Workshop # 4

Workshop # 4- Quilts and Palaces

My view on the role of geometry in grades K-8 is that it is important , essential and is equivalent to arithmetic in importance. Currently, when I substitute in kindergarten and first grade classes, I am
pleased to see that most of the schools ( where I am assigned) use Everyday Calendar Math. Therefore, the students are identifying patterns and shapes on a daily basis. It is built into the curriculum. This eliminates the problem of children receiving a "traditional" geometry unit at the end of the year. As was mentioned, during the panel discussion in Workshop # 4 this has often happened in the elementary grades. With Everyday Calendar Math , the children spend time identifying patterns (colors or shapes). Also, they discuss the different geometric shapes and the properties of those shapes on a daily basis. Therefore, even if the kindergarten students are not taught a geometry unit during the year, they receive exposure to patterns and shapes for at least 20 minutes every morning. Of course, Calendar Math does not replace a unit on geometry for Kindergarten and first grade but it is better than no exposure at all.
A foundation for geometry should be provided in kindergarten and 1^st grade and should be taught at all levels in the primary grades. With the technology boom, space exploration/ discoveries and nuclear energy/war etc., the manner in which we describe our world and universe has changed. Mathematics is another form of communication and it has changed dramatically also. Students must be prepared in elementary school to learn the language of mathematics. Also, as for the secondary education, when students reach high school, they will only have 4 years to move from elementary level algebra and geometry to college level trigonometry, algebra or calculus depending on their career path and interest. I do not work in secondary schools ; however, it is my understanding that traditionally many students must take a math course in the summer if they wish to take college level calculus or math in high school. In other words, if a student takes Algebra I, Algebra II, Geometry, Trigonometry and Calculus, these students must complete five math courses in four years. Also, they will need college level algebra such as vector algebra and matrix algebra before calculus . Therefore, students who enroll in advanced and college level math and science courses in high school cannot complete all of the requirements in four years without an extra course load. Also, in the secondary grades, second year physics and college physics cannot be taught without calculus. I know because I took second year physics in high school and our instructor taught us pre-calculus before we could complete the course. It was enough to get me an A my first month in college calculus until my grades dropped drastically. Geometry should not be put off until middle school or fifth grade because it puts a tremendous pressure on students to "catch up" with prerequisite courses in high school and college.
In order for students to experience success in secondary level math and science, they must have a solid foundation in elementary science and math . Physics depends on calculus and chemistry uses algebra. In elementary school, often students are taught to memorize facts but not the connections with math and science. Also, certain topics are not seen as important in elementary education. However, if a student selects a career in science, they will have to work very hard and "double up on math" in secondary grades and this is not fair. Therefore, it is essential that students receive geometry from K5-through the secondary grades. Students may not know their career choice in elementary school and therefore they need exposure to all branches of mathematics in the primary grades.
As far as an example of "inductive reasoning" in mathematics, the seventh grade math class where students where students developed their own formulas for the volume of a cylinder is an example of inductive reasoning. The students were shown a circle ( a lid) and the instructor moved the lid through space to create a cylinder and then asked the students to develop formulas for a cylinder. She did provide prompts and the students completed this activity as a group. However, if the students had completed this assignment independently and made observations about cylinders while recording the observations in a journal that process would have involved inductive reasoning. Basically, when students make observations about things in nature or the real world and then record these  as generalizations in a journal that is inductive reasoning. The generalizations made from these observations are part of the process of inductive reasoning. An example of deductive reasoning is providing a student with algorithms and asking them to compute the correct answer. It is interesting because I have never allowed students to create solids from nets and perform computations without giving them the formulas beforehand. I usually provide the students with the formulas. Usually, I would have given the students the formula V= , the dimensions of the cylinder and ask them to compute the volume. However, I can see how allowing students to create space figures and develop formulas for this figures will assist students to better understand the concepts.
Finally, if students wish to have careers in certain branches of the military , engineering or science they will need geometry as well as arithmetic. One cannot do vector algebra or vector calculus without it. It is very important for educators to help students to make connections between science and math and to understand the real world applications for this knowledge.

