# The Mean and the Median

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1 The Mean and the Median Objectives To introduce the mean of a set of data; and to review the median of a set of data. epresentations etoolkit Algorithms Practice EM Facts Workshop Game Family Letters Assessment Management Common Core State Standards Curriculum Focal Points Interactive Teacher s Lesson Guide Teaching the Lesson Ongoing Learning & Practice Differentiation Options Key Concepts and Skills Order whole numbers. [Number and Numeration Goal ] Use data to complete a bar graph. [Data and Chance Goal ] Find the median and mean of data sets. [Data and Chance Goal ] Use graphs to ask and answer questions. [Data and Chance Goal ] Key Activities Children make bar graphs for given sets of data. They model the bar graphs with pennies and then rearrange the pennies to determine the mean (average) for each data set. They compare the mean and median of data sets. Recognizing Student Achievement Use journal page. [Data and Chance Goal ] Informing Instruction See page. Key Vocabulary mean average median Materials Math Journal, pp. and Student Reference Book, pp. and Home Link ruler or straightedge tool-kit pennies or counters ( per child) stick-on notes (optional) Interpreting a Pictograph Math Journal, pp. A and B Student Reference Book, pp. and 9 Children use data from a pictograph to answer questions. Then they create a bar graph to show the same data. Math Boxes Math Journal, p. Children practice and maintain skills through Math Box problems. Home Link Math Masters, p. Children practice and maintain skills through Home Link activities. READINESS Graphing Dice Rolls Math Masters, p. 9 per partnership: dice Children make a bar graph to record dice-roll results. ENRICHMENT Making a Data Set Math Masters, pp. and per partnership: counters Children determine a set of data based on given landmarks and graph the data. ELL SUPPORT Building a Math Word Bank Differentiation Handbook, p. Children add the term mean to their Math Word Banks. Advance Preparation Teacher s Reference Manual, Grades pp. Lesson

2 Getting Started Mental Math and Reflexes Pose fact and fact extension problems like the following. Have children record the facts on slates and share their strategies for solving the fact extensions. =, 9 = = 9 = 9 = = 9 = =, 9 =, Math Message Make a bar graph of the data in the table on journal page. Home Link Follow-Up Have partners share answers. Teaching the Lesson NOTE Everyday Mathematics does not draw a distinction between bar graphs and histograms. For a discussion on how some people contrast them, see section.., Organizing and Displaying Data in the Teacher s Reference Manual. Math Message Follow-Up (Math Journal, p. ) WHOLE-CLASS DISCUSSION Check that children have completed the bar graph. Ask: What does the height of each bar represent? The number of children in that family A Mean, or Average, Number of Children Activity Make a bar graph of the data in the table. Number of Children Kugel Abuka Lauer Miller Ellis Bosnak Families The mean, or average, number of children in the six families in the table is. Family Sizes Family Activity (to be done later) Use the table above. List the number of children in order. Number of Children Kugel Abuka Lauer Miller Ellis Bosnak The median number of children in the six families in the table is. Finding the Mean Number of Children (Math Journal, p. ; Student Reference Book, pp. ) WHOLE-CLASS Ask whether anyone can remember how the class found an average class shoe length in an earlier unit. (In Lesson -, children lined up along a paper tape, each placing one foot heel-to-toe on the tape. After cutting off the leftover piece of tape, children folded the tape into equal parts. Each part represented an average class shoe length.) Explain to the class that they are going to find the average number of children per family shown in the table using another method. Lead them in the following routine:. Cover each shaded cell in the bar graph with a penny (or other counter). What does each penny represent? A child. Level off the pennies; that is, rearrange the pennies so each column in the graph has the same number of pennies.. With a straightedge, draw a horizontal line just above the top row of pennies.. Remove the pennies. ELL Math Journal, p. Unit Measurement and Data

