Grades 3 to 6 - Math
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Eatontown Private School : 3-5 Math

The 3-6 math program covers a wide range of topics, staggered to accommodate learners at all levels and all stages of readiness. An emphasis is placed throughout the year on building fluency in computation, which is not dependent on memorization, but on an in-depth understanding of how numbers work in our base 10 number system. Additionally, students are introduced and develop an understanding of fractions, 2 and 3D geometry, beginning algebraic concepts, and numbers and data. Learning is built on mathematical reasoning and discourse where students are guided to not only find solutions but explain their processes in getting there as well.

The Teachers K-2, 3-6 classroom utilize a variety of curricular components including “Investigations in Number, Data, and Space” math curriculum, in addition to a variety of activities for young learners in order to provide constructivist based learning opportunities and real-world, meaningful situations that require mathematical solutions. An emphasis is placed on helping the students master basic numeracy concepts and building the solid foundation for future math learning. Teachers rely on instruction that is differentiated, direct, experiential, and cooperative. The activities and experiences offered are hands-on, multisensory, visual and kinesthetic.

Currently, the students in the 3-6 classroom set up and run the Voyagers’ School Store. This ongoing responsibility is engaging, exciting and mathematically challenging as the students decide on inventory, marketing and advertising, markups, discounts, store hours of operation and all monetary details.

Purpose

Students will…

  • Participate in some aspect of the daily operations of the school store
  • Learn and perform real-world monetary transactions
    (Depending on level) solidify understanding of the base-ten number system as they
  • learn to use and understand numbers in the 100s and 1,000s, 10,000, 100,000, or 1,000,000.
  • Build conceptual understanding of multiplication and division by employing various strategies (skip counting, multiplication arrays, story contexts, 100s charts) to – depending on level- commit basic or more difficult multiplication/division combinations to memory; solve multidigit problems.

Process

As the operation of the school store is a primary component of our 3-6 math program, most of our learning to date has revolved around staffing and setting up the school store. Students now serve as general manager, accountant, order and receiving Managers; members of the marketing and advertising team, and sales clerks. Some of the initial challenges have included placing orders for the teachers’ at the beginning of the school year supplies, setting up a record keeping systems for order and receiving and organizing accounting. Establishing the routines for the sales clerks including counting the cash in the cashbox and setting up the school store during hours of operation. The marketing and advertising team worked on creating a commercial to spotlight specific items in the school store. They also met with mentors to discuss ideas for marketing merchandise and surveyed the student body for healthy snacks suggestions that can be sold during Explore and Wonder on Thursdays.

Real world math learning has continued to surface via the school store as students have begun learning how to make change by counting up. In pairs, students have assumed the role of customer and cashier and practiced polite customer service, filling out receipts, and counting back change. Learning is differentiated as some students build their understanding basic money skills, e.g., identify coins and learning value, adding simple coin combinations and counting to $1.00 using quarters, dimes, and/or nickels; others learn how to use the various operations to calculate problems involving money, e.g., calculate cost per unit, calculate the total cost of a class order form.

We have also begun concept groups where students continue to deepen their understanding of place value and our base ten number system. Currently, students are investigating what happens to the digits of a number when a multiple of 10 or 100 is added or subtracted from a number. Some students have constructed 1,000 books to get a better understanding of the number and better sense of the location of the numbers within it.

Students also continue to build their repertoire of tools and strategies in adding and subtracting numbers in 10s, 100’s, and/or 1,000s place including 100s charts, 1,000s books, number lines and stacking cubes. They then use these tools to both solve problems and to assist them in explaining the solutions.

Students continue to use math in real-world situations through the running of the school store. Most students are now helping to run the Store during Explore and Wonder by fulfilling the role of cashiers where they are practicing money skills. Students are required to count the cash drawer before and after the Store’s opening. A group of students recently looked through October and November’s receipts in search of errors and to determine if the daily sales balanced with the end of the day cash box counts. They then worked with the students to discuss errors that make it difficult to keep track of the bookkeeping. Finally, the new sales list was completed. Many students contributed to this effort by dividing the cost of bulk supplies by the number of units to determine the cost per unit. Our store manager finished the task by entering these amounts into Google Sheets. Rather than manually enter the data, he learned how to create and enter a specific formula into a designated cell to get an exact calculation.

