations inside parentheses first.Evaluate exponents.Then complete all multiplication and division, working from left to right.Then complete all addition and subtraction, working from left to right.These conventions allow expressions with more than one operation to be evaluated in the same way by everyone. Because of these conventions, it is important to use parentheses when writing expressions to indicate which operation to do first. If there are nested parentheses, the operations in the innermost parentheses are evaluated first. Understanding the use of parentheses is especially important when interpreting the associative and the distributive properties.Goals and Learning ObjectivesEvaluate numerical expressions.Use parentheses when writing expressions.Use the order of operations conventions.
Students do a card sort in which they match expressions in words with their equivalent algebraic expressions.Key ConceptsA mathematical expression that uses letters to represent numbers is an algebraic expression.A letter used in place of a number in an expression is called a variable.An algebraic expression combines both numbers and letters using the arithmetic operations of addition (+), subtraction (–), multiplication (·), and division (÷) to express a quantity.Words can be used to describe algebraic expressions.There are conventions for writing algebraic expressions:The product of a number and a variable lists the number first with no multiplication sign. For example, the product of 5 and n is written as 5n, not n5.The product of a number and a factor in parentheses lists the number first with no multiplication sign. For example, write 5(x + 3), not (x + 3)5.For the product of 1 and a variable, either write the multiplication sign or do not write the "1." For example, the product of 1 and z is written either 1 ⋅ z or z, not 1z.Goals and Learning ObjectivesTranslate between expressions in words and expressions in symbols.
Lesson GuideAllow students who have a clear understanding of the content thus far in the unit to work on Gallery problems of their choosing. You can then use this time to provide additional help to students who need review of the unit's concepts or to assist students who may have fallen behind on work.Gallery DescriptionsBuilding BridgesStudents will examine a pattern and use expressions to show how to continue the pattern.Patterns in a TableStudents will complete a table by noticing relationships within the table and using those relationships to fill in empty cells in the table.Expressions for Perimeter and AreaStudents will write equivalent expressions for the perimeters and areas of various rectangles.Multiplication TableStudents will complete an unusual multiplication table by writing the algebraic expression that results from multiplying the terms given in the top row by the ones given in the left column.Garden BedsStudents will find the number of square tiles needed to pave around various configurations of rectangular garden beds. Then, students will write an algebraic equation to represent the number of square tiles needed to go around any number of plants in a single row.Telephone TreeStudents will solve problems about a telephone tree and use expressions to show the number of calls completed after a given number of rounds of calling.Stacks of DVDsStudents will write an expression to describe the width of a stack of DVDs, and then they will evaluate the expression for different numbers of DVD cases and boxed sets.Exponent Card SortStudents will complete a card sort that will give them practice working with exponents. Then they will use a set of blank cards to complete sets that purposely have one or two representations missing.Matching Words and ExpressionsStudents will match a verbal statement with its expression in this card sort.Investigating Factors and MultiplesStudents will investigate an interesting property of numbers involving the greatest common factor and the least common multiple.Fourth RockStudents will solve a problem about how long it will take for two imaginary planets in an imaginary solar system to align so that they are at their closest distance from each other.Factors of a NumberStudents will decide whether a mathematical claim about factors and multiples is true or false based on given criteria.Common FactorsStudents will look at two unknown numbers with a greatest common factor of 20 and determine what other factors must be common to the two unknown numbers. Students will use their answer to make a generalization.History of VariablesStudents will research the history of variables. When were they first used? Where were they first used? Who used them?Create a VideoStudents will use their creative powers to produce a video about expressions.
Students explore what makes a math trick work by analyzing verbal math expressions that describe each step in the trick.Key ConceptsWords can be used to describe mathematical operations.In a math trick, a person starts with a number, follows mathematical directions given in words, and ends up with the original number.Math tricks can be explained by examining the mathematical expressions that represent the verbal directions.Goals and Learning ObjectivesExplore verbal expressions.Predict and test which sets of expressions will result in the original number.