3 The horizontal line represents the mean, or average, of this set of data. In this example, the mean number of children per family is. To support English language learners, discuss the social and mathematical meanings of the words mean and average. Emphasize that in this mathematical context, mean and average describe the same thing. Ask children to count the number of shaded cells above the line and the number of unshaded cells below the line. Are the number of cells the same? Yes. Each time a penny was moved, a shaded cell was matched with an unshaded cell. How could you find the place to draw the horizontal line without using pennies? By trial and error, draw a horizontal line so that the number of shaded squares above the line is the same as the number of unshaded squares below the line. Another method of finding the mean is to model each family by drawing a medium-size circle for each family and putting in the required number of pennies or counters to represent the number of children. The counters are then redistributed among the families (without adding or subtracting any) so each family has the same number. Have children read the essay on The Mean (Average) on pages in the Student Reference Book. A Mean, or Average, Number of Eggs Activity Make a bar graph of the data in the table. Number of Eggs 9 a b c d e Ostrich Clutches Math Journal, p. Ostrich Clutches Clutch The mean, or average, number of eggs in the five clutches is. Activity (to be done later) List the number of eggs in the clutches in order. Number of Eggs a b c d e The median is eggs per clutch. Finding the Mean of Ostrich Egg Clutches (Math Journal, p. ) NOTE Although the mean for this set of data is a whole number, the mean for other data sets will likely be between two whole numbers. A more efficient way to find the mean will be introduced in Lesson -. Children make bar graphs of the data set about ostrich egg clutches. They use one of the methods from the previous activity to find the mean number of eggs. Recognizing Student Achievement Journal Page Use journal page to assess children s ability to complete a bar graph. Children are making adequate progress if they successfully complete the bar graph. Some children may be able to find the mean number of eggs. [Data and Chance Goal ] Interpreting a Pictograph Big Bend National Park is located in southwestern Texas. It contains, acres of protected wilderness. About, people visit the park each year. The pictograph below shows attendance at Big Bend National Park for one week in August. Use the data to answer the questions below. Day Day Day KEY: = people Day Day Day Links to the Future The activities in this lesson are an early exposure to finding the mean of a set of data. Finding the mean of a data set is a Grade Goal. Day. How many people visited the park on Day? people. How many people visited the park on Day? people. How many people visited the park on Day? people. How many more people came to Big Bend National Park on Day than Day? more people. How many fewer people came on Day than Day? fewer people Math Journal, p. A -_EMCS_S_MJ_G_U_.indd A // : PM Lesson

4 Making a Bar Graph from a Pictograph Use the information shown in the pictograph to make a bar graph. Remember to add labels and a title to your graph. Sample answer: Number of Visitors Day Day Day Day Day Day Day KEY: = people Sample answer: Days of Visits Math Journal, p. B B Finding the Median of Sets of Data (Math Journal, pp. and ; Student Reference Book, p. ) Have children complete Activity on journal pages and. Children find the median number of children per family and the median number of eggs per clutch. If necessary, have them read page in the Student Reference Book to review how to find the median, or middle value, of a set of data. Informing Instruction Watch for children who have difficulty finding the median. They can write the number of eggs in each nest on a small slip of paper or a stick-on note. Have them put the numbers in order and then find the middle number(s). Ongoing Learning & Practice Interpreting a Pictograph and Creating a Bar Graph (Math Journal, pp. A and B; Student Reference Book, pp. and 9) Math Boxes. Measure each side of the triangle to the nearest centimeter. Perimeter = 9. James built a rectangular prism out of base- blocks. He used cubes to make the base. He put more layers of cubes on top of that. What is the volume of the prism he built? cubic centimeters 9. Molly is playing with toy cars. This is only _ of her set of cars. How many cars are in her complete set? Fill in the circle next to the best answer. A _ cars C cars B cars D cars Math Journal, p.. There are blocks in a bag. blocks are red, blocks are blue, and block is green. What are the chances of pulling out a red block? out of chances. Complete. gallon = quarts gallons = quarts pint = cups pints = cups liter =,milliliters liters =, milliliters. Color the circle so that it matches the description. _ blue _ green _ yellow Which color would you expect the spinner to land on most often? blue yellow blue green 9 9 Have children examine the pictograph on journal page A. Ask them to explain what each smiley face means. people Next ask them to explain what one-half of a smiley face means. people Have children work independently or with a partner to complete journal page A. When they have finished, children create a bar graph on journal page B to show the same data. Discuss the scale for the bar graph. Since the pictograph key shows that each smiley face represents people, the scale for the bar graph could be in increments of. Math Boxes (Math Journal, p. ) Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson -. The skill in Problem previews Unit content. Writing/Reasoning Have children write an answer to the following: Choose one of the units of capacity in Problem. Describe the unit of capacity by using one of the personal references for units of capacity from Lesson -. For example, describe an object that has about the same measure as a liter (or the unit of capacity you chose). Sample answer: I have a water bottle that holds liter of water. Unit Measurement and Data

