Addition Story Problems Lesson Plan & Critique Page 4

Math Portfolio

Mary Vause CRIN E07 December 12, 2008 Table of Contents

Overview – page 2

Addition Story Problems Lesson Plan & Critique – page 4

Money Counting Problem & Critique – page 14

Literature Connections – page 17

Web Resources – page 23

Reflection – page 29

2 Overview

I believe that a good math teacher makes math concepts as concrete as possible for students; caters instruction and practice to students’ developmental level; and helps students make connections among math, other academic areas, and their own lives.

Making math as concrete as possible is vital for students, especially at the elementary age when, according to Piaget’s developmental theory, many children struggle with abstract thinking. Manipulatives such as Diene’s blocks, Cuisenaire rods, fake money, popsicle sticks, etc., should be used to help children grasp concepts in number sense, patterns, probability, geometry, algebra, and so on. For example, Diene’s blocks can help make the concept of place value more concrete and real for students, providing a hands-on starting point from which they can leap into more abstract understanding. Beyond actual manipulatives, encouraging students to draw pictures and graphically represent computations is also beneficial and can provide students with a halfway point between hands-on manipulatives and abstract understanding.

Targeting instruction to a student’s developmental level is also important.

Through observation and assessment a good math teacher will be able to deduce which problem solving level a student is at for cognitively guided instruction, which van Hiele level the student is at for geometric instruction, etc. A mix of homogeneous and heterogeneous grouping in math can allow students to receive targeted instruction at their ability level in addition to learning from stronger students and mentor weaker students.

Finally, it is important to help students make connections between math and other academic subjects as well as between math and their everyday lives. Interdisciplinary lessons – such as combining math and science or math and literature – can achieve this

3 goal, as can mixing math concepts into day-to-day experiences in the classroom. For example, the teacher could ask children to name different geometric shapes they see around the room, or they could have students graph their grades in other subjects throughout the semester (e.g., keeping a private log of their grades on social studies assignments and creating a line graph to show their performance over time).

4 Addition Story Problems Lesson Plan & Critique

What It Is and Why I Chose It

This is an original lesson plan I wrote for the purpose of teaching addition “story problems” (i.e., word problems) to a K-1 multiage class. The lesson plan includes step- by-step instructions for carrying out the lesson as well as examples to use during whole group instruction. There are also three original story problem worksheets designed for Grade 1 students (the kindergarteners did math centers at that point because they’re not ready to read problems independently). One worksheet is for on-grade-level students, another for below-grade-level, and the third for above-grade-level. The worksheets include the names of students in the class in order to motivate them, so if I taught this lesson again I would substitute the names with students in the new class. I selected this piece for the portfolio because it was effective and engaging for the students when I taught it in the classroom, it shows my ability to create original worksheets for students, and it includes a broad range of adaptations for different ability levels in the classroom.

Critique of the Work

The lesson was successful overall, with students attentive and engaged throughout the instruction. (Many of the kindergarteners expressed disappointment that they had to go to centers while the first-graders completed additional story problems.) Using student names in the story problems was an effective motivator. Having them work in heterogeneous pairs allowed weaker students to learn from stronger students. One area I would improve upon when reteaching this lesson is making sure all students completed the final task of writing their own story problem on a notecard and turning it in to me. I designed this final step as a form of feedback to show me who “got” the lesson and who did not. Unfortunately I did not flip through the cards when I received them to make sure students had finished all parts of the task, and so some data was missing when I went back to analyze the cards. (A full critique is available after the lesson.)

What I Learned from Doing It

Through this lesson, I learned how using students’ names in story problems can make the lesson more engaging for them. I also learned to adapt an on-grade-level worksheet for both below-grade-level and above-grade-level students. Teaching this lesson also made me feel more confident about my abilities as a math educator.

5 Addition Story Problems Learning Plan

Title: Addition Story Problems Grade: K-1

Objective: 1) Given manipulatives and teamwork with a partner, the students will solve at least 6 result-unknown addition story problems. 2) Given manipulatives and teamwork with a partner, the students will create and solve their own story problems. Related SOLs: 1.9 The student will create and solve story and picture problems involving one-step solutions, using basic addition and subtraction facts.

Materials: Miniature bears or other manipulatives.

Procedures: * Beginning: Bring the students up to the carpet for a whole group lesson. - Opening lines: “Today we’re going to work on addition story problems. We’re going to draw pictures and write number sentences to help us solve these problems.” - Draw their attention to the story problems that are already written on the writing easel, and let them discover as you read aloud together that the problems include their names. - After reading each problem, draw small pictures/symbols underneath in order to make the concepts more concrete. (If students struggle with this step, line up manipulatives along the bottom of the writing easel in addition to drawing the pictures.) - Have individual students come up and fill parts of the number sentence in, starting with single numbers and building to the entire number sentence. (15 minutes)

* During: Pair students heterogeneously (stronger readers with weaker readers, if possible) and give each pair a worksheet to complete, along with a cup of manipulatives to help them. Tell students to use the manipulatives or draw pictures about the problem, and then write a number sentence and solve each problem. (Students will turn these worksheets in at the end of the lesson.) (10 minutes)

* Ending: Have each pair of students create and solve their own story problem on a note card to turn in. If there is time, bring the class back to the carpet and share some of the word problems with the class, writing number sentences together to solve the problems. (5 minutes)

Differentiation: For teams that are struggling, provide an adapted form of the worksheet as well as additional scaffolded instruction. For teams that finish early, provide a “challenge” worksheet with more complex problems.

