Probability and Statistics
Underlying Processes and Mathematical Tools
15 days: 1 – 1 ½ hours per day
4.13 Solve problems by collecting, organizing, displaying, and interpreting sets of data.
4.14 Apply Grade 4 mathematics to solve problems connected to everyday experiences and activities in and outside of school.
4.15 Communicate about Grade 4 mathematics using informal language.
4.16 Use logical reasoning.
TEKS / TAKS Obj. / Instructional Scope / Possible Resources /
Region 4 Instruction / Region 4 Assessment / KISD Suggested Resources /
4.13A
Use concrete objects or pictures to make generalizations about determining all possible combinations of a given set of data or of objects in a problem situation.
4.14D
Use tools such as real objects, manipulatives, and technology to solve problems.
4.14C
Select of develop and appropriate problem-solving plan or strategy, including drawing a picture, looking for a pattern, systematic guessing and checking, acting it out, making a table, working a simpler problem, or working backwards to solve a problem. / 5
6
6 / Make Generalizations About Determining Possible Combinations
· Determine ways to represent possible combinations, such as tree diagrams, organized lists, and concrete objects.
Example (Concrete Objects) :
Maddie has the following number tiles:
3 / 4 / 5
What are all of the 3-digit numbers that Maddie can create with these number tiles?
Prompt the students to use number tiles to represent all possible 3-digit numbers.
3 / 4 / 5
3 / 5 / 4
4 / 5 / 3
4 / 3 / 5
5 / 4 / 3
5 / 3 / 4
Answer: Maddie can create 6 different 3-digit numbers with the number tiles given.
Example (Tree Diagram):
Sabrina is planning her summer activity schedule. She can choose 2 different activities from the following list.
Ballet
Modern Dance
Swimming
Karate
Cheerleading
What are all of the possible combinations of activities that Sabrina could choose?
Prompt the students to use a tree diagram to find all possible combinations of activities that Sabrina could choose.
Possible Answer:
Answer: Since the order of the activities does not matter (ballet and swimming represents the same combination as swimming and ballet), there are
10 combinations possible. / TEXTEAMS Mathematics Institute Grades 3-5, “Number Fun.” / www.mathbenchmarks.org / TAKS Informational Booklet: http://www.tea.state.tx.us/student.assessment/taks/booklets/math/g4e.pdf
TEKS Clarifying Activities:
http://www.utdanacenter.org/mathtoolkit/instruction/activities/4.php
Math TEKS Connections Lessons
http://www.tea.state.tx.us/math/training/materials/MTC/index.htm
TMSDS
https://www.tmsds.org/
Illuminations: http://illuminations.nctm.org/
GIZMOS
http://www.explorescience.com/
Probability
Macmillan Texas Mathematics:
Chapter 3
4.13B
Interpret bar graphs. / 5 / Interpret Bar Graphs
· Use the information on a bar graph to answer questions.
Example:
Prompt the students to interpret information from the bar graph.
Ask the students, “Which topping was chosen by the greatest number of students?”
Answer: Pepperoni
Ask the students, “How many more students chose cheese than sausage as their favorite topping?”
Answer: 20 students
Example:
Prompt the students to interpret information from the bar graph.
Ask the students, “What is the difference in the number of students who chose blue as their favorite color versus those who chose red?”
Answer: 15 students
Ask the students, “Which 2 colors were chosen by a total of 110 students?”
Answer: Red and black
Ask the students, “At Wilson Elementary the number of students who chose red as their favorite color was 2 times the number who chose red at Hartman Elementary. How many students at Wilson Elementary chose the color red?”
Answer: 90 students / TAKS Mathematics Preparation Grade 4, “Graphs.” / TAKS Mathematics Preparation Grade 4, “Graphs - Evaluate.”
TAKS Mathematics Preparation Grade 4, “Objective 5 Probability and Statistics Selected Response Questions.”
www.mathbenchmarks.org
Probability and Statistics
Underlying Processes and Mathematical Tools
*Curriculum-Based Assessment 7
*For each student expectation (SE) that incorporates the use of a concrete model/object and/or a mathematical tool, students should be encouraged to use a concrete model/object and/or a mathematical tool to model items on the curriculum-based assessment (CBA).
Probability
NOTE: Before starting this topic, you may want to look at the TAKS Information booklet page 28 and 29. This explains the expectations for this TEKS at 4th grade.
Suggestions for daily problem solving:
Read It! Draw It! Solve It! DOM (Daily oral Math) Problem solving activities utilizing graphs. Pages: 10, 22, 32, 41, 76, 93, 106, 116, 126, 137, 155, 158 and 160.
In the Problem Solver, problem #5 page 5 involves making an organized list. This can be extended to the tree diagram format. If you look in the front of the book a list is available showing what strategies are involved in each activity.
Also please see Groundworks Algebraic Thinking Grade 4 pages iv-7.
Roads to Reasoning - Use for morning work to get the students back into the problem solving mode after the winter break. This books works best when taught sequentially.
ACTIVITY 1. About Teaching Mathematics: (Marilyn Burns) Section on probability: Good teacher information. The activity "How Many Throws and How Many Ways" in the new edition shows how the data can be taken to a tree diagram. Be sure to include activities from this book showing all possible outcomes because the TERC book does not include this. Use these activities as work stations since there are only 2 weeks available for probability. To fulfill the TEKS 4.13c, be sure to graph data. This is a good time to predict outcomes from experiments which will serve as an introduction to TEKS 5.12b.