Students play an Expressions Game in which they describe expressions to their partners using the vocabulary of expressions: term, coefficient, exponent, constant, and variable. Their partners try to write the correct expressions based on the descriptions.Key ConceptsMathematical expressions have parts, and these parts have names. These names allow us to communicate with others in a precise way.A variable is a symbol (usually a letter) in an expression that can be replaced by a number.A term is a number, a variable, or a product of numbers and variables. Terms are separated by the operator symbols + (plus) and – (minus).A coefficient is a symbol (usually a number) that multiplies the variable in an algebraic expression.An exponent tells how many copies of a number or variable are multiplied together.A constant is a number. In an expression, it can be a constant term or a constant coefficient. In the expression 2x + 3, 2 is a constant coefficient and 3 is a constant term.Goals and Learning ObjectivesIdentify parts of an expression using appropriate mathematical vocabulary.Write expressions that fit specific descriptions (for example, the expression is the sum of two terms each with a different variable).
Students critique the work of other students and revise their own work based on feedback from the teacher and peers.Key ConceptsConcepts from previous lessons are integrated into this unit task: rewriting expressions, using parentheses, and using the distributive property. Students apply their knowledge, review their work, and make revisions based on feedback from you and their peers. This process creates a deeper understanding of the concepts.Goals and Learning ObjectivesApply knowledge of expressions to correct the work of other students.Track and review the choice of strategy when problem solving.
Students use a geometric model to investigate common factors and the greatest common factor of two numbers.Key ConceptsA geometric model can be used to investigate common factors. When congruent squares fit exactly along the edge of a rectangular grid, the side length of the square is a factor of the side length of the rectangular grid. The greatest common factor (GCF) is the largest square that fits exactly along both the length and the width of the rectangular grid. For example, given a 6-centimeter × 8-centimeter rectangular grid, four 2-centimeter squares will fit exactly along the length without any gaps or overlaps. So, 2 is a factor of 8. Three 2-centimeter squares will fit exactly along the width, so 2 is a factor of 6. Since the 2-centimeter square is the largest square that will fit along both the length and the width exactly, 2 is the greatest common factor of 6 and 8. Common factors are all of the factors that are shared by two or more numbers.The greatest common factor is the greatest number that is a factor shared by two or more numbers.Goals and Learning ObjectivesUse a geometric model to understand greatest common factor.Find the greatest common factor of two whole numbers equal to or less than 100.
Students express the lengths of trains as algebraic expressions and then substitute numbers for letters to find the actual lengths of the trains.Key ConceptsAn algebraic expression can be written to represent a problem situation. More than one algebraic expression may represent the same problem situation. These algebraic expressions have the same value and are equivalent.To evaluate an algebraic expression, a specific value for each variable is substituted in the expression, and then all the calculations are completed using the order of operations to get a single value.Goals and Learning ObjectivesEvaluate expressions for the given values of the variables.
Students write an expression for the length of a train, using variables to represent the lengths of the different types of cars.Key ConceptsA numerical expression consists of a number or numbers connected by the arithmetic operations of addition, subtraction, multiplication, division, and exponentiation.An algebraic expression uses letters to represent numbers.An algebraic expression can be written to represent a problem situation. Sometimes more than one algebraic expression may represent the same problem situation. These algebraic expressions have the same value and are equivalent.The properties of operations can be used to make long algebraic expressions shorter:The commutative property of addition states that changing the order of the addends does not change the end result:a + b = b + a.The associative property of addition states that changing the grouping of the addends does not change the end result:(a + b) + c = a + (b + c).The distributive property of multiplication over addition states that multiplying a sum by a number gives the same result as multiplying each addend by the number and then adding the products together:a(b + c) = ab + ac.Goals and Learning ObjectivesWrite algebraic expressions that describe lengths of freight trains.Use properties of operations to shorten those expressions.
Students represent problem situations using expressions and then evaluate the expressions for the given values of the variables.Key ConceptsAn algebraic expression can be written to represent a problem situation.To evaluate an algebraic expression, a specific value for each variable is substituted in the expression, and then all the calculations are completed using the order of operations to get a single value.Goals and Learning ObjectivesDevelop fluency in writing expressions to represent situations and in evaluating the expressions for given values.
Fractions and Decimals
Type of Unit: Concept
Students should be able to:
Multiply and divide whole numbers and decimals.
Multiply a fraction by a whole number.
Multiply a fraction by another fraction.
Write fractions in equivalent forms, including converting between improper fractions and mixed numbers.
Understand the meaning and structure of decimal numbers.
This unit extends students’ learning from Grade 5 about operations with fractions and decimals.