### Assessment Management

Area Objectives To provide experiences with the concept of area, distinguishing between area and perimeter, and finding areas of rectangular figures by partitioning and counting squares. www.everydaymathonline.com

### Multiplication Arrays

Multiplication Arrays Objective To provide opportunities to use arrays, multiplication/ division diagrams, and number models to represent multiplication number stories. www.everydaymathonline.com epresentations

### Measuring with Yards and Meters

Measuring with Yards and Meters Objectives To provide review for the concept of nonstandard units of measure; and to introduce yard and meter. www.everydaymathonline.com epresentations etoolkit Algorithms

### Volume of Rectangular Prisms Objective To provide experiences with using a formula for the volume of rectangular prisms.

Volume of Rectangular Prisms Objective To provide experiences with using a formula for the volume of rectangular prisms. www.everydaymathonline.com epresentations etoolkit Algorithms Practice EM Facts

The Addition/Subtraction Facts Table Objectives To provide experience exploring patterns in sums of two dice; and to introduce the Addition/Subtraction Facts Table. www.everydaymathonline.com epresentations

### Capacity. Assessment Management

Capacity Objective To review units of capacity. www.everydaymathonline.com epresentations etoolkit Algorithms Practice EM Facts Workshop Game Family Letters Assessment Management Common Core State Standards

### Assessment Management

Areas of Rectangles Objective To reinforce students understanding of area concepts and units of area. www.everydaymathonline.com epresentations etoolkit Algorithms Practice EM Facts Workshop Game Family

### Analyzing and Displaying Data

Analyzing and Displaying Data Objectives To review data concepts; to provide opportunities to find data landmarks and use graphs to draw conclusions. www.everydaymathonline.com epresentations etoolkit

### Middle Value (Median) of a Set of Data

Middle Value (Median) of a Set of Data Objectives To guide children as they sort numerical data and arrange data in ascending or descending order, and as they find the middle value (median) for a set of

### Exploring Length, Area, and Attributes

Explorations Exploring Length, Area, and Attributes Objectives To guide children as they measure lengths and distances to the nearest inch and centimeter, explore area by tiling surfaces, and sort attribute

### Baseball Multiplication Objective To practice multiplication facts.

Baseball Multiplication Objective To practice multiplication facts. www.everydaymathonline.com epresentations etoolkit Algorithms Practice EM Facts Workshop Game Family Letters Assessment Management Common

### Objective To guide the development and use of a rule for generating equivalent fractions. Family Letters. Assessment Management

Equivalent Fractions Objective To guide the development and use of a rule for generating equivalent fractions. www.everydaymathonline.com epresentations etoolkit Algorithms Practice EM Facts Workshop Game

### Volume of Right Prisms Objective To provide experiences with using a formula for the volume of right prisms.

Volume of Right Prisms Objective To provide experiences with using a formula for the volume of right prisms. www.everydaymathonline.com epresentations etoolkit Algorithms Practice EM Facts Workshop Game

### Subtraction Strategies

Subtraction Strategies Objective To review solution strategies for subtraction of multidigit numbers. www.everydaymathonline.com epresentations etoolkit Algorithms Practice EM Facts Workshop Game Family

### A Coin-Toss Experiment

A Coin-Toss Experiment Objective To guide children as they develop intuition equally likely events. www.everydaymathonline.com epresentations etoolkit Algorithms Practice EM Facts Workshop Game Family

### Sunrise-Sunset Line Graphs

Sunrise-Sunset Line Graphs Objectives To guide children as they analyze data from the sunrise-sunset routine; and to demonstrate how to make and read a line graph. www.everydaymathonline.com epresentations

### Doubles Facts Objective To review and provide practice for doubles facts.

Doubles Facts Objective To review and provide practice for doubles facts. www.everydaymathonline.com epresentations etoolkit Algorithms Practice EM Facts Workshop Game Family Letters Assessment Management

### The Inch. Objectives To introduce the inch as a standard unit of length; and to provide practice measuring to the nearest inch.

