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.
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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.
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 work with a set of cards showing different ways of expressing ratios, including both part-part statements and part-whole statements. They group the cards that show the same ratio of boys to girls, but without the explicit use of the term equivalent.Key ConceptsRatios can be represented in a:b form, as fractions, as decimals, as factors, and in words; they can be expressed in part-part statements or in part-whole statements.Goals and Learning ObjectivesGroup cards showing ratios that are equivalent but expressed in different forms.
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.
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.