Document 11663990

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 Early Mathematics
A Resource for Teaching Young
Children
Prekindergarten
A publication of
The Charles A. Dana Center at
The University of Texas at Austin
2012
Early Mathematics—A Resource for Teaching Young Children Frontmatter Copyright 2012, 2011, the Charles A. Dana Center at The University of Texas at Austin
Unless otherwise indicated, the materials in this resource are the copyrighted property of the Charles A. Dana Center
at The University of Texas at Austin (the University). No part of this resource shall be reproduced, stored in a
retrieval system, or transmitted by any means—electronically, mechanically, or via photocopying, recording, or
otherwise, including via methods yet to be invented—without express written permission from the University. We
use all funds generated through use of our materials to further our nonprofit mission. Please send your permission
requests or questions to us at this address:
Charles A. Dana Center
The University of Texas at Austin
1616 Guadalupe Street, Suite 3.206
Austin, TX 78701-1222
Fax: 512-232-1855
[email protected]
www.utdanacenter.org
The Charles A. Dana Center and The University of Texas at Austin, as well as the authors and editors, assume no
liability for any loss or damage resulting from the use of this resource. We have made extensive efforts to ensure the
accuracy of the information in this resource, to provide proper acknowledgement of original sources, and to
otherwise comply with copyright law. If you find an error or you believe we have failed to provide proper
acknowledgment, please contact us at [email protected]
This edition was developed in Microsoft Word.
October 2012 release.
As always, we welcome your comments and suggestions for improvements. Please contact us at
[email protected] or at the mailing address above.
About the Charles A. Dana Center at The University of Texas at Austin
The Dana Center strengthens our nation’s education systems to provide a reliable path to upward mobility for all
students. Our work focuses on mathematics and science education, with an emphasis on strategies for improving
student engagement, motivation, and persistence. We are dedicated to nurturing students’ intellectual passions and
ensuring that every student leaves school prepared for success in postsecondary education and the contemporary
workplace—and for active participation in our modern democracy.
We advocate for high academic standards, and we collaborate with local partners to build the capacity of education
systems to ensure that all students can master the content described in these standards. We help our partners adapt
promising research to meet their local needs.
We develop innovative curricula, tools, protocols, instructional supports, and professional development systems that
we implement through multiple channels, from the highly local and personal to the regional and national. We
provide long-term technical assistance to school and district leadership teams, advise community colleges and states,
and collaborate with national partners on work such as our Urban District Leadership Networks, Academic Youth
Development project, and Advanced Mathematical Decision Making course.
We have significant experience and expertise in the following:
•
•
Standards development and implementation,
systemic reform, and district capacity
building
Education leadership, instructional coaching,
and teaching
•
•
K–14 course design and development, learning
networks, and programs for bridging critical
transitions
Research, content development, and publishing
The Center was founded in 1991 at The University of Texas at Austin. Our staff of nearly 80 researchers and
education professionals has worked with dozens of school systems in nearly 20 states and with 90 percent of Texas’s
more than 1,000 school districts. We are committed to ensuring that the accident of where a child attends school
does not limit the academic opportunities he or she can pursue. For more information about our programs and
resources, see our homepage at www.utdanacenter.org.
The Charles A. Dana Center at the University of Texas at Austin Page | ii 2012 Early Mathematics—A Resource for Teaching Young Children Frontmatter About the Common Core State Standards for Mathematics
This resource is aligned to the Common Core State Standards for Mathematics.
The CCSS for Mathematics and for English Language Arts are copyrighted by the National Governors Association
Center for Best Practices (NGA Center) and the Council of Chief State School Officers (CCSSO) and are available
at www.corestandards.org/the-standards; these CCSS are being used under the NGA Center and CCSSO Public
License, available at www.corestandards.org/public-license.
Any excerpts of the Common Core State Standards included in this resource are
© Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State
School Officers. All rights reserved.
About the development of this resource
This new revised and expanded edition of Early mathematics: A resource for teaching young children consists of
materials for 20 sessions for each of four grades—prekindergarten, kindergarten, grade 1, and grade 2—for a total of
80 sessions. We were able to develop these materials because of a generous December 2010 grant from the
Noyce Foundation.
First edition (2011)
Twenty of these sessions (10 for prekindergarten and 10 for second grade) were initially developed in spring and
summer 2011 by early mathematics education experts Brian Mowry (prekindergarten) and Carolyn Moore (second
grade), and reviewed in summer and early fall 2011 by ACE: A Community for Education leaders Chetan Makan
and Mary Ellen Isaacs, both of whom are experts in designing and implementing early childhood tutoring programs
that can be implemented at scale. The materials were also reviewed by Patti Bridwell, who has expertise in
professional supports for teachers and tutors.
This first edition was released in fall 2011 as a proof-of-concept resource titled Early mathematics: Resources for
tutoring young children. These initial 20 sessions were field-tested in fall 2011 by tutors from the Dana Center’s
ACE: A Community for Education (www.utdanacenter.org/ace) program in Austin, Texas (prekindergarten
sessions), and by tutors from Experience Corps (www.experiencecorps.org) in Philadelphia, Pennsylvania (secondgrade sessions).
Second edition (2012)
A key finding from the fall 2011 proof-of-concept field testing was that the material as written was probably too
complex for paraprofessionals (e.g., tutors) to deliver, but that it could be very effective if delivered by classroom
teachers. Based on this feedback, we have substantially revised the initial 20 sessions for this new edition, changing
the intended users of this resource from paraprofessionals to classroom teachers.
All 80 sessions are built on recommendations in the 2009 National Research Council report Mathematics Learning
in Early Childhood: Paths Toward Excellence and Equity (Committee on Early Childhood Mathematics;
Christopher T. Cross, Taniesha A. Woods, and Heidi Schweingruber, editors). Center for Education, Division of
Behavioral and Social Sciences and Education. Washington, D.C: The National Academies Press.
In particular, these session materials speak to the recommendation that:
Mathematics experiences in early childhood settings should concentrate on (1) number (which includes whole
number, operations, and relations) and (2) geometry, spatial relations, and measurement, with more
mathematics instruction time devoted to number than to other topics.
Accordingly, our materials focus primarily on number.
The Charles A. Dana Center at the University of Texas at Austin Page | iii 2012 Early Mathematics—A Resource for Teaching Young Children Frontmatter About the Noyce Foundation
The Noyce Foundation1 aims to help young people become curious, thoughtful, and engaged learners. The Noyce
Foundation focuses on a few key areas:
• improving the teaching of math and science in public schools;
• developing leadership to support student achievement;
• supporting education policy and research; and
• expanding opportunities for students to experience hands-on science in out-of-school settings.
The Noyce family created the Noyce Foundation in 1990 to honor the memory and legacy of Dr. Robert N. Noyce,
cofounder of Intel and inventor of the integrated circuit—which fueled the personal computer revolution and gave
Silicon Valley its name.
Although he was an individual of daunting talents and intellect who was honored by two presidents as well as by his
academic and industry peers around the world, Bob Noyce also remained a humble and approachable man who
believed fervently in democracy. In everything the Noyce Foundation undertakes, it remains committed to
promoting the qualities that Bob Noyce embodied: optimism, creativity, risk taking, and determination.
In recognition of Bob’s concern about the narrowing pipeline of students interested in—and committed to—sciencerelated careers, the Noyce Foundation has focused on mathematics, science, and associated work in research and
policy. Much of the Foundation’s focus has been on improving instruction in mathematics, science, and early
literacy in public schools.
As schools began to intensify their focus on math and literacy in response to No Child Left Behind—leaving science
behind in the process—the Noyce Foundation emphasized support for out-of-school science programs that show
promise of sustaining and engaging student interest through middle school, a time when students tend to make
critical decisions about what subjects they want to pursue in the future. The Noyce Foundation informal science
initiative includes support for leadership development in science centers.
For more information about the Noyce Foundation, visit its website at www.noycefdn.org. For more information
about the Silicon Valley Mathematics Initiative, see www.svmimac.org.
1
This description of the Noyce Foundation’s mission and history was adapted from content retrieved from its homepage
(www.noycefdn.org) and its About Us page (www.noycefdn.org/aboutus.php) on October 9, 2012.
The Charles A. Dana Center at the University of Texas at Austin Page | iv 2012 Early Mathematics—A Resource for Teaching Young Children Frontmatter Acknowledgments
With special thanks . . .
The Dana Center thanks the Noyce Foundation for its generous support of this project. The Noyce family
created the Noyce Foundation in 1990 to honor the memory and legacy of Dr. Robert N. Noyce,
cofounder of Intel and inventor of the integrated circuit—which fueled the personal computer revolution
and gave Silicon Valley its name. For more information about the Noyce Foundation, visit its website at
www.noycefdn.org.
All individuals listed below are affiliated with the Dana Center unless otherwise noted.
This edition
Project Lead
Patti Bridwell, senior program coordinator
Authors
Brian Mowry, M.A., consultant (prekindergarten and kindergarten sessions)
David Hughes, M.A., consultant (grade 1 and grade 2 sessions)
Reviewers
Patti Bridwell, senior program coordinator
Editing and Production Staff
Steve Engler, lead editor and production editor
Rachel Jenkins, consulting editor
Phil Swann, consulting designer
Dawn Watkins, freelance illustrator
First edition, 2011
Project Leads
Uri Treisman, Dana Center executive director
Lester Strong, Experience Corps chief executive officer
Project Managers
Mary Ellen Isaacs, Ph.D., ACE program director
Patti Bridwell, senior program coordinator
Authors
Brian Mowry, M.A., consultant (prekindergarten session materials and page design for all sessions)
Carolyn Moore, M.A., consultant (grade 2 tutoring session materials)
Experience Corps
Amy Zandarski-Pica, vice president, education & strategy
Deborah Stiller, Washington, DC Experience Corps executive director
Evette Lucas-Mathis, Philadelphia Experience Corps director, academic tutoring & mentoring
Lester Strong, chief executive officer
Reviewers
Patti Bridwell, senior program coordinator
Chetan Makan, program coordinator, ACE
Editing and Production Staff
Cara Hopkins, editor (prekindergarten)
Rachel Jenkins, editor (grade 2) and production editor
Phil Swann, consulting designer
Norma Salas, print production manager
Fall 2011 Proof-of-Concept Test Sites
Austin, Texas
Philadelphia, Pennsylvania
The Charles A. Dana Center at the University of Texas at Austin Page | v 2012 Early Mathematics—A Resource for Teaching Young Children The Charles A. Dana Center at the University of Texas at Austin Page | vi Frontmatter 2012 Early Mathematics—A Resource for Teaching Young Children Introduction Introduction Background Early Mathematics—A Resource for Teaching Young Children provides a series of instructional tasks, aligned with the Common Core State Standards for Mathematics, that teachers can use to instruct children in prekindergarten, kindergarten, grade 1, and grade 2. The complete resource includes content for 20 sessions for each of these four grade levels. The tasks were developed for whole-­‐class instruction with some small-­‐group work, but they are also easily adaptable for tutoring sessions. The estimated timeframe for each session is as follows: Grade level Prekindergarten Estimated time per session 30 minutes Kindergarten 45 minutes Grade 1 45 minutes Grade 2 45 minutes Most sessions have a literature focus to draw children into the content and/or to keep them connected to a context. These session materials do not provide everything a child needs to know about a given topic, such as number. Rather, each session provides a series of instructional tasks to help you teach selected content and practices described in the Common Core State Standards for Mathematics. You should feel free to modify the sessions as appropriate to meet the individual needs of children in your classroom. Alignment We have embedded key Common Core State Standards for Mathematical Practice in each session to help bring out crucial ideas. In most sessions, though, additional Standards for Mathematical Practice beyond those selected may also be relevant. We chose the content for these sessions based on what content we believe will have the most significant effect on student learning. The language below is drawn from the National Council of Teachers of Mathematics 2006 publication, Curriculum Focal Points for Prekindergarten Through Grade Eight Mathematics: A Quest for Coherence (prekindergarten) and the Common Core State Standards for Mathematics (kindergarten onward). Prekindergarten (1) developing an understanding of whole numbers, including concepts of correspondence, counting, cardinality, and comparison. Kindergarten (1) representing, relating, and operating on whole numbers, initially with sets of objects; The Charles A. Dana Center at the University of Texas at Austin Page | vii 2012 Early Mathematics—A Resource for Teaching Young Children Introduction Grade 1 (1) developing understanding of addition, subtraction, and strategies for addition and subtraction within 20; (2) developing understanding of whole number relationships and place value, including grouping in tens and ones; (3) developing understanding of linear measurement and measuring lengths as iterating length units. Grade 2 (1) extending understanding of base-­‐ten notation; (2) building fluency with addition and subtraction. Structure Each session is divided into three instructional formats—Activate, Engage, and Develop. The activate portion introduces the content in the session and objectives that will be developed in the forthcoming session. In prekindergarten and kindergarten, this section can often occur as a part of the morning circle routine (e.g., calendar, morning message), or it can serve as a transition activity that incorporates songs, movement, and other instructional activities developed to capture the interest and attention of younger students with emerging attention spans. Then children will engage in the content through an activity centered on the content and practices in the standard(s) being addressed. In prekindergarten and kindergarten, this time is spent mostly in whole (or large) group so that the teacher can model the mathematical thinking that children will apply in the Develop section. For younger children, keep in mind that whole-­‐
