Appropriate Use of Technology

Category 2: Lessons 6-8 NCTM

Here is the link for the numbers and operations lesson I chose: http://illuminations.nctm.org/LessonDetail.aspx?id=L252. The lesson title is "Too Big or Too Small?"

Learning Objectives

Students will:
develop intuition about number relationships
estimate computational results
develop skills in using appropriate technology

Materials

One thousand or more fake dollar bills (play money or rectangular sheets of paper the approximate size of a dollar bill) --> Will have to make if one does not have fake bills.
Scissors
One copy of Circle Template (on colored cardstock) for each student --> Can be downloaded from the above link.
Calculators
Decimal Maze Activity Sheet --> Can be downloaded from the above link.

The teacher definitely used NCTM standards in the objective.

There are 3 activities within the lesson: Activity 1: Exploring The Size of a Million Dollars

This activity explores whether one million dollars will fit into a standard suitcase. If so, how large would the suitcase need to be? How heavy would it be? You may have students work in small groups (2 or 3 students per group) to explore these questions.


Begin the investigation by telling the following story:

Just as you decide to go to bed one night, the phone rings and a friend offers you a chance to be a millionaire. He tells you he won $2 million in a contest. The money was sent to him in two suitcases, each containing $1 million in one-dollar bills. He will give you one suitcase of money if your mom or dad will drive him to the airport to pick it up. Could your friend be telling you the truth? Can he make you a millionaire?
Involve students in formulating and exploring questions to investigate the truth of this claim. For example:

Can $1,000,000 in one-dollar bills fit in a standard-sized suitcase? If not, what is the smallest denomination of bills you could use to fit the money in a suitcase?
Could you lift the suitcase if it contained $1,000,000 in one-dollar bills? Estimate its weight.
Calculators should be available to facilitate and expedite the computation for analysis.

Note: The dimensions of a one-dollar bill are approximately 6 inches by 2.5 inches. Twenty one‑dollar bills weigh approximately 0.7 ounces.

You may wish for students to locate these facts about dollar bills on their own, using internet or other appropriate resources. The students will also need to determine the dimensions of a "standard" suitcase.



Activity 2: Estimating Fractions Between 0 and 1

The model suggested here is easy to make and will help you evaluate your students' understanding of fractions between 0 and 1. Encourage students to make estimates using familiar benchmarks (e.g., ½, ¼, ¾).

Copy the Circle Template (download it) onto light-colored cardstock.


Give each student a copy and ask them to cut out the circles and make a cut in the radius of each.

Have students put the circles together so that they can see the fractions printed on one side of one circle. Ask questions such as these:

Show a small part of the shaded circle (less than ¼). Can you name the part represented?
Show a large part of the shaded circle (greater than ¾). Can you name the part represented?
Ask students to reverse the circle with the printed fractions so that they cannot see the fractions. Ask students if they can:

Show a fraction that is a little bigger than ½. What name can you give it?
Show a fraction that is between ½ and ¾. What name can you give it?
Continue asking questions that allow students to show their understanding of the fractions represented.

Other fraction models should also be used to evaluate students' understanding of fractions.


Activity 3: Exploring The Effect of Operations on Decimals

This activity provides an opportunity for students to explore the effect of addition, subtraction, multiplication, and division on decimal numbers.

Write the problem (as described next) on the chalkboard or overhead. Ask students to discuss what they notice. Lead a discussion that focuses on these key points:

In computing the product of 4.5 and 1.2, a student carefully lined up the decimals and then multiplied, bringing the decimal point straight down and reporting a product of 54.0.


Reflection on the answer should have caused the student to realize the product was too big. Multiplying 4.5 by a number slightly greater than 1 produces an answer a little more than 4.5. Instead, this student applied an incorrect procedure (line up the decimals in the factors and bring the decimal point straight down) and did not reflect on whether the resulting answer was reasonable.

Tell students that they will be playing a game to practice decimal operations and their effects. Encourage students to trace several paths through the maze while always looking for the path that will yield the greatest increase in the calculator's display.