ACTIVITY 2. Three Out of Four Like Spaghetti (TERC) Investigation 1:
Session 1 “Playing Guess My Rule”
Students play Guess My Rule as a way of introducing the fraction language and notation used throughout this unit – for example, “14 people out of 26, or 14/26, of the students in the class are wearing long pants.” Their work focuses on:
* dividing the class into categories
* identifying the fraction of the class in each category
Materials:
Strip of paper to make class strip
Session 2 “Finding Familiar Fractions”
Student review the fractions ¼, 1/3, ½ 2/3, and ¾ by folding strips of paper into fractional parts. They use Class Strips to illustrate data about a group of people. By folding the Class Strips into halves, quarters, or thirds, they are able to describe such fractions as 14/26 in more familiar terms – “a little more than ½.” Their work focuses on:
* reviewing the size of familiar fractions relative to one another
* understanding the size of unfamiliar fractions by identifying which familiar fractions they are close to
Materials:
Strips of adding machine tape 18 inches long
Interlocking cubes or counters
Markers or crayons, scissors, tape
Class Strips
Transparency of Class Strips
Student Sheet 1
Large fraction strip
Overhead projector
Session 3 “Comparing Data with Familiar Fractions”
Students use familiar fractions to represent data about themselves and their families and to compare themselves with the country as a whole. Their work focuses on:
* collecting and organizing data
* representing and comparing data using fractions
Materials:
Interlocking cubes or counters
Markers or crayons, scissors, tape
Class Strips
Student Sheet 2
Transparencies of Student Sheet 2
Clear container filled with two colors of objects of similar size and shape
Overhead projector
Session 4 “Using Fractions to Compare Data"
Students discuss and compare their class data and the national data using their work from the previous session. Students do word problems that require them to relate fractions to real situations - for example, “How many people are in three-quarters of the class?” Their work focuses on:
* comparing data from two different-sized groups
* using data and fractions in word problems
Materials:
Cubes and Class Strips
Student Sheet 3
Overhead projector
NOTE: This connects fractions and probability. The students will also organize and compare data in real life situations.
ACTIVITY 3. Three Out of Four Like Spaghetti (TERC) Investigation 2:
Session 1 “Games We Play”
Students think of games they like to play and find ways to categorize them. They make graphs of the data using different categorizations and describe what they can see from their graphs. Students are encouraged to use fraction statements as part of the description of their data. Their work focuses on:
* collecting and recording categorical data
* organizing data into categories
* graphing categorical data
Materials:
Data Cards
Scissors
Index cards and tape, or stick-on notes
Chart paper
Glue or glue sticks
Large paper
Student Sheet 4
Session 2 “More Games, and What Have We Eaten?”
Working as a whole class, students make graphs of their favorite games, using two different sets of categories. Then in small groups of two or three, students devise categories for and graph foods they’ve eaten in the last 24 hours. Their work focuses on:
* categorizing data
* graphing categorical data
Materials
Index cards and tape, or large stick-on notes
Scissors
Glue or glue sticks
Large paper
Data Cards (leftover from Session 1)
Student Sheet 5
Session 3 “What Do You Want to Be When You Grow Up?
Students collect data about what they would like to be when they grow up. Working in small groups, students sort ten of these responses into categories that they make up and predict whether data from the first grade will be different from their own. They make plans to collect first grade data before the next session. Their work focuses on:
* collecting data
* organizing data into categories
* thinking about possible differences in categorical variables between the two groups
Materials:
Index cards or blank pieces of paper
Scissors
Paper clips
Envelopes
Data Cards
Overhead projector
Session 4 “Organizing Some First and Fourth Grade Data”
Students make data cards with the career data from ten of the first graders. They combine these cards with the ten data cards they already categorized from the fourth grade. As they add the first grade data, students make the necessary adjustments in their categories to accommodate the new data. Then groups of pairs meet to share how they made categories and to discuss what issues came up in doing this. Their work focuses on:
* constructing categories for categorical data
* comparing different sets of categories
* communicating their system for categorizing the data to other students
* looking at the same data categorized in a few different ways
Materials:
Paper clips
Scissors
Data Cards
Blank paper for labels
Overhead projector
Sessions 5, 6, and 7
Students work with the complete set of career data from the first and fourth grades. Starting with the categories they made for the first 20 pieces of information, they categorize the remaining data. Some students will find they need to make additional categories or may want to combine categories. Students graph their data and write about their observations. They also compare the fraction of fourth graders and the fraction of first graders in some of the categories. Finally students share conclusions about the data based on their graphs. Their work focuses on:
* adjusting categories to accommodate all the data
* graphing the data
* using fractions to compare first and fourth graders' career choices
* communicating about their findings
* comparing their work with alternative approaches to classifying the data
Materials:
Making Comparisons with All the Career Data
Lists of the fourth and first grade data to post or hand out
Large paper for making data displays
Crayons or markers, glue, scissors
Index cards or stick-on notes, and tape
Cubes, counters, and class strips
Data Cards
Student Sheets 6 and 7
NOTE: This activity will include sorting into categories. This will also prepare students for the reading TAKS test where they will have to compare two pieces of information by what they have in common. The teacher note after session 2 discusses good categorization. The ten-minute math includes information on what is likely. This activity is very high level and will also be great preparation for ratio. Objective 6 from the TAKS Information Booklet shows a table as a means to organizing information. During this activity encourage the students to use a table as one of the ways to organize data in order to prepare students for the TAKS test.
Note: Graphs need to be shown horizontally and vertically. The information on the graphs and charts need to be taken to the problem solving level. Recording of the answers should be practiced as griddables.
Curriculum Cluster 7
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