group sessions are designed to last no longer than 20 minutes. Each session ends with develop, which provides children an opportunity to share and analyze their understandings and/or methods. In prekindergarten and kindergarten, the activities in this section can take place during centers, small group, or math station time. Throughout, the role of the teacher will primarily be to ask probing questions to help children make sense of the content in the session. The Charles A. Dana Center at the University of Texas at Austin Page | viii 2012 Early Mathematics A Resource for Teaching Young Children Prekindergarten Session 1: Benny’s Pennies (Introduction) ..................................................................................... 1 Session 2: Identifying and Making Sets of 5 ................................................................................... 4 Session 3: Identifying and Making Sets of 4 ................................................................................... 7 Session 4: Identifying and Making Sets of 3 ................................................................................. 11 Session 5: Identifying and Making Sets of 2 ................................................................................. 14 Session 6: Counting and Comparing Quantities 0–5 .................................................................... 19 Session 7: Ordering and Labeling Quantities with Numerals 1–5 ................................................ 23 Session 8: Labeling Sets with Numerals 0–5 ................................................................................ 31 Session 9: Combinations of 5 ....................................................................................................... 38 Session 10: Joining and Separating Sets of 5 ................................................................................ 42 Session 11: The Hungry Caterpillar .............................................................................................. 46 Session 12: Strategies for Making Equal Sets ............................................................................... 51 Session 13: Conceptualizing 10 (the Tens-­‐Frame) ........................................................................ 59 Session 14: Ordering Numerals 1–10 ........................................................................................... 63 Session 15: Counting and Creating Sets of 6 ................................................................................ 71 Session 16: Counting and Creating Sets of 7 ................................................................................ 79 Session 17: Counting and Creating Sets of 8 ................................................................................ 83 Session 18: Counting and Creating Sets of 9 ................................................................................ 90 Session 19: Joining and Separating Sets Up to 10 ........................................................................ 95 Session 20: Comparing and Ordering Sets to 10 ........................................................................ 100 A publication of The Charles A. Dana Center at The University of Texas at Austin 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Session 1: Benny’s Pennies (Introduction) At A Glance This session begins with children reciting the counting word sequence to 5. Then, the teacher introduces Benny’s Pennies by Pat Brisson. Each time Benny spends 1 penny (every item he buys costs 1 penny), children are able to visualize a one-­‐to-­‐one relationship—1 penny for 1 gift. The teacher uses linking cubes to highlight the equal sets of 5 presented in the story. After each exchange (e.g., 1 penny for 1 cookie), children count the number of pennies Benny has spent and the number of gifts he has bought. The focus is on recognizing and naming quantities to 5 using counting and noncounting strategies (e.g., eyeballing how many). Materials Benny’s Pennies by Pat Brisson ● Linking cubes ● Empty jar and 5 pennies ●
Mathematics Addressed ●
Use words to rote count to 30. ●
Count 1–10 items, with one count per item. ●
Without counting, verbally identify the number of objects 1–5. The Charles A. Dana Center at the University of Texas at Austin Page | 1 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Activate Display 5 pennies in the palm of your hand. •
Ask, “How many pennies do you see?” Confirm children’s responses by counting out loud as you touch each penny. Display the empty jar. Direct children to count out loud and lift one finger for each penny you drop inside. Congratulate children for their counting. Go around the circle and giving each child a “high five.” Display the penny jar. Explain how you plan to use it for the duration of the next 10 sessions. •
“I am going to collect pennies in this jar. We will count them together every time we meet. When I get 25 pennies, I’ll exchange them for a quarter.” Transition to the next activity by discussing some things that you could purchase with a quarter. Engage Introduce Benny’s Pennies. Draw attention to the book’s cover. •
“Today we are going to read a story about a boy who wants to use pennies to buy gifts for his family. How many pennies do you see?” Begin reading the story. Pause from time to time to monitor children’s comprehension. •
For example, “How many pennies has he spent now? How many (which) gifts has he bought so far? Which family member will get this gift?” The Charles A. Dana Center at the University of Texas at Austin Page | 2 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Develop Count 5 pennies (from the penny jar) onto the carpet. Have children count with you. ● Direct children to lift one finger for each family member shown in the story. ● Match each penny displayed on the carpet to all 5 fingers on your hand. Emphasize the one-­‐to-­‐one relationship. ● Once again, count the number of pennies and then the number of family members. Emphasize the last number you say in both counts to help children see that the number of pennies is the same as the number of people in Benny’s family. ●
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Take a red, brown, white, yellow, and green cube. Tell children which gift from the story each color cube represents: (red = the rose, brown = the cookie, white = the hat, yellow = the bone, green = the fish). Distribute the cubes to different children. (Some children may get more than 1 cube if the group is smaller than 5). Take the 5 pennies from the carpet and arrange them in the palm of your hand. Call on each child to exchange his/her cube for 1 penny. ● Arrange all 5 cubes on the carpet. Have children place the penny they took from the exchange and match it to a corresponding cube. Count out loud for each match. Emphasize how both amounts are the same. ● Distribute a red, brown, white, yellow, and green cube train to each child. Have children use their fingers to represent 5 people and then count 5 cubes for each finger. ●
The Charles A. Dana Center at the University of Texas at Austin Page | 3 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Session 2: Identifying and Making Sets of 5 At A Glance This session begins with children comparing 1 penny to a collection of 5, discussing which set is more or less, which is easier to count, and why. The teacher then makes different arrangements with a set of 5 pennies. Children count the pennies with the teacher to verify that the total stayed the same. The focus is on understanding that a quantity does not change when it is rearranged. Next, the teacher makes an arrangement with 5 sticks on a paper plate, which he/she shows for 5 seconds. The teacher then hides the images as children try to recreate them from memory. The focus is on quickly recognizing a set of 5 or seeing it as a sum of parts (e.g., 2 sticks on top and 3 on the bottom). Materials ●
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Collection jar 6 pennies Container of sticks (e.g., popsicle sticks) 1 sheet of paper per child crayons Mathematics Addressed ●
Demonstrate that the order of the counting sequence is always the same regardless of what is counted. ●
When counting, demonstrate understanding that items can be chosen in any order. ●
Verbally identify without counting the number of objects 1–5. The Charles A. Dana Center at the University of Texas at Austin Page | 4 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Activate Display 1 penny in the palm of your left hand and a set of 5 pennies in your right hand. Compare which hand has more/less pennies. ● Close the hand with 5 pennies, leaving the hand with 1 penny open. Prompt children to tell how many. ● Close the hand with 1 penny and reopen the hand with 5. Challenge children to determine how many are in that hand. ● Display both hands. Extend the hand with 1 penny. Discuss why it was easier to quantify that amount. ●
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Place the 1 penny inside the collection jar and display the set of 5 pennies on the carpet/table in front of you. Explain the purpose of today’s session. o Say, “When there is more than just one, it is hard to tell how many just by looking. Sometimes we have to count.” Engage Count the pennies on the carpet/table in front of you. Confirm that there are 5. ● Shake the coins inside the cup and spill them onto the carpet. Encourage children to help you verify that there are still 5. ● Arrange the pennies so that there are two rows—1 on the top and 4 on the bottom. o “Are there still 5 pennies?” ● Make a circle with the pennies. Count them one more time and point out that the total has not changed even though they were rearranged. ●
The Charles A. Dana Center at the University of Texas at Austin Page | 5 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Develop ●
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Explain the goal for today’s session. o “Today, you are going to make your own story like Benny’s Pennies. Instead of being about 5 pennies, you will create a story about something you can build with 5 sticks.” Display a container of sticks on the carpet/table in front of you. Invite children to come up to the container one at a time and count out 5 sticks. Model how to use the fingers on their hand as a way to make an equal set of 5. Give each child a sheet of paper to use as a placemat for his/her set of sticks. (Note: Build the house so that it is facing children.) Instruct children to watch closely as you make the shape of a house with your set of sticks. Display the image for 5 seconds and then hide it with a sheet of paper. Instruct children to recreate the image from memory. Wait until children feel they have recreated the picture you made as best as they could with their set of sticks. Uncover the image. Encourage children to make changes to their picture so that it looks exactly like yours. Make another image with the same set of sticks. For example, create a tally mark image of 5 with 4 vertical lines and 1 stick arranged diagonally across the top. Cover up the image. Instruct children to recreate the image from memory. As children finish, uncover the image. Encourage them to make changes to their picture so that it looks exactly like yours. Encourage children to discuss how they remembered seeing the image. o “The first picture looked like a house. Describe the new image you just copied.” ● When children finish their stick images, model how to write the number 5 on their picture. o “Use your marker to write the numeral 5. Go straight down, around and down, back to the top, and straight across (to the right).” ●
The Charles A. Dana Center at the University of Texas at Austin Page | 6 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Session 3: Identifying and Making Sets of 4 At A Glance The teacher adds 1 penny to the set of 5 that were already in the collection jar as of Session 1, bringing the total to 6. Children practice clapping and saying the counting words in sequence to 6 as the teacher drops 1 penny at a time into the collection jar. The teacher then shows a group of 4 pennies, which he/she will add to the penny collection jar during Session 4. Children are encouraged to identify how many pennies they see without counting the coins one by one. Children then sort themselves by age (4-­‐year-­‐olds and 5-­‐year-­‐
olds) and begin to make preparations for celebrating Benny’s upcoming fourth birthday. The teacher models how to make equivalent sets, matching 4 fingers to 4 cubes. Children are encouraged to make various arrangements of 4 candles and then draw a pictorial representation of their pretend birthday cake. Mathematics Addressed ●
Use words to rote count to 30. ●
Count up to 10 items and demonstrates that the last count indicates how many items were counted. ●
Verbally identify, without counting, the number of objects from 1–5. Materials ●
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Collection jar with 6 pennies 4 extra pennies “How Old Are You?” chart (chart paper divided in two columns—
one labeled 4-­‐year-­‐olds and the other 5-­‐year-­‐
olds) Linking cubes 1 box of crayons per pair of children Birthday Cake blackline master The Charles A. Dana Center at the University of Texas at Austin Page | 7 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Activate ●
Remind children that during the last session you added 1 penny to the set of 5 that were already in the collection jar. Remove the pennies from the jar. Direct children to count out loud and clap as you drop each of the 6 pennies back inside the jar. Confirm that there are 6 altogether. ● Remind children of the purpose of the penny collection jar. o Say, “When there are 25 altogether, I will exchange the pennies for a quarter.” ●
Engage ●
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Arrange a set of 4 pennies in the shape of an L in the palm of your hand. Prompt children to identify how many they see. Model how to identify the parts (3 and 1) in relation to the whole (4). Then count all the pennies to verify that there are 4. Draw upon what children already know about four—their age. Connect this information to a birthday scenario, which is the focus of today’s session. Sort children into separate lines by age. Count the number of children in each line. Display a sheet of chart paper divided into two categories—4-­‐year-­‐olds/5-­‐year-­‐olds. As children return to their seats, record their names in the appropriate category. o “Which line has more children? Which group has fewer children?” The Charles A. Dana Center at the University of Texas at Austin Page | 8 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Develop Introduce a pretend scenario that sets the stage for today’s session. o “Benny will be turning 4 soon. How many candles will he have on his cake?” ● Distribute a handful of cubes. Show a configuration of 4 fingers. Instruct each child to make an equivalent set of fingers. ● Display a set of 4 cubes in a 2-­‐by-­‐2 arrangement on the Birthday Cake blackline. ● Distribute cubes and a Birthday Cake blackline to each child. Cover the cube arrangement you made on your cake with a sheet of paper. Encourage children to use their cubes to recreate the image. ●
