3rd Grade Operations and

Algebraic Thinking

Jean Donley, Alicia Klaich, Shanlee

Meyers, Rachel Miller, Lauren Robinson

Focus

We will be focusing on four critical areas:

Developing understanding of multiplication and division and strategies for multiplication and division within 100

Developing understanding of fractions, especially unit fractions (fractions with numerator 1)

Developing understanding of the structure of rectangular arrays and of area

Describing and analyzing two-dimensional shapes

Prior Knowledge

Addition and Subtraction within 100 using one and two step word problems (showing representation)

Fluently add and subtract within 20 using mental strategies

Use repeated addition to describe arrays as equal groups

Represent and solve problems involving multiplication and division

3.OA.A.1

Represent and solve problems involving multiplication and division: Interpret products of whole numbers, e.g., interpret 5 x 7 as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed at 5 x 7.

3.OA.A.2

Represent and solve problems involving multiplication and division: Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each. For example, describe a context in which a number of shares or a number of groups can be expressed as 56 ÷ 8.

3.OA.A.3

Represent and solve problems using multiplication and division: 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.

3.OA.A.4

Represent and solve problems involving multiplication and division: Determine the unknown whole number in a multiplication or divison equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 x ? = 48, 5 = _ ÷ 3, 6 x 6 = ?

Gifts From Grandma

a. Juanita spent $9 on each of her 6 grandchildren at the fair. How much money did she spend?

b. Nita bought some games for her grandchildren for $8 each. If she spent a total of $48, how many games did Nita buy?

c. Helen spent an equal amount of money on each of her 7 grandchildren at the fair. If she spent a total of $42, how much did each grandchild get?

Understand properties of multiplication and the relationship between multiplication and division.

3.OA.B.5Understand properties of multiplication and the relationship between multiplication and division: Apply properties of operations as strategies to multiply and divide. Examples: If 6 x 4 = 24 is known, then 4 x 6 =24 is also known (Commutative property of multiplication). 3 x 5 x 2 can be found by 3 x 5 = 15, then 15 x 2 = 30, or by 5 x 2 = 10, then 3 x 10 = 30 (Associative property of multiplication). Knowing that 8 x 5 = 40 and 8 x 2 = 16, one can find 8 x 7 as 8 x (5 + 2) = (8 x 5) + (8 X 2) = 40 + 16 = 56 (Distributive property). *Students do not use formal terms for these properties. Commutative

Distributive

3.OA.B.6

Understand properties of multiplication and the relationship between multiplication and division: Understand division as an unknown-factor problem. For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8.

“Build It and Break It”

The parade had 9 clowns. Each clown carried 6 balloons. How many balloons were the clowns carrying in the parade?

Build an array to show as many ways possible to “slice”, or decompose, the array into pieces using the distributive property.

Multiply and divide within 100.

3.OA.C.7

Multiply and divide within 100: Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 x 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one- digit numbers.

Our students enter school with the

misconception that the goal in math is to do it

fast and get it right.Do we promote that thinking in our teaching without realizing it?

Do we praise students who get the right answer quickly?

Do we become impatient with students who need a little more time to think?

Traditional approaches do not work.

What is fluency?

Rote memorization of basic facts is not fluency.

Principles and Standards for School Mathematics states, “Computational fluency refers to having efficient and accurate methods for computing. Students exhibit computational fluency when they demonstrate flexibility in the computational methods they choose, understand and can explain these methods, and produce accurate answers efficiently. The computational methods that a student uses should be based on mathematical ideas that the student understands well, including the structure of the base-ten number system, properties of multiplication and division, and number relationships” (p. 152). (NCTM 2000, p. 144).

Fluency

Phase I: Constructing meaning and counting strategiesPhase II: Reasoning strategiesPhase III: Working toward quick recall

How do we build fluency?

Introducing the concepts through problem situations and linking new concepts to prior knowledge.

Providing concrete experiences and semi- concrete representations prior to purely symbolic notations.

Teaching rules explicitly.

Providing mixed practice.

Fact Mastery

Timed Tests

Do we promote that thinking in our teaching without realizing it?

Do we praise students who get the right answer quickly?

Do we become impatient with students who need a little more time to think?

Multiplication Top-It

Solve problems involving the four operations, and identify and explain patterns in arithmetic.

3.OA.D.8Solve problems involving the four operations, and identify and explain patterns in arithmetic: Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding. *This standard is limited to problems posed with whole numbers and having whole-number answers; students should know how to perform operations in the conventional order when there are no parentheses to specify a particular order.

“Students often have difficulty with multistep problems”. (Van De Walle p. 167)

Your turn…Solve your problem in your notebook. Use the answer to the first problem to create a second problem.

http://www.youtube.com/watch?v=Na5-hXG1-Mw Using rounding in a two-step word problems

3.OA.D.9

Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations.

What happens when multiply any number by 2?

How can 4 times a number be decomposed into two equal addends?

http://www.youtube.com/watch?v=f-9cYVAThn0 finding patterns using the multiplication table

Students’ Misconceptions

https://mathreasoninginventory.com/Home/VideoLibrary

Students attempt different operations (+, -)

Students will try what they already know, and not think about solving for the groups

Students misinterpret number placement in division (15/3 = 5 vs. 3/15 = 5)

Students may have difficulty knowing where numbers go when creating visual representations

Understanding properties can be abstract and confusing. This is why you would show properties, instead of give them a title for the property.

After conceptual knowledge is built, students still have difficulty memorizing facts.

Students will attempt one single step instead of multiple steps to solve an equation.

Students’ will make computational errors during skip counting (i.e. patterning).

Assessments

Everyday Math has differentiated assessment options through the ADS Management System

Feedback

Considering the math practices, what went well in this lesson?

What would you change in this lesson while using the math practices as a guide?

Did we include cognitively demanding and worthwhile tasks?

Math Practices Used

1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively

3. Construct viable arguments and critique the reasoning of others

4. Model with mathematics.

5. Use appropriate tools strategically.

6. Attend to precision.

7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.

Resources

Everyday Math 3rd Grade

www.learnzillion.com

NCTM Illuminations

University of AZ Progression Documents

WCSD Flipchart

www.EngageNY.org

www.mathreasoninginventory.com

Van de Walle

Learning Trajectories in Mathematics

C&I Washoe County School District