Probability and Statistics (Grades 3-5) Workshop DAY 1 Dr. Leah Shilling-Traina

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Probability and Statistics (Grades 3-5) Workshop DAY 1 Dr. Leah Shilling-Traina

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Probability and Statistics (Grades 3-5) Workshop DAY 1 Dr. Leah Shilling-Traina

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Probability and Statistics (Grades 3-5) WorkshopDAY 1Dr. Leah Shilling-Traina

- Complete 3 X 5 notecard:
- Name
- Where do you teach?
- Number of years teaching & grade levels
- Favorite mathematics topic
- Why are you here?

- Introductions – Introduce another person in our class to everyone!

- Navigating through Data Analysis and Probability in grades 3-5 by Chapin, Koziol, MacPherson, Rezba (ISBN 978-0-87353-521-2) published by NCTM
- Exploring Statistics in Elementary Grades Book 1 by Bereska, Bolster, Bolster, Schaffer (ISBN 1-57232-344-2) published by Dale Seymore in 1998

- Math By All Means by Marilyn Burns (ISBN 0-941355-12-8)
- Mathematics for Elementary Teachers Activity Manual by Sybilla Beckman (ISBN 978-0-321-64696-5) published by Addison-Wesley

- Probability and Statistics for Elementary and Middle School Teachers: A Staff Development Training to Implement the 2001 Virginia Standards of Learning (PSEMT) (http://www.doe.virginia.gov/testing/sol/standards_docs/mathematics/2001/resources/elementary/probability_module/mprobstatentire.pdf)

- 2009 Mathematics Standards of Learning (SOL) (http://www.doe.virginia.gov/testing/sol/standards_docs/mathematics/2009/stds_math.pdf)
- Mathematics Curriculum Framework for grades 3 - 5

- Set of blank dice
- Overhead dice
- Overhead color tiles
- Overhead circle spinners

- The packet of print-out materials include:
- Documents and materials related to the NCTM Standards and VA SOLs
- VDOE’s Mathematics Word Wall Vocabulary Cards for Grades 3-5 Probability and Statistics (all cards can be found at http://www.doe.virginia.gov/instruction/mathematics/resources/vocab_cards/index.shtml)

- A copy of activities from Exploring Statistics in the Elementary Grades by Bereska, Bolster, Bolster, and Schaeffer (ISBN 1-57232-344-2)
- Some additional lesson plans and handouts
- Copies of overheads/handouts for all activities in the NCTM and Burns texts

- Documents and materials related to the NCTM Standards and VA SOLs
- You should also take notes in your binder!

- Remember, you are not be graded during this workshop!
- Please answer the questions to the best of your ability.
- At the end of our third day together, you will take a post-workshop assessment to see how this workshop has impacted your knowledge of Grades 3-5 probability and statistics!

Standards in mathematics education represent the goals we set for our students. They are value judgments about what we would like our students to know and be able to do.

They are chosen through a complex process that is fed by societal expectations, past practice, research information, and visions of the professionals in the field… They represent our priorities.

--- Hiebert, J. (2003). What Research Says About the NCTM Standards

- In small groups, review the VA probability and Statistics Stands for grades 3-5. Classify each as addressing probability OR statistics.
- Why did you make your choices? In your own words, what is probability? What is statistics? How do you distinguish them in your mind?

- 3.17 The student will
a) collect and organize data, using observations, measurements, surveys, or experiments;

b) construct a line plot, a picture graph, or a bar graph to represent the data; and

c) read and interpret the data represented in line plots, bar graphs, and picture graphs and write a sentence analyzing the data.

- 4.14 The student will collect, organize, display, and interpret data from a variety of graphs.
- 5.15 The student, given a problem situation, will collect, organize, and interpret data in a variety of forms, using stem-and-leaf plots and line graphs.
- 5.16 The student will
a) describe mean, median, and mode as measures of center;

b) describe mean as fair share;

c) find the mean, median, mode, and range of a set of data; and

d) describe the range of a set of data as a measure of variation.

3.18 The student will investigate and describe the concept of probability as chance and list possible results of a given situation.

4.13 The student will

a) predict the likelihood of an outcome of a simple event; and

b) represent probability as a number between 0 and 1, inclusive.