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Remove the sheet of paper and instruct children to compare their arrangement to yours. Prompt children to discuss how they see the arrangement of cubes. Encourage children to make a new arrangement. Have children count to verify that there are still 4 cubes altogether (i.e., the amount remained the same even though the cubes were rearranged). Distribute a box of crayons to pairs of children. Instruct them to count out 4 crayons. Direct children to use 4 different-­‐colored crayons to draw a set of 4 candles on top of the cake. Children then take a yellow crayon and count out loud as they light the candles on their cake picture. Model how to write the numeral 4 on the picture of the cake. o “This is how you write the number 4. Go down halfway. Stop. Go across to the right. Now go back to the top, and go straight down.” The Charles A. Dana Center at the University of Texas at Austin Page | 9 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten The Charles A. Dana Center at the University of Texas at Austin Page | 10 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Session 4: Identifying and Making Sets of 3 At A Glance The teacher adds the 4 pennies introduced in Session 3 to the existing set of 6 already inside the collection jar. The teacher then models counting on from 6 all the way up to 10. The focus is on reciting the sequence of counting words to 10. Next, the teacher establishes the following scenario as the focus for today’s session: “Benny plans to invite 3 friends to his birthday party. How many placemats, cups, spoons, and plates will Benny need for each friend that comes to his birthday party?” Children count out a set of 3 cubes and find a matching set of 3 placemats, cups, spoons, and plates in preparation for Benny’s party. Children then make a “3 collage” out of dot stickers, cotton balls, and toothpicks. The focus is on quickly recognizing and making sets of 3. Materials ●
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Collection jar with 10 pennies Linking cubes Paper cups Tongue depressors Paper Mathematics Addressed ●
Use words to rote count to 30. ●
Count up to 10 items and demonstrate that the last count indicates how many items were counted. ●
Without counting, verbally identify the number of objects 1–5. The Charles A. Dana Center at the University of Texas at Austin Page | 11 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Activate Display the penny collection jar containing 6 pennies. Take them out of the jar and count them to verify that there are 6. ● Place the 6 pennies back inside the jar. Display the 4 additional pennies that were introduced in Session 3. o Ask, “How many pennies did I find last time?” ● Model counting on from 6 as you drop the 4 new pennies inside the jar. Emphasize the new total. ●
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Direct children to count out loud and clap as they recite the counting sequence to 10. Engage ●
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Ask a volunteer to take 3 pennies from the jar. Invite children to count out loud as you point to and count each penny in your hand. Place the 3 pennies back inside the jar, and then put the collection jar aside. Introduce a pretend scenario about the number 3. o “Benny celebrates his birthday today. He wants to invite 3 friends to his party, just like the number of pennies we just counted.” Have children show 3 fingers. Distribute a handful of cubes to each child. Tell them to place 1 cube on each finger. Instruct children to count out loud as they remove each cube from their fingers and build a tower of 3. Model the counting and connecting process. ●
Turn around, facing the same direction as children are watching, and model how to write the numeral 3. o “To write three, you go down, around, and back; then down, around, and back.” The Charles A. Dana Center at the University of Texas at Austin Page | 12 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Develop Set out a collection of paper placemats, cups, and tongue depressors. ● Explain to children that they will use the tower of 3 cubes they just made to help them count out as many table supplies as there are friends coming to the party. ● Assign each child a job to help prepare for the pretend birthday party (i.e., someone counts out 3 tongue depressors; another child counts out 3 cups; another child counts out 3 placemats; and one child can be the checker who makes sure everyone got the right amount of items). ●
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Model how to use the cube tower to make an equal set of 3 table items. Take 1 cube from the tower and place it on top/inside of each matching item. Direct children where to place their table items. Count the table items to verify that each set has 3. Have children drop their cubes one at a time inside their cups. Have children sing “Happy Birthday” to Benny. o “Happy birthday to you; happy birthday to you. Happy birthday dear Benny; happy birthday to you.” The Charles A. Dana Center at the University of Texas at Austin Page | 13 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Session 5: Identifying and Making Sets of 2 At A Glance Children recite the counting poem, “One, Two, Buckle My Shoe,” and then continue the counting word sequence to 12 after the teacher adds 2 additional pennies to the collection jar. Following this activity, the teacher takes a collection of 2 sticks, which he or she uses to make capital letters that have two straight lines. The focus is on identifying a set of two objects without counting. Children then practice making sets of 2 as they make letters in the names of Benny’s fictional pets—
V.J. (the dog) and E.T. (the cat). The teacher introduces the words straight and curved to help children distinguish between straight-­‐sided and rounded edges. Children then use their tongue depressors to build all of the letters that have two straight lines—V, T, L, and X. Materials ●
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Collection jar with 10 pennies 2 extra pennies Chart paper and marker Tongue depressors Blackline master, “Two-­‐
Line Letter Cards” Mathematics Addressed ●
Use words to rote count to 30. ●
Demonstrate that the order of the counting sequence is always the same regardless of what is counted. ●
Without counting, verbally identify the number of objects 1–5. The Charles A. Dana Center at the University of Texas at Austin Page | 14 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Activate ●
Introduce the counting poem, “One, Two, Buckle My Shoe.” Act out each verse. One, two, buckle my shoe; (Pretend to tie your shoe.) Three, four, open the door; (Pretend to open an imaginary door.) Five, six, pick up sticks; (Pretend to gather sticks off the ground.) Seven, eight, lay them straight; (Pretend to lay the sticks straight.) ●
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Nine, ten, a big fat hen; (Stand tall and extend your hands out in front of your belly.) Allow children to recite and act out the verses in the poem introduced. Display the penny collection jar and remind children that the last time they counted the pennies inside, there were 10 altogether. Display 2 pennies in the palm of your hand. Ask children to identify how many they see. Confirm that there are 2 pennies. Drop each penny inside the jar. Model how to count on from 10. Engage Prompt children to visualize a mental image of 2 by asking them to remember how many pets Benny had. ● Invent fictional names for Benny’s pets. Write each pet’s name on a separate sheet of paper as you introduce them. o “Benny calls his dog V.J., which he spells with a capital V and capital J. His cat’s name is E.T., which he spells with a capital E and capital T.” ●
Display a pile of tongue depressors. Ask each child to take exactly 2. ● Prompt children to explain how they knew they had taken exactly 2 sticks. o Ask, “How did you know you had 2 sticks? Did you count them? Did you just see 2 or know what 2 looked like in your head?” ●
The Charles A. Dana Center at the University of Texas at Austin Page | 15 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Refer back to the pile of tongue depressors. Explain why it is more difficult to determine how many when a set is larger than 2. o “It is harder to tell how many when there are a lot of sticks in a pile. You may be able to guess (estimate) how many there are, but you have to count each stick to be sure.” ● Explain that a set of 2 is an easy quantity to see without having to count each item one by one. ● Turn around so that you are facing the same direction as children are watching and model how to write the numeral 2. Allow them to practice writing the numeral 2 in the air or on the palm of their hand. o “This is the way you write the number 2: Go around, down, and straight across.” ●
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Refer back to the sheet of paper on which you wrote the name V.J. Cover up the letter J so only the letter V is visible. Prompt children to make a capital V with 2 sticks. Uncover the letter J. Draw children’s attention to the curved line in that letter and why it is impossible to recreate with a set of 2 straight-­‐sided sticks. o “What do you notice about the J? Can you make a capital J with your two sticks without bending them? Why not?” ● Explain and compare the terms straight, slanted, and curved. ●
o “The V has two straight lines that are slanted up and down. The J has one line that goes straight down and then curves around at the bottom.” ● Refer back to the sheet of paper on which you wrote the name E.T. Cover up the letter T so only the letter E is visible. Challenge children to estimate how many sticks they would need to make the letter E without breaking any of them apart. ● Allow children to combine their 2 tongue depressors with a partner. ● Describe the combination of lines that form the capital E. o “The capital E has four straight lines—three that go across—one on top, one in the middle, and one on the bottom—and one on the side that goes up and down.” The Charles A. Dana Center at the University of Texas at Austin Page | 16 2012 Early Mathematics—A Resource for Teaching Young Children ●
Prekindergarten Allow children to work with their partner to recreate the capital E using a combination of 4 sticks. Uncover the letter T. Direct children to build the capital T with their sticks. o “How many sticks would you need to make the capital T?” ● Display letter cards for the capital letters E, L, T, and X. Allow children to use their two sticks to recreate each letter. Each time children make a new letter, direct them to count the sticks to see that there are still 2. ●
The Charles A. Dana Center at the University of Texas at Austin Page | 17 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Two-­‐Line Letter Cards The Charles A. Dana Center at the University of Texas at Austin Page | 18 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Session 6: Counting and Comparing Quantities 0–5 At A Glance This session begins with children reciting the counting word sequence to 12 in a patterned arrangement, pausing after every two counting words (e.g., One, two [pause]; three, four [pause]; five, six [pause], etc.). The teacher then presents various configurations of dots 0–5 as shown on a series of cards. Children use linking cubes to recreate each of these dot images. The focus is on developing strategies for recognizing quantities to 5 without counting. Next, each child in the group takes a dot card and builds a cube tower that corresponds to the number of dots on his/her assigned card. Afterward, children compare their towers to determine which group member has more, less, or the same number of cubes. Children then connect their towers and, with the assistance of the teacher, count how many cubes there are altogether. After one additional round, the group compares which round yielded the longer/shorter train. The focus is on counting, comparing, and making equivalent sets to 5. Materials ●
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Collection jar with 12 pennies 3 extra pennies 4-­‐in. by 4-­‐in. paper squares for each child Linking cubes Blackline master, “0–5 Dot Cards” Prepare ahead of time: Copy and cut apart the 0–5 dot cards. Mathematics Addressed ●
Use words to rote count to 30. ●
Count up to 10 items and demonstrate that the last count indicates how many items were counted. ●
Without counting, verbally identify the number of objects 1–5. The Charles A. Dana Center at the University of Texas at Austin Page | 19 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Activate Display the penny collection jar, which should have 12 pennies as of Session 5. ● Model how to count in groups of 2. Explain the reasoning behind this patterned counting procedure: o Say, “There are a lot of pennies, so we will try counting and pausing after every two counts so that no one gets behind or too far ahead in their counting.” ● Allow children to practice the new “counting by twos” pattern. ●
Introduce the 3 new pennies that you will add to the collection jar. Arrange the pennies in your hand and show them to children for 5 seconds. ● Shut your hand and direct children to discuss how many pennies they saw and how they remembered seeing the arrangement. ● Drop the 3 additional pennies inside the collection jar. Model how to count on from 12. o “Twelve … thirteen, fourteen, fifteen.” ●
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Display the 0–5 dot cards. Prompt children to identify the number of dots on each card. Show the side with 0 dots and lead children to understand that there are no dots to count—an empty set. ● Show the side with 2 dots, and then the side with 1 dot. Prompt children to identify how many they see. ● Discuss which quantities are easy to identify without counting one by one. ●
The Charles A. Dana Center at the University of Texas at Austin Page | 20 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Develop ●
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Show the side of the number cube that has 5 dots. Distribute a 4-­‐in. by 4-­‐in. paper square and handfuls of cubes to each child. Direct them to make an arrangement of cubes on top of their paper square that looks just like the 2-­‐1-­‐2 dot configuration they see on the dot card. Repeat the same procedures for both the 3-­‐ and 4-­‐dot arrangements as shown on the other cards. Gather the paper squares. Model how to make a tower with as many cubes as there are dots on the 5-­‐dot card. If children have difficulty keeping track of the number of cubes they have counted, model how to place one cube on top of each dot. Assign each child a particular color of cubes and one dot card. Have them make a tower with a matching number of cubes. Direct each child to compare his/her cube tower with other group members’ towers. o “Whose tower has more/less/the same number of cubes?” Connect all the towers from the first round of building. Count how many cubes there are altogether in the newly combined tower. ●
Redistribute the cards. Repeat the same procedures for making towers. Place each cube train side by side so that they are aligned at their bases. Compare their lengths to determine which is shorter/longer. The Charles A. Dana Center at the University of Texas at Austin Page | 21 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten 0–5 Dot Cards The Charles A. Dana Center at the University of Texas at Austin Page | 22 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Session 7: Ordering and Labeling Quantities with Numerals 1–5 At A Glance Children recite the counting word sequence to 15. First, they count by threes and then once again by fives (with pauses). The focus is on understanding that the sequence of the words and the end count stay the same regardless of how the items in a set are grouped. Children then revisit the activity introduced in Session 6 in which they built a tower using as many cubes as there were dots on their assigned number cards. The focus is on recognizing and counting quantities to 5. Next, the teacher calls on a volunteer to construct a staircase pattern by arranging each cube tower children built in ascending order from 1 to 5 so as to emphasize how the least-­‐to-­‐
greatest ordering of the towers corresponds to the standard number sequence. The teacher then introduces a scenario in which Benny goes shopping at a 5-­‐story department store. Children assist the teacher in arranging the buttons for the store elevator in numerical order. The focus is on sequencing numerals 1–5. Mathematics Addressed ●
