1) The document discusses standards and instructional focus areas for Grade 4 mathematics, including multi-digit multiplication and division, fractions, and geometric shapes. 2) It provides examples of case studies to solve, including arranging fractions in order and word problems involving fractions of amounts. 3) Theories of learning and representation discussed include Bruner's CPA approach and Skemp's classifications of understanding in mathematics.

1. BLAKEINSTITUTE2015 | Grade 4 Institute on THURSDAY
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Standards and Differentiation
In Grade 4, instructional time should focus on three critical areas: (1) developing
understanding and fluency with multi-digit multiplication, and developing
understanding of dividing to find quotients involving multi-digit dividends; (2)
developing an understanding of fraction equivalence, addition and subtraction of
fractions with like denominators, and multiplication of fractions by whole numbers; (3)
understanding that geometric figures can be analyzed and classified based on their
properties, such as having parallel sides, perpendicular sides, particular angle
measures, and symmetry.
Source | Common Core State Standards
Case Study A |
Grace arranged
2
1
,
3
2
,
4
1
in order, from the greatest to the smallest.
Is she correct?
Method 1 Method 2

2. Basic Lesson Format
 Anchor Task – Explore, Structure, Journal, Reflect
 Guided Practice
 Independent Practice
Bruner’s idea on representation (CPA Approach) classifies representations as
enactive, iconic or symbolic. Students use concrete experiences and pictorial
representations to understand abstract ideas.
Skemp classified understandings in mathematics into three types – instrumental,
relational and conventional understanding.
Journal
Journal   June 2015
Title
Problem
__________________________________________________________________

3. Visualization – Equivalent Fractions, Mixed Numbers, Improper
Fractions
Visualization is the ability to see with your mind what your eyes cannot
see.
Case Study B |
Case Study C |

4. Case Study D |
Ben cut each of 3 same-sized cakes into 6 equal pieces.
Cheryl took 7 pieces. David took 9 pieces. Ben took the rest
Does this make sense?
Case Study E |
3
1
3

5. Case Study F|
3
2
12
Building Upon Prior Understanding
Case Study G |
Learning Outcome To multiply multi-digit numbers with a 1-digit number
Anchor
Task
x= where all the five digits
are different.
Given 15 x 4 = 60, find 215 x 4 and
such.
digit tiles
Assessment
Runway Indicator
 if they are able to multiply
small numbers
Target Indicator
 if they are able to use
associative property to relate
15 x 4 = 60 and 215 x 4 and
multiply two-digit number by
one-digit numbers
Observation
Support …
…
…
…
…
…
…
…
…
…
Challenge

6. Bar Model

7. Gary spent
5
1
of his savings on a gift and
8
3
of the remainder on a book.
As a result, he had $60 left. How much was the gift?
Gary spent
5
1
of his savings on a gift and
4
3
of the remainder on a book.
As a result, he had $60 left. How much was the gift?
Gary spent
5
1
of his savings on a gift and
7
3
of the remainder on a book.
As a result, he had $60 left. How much was the gift?

8. Key Theories
1. Piaget
2. Bruner
3. Dienes
4. Vygotsky
5. Skemp
Challenging Advanced Learners
1. Solve Using Another Method (Polya)
2. Write a Note to a Friend (21st Century Competencies)
3. Make Up Another Problem (Silver)
Traits of Advanced Learners
1. Real-World Model
2. Visual Model
3. Oral Explanation
4. Written Explanation
5. Initiative