5.14 The student will make predictions and determine the probability of an outcome by constructing a sample space.

- The NCTM Standards tell us what the government is expecting us to do in the classroom—how do the SOL compare?
- What aren’t we doing that we SHOLD be doing? Where should we be going?
- Together, can we list the NCTM’s 5 Process Standards? How do they relate to probability and statistics?

- Although our focus is on the probability and statistics learned in Grades 3-5, as teachers it is important to know what has come before and what will come after!
- Let’s look at these expectations together—PSEMT, pages 15-18.

Focusing on the big ideas also means that teachers use strategies for advancing all students’ mathematical thinking (Fraivillig, 2001) by:

- eliciting from students a variety of solution methods through appropriate prompts, collaborative learning, and a positive, supportive classroom environment;
- helping students develop conceptual understanding by attending to relationships among concepts;
- extending students’ mathematical thinking by (a) encouraging them to try alternative ways of finding solutions and to generalize, and (b) setting high standards of mathematical performance for all students.

- Probability and statistics are NOT THE SAME…but they are closely related and each depends on the other in a number of different ways. They have been traditionally studied together (stochastics) and justifiably so.
- The relationship cuts both ways – statistical analyses makes use of probability and probability calculations makes use of statistical analyses.

- Break into small groups of 3 or 4; each group should get one piece of chart paper and a marker.
- Come up with a list of words and phrases that you (and potentially, your students) associate with the term statistics.
- When you’re done, hang them in front of the room? What commonalities do we see? Can we agree on some of the BIG IDEAS of statistics?

- A problem-solving process that has 4 major components:
- Ask a question
- Collect the appropriate data
- Analyze the data.
- Interpret the results.

- In short, using data to answer questions!

- GAISE offers as a central part of their framework 4 components of statistical problem-solving that include:
1. formulating questions

2. collecting data

3. analyzing data

4. interpreting results

*Guidelines for Assessment and Instruction in Statistics Education for Pre K-12 Education

- In statistics, it all begins with THE QUESTION! It’s always where the process begins. Think of a question that you would like to know the answer to! Write it on your post-it!
- Let’s think about the purposes/reasons for conducting an investigation, as well practice formulating questions that have meaning related to those purposes.

- To describe or summarize what was learned from a set of data
- To determine preferences or opinions from a set of data
- To compare and contrast two or more sets of data
- To generalize or make predictions from a set of data

- In attempts to answer this question, let’s break into small groups. Use 3 different measuring tools to collect your data: (1) a ruler, (2) a shoe, and (3) an arm span (stretch arms out like a “T” and it’s the length from fingertip to fingertip).
- Once collected, add your data to the whole class data at the front of the room.
- Which tool was best? Why? What happened with the other tools?
- What is the lesson here?

- Question: What TV shows do Americans watch the most?
- In most circumstances, collecting information from every member of a population is impossible. Therefore, we collect data from a sample of the population and use the sample to make inferences about the population.
- Samples can be very accurate in describing the population characteristics. However, for samples to be accurate, they must represent the population.
- If the sample is not representative of the entire population, the sample is considered biased because it does not accurately reflect the population being studied.

- It’s summertime, and there is really only one snack to help beat the heat—ice cream! Before you start dreaming about your favorite flavor, let’s do an activity together. As I pass out some papers, read over the description. Don’t turn over your paper until I say to!
- Let’s answer some questions!
- Now…what is YOUR favorite flavor?? Let’s collect data, and then think of some ways to represent it!

- I want to know the eye colors of the students in the classroom-how can we figure this out?
- Now, suppose you can’t use any written communication—how can we organize our data?
- My curiosity is unbounded! Now I want to know during what month you were born! Let’s get data from the whole class and make some graphs and observations!
- What do these questions have in common?

- The past two activities have used categorical data-values that correspond to a particular category or label. Can we think of other questions that would require the use of categorical data to answer?
- Graphical representations of categorical data include:
- Pictograph-- uses a picture or symbol to represent an object. If there is more than one of the objects, multiple representations of the symbol are used. A key should be included that states the value of the symbol.
- Bar graph -- uses parallel horizontal or vertical bars to represent counts for several categories. One bar is used for each category, with the length of the bar representing the count for that category
- Circle graph -- shows the relationship of the parts to the whole

- Together, let’s interpret these graphs representing categorical data.
- For each graph we discuss, let’s also identify the graph type!

