The Mean and the Median

Transcription

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

5 Home Link (Math Masters, p. ) Home Connection Children find the mean of a data set. Differentiation Options READINESS Graphing Dice Rolls (Math Masters, p. 9) Min To provide experience with making a bar graph, have children predict how many times they think they will roll a and then make a graph to record their results. When children have completed their graphs, have them discuss their predictions and results. Name HOME LINK Name Number of Goldfish Reba Bill Lucy Meg Nate Pat Mean, or Average, Number of Fish Family Many of us learned that to find the mean (average) of a set of numbers, we add all the Note numbers and then divide the total by how many numbers we added. In today s lesson, the class tried a different method of finding the mean. After your child has completed the page, ask him or her to explain how this method works. In the next lesson, we will introduce finding the mean by adding the numbers and dividing to find the answer. Please return this Home Link to school tomorrow. The table below lists how many goldfish each child won at the school fun fair.. Put a penny over each shaded square in the bar graph. Home Link Master Number of Goldfish. Move the pennies so that each column has the Reba Bill Lucy Meg same number of pennies. Names Nate Pat. Draw a horizontal line across your graph to show the height of the pennies when all of the columns are the same height.. The mean (average) number of goldfish won by children at the fun fair is. Math Masters, p. 9-9_EMCS_B_MM_G_U_9.indd // : PM ENRICHMENT Making a Data Set (Math Masters, pp. and ) Min To apply children s understanding of landmarks (maximum, minimum, range, mode, and median), have them use the given median, range, and mode to display a possible data set when children share cookies. Children record their work on Math Masters, page and make a bar graph of their data set on Math Masters, page. Teaching Master Name ELL SUPPORT Building a Math Word Bank (Differentiation Handbook, p. ) SMALL-GROUP Min To provide language support for landmarks, have children use the Math Word Bank template found on Differentiation Handbook, page. Ask children to write the term mean, draw a picture representing the term, and write other related words. See the Differentiation Handbook for more information. Altogether, children have cookies in their lunch bags. The median number of cookies is. The range of the number of cookies is. Making a Data Set The mode of the number of cookies is. How many cookies could each child have? Use counters and the drawings of lunch bags below to organize your data. Draw cookies in the lunch bags to match the description above. Sample answer: Lunch bag A Lunch bag B Lunch bag C Lunch bag D Lunch bag E 9 Graph your results on Math Masters, page. Remember to include labels and a title. Math Masters, p. 9-9_EMCS_B_MM_G_U_9.indd // : PM Lesson 9