Making Sense of Math: Early Number Concepts Amy Lewis Math Specialist IU1 Center for STEM Education

Goals 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. Participate collaboratively in solving problems in other base systems to strengthen reasoning skills. Connect new understandings of numbers to classroom practice.

Day 2: Examine the defining characteristics of our base-ten number system by exploring other number bases. Consider place value understanding in alternative algorithms for addition, subtraction, multiplication, and division. Explore activities for deepening place-value understandings.

Homework Examine the student responses. Which artifact demonstrates the strongest understanding of place value? Why? Which artifact demonstrates the weakest understanding of place value? Why? How would you address one of the misconceptions demonstrated by one of these responses?

Question #1 Think carefully about the following question. Write a complete answer. You may use drawings, words, and numbers to explain your answer. Be sure to show all of your work. Laura wanted to enter the number 8375 into her calculator. By mistake, she entered the number 8275. Without clearing the calculator, how could she correct her mistake? Without clearing the calculator, how could she correct her mistake another way?

Student #1 Partial

Student #2 Extended

Student #3 Satisfactory

Student #4 Minimal

Student #5 Satisfactory

Student #6 Extended

Question #1 1992 Mathematics Assessment 4th Grade Minimal – 9% ECR – Extended Constructed Response “Hard” Minimal – 9% Partial – 10% Satisfactory – 13% Extended – 7% Omitted – 61%

Question #2 In a game, Carla and Maria are making subtraction problems using tiles numbered 1 to 5. The player whose subtraction problem gives the largest answer wins the game. Look at where each girl placed two of her tiles.  Who will win the game?  Explain how you know this person will win.

Student #1 Satisfactory

Student #2 Extended

Student #3 Partial

Student #4 Minimal

Student #5 Extended

Student #6 Partial

Question #1 1996 Mathematics Assessment 8th Grade ECR – Extended Constructed Response “Hard” Incorrect – 29% Minimal – 32% Partial – 19% Satisfactory – 14% Omitted – 5% Off-Task – 1%

Literature Connections The King’s Commissioners (Friedman, 1994) A book about place value concepts, including grouping and different counting methods, large numbers and early computation. How Much is a Million? (1985), If You Made a Million (1989), On Beyond a Million (1999), Millions to Measure (2001), (Schwartz) Exploring and visualizing large numbers and counting by powers of ten.

Literature Connections Anno’s Counting Book (1977), Anno’s Mysterious Multiplying Jar (1983),(Anno) Beautifully illustrated books exploring counting and factorials. Can You Count to a Googol? (Wells, 2000) Visually builds the number system by powers of ten to a googol.

Orpda Number System What is the Value of @*#?

Number of Objects and Representative Symbol in the Orpda Number System No objects ~ 1 object * 2 objects @ 3 objects # 4 objects ^

Orpda Number System With your group, represent a group having 5 objects. Be ready to explain why that representation is correct.

Orpda Number System In your teams, complete the @~~ Chart. How does the @~~ Chart compare to a Hundreds Chart? What did you learn about the Base-10 System when completing the @~~ Chart?

Orpda Number System Using the multi-link cubes (or any other manipulatives that you are comfortable with), represent sets of objects up to 30 within the Orpda number system. Be sure to connect the model representation to the symbolic representation for each number.

Orpda Number System Extensions: Understanding 0 as a place holder Orpda games Make a flat! How many ways? How could we create our own number system?

Orpda Number System What is the value of this task? How could you think about other bases with your students?

Alternative Algorithms

Alternative Addition Strategies Without using pencil and paper, complete the following addition problem: 37 + 56 What strategy did you use to find this sum? Can you generalize this strategy to apply to ANY sum? How did you use place-value understandings when applying this strategy?

Alternative Subtraction Strategies Without using pencil and paper, complete the following subtraction problem: 82 - 45 What strategy did you use to find this difference? Can you generalize this strategy to apply to ANY sum? How did you use place-value understandings when applying this strategy?

Alternative Multiplication Strategies Without using pencil and paper, complete the following multiplication problem: 16 x 23 What strategy did you use to find this product? Can you generalize this strategy to apply to ANY sum? How did you use place-value understandings when applying this strategy?

Alternative Division Strategies Without using pencil and paper, complete the following addition problem: 154 ÷ 13 What strategy did you use to find this sum? Can you generalize this strategy to apply to ANY sum? How did you use place-value understandings when applying this strategy?

Using Alternative Strategies Why would we encourage students to use alternative strategies? How could we encourage students to use alternative strategies? What student benefits might you anticipate from using alternative strategies?

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! Use a clicker for the first couple of rounds. When doing this with students, teachers need to remind students that they need to use the smallest number of blocks in each column on the place-value mat as possible. After the participants are used to the process, switch to either a spinner (1-9) or a pair of dice.

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? 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? Given any number, what do you have to do to make one of its neighbors?

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.

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 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? 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. 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. 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. 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?

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 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 is the value of this task? When should it be used?

Plain and Fancy Counting Counting by Ones Forwards Backwards Starting from any number Skip Counting Forwards by 10’s, 2’s, etc. Backwards by 10’s Starting from any number Have participants practice counting and skip counting using the number line. The facilitator should randomly pick a starting number, an a number to count by. Make sure that examples are given counting in both directions on the number line. When participants count backwards for the first time, they tend to stop when they reach zero. Wait and encourage the teachers to continue the counting below zero. Be sure the connect these counting activities to algebraic concepts – skip counting by 3 starting at 7 connects to understanding the expression 3n+7. What makes these counting activities worthwhile?

Looking Back What mathematics have we explored today? How do the activities that we completed today match with what we did last week? How have these activities shaped your understanding of place value? How would you describe the cognitive demand of the tasks we explored today?

Post-Test Please take a few minutes to complete the post-test. This this will serve as a measure of growth from the start of the course until the end.

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. Due Date: January 7, 2011

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 number sense strategies into your mathematics instruction. Describe an action plan for implementation.

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. 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.

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 order to receive permission to pursue your own line of study.

Where to send your Final Project alewis@washjeff.edu Amy Lewis Science Matters 60 S. Lincoln St. Washington, PA 15367

Post-Test