defining a collaborative classroom characteristics of...

The Collaborative Classroom

Defining a Collaborative Classroom A collaborative classroom is an environment in which knowledge and authority are shared between the teacher and the students. In a collaborative classroom, teachers are facilitators and students are active participants. All students, not segregated by ability level, interest, or achievement, benefit from the environment created in the collaborative classroom. Teachers in the collaborative classroom combine their extensive knowledge about teaching and learning, content, and skills with the informal and formal knowledge, strategies, and individual experiences of their students. The collaborative classroom differs from the traditional classroom in which the teacher is seen as an information giver (Tinzmann, M.B.; Jones, B.F.; Fennimore, T.F.; Bakker, J; Fine, C.; and Pierce, J., 1990).

Characteristics of the Collaborative Classroom The collaborative classroom is identified by discussion, with in-depth accountable talk and two-way interactions, whether among members of the whole class or small groups. It is a well-structured environment in which questioning and dialogue are valued and appropriate parameters are set so that active learning can occur. Careful planning by the teacher ensures that students can work together to attain individual and collective goals and to develop learning strategies. In the collaborative classroom, students are encouraged to take responsibility for their learning through monitoring and reflective self-evaluation. The collaborative classroom is one in which teachers spend more time in true academic interactions as they guide students to search for information and help students to share what they know. As facilitators, teachers have the opportunity to provide the correct amount of help to individual students by providing appropriate hints, probing questions, feedback, and help in clarifying thinking or the use of a particular strategy.

Effective Collaboration and Teamwork Critical to teaching and learning in the collaborative environment is the ability to define the responsibilities of the teacher and students. For effective collaboration and teamwork, teachers and students must agree to certain responsibilities that support the learning process. The table shown reflects the parallel responsibilities of teachers and students.

Teacher Responsibilities Student Responsibilities

Monitor student behavior. Develop the skills to work cooperatively.

Provide assistance when needed. Learn to talk and discuss problems with each other in order to accomplish the group goal.

Answer questions only when they are group questions.

Ask for help only after each person in the group has considered the problem and the group has a question for the teacher.

Interrupt the process to reinforce cooperative skill or to provide direct instructions to all students.

Believe that all members of the group work together toward a common goal. Understand that the success or failure of the group is to be shared by all members.

Provide closure for the lesson. Reflect on the work of the group. Evaluate the group process by discussing the actions of the group members.

Appreciate that working together is a process and encourage each group member to interact with, and relate to the rest of the group members.

Help students to become individually accountable for learning and reinforce this understanding regularly.

Realize that each member must contribute as much as he or she can to the group goal. Understand that the success of the group is dependent on the individual work of each member. Understand that group members are individually accountable for their own learning.

The Learner-Centered Classroom To understand where you are in the transition process of creating a learner-centered classroom, consider the questions and behaviors shown (modified from the work of Jaclyn Snyder, Jan Sinopoli, and Vince Vernachhio, Pittsburgh Public Schools).

A Reflective Teacher Tool

Reflective Questions Behaviors in a Learner-Centered Classroom

How is the classroom arranged? The students are typically paired or grouped to work together while the teacher facilitates the process.

Who is directing the classroom activities?

The teacher facilitates classroom activity.

Most activities require only guidance from the teacher.

Where is the teacher spending his or her time?

The teacher walks around the classroom during all activities, watching and listening to student-to-student discourse.

The teacher monitors the students to keep them on task, while the students actively work together on an activity.

Who is doing the math? The students do the thinking and the work.

What type of questions is the teacher asking?

The teacher asks thought-provoking questions that require students to explain their thinking and processes.

Who is explaining and justifying answers?

Students are required to make presentations, explaining their progress within the activity.

The majority of discourse is student-to-student discourse.

When and how does the teacher provide information to the students?

The teacher encourages students to construct knowledge. Prior knowledge is assessed as new concepts emerge.

During the lesson, the teacher systematically brings the class together on several occasions, assuring that the mathematics of the lesson are understood.

When a student struggles, how does the teacher respond?

If a student is having difficulty understanding something, even after consulting with his or her group members, the teacher asks the group leading questions to guide them to the desired outcome.

31Workshop Resources

WoRkshop ResouRces • couRse 1 • couRse 2 • couRse 3

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Student work displayedDesks or tables arranged to provide teacher access to all students and students access to each otherNecessary materials and tools readily availableWord walls (optional) Warm-up or bell-ringer (optional)A structured routine for engaging in mathematical tasks, including:

A launch of the problem scenario to check for student understandingStudent work time for engagement in mathematical problems and communicating understandingsA lesson closure and summary of primary mathematical objectives from the day.