- I’m still curious about you all! You can tell a lot about a person from their shoes…everyone take off your right shoe and put it on the table. In small groups, first come up with some questions we could ask about the shoes worn by people in the class.
- Which questions will we answer using categorical data? What graphs could you use to organize this data?
- What kind of data is needed to answer the other questions?
- Let’s look at some of these other kinds of questions together….

- The last activity elicited some questions that used numerical data- data that consist of numerical measures or counts. Can we think of other questions that would require the use of numerical data to answer?
- Graphical representations of numerical data include:
- Line plot-- uses stacked x ’s to show the distribution of values in a data set.
- Line graph-- uses a line to show how data changes over time.
- Stem-and-leaf plot-- a table showing the distribution of values in a data set by splitting each value into a “stem” and a “leaf”; this is a useful way to display data that range over several tens (or hundreds).

- Is it because we just don’t like it as much as the other graphs?
- VDOE’s Enhanced Scope and Sequence Sample Lesson Plans: http://www.doe.virginia.gov/testing/sol/standards_docs/mathematics/index.shtml
- Hot or Cold Lesson and www.weather.com!

- Let’s try to determine the typical length of shoe worn by people in this class! There are a couple ways we could do this…suggestions?
- For now, let’s all just measure our shoe. What kind of questions do we need to consider first?
- Put your name and measurement on a post-it. How can we organize and display this data?
- How would this activity be different if we asked about the typical shoe size of students in the class?

- Look at the “Can We Count on You” Survey—what do you notice?
- Would you like to add any questions?
- Break into small groups; each of you are responsible for answering 1-2 of these survey questions!
- Display your data in a way that makes sense based upon your data type to share with the class.

- For the last 30 minutes (or so) of our day, let’s break up into small groups according to the grade we teach! Where are the 3rd grade teachers? 4th grade? 5th grade?
- Reflect on the different activities in which you engaged today. Which activities could/would you do with your students? How would you implement those activities? Be sure to refer to all of the documents (i.e. the Curriculum Framework)!

Probability and Statistics (Grades 3-5) WorkshopDAY 2Dr. Leah Shilling-Traina

- How are you feeling today? What kind of graph could we use to represent our data if we asked for responses to be “happy,” “sad,” “angry,” or “indifferent”? What kind of data is this?
- Any questions/comments about what we’ve done so far?
- Let’s begin where we left off…

- Think about possible responses to the following questions and decide which responses will provide categorical data and which will provide numerical data:
- How many pets do students in our class have?
- How many hours a week do we spend watching TV?
- What is the typical monthly rainfall in Seattle?
- What kind of music do we like best?
- How many hours a week do we talk on the phone?
- What kinds of snacks do we like?
- How many kids eat in the cafeteria each day?
- How much candy do we eat each week?

- What kind(s) of graphs would you use to display each data set?

- Look at the three graphs and conclusions. In your group, discuss whether the conclusion is accurate and what factors about the graph may have lead to inaccurate conclusions.
- Be prepared to discuss your thoughts with the whole class!

- We answered questions like “How many colors are you wearing?” , “What is the length of your foot?” and “How tall are you?”and looked at how to display the data for the entire class.
- What other questions might we ask about the different colors we are all wearing, or the length of our feet, or how tall we are? What else might be good to know?

- There are three measures of center:
- Mean-- the sum of the values in a data set divided by the number of values; also known as the average. The mean can also be interpreted as fair-share.
- Median-- the middle value in equal distribution a data set when the values are arranged in order; if there is no “middle” value, the average of the two middle values.
- Mode-- the value(s) that occur most often in a data set.

- Other important terms
- Range-- the difference between the least and greatest numbers in a data set.
- Outliers-- a value widely separated from the others in a data set.

- Take one unifix cube for each letter of your first name and make a vertical stack. Also, write your number on a post-it.
- What kind of graph(s) can we make with our data? Let’s use our post-its AND ourselves to graph this data!
- What is the range of our data? Are there any outliers? What is the median? Mode? Mean?

- Let’s explore more the idea of mean as fair-share—everyone look to the front at the balance! Let’s think through some questions.
- Cut out your own scale and take some paperclips. Let’s balance!
- In pairs, discuss why mean is called a center of measure, and why it can be thought of as fair-share (reflecting on this activity)!