Give each student a calculator and a copy of the Maze Playing Board activity sheet.
Maze Playing Board Activity Sheet --> download!
Students are to choose a path through the maze. To begin, have the students enter 100 on their calculator. For each segment chosen on the maze, the students should key in the assigned operation and number. The goal is to choose a path that results in the largest value at the finish of the maze. Students may not retrace a path or move upward in the maze.

In pairs or in groups of three, students should discuss their strategies (after playing the game) and what worked best for them.

Students should be able to achieve a score in the thousands. The path highlighted below gives a result of roughly 6332.


Possible follow-up activities include finding the path that leads to the smallest finish number or finding a path that leads to a finish number as near the start number (100) as possible.


I think the first activity where students explore the size of a million dollars is the best in regards to critical thinking. People are always drawn by the topic of money and I think because of that students are more willing to invest time in figuring out the problem. This activity helps with the 8.NS.1 standard. It addresses the need to understand that informally that every number has a decimal expansion. The dimension of the bills is a great example of this (The dimensions of a one-dollar bill are approximately 6 inches by 2.5 inches. Twenty one‑dollar bills weigh approximately 0.7 ounce). The other activities in the lesson does address the standards but seem to be more "math activities" and does not use more of a real life scenario like the first.

Questions to consider when the activity is in progress are: Are the students were engaged in the activity? Was the activity more of a fun activity without meaning or was it properly helping develop the understanding of mathematical concepts. Did the students meet the objectives of the lesson? If not, how can I change the lesson for the better?


If you were were to teach this lesson, I probably would not do all three activities. I would use probably the first activity/scenario where a friend calls about the million dollars. I would use that as a sort of group warm-up to get the math juices flowing and then get into another lesson. I think warm-ups can serve as review in math and also allows students to practice what they have learned. I think this could be a great warm-up.

I think the activity uses a very constructivist approach. The students are not being lectured to directly. They are suppose to critically think on their own to figure out the answers. I think this sometimes is very helpful in math where people usually associate teaching math with direct instruction. Students in these activities are constructing their own understanding of money and how it relates to math. I think the activity has some great aspects to it!

About Me

My name is Maria David. This blog is for my math methods class (educ 533) at Willamette University. I finished my BA at Portland State University but spent some time at the University of Portland as well. I graduated June 2010 with a degree in political science and sociology and have been out of school for 1 year before entering the MAT program. I hope to teach high school or middle school social studies (particularly government and history, not economics) or math. I am really excited about this year and can't wait to see what it has to offer.

I spent my year off working for a big, bad bank. I do not think it was mentally stimulating and was pretty monotonous so I am really excited that I am doing something that is different everyday.

In my free time I like to read (although that has not happened in awhile, text books and academic material are consuming my life) and watch sports. I am really disappointed that it looks like there will be no NBA season, but football has started so that lessens the blow a little.

So, those are the basics. Feel free to ask if you want to know something. :)

Task 1-3: Best Practices Research

Best practices in education:

When I googled "best practices in education" one of the sites I found was: http://www.ctserc.org/s/index.php?option=com_content&view=section&id=8&Itemid=28. They defined best practices as "what works in a particular situation or environment". I would have to agree with that vague statement. We will never have the same class (even though it might be the same section of algebra) nor the same students. We must adjust our teaching styles and even curriculum to fit the needs of the class and students. Some classes may figure out the concepts at a quicker pace or some classes might have more constructivist type learners. As teachers we need to recognize this and work towards teaching to the students and not simply the material. We must get our students to understand the material and this might have to be done in multiple ways for different students and classes.

That site also gave nine standards for best practices in education. They are:

1: A Clear and Common Focus
In high-performing schools, administrators, teachers, students, and parents share and commit to clearly articulated and understood common goals based on the fundamental belief that all students can learn and improve their performance. There is clear evidence of school practices to support this belief.

2: High Standards and Expectations
High-performing schools show evidence that each teacher believes “all students can learn and I can teach them.” Staff members are dedicated to helping every student achieve challenging state and local standards. All students are engaged in an appropriately ambitious and rigorous course of study in which the high standards of performance are clear and consistent and the conditions for learning are modified and differentiated. This results in all students being prepared for success in the workplace, postsecondary education, and civic responsibilities.