Maintaining the pace of the lesson to achieve the day’s objective(s)Initiating and facilitating student discourseWell-planned and prepared to:

Extend tasksAccommodate individual differencesConnect classroom work to lab workSummarize mathematical concepts of the lesson

Connecting class work to work in the softwareUsing effective questioning strategies to:

Access prior knowledge Clarify student thinkingAssess student understanding of the mathematicsEnsure individual accountability Connect multiple representations

Making meaningful connections to prior knowledge, real-world experiences, and tools and/or technologyDiscussing mathematical concepts with each otherWorking in groups or pairs, focusing on mathematics as they manage their own learning as well as take an active roleCommunicating effectively within a whole class, small group, partner, or individual activitiesDetermining the correctness of answersDescribing their mathematical thinking or processChoosing a variety of methods to communicate (i.e. tables, graphs, models, diagrams, oral, written, technology)Using precise mathematical vocabularyHolding each other accountable during discussionsEngaging in class work, taking full ownership of learning activities, displaying high levels of energy, and showing a willingness to ask questions and take risks

The classroom will have: Teacher will be: Students will be:

NOTES:NOTES:NOTES:

CLASSROOM LOOK-FORS

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A person who weighs 100 pounds on Earth would weigh only about 40 pounds on the planet Mercury and about 91 pounds on Venus. In fact, there are only three

planets in our solar system where a 100-pound person would weigh more than 100 pounds: Jupiter, Saturn, and Neptune. On Saturn, a 100-pound person would weigh about 106 pounds, on Neptune, about 113 pounds, and on Jupiter, about 236 pounds! On Pluto——which is no longer considered a planet—–a 100-pound person would weigh less than 7 pounds.

But what if a 100-pound person could stand on the surface of the Sun? If that were possible, then that person would weigh over 2700 pounds! More than a ton! What causes these differences in weight?

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In this lesson, you will:

• Write ratios as part-to-part and part-to-whole relationships .

• Represent ratios using models .• Use models to determine equivalent ratios .

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A Trip to the MoonUsing Tables to Represent equivalent Ratios

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A Picture Is Worth a Thousand WordsUnderstanding Quantities and Their Relationships

How interesting would a website be without pictures or illustrations? Does an inviting image on a magazine cover make you more likely to buy it? Pictures

and images aren’t just for drawing your attention, though. They also bring life to text and stories.

There is an old proverb that states that a picture is worth a thousand words. There is a lot of truth in this saying—and images have been used by humans for a long time to communicate. Just think: would you rather post a story of your adventure on a social media site, or post one picture to tell your thousand-word story in a glance?

In this lesson, you will:

• Understand quantities and their relationships with each other .

• Identify the independent and dependent quantities for a problem situation .

• Match a graph with an appropriate problem situation .• Label the independent and dependent quantities on

a graph . • Review and analyze graphs .• Describe similarities and differences among graphs .

LeARnIng goALS Key TeRMS

• dependent quantity• independent quantity

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4 Chapter 1 Quantities and Relationships

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Problem 1 What’s the Dependency?

Have you ever planned for a party? You may have purchased ice, gone grocery shopping, selected music, made food, or even cleaned in preparation . Many times, these tasks depend on another task being done first . For instance, you wouldn’t make food before grocery shopping, now would you?

Let’s consider the relationship between:

• the number of hours worked and the money earned .

• your grade on a test and the number of hours you studied .

• the number of people working on a particular job and the time it takes to complete a job .

• the number of games played and the number of points scored .

• the speed of a car and how far the driver pushes down on the gas pedal .

There are two quantities that are changing in each situation . When one quantity depends on another in a problem situation, it is said to be the dependent quantity . The quantity that the dependent quantity depends upon is called the independent quantity .

1. Circle the independent quantity and underline the dependent quantity in each statement .

2. Describe how you can determine which quantity is the independent quantity and which quantity is the dependent quantity in any problem situation .

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1.1 Understanding Quantities and Their Relationships 5

3. Read each scenario and then determine the independent and dependent quantities . Be sure to include the appropriate units of measure for each quantity .

Something’s Fishy

Candice is a building manager for the Crowley Enterprise office building . One of her responsibilities is cleaning the office building’s 200-gallon aquarium . For cleaning, she must remove the fish from the aquarium and drain the water . The water drains at a constant rate of 10 gallons per minute .

