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Welcome to
Making Sense of Math:
Early Number Concepts
• A number of different graphs are posted
around the room. Put “yourself” on each around the room. Put “yourself” on each
graph.
• Look around the room. Introduce yourself to
three people that you don’t know.
• Introduce yourself to the people at your table.
Making Sense of Math:
Early Number ConceptsEarly Number Concepts
Amy Lewis
Math Specialist
IU1 Center for STEM Education
Goals for the courseGoals for the course
• Use a variety of tools to deepen understanding
of place value and explore number relationships
to connect number concept meanings and
representations.representations.
• Participate collaboratively in solving problems in
other base systems to strengthen reasoning
skills.
• Connect new understandings of numbers to
classroom practice.
Who are we?Who are we?
• Look at the graphs about us.
• What do you notice?
• With a partner, write a single sentence • With a partner, write a single sentence
that best describes the group.
Pre-Test
• Please take a few minutes to complete the
pre-test.
• Although you should do the best that you can,
please do not feel pressure to get all of the please do not feel pressure to get all of the
questions perfect.
• This is only a measure of growth from the
start of the course until the end.
Day 1:
• Use physical models to better understand
numbers in the Base-10 system through
number construction and deconstruction.
• Examine the Hundred Chart and use it to • Examine the Hundred Chart and use it to
deepen understanding of place value.
• Reflect upon how to use these strategies in
the classroom.
What does it mean for students
to understand place value?
• Common understandings:
– The student can locate and indicate ones, ten, and
hundreds columns in three-digit numbers.
– The student knows the values of numbers according to the – The student knows the values of numbers according to the
column in which they are placed, e.g. the student says,
“The eight is in the hundreds column, so it means 800.”
– The student can line digits up in columns in the manner of
the formal algorithm.
• Are these really indicators of a deep
understanding of place values and numbers?
How can we make the concept of
place-value visible, concrete, and
relevant?relevant?
Groups and Leftovers
• For each row of the chart
– Grab a handful of beans (less than 100 beans)
– Group the beans into groups of whatever is in the
first column of the table.first column of the table.
• For example, if the number in the first column is 7, put
your beans into groups of 7 and then fill in the rest of
the information.
– Grab a new handful of beans for the next row.
• What patterns do you notice?
Two-Handed Math
• A tool that we use in the early years is to have
students show values with their fingers.
– Show me 4 fingers
– 7 fingers– 7 fingers
• Let’s extend this…
– Show me 12 fingers.
Two-Handed Math
• Preferred method for this activity:
• How would we show 1, 3, and 7 using this
method?
Two-Handed Math
• What about…
– 14 fingers?
– 16 fingers?
– 18 fingers?
• How many more from 18 are needed for 20?
• What about…
– 24 fingers?
– 34 fingers?
• The Wave! �
Two-Handed Math
• What are possible extensions?
– If you wanted to make 73, how many people
would you need?
– Musical numbers– Musical numbers
– 34 + 20
– 43 + 25
– 31 – 12
– 48 + 37
• Other possibilities?
Two-Handed Math
• What mathematics is involved in this task?
• What is the value of this task?• What is the value of this task?
• When should it be used?
Looking at Models
• Each group will examine examples of physical
models that can be used to represent the
base-10 number system.
• For each one,
– How does this model work?
– How can the model be used to deepen students’
understanding of number?
– What are the benefits and limitations of this
model?
• Place your answers on a poster paper.
Looking at Models
• Two Hands
• Counters and Cups
• Bundles of Sticks
• Interlocking cubes
• Pre-grouped Base-Ten Models
• Base-Ten Blocks
• Bean Sticks
Looking at Models
• How do we know when to use the different
models with our students?
• How can we make these models available to
students?students?
Three Other Ways
463• Show this using your pre-grouped base-10
models.models.
• Find and record at least three other ways to
show this number.
Make a Square
• Materials needed:
– Pre-Grouped Base-Ten pieces
– Place-Value Mat (optional)
• Listen to the number of claps that you hear.
• For each number of claps, add that many ones
to your place-value mat.
• Your goal? Make a square!
I have… Who has…?
Standard Game
Challenge Game
• How is the mathematics different in the
two games?
• Why is the Challenge Game important?
Hundreds Charts
• Using your (essentially) blank hundreds chart,
fill in the numbers that are to the left, right,
above, and below the printed numbers.
• How do you know which numbers go in the • How do you know which numbers go in the
empty spots?
• What do you notice about neighbor numbers?
• Create a model for one of the printed
numbers on the chart.
Hundreds Charts
• Make all of the numbers in that row.
– How are all of the numbers alike?
– How are they different?