6 Adaptations: Seat students with ADHD and/or behavioral problems in front during the whole group segment and make sure to pair them with calm, focused students during group work. Monitor students with ADHD and/or behavioral problems closely during group work. Large-font versions of the worksheets can be created for students with visual impairments. Simpler, less cluttered versions of the worksheets can be created for learning disabled students.

Evaluation: * Formative: Teacher will listen to students’ answers during the whole group discussion and observe the students as they work in pairs. * Summative: Each pair of students will turn in their story problem worksheet(s) as well as their card with an original story problem and solution.

Sources:

Kliman, M., and Russell, S. J. (2004). Number Games and Story Problems: Addition and Subtraction (Grade 1). U.S.A.: Scott Foresman.

Mason, M. (2008). EDUC 407/CRIN E07: Elementary mathematics curriculum and instruction. Williamsburg: College of William & Mary.

7 Story Problems on Writing Tablet [personalized with students’ names]

1) Zechariah has 7 video games. Shauna gives him 3 more video games. How many video games does Zechariah have in all?

___ + ___ = ___

2) Ilissa has 5 necklaces. Ciara also has 5 necklaces. How many necklaces do they have in all?

___ + ___ = ___

3) Kenan has 6 comic books. Jalen also has 6 comic books. How many comic books do they have in all?

______

4) Amanda has 5 Gummi Bears. Brayden gives her 4 more Gummi Bears. How many Gummi Bears does Amanda have in all?

______

Challenge Problems:

1) Arielle has 3 books, Leah has 2 books, and Ms. Vause has 1 book. How many books do they have all together?

___ + ___ + ___ = ___

2) Shauna had 4 pencils. Brayden gave her some more pencils. Then Shauna had 6 pencils. How many pencils did Brayden give Shauna?

___ + ___ = ___

8 Story Problem Worksheet

Names: ______Date: ______

1) Shauna has 5 cookies. Amanda has 4 cookies. How many cookies do they have in all?

__ + __ = __

2) Jalen has 5 pennies. Brayden has 6 pennies. How many pennies do they have in all?

__ + __ = __

3) Leah has 8 birthday presents. Ilissa gives her 4 more birthday presents. How many birthday presents does Leah have in all?

4) Kenan has 6 toy cars. Zechariah gives him 8 more toy cars. How many toy cars does Kenan have in all?

5) Arielle has 10 rings. Ciara has 12 rings. How many rings do they have in all?

6) Jalen has 5 stickers. Ms. Gatski gives him 4 stickers. Ms. Vause gives him 3 stickers. How many stickers does Jalen have in all?

9 Adapted Story Problem Worksheet

Names: ______Date: ______

1) Shauna has 5 cookies. Amanda has 4 cookies. How many cookies do they have in all?

5 + 4 = __ ☺☺☺☺☺‌ + ☺☺☺☺ = __

2) Jalen has 5 pennies. Brayden has 6 pennies. How many pennies do they have in all?

5 + 6 = __ ☺☺☺☺☺ + ☺☺☺☺☺☺ = __

3) Leah has 8 birthday presents. Ilissa gives her 4 more birthday presents. How many birthday presents does Leah have in all?

8 + __ = __ ☺☺☺☺☺☺☺☺ + ☺☺☺☺ = __

4) Kenan has 6 toy cars. Zechariah gives him 4 more toy cars. How many toy cars does Kenan have in all?

6 + __ = __

5) Arielle has 10 rings. Ciara has 3 rings. How many rings do they have in all?

__ + __ = __

6) Jalen has 5 stickers. Ms. Gatski gives him 4 stickers. Ms. Vause gives him 3 stickers. How many stickers does Jalen have in all?

__ + __ + __ = __ ☺☺☺☺☺ + ☺☺☺☺ + ☺☺☺ = __

10 Challenge Story Problems

Names: ______Date: ______

1) Arielle has 5 books, Kenan has 5 books, and Ilissa has 6 books. How many books do they have in all?

2) Brayden has 4 apples and 2 oranges. Ms. Gatski gives him 4 more apples. How many apples does he have in all?

3) Leah has 5 crayons. Amanda gives her some more crayons. Now she has 8 crayons. How many crayons did Amanda give her?

5 + __ = 8 ☺☺☺☺☺ + ______= ☺☺☺☺☺☺☺☺

4) Jalen has 6 lollipops. Zechariah also has some lollipops. They have 10 lollipops in all. How many lollipops does Zechariah have?

6 + __ = 10

5) Shauna has 7 blue marbles and 3 red marbles. Ciara has 5 blue marbles. How many blue marbles do they have in all?

6) Kenan has 8 blocks. Amanda gave him some blocks. Now Kenan has 13 blocks. How many blocks did Amanda give him?

11 Addition Story Problems Learning Plan Critique

Description

To summarize, I taught a lesson on addition story problems. The relevant SOL is for first grade, but since my cooperating teacher has a K-1 multiage classroom, kindergarteners also participated. I started out with whole group instruction. Students came and sat on the carpet while I guided them through addition story problems on the large writing easel. The story problems included the students’ names in order to increase motivation. An example was “Ilissa has 5 necklaces. Ciara also has 5 necklaces. How many necklaces do they have in all?” I had students read the problems with me, and then together we circled the most important information in each problem, drew pictures to help us understand the problem more concretely, and then wrote a number sentence to describe the problem. I provided less and less scaffolding with each successive problem and called on a variety of different students to come to the writing easel to fill in the information. After the direct instruction portion, the kindergarteners were assigned to math centers because their teacher felt the worksheets would be too hard for them. The first graders were placed in heterogeneous pairs based on math ability. (The teacher and I also made sure that there was at least one able reader in each group since they would be reading the story problems independently.) Each pair received a story problem worksheet with six addition problems, and then each group worked in a comfortable spot around the room. I walked around the room monitoring, checking answers, and helping to guide them to better understanding when I saw wrong answers. After they completed their worksheet, I gave each student a 3x5 inch note card and asked them to write their own story problem and solve it. I said that they could look at the worksheet for ideas but needed to come up with their own. Then I collected the note cards at the end as a form of assessment. The lesson lasted about one hour.

