math basics amy lewis math specialist. day 1: use physical models to develop number sense in our...
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Math Basics
Amy Lewis
Math Specialist
Day 1: Use physical models to develop number sense
in our Base-10 system through number construction and deconstruction.
Consider alternative algorithms for adding and subtracting numbers.
Use a number line to add and subtract.
Reflect upon how students think about numbers.
Number Meaning Relationships Magnitude Operation Sense Real Life Number Sense - Applications
Howden, 1989
What does it mean for people to have “number sense”?
Sense of Number…
…in its most fundamental form, entails an ability to immediately identify the numerical value associated with small quantities;
…should extend to numbers written in fraction, decimal, and exponential forms.
…when lacking, interferes with learning algorithms and number facts and prevents use of strategies to verify if solutions to problems are reasonable.
NMAP, page 27, March 2008
Big Idea
How can we make the concept of our place-value system
visible, concrete, and relevant to give students a better sense
of number?
The Base-10 Positional System
How Many Stars?
How Many Stars?
What does this response tell us?
How Many Stars?
What does this response tell us?
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. 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
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
34 + 20
43 + 25
48 + 37
31 – 12
Other possibilities?
Three Other Ways
72 Show this using your base-10 blocks. Find and record at least three other
ways to show this number.How many 1s are in this number?
How many 10s are in this number?
Three Other Ways
463 Show this using your base-10 blocks. Find and record at least three other
ways to show this number.How many 1s are in this number?
How many 10s are in this number?How many 100s are in this number?
7536 How many 1000s are in this number? How many 100s are in this number? How many 10s are in this number? How many 1s are in this number?
How is this understanding richer than,“What is the digit in the 10s place?”
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 empty spots?
What do you notice about neighbor numbers? Using the base-10 blocks, 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?
Say It/Press It Directions:
Say the number in base-10 language. Say the number in standard language. 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.
Say It/Press It Directions:
Say the number in base-10 language. Say the number in standard language. Enter the number into your calculator.
Wipe Out Enter the number 45,673 into your calculator.
What is this number?
Your challenge is to make your screen become 40,673 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.
Wipe out the number in the hundreds place by taking away a number.
Change the ones digit to a 7 by adding a number.
Wipe Out Wipe out the number in the ten thousands
place.
Wipe out the ones.
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?
When should it be used?
Considering Alternative Algorithms
“Children naturally have mathematical ideas. If they are encouraged to articulate them, they become aware of their own ideas and continue to have more. When classrooms are organized to encourage children to analyze their own, their classmates’, and their teacher’s ideas, then they will develop strong and more refined concepts.”
Authors of Developing Mathematical Ideas
Alternative Addition Strategies
Without using pencil and paper, complete the following addition problem:
Derrick has 57 pennies in his pocket. He finds 24
more pennies and put them in the same pocket. How
many pennies does Derrick now have in his pocket?
Alternative Addition Strategies
What strategy did you use to find this sum?
How can you use your base-10 blocks to model this problem?
How can you use a number line to model this problem?
How can you state your addition strategy in general terms?
Alternative Addition Strategies
Sarina has 17 Barbie dolls. Her best friend comes over and brings 24 Barbie dolls. How
many Barbie dolls do the girls have altogether?
Solve using mental math. Model with base-10 blocks. Demonstrate with a number line. State your strategy in general terms.
Alternative Addition Strategies Use base-10 blocks to model the following
addition problems. 21 + 13
47 + 33
58 + 71
103 + 19
161 + 205
Alternative Addition Strategies Use an empty number line to model the
following addition problems. 37 + 26
48 + 21
74 + 39
157 + 25
553 + 38
Alternative Addition Strategies
Without using pencil and paper, complete the following addition problem:
Derrick has 57 pennies in his pocket. 24 pennies fall
out of his pocket. How many pennies does Derrick
now have in his pocket?
Alternative Addition Strategies
What strategy did you use to find this difference?
How can you use your base-10 blocks to model this problem?
How can you use a number line to model this problem?
How can you state your subtraction strategy in general terms?
Alternative Addition Strategies
Sarina made 62 cookies to take to school. Overnight, the dog finds the cookie and eats 27 of them. How many cookies are
left for Sarina to take to school?
Solve using mental math. Model with base-10 blocks. Demonstrate with a number line. State your strategy in general terms.
Alternative Addition Strategies Use base-10 blocks to model the following
subtraction problems. 53 – 17
37 – 19
83 – 56
123 – 76
387 – 19
Alternative Addition Strategies Use an empty number line to model the
following addition problems. 99 – 17
75 – 26
52 – 49
167 – 78
433 – 251
The Traditional Algorithm Complete the following problem using the
traditional algorithm: 125– 67
What are the mathematics involved in this problem?
Thinking Bigger
Where do we encounter bigger numbers in daily life?