Activities that I will Incorporate Into My Lessons
I will incorporate the suggested activity –The Platonic Solids. I have taught lessons with 3-5 grade classes using nets to create geometric solids. However, after reviewing the suggested activities in the workshop, I would like to use polyhedron nets to create hexahedrons (cubes), tetrahedron ( triangular pyramids , and dodecahedrons ( such as decagonal prism or elongated square dipyramid). The students would be able to identify the Platonic Solid after constructing the solid using nets and then identify the properties of the solids.

Workshop # 8 –ITV Course Elementary Math

 

 

Workshop #8

The Future of Mathematics: Ferns and Galaxies

I feel that allowing students to use the – state- of- the- art technologies in math class is a great way to engage them. Research shows that students learn more quickly and easily with instruction through a variety of media, or by using multiple modalities.

As far as allowing students to use graphing calculators or any calculator, the National Council of Teachers of Mathematics recommends the integration of the calculator into the school mathematics program at all grade levels in classwork, homework, and evaluation. Calculators free large amounts of time that students currently use to practice computation. This time should be spent helping students to understand mathematics, to develop reasoning and problem- solving strategies. The calculator should be an integral part of the mathematics curriculum and not used solely to check calculations already performed. There are many units and activities available to assist teachers who do not have proper training on the use of the calculator in primary classrooms.

Today, most of the attention which was focused on the use of the calculator in primary classrooms, has switched to the use of computers, Wiis, mobile devices, iPads, iPods, and interactive white boards in the classroom. In a recent survey of secondary students, two –thirds of secondary students want to use laptops, cell phones, or other mobile devices at school. Of course, the schools are concerned with internet safety, bullying and the funds and resources to provide the technology. Therefore, the debate still continues as to the usefulness and necessity of these devices in schools.

As an educator, I have had the opportunity to attend training iPods in the primary classroom. I have also received some training on how to use mobile devices as well as many software programs. And, I have received some training on the interactive white boards. I would like to receive additional training on the use of educational computer games such as the Wii. Wii's Big Brain Academy, for kindergarten studens, can be used to challenge students on logical thinking and math. The game tests students' abilities and speed at solving various problems, involving skills like counting, recognizing patterns, and size comparisons. Also, third grade students can use the sports games as an educational activity to teach mental math and estimation, as well as double digit addition. Of course, teacher training and education in terms of how to use educational games effectively is important.

The advent of new technologies is changing what and how we teach in the mathematics classroom. Many new technologies are available for students today. It is impossible to discuss all of them. Students must be taught when and how to use the technologies appropriately. When used appropriately, this new technology can be used effectively at all grade levels. The technology must be used as an aid to understanding. Computer technology is changing the ways we use math, and therefore the curricula for instruction as well as methods of instruction is changing.

The new technology and mobile devices are expensive. Many districts are struggling to provide adequate school buildings and textbooks. Therefore, funding for iPods, Wiis and other mobile devices is not available. The district in which I work has interactive white boards in the majority of the primary classrooms. Each school provides different technology for their students. Some schools may provide mobile laptop labs , and others have 5 or 6 computers in each elementary classroom as well as access to computer labs. Therefore, calculators and computers are readily accessible in primary classrooms. As long as, students use the technology to better understand math concepts and problem solving concepts they are a worthwhile aid for the primary classroom.

 

 

 

 

Activities That I Will Use In My Lesson

 

I will use Pascal's Triangle to teach integer sequences and number patterns to grades 2 and 3. For 4^th and 5^th grade, I will use Pascal's Triangle for activities using Fibonacci sequences and patterns. Also, I will also use Sierpinski Triangles and Geometric Fractals with grades 4 and 5. And for an educational game, I will use the Chaos Game.