The Inch Objectives To introduce the inch as a standard unit of length; and to provide practice measuring to the nearest inch. www.everydaymathonline.com epresentations etoolkit Algorithms Practice EM

### Review of Basic Fraction Concepts

Review of asic Fraction Concepts Objective To review fractions as parts of a whole (ONE), fractions on number lines, and uses of fractions. www.everydaymathonline.com epresentations etoolkit lgorithms

Subtraction from Addition Objectives To review the and 1 shortcuts; and to guide children to identify the subtraction facts related to given addition facts. www.everydaymathonline.com epresentations etoolkit

### Assessment Management

Facts Using Doubles Objective To provide opportunities for children to explore and practice doubles-plus-1 and doubles-plus-2 facts, as well as review strategies for solving other addition facts. www.everydaymathonline.com

### Comparing and Ordering Fractions

Comparing and Ordering Fractions Objectives To review equivalent fractions; and to provide experience with comparing and ordering fractions. www.everydaymathonline.com epresentations etoolkit Algorithms

### Fractions and Decimals

Fractions and Decimals Objectives To provide experience with renaming fractions as decimals and decimals as fractions; and to develop an understanding of the relationship between fractions and division.

### Exploring Arrays, Coins, and Division

Explorations Exploring Arrays, Coins, and Division Objectives To develop readiness for multiplication; to guide children in finding coin combinations equivalent to \$1.00; and to explore one meaning of

### The Distributive Property

The Distributive Property Objectives To recognize the general patterns used to write the distributive property; and to mentally compute products using distributive strategies. www.everydaymathonline.com

### Constructing Geometric Solids

Constructing Geometric Solids www.everydaymathonline.com Objectives To provide practice identifying geometric solids given their properties; and to guide the construction of polyhedrons. epresentations

### The Median. Objectives To review how to display a set of data with a line plot; and to review how to find the median of a set of data.

The Median Objectives To review how to display a set of data with a line plot; and to review how to find the median of a set of data. www.everydaymathonline.com epresentations etoolkit Algorithms Practice

### Objective To guide exploration of the connection between reflections and line symmetry. Assessment Management

Line Symmetry Objective To guide exploration of the connection between reflections and line symmetry. www.everydaymathonline.com epresentations etoolkit Algorithms Practice EM Facts Workshop Game Family

Addition Number Stories Objective To guide children as they make up, represent, and solve addition number stories. www.everydaymathonline.com epresentations etoolkit Algorithms Practice EM Facts Workshop

### Surface Area. Assessment Management

Surface Area Objective To introduce finding the surface area of prisms, cylinders, and pyramids. www.everydaymathonline.com epresentations etoolkit Algorithms Practice EM Facts Workshop Game Family Letters

### Subtracting Mixed Numbers

Subtracting Mixed Numbers Objective To develop subtraction concepts related to mixed numbers. www.everydaymathonline.com epresentations etoolkit Algorithms Practice EM Facts Workshop Game Family Letters

### Objective To introduce a formula to calculate the area. Family Letters. Assessment Management

Area of a Circle Objective To introduce a formula to calculate the area of a circle. www.everydaymathonline.com epresentations etoolkit Algorithms Practice EM Facts Workshop Game Family Letters Assessment

### Number Models for Area

Number Models for Area Objectives To guide children as they develop the concept of area by measuring with identical squares; and to demonstrate how to calculate the area of rectangles using number models.