How Much Is It?On October 12, 2009, at 11:15:26 p.m EST, the
US National Debt Clock read:
$12,516,385,125,567.99
How do you read that number?How big is this number?
How Much Is It?The National Debt has increased an average
of
$3.95 billionper day since September 28, 2007.
Write this number out. Is it:
$3,000,000,000.95? $30,000,000.95? $3,950,000,000.00? $3,000,000,095.00?
How Much Is It?
100 years ago—July 1, 1910 —the National Debt was:
$2,652 million How do you write that number? How does it compare to our current debt of
almost $12 trillion? A lot less? Less, but only a little less? About the same? More, but only a little more? A lot more?
Is This Possible?Ms. Hope E. Ternal entered her third grade class on Monday morning counting:
999,997. . . 999,998 . . . 999,999. . . 1,000,000!
“Whew, I made it,” she said. “After school on Friday I started counting, and did not stop all weekend. I wanted to see if I could get to one million—and I made it!”
Is this possible? Could Ms. Ternal have started counting on Friday and gotten to 1 million by Monday? Assume that she counts at a rate of one number per second.
How Different Is It?
Jim was thinking about counting from 1 to different numbers:
A thousandA millionA billion
“They’re all big numbers. They come in order—thousands, millions, billions — so I think counting to
each one will take a little bit longer than the one before, but not a lot.”
Is Jim correct? Why or why not?
How Different Is It?
Assume you count at a rate of one number per second.
How long will it take to count from 1 to one thousand?
How long will it take to count from 1 to one million?
How long will it take to count from 1 to one billion?
Counting Bigger
If you start counting at a rate of one number per second:
It will take about 17 minutes to get to a thousand;
It will take you 12 days to count to a million;
It will take you 31 years to count to a billion.
Understanding Positional Systems
Smart by Shel Silverstein
My dad gave me one dollar bill'Cause I'm his smartest son,And I swapped it for two shiny quarters'Cause two is more than one!
And then I took the quartersAnd traded them to LouFor three dimes -- I guess he don't knowThat three is more than two!
Smart by Shel Silverstein
Just then, along came old blind BatesAnd just 'cause he can't seeHe gave me four nickels for my three dimes,And four is more than three!
And I took the nickels to Hiram CoombsDown at the seed-feed store,And the fool gave me five pennies for them,And five is more than four!
Smart by Shel Silverstein
And then I went and showed my dad,And he got red in the cheeksAnd closed his eyes and shook his head--Too proud of me to speak!
Big Ideas of the Base-Ten System
The position of the digits in numbers determines what they represent—their value.
Each place value to the left of another is ten times greater than the one to the right. (e.g., 10 x 10 = 100)
There are standard “trade rules”: Right to left: 10 for 1 Left to right: 1 for 10
Thinking Even Bigger
Representing Large Numbers
Thinking Even Bigger
In this activity you will try to extend what you know about representing place values for smaller numbers—ones, tens, hundreds—to larger numbers—up to millions, billions, and even trillions.
Representing Relatively Small NumbersTo start, think about the relationships among your
base-ten blocks:
How many units cubes are need to make a long?What are the dimensions of a long?
How many longs are needed to make one flat? How many unit cubes are in one flat?What are the dimensions of a flat?
How many flats are needed to make a super cube? How many unit cubes in one super cube?What are the dimensions of a super cube?
Thinking Even Bigger
Ms. Take thinks that having only four types of blocks in a set is too limiting for her gifted students. “I can’t imagine why no one has ever expanded the set to include larger blocks.”
Thinking Even Bigger
With a partner or group, design an extended set of base-ten blocks.
What would the next blocks in the set look like? What does each new block look like? Build or draw a
diagram. What does each represent? What should you name each one? What are the dimensions of each?
Extend the set as far as you can. What patterns do you notice?
Record your results on poster paper.
Poster Presentations
Place Value ChartWhat patterns do you notice?
Billions Millions Thousands Ones
10 11 10 10 10 9 10 8 10 7 10 6 10 5 10 4 10 3 10 2 10 1 10 0
extra-super-
duper flat
extra-super-duper long
extra-super-duper cube
super-duper flat
super-duper long
super-duper cube super-flat
super-long
super-cube flat long cube
Billions Millions Thousands Ones
Thinking Bigger
A millionpennies
Wall 5 ft x 4 ft x 1 ft thick with a 9 in cube stepstoolHeight stacked: 0.99 miles
Thinking Bigger
Ten millionpennies
6 ft x 6 ft x 6 ftHeight stacked: 9.88
miles
10-Billion Pennies
90 ft x 11 ft x 250 ftIf pennies stacked, height: 9,864 miles
Thinking Bigger
Hundred billion pennies
126.72 ft x 126.72 ft x 127.72 ftHeight Stacked 98,660 miles8,969 acres laid flat