What’s the Big Idea ?- Elementary Math Recertification Course

Workshop # 7

Algebra: It Begins in Kindergarten

 

This workshop was interesting because it covered information for all grade levels in a short amount of time. Mrs. Neagoy explained the methods for conveying algebraic concepts to primary students. In addition , we compared our "traditional concept of algebra" from our high school years to a broader perspective which includes elementary instruction.
During the workshop, Mrs. Buckley, asked us to think about the paths that we have taken with our students "through the worlds of algebra" as elementary teachers. By "worlds of algebra", Mrs. Buckley was referring to the pictorial, tabular, symbolic, verbal and graphical methods presented to us in this workshop.
I have taken all of these paths as a k-5 substitute teacher. In grades k-2, my instruction usually follows a pictorial path.  In grades K-2 we could have the students drawing 2 eyes for one student, and four eyes for 2 students, as well as 8 eyes for 4 students to represent this relationship. In addition, to the pictorial path, with older students (grades 3-4), I have conveyed many concepts using graphs. With this age group, I have used bar graphs, circle graphs and pictographs to represent this idea. Pictographs can also be used in grades k-2.
In grades 3-5 , my path has also been tabular. I have used the tabular path with students, using tally sheets to represent various ideas. I have used tally sheets when conducting surveys with students, and before transferring this information to graphs. Fifth grade students understand symbolic representation of information, and many of the concepts of elementary algebra. The symbolic world of elementary algebra has also been of part of the "worlds of algebra" that I have used in math instruction.
I felt that the definition, which Ms. Neagoy provided at the beginning of the workshop was important. As teachers in the primary grades, we are attempting to teach students how to look for patterns, observe data and search for relationships when solving problems. We should use manipulatives, visuals, pictorials, graphs and tabular strategies to assist students, as they learn how to express themselves using the language of math. Therefore, my path in teaching these concepts has taken me through all of the "worlds of algebra" used in primary instruction.
I also found it interesting that President Clinton's education agenda called for all American students to be competent in algebra by the end of the eighth grade. This requirement is one reason that algebraic concepts must be taught beginning in kindergarten.




Activities That I Will Use In My Lessons
I will use the concepts of developing algebraic reasoning through literature (k-2). I have included a shared reading activity in my final course project for this course. I will also use the What Comes Next? activity with my 2^nd and 3^rd grade classes. I will use the Cube Counting Problem with (grades 3-5). For the 5^th grade students, I will use graphing calculators and linear equations.

Math Lesson Plan- Multiplication Squares

Final Course Project

Mathematics: What's The Big Idea ?

Grades K-8

Gabriella Robinson

July 23, 2011

Bunches and Bunches of Bunnies

Multiplication Squares

4^th Grade

The handouts for this lesson can be found at classwik45   on the professional development page.

Multiplication, Square Numbers
Lesson Overview


This lesson is for fourth grade students. The students will learn basic multiplication facts and will demonstrate an understanding of multiplication. The teacher will use the story Bunches and Bunches of
Bunnies to introduce the concept of square numbers. The students will listen to Bunches and Bunches of Bunnies as a shared reading activity. The teacher will then introduce two math activities to reinforce the concept of multiplication squares.
Summary : The students will create multiplication squares to help them understand the concept of square numbers. This activity will help students to understand multiplication's relationship to repeated addition.
SC Curriculum Standards-Math
4-2 Number and Operations
4-2.3 Apply an algorithm to multiply whole numbers fluently.

• Multiplication facts to 12 (D.1)
• Choose the multiples of a given number up to 12(D.2)
• Choose numbers with a particular product (D.11)

4-3 Algebra

4-3.4 Translate among letters, symbols, and words to represent quantities in simple mathematical expressions or equations.

• Write variable equations to represent word problems (G.5 )

Materials
Bunches and Bunches of Bunnies by Louise Matthews
Grid Paper
Scissors
Paste
Crayons
Construction Paper
Square Facts Worksheet
Counters
Activity Sheet 16
Cards
Calculators




Background for Teachers
A squared number is a number that is a result of multiplying an integer by itself. Any squared number can be represented in a square array. You can write each squared number as a product using an exponent.
4² =16
Intended Learning Outcomes

1. Demonstrate a positive learning attitude toward mathematics.
2. Become mathematical problem solvers.
3. Reason mathematically.
4. Communicate mathematically.
5. Make mathematical connections.
6. Represent mathematic situations.