• independent quantity:

• dependent quantity:

Smart Phone, but Is It a Smart Deal?

You have had your eye on an upgraded smart phone . However, you currently do not have the money to purchase it . Your cousin will provide the funding, as long as you pay him interest . He tells you that you only need to pay $1 in interest initially, and then the interest will double each week after that . You consider his offer and wonder: is this really a good deal?



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1Can’t Wait to Hit the Slopes!

Andrew loves skiing—he just hates the ski lift ride back up to the top of the hill . For some reason the ski lift has been acting up today . His last trip started fine . The ski lift traveled up the mountain at a steady rate of about 83 feet per minute . Then all of a sudden it stopped and Andrew sat there waiting for 10 minutes! Finally, the ski lift began to ascend up the mountain to the top .



It’s Magic

The Amazing Aloysius is practicing one of his tricks . As part of this trick, he cuts a rope into many pieces and then magically puts the pieces of rope back together . He begins the trick with a 20-foot rope and then cuts it in half . He then takes one of the halves and cuts that piece in half . He repeats this process until he is left with a piece so small he can no longer cut it . He wants to know how many total cuts he can make and the length of each remaining piece of rope after the total number of cuts .



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Baton Twirling

Jill is a drum major for the Altadena High School marching band . She has been practicing for the band’s halftime performance . For the finale, Jill tosses her baton in the air so that it reaches a maximum height of 22 feet . This gives her 2 seconds to twirl around twice and catch the baton when it comes back down .



Music Club

Jermaine loves music . He can lip sync almost any song at a moment’s notice . He joined Songs When I Want Them, an online music store . By becoming a member, Jermaine can purchase just about any song he wants . Jermaine pays $1 per song .



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1A Trip to School

On Monday morning, Myra began her 1 .3-mile walk to school . After a few minutes of walking, she walked right into a spider’s web—and Myra hates spiders! She began running until she ran into her friend Tanisha . She stopped and told Tanisha of her adventurous morning and the icky spider’s web! Then they walked the rest of the way to school .



Jelly Bean Challenge

Mr . Wright judges the annual Jelly Bean Challenge at the summer fair . Every year, he encourages the citizens in his town to guess the number of jelly beans in a jar . He keeps a record of everyone’s guesses and the number of jelly beans that each person’s guess was off by .



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Problem 2 Matching graphs and Scenarios

While a person can describe the monthly cost to operate a business, or talk about a marathon pace a runner ran to break a world record, graphs on a coordinate plane enable people to see the data . Graphs relay information about data in a visual way . If you read almost any newspaper, especially in the business section, you will probably encounter graphs .

Points on a coordinate plane that are or are not connected with a line or smooth curve model, or represent, a relationship in a problem situation . In some problem situations, all the points on the coordinate plane will make sense . In other problem situations, not all the points will make sense . So, when you model a relationship on a coordinate plane, it is up to you to consider the situation and interpret the meaning of the data values shown .

1. Cut out each graph on the following pages . Then, analyze each graph, match it to a scenario, and tape it next to the scenario it matches . For each graph, label the x- and y-axes with the appropriate quantity and unit of measure . Then, write the title of the problem situation on each graph .

What strategies

will you use to match each graph with one of the

eight scenarios?

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x

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Graph B

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Graph C

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Graph G

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Problem 3 oh, Say, Can you See (in the graphs)!

Now that you have matched a graph with the appropriate problem situation, let’s go back and examine all the graphs .

1. What similarities do you notice in the graphs?

2. What differences do you notice in the graphs?

3. How did you label the independent and dependent quantities in each graph?

4. Analyze each graph from left to right . Describe any graphical characteristics you notice .

Look closely when analyzing

the graphs. What do you see?

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1 5. Compare the graphs for each scenario given and describe

any similarities and differences you notice .

a. Smart Phone, but Is It a Smart Deal? and Music Club

b. Something’s Fishy and It’s Magic

c. Baton Twirling and Jelly Bean Challenge

6. Consider the scenario A Trip to School .

a. Write a scenario and sketch a graph to describe a possible trip on a different day .

Scenario Graph

b. Compare your scenario and sketch with your classmates’ scenarios and sketches . What similarities do you notice? What differences do you notice?

Be prepared to share your solutions and methods .

Think about all the different

graphical characteristics you just identified.

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