Illustrative Math Alignment: Grade 6 Unit 8

December 2022. In this issue of Math in the News we look at box office hits and misses from Disney. The House of Blockbosters every now and then misfires. It's useful to analyze box office data to see what we can learn from the hits and misses. 

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This is a collection of definitions related to the concept of charts, graphs, and data displays.

This is a collection of definitions related to the concept of charts, graphs, and data displays.

This is a collection of definitions related to the concept of charts, graphs, and data displays.

This is a collection of definitions related to the concept of charts, graphs, and data displays.

This is a collection of definitions related to the concept of charts, graphs, and data displays.

This is a collection of definitions related to the concept of charts, graphs, and data displays.

This is a collection of definitions related to the concept of charts, graphs, and data displays.

This is a collection of definitions related to the concept of charts, graphs, and data displays.

This is a collection of definitions related to the concept of charts, graphs, and data displays.

This is a collection of definitions related to the concept of charts, graphs, and data displays.

This is a collection of definitions related to the concept of charts, graphs, and data displays.

In this tutorial we walk students through the steps of doing a quadratic regression on a Desmos graphing calculator. The steps include showing how to create pseudo-quadratic data on a spreadsheet.

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In this application of arithmetic, students analyze the changing population of bison in the Americas. Using real-world data and applying key math concepts, students will do some sophisticated analysis using basic skills.  These are the key math concepts covered: Place value Skip counting Doubling Data analysis Given the wide disparity in the population of bison over several centuries, students will be analyzing data in the millions but using place value to gain a clear understanding of the numbers involved. Skip counting will be used to analyze changes to populations, including patterns than involve doubling. Students will be actively engaged with the subject matter and will develop a mathematical mindset around data. This is an excellent preparation for the data analysis developed in upper elementary and middle school.

August 2021. In this issue of Math in the News we look at various charts and statistics about the Tokyo Olympics. Students are shown a series of charts and are then asked questions to encourage them to analyze the data. This is an excellent back-to-school activity with the focus on real-world data.

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Topic

Sequences and Series

Definition

A sequence is an ordered list of numbers that follow a particular pattern or rule.

Description

Sequences are fundamental concepts in mathematics, representing ordered lists of numbers that follow a specific pattern or rule. This concept is crucial in various mathematical applications, including algebra, calculus, and computer science.

Topic

Sequences and Series

Definition

A finite sequence is a sequence that has a limited number of terms.

Description

A finite sequence is a sequence with a specific number of terms, making it a crucial concept in the study of sequences and series. This concept is fundamental in various mathematical applications where the number of elements is limited, such as in statistics and computer science.

In real-world applications, finite sequences are used in data analysis, where datasets have a specific number of entries. They are also used in computer algorithms that process a fixed number of elements. Algebraically, a finite sequence can be represented as

Topic

Sequences and Series

Definition

An infinite sequence is a sequence that continues indefinitely without terminating.

Description

An infinite sequence is a fundamental concept in mathematics, representing a sequence that continues indefinitely without terminating. This concept is essential in various mathematical and scientific applications, including calculus and analysis.

In real-world applications, infinite sequences are used in mathematical modeling and simulations where processes continue indefinitely, such as in population dynamics or financial forecasting. Algebraically, an infinite sequence can be represented as  

Topic

Sequences and Series

Definition

A series is the sum of the terms of a sequence.

Description

Series are significant concepts in mathematics, representing the sum of the terms of a sequence. This concept is essential in various mathematical and scientific applications, including calculus, analysis, and financial modeling.

In real-world applications, series are used in financial calculations, such as determining the total amount of payments or investments over time. They are also used in data analysis to find the sum of a dataset. Algebraically, series can be represented in various forms, such as arithmetic series, geometric series, and more.

Topic

Sequences and Series

Definition

A finite series is the sum of the terms of a finite sequence.

Description

A finite series is a significant concept in mathematics, representing the sum of the terms of a finite sequence. This concept is essential in various mathematical and scientific applications, including statistics, finance, and computer science.

In real-world applications, finite series are used in financial calculations, such as determining the total amount of payments over a fixed period. They are also used in data analysis to find the sum of a dataset. Algebraically, a finite series can be expressed as  

Topic

Sequences and Series

Definition

An infinite series is the sum of the terms of an infinite sequence.

Description

An infinite series is a significant concept in mathematics, representing the sum of the terms of an infinite sequence. This concept is essential in various mathematical and scientific applications, including calculus and analysis.

In real-world applications, infinite series are used in mathematical modeling and simulations where processes continue indefinitely, such as in signal processing or financial forecasting. Algebraically, an infinite series can be represented as 

Topic

Sequences and Series

Definition

A convergent series is a series whose terms approach a specific value as the number of terms increases.

Description

A convergent series is a critical concept in the study of sequences and series, where the sum of its terms approaches a finite limit as the number of terms increases. This concept is essential in various mathematical and scientific applications, including calculus and analysis.

Topic

Sequences and Series

Definition

A divergent series is a series whose terms do not approach a specific value as the number of terms increases.

