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?

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     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.     

Alternative Addition Strategies  Use an empty number line to model the following addition problems.     

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, 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 billion per day since September 28,  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, , , ,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 thousand A million A 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 quarters And traded them to Lou For three dimes -- I guess he don't know That three is more than two!

Smart by Shel Silverstein  Just then, along came old blind Bates And just 'cause he can't see He gave me four nickels for my three dimes, And four is more than three!  And I took the nickels to Hiram Coombs Down 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 cheeks And 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 Numbers To 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 Chart What patterns do you notice?

Thinking Bigger  A million pennies Wall 5 ft x 4 ft x 1 ft thick with a 9 in cube stepstool Height stacked: 0.99 miles

Thinking Bigger  Ten million pennies 6 ft x 6 ft x 6 ft Height stacked: 9.88 miles

10-Billion Pennies 90 ft x 11 ft x 250 ft If pennies stacked, height: 9,864 miles

Thinking Bigger  Hundred billion pennies ft x ft x ft Height Stacked 98,660 miles 8,969 acres laid flat

Questions? Amy Lewis (724)