Student Performance Data

The worksheets that the student pairs turned in were all completely correct because I was walking around monitoring and helping them with the ones they had trouble with. A couple of the groups sailed through the problems easily without assistance while a couple of the groups needed repeated interventions, but I did not jot down which students excelled and which struggled, and so now the note cards are the only accurate records of whether or not they “got it” on their own. The note cards were as follows:

Shashauna  “Leah has 12 dalls. Ilissa gaves Leah 100 5 more. How many dalls does Leah have in all? 12 + 5 = 17”

Jalen  “Amanda has 7 dogs. Zechariah has 10 fish. Leah gave 8 more. Shashuna has 14 raser (??) Jalen gav 6 more” [He forgot to solve the problem.]

Brayden  “Brayden has 12 x boxses Dad hav 1 mor How man e do Brayden hav in all 12 + 1 = 13.”

12 Arielle  “Arielle has 12 ring Leah has 11 ring 12 + 11 = 23” [with pictures drawn to help her]

Ilissa  “Kailani have 4 cakes ad Leah have 8 cakes How many in all. 4 + 8 = 12”

Ciara  “Ciara haves 13 books and Ciara mommy 4 book 13 + 4 = 10”

Leah  “Ilissa has 12 dogs Amanda givs her 4 more how mene in all?” [She forgot to solve the problem.]

Zechariah  “Zechariah have 8 dog Amanda gav 2 dog How many dog in all?” [He forgot to solve the problem.]

Amanda  “Arielle has 4 lolypops. Ilissa gave her 4 more. how many did she get. 4 + 4 = 8”

Six of the nine first-grade students correctly formulated a story problem complete with two addends and a question. Four of the nine students wrote and correctly solved an equation or “number sentence” to accompany their problem; three students forgot to solve the story problem that they wrote so I have no way to assess their mathematical understanding. Only four of the nine students both adequately formulated a story problem (complete with a question) and correctly solved it.

Critique

One issue to note is that the note card assessments could have unfairly hurt weak readers and writers. The word problems that the students wrote are in some ways better assessments of their literacy skills than their math skills. I did not double check students’ note cards to make sure they completed all parts of the assigned task (both to write a story problem AND solve it). I also did not take advantage of the three levels of worksheets I had developed for different levels of students (high-, medium-, and low- difficulty) because I realized mid-lesson that I had not planned a way to assess which worksheet was most appropriate for each of the heterogeneous pairs of students. Despite these shortcomings, I do feel that the lesson was successful overall. My cooperating teacher told me that she was pleased with my performance. I had worried about adequately holding students’ attention during the direct instruction portion of the lesson, but I found them to be very attentive and engaged throughout. I think that they were motivated by seeing their names and their classmates’ names in the story problems, because the story problems were interesting to kids (with questions about video games, pets, etc.), and because I had student volunteers come up frequently to circle important parts of the question, complete drawings, and fill in parts of the number sentence. Throughout the direct instruction lesson, many students eagerly raised their hands to answer questions. The sample problems that we worked on together as a group progressed to increasingly more challenging problems, until by the end the problems

13 included missing addends or addition of a series of three items, etc. I think this increased challenge also helped to keep them engaged. I had worried that the story problems might prove too difficult for the first graders to read. However, they turned out to be at just the right level for pairs of students to figure out together. The children asked for my help sounding out a word only a few times. I had also worried that it might be too difficult for a first grader to write their own story problem on the assessment note card. Most of the children proved to be very capable of this, and out of all the note cards there was only one word that I couldn’t understand.

Reflections & Modifications for Future Use

In the future, I should probably craft assessment problems that are less literacy- intensive. Next time I should also check that the students have adequately completed all aspects of their assessment so that I will have all of the data I need when planning a follow-up lesson. For example, three students wrote a word problem but forgot to solve it. In two of those cases, their word problem suggested a tenuous understanding of the concept, and a number sentence with solution would have let me know whether they did actually understand the math despite their writing difficulties. I spent a lot of time creating different levels of worksheets and it was a shame not to take advantage of them. In the future, I should conduct a very quick pretest for each pair of students to work on in order to determine which worksheet to give that group. Things that I would like to continue in the future is writing original story problems that include students’ names, and incorporating heterogeneous group work since that seemed to pull up weaker students while giving stronger students the opportunity to act as leaders. I noticed that with the students working together in pairs, they asked much fewer questions of teachers and instead talked through their difficulties with each other.

What I Learned

By teaching this lesson, I learned that incorporating whole group instruction, group work, and individual assessment into a single lesson can be an effective way to impart information and model skills, encourage interpersonal and teamwork skills, and check individual student understanding. I also learned that direct instruction can be engaging for students if I make sure to involve them frequently with questions and if I make the examples interesting and relevant to them. I also learned that pre-assessment data can be useful to determine which worksheet a group of students will be able to do independently, and post-assessment data needs to be complete so that I can effectively assess students’ understanding and plan for future instruction.