Both Concepts Groups continue to learn and utilize addition and subtraction tools and strategies.

Group 1 recently learned math games like Capture 5 and Close to 2.00 where they practice these strategies in a fun nonstressful environment. They have also been finding and vocalizing their solutions to word problems that focus on the difference between a number and 100.

Group 2 has been solving word problems involving distance. Using their work on these problems they have created official addition strategy charts that document all the different strategies they can use when approaching addition and subtraction problems, including breaking apart one or more numbers, changing one number to a landmark number, US algorithm, Adding by Place

In the New Year, West 2 students have been working hard in the School Store. In their first meeting of 2016, the store manager received input from other students to plan and tentatively schedule all of January’s work goals. They first tackled inventory, recounting every item in the school store. While in the process, they also reorganized supplies and labeled each bin in the VCSS storage closet. As our Inventory and Receiving team began entering the inventory counts, our Marketing and Advertising team began creating a survey to gather data as to what supplies students, teachers, and staff would like to see in the school store. Meanwhile, the cashiers busied themselves organizing and labeling the School Store Cabinet, and our accountant started the work of balancing teacher budgets and writing letters to each room to notify them of their remaining budget. Our Store Manager has been wearing many hats. He has worked with the Marketing and Advertising team to give guidance in creating the survey, helped Order and Receiving to enter the Inventory Counts, and run meetings each day to help teams set daily work goals. Real-world learning at its best.

After finishing the survey, Marketing and Advertising brought the survey to the community meeting where students and teachers in the community completed the survey. West 2 students worked together to record the data from the survey. After discovering that they did not receive the information needed from the teachers, Order and Receiving re-surveyed the teachers and added their suggestions to the order. Students in groups of 2 or 3 tallied the choices of a particular class and organized the data into a bar graph. During Mothers and Others (1/29), each group presented their findings to students and guests.

With all the data in, students finalized the order and began product research. Working in groups of 2 or 3, students were assigned 3-4 items from the list for which they had to research prices at 3 separate companies. For each item, they had to calculate the cost per unit. When they had completed their product research, they identified the best buy for each product and completed the purchase orders. Throughout this process students utilized 3 of the 4 operations: addition, multiplication, and division with and without decimals. Finally, Christi placed the order before Winter Break.

The new inventory awaited us as we returned from Mid-Winter Break. We added each new item to the stock record, reorganized the cabinet to accommodate the new inventory, and stored them away. Then we began the task of figuring retail price. Students again applied division skills to determine the price per unit and were introduced to the percentage-to-decimal conversions as they figured the 10% mark-up for the retail price. Some students were introduced to algebraic notation as they plugged the numbers into a standard equation. Finally, students created the new price list on a google doc.

With the school store math project finished we have now begun a math unit on multiplication. Students are divided into 3 groups according to their current understanding of the concept.

Multiplication Concepts Group
Emerging Understanding of Multiplication Skills and Concepts
Students in this program are just beginning to build their understanding of multiplication concepts are working on creating pictures of groups of items and using repeated addition, skip counting, and multiplication to solve problems of their own imagining.

Students continue to build and solidify understanding of multiplication concepts and skills. Recently we have been focusing on using the strategy of skip counting to find multiplication solutions. Students began by highlighting multiples of 10 and 5 and then moved onto 3s and 6s. They then looked for connections and patterns between the aforementioned multiple pairs. Next, we turned our attention to related story problems involving 3s and 6s. Students applied a variety of strategies including, visualizing story problems, utilizing skip counting, and drawing pictures to find solutions. Additionally, the teacher facilitated discussions to help students to connections between 3s and 6s, specifically doubling one number in a multiplication problem, doubles the answer.

Basic Understanding of Multiplication Skills and Concepts
Students in this program have a basic understanding of multiplication concepts. They began their work by finding possible arrays for pairs of related numbers e.g., 16 and 64 and were encouraged to arrange tiles in different configurations and then to copy them onto grid paper and to create a poster representing all the possible ways to make that number, e.g., Ways to Make 64.

