Properties of Operations Grade 3

Find a Place Find a Place Rules: No changing of numbers once placed. You must play the number on its turn. Do not take advise from your partner.

Small Group Discussion What strategies did you use in playing this game? After playing this first round, is there a new strategy you plan to try the next time we play? If you were allowed to make one change to your game card, what would it be and why?

Is It Closer? Instructions: Shuffle deck of cards and place them face down. Assign a team leader for each group. The team leader will turn over one card and announce the number to his or her group. The team will then decide which box the digit will be recorded (under the “Number Created” column of the game board). The team must record the digit before the next card is drawn or announced; once the digit is written, the placement of the number cannot be changed. The team leader will continue to turn over one card at a time until all eighteen boxes are filled. (Note: each team will have 2 unused digit cards in their deck). As a whole class compare each of the team’s numbers to the goal number. The team with the closest number wins a point. Each team will add up their points to find their score. The team with the most points wins the game.

Small Group Discussion What strategies did you use in playing this game? How did your group determine where to place a digit? Share a mistake your group made while playing this game and explain how you could prevent making the mistake next time.

The Fundamental Components of Place Value Base-ten units: Ones, tens, hundreds, thousands, and so on. Students have to get a feel for the sizes of these units (up to one hundred thousand in grade 3). Flexible bundling and unbundling: Base-ten units can be broken down and built back up in different ways. By definition, ten ones make a larger unit called “a ten.” By definition, ten tens make a larger unit called “a hundred.” And by definition, ten hundreds make a larger unit called “a thousand.” Therefore it is equally true that a hundred ones make a hundred; a thousand ones make a thousand; and a hundred tens also make a thousand. Bundling and unbundling are central ideas in developing computation algorithms. Longs, flats, and cubes are good for illustrating the ways to compose and decompose base-ten units. Positional notation: According to convention, the location of each digit in a multi- digit number corresponds to a particular base-ten unit. Also, the digit itself tells how many copies of that unit are in the number.† So, 908 is 9 hundreds and 8 ones. Understanding place value means understanding all three of these things in connection with one another. So for example, working with 10-longs, 100-flats, and 1000-cubes alone won’t by itself teach place value, because these manipulatives do not have any connection to positional notation. The place value system belongs to the art of writing. Working with place value requires writing numbers with an understanding of how they name quantities. Understanding place value is having number sense of the base ten units, understanding how these units are bundled and unbundled at will, and connecting this understanding to the positional notation system. All of this becomes a target of operational thinking as students develop computational algorithms.

Looking at the Landscape of Numbers It is important for students to work with patterns beyond 100. Look at place value patterns using the chart. 186 and 10 more; 105 and 10 less; 154 and 1 more; 200 and 1 less. Distribute 1- 1000 number chart. Have teachers attach the charts to make a continued 1-1000 chart. Using the 1000 chart: Use counters to name a number and represent 10 more or 10 less; use the same method to represent 100 more or 100 less. What patterns do you see? What place does it effect? Is that always true? Can you give me an example of when it may not follow that rule?

Small Group Discussion How will understanding place value increase student understanding with addition and subtraction of multi-digit numbers? What do you currently have in place to build place value understanding in your classroom?

Conceptual: Properties of Operations Addition Commutative Associative Distributive Identity Multiplication Zero Wow! What a mouth full of words, but the concepts are simple. Distribute Note Page for Properties of Operations. Using manipulative build each property to provide a concrete representation. We will use fabric tape measures and unit blocks. Use the note page to create a pictorial. Distribute MathFactsOh! (Properties) – allow time for teachers to read the instructions and to play a round as a table. Then give them a basic facts game of MathFactsOh!

Types of Understandings Procedural – Student can perform a computation or algorithm by following a series of prescribed steps. Conceptual – Student understands the basis of why a computation or algorithm works. They can apply it later without re-teaching. Student can identify, describe, and explain the big idea related to a topic or a class of problems. Problem Solving – Student can solve a problem when there is no specific solution pathway or algorithm. Disperse samples to allow small groups to discuss.

Word Problem BINGO Instructions: Write FREE in one of the spaces and the numbers: 21, 981, 361, 10, 172, 88, 82, 128, 215, 56, 228, 354, 140, 5, 91, 1546 randomly in the spaces that are left on your BINGO card. You will have a number left over. Mix up the BINGO Word Problem Cards and place the deck face down. Player one turns over a card and reads it to the group. The team works together to find the solution and puts an X on the answer on their BINGO card. The first team to get four Xs or three Xs and a free space in a row says, BINGO. A BINGO can be horizontal, vertical, or diagonal. As a class the team’s BINGO card will be check.

Writing Multi-Step Word Problems As a table, write a story problem that represents the model or pictorial you received. Your problem must include at least two different operations. Record your problem on chart paper (do not solve the problem).

Two Primary Themes of Grade 3 Addition and Subtraction 3.4A solve with fluency one-step and two-step problems involving addition and subtraction within 1,000 using strategies based on place value, properties of operations, and the relationship between addition and subtraction; 3.5A represent one- and two-step problems involving addition and subtraction of whole numbers to 1,000 using pictorial models, number lines, and equations In a sense, 3.5A is “applied math” while 3.4A is “pure math.” Both are important, and indeed both can work together. Word problems reinforce what addition and subtract mean, both in the abstract as well as in terms of what kinds of quantitative relationships these operations can model. Real-life situations also provide context and empirical support for the mathematical properties of addition (commutativity and associativity, which combine to make the so-called “any which way rule”) and for the mathematical relationship between addition and subtraction. But it isn’t enough to understand what addition and subtraction mean and how they can be applied to solve word problems. Grade 3 must also learn general and efficient methods for expressing the sum or difference of two multi-digit numbers as another multi-digit number.

The Myth of Keywords Keywords do not – Develop a sense of making or supporting meaning Build structures for more advanced learning Appear in many problems Students use keywords inappropriately Multi-step problems are impossible to solve with key words.

The Busiest Place in the House!