14 Money Counting Problem & Critique

This problem is aimed at helping students count money. They are challenged to get to 12 cents in as many ways as possible, using plastic penny, nickel, and/or dime manipulatives. Ideally, students discover four different ways to get to 12 cents: 12 pennies, 1 dime and 2 pennies, 2 nickels and 2 pennies, or 1 nickel and 7 pennies. I included this problem in the portfolio because I am proud of coming up with an interesting money problem on my own. I also feel that I taught this problem effectively in the classroom and that it could be used again successfully.

I taught this lesson to a group of three average-ability first graders and one high- ability kindergartener. Overall, the problem was successful – the students were engaged and enjoyed the opportunity to work with the coin manipulatives. If I teach this problem again in the future, I will probably extend it by having students draw, label, and write a number sentence about each 12-cent combination in their math journal. I think this would make students’ understanding of counting money deeper and longer-lasting than simply having them talk through the solution. (A full critique is available after the lesson.)

Through this problem, I learned that I can come up with interesting problems for students on my own. When teaching this problem, I learned the importance of using manipulatives as well as the enjoyment that students get out of using manipulatives. I also learned that it can be challenging to hold students’ attention when they have manipulatives in their hands and that it is important to explicitly tell students to put down the manipulatives and look at you when you are providing direct instruction.

15 Money Counting Problem

The Topic:

Counting Money

The Expected Heuristic(s):

 Act it out  Use a picture

The Age or Grade Range Appropriate for the Problem:

First grade.

The Problem:

How many different ways can you get to 12 cents?

The Answer to the Problem:

There are 4 ways to create 12 cents: - 12 pennies - 1 dime and 2 pennies - 2 nickels and 2 pennies - 1 nickel and 7 pennies

At Least One Possible Method or a Very Brief Discussion of the Solution:

If students struggle with this problem, provide fake money and let them make groups of coins that each add up to 12 cents. Students may need a review of how much a nickel is worth and how much a dime is worth. If they are still struggling, the teacher may help by starting with 12 pennies (the most basic way to get to 12 cents) and taking away 5 pennies and replacing it with a nickel, or taking away 10 pennies and replacing it with a dime. Students may write and draw their answers in their math journals to solidify the concept.

The Source of the Problem:

Original problem by Mary Vause.

16 Money Counting Problem Critique

The Problem: How many different ways can you create 12 cents?

What Happened: I pulled a group of three average-ability first graders and one high- ability kindergartener to work this problem. I put plenty of “fake money” (pennies, nickels, and dimes) on the table in front of the students and invited them to get to 12 cents as many different ways as possible, and to slide each stack of 12 cents over to the side as they finished so that they could share them at the end. At the very end, I asked students how many combinations to 12 cents we had discovered and they correctly counted up all four.

How the Children Thought about the Problem: Twelve pennies was the first way to 12 cents that the students tried. Next came a dime and two pennies. The students were all able to achieve those two combinations on their own. However, I had to urge three of the four students to try using the nickels. After my urging, the first graders were all able to put together two nickels and two pennies to equal 12 cents, and the kindergartener was able to do so after a review that nickels are worth 5 cents, so one nickel plus one nickel equals 10 cents. One first-grader was able to get the one-nickel-plus-seven-pennies combination on her own without help. To the rest of the students, I said, “See if you can get to 12 cents with just one nickel.” After working with the manipulatives, the other two first graders were able to get this combination. The kindergartener was unable to get it on her own, but when I helped her line up a nickel and 7 pennies and asked what does 5 cents plus 7 cents equal, she immediately said “5 + 7 = 12.” (This is one of the facts she has memorized at home, which is more advanced than most of the kindergarteners are currently capable of.)

Modifications Needed: Depending on the students’ math ability level, a quick review of the names and values of each coin might have been appropriate for some students before beginning the problem. I also think that having the students draw, label, and write a number sentence about each combination to 12 cents in their math journal would be a good way to make the concept “stick.” It was somewhat difficult to hold students’ attention during the explanation portions; perhaps magnetic coin manipulatives on a magnetic white board would be helpful to focus attention during those segments.

What I Learned from the Interaction: Unusual combinations (such as one nickel plus 7 pennies to make 12 cents) can be very difficult for students in problems like these, and most of them will need some scaffolding in that area. I also learned that manipulatives are very useful, but they can sometimes cause students to have difficulty paying attention, so the teacher should have a plan for keeping the manipulatives from becoming distracting (e.g., “hands in your laps and eyes on me” when she is about to share an explanation).

17 Literature Connections

Literature Connections is a collection of five math-related books that can be read aloud to students in order to supplement their math lessons. Each book write-up includes title, author, illustrator, ISBN number, description of contents, how teachers can use the source, and a bibliographic citation. The books can be used to help teach fractions, skip counting, division remainders, patterns, and geometry. I chose this for my portfolio because the write-ups are detailed and well-written, and I think that all of these books would make valuable contributions to math lessons.

These books were carefully selected from a large collection of math-related books that my cooperating teacher has amassed over 37 years of teaching. Therefore these were not the first five books I could find with math connections, but rather a list that was carefully compiled from many options. When writing up the description of each book, I paid attention to many features such as pictures, layout, and special sections. I took my time writing the sections on how to use the books so that I could come up with creative ways to integrate the books into math lessons.

I learned how much literature can contribute to math lessons. In the future I would like to use books like these as introductions to new math concepts and also as reviews of old math concepts. I also learned that it is useful to write down information and descriptions of quality books with math connections so that I can remember to go back and incorporate them into relevant lessons later.