- Think of everything we’ve done so far…can we go back and describe the mean, median, and mode of our data?
- Ice Cream Preferences
- Eye Color and Birth Month
- Length of Foot

- What if we have categorical data? Can you find all three measures of center?
- What if we have numerical data?

- Mean works well for sets of data with no very high or very low numbers (no outliers).
- Median is a good choice when data sets have a couple of values much higher or much lower than most of the others (a data set with outliers).
- Mode is a good descriptor in two situations:
- When the set of data has some identical values.
- When using categorical data.

- Maddie scored 7, 15, 5, 6, 3, 15, and 9 points in seven basketball games. She wants to show that she is a valuable player.
- Mr. and Mrs. Rodriquez collected donations of $50, $125, $10, $50, and $175 for charity. They want to show that they are good fundraisers.
- Andy's last 5 phone calls lasted 20 min., 7min., 9 min., 12min., and 7min. He wants to show how long a typical phone call is.

- How much time do you think teachers typically spend on school work outside of school? More or less time than you do? Make a prediction, then let’s find out!
- What kinds of things do we need to consider to answer this question? Who do we ask? What unit of time do we use? Do we round?
- Using the data collected with our survey, create a stem-and-leaf plot to represent the data.
- Let’s analyze and interpret our findings! What do you see? What is the range for the number of hours spent working outside of class? What is the mean, median, and mode? Are there any outliers?

- Up until this point, we have been focusing primarily on statistical concepts…but we can’t ignore the other half of the title for the workshop!
- How are statistics and probability connected? How are they different? These are questions for us to ponder as we engage in the upcoming activities!

- Break into small groups of 3 or 4; each group should get one piece of chart paper and a marker.
- As we did with statistics, come up with a list of words and phrases that you (and potentially, your students) associate with the term probability.
- When you’re done, hang them in front of the room? What commonalities do we see? Can we agree on some BIG IDEAS?

- Likely
- Unlikely
- Chance
- Probable
- Possible
- Impossible

- Counting
- Predictions
- Dice, cards
- Weather, lottery
- Experiments
- Fairness

- GAMES!

- Although there is often an omission from the curriculum, probability has great relevance to society/life!
- A major advantage to teaching probability: children intuitively have some understanding of the concept from their real-life experiences!!
- Ex.Weather reports (i.e. what is the chance of rain today?)

*

- As teachers, we can build on our students’ past experiences by engaging them in games, activities, and hands-on investigations that allow them to predict outcomes and test predictions.
- The aim should be to provide investigations that engage the student to think and reason mathematically, and to have them ENJOY doing it!

- Experimental probability is a probability based on the statistical results of an experiment.
- The larger the number of trials, the better the estimate will be. The results for a small number of trials may be quite different than those experienced in the long run.
- This is a focus in Grades 3-5.

- Theoretical probability is the ratio of the number of ways an event can happen to the total number of possible outcomes.
- What is the probability of getting a heads when flipping a coin? Of rolling a 2 with a regular die?
- This is not a major focus in Grades 3-5, but we can start to think about it!

- A problem-solving process that has 4 major components:
- Ask a question
- Collect the appropriate data
- Analyze the data.
- Interpret the results.

- How does probability relate to this process of statistics?

- Although our focus is on the probability and statistics learned in Grades 3-5, as teachers it is important to know what has come before and what will come after! WHY??
- Let’s look at these expectations together—PSEMT pages 15-18.

- Some events are more likely than others, while some events are equally likely.
- It’s possible to measure the likelihood of events.
- A sample set of data can be useful for making a prediction about an outcome.
- Larger sample sizes of data give more reliable information than do smaller sample sizes.
- Sometimes an experiment produces data that do not exactly match a theoretical probability.

- Together, let’s take some time to familiarize ourselves with the text and its organization.
- Whole-class activities (WC)
- Small-group activities (SG)
- From the Classroom
- Assessment

- So when describing what we think probability is, many of us described the chance of various events occurring, or the likelihood of the events.
- Do we know what we mean when we talk about events that are certain? Or impossible? Or likely? Or unlikely? Can we think of some examples of each?
- Let’s look at the list of events—where would you place each of these activities on the scale at the bottom?
- Looking at this scale, can we describe what “0” means in terms of the probability? What does “1” mean? Can we determine some other numerical values?