Demonstrate that the order of the counting sequence is always the same regardless of what is counted. ●
Count up to 10 items and demonstrate that the last count indicates how many items were counted. ●
Use the verbal ordinal terms. Materials ●
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Collection jar with 15 pennies Blackline master, “0–5 Dot Cards” Linking cubes Blackline master, “0–5 Numeral Cards” Blackline master, “Benny’s Department Store” Blackline master, “Elevator” Prepare ahead of time: Cut apart a set of 1–5 numeral cards for each child. The Charles A. Dana Center at the University of Texas at Austin Page | 23 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Activate ●
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Display the penny collection jar. Remove the 15 pennies that you have collected so far. Group the pennies by threes. Count and pause accordingly. Invite children to recite the “counting in groups of threes” pattern you just modeled. Reorganize the pennies in groups of 5. Then model the corresponding counting pattern—
counting and pausing after each fifth coin you tally. Invite children to recite the “counting by fives” pattern you just modeled. Compare and discuss the results of each count. o Ask, “What number did we land on after the first count? What number did we land on after the second count?” Place the pennies back inside the jar. Emphasize how the number of pennies remained unchanged. Engage Depending on the number of children in the group, assign each child one or two 0–5 dot cards. If there are three children, give two cards apiece. If there are four children, give one child the 5-­‐dot card, one child the 4-­‐dot card, one child the 3-­‐ and 0-­‐dot cards, and the remaining child the 1-­‐ and 2-­‐dot cards. ● Instruct children to build a cube tower that matches the number of dots on their assigned card(s). ●
The Charles A. Dana Center at the University of Texas at Austin Page | 24 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Develop ●
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Call on a volunteer to locate the tower that has the fewest number of cubes. If the child does not understand the word fewest, emphasize height. o “Which tower has the fewest cubes? (Which one is the shortest?)” Call on another volunteer to arrange the towers from fewest to greatest (i.e., like a staircase). Ask a volunteer to identify the number of cubes in the first tower and then find the matching numeral card. Continue prompting volunteers to identify or count the number of cubes in each remaining tower within the staircase arrangement. After each count, label the respective tower with its corresponding numeral card. Highlight the 1–5 arrangement of index cards. Point out how numerals are arranged in standard order to correspond with the “one more” pattern of the counting sequence. Introduce a pretend scenario in which Benny (from the story Benny’s Pennies) is going to shop at a 5-­‐story department store. ● Display the Benny’s Department Store blackline. Discuss on which floors items are located. o “On the first floor, Benny will find flower arrangements. On the second floor, there is a bakery where he can buy cookies. Clothing is located on the third floor. Dog food is on the fourth floor and cat food on the fifth floor.” ●
The Charles A. Dana Center at the University of Texas at Austin Page | 25 2012 Early Mathematics—A Resource for Teaching Young Children ●
Prekindergarten Display the Elevator blackline. Point to a numeral on the display panel located above the elevator doors. Call on a volunteer to go to the Department Store blackline and point to the floor that corresponds to the numeral you highlighted on the elevator display panel. o “Show me on which floor you would be if this numeral lights up.” ●
Distribute a set of 1–5 numeral cards to each child. Allow them to order the cards in numerical order. o “Which numeral goes first/second/third/fourth/last?” The Charles A. Dana Center at the University of Texas at Austin Page | 26 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten 0–5 Dot Cards The Charles A. Dana Center at the University of Texas at Austin Page | 27 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten 0–5 Numeral Cards 0 4 2 5 1 3 The Charles A. Dana Center at the University of Texas at Austin Page | 28 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten The Charles A. Dana Center at the University of Texas at Austin Page | 29 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Elevator
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The Charles A. Dana Center at the University of Texas at Austin Page | 30 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Session 8: Labeling Sets with Numerals 0–5 At A Glance After adding 4 pennies to the collection jar, the teacher models the counting progression from 15, emphasizing how the subsequent number names—sixteen, seventeen, eighteen, and nineteen—
adhere to a more predictable counting pattern than the nonsensical wording of the preceding 11–15 sequence. The teacher then shows children a box of 8 crayons (or markers) to point out how the numeral printed on the box tells how many crayons are inside. The focus is on identifying numerals in the environment and understanding how they can be used as symbols to label quantities. Next, the teacher introduces a counting poem about a collection of 10 pennies used to purchase 10 gumballs. First, children act out the poem, counting out a set of 10 pretend gumballs (linking cubes), sorting them by color—red, yellow, and blue—and then labeling each group with a respective numeral. Finally, children illustrate the poem and the data from their gumball sort in a counting book, which they compile (with assistance from the teacher) and take home. Materials ●
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Collection jar with 15 pennies 4 extra pennies Linking cubes Box of 8 crayons 1–10 Number Line blackline (1 per child) Counting Book blackline (1 per child) Mathematics Addressed ●
Use words to rote count to 30. ●
Count up to 10 items and demonstrate that the last count indicates how many items were counted. ●
Recognize one-­‐digit numerals 0–9. The Charles A. Dana Center at the University of Texas at Austin Page | 31 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Activate Arrange a set of 4 pennies in the shape of an inverted T in the palm of your hand. Show the arrangement for 5 seconds. Have a set of cubes close by for children to use to recreate the penny arrangement you made in your hand. ● Close your hand. Distribute cubes and invite children to recreate the arrangement of 4 pennies they saw displayed in your hand. ●
When children finish arranging their cubes, open your hand. Allow them to rearrange their cubes to look exactly like the penny arrangement in your hand. Invite children to discuss how the pennies are configured. ● Display the 4 pennies in front of the collection jar. Take out the 15 pennies that are already inside and invite children to count out loud as you drop each coin back into the jar. • Add the new set of 4 pennies. Count in rhythm to emphasize the number word pattern. o Say, “Now, I am going to add four more to fifteen. Listen closely to the counting words I say and the pattern they make: sixteen, seventeen, eighteen, and nineteen. All the words I said ended in ‘teen.’ ” ●
The Charles A. Dana Center at the University of Texas at Austin Page | 32 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Engage Display a box of 8 crayons/markers. Point to the numeral 8 and call on volunteers to identify how many crayons are inside the box. ● Explain how numerals are used to label quantities. o “When you go to the store and want to know how many things are inside a package—
like this box of crayons—you cannot just open it and count what is inside. The numerals printed on the package tell you how many items there are.” ● Display the 1–10 number line. Start at 1 and lay one crayon at a time on top of each numeral. Count as you match each crayon to a numeral. Discuss how the last numeral on the number line and the numeral on the box of crayons match. ●
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Inform children that they will make a book about 10 gumballs. Develop ●
Make a train of 3 yellow, 4 red, and 3 blue cubes and display it on the carpet/table. Remove 10 pennies from the collection jar and arrange them in your hand. Recite the poem, “With These 10 Pennies,” printed on the Counting Book blackline. Take apart the train. Match each penny in your hand to 1 cube. ● Display the Counting Book blackline. Underline the color words with a matching crayon/ marker. Then, sort each set of cubes by color on their respective page of the blackline. ● Count the number of cubes in each group. ● Remove 1 yellow cube at a time from the square on the blackline where the yellow cubes are grouped together. Model how to draw a yellow circle for each cube you remove. ●
The Charles A. Dana Center at the University of Texas at Austin Page | 33 2012 Early Mathematics—A Resource for Teaching Young Children ●
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Prekindergarten Arrange the yellow cubes you removed from the blackline going from left to right on the number line card. Locate the numeral on the number line where the row of yellow cubes ends. Write that numeral on the blackline page to show how many yellow gumballs you counted. Repeat these steps for the group of red cubes and then the group of blue cubes. Distribute the following items to each child: o One Counting Book blackline o One 1–10 number line card o One box of crayons/markers o A train with different combinations of red, yellow, and blue cubes (10 altogether) Direct children to make their counting books, following the procedures you modeled. (Note: Given the number of steps you modeled, you may need to go back and perform each task alongside the children.) Move about the group, assisting each child as necessary. When the child is finished, read his/her book and discuss the gumball/cube data. o “How many red/yellow/blue gumballs did you have? Are there more/fewer red/ yellow/blue?” The Charles A. Dana Center at the University of Texas at Austin Page | 34 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten   1 2 3 4 5 6 7 8 9 10 __________________‘s Number Line The Charles A. Dana Center at the University of Texas at Austin Page | 35 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten The Charles A. Dana Center at the University of Texas at Austin Page | 36 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten The Charles A. Dana Center at the University of Texas at Austin Page | 37 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Session 9: Combinations of 5 At A Glance The teacher adds 1 penny to the collection jar, bringing the total to 20. Children count the pennies by ones and clap after each fifth count. The focus is on helping children to visualize the counting word sequence as a predictable pattern with structure rather than as a haphazard string of random numbers. Next, the teacher introduces a song/poem about shaking and spilling a piggy bank containing 5 pennies. Children confirm that the total number of pennies remains the same when the coins are dispersed and rearranged after one spill. Then, as children practice counting out a set of 5 pennies into a cup and shaking and spilling the coins, the teacher makes a table on a sheet of chart paper to represent and record the results of each child’s spill. The focus is on identifying and recognizing different combinations of 5 (e.g., 3 and 2, 4 and 1, 5 and 0). Materials ●
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Collection jar with 19 pennies 1 extra penny Chart paper Crayons (1 box per child) 5 pennies per child 1 cup per child Blackline master, “Shake the Piggy Bank” (1 per child) Markers Mathematics Addressed ●
Use words to rote count to 30. ●
Count up to 10 items and demonstrate that the last count indicates how many items were counted. ●
When counting, demonstrate understanding that items can be chosen in any order. The Charles A. Dana Center at the University of Texas at Austin Page | 38 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Activate Display the penny collection jar and an extra penny in your hand. Ask children to predict how many pennies there will be when you add one more. ● Arrange all 20 pennies in groups of 5. ● Model how to count in groups of 5. Shout as you say each fifth counting word (i.e., five, ten, fifteen, twenty). o Say, “Let’s count our groups. Clap when you reach the end of each group. Ready,—
one, two, three, four, five [clap]; six, seven, eight, nine, ten [clap]; eleven, twelve, thirteen, fourteen, fifteen [clap]; sixteen, seventeen, eighteen, nineteen, twenty [clap].” ●
Engage Display the Shake the Piggy Bank blackline. ● Take a cup and one of the four groups of 5 pennies that you just counted from the collection jar. Distribute 1 penny to each child, and allow them to observe its top and bottom. Introduce the words heads and tails. o “The top of the coin with the face is ‘heads.’ The bottom with the building is ‘tails.’ When a penny falls, it lands on heads or tails.” ● Invite children to count with you as you drop 1 penny at a time inside the cup (the pretend piggy bank). ● Sing the song shown on the Shake the Piggy Bank blackline to the tune of “Row, Row, Row Your Boat.” Shake the cup as you sing and then spill the pennies on the carpet. ●
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Count the pennies to confirm that there are still 5. The Charles A. Dana Center at the University of Texas at Austin Page | 39 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Develop ●
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Display the Shake the Piggy Bank blackline sorting mat. Underline the word heads with a blue crayon/marker and the word tails with a red crayon/marker. Call on volunteers to help you sort each of the 5 pennies you just spilled. Use the blackline to sort. Count the number of pennies in each category. Make a table on a sheet of chart paper. Label the columns Heads and Tails. Make blue and red hash marks to record the results of the first spill. Make a numerical representation to correspond with the hash mark data. Distribute a set of 5 pennies, one cup, a Shake the Piggy Bank blackline, and a box of crayons to each child. Give instructions for the practice activity. Consult with each child about his/her penny spill data. Use the table you created on chart paper to record the results of each spill. As time allows, have children continue shaking and spilling to find new combinations of 5. Record the new combinations. o “How many of your pennies landed on heads/tails?” The Charles A. Dana Center at the University of Texas at Austin Page | 40 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten The Charles A. Dana Center at the University of Texas at Austin Page | 41 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Session 10: Joining and Separating Sets of 5 At A Glance The teacher adds 5 new pennies to the collection jar and organizes all 25 pennies in groups of 5 so that children can practice the counting and clapping pattern that was introduced in Session 9. The focus is on helping children to internalize the repetitive pattern within the counting word sequence to 25. Children then count out a set of 5 onto a piggy bank storyboard as they prepare to review what they know about counting and identifying sets up to 5. After discussing and acting out the joining and separating scenarios introduced by the teacher, children create and illustrate their own number stories showing what they would do with a set of 5 pennies (e.g., How many pennies would you spend and how many would you save?) The focus is on making verbal word problems for adding up to 5 objects and subtracting 1–5 from a set. Materials ●