18 Literature Connections Apple Fractions By Jerry Pallotta Illustrated by Rob Bolster

ISBN #: 0-439-38901-1

Description of contents: Apple Fractions is an engaging, beautifully illustrated nonfiction book that shows different types of apples (McIntosh, Golden Delicious, Gala, etc.) divided up into fractions. I found it superior to other fraction books because it helps students visualize fractions on three-dimensional shapes. On each page tiny elves divide up an apple into halves, thirds, quarters, etc. The book does a good job of not just showing a fraction like 1/3, but also 2/3 and 3/3. The book is also innovative in that it does not stick exclusively to apple slice fractions. One page shows a group of bees hovering around an apple blossom while one bee pollinates the flower, and the page reads “One-sixth of the bees is busy working. Five-sixths of the bees are looking for another apple tree….” On another page, 1/7th of a Cortland apple is shown by cutting the apple into horizontal slices in addition to the traditional vertical slices. One-tenth and nine-tenths are shown with one apple set apart from nine neighboring apples. The little elves fill cups of apple cider and apple juice to varying levels to show 1/4th, 1/2, etc., with liquid. The final page shows a traditional representation of fractions as pieces of an apple pie. There is even some scientific information included about the parts of an apple, characteristics of each type of apple, where each type of apple is found, how apples grow, how apples are harvested, and how to make apple juice. The book pulls everything together and shows how to discuss fractions as words by informing us that out of every ten apples picked, six out of ten are eaten fresh, two out of ten are squeezed into cider or juice, and two out of ten are made into canned apples, pie filling, jams, jellies, dried apples, or apple butter.

How you can use this source: This book can be used as a teacher read aloud for grades 1-5 to complement a math lesson on fractions. The pages are fairly large, the fractions are in large font, and the pictures are big, so students will be able to look on during the read aloud and better visualize the division of three-dimensional shapes into fractional pieces. Younger students will probably enjoy the comical pictures of the elves pouring cider, classifying apples, and struggling to take apart and piece together the apple slices. Older students will probably be interested in the scientific information about apples and can benefit from the basic fraction number sentences in the second half of the book (“1/5 + 4/5 = 5/5,” “5/5=1/1=1,” etc.). The book is roughly a Grade 3 reading level and can be enjoyed independently by most students in Grades 3-6. After students read the book, they can complete a fraction worksheet in which they label the fractions represented by pictures of apple slices, apple juice, and apple pie. As an extension activity, students can choose a fruit or vegetable of their liking and represent it in fractions in their math journals.

Pallotta, J. (2002). Apple fractions. New York: Scholastic.

19 One Hundred Ways to Get to 100 By Jerry Pallotta Illustrated by Rob Bolster

ISBN #: 0-439-38913-5

Description of contents: This book shares the same author, illustrator, and publisher as Apple Fractions, and it is similarly engaging in its use of pictures and text. The opening page assures the reader, “If you can count to ten, you can count to one hundred.” The book then illustrates 100 different ways to get to 100, from skip counting to addition, subtraction, multiplication, and division.The ways to get to 100 start simple and then build in complexity. Students begin by counting by 1s, and 100 colorful objects are shown on the page (kid friendly- items like basketballs, butterflies, cheeseburgers, ice cream cones, etc.). Next, students skip count by twos with pairs of cherries, skip count by fours with hamburgers, count by fives with toothbrushes, count by tens with fish, by twenties with crabs, by twenty-fives with dinosaurs, and by fifties with rocket ships. (Each pages has a “Where’s Waldo?” quality with a single fish in lipstick, a single ice cream cone with chocolate sprinkles, etc.) Next the author reviews the skip counting by representing it in multiplication terms (“1x100=100,” “2x50=100,” “4x25=100,” etc.) The next section shows students how to reach 100 through addition, depicting computations such as “100+0=100,” “97+3=100,” “81+19=100,” etc. The use of color on these pages is effective, with the addends shown in different colors (e.g., 32 red robins and 68 blue jays, 98 basketballs and 2 volleyballs, etc.). Color is also utilized in the subtraction section, which includes problems like “101- 1=100,” “104-4=100,” “200-100=100,” etc. Division number sentences round out the book, with problems such as “200/2=100,” and “1000/10=100.” Simple definitions of each mathematical process are provided throughout the book, such as “Dividing a number into equal amounts is called division. It is the opposite of multiplication.”

How you can use this source: This book would be an excellent teacher read aloud for Grades k-5 on the 100th day of school, which many elementary schools now celebrate. Teachers of primary grade students can focus on the addition and subtraction sections and can lead students in actually counting along by 1s, 2s, etc., leaving the sections on multiplication and division as a brief introduction of skills they will master in the future. Upper elementary teachers will probably want to move fairly quickly through the addition and subtraction pages and focus the most time on the multiplication and division pages. The book is roughly a Grade 3 reading level and can be enjoyed for independent reading by most students in Grades 3-6. After students read the book, they can complete any number of worksheet assignments involving, for example, circling groups of 2s, 4s, 5s, etc., in pictures of 100 objects. As an extension activity, students can design and color two additional pages for the book with two ways to get to 100 that were not illustrated.

Pallotta, J. (2003). One hundred ways to get to 100. New York: Scholastic.

20 A Remainder of One By Elinor J. Pinczes Illustrated by Bonnie Mackain

ISBN #: 0-590-12705-5

Description of contents: This is a fiction story about a bug named Joe who is part of a squadron of 25 bugs who are marching in a parade for their queen. They want to make the queen proud by marching past her in equal rows, but somehow Joe keeps ending up the odd bug out. First, they march past her in 2 rows, and Joe ends up marching by himself behind 2 rows of 12 bugs. A bee angrily informs Joe that “The queen likes things tidy” and that he has brought shame upon himself by being “the remainder of one.” The next day Joe determines to make things right and organizes his squadron into 3 rows. Alas, Joe is the odd bug out again behind three rows of 8, and this time the queen sends a mosquito to voice her displeasure. The next day, poor Joe tries rows of four as they march past the queen. Alas, he is once again the remainder of one as he marches behind 4 rows of 6. This time, a dragonfly delivers the queen’s consternation. Joe has become more and more anxious throughout the book, but he decides to give it one last try. The next day, sleep- deprived Joe arranges the squadron in rows of 5. He is no longer the remainder of one as the troupe march past in exactly 5 rows of 5. “Good show,” says the queen. “Your rows are divine. We see no remainder to ruin your line.” The pictures are humorous and engaging, and rhyme is used throughout the text.

