Task: Test Scores - nctm. · PDF fileTask: Test Scores In Mrs. Smith’s classroom, more people sit on the left side of the room than the right side of the room. Mrs. Smith wants to

Task: Test Scores In Mrs. Smith’s classroom, more people sit on the left side of the room than the right side of the room. Mrs. Smith wants to know which side of the room is more successful on tests. The students on the left side of the room received the following scores on their 100 point test:

The students on the right received these test scores:

a) Create an appropriate graph(s) to represent the data.

b) Which side of the room do you believe is more successful? Make a statistical argument for your opinion.

c) Pretend you are a student from the side of the room you deemed “unsuccessful.” Make a statistical argument for why your side of the room is actually the best.

This task is adapted from the Illustrative Mathematics 6.SP Comparing Test Scores task.

Grade Level

Kindergarten 1st 2nd 3rd 4th 5th

Domain Measurement and Data

Measurement and Data





Cluster Standards

Classify objects and count the number of objects in each category. CCSS.MATH.CONTENT.K.MD.B.3 Classify objects into given categories; count the numbers of objects in each category and sort the categories by count.

Represent and interpret data. 1.MD.C.4 Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.

Represent and interpret data. 2.MD.D.9 Generate measurement data… Show the measurements by making a line plot, where the horizontal scale is marked off in whole-number units. 2.MD.D.10 Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put-together, take-apart, and compare problems using information presented in a bar graph.

Represent and interpret data. 3.MD.B.3 Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one- and two-step "how many more" and "how many less" problems using information presented in scaled bar graphs. 3.MD.B.4 Generate measurement data… Show the data by making a line plot, where the horizontal scale is marked off in appropriate units— whole numbers, halves, or quarters.

Represent and interpret data. 4.MD.B.4 Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Solve problems involving addition and subtraction of fractions by using information presented in line plots.

Represent and interpret data. 5.MD.B.2 Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Use operations on fractions for this grade to solve problems involving information presented in line plots..

Grade Level

6th 7th 8th

Domain Statistics & Probability Statistics & Probability Statistics & Probability

Cluster Standards

Develop understanding of statistical variability. 6.SP.A.1 Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. 6.SP.A.2 Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape. 6.SP.A.3 Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number. Summarize and describe distributions. 6.SP.B.4 Display numerical data in plots on a number line, including dot plots, histograms, and box plots. 6.SP.B.5 Summarize numerical data sets in relation to their context, such as by: A) Reporting the number of observations. B) Describing the nature of the attribute under investigation C) Giving quantitative measures of center (mean/median) and variability (interquartile range/mean absolute deviation) as well as describing any overall pattern and any striking deviations from the overall pattern D) Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered.

Use random sampling to draw inferences about a population. 7.SP.A.1 Understand that statistics can be used to gain information about a population by examining a sample of the population 7.SP.A.2 Use data from a random sample to draw inferences about a population with an unknown characteristic of interest Draw informal comparative inferences about two populations. 7.SP.B.3 Informally assess the degree of visual overlap of numerical data distributions w/ similar variabilities 7.SP.B.4 Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. Investigate chance processes and develop, use, and evaluate probability models. 7.SP.C.5 Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. 7.SP.C.6 Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. 7.SP.C.7 Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies

Investigate patterns of association in bivariate data. 8.SP.A.1 Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association. 8.SP.A.2 Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line. 8.SP.A.3 Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. 8.SP.A.4 Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables.

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7

The Spinner: Probability 1. What is the theoretical probability of spinning Blue? Yellow? Cyan? Red? Purple?

2. Spin the spinner 1 times. How does the experimental probability compare with the theoretical probability?

3. Change the number of spins to 100 and click the spin button. As the spinner continues to spin, what happens?

4. Hit Skip to End. How do the experimental and theoretical probabilities now compare?

5. What would you expect to happen as the number of spins increases?

6. Increase the Number of Spins to a very large number. Hit the Spin button and the Skip to End button. Now how do the experimental and theoretical probabilities compare?

7. Assume we spin the spinner 1,000 times. How many times would you expect it to land on red? Explain.

8. Test your hypothesis from #7. Were you correct? Explain why this might occur.

The Spinner: Statistics Below are the results from 4 groups spinning a new spinner 20 times.

Group A Group B

Group C Group D

1. Draw what you think the spinner looks like. Be as detailed as possible.

2. Justify your drawing.

CONCEPT DEVELOPMENT

Mathematics Assessment Project

CLASSROOM CHALLENGES A Formative Assessment Lesson

Analyzing

Games of Chance

Mathematics Assessment Resource Service

University of Nottingham & UC Berkeley

For more details, visit: http://map.mathshell.org © 2015 MARS, Shell Center, University of Nottingham May be reproduced, unmodified, for non-commercial purposes under the Creative Commons license detailed at http://creativecommons.org/licenses/by-nc-nd/3.0/ - all other rights reserved

Student materials Analyzing Games of Chance S-1© 2015 MARS, Shell Center, University of Nottingham

Spinner Bingo

Sally has made a Spinner Bingo game for her class.