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Collection jar with 20 pennies 5 additional pennies 1 quarter Combinations of 5 (heads and tails) chart from Session 9 Piggy Bank blackline (1 per child) Crayons or markers Mathematics Addressed ●
Use words to rote count to 30. ●
Use concrete models or make a verbal word problem for adding up to 5 objects. ●
Use concrete models or make a verbal word problem for subtracting 1–5 objects from a set. The Charles A. Dana Center at the University of Texas at Austin Page | 42 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Activate Remove all 20 pennies from the collection jar. Drop 1 penny at a time back inside the jar and count to verify that the number has not changed. ● Display a set of 5 additional pennies in the palm of your hand. Invite a volunteer to count each penny to determine how many you are showing. ● Arrange all 25 pennies in groups of 5. ● Shout for each fifth counting word. o For example, “One, two, three, four, five [clap], six, seven, eight, nine, ten [clap] …” ●
Engage Take all 25 pennies out of the jar and exchange them for a quarter. Identify which side of the quarter is heads and which side is tails. Then, pass the quarter around for children to examine. ● Discuss ways to spend a quarter. ● Distribute a set of 5 pennies and the Piggy Bank blackline to each child. Lead children to verify that they have the same number of pennies as Benny. ●
The Charles A. Dana Center at the University of Texas at Austin Page | 43 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Develop Instruct children to place the 5 pennies you gave them in their hand. Introduce the word problem that involves the joining of two sets. o Say, “You have 3 pennies inside the piggy bank. Then, your friend gives you 2 more pennies to add to the piggy bank. How many pennies do you have now?” ● Confirm that the result of joining 3 pennies and 2 pennies is 5 altogether. ● Introduce a new word problem that involves an inverse operation—the separation of two sets. o “Here is a new story: You have 5 pennies in your piggy bank, but you spend 1 penny on a piece of candy. How many pennies do you have now?” ●
Allow children to use the back of their Piggy Bank blackline to trace around their hand. ● Encourage children to make up their own story about a set of 5 pennies, some of which they will save and some of which they will spend. Encourage children to refer to the Combinations of 5 chart you created in Session 9 to help them think about how many pennies they would have left if they spent a certain amount. ● Observe children and ask questions about their story. o “How many pennies did you spend/save?” ●
The Charles A. Dana Center at the University of Texas at Austin Page | 44 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten The Charles A. Dana Center at the University of Texas at Austin Page | 45 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Session 11: The Hungry Caterpillar At A Glance Children revisit the counting word list they recited from the previous session, counting all the way up to 25. The focus is on allowing children to practice and commit the rote-­‐counting sequence to memory. The teacher then introduces a new book, The Very Hungry Caterpillar, by Eric Carle. In this story, the caterpillar eats a collection of foods on each consecutive day of the week. The cumulative total follows a one-­‐more pattern 1–5 during the Monday to Friday sequence and then skips to 10 on the final day (Saturday) of the caterpillar’s eating frenzy. The focus is on ordering sets to 5 and then visualizing a quantity of 10. The session concludes with children rolling a dot cube to determine how many cubes to place on a path of up to 25 squares. The objective of the path game is to use one-­‐to-­‐one correspondence to count and keep track of a growing total of objects up to 25. Materials ●
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The Very Hungry Caterpillar by Eric Carle Empty jar Linking cubes 6 index cards labeled by the days of the week, Monday through Saturday 0–5 dot cube Blackline master, “The Hungry Caterpillar Counting Game” (1 per pair of children) Mathematics Addressed ●
Use words to rote count to 30. ●
Count 1–10 items, with one count per item. ●
Count up to 10 items and demonstrate that the last count indicates how many items were counted. The Charles A. Dana Center at the University of Texas at Austin Page | 46 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Activate Refresh children’s memory about the 25 pennies you counted and exchanged for a quarter during Session 10. ● If possible, show children something you could have purchased with 1 quarter (e.g., a box of new crayons). Make the item available for the whole class to share or use during center time. ● Introduce The Very Hungry Caterpillar. o Say, “Today, we are going to read a story about a caterpillar that ate 25 different types of food in 1 week.” ● Count out a set of 25 cubes to show children how many items the caterpillar ate. Display an empty jar and invite children to count out loud, reciting the counting word sequence to 25 as you slowly drop each of the 25 cubes one by one inside. ●
Engage Begin reading the book. If children are unfamiliar with The Very Hungry Caterpillar, conduct a brief picture walk, discussing and making inferences as to what might happen in the story. ● Pause each time the caterpillar munches through a particular group of foods during a specified day of the week and take a matching set of cubes from the jar. Then, match each cube to a corresponding hole on the page spread and display the set of cubes on an index card labeled with the respective day of the week. ●
The Charles A. Dana Center at the University of Texas at Austin Page | 47 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten o For example, “At the point in the story when the Hungry Caterpillar ate the apple, you would remove 1 cube from the jar, match it to the hole on the page, and then place the cube on an index card labeled Monday. As you turn the next page, you would remove 2 more cubes from the jar, match each one to the holes imprinted on that page, connect the 2 cubes together, and then display the tower on an index card labeled Tuesday. Continue this counting, matching, and labeling procedure for the remaining set of cubes. By the time you reach Saturday in the story, you will have 10 cubes remaining for each of the 10 food items shown on the page spread.” Develop ●
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After reading the story, direct children’s attention to the display of cubes stacked on top of the linear arrangement of index cards, each labeled as a corresponding day of the week. Count out loud as you remove the cubes from the index cards and connect them together into a train of 25. Emphasize the final counting word—twenty-­‐five—to help children remember how many foods the caterpillar ate altogether. Instruct children to form a circle so as to make room to display The Hungry Caterpillar Counting Game board on the carpet in the middle of the viewing area. Provide and model the following instructions on how to play the game: 1. Roll a dot cube and take a corresponding set of connecting cubes. 2. Begin at the end of the pathway labeled Start and match each of the cubes you just counted to a corresponding square on the board. If you rolled a 3, your path stops at the final pear on the game board. 3. Continue rolling the dot cube, counting out a matching set of cubes and arranging each cube on the game board until you have reached the end of the pathway. (Note: As you approach the end of the game board, point out how the number of dots you roll must be equal to or less than the number of empty squares on the game board. For example, if 4 squares remain uncovered on the game board, you must roll a 4 or lower to continue to the end of the pathway [rolling a 5 would not do].) Distribute one game board to each pair of children. Instruct them to play the game with their assigned partner. As one partner rolls the dot cube, the other counts out a matching set of cubes and arranges them onto the pathway. After each roll, children switch jobs (i.e., roller and counter). Ask questions to guide children to think about the growing/changing total. o “How many cubes do you have now? How many more do you need to reach the last square with an orange inside?” Observe children’s strategies for determining how many spaces they need to move forward on the game board path. o Counting: Do they count the number of pips on the dot cube and then move that many spaces forward? The Charles A. Dana Center at the University of Texas at Austin Page | 48 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten o Subitizing: Do they look at the arrangement of pips on the dot cube and then know exactly (without counting) how many spaces to move? o Perceptual: Do they perceptually gauge the quantity whereby they look at the arrangement of pips and then make a guess on to how many spaces to move forward? (Note: This strategy does not guarantee accuracy. If the child is inaccurate, scaffold the activity by having him/her place 1 cube on the game board for every pip on the cube.) The Charles A. Dana Center at the University of Texas at Austin Page | 49 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Start
The Hungry Caterpillar Counting Game
The Charles A. Dana Center at the University of Texas at Austin Page | 50 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Session 12: Strategies for Making Equal Sets At A Glance Children review how many items the Hungry Caterpillar ate altogether in 1 week and practice reciting the counting word sequence up to that number—25. The focus is on remembering key transitions between decades (e.g., “after nineteen you say twenty”). Children then use their fingers and remove an equivalent set of cubes from the train of 25 to show how many food items the caterpillar ate each day of the week. The focus here is on using quantification strategies (e.g., one-­‐to-­‐one correspondence, counting, and instantaneous recognition of small numbers) to make a matching set of objects. This session concludes with children practicing and reflecting on these strategies as they play the card game, “Concentration,” with a partner. Materials Blackline master, “Concentration Game Cards” ● 25 linking cubes ● Chart paper ● Blackline master, “Strategies for Making Equal Sets” ●
Mathematics Addressed ●
Use words to rote count from 1 to 30. ●
Count up to 10 items and demonstrate that the last count indicates how many items were counted. ●
Without counting, verbally identify the number of objects 1–5. The Charles A. Dana Center at the University of Texas at Austin Page | 51 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Activate Review what happened in The Very Hungry Caterpillar. ● Sing the following song to the tune of the “Itsy Bitsy Spider.” ●
“The Hungry Caterpillar liked to munch on food” (Make a chewing motion with your fingers.) “Everyday he ate he grew and grew and grew” (Make an expansion motion with your hands.) “When he ate too much he gave a great big sigh” (Expand your chest in and out.) “But soon the caterpillar would become a butterfly” (Join your two hands to make a butterfly.) Show children a train of 25 connecting cubes to refresh their memory of how many different types of food the caterpillar ate during his one-­‐week eating frenzy. ● Encourage children to count along as you point to each cube. Count slowly, emphasizing key transitions in the counting word sequence (e.g., “after nineteen you say twenty, not nineteen-­‐ten”). ●
Engage Randomly display the set of cards showing the different collections of foods that the caterpillar ate each day of the week. ● Invite a volunteer to arrange the cards in the Monday through Saturday order that the caterpillar ate each group of foods. ● Randomly remove each card from the display out of sequential order and call on other volunteers to use the fingers on their hands to show how many foods are on that particular card. Instruct each volunteer to remove as many cubes from the train of 25 to match the number of foods on the selected card. ●
The Charles A. Dana Center at the University of Texas at Austin Page | 52 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Develop Discuss the different strategies that children used to determine how many cubes they needed to take from the train in order to show/match how many foods were on each card. ● Highlight and record each strategy on a sheet of chart paper. Model the strategies to help children reflect on how to use them intentionally to make equivalent sets. o Eyeballing: Make an equal set of fingers/cubes without counting each piece of food on the card. o One-­‐to-­‐one matching: Match one finger/cube to each piece of food on the card. o Counting: Count the number of foods on the card and then make an equal set of fingers/cubes. ●
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Point out that the fewer the number of foods on one of the cards, the easier it was to simply recognize how many without counting each item. (This was likely the strategy used for the apple and pear cards.) Likewise, when there were a greater number of items on one of the cards, the more likely it was that children used a counting or one-­‐to-­‐one strategy to determine how many. (This strategy is most suitable for a quantity such as 10—the number of foods the caterpillar ate on Saturday.) (Note: Most 4-­‐year-­‐olds, however, would use a global/perceptual estimation strategy, responding that there were simply “a lot” or “this many” (flashing many fingers at once). Therefore, the objective is to lead children to understand that counting is a good strategy for achieving accuracy—knowing exactly how many.) ●
Introduce the card game, “Concentration.” Model the following procedures as you play a demonstration game with a volunteer partner. 1. Make two stacks of playing cards—the food cards and the finger cards—and shuffle them separately. 2. Arrange the 6 food cards face down on the carpet in a 2-­‐by-­‐3 array on one side and the 6 number/finger cards in an identical arrangement on the opposite side. 3. One partner starts by turning over a card on one side and another card on the opposite side. If the two cards are a match, that player removes them from the arrangement and pairs them together. If they do not match, the player turns the cards back over face down, allowing his/her partner to take a turn. The Charles A. Dana Center at the University of Texas at Austin Page | 53 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten 4. Children continue taking turns turning over one card from each set and determining if they are a match. 5. The player with the most cards accumulated at the end of the game wins. ● Assign each child a partner, and observe their play. Refer to the Strategies for Making Equal Sets blackline when you see children use one of the highlighted strategies. o “How do you know that the number of items on your food card matches the number of fingers on the number card? Which strategy did you use?” The Charles A. Dana Center at the University of Texas at Austin Page | 54 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Concentration Cards The Charles A. Dana Center at the University of Texas at Austin Page | 55 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Concentration Cards The Charles A. Dana Center at the University of Texas at Austin Page | 56 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Concentration Cards The Charles A. Dana Center at the University of Texas at Austin Page | 57 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Strategies for Making Equal Sets The Charles A. Dana Center at the University of Texas at Austin Page | 58 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Session 13: Conceptualizing 10 (the Tens-­‐Frame) At A Glance The teacher leads children to differentiate between their left and right hands so as to make sense of the 5-­‐and-­‐5 combination of fingers, which makes 10 altogether. Children then transfer their knowledge about the number of fingers on their hands to the formal representation of 10 as displayed in a 2-­‐by-­‐5 tens-­‐
