Minds On: 5 / Learning Goal:
- Calculate, interpret and apply measures of central tendency.
- 10 - 12 Tape Measures
- BLM 3.1.1 to 3.1.2
- Graphing calculators
Action: 50
Consolidate:20
Total=75 min
Assessment
Opportunities
Minds On… / Whole Class Discussion
Lead students in a brainstorming session to discuss what it means to be “average”. What does it mean to be above or below average?
Whole Class Introduction to Activity
Students collect the measurements listed in BLM 3.1.1. / / Students make connections between terms, concepts and principles of central tendency.
The mean card can be held by the student whose value is closest to the calculated mean.
The Golden Ratio is approximately 1.61803399. Discuss with students how this number relates to the results.
Action! / Whole Class The Golden Ratio
Using BLM 3.1.1, students collect individual data and generate class data for the four different ratios. The students calculate measurements of central tendency using technology (TI-83, Fathom 2, Excel) and record the class results in Table 3.1.1a. The students stop when the table has been completed and wait for further instructions from the teacher.
Small Groups Discussion (Home)
Arrange the students in ascending order of L1 ratio. Distribute mean, median, mode, minimum, Q1, Q3, and maximum cards to the appropriate students. Break the students into four groups using the quartiles: each quartile group is assigned one of the four ratios for analysis.
Small Groups Discussion (Expert)
Using numbered heads, break the home groups into smaller expert groups (include representation from each home group) and have the students complete the expert group question.
Process Expectations/Communicating/Observation: Observe groups as they connect their results to the measures of central tendency. Listen to discussions and ideas looking for items that students can share with others during the Consolidate Debrief.
Consolidate Debrief / Whole Class Discussion
Discuss results of the expert question with the whole class, highlighting the differences between the measures of central tendency. Include a description of quartiles, standard deviation and variance.
Exploration
Application / Home Activity or Further Classroom Consolidation
Which measure would you prefer for your grade – mean, median, or mode? Why?
MDM4U: Unit 3 – Statistics (Draft – July 2007)
Last saved 17/11/2018 at 8:06 PM1
3.1.1: The Golden Ratio
Perform the following measurements, standing straight up, with your arms at your sides and relaxed:
- Your height, shoes off!
- Top of your head to your finger tips
- Top of your head to your elbows
- Top of your head to the inside top of your arms
- Your elbow to your fingertips
Now calculate your individual ratios, correct to two decimal places:
- L1 = A / B
- L2 = B / C
- L3 = C / D
- L4 = C / E
Record your L1, L2, L3, L4 ratios on the chalkboard under the appropriate column. Copy the class data set into the table below.
Table 3.1.1a - Student ResultsL1 / L2 / L3 / L4
3.1.1: The Golden Ratio (Continued)
Complete the table below for each of the measures, correct to two decimal places.
Table 3.1.1b – Measures of Central TendencyL1 / L2 / L3 / L4
Mean
Median
Mode
Minimum
Q1
Q3
Maximum
Variance
Standard Deviation
Once you have completed the chart, wait for further instructions from your teacher.
Home Group: Within your assigned group, discuss answers to the following questions.
1)Consider the data set for your assigned ratio (L1, L2, L3 or L4). Which measurement (mean, median or mode) “best represents” this data? Why?
2)Which measurement “least represents” this data? Why?
Expert Group: Within your assigned group, determine the “best overall” measure of central tendency.
3.1.2: Measures of Central Tendency Cards
3.1.2: Measures of Central Tendency Cards (continued)
3.1.2: Measures of Central Tendency Cards (continued)
3.1.2: Measures of Central Tendency Cards (continued)
3.1.2: Measures of Central Tendency Cards (continued)
3.1.2: Measures of Central Tendency Cards (continued)
3.1.2: Measures of Central Tendency Cards (continued)
Unit 3: Day 2: On TargetMinds On: 5 / Learning Goal:
- Calculate, interpret and apply standard deviation as a measure of central tendency.