### Number-Line Posters for Fractions

Number-Line Posters f Fractions Objective To introduce the number line as a model f fractions. www.everydaymathonline.com epresentations etoolkit Algithms Practice EM Facts Wkshop Game Family Letters Assessment

### Number-Line Posters for Fractions

Number-Line Posters f Fractions Objective To introduce the number line as a model f fractions. www.everydaymathonline.com epresentations etoolkit Algithms Practice EM Facts Wkshop Game Family Letters Assessment

### Multiplication Facts Practice

Multiplication Facts Practice Objectives To introduce the 5-facts test; and to provide practice with multiplication facts. www.everydaymathonline.com epresentations etoolkit Algorithms Practice EM Facts

### Measuring with a Ruler

Measuring with a Ruler Objective To guide children as they measure line segments to the nearest inch, _ inch, _ inch, centimeter, _ centimeter, and millimeter. www.everydaymathonline.com epresentations

### Comparing Fractions Objective To provide practice ordering sets of fractions.

Comparing Fractions Objective To provide practice ordering sets of fractions. www.everydaymathonline.com epresentations etoolkit Algorithms Practice EM Facts Workshop Game Family Letters Assessment Management

### Objective To introduce the concept of square roots and the use of the square-root key on a calculator. Assessment Management

Unsquaring Numbers Objective To introduce the concept of square roots and the use of the square-root key on a calculator. www.everydaymathonline.com epresentations etoolkit Algorithms Practice EM Facts

### Addition Strategies That Use Doubles Facts

Addition Strategies That Use Doubles Facts Objective To provide opportunities for children to explore and practice doubles-plus- and doubles-plus- facts. www.everydaymathonline.com epresentations etoolkit

### Assessment Management

Number Stories Objectives To introduce everyday examples of rectangular arrays; and to provide experiences with solving multiplication problems using multiplication diagrams and array models. www.everydaymathonline.com

### Parentheses in Number Sentences

Parentheses in Number Sentences Objective To review the use of parentheses. www.everydaymathonline.com epresentations etoolkit Algorithms Practice EM Facts Workshop Game Family Letters Assessment Management

### Factor Trees. Objective To provide experiences with finding the greatest common factor and the least common multiple of two numbers.

Factor Trees Objective To provide experiences with finding the greatest common factor and the least common multiple of two numbers. www.everydaymathonline.com epresentations etoolkit Algorithms Practice

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Formula for the Area of a Parallelogram Objectives To review the properties of parallelograms; and to guide the development and use of a formula for the area of a parallelogram. www.everydaymathonline.com

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Order of Operations Objectives To review the order of operations; and to evaluate expressions containing parentheses. www.everydaymathonline.com epresentations etoolkit Algorithms Practice EM Facts Workshop

### Change-to-More Number Stories

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### Review: Addition and Subtraction of Positive and Negative Numbers

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### The Lattice Method of Multiplication

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### Simplifying Expressions: Combining Like Terms Objective To simplify algebraic expressions by combining like terms.

Simplifying Expressions: Combining Like Terms Objective To simplify algebraic expressions by combining like terms. www.everydaymathonline.com epresentations etoolkit Algorithms Practice EM Facts Workshop

### The Length-of-Day Project

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### Multiplication and Division Fact Families

Multiplication and Division Fact Families Objectives To review fact families and the Multiplication/Division Facts Table; and to guide children as they practice multiplication and division facts. www.everydaymathonline.com

### 1 Teaching the Lesson

Getting Started Mental Math and Reflexes Children solve division number stories. They write answers on slates and explain their strategies to the class. Provide counters. There are children in the class.

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### Multiplication Facts Survey

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### Partial-Products Multiplication (Part 2)

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### Place Value in Whole Numbers

Place Value in Whole Numbers Objectives To provide practice identifying values of digits in numbers up to one billion; and to provide practice reading and writing numbers up to one billion. www.everydaymathonline.com

### Math Message and Number Sequences

Math Message and Number Sequences Objectives To introduce the Math Message routine; and to review number sequences and number lines. www.everydaymathonline.com epresentations etoolkit Algorithms Practice

### Change Number Stories Objective To guide children as they use change diagrams to help solve change number stories.

Number Stories Objective To guide children as they use change diagrams to help solve change number stories. www.everydaymathonline.com epresentations etoolkit Algorithms Practice EM Facts Workshop Game

### Buying at the Stock-Up Sale

Buying at the Stock-Up Sale Objective To guide children as they multiply using mental math and the partial-products algorithm. www.everydaymathonline.com epresentations etoolkit Algorithms Practice EM

### Family Letters. Assessment Management. Playing Target: 50

The Partial-Sums Algorithm Objectives To guide children as they make ballpark estimates; and to provide opportunities to model and practice the partialsums algorithm for - and -digit numbers. www.everydaymathonline.com

### Objectives To review making ballpark estimates; and to review the counting-up and trade-first subtraction algorithms. materials. materials.