How you can use this source: The book is roughly a Grade 2 reading level and can be enjoyed for independent reading by most students in Grades 2-6. But I think that the optimal use of this book is as a teacher read aloud for students who are in the early stages of learning about remainders in division. During the reading, the teacher can pause before each marching page to elicit from students whether they think there will be a remainder. Then the teacher or a student volunteer can predict what the rows of marching bugs will look like by arranging magnetic manipulatives on a white board, or by drawing pictures on a writing easel. Even better, the students could each have their own white boards and draw how they predict the rows will look. The teacher can have the students hold up their white board predictions before turning the page and revealing how many bugs were in each row and whether Joe was a remainder of one. After the book, the teacher should explain that the remainder of a division problem is not always 1 (there can be remainders of 2, 3, 4, etc.). As an extension activity, students can write and illustrate their own story about a remainder.

Pinczes, E. J. (1997). A remainder of one. New York: Scholastic.

21 Patterns By Karen Bryant-Mole

ISBN #: 0-8368-2618-3

Description of contents: This is a primary-level text presented as a nonfiction book complete with a table of contents, chapter headings, glossary, and photographs. Mortimer the teddy bear (who has his own “Mortimer Math” series) challenges students to recognize patterns. “You can make patterns with almost anything,” Mortimer declares, and indeed he creates patterns with toys, clothespins, blocks, socks, candles, buttons, carrots, etc. Mortimer’s patterns start out simple (e.g., red-blue-red-blue) and gradually build in complexity (big-small- small, plum-grape-grape-strawberry, etc.). Throughout the book, Mortimer alternates two colors, then alternates objects and colors, then shapes, then three colors, and finally ends with his most challenging fruit patterns. The book is meant to be interactive, with Mortimer asking readers, “Can you name the pattern?” Students can guess the pattern, then turn the page to see if they are correct. The layout is effective, with photographs of Mortimer on a white background and Mortimer’s pattern lined up in the foreground. Mortimer speaks to students through quotation bubbles like those in a comic strip. The objects making up Mortimer’s pattern are large, bright, and colorful, helping students pay attention to the pattern exercises.

How you can use this source: This book can probably be read independently by Grade 2 and up. However, primary students from pre-kindergarten through Grade 1 are the students who will get the most out of this book with its practice of simple patterns. Since the text will be inaccessible to most of these students independently, Patterns is ideal as a teacher read aloud. The headings and pictures are large and will be seen easily by students from their seats on the carpet. The teacher should make the reading as interactive as possible, eliciting from students what they think the pattern is. The teacher can ask students what is different among the objects in the pattern (e.g., color, size, shape, or some combination of these). As an extension activity, students can create their own patterns with manipulatives of different shapes, sizes, and/or colors, and then have the students name these patterns (e.g., blue-blue-red, big-big-small, etc.) and draw the patterns in their math journals.

Bryant-Mole, K. (2000). Patterns. Milwaukee: Gareth Stevens Publishing.

22 The Silly Story of Goldie Locks and the Three Squares By Grace Maccarone Illustrated by Anne Kennedy

ISBN #: 0-590-54344-X

Description of contents: This fiction book is a retelling of the “Goldilocks and the Three Bears” story with a geometric twist. The humorous author informs us that this Goldie Locks is the great- great-granddaughter of the original heroine. One day Goldie Locks is walking through the woods and views a “pretty house in the shape of a pentagon.” Inside, three bowls of various sizes are filled with noodles of different shapes. The big bowl has triangles (some with three equal sides, some with two equal sides, and some with no equal sides), the medium-sized bowl has rectangles (some short and wide, some long and skinny), and the small bowl has squares with four equal sides that were just right. So Goldie Locks eats the squares and then wonders through the house intent on finding the perfect spot for a nap. First she tries a triangular chair and falls off it. Then she tries a circular chair but almost rolls away. Finally she tries a rectangular couch and discovers it is just right… until Goldie falls flat because it is a two-dimensional shape. Next Goldie Locks meanders over to some three-dimensional beds that look more promising. She tries a cylinder- shaped bed but cannot stretch out her legs. She tries a triangular prism bed but her feet fall off the sides. Finally she tries a rectangular prism bed and it is – you guessed it – just right. Goldie Locks falls asleep but is soon awoken by the Three Squares, who are irritated that their pentagonal house has been trashed. Things seem dire for Goldie, but suddenly she remembers that “the shortest distance between two points is a straight line,” and she dashes to safety.

How you can use this source: This book can be read independently by most students in Grade 2 and above. The best audience for the book, however, is pre-kindergarten through Grade 2 since the focus is on simple geometric shapes. This makes Goldie Locks and the Three Squares an ideal teacher read aloud book in lower elementary classrooms. The teacher can have students name the shapes on each new page before reading the text, and the teacher can highlight the geometry in the book by drawing and labeling the shapes on a whiteboard or writing easel as they appear in the story. At the end, students can recall which objects in the house were shaped like triangles, squares, rectangles, etc. As an extension activity, younger students can draw a scene from the book and label the shapes, and more advanced students can write and illustrate an additional scene for Goldie Locks and the Three Squares or give another fairytale a geometric twist (e.g., “Snow White and the Seven Shapes,” “Little Rectangular Riding Hood,” etc.).

