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Developing Number Sense through Manipulatives and
Problem-Based LearningCydney BatesOrange County Schools
Our Goals For Today➔ Determine how number talks support our vision of student learning in
a K-1 classroom setting
➔ Review various types of number talks to utilize with k-1 students that help develop students’ number sense
➔ Discuss the concept of subitizing and manipulatives that reinforce this developmental concept
➔ Explore the rekenrek and ten frames as tools to support students’ discourse within number talks
Our Goals For Today➔ Discuss the power of problem-based learning and effective ways to
implement this environment into our classroom
➔ Explore task-based learning activities that develop both procedural fluency and conceptual development of K-1 mathematics standards
➔ Relate the concept of linear measurement to addition and subtraction
➔ Determine how to utilize cuisenaire rods to develop our students’ number sense through quick images and task-based learning
Manipulative Use WithinNumber TalksBuilding Number Sensein the K-1 Classroom
Vision for Student Learning“Mathematics has been viewed as a discrete set of rules and procedures to be implemented with speed and accuracy but without necessarily understanding the mathematical logic.”
-Sherry Parrish
Vision for Student Learning
Turn and Talk: What is your vision for student learning? How is this different from the way in which you may have experienced mathematics instruction
when you were a student?
Vision for Student Learning“Our students must have the ability to reason about quantitative information, possess number sense, and check for the reasonableness of solutions and answers.”
-Sherry Parrish
AccuracyEfficiencyFlexibility
How Do Number Talks Support Student Learning?
How Do Number Talks Support Student Learning?
“By sharing and defending their solutions and strategies, students are provided with opportunities to collectively reason about numbers while building connections to key conceptual ideas in mathematics.”
-Sherry Parrish
Key Components of a Number Talk● Classroom environment and community
○ Norms, Growth Mindset, Productive Struggle
● Classroom discussions○ Talk moves, Color Groups
● The teacher’s role● The role of mental math
Types of Number Talks● Dot Cards● Five Frame / Ten Frame / Double Ten Frame● Rekenreks● Number Strings● Open Number Lines
What does it mean to subitize?
How many dots do you see? How do you see them?
What does it mean to subitize?Subitizing: having an innate ability to see amounts instantly
Perceptual Subitizing: Seeing a very small amount and “just knowing” its total without any other learned mathematical knowledge
Conceptual Subitizing: Quickly putting together smaller amounts OR recognizing previously known arrangements
Let’s Watch!While we watch focus onthe role of the following
throughout the Number Talk:
Kindergarten: The TeacherFirst Grade: The Children
Benefits to Using Manipulatives during number talks● Children can see the addend AND the sum
in the image● They can see numbers composed and
decomposed in more that one way● Moves children beyond counting to figure out
“how many”
Deeper Questioning with Manipulatives● Helps build fluency with targeted numbers● Not only asking “How many did you see?” ● Deeper questions such as:
○ How many more do we need to make 10?○ How many are left after removing _?○ Question Stems
Dot Cards and Ten Frames ActivitiesDot Cards:● Dot Card War● Concentration● I Wish I had...● Add-A-Rama
●
Ten Frames:● Five/Ten Frame Flash● Go Fish - Make 10 or 20● Roll and Build● Ten Frame Difference Challenge
What is a rekenrek?
What is a rekenrek?● An arithmetic rack read from left to right● Start with red beads and then incorporates white beads● Supports the natural mathematical development of children● Helps students develop strategies for addition and subtraction
Let’s Watch!In what ways does the use of a rekenrek in this video support the
learning of:
AutomaticitySubitizingFlexibility
Number Sense
Introducing Rekenreks to studentsExploration activities:● Give students a number to build; facilitate discourse that challenges them to share
and discuss the various number arrangements.● Read a counting book and have students model the action.● Begin building number sentences as students become familiar with the rekenrek.● Rebuild the rekenrek using unifix cubes or some other manipulative to reinforce
number combinations.Using tools efficiently:● Challenge students to make numbers in a certain number of moves.● Set up situations that allow them to look for various ways to make a total.● Practice “Quick Looks” through partner work or within a whole group setting.
Materials to Make and Take● Individual Rekenreks
○ Foam/Laminated Construction Paper/Cardstock, Red & White Beads, Pipe Cleaner, Hole PUncher
● Question Stems/Communication Posters
What Is Problem-Based Learning?
How can I develop my students’ number sense throughproblem-based learning?
Problem-Based LearningWhat words come to mind when you think about problem-based or task-based instruction and learning?
● What is it? ● Why do we need to use it?
Post your responses tohttps://todaysmeet.com/MathSummit
What is the connection between measurement and addition/subtraction?
Problem-Based LearningThis length measurement task provides opportunities: ● to contextualize mathematics and learn how context influences how we think
about and deal with numbers and measurement● for kids to investigate, assess, and refine mathematical rules and
generalizations about measurement● to work with equivalence, build number sense and compare lengths● to engage in math discourse and share ideas through productive struggle● to build reasoning and sense making skills by working with a variety of
manipulatives and tools
“Farms and Fences” Unit: Suggestions adapted from Beyond Pizzas and Pies by McNamara & Shaughnessy (2015)
What’s the Context?Four farmers are working to build a fence to border each of their gardens. After
students determine the appropriate length of each fence, they are then tasked with trying to help the Lumberyard workers provide the
perfect fence for each farmer!
