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

Prior Knowledge

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.

Lesson Flow

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.

Expressions

Type of Unit: Concept

Prior Knowledge

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.

Lesson Flow

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.