Page 1 Building Understanding of the Number System Through Hands-On Experiences Marcia Torgrude K-12 Math Specialist

Page 2 Develop understanding and ideas to promote deeper understanding of the number system within the Common Core Develop hands-on strategies to build understanding of place value. Develop hands-on strategies to help promote understanding of fractions. Use tools to help students work fluently with rational numbers. Experience online tools for the number system Outcomes for Today

Page 3 I Have….. Who Has Let’s play! What does this have to do with learning? Where does it fit the common core standards? What about the Standards of Mathematical Practice? Search I Have…Who Has online.

Page 4 Standards of Mathematical Practice

Page 5 What does it mean to “do mathematics?” What does it mean to “do mathematics?” The Standards of Mathematical Practice are descriptions of the fundamental skills needed to “do” mathematics.

Page 6 What does it mean to “do mathematics?” What does it mean to “do mathematics?” Standards of Practice describe what it means for students to demonstrate proficiency in mathematics. They are our new “basic skills.” Content Standards are the “what” of mathematics

Page 7 We must get past the idea of mathematics as a collection of algorithms, steps, or procedures. Just getting answers, although important, is not “doing mathematics.” “Doing mathematics”

Page 8 Working with Whole Numbers Adding Subtracting Multiplying Dividing With Base 10 Blocks

Page 9 “Doing mathematics” Using Modeling to Make Sense of Mathematical Procedures Modeling addition with Base 10 blocks Place It!

Page 10 “Doing mathematics” Using Modeling to Make Sense of Mathematical Procedures Modeling subtraction with Base 10 blocks ONLY BUILD the beginning number 302 −

Page 11 Multiplication and Division Identify strategies that individuals can use to solve multi-digit multiplication and division problems in sense-making ways Connect concepts to “standard algorithms” Discuss teaching strategies that enhance a child’s understanding

Page 12 Practice Concrete Multiplication What does multiplication look like using base ten blocks?

Page 13 Let’s Try this without the blocks 143x23

Page 14 Making Connections through diagram – 23 x

Page 15 Making Connections – Getting to the algorithm X x x

Page 16 Understanding the abstract Do you think that using base-10 blocks helps to give meaning to the multiplication algorithm? How? One common concern when using models is that students will not make connections between the concrete models, their representations, and the mathematical concept. Did we make the connections? How?

Page 17 Practice Concrete Division What does division look like using base ten blocks?

Page 18 American Idol is back! If they travel to 11 different cities and can only take a total of 132 people to Hollywood, how many people can be selected from each city? How can we use the base ten block and the array model to help us with division?

Page 19 Understanding the abstract Do you think that using base-10 blocks helps to give meaning to the division algorithm? How?

Page 20 Another Strategy for Division Use of friendly or “benchmark” numbers Partial quotient division: Multiplication for division – use what we know

Page 21 Partial Quotient Division Our family took a trip and my dad told me we drove a total of 2,112 miles in 6 days. How many miles per day did we travel on the average? What friendly numbers did you use?

Page 22 Virtual Whole Number Tools –Activities –Calculation Nation Tens Frame - Grouping and Grazing - Adding with base 10 Blocks - html html Subtracting with Base 10 Blocks - html html Primary Krypto - Product Game - Times Table - illuminations.nctm.org/ActivityDetail.aspx?ID=155http:// illuminations.nctm.org/ActivityDetail.aspx?ID=155

Page 23 What were the goals of the activities? What common core standards have we been working on? What Standards of Mathematical Practice were present during the activities? Small Group Discussion

Page 24 Working with Fractions Equivalence Addition Subtraction Multiplication Division

Page 25 Fraction Equivalence, Adding, and Subtracting Using Pattern Blocks

Page 26 Fraction Equivalence, Adding, and Subtracting Using Pattern Blocks

Page 27 Fraction Using Cuisenaire Rods /session8/part_b/modeling.htmlhttp:// /session8/part_b/modeling.html 53http:// 53 If we are trying to work with fourths and thirds what will our new whole need to be? Using Cuisenaire rods model: 1/3 + 1/4 1/3 - 1/4 1/3 x 1/4 1/3 ÷ 1/4

Page 28 Modeling equivalence, adding, subtracting, multiplying, and dividing Using Cuisenaire Rods ttt12/rttt12_int_cuisenaire/index.htmlhttp:// ttt12/rttt12_int_cuisenaire/index.html

Page 29 Array Model for Fractions ndow.html?pid=1853http:// ndow.html?pid=1853 Modeling equivalence, adding, subtracting, multiplying, and dividing

Page 30 Why do we invert and multiply to divide? How does this work? ¾ ÷ 5/6= B How many 5/6 are in ¾?  x 5/6 = ¾ B x 5/6 = ¾ I need to multiply 5/6 by its reciprocal to solve for B or box. B x 5/6 x 6/5 = ¾ x 6/5 B = ¾ x 6/5

Page 31 Fractions, Decimals, and Percents Make sense through grids

Page 32 Virtual Fractions Equivalent Fractions - Fraction Models - Fraction Game - Fraction Pieces - category_g_3_t_1.html category_g_3_t_1.html Fraction Adding html 1.html Fraction Comparing html 1.html Fraction Equivalence html 1.html Fraction Rectangle Multiplication html 1.html

Page 33 What were the goals of the activities? What common core standards have we been working on? What Standards of Mathematical Practice were present during the activities? Small Group Discussion

Page 34 Rational Numbers Integers –Charge Model –Linear Model

Page 35 · Charge Model Use your positive/negative counters to represent the following numbers using at least the number of tiles listed. You can challenge yourself by using more than the minimum number of tiles. Be prepared to share and prove your solution. Ways to build understanding of Integers

Page 36 Model the following problems with your counters and sketch your work using a plus sign for positive and a negative sign for negative counters: (-5) (-5) What do you notice? Make some generalizations about the rules for adding integers. Now consider: -3 - (-5) What generalization can you make? Ways to build understanding of Integers

Page 37 · Linear Model Matt earns merits and demerits at his school. One day he earned 3 merits for his math game, 2 demerits for being late to class, 1 merit for being courteous, 5 demerits for arguing with his teacher, and 2 merits for helping another student. If he began the day with 4 merits, how many did he have at the end of the day? Ways to build understanding of Integers

Page 38 Charge and Linear Model Solve this problem using both methods: He ather started the month with $12. She spent $5 on a game, but realized that she forgot to pay her annual club dues so she wrote a check for $15 because her dad said he would loan her enough money to cover the check. How much does Heather have to borrow from her dad? Ways to build understanding of Integers

Page 39 How is this different from the way students built their understanding of positive/negative integers in the past? What common core standard have we been working on? What Standards of Mathematical Practice were present during the activity? Ways to build understanding of Integers

Page 40 Concrete Algebra Explore Build Add Subtract Multiply Divide

Page 41 Connecting Number System to Algebra

Page 42 Making Connections through diagram – 23 x

Page 43 Virtual Algebra Illuminations Algebra Tiles NLVM algebra Tiles - en=activities&from=category_g_3_t_2.html en=activities&from=category_g_3_t_2.html NLVM Scales -Positives en=instructions&from=category_g_3_t_2.html en=instructions&from=category_g_3_t_2.html NLVM Scales – Negatives en=instructions&from=category_g_3_t_2.htmlhttp://nlvm.usu.edu/en/nav/frames_asid_201_g_3_t_2.html?op en=instructions&from=category_g_3_t_2.html Pan Balance - Numbers Pan Balance - Expressions

Page 44 Thank You - Exit Card I am reaffirmed because I already… The big idea I will work on is… I still need help with….