Objectives To review making ballpark estimates; and to review the counting-up and trade-first subtraction algorithms. Teaching the Lesson materials Key Activities Children make ballpark estimates for -digit

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Properties of Polygons Objective To explore the geometric properties of polygons. www.everydaymathonline.com epresentations etoolkit Algorithms Practice EM Facts Workshop Game Family Letters Assessment

### The Partial-Quotients Division Algorithm, Part 1

The Partial-Quotients Division Algorithm, Part 1 Objectives To introduce and provide practice with a low-stress division algorithm for 1-digit divisors. www.everydaymathonline.com epresentations etoolkit

### Family Letters. Assessment Management. Math Boxes 1 4

Objective To calculate and compare the median and mean of a data set. www.everydaymathonline.com epresentations etoolkit Algorithms Practice EM Facts Workshop Game Family Letters Assessment Management

### Volume of Pyramids and Cones

Volume of Pyramids and Cones Objective To provide experiences with investigating the relationships between the volumes of geometric solids. www.everydaymathonline.com epresentations etoolkit Algorithms

### Calculator Practice: Computation with Fractions

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### Reading and Writing Large Numbers

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### Solving Simple Equations Objective To use trial and error and a cover-up method to

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### Inequalities. Assessment Management. Dividing Decimals by Decimals: Part 1

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### Line Plots. Objective To provide experience creating and interpreting line plots with fractional units. Assessment Management

Line Plots Objective To provide experience creating and interpreting line plots with fractional units. www.everydaymathonline.com epresentations etoolkit Algorithms Practice EM Facts Workshop Game Family

### Objectives To find the absolute value of rational numbers; and to solve problems using absolute value. Family Letters. Assessment Management

Absolute Value Objectives To find the absolute value of rational numbers; and to solve problems using absolute value. www.everydaymathonline.com epresentations etoolkit Algorithms Practice EM Facts Workshop

### Multiplying Fractions by Whole Numbers

Multiplying Fractions by Whole Numbers Objective To apply and extend previous understandings of multiplication to multiply a fraction by a whole number. www.everydaymathonline.com epresentations etoolkit

### Assessment Management

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### Reading and Writing Small Numbers

Reading Writing Small Numbers Objective To read write small numbers in stard exped notations wwweverydaymathonlinecom epresentations etoolkit Algorithms Practice EM Facts Workshop Game Family Letters Assessment

### Multiplication and Division of Positive and Negative Numbers

Multiplication and Division of Positive Objective o develop and apply rules for multiplying and dividing positive and www.everydaymathonline.com epresentations eoolkit Algorithms Practice EM Facts Workshop

### Exponential Notation for Powers of 10

Expntial Notation for Objective To introduce number-and-word notation for large numbers and expntial notation for powers of 10. www.everydaymathonline.com epresentations etoolkit Algorithms Practice EM

### Subtraction of Multidigit Numbers

Subtraction of Multidigit Numbers Objectives To review the trade-first and counting-up methods, and to introduce the partial-differences method of solving multidigit subtraction problems; and to provide

### materials materials materials Technology Teaching the Lesson Ongoing Learning & Practice Differentiation Options

Objectives To guide children as they make ballpark estimates; and to provide opportunities to model and practice the partialsums algorithm for - and -digit numbers. Teaching the Lesson Key Activities Children

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ddition with hree or More ddends Objective o guide children as they solve number stories having three or more addends. www.everydaymathonline.com epresentations eoolkit lgorithms Practice EM Facts Workshop

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Multiplication of Mixed Numbers Objective To introduce multiplication with mixed numbers. www.everydaymathonline.com epresentations etoolkit Algorithms Practice EM Facts Workshop Game Family Letters Assessment

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Points and Line Segments Objective To guide children as they define, name, and draw line segments. www.everydaymathonline.com epresentations etoolkit lgorithms Practice M acts Workshop Game amily Letters