Web Site

Aunty Math (Grades K-5)

Synopsis: A visit to relatives can be a great time to learn math. Aunty Math, has great math challenges for students in K-5. The students will learn the use of multiple intelligences in problem solving. Students submit their solutions to the challenges, and teachers can explore the mathematics behind the challenges.

www.dupagechildrensmuseum.org/aunty

Strategies
Manipulatives- counters, multiplication squares
Modeling
Shared reading
Calculators

Instructional Procedures: Invitation to Learn


Shared Reading


Read the book: Bunches and Bunches of Bunnies by Louise Matthews.
Say, "Which is easier to count the bunnies in large groups, or the bunnies by twos, threes and fours? Discuss their reasoning. Ask students to predict mutiplication sentences for the different groups. Next discuss the squares of numbers. Give children counters. Reread the story, write each multiplication sentence on chart paper and ask children to model it with their counters.


Multiplication Squares


Give pairs of children, Grid Paper. Also provide the students with scissors, paste, crayons and construction paper. Ask children to color and cut out a model for each multiplication sentence( this is the chart paper that was used during the Shared Reading activity, above ). Have children paste these models on construction paper and then write a corresponding multiplication sentence beneath it. The students should be able to describe the shape of each model ( each model forms a square). Examples of the models and multiplication sentences are provided in the handouts section of the lesson plan.






Modeling Multiplication Facts


Provide students with Activity Sheet 16 to make a booklet. The students will also need scissors and paper. Each pair should be assigned a multiplication table, such as the four table. Students will write a multiplication sentence on each page of the booklet, and cut and paste carrot bunches to illustrate the sentence. The students will staple the booklets together and share them with the class. ( An example, of the booklet is included in the handouts section).


Extending the Lesson


Discuss the squares of numbers. Ask the students to fold a sheet of paper. Ask them to half the paper again. Can you fold it into half eight more times? Show the math involved.


Additional Extensions


Students can color in the square numbers on a multiplication chart. Look for patterns in the chart. ( the square numbers will form a diagonal on the chart).


Predictions
Ask the students to predict if the product of any number multiplied by itself will form a square. Ask children to choose a number between 12 and 20 and multiply the number by itself. Then make a model of it on grid paper. Discuss why these numbers are called square numbers.


Calculator
Exploring Area with the TI -10
Activity – The teacher will provide each student with 49 cards.
Ask the students " How many cards cover the desktop?"
"How might your cards be arranged to count with your TI -10 (rows and columns)?"
The following instructions are given for TI-10 calculators. However, your students may use other calculators with this activity.


Press the on button.
Press AC to clear anything previously stored.
Press clear. The screen is blank and the memory is clear.


Press Opl to begin counting.
Press + 7 because you will be counting cards by rows of seven.
Press Opl to let the TI -10 know that you are ready to count.
Press 0 to begin counting at 0.
Press Opl to continue counting.
When you have counted seven rows the TI -10 displays
42 +7
7 49


Answer: Area is 7X7=49 or 7² =49




Pass out two centimeter grid paper.


Question to ask:


What might you say about the size of the squares on the grid paper?
(Answers should suggest they are all the same size)


How might you and your partner find the number of squares to cover the desktop with grid paper?


How might the TI-10 help?


( The two centimeter grid paper and an example for this activity is included in the Handouts section of the lesson plan. In the example in the handout sections I used seven rows and seven columns. However, the number of rows in columns will vary depending on the size of the desks in your classroom.)


Homework and Family Connections
After completing several activities in class, ask the students to complete the Square Facts sheet at home. Ask the students to have a family member complete the game with them. The family member can sign the activity sheet to indicate that they have worked together to enhance their mathematical understanding.
( The Square Facts sheet is included in the Handouts section of the lesson plan)


Encourage students to play any of the mathematics related board games at home to reinforce their math skills: Dominoes, Connect 4, Rummikub, Triple Yahtzee, Yahtzee, Backgammon, Chinese Checkers, Tower of Brahams, UNO.


Assessment Plan:



Journal Activity : Have students explain what square numbers are. Use pictures, words and numbers to explain what 5² is.




Differentiated Instruction


For Auditory and Kinesthetic Learners: Use a multiplication rap such as Hap Palmers' Multiplication Rap to reinforce multiplication facts. Students can listen to these raps at a learning center and learn the facts.


For Visual Learners: Students can complete math activities at file:///C:/Documents%20and%20Settings/Gabriella%20Robinson/My%20Documents/www.funbrain.com or mathworld/wolfram.com in the classroom computer center.