- Can you give an example of an event that would have probability 0? Probability 1?
- Major take-aways:
- Sometimes people use numbers instead of words to describe the chance of an event occurring; this number is between 0 and 1.
- 1 means the same thing as 100% on the line, and 0 means the same thing as 0%.
- “Impossible” events have a probability of 0 and “Certain” events have a probability of 1!

- Now that we have a general idea about probability, let’s start playing some games!
- Let’s make some spinners—let me show you how. Also make sure you have a Spinner Recording Sheet!
- Which number do you think has the best chance (in other words, is more likely) of reaching the top first? Why?
- Do 4 spins. What did you find? Have we done enough experiments (i.e. collected enough data) to answer our question? Form groups of 4 and compare your data—what did you find?
- Do our results support our prediction? How did your results compare to the whole class data?

- What Big Ideas have we addressed?
- Together, let’s discuss the “From the Classroom” section.

- Larger samples are more reliable than smaller samples for making predictions!
- If not already there, let’s include the terms data and sample on our Probability List

- In this case, we found that one event was more likely than others—can we look at the spinner and reason why this was the case? Can you design a different spinner that would yield an event that is more likely than others?
- Let’s look at what happened in Burns’ classroom—p. 28.

- Our last activity looked at how we could use spinners to explore the probability of events. Can we create some spinner faces that would correspond to “impossible,” “very unlikely,” “equally likely,” “very unlikely,” and “certain” on the probability line?

- For the last 30 minutes (or so) of our day, let’s break up into small groups according to the grade we teach! Where are the 3rd grade teachers? 4th grade? 5th grade?
- Reflect on the different activities in which you engaged today. Which activities could/would you do with your students? How would you implement those activities? Be sure to refer to all of the documents (i.e. the Curriculum Framework)!

- Counterintuitive results in probability are found even at very elementary levels. For example, the fact that having obtained a run of four consecutive heads when tossing a coin does not affect the probability that the following coin will result in heads is counterintuitive. (More about coin flipping tomorrow!)
- In arithmetic or geometry, elementary operations (like addition) can be reversed and this reversibility can be represented with concrete materials. In the case of random experiment, we obtain different results each time the experiment is carried out and the experiment cannot be reversed (we cannot get the first result again when repeating the experiment). This makes the learning of probability comparatively harder for children.

Probability and Statistics (Grades 3-5) WorkshopDAY 3Dr. Leah Shilling-Traina

- What is the probability that you are excited that today is the last day of the workshop??
- Any questions/comments about what we’ve done so far?
- Let’s quickly discuss where we left off…

- Break into pairs and be sure you have a pair of dice! What do we observe about these dice?
- Let’s play Pig—here are the rules. Let me demonstrate with some help. Play a round in your pair!
- Now, I wonder…what do you think the chances are of rolling a 1 with one die?
- Do at least 5 experiments each and record your data on our class chart.

- This relates to the Game of Pig activity we’ve just done!
- If time, let’s play!

- Here I have a bag 12 colored tiles in it, some red and some yellow. Can you tell me what combination might be in here? Can we be sure we found them all?
- Let’s gather some data…here’s what we’re going to do.
- Write your predictions!
- Okay, what if I told you that now this bag has 12 tiles, but they are red, yellow, and blue? Let’s do the activity again!

- The last activity used what is called sampling with replacement– when you take out a sample, record it, then return it to the population.
- How would the activity have been different if replacement was not used?

- This activity provides further experience in making predictions based on a sampling of information.
- Read through on your own; use if it seems that your students could use more experience!

- What do we notice about the outcomes of the games we’ve played so far?
- Let’s return to playing with a different spinner—compare this new one to our old one. How are they the same? How are they different?
- Let’s once again record the outcomes of our spin experiments. Make a prediction—which number do you think will be landed on most frequently?
- Do 4 experiments and post your data at the front of the class.
- How do our results compare to our first spinner activity? To our prediction? Does a larger sample size really make a difference?

- Let’s extend what we’ve done with our other spinner activities to think about how different spinner faces can affect outcomes!
- I’ve given you a sheet of paper with 6 blank circles on it. Turn your attention to the “Spinner Puzzles” overhead. Let’s read through this together, and I’ll show you an example.
- Now that you have the hang of it, try it! Be sure to attend to #2! Hold on to your sheet until everyone is done, then we’ll exchange.