- BLM 3.2.1 to 3.2.4
- Timer
- Graphing calculators
- Masking Tape
- Integer chips or flat discs
Action: 50
Consolidate:20
Total=75 min
Assessment
Opportunities
Minds On… / Small Groups Pass the Paper
Students each start with a paper and a title of “mean”, “median” or “mode” in groups of three. Allow 1 minute for students write down what they know about the term, limitations and examples. After 1 minute, instruct students to pass their paper to the person beside them and continue in this way for three turns. After activity is completed, students engage in a discussion regarding the limitations of mean, median and mode as measures of central tendency. That is, they provide a central value, but do not indicate the spread and consistency of the data. / / Students make connections between consistency and standard deviation.
Action! / Whole Class Hitting the Mark!
Using BLM 3.2.1, students collect individual scores for three trials of the game.
Whole Class Discussion
With reference to BLM 3.2.2, lead students in a discussion on the difference between precision and accuracy. Comment on the connection between precision and consistency and how these terms relate to standard deviation.
Process Expectations/Observation/Checklist
Observe groups as they develop their understanding of consistency as a measure of dispersion. Listen to discussions and ideas looking for connections to the next activity, BLM 3.2.3.
Whole Class All Charged Up!
Students complete the performance task BLM 3.2.3.
Process Expectations/Performance Task/Rubric
Assess the students on the All Charged Up activity usingBLM 3.2.4.
Consolidate Debrief / Whole Class Discussion
Lead students in a discussion on the interplay of precision, accuracy, consistency and standard deviation.
Exploration
Application / Home Activity or Further Classroom Consolidation
Think of a situation from everyday life. In this situation, is it better to have high accuracy or high precision? Can you think of a situation in which low precision (or low accuracy) would be acceptable?
MDM4U: Unit 3 – Statistics (Draft – July 2007)
Last saved 17/11/2018 at 8:06 PM1
3.2.1: Hitting the Mark (Scoring Sheet)
Student Name: ______
Scoring Instructions:
Keep a tally chart of your partner’s performance below to calculate their total score.
Trial1 / 2 / 3
Points / 10
5
1
0
Total
For each toss, record the spot where the marker lands on the targets below
Trial #1Trial #2Trial #3
With reference to the grouping of your markers, how did your results change?
3.2.2: Hitting the Mark (Teacher Instructions)
Game Setup: Construct a game board on the floor with masking tape. Use the following dimensions:
Outer square:150 cm by 150 cm
Middle square:100 cm by 100 cm
Inner square:50 cm by 50 cm
Add a starting line that is 2 m away from the outer edge of the target. With the addition of extra starting lines arranged around the target, up to four students can play at once.
Point values: Outer Square (1 point); Middle Square (5 points); Inner Square (10 points); outside of the target area scores no points.
Playing the game: students approach the starting line, and toss each of their 5 markers (integer chips, coins, coloured tiles) into the target area one at a time. A partner records where the chips land on the provided scoring sheet. Each player tries the game three times.
Recording: students record their results on the sheet provided (BLM 3.2.1) in both a table and a diagram.
Observations: Use the target analogy to lead a discussion regarding the class results on “Hitting the Mark”. Comment on the connection between precision and consistency and how these terms relate to standard deviation.
High precision, but low accuracyHigh accuracy, but low precision
3.2.3: All Charged Up!
You have been hired by LowTech Enterprises, a company that manufactures portable MP3 players, to choose a battery supplier. LowTech offers a warranty program that guarantees 200 recharges of their players; that is, LowTech will repair or replace any MP3 player that does not recharge 200 times.
The original supplier of the battery was supplier X. Their competition, Supplier Y, wants to be the new exclusive battery supplier for LowTech. You choose a random sample of twenty batteries from each supplier and experimentally determine the number of recharges for each battery.