Students continue to build and solidify multiplication skills through math explorations including playing number games, using array models, creating visuals, and solving story problems. Students first identified multiplication problems that were challenging for them. They found among the group that multiplying by 6, 7, and 8 were areas where they wanted to focus. Drawing, using array models, using skip-counting strategies, and thinking about problems that they know to help them solve problems they don’t (examples: “I know 7×5 so I can solve 7×6”). The group is constructing their own tools for multiplication and have made connections between operations such as seeing multiplication as repeated addition and identifying the relationship between multiplication and division. Finally, their work with visual models led them to discuss regrouping numbers

These students also recently joined with other students to gain additional practice utilizing related multiplication combinations to solve more advanced problems. They too turned their attention to related story problems involving 3s and 6s. Students applied a variety of strategies including, visualizing story problems, utilizing skip counting, and drawing pictures to find solutions. Additionally, the teacher facilitated discussions to help students make connections  between 3s and 6s, specifically that doubling one number in a multiplication problem, doubles the answer.

Intermediate to Advanced Understanding of Multiplication Skills and Concepts
Students in this program have a solid understanding of multiplication and division concepts are beginning their work on identifying the differences between factors and multiples. Like the previous group, they too are working to find possible arrays of pairs of numbers but are asked to extend their thinking by using their understanding of multiplication combinations (multiplication facts) to generate potential factors for a given product. This work will serve as the basis for their next discussion on composite, prime, and square numbers.

Students continue to develop their understanding of multiplication concepts by building and using arrays to model multiplication problems and by identifying properties of numbers including even, odd, prime, composite and square numbers. Students began this work by examining the posters they had previously created. They identified numbers that had- one multiplication combination, many multiplication combinations, combinations that made a perfect square array. These discussions prompted the introduction of math terms such as prime, composite and square numbers and helped students to formulate visual representations for these abstract ideas.

Additionally, while past work has focused on finding pairs of factors for a given product, students now begin to find all the ways to multiply whole numbers to produce a certain product using 2, 3, or 4 factors. Ongoing math games and activities such as Quick Images encourage students to generate as many multiplication combinations as they can get for a given product, thereby leading them away from traditional models of multiplication where only pairs of factors are associated with a given product. Students also continue to identify and apply number properties through games such as Number Puzzles in which they are charged with the task of identifying a number based on 4 clues, or properties of that number.

Students have also practiced using one multiplication combinations to generate others. They were challenged to find the longest combination for a given product and were asked to make connections between the factors they found for related products such as 18, 180, and 1800 or 21, 210, and 2100. By working on these problems, students called on the associative property, something they will formally explore in future years. Additionally, in finding the longest combination of factors for a given product, students discovered that these combinations could be reduced to a string of prime numbers, thereby beginning their experience with prime factorization.

In late April, students began to turn their attention toward naming strategies in multiplying 2 digit by 2 digit numbers. Again we have led them away from traditional models that would begin this work with a standard algorithm. Alternatively, we have begun to build their visual understanding of multiple digit multiplication problems by requiring them to show solutions through nonnumerical means using tiles, cubes, or by creating multiplication arrays on grid paper.

Authentic Math Experience
West 2 has taken their math learning to the Outdoor Program on Wednesdays. One recent estimation activity challenged students to collect as many sweet gum ball seeds as they could find in a 5 minute period. Students were first asked to estimate how many seed balls they predicted each person could collect in that time period. The class set a goal of collecting 50 balls each, a total of 450 seed balls. After collecting for the 5 minute period, the group was asked to estimate the number of sweet gum balls they thought they had gathered. After visually assessing the basket of sweet gum ball seeds, the class predicted that they had only collected half of their goal and requested an additional 5 minutes in order to collect the full amount. After they had finished collecting, a few ambitious students began organizing the seed balls in groups of 100 in order to count them. The class was surprised to find that they had collected nearly 1000 gum balls.

On another occasion, students were asked to write a multiplication problem and then to collect and arrange natural materials to represent their solution.
Authentic math experiences continue to arise weekly during the opening of the school store on Thursdays and School Store Workshop Thursday afternoon. Recently, we have begun inviting other W2 students to serve as cashiers, so that each student is given an opportunity to improve their knowledge and understanding of operations involving money. Our official cashiers continue to count the cash box and calculate daily sales on Thursdays. Additionally, others have been working hard to balance teacher accounts and record income and expenses to calculate overall revenue during School Store Workshop.

TEACHER:Christi Sandbach