OH No! After visiting the Lumberyard the farmers learn that the fence posts come in many different sizes and colors. The farmers can choose any color that they would like, but they have to make sure that the color posts they choose will fit their fence. Also, each farmer wanted a different arrangement of posts!
● Carrot Farmer wanted a fence made with only orange posts.● Tomato Farmer wanted a fence made with only red posts.● Lemon Farmer wondered whether his fence could be made with only yellow
posts.● Berry Farmer wanted to have the fanciest fences ever! He just wanted a
mix of all the colors!
What are all the different sizes of
each specific fence post?
Sizes and Colors of Fence Posts at the LumberYardWhite 1
Red 2
Light Green 3
Purple 4
Yellow 5
Dark Green 6
Black 7
Brown 8
Blue 9
Orange 10
Wow! We have so many different
choices and designs to make
fancy Fences.
Building a ModelLet’s help the farmers by building a fence. We will make models of the farms so that the farmers can see how the fence sections fit together.
Mathematicians use models to figure out solutions to problems.
Cuisenaire rods can help us serve as a model to solve a problem!
Mathematical ModelingLet’s look at the line on our piece of paper.
0 1 60
Let’s work in groups to determine how you can use thecuisenaire rods to build a fence design.
Fence Posts Designs
CARROT FARMERCarrot Farmer would just like to use orange fence posts.
Is that possible? Let’s try it with orange fence posts.
LEMON FARMERLemon Farmer wants to try just yellow. Lemon Farmer asks,
● Will yellow work? ● Will yellow make a nice fancy
fence?● Do yellow designs first and
then you can finish the rest?
BERRY AND TOMATO FARMERSPredict how many of the purple and red sections these farmers will need to build their fence. Then investigate the actual number to see how close you came with your estimation!
Fancy Fences!The farmers now all want fancy fences? Can you help them?
Create your own fancy fence?
Gallery Walk Take a scavenger hunt looking for strategies that are similar or different from the ones that
your group used.
I agree
I disagree
I’m not sure
White 1
Red 2
Light Green 3
Purple 4
Yellow 5
Dark Green 6
Black 7
Brown 8
Blue 9
Orange 10
● What strategies did you use to help you create your different fence posts designs?
● How did you figure out what lengths and colors would work to build your fence?
● Could you have been more efficient when building your fence?
MATH CONGRESS
Which farmer has the longest fence along the road?
FINAL FENCE DESIGNSBerry Farmer
Used 16 purple fence posts and 16 white fence posts
Lemon Farmer Used 16 yellow fence posts
Tomato Farmer Used 40 red fence posts
Carrot Farmer
Used 4 orange fence posts, 8 light green fence posts, and 8 red fence posts
QUICK IMAGES
CAN YOU GUESS MY Fence Design?My fence is _____ cm long. Can you guess what my fence looks like?
Ex. My fence is 10 cm long. Can you guess what my fence looks like?
● 1 Orange Rod● 2 Yellow Rods● 2 Purple Rods and 1 Red Rod● 1 Black Rod and 1 Light Green Rod● 1 Dark Green Rod and 1 Purple Rod
COMPARING LENGTHSWhich one is longer? How do you know?
Options for visuals:
● Show two rods (Black and Orange Rods)● Show three rods (Black, Brown, and Orange Rods)● Make connections across rods that were previously (Only show Brown, but ask about the Brown
and Orange)
Ask questions such as:
● Can you tell which color is the longest without me showing you? How do you know?● If we ordered the rods shortest to tallest, which would be first, which would be second?● Could you put the rods together like a staircase? What order would they go in? ● Are any rods missing? How do you know?
Flash Fence DesignsHow long is my fence? How do we know?
HOW FAR?How far are we in building our fence? How do we know?
1. 1 Orange Rod2. Plus 1 Yellow Rod3. Plus 1 Dark Green Rod4. Plus 1 Blue Rod5. Plus 1 Purple Rod and 1 Dark Green Rod
Making trainsLaura made a train with the Cuisenaire rods. Her train was made from three rods which were all different colors.It was the same length as three purple rods:
Laura's train looked like this:
Can you make a different train, the same length as Laura's, with three rods that are all different colors?
Can you make one the same length using no colors that Laura used? http://nrich.maths.org/4331
Making trainsRob made a train that was the same length as Laura's using four different colored rods. How did he do it?
Charlene made a train which was as short as it could be using four different colored rods. Can you find a rod that is the same length as Charlene's train?
Ben made a train using four different colored rods the same length as two black rods and a light green one:
He made it without using a white rod. How did he do it? http://nrich.maths.org/4331
Number Talk ResourcesKindergarten Number Talks
First Grade Number Talks
Ten Frame Number Talks with the Number 6Double Ten Frame Number Talks
Dot Images Number TalksAddition/Subtraction Number Talks to 10
Rekenrek Mini-Lessons for Early Addition and Subtraction
Also linked on theOrange County PreK-5 Mathematics Website: Teacher Resources
https://sites.google.com/a/orange.k12.nc.us/prek-5-mathematics/teacher-tools
Additional Classroom ResourcesDot Card and Ten Frame Games/Activities
Ten Frame Playing CardsKindergarten K.NBT.A.1 - Rekenrek Activity
“Making Trains” ReferenceNumber Talk Classroom Posters
Number Talk Communication PostersMath Talk Stems ResourcePrintable Cuisenaire RodsBlank Cuisenaire Rods
Evaluation Survey - Thank You for Your Feedback!!http://tinyurl.com/2017MathSummit