frame. The focus is on helping children to develop a visualization strategy for making and counting sets of 10. Children then practice one-­‐to-­‐one-­‐correspondence, filling the tens-­‐frame in the standardized top-­‐to-­‐bottom, left-­‐to-­‐right directional procedure. The objective is to familiarize children with the tens-­‐frame visual as a strategy for comparing quantities 1–10 to a base-­‐10 benchmark. Materials Linking cubes ● Blackline master, “Tens-­‐frame” ● Paper for each child ● Tempera paint ●
Mathematics Addressed ●
Count 1–10 items, with one count per item. ●
Demonstrate that the order of the counting sequence is always the same regardless of what is counted. ●
Count up to 10 items and demonstrate that the last count indicates how many items were counted. The Charles A. Dana Center at the University of Texas at Austin Page | 59 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Activate Turn with your back facing children. Raise your left and right hands, instructing children to do the same, so that they can differentiate between left and right. ● Allow children to identify how many fingers they have on each hand. Confirm that most people have 5 fingers on each hand. (Note: Be sensitive to those children who might not have 5 fingers on each hand.) ● Show children both of your hands and invite a volunteer to count how many fingers there are altogether. Confirm that there are 10. ● Invite children to stand and perform/sing “The Hokey Pokey,” modeling the following movements: 1st verse—left hand; 2nd—right hand; and 3rd—10 fingers. ●
Engage Display a tens-­‐frame and challenge children to estimate the number of squares they see in the 2-­‐by-­‐5 array. o Ask, “How many squares are on the top/bottom row? How many squares are there altogether?” ● Demonstrate the following steps to help children see how the number of squares in each row corresponds to the number of fingers on each hand. 1. Invite a volunteer to stand and display both hands, spreading their fingers apart so as to accentuate the 5-­‐and-­‐5 combination. 2. Start with the volunteer’s left hand and affix 5 red cubes to each finger, counting and moving from left to right starting with the pinky and ending on the thumb. 3. Affix 5 yellow cubes to the opposite hand, again moving from left to right but starting at the child’s right thumb and ending on her right pinky finger. 4. Transfer the cubes from the volunteer’s hands to the tens-­‐frame displayed on the carpet. Use the red cubes to fill in the top row of the frame, counting out loud 1–5 as you move from left to right. Then, fill the bottom frame with the yellow cubes and count on from 5, starting at 6 and proceeding to 10 as you move from left to right. ●
The Charles A. Dana Center at the University of Texas at Austin Page | 60 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Develop ●
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Distribute ten-­‐frames and piles of cubes to pairs of children. Instruct children to make a set of 10 cubes and then fill the tens-­‐frame as you modeled in the Engage section (i.e., start at the top row and go left to right; then move to the bottom row and proceed in the same direction). Refer back to the Strategies for Making Equal Sets blackline introduced in Session 12 to guide children as they think about how to accurately count out a set of 10 cubes. If children struggle to count and keep track of set of 10, encourage them to work with a partner—the partner matches 1 cube per finger on his/her friend’s hands, and then they switch roles. When children are at centers, work independently with individuals to make a handprint placemat for counting and making sets during snack time. 1. Paint the right and left hand different colors. 2. The child then presses both hands down on a sheet of paper to create handprints. 3. When the paint dries, glue a tens-­‐frame at the top of the paper. Children will use this placemat during Session 14. The Charles A. Dana Center at the University of Texas at Austin Page | 61 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Tens-­‐Frame The Charles A. Dana Center at the University of Texas at Austin Page | 62 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Session 14: Ordering Numerals 1–10 At A Glance The teacher directs children’s attention to the ordering of numerals on the classroom calendar so as to highlight how each numeral in the counting system corresponds to the sequencing of counting words (e.g., the word four matches the numeral 4). Next, a volunteer drops a cube inside an empty jar as the teacher points to each numeral representing 1 day that has passed in the current month. The focus is on leading children to understand how each consecutive numeral in the counting word and number line sequence represents an accumulation of days/objects (e.g., the last word you say tells how many you have so far). Children then create their own number line, focusing on the ordinal value (e.g., first, second, third) of each consecutive numeral. Materials ●
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Count up to 10 items and demonstrate that the last count indicates how many items were counted. ●
Use the verbal ordinal terms. ●
Recognize one-­‐digit numerals 0–9. Empty jar Calendar Cubes Chart paper/pen Blackline masters, “Caterpillar Numeral Sequencing Cards” Number Line blackline master Optional: Sheet of dot stickers per child with numerals 1–10 Handprint placemats from Session 13 The Charles A. Dana Center at the University of Texas at Austin Page | 63 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Activate Instruct children to wrap their arms over their head and pretend that they are a tiny caterpillar hatching from an egg. ● Remind children that the Hungry Caterpillar hatched from his egg on a Sunday. Invite a volunteer to locate the square on the class calendar that corresponds to the first Sunday of the current month. ● Discuss the sequence of numerals on the calendar. o Ask, “What numeral comes first? What comes after the number 1? What is the last numeral posted on the calendar?” ● If children are not familiar with a calendar, explain that the numerals are arranged in order from left to right. ●
Engage ●
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Point to the numeral 1 on the calendar. Invite a volunteer to drop a cube inside an empty jar each time you continue to point to a subsequent numeral on the calendar, starting at the number 1 and stopping at the final numeral that corresponds to the current date. Explain that the last counting word children recited and the last numeral you pointed to on the calendar tell how many days have passed so far in the current month. Start at the left on a sheet of chart paper and begin writing/creating a number line 1–10. Invite children to recite each counting word as you write the corresponding numeral. Take the caterpillar face card from the set of Caterpillar Numeral Sequencing Cards and place it in the center left of the circle/carpet area. Randomly assign and distribute a sequencing card to each child in the group. (Note: If there are more than 10 children in the group, assign one card per student pair. If there are fewer than 10 children, distribute more than one card to each child.) Direct children’s attention to the number line displayed on the chart. Point to the numeral 1 and direct the child who has that same number to place his/her card to the left of the caterpillar face card. The Charles A. Dana Center at the University of Texas at Austin Page | 64 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten ●
Move in numerical order (left to right) across the number line, pointing to each consecutive numeral and soliciting the child who has that matching card. Develop Distribute the following to each child: o The individual handprints that children painted during Session 13. o A sheet of dot stickers, each of which have a numeral (1–10) displayed randomly on the sheet. (Note: If you do not have dot stickers available, distribute individual baggies with a set of numeral cards that have been cut apart ahead of time.) ● Instruct children to create their own number line by sticking a dot sticker numeral above each corresponding finger on their handprint placemat. Remind children to start at the left and move to the right as they stick/glue each numeral above a finger. ● Encourage children to refer to a number line as they sequence the dot stickers 1–10 above each matching finger. ●
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As children complete the task, probe their familiarity with numerals and understanding of the ordinal values of numbers. o For example, “Which numeral comes before/after/between 1 and 3? Which is fourth?” The Charles A. Dana Center at the University of Texas at Austin Page | 65 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten 1 Caterpillar Numeral Sequencing Cards (Color Version) 2 3 4 5 The Charles A. Dana Center at the University of Texas at Austin Page | 66 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten 6 7 8 9 10 The Charles A. Dana Center at the University of Texas at Austin Page | 67 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten 1 Caterpillar Numeral Sequencing Cards (Black and White Version) 2 3 4 5 The Charles A. Dana Center at the University of Texas at Austin Page | 68 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten 6 7 8 9 10 The Charles A. Dana Center at the University of Texas at Austin Page | 69 2012 1 1 3 4 5 6 7 8 9 10 The Charles A. Dana Center at the University of Texas at Austin Page | 70 2 3 4 5 6 7 8 9 10 Cut apart and place in individual baggies for each child. Children order and glue down each numeral to its corresponding finger on their handprint placemat, which they painted during Session 13. (Optional: Use dot stickers instead.)
2 ____________’s Number Line
Early Mathematics—A Resource for Teaching Young Children Prekindergarten 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Session 15: Counting and Creating Sets of 6 At A Glance Children view and discuss the number of legs on a butterfly so that they can develop a mental image of 6. Children then count and compare this quantity to the number of fingers on one hand in order to acquire a beginning understanding of numerical relationships (e.g., 6 is one more than 5). Next, children continue to construct a mental representation of 6 as they count out a set of 6 counters on a number line and then rearrange them inside a tens-­‐frame. The focus is on confirming that a quantity remains the same regardless of how it is arranged. Finally, children practice making different combinations of 6 as they roll a 1–6 dot cube and make a representative domino arrangement (e.g., 5 and 1, 4 and 2, 3 and 3) on a domino mat. Although the focus in this activity is on counting to 6, children are practicing being able to quickly identify smaller quantities to 5. Materials ●
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Demonstrate that the order of the counting sequence is always the same regardless of what is counted. ●
Count up to 10 items and demonstrate that the last count indicates how many items were counted. ●
Without counting, verbally identify the number of objects 1–5. The Very Hungry Caterpillar by Eric Carle Children’s individual handprint placemat (created in Sessions 13 and 14) Chart paper Cubes or chip counters 1–6 dot cube per pair of children Blackline master, “Domino Mat” Blackline master, “Combinations of 6 Domino Cards” The Charles A. Dana Center at the University of Texas at Austin Page | 71 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Activate ●
Sing the following song (introduced during Session 12) to the tune of the “Itsy Bitsy Spider.” The Hungry Caterpillar liked to munch on food (Make a chewing motion with your fingers.) Everyday he ate he grew and grew and grew (Make an expansion motion with your hands.) When he ate too much he gave a great big sigh (Expand your chest in and out.) But soon the caterpillar would become a butterfly (Join your two hands to make a butterfly.) ●
Open the book to the final page spread displaying the butterfly. (The picture shows 2 legs at the top and 4 legs at the bottom, making 6 altogether—the focus of today’s session.) Lead children as they count the legs. ● Compare how many legs an insect has to a human. o Ask, “How many legs do you have? Who has more—the butterfly or you?” ●
Engage Write the numeral 6 on chart paper. Instruct children to copy you as they use their finger to trace the numeral 6 in the air. ● Now guide children as they verify that they have fewer than 6 fingers on one hand. o “If you do not have 6 fingers on one hand, what would you have to do to show a set of 6?” ●
The Charles A. Dana Center at the University of Texas at Austin Page | 72 2012 Early Mathematics—A Resource for Teaching Young Children ●
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Prekindergarten If no one suggests making a set of 5 and 1, flash a set of 5 fingers and invite a volunteer to stand up and add 1 finger to make a group of 6. Lead children to visualize that “6 is one more than 5.” Distribute sets of cubes or chip counters to pairs of children and the handprint counting placemats (which children began in Session 13) to each individual owner. Point to the numeral 6 that you wrote on chart paper and have children count out that many cubes/chip counters/pom-­‐poms. Encourage children to refer to the 1–10 number line they created above each finger on their handprint placemat as they make a set of 6 counters. o “Remember to start at the numeral 1 and count forward one by one until you reach the 6, which is the numeral I am pointing to here on the chart.” Once each child has made an arrangement of 6 on the number line, model how to rearrange the counters inside the tens-­‐frame at the top of their placemat. o “Start at the top and move from left to right. How many cubes were able to fit in the top row?” (5) “How many in the bottom row?” (1) Develop ●
Allow children to practice filling the tens-­‐frame (just as you previously modeled) using the counters on their own placemat. When children finish, gather the handprint placemats, display a 1–6 dot cube, blank domino mat, and set of domino cards on the carpet. Demonstrate how to play the “Make a 6” domino game. 1. Roll the dot cube and place that many counters on one side of the domino. 2. Place the remaining cubes on the other side. 3. Count the total number of cubes/chips on the domino to confirm that there are still 6 altogether (both sides combined). 4. Find a domino card that matches the arrangement you made on your domino mat. ● Distribute the domino mat and a 1–6 dot cube to pairs of children. After children roll and make a corresponding cube/chip arrangement on the domino mat, remind them to find a matching domino card from the group pile. ●
The Charles A. Dana Center at the University of Texas at Austin Page | 73 2012 Early Mathematics—A Resource for Teaching Young Children ●