Maccarone, G. (1996). The silly story of Goldie Locks and the Three Squares. New York: Scholastic.

23 Web Resources

Web Resources is a collection of five websites that can be utilized for different aspects of mathematics education. Some provide kid-friendly, online games to give students additional math practice. Others are searchable databases for math lesson plans and math worksheets, and some permit teachers to create their own worksheets. I am including this in my portfolio to show that I am comfortable navigating math resources on the web.

The “Math Playground” website is very kid-friendly, and children will not need much help from adults to jump in and get started. The “Math Lesson Plans Page” provides some good lesson plans, but some of the lessons on it are of lower caliber. After feedback from my professor, I went back and ordered these web resources according to level of usefulness, with the most effective sites at the beginning.

I learned how many math resources are available online to assist teachers, students, and parents. Working on Web Resources gave me an opportunity to think about how I might incorporate these online resources into my teaching. For example, I may do online searches of math lesson websites to get ideas for my own lessons, go to math worksheet websites for problem ideas or to create my own math worksheets, and have students visit the math game websites at the computer center or share links to the math games with parents to encourage practice at home.

24 Web Resources

Title: Math Playground

Address: http://www.mathplayground.com

Description of Contents:

This site, which was created by teachers, was designed for elementary and middle school students. The left tab leads to sections on “Math Games” (such as “Math Millionaire,” “Number Invaders,” and “Making Change”); “Word Problems” and “Logic Puzzles” (which are organized by grade level); and “Math Videos” (which provide step-by-step instruction in math concepts). The main page includes links to “Computation” (where students can practice basic facts for addition, subtraction, multiplication, division, fractions, percents, etc.); “Flashcards” (where students race the clock to get through as many problems as possible); “Worksheets” (teachers can select from pre-made worksheets or create their own); “Manipulatives” (highly interactive and colorful math games); and “Thinking Blocks” (videos that help students through word problems step- by-step with colorful, movable blocks). There is even an activity to introduce students to basic computer programming! I tried out activities in each category and found the links to be very colorful, engaging, and kid-friendly. These activities are several steps above most of the free math games for kids available on the internet, which are often in drab, unappealing formats and/or have complicated instructions that would require help from an adult. The activities and games are all free, and none of them seemed to require special downloads or plug-ins, unlike at other sites.

How You Can Use This Source:  Teachers can search “Word Problems” and “Worksheets” to get ideas for problems to present in class.  Teachers can also listen to the how-to “Math Videos” for ideas on how to teach various topics.  The “Thinking Blocks,” “Manipulatives,” “Flashcards,” and “Math Games” are engaging, easy-to-use activities that students could enjoy in the classroom’s computer center or at home for extra practice.

APA Citation:

Math Playground. (2008). Math Playground. Retrieved September 9, 2008, from http://www.mathplayground.com.

25 Title: Ed Helper – Math

Address: http://www.edhelper.com/math.htm

EdHelper.com is a resource for free worksheets across the content areas, and its math section is especially extensive. You can find worksheets through a keyword search, and they are also organized by topic and grade level, from preschool through high school math. In the lower grades, printable math books are available to help children with concepts such as patterns and counting by 2s. In addition to traditional math concepts, worksheets are also available on math puzzles, critical thinking puzzles, word problems, and math sequences. Pre-made worksheets are available, or teachers can build their own. Mixed review worksheets are also offered for many grade levels, ranging from Level 1 (for the first two months of school) through Level 4 (for the final months of school). I have used these worksheets before for tutoring and they do not quite conform to the Virginia SOL grade levels, but they are fairly close. Also, while the problems generated are of high quality, the actual format and presentation of the problems are mediocre. Therefore, teachers may not want to use these worksheets in the format presented, but instead look to the worksheets to generate exercises that they can input into their own neater format and then print. Nonetheless, these worksheets are very useful for assignments needed on short notice, such as if plans for the day suddenly change and the teacher needs to generate a math worksheet quickly. Another idea is to print off these worksheets for an individual student who is particularly strong or particularly weak and who needs additional challenge or remediation beyond their regularly assigned work.

How You Can Use This Source:  Teachers can quickly create math worksheets for students. This is especially useful if lesson plans for the day change unexpectedly and the teacher needs to generate a math assignment quickly.  Teachers can print off math puzzles and printable math books.  This is a good resource for parents who want to give their children extra math practice at home.

APA Citation: Ed Helper. Math Worksheets, Puzzles, Printables, Problems, Test Prep. Retrieved September 9, 2008, from http://www.edhelper.com/math.htm.

26 Title: Math Worksheet Wizard

Address: http://www.mathworksheetwizard.com

This site provides thousands of free math worksheets, and unlike most other worksheet sites, the focus is strictly on math. Worksheets are available for kindergarten through third grade, and a more limited selection is available for fourth grade. This site is excellent for lower elementary teachers, who may have trouble finding quality worksheets for primary grade students at other math worksheet sites. The Math Worksheet Wizard main page provides links to the aforementioned grade levels which are further broken down by specific math concepts. For example, kindergarten teachers can choose from topics such as number, shape, color, and money, and third-grade teachers can choose from more advanced concepts such as arithmetic and measurement. In addition to choosing a pre-made worksheets, teachers can create their own. The link to “Arithmetic Worksheets” prompts teachers to choose amongst the arithmetic processes (addition, subtraction, multiplication, division, or mixed review) and select the number of problems, the size of the problems, and whether they would like regrouping or remainders to be required. There are templates for both vertical and horizontal math exercises. The link to “Math Forms and Charts” provides templates for multi-purpose math forms, and teachers can select the title, font size, number of rows, number of titles, and text alignment in order to customize the forms for their purposes. Different templates are recommended for activities such as student surveys, comparing two amounts, money and counting, and multi-step problem solving.