The first lesson informally introduces the idea of dividing a fraction by a fraction. Students are challenged to figure out how many times a 14-cup measuring cup must be filled to measure the ingredients in a recipe. Students use a variety of methods, including adding 14 repeatedly until the sum is the desired amount, and drawing a model. In Lesson 2, students focus on dividing a fraction by a whole number. They make a model of the fraction—an area model, bar model, number line, or some other model—and then divide the model into whole numbers of groups. Students also work without a model by looking at the inverse relationship between division and multiplication. Students explore methods for dividing a whole number by a fraction in Lesson 3, for dividing a fraction by a unit fraction in Lesson 4, and for dividing a fraction by another fraction in Lesson 6. Students examine several methods and models for solving such problems, and use models to solve similar problems.
Students apply their learning to real-world contexts in Lesson 6 as they solve word problems that require dividing and multiplying mixed numbers. Lesson 7 is a Gallery lesson in which students choose from a number of problems that reinforce their learning from the previous lessons.
Students review the standard long-division algorithm for dividing whole numbers in Lesson 8. They discuss the different ways that an answer to a whole number division problem can be expressed (as a whole number plus a remainder, as a mixed number, or as a decimal). Students then solve a series of real-world problems that require the same whole number division operation, but have different answers because of how the remainder is interpreted.
Students focus on decimal operations in Lessons 9 and 10. In Lesson 9, they review addition, subtraction, multiplication, and division with decimals. They solve decimal problems using mental math, and then work on a card sort activity in which they must match problems with diagram and solution cards. In Lesson 10, students review the algorithms for the four basic decimal operations, and use estimation or other methods to place the decimal points in products and quotients. They solve multistep word problems involving decimal operations.
In Lesson 11, students explore whether multiplication always results in a greater number and whether division always results in a smaller number. They work on a Self Check problem in which they apply what they have learned to a real-world problem. Students consolidate their learning in Lesson 12 by critiquing and improving their work on the Self Check problem from the previous lesson. The unit ends with a second set of Gallery problems that students complete over two lessons.
Students determine how many times they would need to fill a quarter cup to measure the ingredients in a recipe.Key ConceptsThis lesson informally introduces the idea of dividing by a fraction. Students must figure out how many times a quarter cup must be filled to measure the ingredients in a recipe. This involves dividing each amount by 14. Here are some methods students might use:Add 14 repeatedly until the sum is the desired amount. Count the number of 14s that were added.Start with the amount in the recipe. Subtract 14 repeatedly until the difference is 0. Count the 14s that were subtracted.Draw a model (e.g., a bar or a number line model) to represent the amount in the recipe. Divide it into fourths and count the number of fourths.Goals and Learning ObjectivesLearn how to divide by a fraction.
Students solve decimal multiplication and division problems related to the basic fact 3 × 7 = 21.Students match cards that represent word problems, visual models, and numerical solutions to problems that include the numbers 0.8 and 0.2 for all four operations.Key ConceptsNo new mathematics is introduced in this lesson. Students apply their knowledge about decimal operations.Goals and Learning ObjectivesUse reasoning and mental math to solve problems.Solve word problems involving simple addition, subtraction, multiplication, and division with decimals.
Students explore methods of dividing a fraction by a unit fraction.Key ConceptsIn this lesson and in Lesson 5, students explore dividing a fraction by a fraction.In this lesson, we focus on the case in which the divisor is a unit fraction. Understanding this case makes it easier to see why we can divide by a fraction by multiplying by its reciprocal. For example, finding 34÷15 means finding the number of fifths in 34. In this lesson, students will see that this is 34 × 5.Students learn and apply several methods for dividing a fraction by a unit fraction, such as 23÷14.Model 23. Change the model and the fractions in the problem to twelfths: 812÷312. Then find the number of groups of 3 twelfths in 8 twelfths. This is the same as finding 8 ÷ 3.Reason that since there are 4 fourths in 1, there must be 23 × 4 fourths in 23. This is the same as using the multiplicative inverse.Rewrite both fractions so they have a common denominator: 23÷14=812÷312. The answer is the quotient of the numerators. This is the numerical analog to modeling.Goals and Learning ObjectivesUse models and other methods to divide fractions by unit fractions
Students use models and the idea of dividing as making equal groups to divide a fraction by a whole number.SWD: Some students with disabilities will benefit from a preview of the goals in each lesson. Students can highlight the critical features or concepts in order to help them pay close attention to salient information.Key ConceptsWhen we divide a whole number by a whole number n, we can think of making n equal groups and finding the size of each group. We can think about dividing a fraction by a whole number in the same way.8 ÷ 4 = 2 When we make 4 equal groups, there are 2 wholes in each group.89÷4=29 When we make 4 equal groups, there are 2 ninths in each group.When the given fraction cannot be divided into equal groups of unit fractions, we can break each unit fraction part into smaller parts to form an equivalent fraction.34 ÷ 6 = ? 68 ÷ 6 = ? 68 ÷ 6 = 18 Students see that, in general, we can divide a fraction by a whole number by dividing the numerator by the whole number. Note that this is consistent with the “multiply by the reciprocal” method.ab÷n=a÷nb=anb=an×1b=an×b=ab×1nGoals and Learning ObjectivesUse models to divide a fraction by a whole number.Learn general methods for dividing a fraction by a whole number without using a model.