- Have you ever heard of a “lucky coin”? What does it mean for it to be “lucky”?
- We can think of these coins as being “fair” or “unfair.” Now, we are going to toss some coins (our penny, nickel, dime, and quarter) to check their “fairness.”
- What do you expect to happen?
- Toss each of your four coins 10 times each and record the number of heads (H) and tails (T) for each coin. Once you’ve done your tosses, come add your data to the table in front of the class.
- What do we notice? Is there anything surprising in the results? How does one group’s probabilities relate to the whole group’s probabilities (look at ratios)?

- All US coins ARE fair coins!
- Did size or weight impact the probability of tossing a heads (or tails)?
- Why might somebody consider a coin to be “lucky”?
- If you toss a dime 200 times, how many heads would you expect? 500 times?
- If we toss a quarter 10 times, will we get 5 heads and 5 tails?

- If I roll two dice, what are all the possible sums we can get? Let’s list them!
- In pairs, let’s play the Sum Game: Student 1 gets a point if a sum of 2, 3, 4, 5, or 6 is rolled. Student 2 gets a point if a sum of 7, 8, 9, 10, or 11 is rolled. No points for a sum of 12. Roll the dice 30 times and keep track of the points for each of you. Do you think the game is fair? Why or why not?
- Can we create a line plot to help us understand this?

- Your friend wants to use a pair of dice to determine which of you gets the last cookie in the bag. Which of the following situations will ensure that the selection is made fairly? Explain your thinking.
- You both roll 1 die. If the sum of the two dice is even, he wins the cookie. If the sum is odd, you win the cookie.
- You both roll 1 die. If the sum of the two dice is prime, he wins the cookie. If the sum is composite, you win the cookie.
- You both roll 1 die. If the product of the two dice is even, he wins the cookie. If the product is odd, you win the cookie.

- Review the lesson “Spin City” (NCTM, p. 73) in your small group and discuss.
- Now, turn to p. 116. Take some time to independently try to match each line plot with a spinner.
- Now, in small groups, compare your answers and explain how you know they match.

- In our probability experiments, we’ve used spinners, dice, and coins.
- Do these tools result in categorical data or numerical data? What kinds of graphs could we use to display our data?
- We want to MAKE CONNECTIONS as often as possible!

- Now that we’ve covered many of the big ideas in probability, let’s look at some situations that address common student misconceptions.
- Taken from Beckmann textbook!
- In small groups, reflect on your experiences thus far to answer the questions in Activity 16B.

- Assessment, while likely not your favorite part of teaching, is a necessary part of instruction.
- In her text, Burns provides 7 different assessments that are placed near the activities from which they evolved (activities with which we’ve ourselves engaged!)
- Let’s take a look at a couple of these together and discuss.

- Equally Likely, p. 83
- Spinner Statements, p. 136
- Probability Vocabulary, p. 197
- Favorite Activity, p. 200

- Think about all of the activities we’ve done during this workshop (feel free to flip back through your notes to help you). What was your favorite activity in the probability (or statistics) unit? Why did you like it so much?
- What did you learn from these activities?
- How will you use these activities with your own students?
- In general, how do you feel about teaching probability and statistics to your students?

- For the last 30 minutes (or so) of our workshop, let’s break up into small groups according to the grade we teach! Where are the 3rd grade teachers? 4th grade? 5th grade?
- Reflect on the different activities in which you engaged today (and over the course of the workshop). Which activities could/would you do with your students? How would you implement those activities? Be sure to refer to all of the documents (i.e. the Curriculum Framework)!
- Share general thoughts/questions!

- How quickly three days pass when you’re having fun! Remember, you are not be graded during this workshop!
- Please answer the questions to the best of your ability so that we can see how this workshop has impacted your knowledge of Grades 3-5 probability and statistics!

- Visit the VDOE website to explore all of the great resources available (i.e. Word Wall Vocabulary Cards, ESS Sample Lesson Plans)
- Also check out the NCTM Illuminations website for great lesson ideas
- Probability and statistics are FUN topics to teach and learn!

Thank you for coming to this workshop and sharing with us your thoughts, ideas, and experiences—but moreover, THANK YOU for everything that you do in the classroom each and every day!

And remember… "Good teaching is more a giving of right questions than a giving of right answers." - Josef Albers