Here are three Bingo cards the players made:

Card A Card B Card C

1. Which card has the best possible chance of winning? Give reasons for your answer.

2. Fill in a card so that it has the best chance of winning. Explain how you chose the numbers foryour card.

14 6 17

7

1

10 4

15 12

4 13 5

12 9 6

8 11 15

5

14

2

15

3

13

4

16

10

Write down 9 different numbers on your card.

I will spin both spinners and add up the two numbers I get. If you have that total on your Bingo card, you cross it off. The first person to cross off all the numbers wins the prize.

Student materials Analyzing Games of Chance S-2 © 2015 MARS, Shell Center, University of Nottingham

3. Would you be guaranteed to win playing with the card containing your chosen numbers? Explain why / why not.


The Horse Race

Fin

ish

Sta

rt

23

45

67

89

1011

12

2 3 4 5 6 7 8 9 10

11

12

1


Rules of Horse Race

Aim of the Game

Eleven horses enter a race. The first one to pass the finish line wins.

Rules

Place counters on the starting squares labeled 2 to 12.

Share out the horses so that each person in your group has three or four horses.

How to play

Roll the two dice and add the scores.

The horse with that number moves one square forward.

Keep rolling the dice.

The horse that is first past the finishing line wins.

1. Before you start the race, write down the order you predict the horses will finish in on

the Race Results sheet. Why do you predict this? Write down your reason next to your prediction in the space provided.

2. Play the race twice. Record the final positions of the horses each time. 3. Try to explain any patterns you find in your data.

• Does the outcome vary very much from race to race? • Which horses are most likely to win? Why? • Which horses are least likely to win? Why? • Could the finishing order have been predicted? • Could the winning distance have been predicted?


Race Results Sheet

Predicted finish positions

Explain why you predict this

Game 1 Actual finish positions Game 2 Actual finish positions

Explain your results. Are they different from what you expected? Why is this?

FinishStart

2 3 4 5 6 7 8 9101112

2

3

4

5

6

7

8

9

10

11

12

1

FinishStart

2 3 4 5 6 7 8 9101112

2

3

4

5

6

7

8

9

10

11

12

1

FinishStart

2 3 4 5 6 7 8 9101112

2

3

4

5

6

7

8

9

10

11

12

1


Sample Student Reasoning: Tom

1. What has Tom done well?

2. What errors has Tom made? Can you correct them?

3. Try to improve and extend Tom’s work.


Sample Student Reasoning: Jessica

1. What has Jessica done well?

2. What errors has Jessica made?

3. Try to improve and extend Jessica’s work.


Other Horse Races

Analyze one of the following races in a similar way.

First, analyze it theoretically, using what you have learned from the sample approaches.

Then play out the race using the dice and see what happens!

The Difference Race

Six horses enter a race. The first one to pass the finishing line wins.


Roll the two dice and find out the difference between the scores.


The Maximum Race

Six horses enter a race. The first one to pass the finishing line wins.


Roll the two dice and find the highest number on the two dice.


The Multiples Race

Five horses enter a race. The first one to pass the finishing line wins.

Place counters on the starting squares labeled Multiple of 2 to Multiple of 6.

Roll the two dice and multiply the numbers together.

The horses with this number as a multiple moves one square forward.


Number Game

Karl’s Math class is playing a number game.

Each student is given a number card containing the numbers 1 to 6:

The rules of the game are that each student must put a cross through two numbers on the card and hand it in to the teacher.

The teacher has a bag containing six balls numbered 1 to 6:

When all the number cards have been handed in the teacher draws out two balls from the bag.

Every student who has chosen the same two numbers shown on the balls wins a prize.

If there are 30 students in Karl’s class, how many students are likely to win a prize? Describe your reasoning.

1 2 3

4 5 6

Mathematics Assessment Project

CLASSROOM CHALLENGES

This lesson was designed and developed by the

Shell Center Team

University of Nottingham, England

Malcolm Swan, Clare Dawson, Sheila Evans,

Colin Foster and Marie Joubert

with

Hugh Burkhardt, Rita Crust, Andy Noyes, and Daniel Pead

It was refined on the basis of reports from teams of observers led by

David Foster, Mary Bouck, and Diane Schaefer

based on their observation of trials in US classrooms

along with comments from teachers and other users.

This project was conceived and directed for

MARS: Mathematics Assessment Resource Service

by

Alan Schoenfeld, Hugh Burkhardt, Daniel Pead, and Malcolm Swan

and based at the University of California, Berkeley

We are grateful to the many teachers, in the UK and the US, who trialed earlier versions

of these materials in their classrooms, to their students, and to

Judith Mills, Mat Crosier, Anne Floyde, Michael Galan, Nick Orchard, and Alvaro Villanueva who contributed to the design.

This development would not have been possible without the support of

Bill & Melinda Gates Foundation

We are particularly grateful to

Carina Wong, Melissa Chabran, and Jamie McKee

© 2015 MARS, Shell Center, University of Nottingham This material may be reproduced and distributed, without modification, for non-commercial purposes, under the Creative Commons License detailed at http://creativecommons.org/licenses/by-nc-nd/3.0/

All other rights reserved. Please contact [email protected] if this license does not meet your needs.

Documents

Task: Test Scores - nctm. · PDF fileTask: Test Scores In Mrs. Smith’s classroom, more people sit on the left side of the room than the right side of the room. Mrs. Smith wants to