How You Can Use This Source:  Teachers can get access to free math worksheets organized by grade level and math concept.  Teachers could steer parents towards this website as a good source for at-home practice.

APA Citation:

Math Worksheet Wizard. (2008). Make Math Worksheets! – Math Worksheet Wizard. Retrieved September 9, 2008, from http://www.mathworksheetwizard.com.

27 Title: New Hampshire Estates Elementary School – Useful Links

Address: http://www.montgomeryschoolsmd.org/schools/nhees/useful%20links.htm

This is a large collection of online resources for grades k-2, mainly in math and reading, that was put together by the technology teacher at a school in Montgomery County, MD, where I worked as a teacher assistant. Teachers and students alike at the school loved this site, which has links to high-quality games and resources organized by grade level and subject. What impresses me most about the math and reading game links is how engaging and child-friendly they are, allowing students to practice independently. Even kindergarteners at the school were adept at navigating to this page from the school main page and clicking on links to websites with their favorite games and activities, such as “Rainforest Math,” “Cool Math 4 Kids,” and “Funbrain.” There is also a special section on resources geared to second-language learners. The activities and games are all free, and none of them seemed to require special downloads or plug-ins, unlike at other sites.

How You Can Use This Source:  The children can access and complete these math games and activities independently at the classroom computer learning center. If their parents are given the site link to bookmark, students can also access these resources and get extra practice at home.  Teachers can take advantage of the teacher links to various content areas when looking for lesson plan ideas or other teaching resources.

APA Citation:

New Hampshire Estates Elementary School. (2008). Useful links. Retrieved September 9, 2008, from http://www.montgomeryschoolsmd.org/schools/nhees/useful %20links.htm.

28 Title: Math Lesson Plans Page

Address: http://www.lessonplanspage.com/Math.htm

The general Lesson Plans Page website provides free lessons by grade level across the subject areas, and this special math section is very extensive in its own right. On the main math page, teachers are prompted to choose their grade level, which is broken into PreK- 1, Grades 2-3, and Grades 4-5, and goes all the way up to high school math. On many lesson plan websites, I found the options for the lower elementary grades very sparse, but this site has ample lesson plans across every age group. Within each grade level range, a long list of lesson plans are organized by math topics, and then there are numerous lesson plans for interdisciplinary teaching, arranged by content area (for example, math lesson plans with art connections, with computer connections, with social studies connections, etc.). In addition, the main math page provides links to math worksheets, seasonal lesson plans for each month, and a “Recent Additions” page where teachers who have already combed through all the math lesson plans for their grade level can check back periodically for newly uploaded lessons. The site’s left tab also provides links to education articles, an extensive teacher discussion forum, inspirational teacher stories, a 10-step lesson plan guide, and the Lesson Plans Page newsletter.

How You Can Use This Source:  Teachers can use this to find specific lesson plan ideas arranged by grade level and math concepts.

APA Citation:

Math Lesson Plans Page. (2008). The Lesson Plans Page – Math Lesson Plans, Math Ideas, and Math Activities. Retrieved September 9, 2008, from http://www.lessonplanspage.com/Math.htm.

29 Reflection

I believe that I am prepared and ready to be a good math teacher. There will be many opportunities for professional growth over the course of my career, but this portfolio shows that I already have the basic foundations needed to be an effective math educator at the elementary level. As my sample work proves, I am capable of incorporating manipulatives into my math instruction, differentiating based on ability level, and making connections between math and other areas such as literature. In addition, my web resources show that I am savvy about online resources and have thought about ways to incorporate technology such as online math games into the classroom.

My money counting problem (page 14) reveals my desire to incorporate manipulatives into the classroom as much as possible. I had first-grade and kindergarten students figure out all the possible coin combinations to get to 12 cents by working hands-on with plastic money coins. My lesson plan on addition story problems (page 4) also shows my interest in using manipulatives because during the direct instruction portion of the lesson I showed students how to model each problem with manipulatives, and during their independent work I encouraged them to either use manipulatives or drawings to figure out each problem.

I am capable of differentiating instruction to different ability levels, as shown by my story problem lesson plan (page 4). During the instruction portion, if volunteers who came to the writing easel to fill in answers struggled, I used manipulatives to help them better understand the problem. I allowed stronger students to simply draw pictures to figure out the answer, or even to do it in their head if they were able. I also provided a

30 range of addition story problems, including result unknown and change unknown as well as two addends versus three addends in order to adequately challenge all levels of students. For students’ independent work, I differentiated even more, creating three different worksheets for varying ability levels. The easiest story problem worksheet provided drawings on many of the problems in order to provide scaffolding for students, while the most difficult story problem worksheet included change unknown problems in order to challenge students.

Finally, my literature connections (page 17) show my enthusiasm for interdisciplinary teaching that ties math to other subjects. I am very interested in using read aloud literary texts in order to help make math more accessible to students and to help students see the connection between math and their daily lives. For example, the book Apple Fractions applies fractions to apple slices, and there are connections to science throughout the book. The book The Silly Story of Goldie Locks and the Three

Squares relates geometry to the popular children’s story and encourages children to look for geometric shapes in the book’s illustrations (and by extension in their own surroundings).