The data from your experiment is as follows (the number given is how many times each battery was capable of being recharged):
Supplier X:
254, 259, 256, 253, 252, 250, 250, 249, 256, 254,
250, 251, 250, 248, 248, 254, 258, 255, 258, 255
Supplier Y:
257, 306, 179, 245, 192, 164, 325, 283, 289, 293,
287, 305, 155, 267, 331, 192, 265, 279, 312, 274
X claims that their batteries will last for an average of 253 recharges, while Y claims that their batteries will last for an average of 260 recharges. Which battery supplier would you recommend? Justify your choice by considering appropriate measures of central tendency.
3.2.4: All Charged Up! Rubric
Reasoning and ProvingCriteria / Level 1 / Level 2 / Level 3 / Level 4
Making inferences, conclusions and justifications / Justification of the answer presented has a limited connection to the problem solving process and models presented / Justification of the answer presented has some connection to the problem solving process and models presented / Justification of the answer presented has a direct connection to the problem solving process and models presented / Justification of the answer presented has a direct connection to the problem solving process and models presented, with evidence of reflection
Connecting
Criteria / Level 1 / Level 2 / Level 3 / Level 4
Making connections among mathematical concepts and procedures / Makes weak connections / Makes simple connections / Makes appropriate connections / Makes strong connections
Unit 3: Day 3: Graph It!
Minds On: 5 / Learning Goal:
- Generate a graphical summary (box and whisker plot, histogram) of a one variable data set.
- BLM 3.3.1 to 3.3.3
- Rulers
- Technology (Fathom 2, Excel, TI-83)
Action: 55
Consolidate:15
Total=75 min
Assessment
Opportunities
Minds On… / Pairs Picture Perfect
Student work in pairs and discuss the questions in BLM 3.3.1 regarding the graphical representation of data. Why is it important to represent data in a graphical format? / / If technology is not available, the students generate the representations by hand.
Action! / Whole Class Reaching New Heights
Using BLM 3.3.2, students generate a box and whisker plot and a histogram for a given data set.
Small Groups Discussion
With reference to BLM 3.3.2, students discuss their response to the last question regarding which representation (box and whisker or histogram) “best” represents this data set.
Process Expectations/Communicating/Observation: Observe groups as they develop their understanding of graphical representations of data. Listen to discussions and ideas looking for connections to the next activity, BLM 3.3.2.
Whole Class Between Friends!
If available, students use technology (Fathom 2, Excel, TI-83) to complete Between Friends (BLM 3.3.3).
Process Expectations/Representing/Observation: Observe students as they generate graphical representations of data; check the box and whisker plots and histograms for accuracy and completeness.
Consolidate Debrief / Whole Class Discussion
Lead students in a discussion on the challenges of graphical representations of data (how are scales chosen, what representations are most appropriate).
Skill Development / Home Activity or Further Classroom Consolidation
Describe how a histogram can be converted into a box and whisker plot. Is it possible to convert a box and whisker plot to a histogram? Why?
3.3.1: Picture Perfect
- Which has more variability – A or B? Why?
Graph A Graph B
- Which class did better? How do you know?
Blue Class Yellow Class
3. Are there the same number of raisins in each box? How can you tell?
MDM4U: Unit 3 – Statistics (Draft – July 2007)
Last saved 17/11/2018 at 8:06 PM1
3.3.2: Reaching New Heights
4. The following are jump heights (in cm) from eleven different cats. Illustrate the data with a box and whisker plot using the number line below.
72, 40, 95, 58, 62, 35, 56, 65, 74, 68, 90
5. Determine appropriate intervals and represent the jumping heights in a histogram. Properly label your axes and provide a title.
6. Which tool is the better graphical representation of the data? Why?
3.3.3: Between Friends
Pick one of the questions below and survey your classmates.
- What is your birth month by number (January = 1, February = 2, …)?
- What is the last digit of your phone number?
- How many hours of television did you watch last week?
- How many books have you read this year?
- How many letters are in your last name?
Record the responses below.
______
______
______
Prepare a box and whisker plot of your data. Be sure to indicate the scale and label the important data points (minimum, Q1, median, Q3, maximum).
Determine apropriate intervals and represent your data in a histogram. Properly label your axes and provide a title.
