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Welcome Introduce yourself to your tablemates What is it you would like to learn today about place value? Please write down 2-3 goals. Round robin with your teammates and then come up with 2 questions from your table to share with the whole group. Thank you!

Teaching to the big ideas … Some sources for helping teachers think about the big ideas in mathematics: Randall Charles, Big Ideas & Understandings as the Foundation for Elementary and Middle School Mathematics Randall Charles, Big Ideas & Understandings as the Foundation for Elementary and Middle School Mathematics Marian Small, Big Ideas from Dr. Small Marian Small, Big Ideas from Dr. Small Fosnot & Dolk, Young Mathematicians at Work Fosnot & Dolk, Young Mathematicians at Work

Place Value Resources NYC DOE Early Childhood Assessment in Mathematics (ECAM) Organizing and Collecting: The Number System, Nina Liu, Maarten Dolk, Catherine Twomey Fosnot, Contexts for Learning, Heinemann (firsthand.heinemann.com) Coming To Know Number, Grayson Wheatley

What is place value? Write down everything you know about place value. Have a table talk and construct a chart that lists the main ideas related to understanding place value

Randall Charles-Base Ten System The base ten numeration system is a scheme for recording numbers using digits 0-9, groups of ten, and place value Numbers can be represented using objects, words, and symbols For any number, the place of a digit tells how many ones, tens, hundreds, and so forth are represented by that digit.

Place Determines Value You can put numbers on a place value chart. You can determine the value of the digit by reading its place. How many tens are in 345? HundredsTensOnes 345

Randall Charles-Base Ten System Each place value to the left of another is ten times greater than the one to the right. Sets of ten, one hundred and so forth must be perceived as single entities when interpreting numbers using place value. Decimal place value is an extension of whole number place value The base-ten numeration system extends infinitely to very large and very small numbers

Which of these ideas are often underdeveloped? Sets of ten, one hundred and so forth must be perceived as single entities when interpreting numbers using place value. Each place value to the left of another is ten times greater than the one to the right.

Unitizing is a Big Idea Flexibility with units or unitizing is one of the big ideas in mathematics. It is one of the ideas that many children do not construct, which is why so many students have difficulty with fractions, decimals and percents and proportional reasoning.

How many tens are in 345? Additively 4 Multiplicatively 34 Precisely 34.5

What is multiplicative growth? How big is 1,000 square centimeters? Can you picture a rectangle that has the area of 1,000 square centimeters? You task is to work in groups of 4 and create as many different rectangles using centimeter square graph paper that have an area of 1,000 square units. How will you know if you have found and created them all? Compare the relative lengths of each rectangle and make comparisons (e.g. this one is 10 times as long as that one)

How big is…. Can you picture a 10,000 sq. cm. rectangle? 100,000 sq. cm. rectangle? Have we created enough rectangles as a group to make a 100,000 sq. cm. rectangle? Is there space in this room to lay out 100,000 sq. rectangles?

How big is 1,000,000? Is the floor space in this room large enough for us to lay down a 1,000,000 square centimeter rectangle?

Adding A Place or Multiplying by 10? ,000 1,000 10,000 10, , ,000 In an additive system, you are “putting” another place. 100 has “one more 0 than 10.” In a multiplicative system you are are multiplying 10 by 10. “100 is ten times as great as 10.”

Units, Measurement and Place Value Why is it important for children to be able to visualize number in terms of units and measurement? How are the big ideas in place value and measurement related to and supportive of proportional reasoning?

Measurement, Ratio and Multiplicative Reasoning Measurement is ratio! It is a multiplicative comparison. There is a big difference between thinking about linear measurement as the number of things [how many centimeters] versus thinking about measurement as a ratio comparison. 10 little lengths, called “centimeters The total length of this unit ( ) is 10 times as large as the length of this ( ).

Visualizing Relationships This unit is 1. What would.1 of this unit be? Can you visualize how long.01 of this unit would be? Would a.001 of a unit be visible to the naked eye?

Exponents Represent Multiplicative Growth In our place value system, the magnitude of the place is directly related to multiplication and division. 10 is 10 times great than is 10 times greater than ,000 is 10 times greater than ,000 is 10 times greater than 1,

Big Ideas in Place Value In a multiplicative system you are are multiplying 10 by 10 to get 100. You are not “putting a 0,” “adding a 0,” or adding 10 ten times.

Big Ideas in Place Value In our place value system, the magnitude of the place is directly related to multiplication and division. 1 is 1/10 the size of is 1/10 the size of is 1/10 the size of 1,000. 1,000 is 1/10 the size of 10,000.

Big Ideas in Place Value In our place value system, the magnitude of the place is directly related to multiplication and division and these relationships are inverse..1 is 1/10 the size of 1, but 10 times the size of.01.1 is 1/10 the size of 1, but 10 times the size of is 1/10 the size of.1, but 10 times the size of is 1/10 the size of.1, but 10 times the size of is 1/10 the size of.01, but 10 times the size of is 1/10 the size of.01, but 10 times the size of.0001

Big Ideas in Place Value But all of this is affected by the unit. So if 100 is the unit, what is the size of 100?10?1?.1?

Unitizing is one of the central big ideas in place value. Sets of ten, one hundred and so forth must be perceived as single entities when interpreting numbers using place value. Randall Charles

Big Ideas in Place Value: Unitizing How does not having this mental structure affect students’ mathematical development?

The Matchbox Factory The Matchbox Factory makes toy cars, which are packaged in boxes of ,358 cars were made today. How many boxes of 100 will be filled? Please solve this problem. Consider how students might solve this problem.

Is Computation Necessary? The Matchbox factory makes cars, which are packaged in boxes of ,358 cars were made today. How many boxes of 100 will be filled? The Matchbox factory makes cars, which are packaged in boxes of ,358 cars were made today. How many boxes of 100 will be filled? How many of our students would perform an operation to solve this problem? How many of our students would perform an operation to solve this problem?

The Matchbox factory makes cars, which can also be packaged in boxes of 10. 6,358 cars were made today. How many boxes of 10 will be filled? The Matchbox factory makes cars, which can also be packaged in boxes of 10. 6,358 cars were made today. How many boxes of 10 will be filled? How many of our students solve this “new” problem as a totally different problem? How many of our students solve this “new” problem as a totally different problem? Is this a totally different problem?

Student Work Let’s analyze student work through the lens of big ideas Let’s analyze student work through the lens of the CCSS math practices—specifically: Reasoning Sense-making Precision in Language Models (e.g. representing mathematical ideas) Look for and make use of structure

Eight CCSS Math Practices Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.

Assessing for Understanding How do children develop an understanding of place value?

Complements of 10 and 100 How does knowing the combinations of 10 assist you in knowing combinations of 100? Play the game, Close to 100 until you can articulate a strategy for winning. What does this have to do with place value? With computational fluency?

Complements of 100 How does knowing = 100 help you solve ? Solve each of the problems on both sides of the handout. What might students begin to think about as they work through this worksheet? How is this worksheet different from a typical “practice” worksheet?

Mathematical Models— The Number Line How might a number line assist students in making meaning of the magnitude of number? How might a number line assist students with estimation and “rounding?” How might a number help students come to understand the relationship between numbers? How might a number line be a tool for helping students represent their thinking? How might teachers and students use a number line to think with?