algebraic thinking
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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. - PowerPoint PPT PresentationTRANSCRIPT
Algebraic Thinking
CCSSM in the Third Grade
Oliver F. JenkinsMathEd 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
Number and Operations:
Fractions
Operations and Algebraic Thinking
The Number System
Expressions and
Equations
Algebra
K – 5
3 – 5
6 – 8 9 – 12
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 TargetsKnowledge 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
EqualSet
EqualSet
EqualSet
1
2
3
n
. .
.
Product
(Whole)
Number of sets
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
EqualSet
EqualSet
EqualSet
1
2
3
n
. .
.
Product
(Whole)
Number of sets
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
EqualSet
EqualSet
EqualSet
1
2
3
n
. .
.
Product
(Whole)
Number of sets
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
The Pet Shop
Two Interpretations of Division
Gifts from Grandma
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
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:
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”
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
Shirt
s
Problem-Solving Lesson Format
Pose a problemStudents’ problem solvingWhole-class discussionSumming upExercises 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!