Unit 3: Day 4: Dazed by DataMinds On: 10 / Learning Goal:
- Explore different types of data (numerical, categorical - ordinal, nominal, interval, continuous, discrete)
- Establish the attributes that information must have to be meaningful.
- BLM 3.4.1 to 3.4.5
- Acetates
- Overhead projector
Action: 25
Consolidate:40
Total=75 min
Assessment
Opportunities
Minds On… / Pairs Data in the Real World
Pairs choose one of the occupations suggested on BLM 3.4.1. Students interview each other about the kinds of data used in their work. Students discuss the data in terms of the two attributes that information must have to be meaningful: numerical data (the number or scalar) and categorical data (the labels or units telling us what the numbers are measuring).
In a 2-Dimensional graph which axis is usually numerical and which is usually categorical? Think of a 2-D graph where both axes are numerical. How much information does it convey? / / Example:
7314
7314 Km
7314 Km from Victoria, B.C. to St. John’s, Nfld.
Answers:
Line graph:
Horizontal axis must be categorical.
Histogram:
Either horizontal or vertical axis can be categorical.
Scatter Plot:
Both axes are numerical.
Action! / Whole Class On The Road Again
Using BLM 3.4.2, students attempt to establish a relationship between data points provided on a graph without numerical or categorical descriptors.
Small Groups Discussion
With reference to BLM 3.4.2, students discuss their response to the three questions.
Process Expectations/Communicating/Observation: Observe groups as they develop their understanding of graphical representations of data.
Whole Class On The Road Again
Using BLM 3.4.2 (Hints) provide a hint to the students.
Whole Class Discussion
Using BLM 3.4.2 (Teacher Notes) present the solution to the students. Students engage in a discussion of the three questions asked. Using BLM 3.4.3 on acetate show the overlay naming the capital cities.
Consolidate Debrief / Small Groups What’s My Word?
Students engage in a discussion on the challenges of data representation. Students create word association cards to help distinguish between continuous data and discrete data and the three types of categorical scales: nominal, ordinal and interval. Students use BLM 3.4.4 as a guide to the activity if word association cards have not been created before.
Process Expectations/Communicating/Observation: Circulate and assess for understanding making mental notes of incomplete or incorrect illustrations and definitions.
Practice / Home Activity or Further Classroom Consolidation
Complete BLM 3.4.4.
Classify the graphs on BLM 3.4.5 according to data type..
3.4.1: Data in the Real World
Choose an occupation. Interview your partner with the questions provided. Discuss the types of data you use in your line of work and sort them into numerical or categorical data sets.
Environment
/Public Sector
/Sciences and Engineering
/Business
/Transportation
Occupations
/Meteorologist
/Policeman
/Forensic Scientist
ArchitectChemical Engineer /
Accountant
Stockbroker /Air Traffic Controller
Interview Questions:
What do you find challenging in your job?
What kinds of data do you use in your work?
How is the data collected?
What types of tools do you use to work with the data?
Numerical Data / Categorical Data3.4.2: On The Road Again
Working in groups of three, determine a possible pattern or relationship between the data points that would account for the scatter plot shown below.
Within your assigned groups, discuss answers to the following questions:
1)Without a scale, how much information is this scatter plot conveying?
2)What possible types of relationships could these data points have?
3)Is it necessary to have a predetermined scale to establish a relationship, assuming the dots are placed according to some representative scale?
3.4.2: On The Road Again (Hints)
Hint: Scale factors have been added and the scatter plot has been turned into a line graph. Does this help you establish a relationship between the data points? Has categorical data been added yet?
3.4.2: On The Road Again (Teacher Notes)
Does this map overlay help you establish a relationship? The dots represent Canada’s provincial and territorial capital cities and the scale factors are the distance (in thousands of kilometres) between the cities.
Whole Class Discussion:
What was the categorical data that was added? How did it very quickly allow you to establish a relationship between the data points? Can you name the city that each dot represents?
3.4.3: On The Road Again (Supplemental)