Prekindergarten After one round, discuss and highlight all of the different combinations of cubes/chips (e.g., 6 and 0, 5 and 1, 4 and 2, 3 and 3). The Charles A. Dana Center at the University of Texas at Austin Page | 74 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Domino Mat The Charles A. Dana Center at the University of Texas at Austin Page | 75 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Combinations of 6 Domino Cards The Charles A. Dana Center at the University of Texas at Austin Page | 76 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Combinations of 6 Domino Cards The Charles A. Dana Center at the University of Texas at Austin Page | 77 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten 1-­‐6 Dot Cube Cut apart the along the edge of the outline. Then, fold on the dotted lines and tape together to make a cube.                      The Charles A. Dana Center at the University of Texas at Austin Page | 78 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Session 16: Counting and Creating Sets of 7 At A Glance Children develop a mental representation of 7 by linking the quantity to the number of days in a week. The teacher leads children to think of another way to visualize 7 in terms of its relationship to 5 (e.g., 7 is two more than 5). Next, children are challenged to focus on the attribute of number versus size (conservation of number) as they help the teacher count and compare three sets of 7 blocks, each of which is larger or smaller in relation to the others. Children then begin to explore the concept of patterning as they assist the teacher in arranging 3 mealtimes—breakfast, lunch, and dinner—in the order they occur as a daily cycle within a week. The focus is on helping children develop an emerging awareness of multiplicative reasoning (e.g., how a group can be counted as an individual unit using the same one-­‐to-­‐one numeration system applied to discrete objects). Materials ●
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Classroom calendar Seven 2-­‐by-­‐3 paper squares Pattern blocks Linking cubes Markers Blackline master, “Strategies for Making Equal Sets” (from Session 12) Handprint placemat from Session 13 Mathematics Addressed ●
Understand that objects or parts of an object can be counted. ●
Demonstrate that the order of the counting sequence is always the same regardless of what is counted. ●
Count up to 10 items and demonstrate that the last count indicates how many items were counted. The Charles A. Dana Center at the University of Texas at Austin Page | 79 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Activate Locate the square on the classroom calendar that marks the day when children engaged in the activities about the number 6 presented during Session 15. o Say, “When you turn 6, how many fingers will you need to hold up to show how old you are? Which animal has 6 legs?” ● Highlight one row on the calendar, which represents a week, and invite children to count out loud as you move left to right and point to each square. ● Explain that there are 7 days in a week. Model how to make a 7 in the air with your finger and invite children to hold up 7 fingers as they sing about the days of the week to the tune of “Oh My Darling.” ●
Engage Count out 5 paper squares of the same color and 2 of another color. Arrange the 7 squares in a row on the carpet, placing the 5 of one color in the middle and the remaining 2 on opposite ends. ● Count the 5 squares in the middle of the row and explain that they each represent a weekday—Monday, Tuesday, Wednesday, Thursday, and Friday. Then, point out that the two squares at each end represent days (Saturday and Sunday) called the weekend. ● Lead children to see the relationship between 6 and 7 and the benchmark number 5 (e.g., 6 is one more than 5, and 7 is two more). ● Introduce a pretend scenario whereby children think about the structure of a day and the predictable order of its constituent parts—morning, afternoon, and evening. ●
The Charles A. Dana Center at the University of Texas at Austin Page | 80 2012 Early Mathematics—A Resource for Teaching Young Children ●
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Prekindergarten Explain that the Hungry Caterpillar eats three meals a day, always in the same order, but different proportions at each mealtime. Use pattern blocks to represent each meal—a hexagon for breakfast, trapezoid for lunch, and triangle for dinner. Display each representative pattern block as you explain the caterpillar’s eating pattern. o “In the morning he eats a big breakfast, a regular size lunch in the afternoon, and a light/small dinner in the evening.” Count out 7 of each block to represent the weekly cycle. Invite children to count out loud as you arrange each set in a separate pile on the carpet. Discuss the apparent size differences among the three sets of blocks. o “Does each set have the same amount? Why does the group of hexagons appear to have more than the trapezoids and triangles? What can we do to make sure there are 7 blocks in each set?” Pair each hexagon in one-­‐to-­‐one correspondence with a matching trapezoid and triangle to accentuate the equivalency among the three sets. Count the number of blocks in each set, emphasizing the last count so as to confirm that there are as many hexagons as there are trapezoids and triangles. Develop Start with the paper square on the carpet that represents Sunday and begin to arrange the pattern blocks corresponding to the order of their representative mealtime—the hexagon (breakfast) first, the trapezoid (lunch) second, and the triangle (dinner) last. ● Move to the second paper square (representing Monday) and prompt children to describe how to arrange another set of blocks according to the order you began on the first square (Sunday). o “Which meal comes first/second/third?” ● For the each remaining day of the week, invite a volunteer to come up to the arrangement of paper squares on the carpet and continue the hexagon/trapezoid/ triangle pattern by selecting a block that comes next in the daily sequence. ●
The Charles A. Dana Center at the University of Texas at Austin Page | 81 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Once the pattern is complete, invite children to read it as you point to each block in the sequence (i.e., breakfast, lunch, and dinner/breakfast, lunch, and dinner …). ● Model and provide instructions on how to construct a night/day pattern at the art center. 1. Use the handprint placemat to count out a set of 7 blue cubes (day) and an equal set of 7 black cubes (night). 2. Refer to the Strategies for Making Equal Sets blackline to decide on a way to make sure you get 7 of each color. (Note: If children choose use an “eyeball” strategy, remind them to use another strategy [e.g., counting or one-­‐to-­‐one matching] to double-­‐check their eyeball estimate.) 3. Arrange the cubes in a one-­‐week night and day pattern. 4. Use markers to record the pattern on paper. 5. Make a circle around each day/night unit to show that there are 7 days in the entire week. ●
The Charles A. Dana Center at the University of Texas at Austin Page | 82 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Session 17: Counting and Creating Sets of 8 At A Glance Children assist the teacher in ordering numerals and matching picture cards 1–8 from bottom to top on a sheet of chart paper called the “Number Wall.” Children use this representation at center time to help them count and develop mental images of each quantity to 8. Children then practice making a spider with 10 fingers, joining 2 thumbs together to make the head and extending the remaining 4 fingers on each hand for the legs. The objective is to help children begin to understand how 8 is related to other numbers (e.g., 4 and 4 makes 8; 8 is 2 less than 10). Finally, the teacher introduces a new counting game whereby children try to grab a set of 8 counters and match each one to corresponding tentacle on an octopus game board. The focus is on estimating how much 8 is and then confirming that estimate through comparison and one-­‐to-­‐one correspondence (e.g., I need to add 2 more/remove 2 to have exactly 8). Materials ●
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Chart paper Tape/markers Tongue depressors Circle paper Chenille sticks Sting Counters Blackline master, “Number Wall Cards” Blackline master, “Octopus Match” Mathematics Addressed ●
Count up to 10 items and demonstrate that the last count indicates how many items were counted. ●
Use concrete models or make a verbal word problem for adding up to 5 objects. ●
Use concrete models or make a verbal word problem for subtracting 1–5 objects from a set. The Charles A. Dana Center at the University of Texas at Austin Page | 83 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Activate Display a sheet of chart paper and arrange numeral cards from the Number Wall Cards blackline on the carpet. ● Explain to children that that you need their help in ordering the numerals going from top to bottom on the chart, which you will post as a number wall to be used at center time. ● Start at the bottom of the sheet and invite a volunteer to show which numeral goes first. Invite additional volunteers to locate the remaining numerals as you continue making (adhering to the chart with tape) the vertical (bottom-­‐to-­‐top) number line 1–8. ● Introduce the picture cards that correspond to each numeral, pairing the face card with the numeral 1, the eyes with the 2, the triangle with the 3, etc. ●
As you pair each picture to its corresponding numeral, be sure to explain why it matches: o Say, “You have 1 face; most people have 2 eyes; a triangle has 3 sides/corners; a square has 4 sides/corners; a hand has 5 fingers; a hexagon has 6 sides/corners; a week has 7 days; and a stop sign (an octagon) has 8 sides/corners.” ● Explain that a stop sign is shaped like an octagon, which has 8 sides. ● Invite a volunteer to help you count out 8 tongue depressors. ● Arrange the tongue depressors so that they resemble the shape of the stop sign on the number wall. Count the sides of the octagon to confirm that there are 8 sides altogether. ●
The Charles A. Dana Center at the University of Texas at Austin Page | 84 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Engage ●
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Explain to children that you will make a spider that they can use to point with as they practice counting and crawling up the number wall at center time. Model how to make a spider with your hands. Hold up all 5 fingers on each hand. o “How many fingers do I have altogether?” Join the two thumbs together and bend them down to create the spider’s head. Explain that the fingers that are still extended on both sides are the spider’s legs. o “How many legs does a spider have on each side of its body? How many legs does it have altogether?” Allow children to practice making a spider with their hands just as you modeled (e.g., making a set of 10 fingers, joining together 2 thumbs to make the head, and wiggling the other 8 fingers that remain extended). Challenge children to visualize how 8 is related to the benchmark number 10: o “How many fingers did I have to bend down to make 8 legs on the spider? How many fingers would be showing if I were to raise both thumbs? How is 8 different from 10? (or) How far away is 8 from 10?” ● If necessary, make a representation of 8 counters inside a tens-­‐frame to highlight how 8 is only two less than 10. ● Use a paper circle to make the spider’s body and cut apart 8 chenille sticks to make the legs. Then, attach the spider to a string. ● Model how to drape the string over the number wall so that the spider is dangling at the bottom facing toward children. Pull down on the string from the back of the chart so that the spider crawls up the number wall as you and children count from 1 to 8. ●
The Charles A. Dana Center at the University of Texas at Austin Page | 85 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Develop ●
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In addition to using the spider puppet to climb and count up the Number Wall at the gross motor center and making an 8-­‐legged spider at the art center, inform children that they will also be playing a grab-­‐and-­‐match game with the number 8 at the math center. Display the Octopus game board and invite children to count the number of tentacles so that they understand that there are 8. Model and explain how to play the game. 1. Use an eyeball strategy to grab what looks like 8 counters. 2. Match each counter to 1 tentacle. 3. Decide if there are too many or too few. 4. If there are too many, take some away so that each tentacle has exactly 1. Likewise, if there are too few, add enough counters so that each tentacle has exactly 1. Explain that the goal is to grab exactly 8 counters, just enough to attach to each of the octopus’s tentacles. Observe children as they play the game and prompt them to reflect on how they know they have exactly 8 counters. o “How do you know that you have exactly 8? How many more/fewer counters do you need to add/take away so that you will have exactly 8? What could you do next time to be sure you get exactly 8 counters?” The Charles A. Dana Center at the University of Texas at Austin Page | 86 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Number Wall Cards The Charles A. Dana Center at the University of Texas at Austin Page | 87 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Number Wall Cards
The Charles A. Dana Center at the University of Texas at Austin Page | 88 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Octopus Match Grab 8 counters and match each one to a tentacle. The Charles A. Dana Center at the University of Texas at Austin Page | 89 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Session 18: Counting and Creating Sets of 9 At A Glance The teacher introduces a 3-­‐by-­‐3 “Tic-­‐Tac-­‐Toe” game board as a visual representation for the quantity/number 9. After demonstrating the traditional game, the teacher introduces a variation whereby players take turns placing their respective color (red or yellow) inside any square on the grid. Once the grid is filled, the teacher arranges and compares the counters in a row of 5 and 4 on a tens-­‐frame to lead children to see the unequal distribution of turns and how 9 is one less than 10. Children then practice with a partner counting out a set of 9 counters and trying to divide the items evenly among each other. In addition to familiarizing children with how 9 is related to other numbers, the goal is to allow them to think about ways to use one-­‐to-­‐one correspondence in order to distribute a quantity evenly and then add or remove one if the two divided parts are not equal. Materials ●
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Chart paper Markers Sticky notes Yellow and red counters Double-­‐sided red and yellow counter Blackline master, “Tens-­‐Frame” Blackline master, “Tic-­‐
Tac-­‐Toe” game board Mathematics Addressed ●
Count 1–10 items, with one count per item. ●
Count up to 10 items and demonstrate that the last count indicates how many items were counted. ●
Use informal strategies to share or divide up to 10 items equally. The Charles A. Dana Center at the University of Texas at Austin Page | 90 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Activate Refer to the Number Wall introduced in Session 17. Gather suggestions from children about images similar to the ones on the Number Wall that would help them think about the number/quantity 9 (e.g., a cat has 9 lives). ● If no one suggests/refers to the game Tic-­‐Tac-­‐Toe, draw a 3-­‐by-­‐3 grid on a sheet of chart paper. Count the number of squares to confirm that there are 9 altogether. ● Arrange 3 sticky notes in a row across the top of the grid. Draw a red circle on each square. o Ask, “How many circles do you see going across the top?” ● Make two additional arrangements with the sticky notes—one running vertically up and down the middle row and another going across the grid diagonally. After each rearrangement, prompt children to confirm that there are still 3 circles. ●
Engage Explain the rules for playing Tic-­‐Tac-­‐Toe for those children who might be unfamiliar with the game. 1. Two players take turns—one person marking Xs inside any square and the other player making only Os. 2. The player who succeeds in placing three of his/her assigned marks in a horizontal (straight across), vertical (straight up and down), or diagonal row wins the game. ● If time permits, play a demonstration round with you competing against children. (Note: Try to let them win so that they feel successful.) ●