– What happens at the end of each row?
• Make all of the numbers in that column?
– How are the numbers in the columns alike?
– How are they different?
• Given any number, what do you have to do to
make one of its neighbors?
Model Math
Think about the last 4 activities.
• What mathematics is involved in these tasks?• What mathematics is involved in these tasks?
• What is the value of these tasks?
• How should they be used and sequenced?
Human Number Line
• Each person will receive one number.
• Please come up and stand in a straight line.
What number did we form?
• Who is in our three-digit number period (aka
“number family”)?
• Re-order.
Say It/Press It
• Directions:
– Say the number in base-10 language.
– Say the number in standard language.
– Enter the number into your calculator.– Enter the number into your calculator.
Say It/Press It
• Directions:
– Say the number in base-10 language.
– Say the number in standard language.
– Enter the number into your calculator.– Enter the number into your calculator.
Say It/Press It
• Directions:
– Say the number in base-10 language.
– Say the number in standard language.
– Enter the number into your calculator.– Enter the number into your calculator.
Say It/Press It
• What mathematics is involved in this task?
• What is the value of this task?
• When should it be used?
• What are possible extensions?
Wipe Out
• Enter the number 45673.189 into your
calculator.
• What is this number?• What is this number?
• Your challenge is to make your screen become
40673.189 by taking away one number.
Wipe Out
• Wipe out the number in the tens place by
taking away a number.
• Change the number in the ten-thousands
place to a 6 by adding a number.place to a 6 by adding a number.
• Wipe out the number in the tenths place by
taking away a number.
Wipe Out
• Wipe out the number in the hundreds place
by taking away a number.
• Change the hundredths digit to a 7 by
subtracting a number.subtracting a number.
• Wipe out the number in the thousandths
place by taking away a number.
Wipe Out
• Wipe out the number in the ten thousands
place.
• Wipe out the ones.
• Wipe out the hundredths place.• Wipe out the hundredths place.
• Are you wiped out?
Wipe Out
• What big ideas of the base-ten system did you
use in Wipe Out?
• What is the value of this task?• What is the value of this task?
• When should it be used?
Spin to Win!
• Create 4 connecting circles and one “Reject”
circle on your white board.
Reject
Spin to Win!
• The spinner will spin 5 times.
• After each spin, place each number in one of
the five game circles.
• Your goal is to make the largest 4-digit whole • Your goal is to make the largest 4-digit whole
number that you can without moving or
erasing a number.
• Now, let’s create the smallest number
possible.
Spin to Win!
• What strategies did you use to create the
largest number possible? The smallest?
• What mathematics is involved in this task?• What mathematics is involved in this task?
• What is the value of this task?
• When should it be used?
Finding Meaning
Answer the following problem:
Derrick has 23 play cars. For his birthday, he gets
34 more cars. How many play cars does he
have now?have now?
What does the 7 digit mean in the answer?
What does the 5 digit mean in the answer?
How would your students answer that question?
Finding Meaning
Let’s visit NCTM Illuminations to see more.
http://illuminations.nctm.org/Reflections_preK-2.html
Looking Back
• What mathematics have we explored today?
• How have these activities shaped your
understanding of place value?
• How would you describe the cognitive • How would you describe the cognitive
demand of the tasks we explored today?
Homework
• Complete the following problem.
• Look at how the students solved the problem.
• What place value concepts do they seem to
struggle with?struggle with?
• How would you address these struggles?
Making Sense of Math Wiki
http://makingsenseofmath.iu1.wikispaces.net
• Join the wiki to ensure that you can post
comments on the discussion board.comments on the discussion board.
Final Project
• To receive 1 CPE Credit for this course,
participants must complete a Final Project.
• Each participant can choose a Final Project
from the following three choices.from the following three choices.
Final Project
• Unit of Instruction
– How might you teach early number concepts
different based on your learning from this course?
– Write a unit of instruction to incorporate the – Write a unit of instruction to incorporate the
number sense strategies into your mathematics
instruction.
– Describe an action plan for implementation.
• Rubric is posted on the wiki.
Final Project• Student Work
– Collect 5 pieces of student work that demonstrate varying levels of place value misunderstandings.
– Identify the mathematical misconceptions in the work.work.
– For each artifact, write and implement an action plan that describes how you are going to use the strategies used in this course to address the misunderstanding.
– Reflect on the successes and challenges faced when implementing each action plan.
– Rubric is posted on the wiki.
Final Project
• Self-Study
– Do you have an idea/topic for a project you’d like
to explore which is not listed above?
– Please discuss your idea with the instructor in – Please discuss your idea with the instructor in
order to receive permission to pursue your own
line of study.