Description

A divergent series is a significant concept in the study of sequences and series, where the sum of its terms does not approach a finite limit as the number of terms increases. This concept is essential in various mathematical and scientific applications, including calculus and analysis.

Topic

Sequences and Series

Definition

The binomial series is the expansion of a binomial raised to any integer power.

Description

The binomial series is a powerful tool in mathematics, particularly in the study of sequences and series. It represents the expansion of a binomial expression raised to any integer power, which is essential in various mathematical and scientific applications, including probability, algebra, and calculus.

Topic

Sequences and Series

Definition

The binomial theorem describes the algebraic expansion of powers of a binomial.

Description

The binomial theorem is a fundamental theorem in algebra that provides a formula for expanding binomials raised to any positive integer power. This theorem is essential in various mathematical disciplines, including algebra, calculus, and probability theory.

Topic

Sequences and Series

Definition

Pascal's Triangle is a triangular array of binomial coefficients.

Description

Pascal's Triangle is a fundamental concept in mathematics, representing a triangular array of binomial coefficients. This concept is essential in various mathematical and scientific applications, including combinatorics, algebra, and probability theory.

In real-world applications, Pascal's Triangle is used in probability calculations, particularly in binomial expansions and combinatorial problems. It is also used in computer science for algorithm design. Algebraically, the elements of Pascal's Triangle can be expressed as  

Topic

Sequences and Series

Definition

A recursive formula defines each term of a sequence using the preceding term(s).

Description

A recursive formula is a fundamental concept in mathematics, representing a way to define each term of a sequence using the preceding term(s). This concept is essential in various mathematical and scientific applications, including computer science and algorithm design.

Topic

Sequences and Series

Definition

An explicit formula for a sequence provides a direct way to calculate any term in the sequence.

Description

An explicit formula for a sequence is a mathematical expression that allows for the direct computation of any term in the sequence without needing to know the previous terms. This concept is fundamental in the study of sequences and series and is widely used in various mathematical applications.

Topic

Sequences and Series

Definition

The common difference is the difference between consecutive terms in an arithmetic sequence.

Description

The common difference is a key concept in arithmetic sequences, representing the consistent interval between consecutive terms. This concept is fundamental in understanding linear patterns and is widely used in various mathematical applications.

Topic

Sequences and Series

Definition

The common ratio is the ratio between consecutive terms in a geometric sequence.

Description

The common ratio is a fundamental concept in geometric sequences, representing the consistent factor by which each term is multiplied to obtain the next term. This concept is crucial in understanding exponential growth and decay, which are prevalent in various mathematical and scientific applications.

Topic

Sequences and Series

Definition

Summation notation is a mathematical notation used to represent the sum of a sequence of terms.

Description

Summation notation is a powerful tool in mathematics, representing the sum of a sequence of terms in a compact form. This concept is essential in various mathematical and scientific applications, including calculus, statistics, and computer science.

Topic

Sequences and Series

Definition

The Fibonacci sequence is a sequence where each term is the sum of the two preceding ones, starting from 0 and 1.

Description

The Fibonacci sequence is a famous sequence in mathematics, where each term is the sum of the two preceding ones, starting from 0 and 1. This sequence is significant in various mathematical and scientific applications, including computer science, biology, and art.

Topic

Sequences and Series

Definition

Triangular numbers are numbers that can be represented by an equilateral triangle.

Description

Triangular numbers are fundamental concepts in mathematics, representing numbers that can form an equilateral triangle. This concept is crucial in various mathematical applications, including algebra, geometry, and number theory.

In real-world applications, triangular numbers are used in combinatorial problems and in determining the number of connections in a network. They are also used in computer algorithms and game design. Algebraically, triangular numbers can be represented as  

Topic

Sequences and Series

Definition

Square numbers are numbers that are the square of an integer.

Description

Square numbers are fundamental concepts in mathematics, representing numbers that are the square of an integer. This concept is crucial in various mathematical applications, including algebra, geometry, and number theory.

In real-world applications, square numbers are used in area calculations, particularly in determining the area of squares. They are also used in computer algorithms and cryptography. Algebraically, square numbers can be represented as n2 , where n is an integer.

Topic

Sequences and Series

Definition

Graphs of arithmetic sequences form a linear graph on a coordinate plane.

Description

Graphs of arithmetic sequences are a visual tool used in mathematics to represent the terms of an arithmetic sequence on a coordinate plane. This concept is crucial for understanding linear patterns and relationships in sequences and series.

Topic

Sequences and Series

Definition

Graphs of geometric sequences visually represent the terms of a geometric sequence on a coordinate plane. These are nonlinear graphs.

Description

Graphs of geometric sequences are a visual tool used in mathematics to represent the terms of a geometric sequence on a coordinate plane. This concept is crucial for understanding exponential patterns and relationships in sequences and series.

Topic

Statistics

Definition

The average is a measure of central tendency, calculated by dividing the sum of values by their count.

Description

In statistics, the average is crucial for analyzing data sets, revealing trends, and providing insight into overall performance. It’s applicable in various fields, from school grades to business metrics. For example, if a student scores 80, 90, and 100 on three exams, the average can be calculated as follows: Average = (80 + 90 + 100) / 3 = 90. The average is essential in math education as it forms a foundational concept for more advanced statistical analyses.