3: Strong Leadership
School leadership is focused on enhancing the skills, knowledge, and motivation of the people in the organization and creating a common culture of high expectations based on the use of skills and knowledge to improve the performance of all students. Leadership fosters a collaborative atmosphere between the school and the community while establishing positive systems to improve leadership, teaching, and student performance.

4: Supportive, Personalized, and Relevant Learning
In high-performing schools, supportive learning environments provide positive personalized relationships for all students while engaging them in rigorous and relevant learning.

5: Parent/Community Involvement
In high-performing schools, parents and community members help develop, understand, and support a clear and common focus on core academic, social, and personal goals contributing to improved student performance and have a meaningful and authentic role in achieving these goals. The school community works together to actively solve problems and create win-win solutions. Mentoring and outreach programs provide for two-way learning between students and community/business members.

6: Monitoring, Accountability, and Assessment
In high-performing schools, teaching and learning are continually adjusted on the basis of data collected through a variety of valid and reliable methods that indicate student progress and needs. The assessment results are interpreted and applied appropriately to improve individual student performance and the instructional program.

7: Curriculum and Instruction
High-performing schools have aligned curriculum with core learning expectations to improve the performance of all students. Students achieve high standards through rigorous, challenging learning. Staff delivers an aligned curriculum and implements research-based teaching and learning strategies. Students are actively involved in their learning through inquiry, in-depth learning, and performance assessments.

8: Professional Development
Ongoing professional development aligned with the school’s common focus and high expectations to improve the performance of all students is critical in high-performing schools. These professional development offerings are focused and informed by research and school/classroom-based assessments. Appropriate instructional support and resources are provided to implement approaches and techniques learned through professional development.

9: Time and Structure
High-performing schools are flexibly structured to maximize the use of time and accommodate the varied lives of their students, staff, and community in order to improve the performance of all students. The structure of programs extends beyond the traditional school day and year as well as beyond the school building. The program draws on the entire community’s resources to foster student achievement.

I think all nine are very important in terms of best practices in education. The one that stands out to me the most is number 8, professional development. I think it is easy to fall into a routine with jobs and that teachers sometimes think that since they are the ones in front of the students that they know best, but this one emphasizes that teachers are also learners. Teachers should learn new methods and see if they can work, and if they don't, that is okay too. Teachers that seek professional development I think try and stay above the curve and are not seeking to be sedentary in their careers and search for methods to improve their teaching which in return will help their students.

Best practices in instruction:

Marzanohas instructional strategies that I found to be pretty useful. They are Identifying Similarities and Differences, Nonlinguistic Representations,
Summarizing and Note Taking, Setting Objectives and Providing Feedback, Reinforcing Effort and Providing Recognition, Generating and Testing Hypotheses, Homework and Practice, Cues, Questions, and Advanced Organizers, and Cooperative Learning.

I think summarizing and note taking and objectives and providing feedback are the instructional strategies that stand out. I think note taking and and summarizing are extremely important even for a math class. I think these help students remember the information that they learned better and that they have a go to reference guide that is theirs forever. High school and middle school students do not typically get to keep their textbooks or write in them so having a journal or a place to take notes and summarize the lectures and text they've read can be highly valuable.

The instructional strategy of setting objectives and providing feedback I think are also very important because it establishes the direction teachers want to go in their class. When students enter a classroom they expect teachers know what they are talking about and that they have a plan for the class and their students. Teachers that go into a classroom without a game plan for the most part do not do as good of a job (in my opinion) as teachers with direction. In addition to having objectives teacher providing students feedback on their homework, tests and papers (maybe not papers for a math class) help students identify areas where they are succeeding and also areas that need to be worked on. Also if multiple students are struggling in the same area teachers can focus more attention in that area or even potentially try and spin the material in a different way so that students understand it because the teacher might not be clear in teaching that particular area.

I think as a teacher it is important to work towards best practices in our teaching styles and material. Best practices are not set in stone and can be adapted and differentiated in different subject areas, classes, and even students.

Citations:
http://www.ctserc.org/s/index.php?option=com_content&view=section&id=8&Itemid=2

http://www.tltguide.ccsd.k12.co.us/instructional_tools/Strategies/Strategies.html