Develop Display a Tic-­‐Tac-­‐Toe game board on the carpet. ● Inform children that they are going to help you play a variation of Tic-­‐Tac-­‐Toe. 1. Instead of making Xs and Os, players take turns placing an assigned color (red or yellow) inside any square on the grid. 2. Players flip a double-­‐sided counter (red and yellow) to determine who goes first. 3. The goal is not to make a row of one color, but rather to see who has the final turn filling in the grid and who has the most of his/her color on the game board. ● Play a demonstration round with a volunteer. ●
The Charles A. Dana Center at the University of Texas at Austin Page | 91 2012 Early Mathematics—A Resource for Teaching Young Children ●
Prekindergarten At the end of the game, count to confirm that there were a total of 9 counters used to fill the grid. Then, remove the counters and use a tens-­‐frame to compare how many there are of each color. o “Is this a fair game? Why/why not?” Discuss the arrangement of red and yellow counters inside the tens-­‐frame so as to prompt children to think about the relationship of 9 to the benchmark numbers 5 and 10 (e.g., 9 is one less than 10 and four more than 5). o “How many red/yellow counters? How many more reds would there have to be so that there is the same number of red counters as yellow?” ● Arrange the counters in an alternating pattern so that children can visualize the unequal distribution/order of turns. Point out how the first color in the sequence is the same color at the end. ●
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Prompt a volunteer to add a counter to the end of the sequence that would complete the pattern. Then, count and compare the new total. Suggest another way to make the game even—removing the last counter at the end of the pattern so that there are 8. ● Point out how 9-­‐-­‐unlike the numbers before (8) and after (10)—is odd because you cannot divide it evenly. ● Before dismissing to centers (or while working with small groups at centers), pair children with a partner and allow them to first count out a set of 9 counters and then determine if it is possible to divide that many items evenly among each other. Allow children to use/fill the 3-­‐by-­‐3 game board to make sure they have counted out a set of 9. ●
The Charles A. Dana Center at the University of Texas at Austin Page | 92 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Tic-­‐Tac-­‐Toe The Charles A. Dana Center at the University of Texas at Austin Page | 93 2012 Tens-­‐Frame Early Mathematics—A Resource for Teaching Young Children Prekindergarten The Charles A. Dana Center at the University of Texas at Austin Page | 94 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Session 19: Joining and Separating Sets Up to 10 At A Glance The teacher uses a calendar to measure the time frame of the Hungry Caterpillar’s metamorphosis from caterpillar to butterfly. First, children practice reciting the counting word sequence to 31 as the teacher marks off each day/square with a representative numeral. The teacher then draws a picture in each of the representative squares to help children visualize and sequence the growing accumulation of foods the caterpillar ate and the number of days he spent inside the cocoon before becoming a butterfly. Next, the teacher models the joining of two sets (7 and 7) to represent the duration of 2 weeks. Finally, during center time, children work in small groups using cubes and a number line to directly model the daily combinations of foods that the caterpillar ate in the story. The focus is on solving word problems that involve the joining and separation of sets. Materials ●
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Note: Prepare a 5-­‐by-­‐7-­‐
grid calendar created on poster board or chart paper ahead of time. The Very Hungry Caterpillar by Eric Carle Linking cubes Blackline master, “Number Line” Handprint placemats (from Sessions 13 and 14) Mathematics Addressed ●
Use words to rote count to 30. ●
Use concrete models or makes a verbal word problem for adding up to 5 objects. ●
Use concrete models or makes a verbal word problem for subtracting 1–5 objects from a set. The Charles A. Dana Center at the University of Texas at Austin Page | 95 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Activate ●
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Display a 5-­‐by-­‐7-­‐grid calendar created on poster or chart paper. Inform children that you will use the calendar to measure in days and weeks the amount of time that it took the Hungry Caterpillar to transform from larva to a mature butterfly. Remind children that 1 month is typically 31 days. Start at the very top left-­‐hand square on the grid and begin counting left to right (with a return sweep from each top to bottom shift) until you reach the 31st square. Write a corresponding numeral in the top corner of each square as you count forward. Encourage children to count out loud as you point and label. Emphasize the transition patterns between decades (e.g., after 20 comes after 19, and 30 comes after 29). Engage ●
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Prepare children for a second reading of The Very Hungry Caterpillar. As you read, pause after each day that the caterpillar stopped to eat food. Make a quick drawing/representation of the amount of food that the caterpillar ate that day inside the representative square on the calendar. Stop drawing once you get to the part of the story where the author states that the caterpillar spent a total of 2 weeks inside the cocoon. Challenge children to problem-­‐solve the following calculation. o Ask, “How many days are in 1 week? How many days would there be altogether if you added 7 more?” Invite two volunteers to make one train of 7 cubes apiece. Remind volunteers to compare their trains to be sure they each have exactly 7. Join the two trains together and count the new total—14 cubes. Discuss how the quantity changed. o “Did the number of cubes grow (get larger) or decrease (get smaller)? What happened to make the total get larger?” Count on 14 additional squares from the second Sunday on the calendar where you made a drawing to show the caterpillar eating a leaf. Draw a cocoon inside each of these 14 squares. The Charles A. Dana Center at the University of Texas at Austin Page | 96 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Turn to the proceeding page in the book showcasing the caterpillar as a butterfly. Draw a butterfly inside the square on the calendar that follows the sequence of cocoons. ● Go back to the starting point on the calendar and count the number of days it took the caterpillar to transform into a butterfly. ● Emphasize the last count to confirm that the entire transformation cycle lasted 21 days. ●
Develop During center time, convene a small group of children in front of the calendar displaying the caterpillar-­‐to-­‐butterfly transformation. ● Begin with the caterpillar’s first Monday feeding and instruct children to determine how many fruits he had eaten by the end of Tuesday. o “If the Hungry Caterpillar ate 1 apple on Monday and 2 pears on Tuesday, how many fruits did he eat altogether by the end of Tuesday?” ● Allow children to use counters to model the joining of the two sets presented in the above. Some children, however, might be able to figure out the total in their head simply by looking at the two sets, which are relatively small, and visually counting 3 altogether. ● When children have determined the total, arrange a red cube (the apple) and two green cubes (the pears) on a number line to show/model the joining of the two sets and the resulting increment from 1 (Monday’s amount) to 3 (the accumulation of both days). ●
The Charles A. Dana Center at the University of Texas at Austin Page | 97 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten With the combination of 3 cubes still in place, arrange an additional set of 3 purple cubes on the number line to demonstrate the new total after Wednesday’s feeding. o “If the Hungry Caterpillar had eaten 3 fruits altogether by Tuesday and then 3 prunes on the following day, how many fruits had he eaten by the end of Wednesday?” ● Model how to count on from 3 up three spaces on the number line to 6. ●
o “That’s 3 here and then 4, 5, and 6; he ate 6 fruits by the end of Wednesday.” Distribute cubes and the handprint placemats children created in Sessions 13 and 14. ● Instruct children to arrange the cubes on the number line on their placemat to determine the total number of fruits the caterpillar had eaten by the end of Thursday. ● Observe children’s counting strategy for determining the new combination. o Do they represent each day’s feeding, making a separate set of cubes for each group, and then start back at 1, counting how many there are altogether? o Do they start at 6 (the last total you modeled) and count up the number line 4 spaces to 10? If so, do they correctly recite the counting word sequence (e.g., “Six … seven, eight, nine, and ten”). ● Throughout the week, continue meeting with small groups of children and guide/observe them as they model additional joining and separating situations. o Joining “If the caterpillar had eaten 3 bananas on Monday morning and then 2 grapes in the afternoon, how many fruits would he have eaten by the end of the day?” o Separating “The caterpillar found 5 apples. He ate 3 of them, but then was too full to eat anymore. How many apples were left (not eaten)?” ●
The Charles A. Dana Center at the University of Texas at Austin Page | 98 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten   1 2 3 4 5 6 7 8 9 10 Number Line The Charles A. Dana Center at the University of Texas at Austin Page | 99 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Session 20: Comparing and Ordering Sets to 10 At A Glance Children revisit the calendar you created during Session 19 illustrating the Hungry Caterpillar’s metamorphosis from larva to butterfly. Children practice reciting the counting word sequence (paying close attention to accuracy and order) as they watch and listen to the teacher touch and count each square on the calendar up to the designated stopping point—
22. Children then revisit their learning with selecting appropriate quantification strategies for making an equal set of objects as they estimate which color path along a grid is the shortest to the center. After children make an eyeball estimate, they confirm their visual intuition by using a one-­‐to-­‐
one matching strategy to count and compare the number of squares along each path. Finally, children assemble the cubes they matched to the grid into separate towers, which they transfer to a graph in order to determine the numerical value of each color path. The focus is on using counting strategies and numerals to determine how many. Materials ●
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The Hungry Caterpillar Calendar created during Session 19 Pathway Grid blackline master Blackline master, “Pathway Graph” Blackline master, “Strategies for Making Equal Sets” Color cubes (green, yellow, blue, and red) Mathematics Addressed ●
Use words to rote count to 30. ●
Count up to 10 items and demonstrate that the last count indicates how many items were counted. ●
Recognize one-­‐digit numerals 0–9. The Charles A. Dana Center at the University of Texas at Austin Page | 100 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Activate ●
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Display the calendar you illustrated during Session 19 to help children visualize the amount of time that elapsed (as measured in days and weeks) from the day that the Hungry Caterpillar hatched from his egg to the day he became a butterfly. Count the squares on the calendar with illustrations on them to refresh children’s memory about the number of days the larva-­‐to-­‐butterfly metamorphosis lasted. Instruct children to squat on their knees and make their bodies look tiny like the Hungry Caterpillar did on the day he hatched from the egg. Tell children to gradually stand up and make their bodies look bigger as they help you count up from 1 to 21—the number of days it took the caterpillar to grow into a butterfly. Once you reach 22, have children open up their wings and shout the number as they fly away into the sky. Congratulate children for carefully counting up to such a big number. Engage ●
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Display a color copy of the Pathway Grid so that all children can easily see it. Instruct children to look closely at each path, which all lead to a leaf in the center of the grid. Explain that the Hungry Caterpillar would like to get to the leaf as quickly as possible. o Ask, “Which would be the shortest path for the Hungry Caterpillar to take to get to the center where the leaf is located? How do you know? If you are not sure, what could you do to figure which path would be the shortest distance to path?” Discuss how it might be difficult to estimate which path is the shortest because of the way each one bends. If no one suggests, point out that counting the squares in each path would help confirm which one was the shortest/longest. Highlight the blue path and invite a volunteer to estimate how many cubes long it appears to span from start to finish. Instruct the volunteer to count out the number of cubes he estimated. Refer to the Strategies for Making Equal Sets blackline and emphasize to the rest of the group how the volunteer used an eyeball (estimation) strategy to make a tower that had as many cubes as there are on the path. The Charles A. Dana Center at the University of Texas at Austin Page | 101 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Direct the volunteer to confirm his/her eyeball estimation by matching one blue cube to each square. o “Does [volunteer’s name] have too many/too few/just the right number of cubes? How many more should he add/take away so that the number is the same?” ● Count the cubes and emphasize that the last counting word tells how many. ●
Develop During centers, form a small group and distribute a color copy of the Pathway Grid to pairs of children. ● Instruct children to select a path on the grid and count the number of squares, starting at the point of origin all the way to the center. ● Allow children to use connecting cubes that match the color of the squares along their assigned path, placing them one by one on top of each square as they count. Emphasize that the last counting word children say as they enumerate the squares and cubes tells how many (or how long the path is). ● Direct children to reassemble the cubes they arranged along their assigned path and then make a linear tower. Gather each tower and order/compare them by height. o “Which is tallest/shortest? How does this information help you decide which path would provide the quickest route to the park?” ●
Emphasize the one-­‐more/one-­‐less relationship between each quantity (e.g., 6 is one less than 7; 9 is one more than 8). ● Direct children to color in the Pathway Graph that accompanies the Pathway Grid to represent and compare the measurement data. ● Point out how the vertical (up and down) display of the numerals 1–10 in the left-­‐hand column on the graph correspond to the height (distance) recorded for each color path. ●
The Charles A. Dana Center at the University of Texas at Austin Page | 102 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten o “How can you use the numerals on the side of the graph to help you know how many cubes long the tower is?” The Charles A. Dana Center at the University of Texas at Austin Page | 103 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten The Charles A. Dana Center at the University of Texas at Austin Page | 104 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten The Charles A. Dana Center at the University of Texas at Austin Page | 105 2012 Early Mathematics—A Resource for Teaching Young Children Prekindergarten Strategies for Making Equal Sets The Charles A. Dana Center at the University of Texas at Austin Page | 106 2012 
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