Students solve word problems that require dividing and multiplying with fractions and mixed numbers.Key ConceptsStudents apply their knowledge about multiplying and dividing fractions to solve word problems. This includes applying the general methods for dividing fractions learned in previous lessons:Rewrite the dividend and the divisor so they have a common denominator. The answer to the original division will be the quotient of the numerators.Multiply the dividend by the reciprocal of the divisor.Goals and Learning ObjectivesApply knowledge of fraction multiplication and division to solve word problems.
Gallery OverviewAllow students who have a clear understanding of the content in the unit to work on Gallery problems of their choosing. You can use this time to provide additional help to students who need review of the unit's concepts or to assist students who may have fallen behind on work.Gallery DescriptionStew RecipeStudents use fraction operations to help Molly figure out if she has enough potatoes to make stew for all the guests at her party.Multiply or Divide?Students match descriptions of situations to multiplication and division situations.Card SortStudents find the diagram, expression, and answer that match given word problems.Complex FractionsStudents learn about complex fractions and how they are useful for dividing fractions.
Gallery OverviewAllow students who have a clear understanding of the content in the unit to work on Gallery problems of their choosing. You can use this time to provide additional help to students who need review of the unit's concepts or to assist students who may have fallen behind on work.Gallery DescriptionTiling a FloorStudents determine which size tiles are cheaper to use to tile a floor with given dimensions.Adam's HomeworkStudents find and correct an error in a whole number division problem.Then and NowStudents solve comparison problems involving census data from 1940 and 2010.Graphical MultiplicationGiven points m and p on a number line, students must locate m × p.When Does Zero Matter?Students must determine how the placement of 0 affects the value of a number.
Students explore whether multiplying by a number always results in a greater number. Students explore whether dividing by a number always results in a smaller number.Key ConceptsIn early grades, students learn that multiplication represents the total when several equal groups are combined. For this reason, some students think that multiplying always “makes things bigger.” In this lesson, students will investigate the case where a number is multiplied by a factor less than 1.Students are introduced to division in early grades in the context of dividing a group into smaller, equal groups. In whole number situations like these, the quotient is smaller than the starting number. For this reason, some students think that dividing always “makes things smaller.” In this lesson, students will investigate the case where a number is divided by a divisor less than 1.Goals and Learning ObjectivesDetermine when multiplying a number by a factor gives a result greater than the number and when it gives a result less than the number.Determine when dividing a number by a divisor gives a result greater than the number and when it gives a result less than the number.
Students critique and improve their work on the Self Check.Key ConceptsNo new concepts are introduced in this lesson. To solve the problems in the Self Check, students use fraction division and operations with decimals.Goals and Learning ObjectivesUse knowledge of fraction division and decimal operations to solve problems.
Students review the standard long-division algorithm and discuss the different ways the answer to a whole-number division problem can be expressed (as a whole number plus a remainder, as a mixed number, or as a decimal).Students solve a series of real-world problems that require the same whole number division operation, but have different answers because of how the remainder is interpreted.Key ConceptsStudents have been dividing multidigit whole numbers since Grade 4. By the end of Grade 6, they are expected to be fluent with the standard long-division algorithm. In this lesson, this algorithm is reviewed along with the various ways of expressing the answer to a long division problem. Students will have more opportunities to practice the algorithm in the Exercises.Goals and Learning ObjectivesReview and practice the standard long-division algorithm.Answer a real-world word problem that involves division in a way that makes sense in the context of the problem.
