Algebraic Thinking

Algebraic Thinking CCSSM in the Third Grade Oliver F. Jenkins MathEd Constructs, LLC www.mathedconstructs.com

Grade 3 CCSSM Domains • Operations and Algebraic Thinking • Represent and solve problems involving multiplication and division. • Understand properties of multiplication and the relationship between multiplication and division. • Multiply and divide within 100. [Fluency standard] • Solve problems involving the four operations, and identify and explain patterns in arithmetic. • Number and Operations in Base Ten • Use place value understanding and properties of operations to perform multi-digit arithmetic. • Number and Operations – Fractions • Develop understanding of fractions as numbers. • Measurement and Data • Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects. • Represent and interpret data. • Geometric measurement: understand concepts of area and relate area to multiplication and to addition. • Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures. • Geometry • Reason with shapes and their attributes.

Algebraic Thinking Stream Number and Operations in Base Ten The Number System Algebra Number and Operations: Fractions Operations and Algebraic Thinking Expressions and Equations 9 – 12 K – 5 6 – 8 3 – 5

Domain: Operations and Algebraic Thinking • Cluster: • Represent and solve problems involving multiplication and division • Content Standard 3.OA.3: • Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

What must students know and and be able to do in order to master this standard? • Content Standard 3.OA.3: • Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

Unwrapping Content Standards Instructional Targets Knowledge and understanding (Conceptual understandings) Reasoning (Mathematical practices) Performance skills (Procedural skill and fluency) Products (Applications)

What is the significance of . . . . . . situations involving equal groups, arrays, and measurement quantities . . . . . . in content standard 3.OA.3?

Problem Structures for Multiplication and Division Equal Groups, Arrays, and Measurement Quantities

Equal-Group Problems • Whole Unknown • Mark has 4 bags of apples. There are 6 apples in each bag. How many apples does Mark have altogether? (repeated addition) • If apples cost 7 cents each, how much did Jill have to pay for 5 apples? (rate) • Peter walked for 3 hours at 4 miles per hour. How far did he walk? (rate) • Lucy needs 5 feet of material to make a scarf. She plans to make 8 scarfs. How many feet of material does she need? (measurement quantities) EqualSet 1 EqualSet Product (Whole) 2 EqualSet 3 . . . EqualSet Number of sets n

Equal-Group Problems • Size of Groups Unknown (Partition Division) • Mark has 24 apples. He wants to share them equally among his 4 friends. How many apples will each friend receive? (fair sharing) • Jill paid 35 cents for 5 apples. What was the cost of 1 apple? (rate) • Peter walked 12 miles in 3 hours. How many miles per hour (how fast) did he walk? (rate) • Lucy has 40 feet of material for making scarfs. She plans to make 8 scarfs. How many feet of material will she use for each scarf? (measurement quantities) EqualSet 1 EqualSet Product (Whole) 2 EqualSet 3 . . . EqualSet Number of sets n

Equal-Group Problems • Number of Groups Unknown (Measurement Division) • Mark has 24 apples. He put then into bags containing 6 apples each. How many bags did Mark use? (repeated subtraction) • Jill bought apples at 7 cents apiece. The total cost of her apples was 35 cents. How many apples did Jill buy? (rate) • Peter walked 12 miles at a rate of 4 miles per hour. How many hours did it take Peter to walk the 12 miles? (rate) • Lucy has 40 feet of material to make scarfs. Five feet of material is needed to make a scarf? How many scarfs can she make? (measurement quantities) EqualSet 1 EqualSet Product (Whole) 2 EqualSet 3 . . . EqualSet Number of sets n

Array Problems • Product Unknown • Paul placed some apples into 5 rows. If there are 7 apples in each row, how many apples are there altogether? (equal groups language) • Sally arranged hats in the display window into 4 rows and 6 columns. How many hats are there? (row and column language) rows columns

Array Problems • Factor Unknown • If Paul placed 35 apples into 5 equal rows, how many apples are in each row? (equal groups language) • Sally arranged 24 hats in the display window into an array with 4 rows. How many columns of hats are there? (row and column language) • Sally arranged 24 hats in the display window into an array with 6 columns? How many rows of hats are there? (row and column language) rows columns

Progressions of Note Measurement examples are more difficult than examples about discrete objects, so these should follow problems about discrete objects. Problems where regions are partitioned by unit squares are foundational to developing understandings of area. Students in grade 3 begin the step to formal algebraic language by using a letter for the unknown quantity in expressions or equations for one- and two-step problems. Engaging students in array problems facilitates the generalization to the commutative property of multiplication.

Problem Solving Tasks Desirable Features, Types, and Examples with Reference to Applicable Problem Situations

Desirable Features ofProblem-Solving Tasks Genuine problems that reflect the goals of school mathematics Motivating situations that consider students’ interests and experiences, local contexts, puzzles, and applications Interesting tasks that have multiple solution strategies, multiple representations, and multiple solutions Rich opportunities for mathematical communication Appropriate content considering students’ ability levels and prior knowledge Reasonable difficulty levels that challenge yet not discourage

Problem Types Contextual Problems. Context problems are connected as closely as possible to children’s lives, rather than to “school mathematics.” They are designed to anticipate and to develop children’s mathematical modeling of the real world. Model Problems. The model is a thinking tool to help children both understand what is happening in the problem and a means of keeping track of the numbers and solving the problem.

Sample Problems Analyzing Word Problems Involving Multiplication Gifts from Grandma Two Interpretations of Division The Pet Shop

Implications for Instruction

Bruner’s Stages of Representation • Enactive:Concrete stage. Learning begins with an action – touching, feeling, and manipulating. • Iconic:Pictorial stage. Students are drawing on paper what they already know how to do with the concrete manipulatives. • Symbolic:Abstract stage. The words and symbols representing information do not have any inherent connection to the information.

Lessons Built on Context or Story Problems • Build multiplicative lessons around only two or three problems • Students should not just solve the problems but also use words, pictures, and numbers to explain: • How they went about solving the problem • Why they think they are correct • Explanations provided by students should communicate what they did well enough to allow someone else to understand their thinking • Children should be allowed to use whatever physical materials they feel are needed • Don’t be afraid to use “large” numbers (e.g., 14 × 8)

Introducing Symbolism Guidelines Issues Before learning multiplication symbolism, students will likely write repeated addition equations to represent what they did Do not use the phrase “six goes into twenty-four” • Use students’ repeated addition representations as an opportunity to introduce the multiplication sign • Explain what the two factors mean • Make sure students understand the equivalence of:

Lessons Built onModel-Based Problems • Arrays are extremely important and widely used models for multiplication and division • Other useful models are equal sets and the number line Pants Shirts

Problem-Solving Lesson Format Pose a problem Students’ problem solving Whole-class discussion Summing up Exercises or extensions (optional)

Your Turn • Build a problem-solving lesson based on “Analyzing Word Problems Involving Multiplication,” “Gifts from Grandma,” “Two Interpretations of Division,” or “The Pet Shop” • Identify lesson objectives aligned with standard 3.OA.3 • Describe how the class and lesson materials will be organized • Write three questions that you will ask students during each of the first four lesson phases Solving problems is fun!

Algebraic Thinking