Topic

Statistics

Definition

A weighted average is an average that takes into account the relative importance of each value, calculated by multiplying each value by its weight and summing the results.

Description

The weighted average is used when different data points contribute unequally to the final average. It is commonly applied in finance to calculate portfolio returns, in education to compute weighted grades, and in various fields where data points have different levels of significance. The weighted average provides a more accurate representation of data by considering the relative importance of each value.

Topic

Statistics

Definition

The weighted mean is the average of a data set where each value is multiplied by a weight reflecting its importance.

Description

The weighted mean is used when different data points contribute unequally to the final average. It is commonly applied in finance to calculate portfolio returns, in education to compute weighted grades, and in various fields where data points have different levels of significance. The weighted mean provides a more accurate representation of data by considering the relative importance of each value.

Topic

Statistics

Definition

Average speed is the total distance traveled divided by the total time taken.

Description

This concept finds application in areas such as physics, transport, and everyday scenarios like calculating travel time. For example, if a car travels 300 km in 3 hours, the average speed is Average Speed = 300 km / 3 hours = 100 km/h. Understanding average speed is key in mathematics as it helps contextualize rate and distance problems in real-life situations.

Topic

Statistics

Definition

The sample mean is the average of a sample, calculated by summing the sample values and dividing by the sample size.

Description

The sample mean is a measure of central tendency that provides an estimate of the population mean based on a sample. It is widely used in statistics for making inferences about populations from samples, playing a crucial role in hypothesis testing and confidence interval estimation. The sample mean is used in fields such as economics, biology, and psychology to analyze data and draw conclusions about larger populations.

Topic

Statistics

Definition

The population mean is a measure of central tendency that provides an average representation of a set of data.

Description

The Population Mean is an important concept in statistics, used to summarize data effectively. It is meant to represent the mean for a given statistic for an entire population. For example, the mean length of a salmon.

Topic

Statistics

Definition

The geometric mean is the nth root of the product of n numbers, used to calculate average rates of growth.

Description

The geometric mean is particularly useful in finance and economics for calculating compound interest and growth rates. Unlike the arithmetic mean, it is appropriate for data sets with values that are multiplicatively related. For example, the geometric mean of 2, 8, and 32 is calculated as (2 × 8 × 32)1/3 = 8. In mathematics, the geometric mean is essential for understanding exponential growth and decay.

Topic

Statistics

Definition

Quartiles divide a ranked data set into four equal parts.

Description

Quartiles are used to summarize data by dividing it into four parts, each representing a quarter of the data set. They provide insight into the spread and center of data, helping to identify the distribution and variability. Quartiles are used in box plots to visually represent data distribution, making them valuable in fields such as finance and research for analyzing data trends.

Topic

Statistics

Definition

The lower quartile (Q1) is the median of the lower half of a data set, representing the 25th percentile.

Description

The lower quartile is a measure of position, indicating the value below which 25% of the data falls. It is used in conjunction with other quartiles to understand the distribution and spread of data. In real-world applications, the lower quartile is used in finance to assess the performance of investments and in education to evaluate student achievement levels.

Topic

Statistics

Definition

The upper quartile (Q3) is the median of the upper half of a data set, representing the 75th percentile.

Description

The upper quartile is a measure of position that indicates the value below which 75% of the data falls. It is used in conjunction with other quartiles to understand the distribution and spread of data. In real-world applications, the upper quartile is used in finance to assess investment performance and in education to evaluate student achievement levels.

Topic

Statistics

Definition

The interquartile range (IQR) is the range between the first and third quartiles, representing the middle 50% of a data set.

Description

The IQR is a measure of statistical dispersion, indicating the spread of the central portion of a data set. It is particularly useful for identifying outliers and understanding the variability of data. In real-world applications, the IQR is used in finance to assess investment risks and in quality control to monitor process stability.

Topic

Statistics

Definition

A box-and-whisker plot is a graphical representation of data that displays the distribution through quartiles.

Description

Box-and-whisker plots are useful for visualizing the spread and skewness of a data set, highlighting the median, quartiles, and potential outliers. They are particularly valuable in comparing distributions across different groups. In real-world applications, box plots are used in quality control processes and in analyzing survey data to identify trends and anomalies.

Topic

Statistics

Definition

The median of an odd data set is one of the terms in the data set.

Description

The Median is the middle term of a data set. If the data set consists of an odd number of terms, no matter how many terms there are, the Median will be the middle term of that set.

In mathematics education, understanding median of an odd data set is crucial as it lays the foundation for more advanced statistical concepts. It allows students to grasp the significance of data analysis and interpretation. In classes, students often perform exercises calculating the mean of sets, which enhances their understanding of averaging techniques.

Topic

Statistics

Definition

The median of an even data set is the mean of two of the terms.

Description

The Median is the middle term of a data set. If the data set consists of an even number of terms, then the Median won't be one of ther terms in the set. In such a case the Median is the Mean of the two middle terms.