Students explore methods of dividing a fraction by a fraction.Key ConceptsStudents extend what they learned in Lesson 4 to divide a fraction by any fraction. Students are presented with two general methods for dividing fractions:Rewrite the dividend and the divisor so they have a common denominator. The answer to the original division will be the quotient of the numerators.Multiply the dividend by the reciprocal of the divisor.These two methods will work for all cases, including cases in which one or both of the numbers in the division is a fraction or whole number.Goals and Learning ObjectivesUse models and other methods to divide fractions by fractions.
Students use estimation or other methods to place the decimal points in products and quotients. They review the algorithms for the four basic decimal operations and solve multistep word problems involving decimal operations.Key ConceptsThe algorithms for whole-number operations can be extended to decimal operations. Students learned the algorithms for decimal operations in Grade 5. By the end of Grade 6, they should be fluent with these operations.For decimal addition and subtraction, once the decimal points of the addends are aligned (which aligns like place values), the algorithms are the same as for whole numbers. The decimal point in the sum or difference goes directly below the decimal point in the numbers that were added or subtracted.For decimal multiplication and division, one method is to ignore the decimal points and apply the whole-number algorithms. Then use estimation or some other method to place the decimal point in the answer.Goals and Learning ObjectivesReview and practice the algorithms for all four decimal operations.Solve real-world problems involving decimal operations.
Students explore methods for dividing a whole number by a fraction.Key ConceptsIn earlier grades, students learned to think of a whole number division problem, such as 8 ÷ 4, in terms of two types of equal groups.Divisor as the Number of Groups Divide 8 into 4 equal groups and find the size of each group.Divisor as the Group Size Divide 8 into groups of 4 and find the number of groups.To divide a fraction by a whole number in Lesson 2, students used the first interpretation. For example, to find 89 ÷ 4, they divided 8 ninths into 4 equal groups and found that there were 2 ninths in each group.To divide a whole number by a fraction, the second interpretation is most helpful. For example, to find 3 ÷ 34, we find the number of groups of 3 fourths in 3 wholes. The diagram in the Opening shows that there are 4 groups, so 3 ÷ 34 = 4.Just as with whole number division, the quotient when a whole number is divided by a fraction is not always a whole number. Below is a model for 2 ÷ 35. The model shows that there are 3 groups of 3 fifths in 2 wholes plus 13 of another group (13 of a group of 3 fifths is 1 fifth). Therefore, 2 ÷ 35 = 313. Notice that once we have divided the 2 wholes into fifths, we are finding the number of groups of 3 fifths in 10 fifths. This is simply 10 ÷ 3.These models can help explain that the “multiply by the reciprocal” method of dividing a whole number by a fraction works. To find 2 ÷ 35, we can multiply 2 by 5 to find the total number of fifths in 2 and then divide the result (10) by 3 to find the number of groups of 3 of these fifths in 2. So, 2÷35=2×53=2×53.ELL: Encourage students to verbalize their explanations. To help students gain confidence and increase their understanding, allow those that share the same language of origin to speak in small groups using their prefered language.Goals and Learning ObjectivesUse models and other methods to divide a whole number by a fraction.
Type of Unit: Introduction
Students should be able to:
Solve and write numerical equations for whole number addition, subtraction, multiplication, and division problems.
Use parentheses to evaluate numerical expressions.
Identify and use the properties of operations.
In this unit, students are introduced to the rituals and routines that build a successful classroom math community and they are introduced to the basic features of the digital course that they will use throughout the year.
An introductory card sort activity matches students with their partner for the week. Then over the course of the week, students learn about the lesson routines: Opening, Work Time, Ways of Thinking, Apply the Learning, Summary of the Math, and Reflection. Students learn how to present their work to the class, the importance of taking responsibility for their own learning, and how to effectively participate in the classroom math community.
Students then work on Gallery problems to further explore the program’s technology resources and tools and learn how to organize their work.
The mathematical work of the unit focuses on numerical expressions, including card sort activities in which students identify equivalent expressions and match an expression card to a word card that describes its meaning. Students use the properties of operations to identify equivalent expressions and to find unknown values in equations.