This document provides an overview of a middle school mathematics institute that will take place on Saturday. It discusses the basic lesson format, which includes an anchor task, guided practice, and independent practice. It also references Bruner's idea of using concrete experiences and pictorial representations to help students understand abstract ideas. The document then provides several case studies as examples of lessons that could be used to develop, apply, and practice various mathematical concepts involving fractions, algebra, geometry, and more.

1. BLAKEINSTITUTE2015 | Grade 6 Institute on SATURDAY
banhar.yeap@pathlight.org.sg | www.banhar.blogspot.com
Middle-School Mathematics
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
__________________________________________________________________

2. Lesson for Development of Basic Concept
Case Study A | Fraction Division
 2
4
3


2
1
4
3


4
3
2
1


3
1
4
3


3. Lesson for Development of Basic Concept
Case Study B | Algebraic Expression
Learning Outcome To write algebraic expressions for given situations
Anchor
Task


 using consecutive 1-digit
numbers so that vertical sum equals
horizontal sum
number cut-outs


 template
Assessment
Runway Indicator
 if they are able to generalize
Target Indicator
 if they are able to write
algebraic expressions to
generalize
Observation
Support …
…
…
…
…
…
…
…
…
…
Challenge
Lesson for Development of Basic Concept
Case Study C | Geometry

4. Lesson for Application and Extension of Concepts
Case Study D | Geometry
Is it possible to find the sum of the angles at the five corners of the star?
Lesson for Application and Extension of Concepts
Case Study E | Four Operations of Whole Numbers
105 red and white beads are used to make a necklace. There must be at least 3
white beads between any 2 red beads. Find the greatest number of red beads in
this necklace.

5. Lesson for Practice
Case Study F |
How many dots are there in this diagram?
Write an equation to show how the number is obtained.
Lesson for Practice
Case Study G |
Use the four given numbers as well as any basic operations and at most one pair of
( ) to make an expression that has a value of n, where 1 < n < 10.

6. Bar Model
Newman’s Procedures
Helps teachers to identify difficulties in a mathematics word problem.
 Read - Can the student read the problem?
 Comprehend - Can the student answer basic questions? “How many boys are
there? How man coins does each girl have?”
 Knowledge of Strategies - The student knows possible strategies.
 Transform - Can the student translate the story into an appropriate
mathematical form e.g. a diagram or an equation?
 Process/Compute – Can the student perform the necessary computation?
 Encoding/Interpreting - The student can interpret the computation results to
solve the problem

7. Problem 1
Find the value of 12
5
4
 .
Give your answer as a fraction in the simplest form.
PSLE Singapore 2013
Problem 2
The total cost of a pen and a book is $32. The cost of the pen is
5
3
the cost of the
book. What is the cost of the book?
PSLE Singapore 2013
Problem 3
Kai Li spent
3
1
of her money on 5 magnets and 11 postcards.
The cost of each magnet is 3 times the cost of each postcard.
She bought some more magnets with
4
3
of her remaining money.
How many magnets did Kai Li buy altogether?
PSLE Singapore 2013
Problem 4
Mr Lim had a total of 540 long and short rulers. After selling an equal number of
both types, he had
3
1
of the long rulers left and
6
1
of the short ones left.
What was the total number of rulers left?
PSLE Singapore 2014

8. Problem 5
Aini and Usha each had a piece of dough of the same mass for making buns.
The same mass of dough was used for each bun. Aini made 40 buns and had 50 g
of dough left. Usha made 10 buns and had 1.1 kg of dough left.
With the remaining dough from both girls, how many more such buns can be made
at most?
Problem 6
Yulin wants to make 14 small identical stars and 20 large identical stars using wire.
She has made 12 small stars and 7 large ones using 960 cm of wire. The length of
the wire she used for 3 large stars is the same as that for 4 small stars.
What is the length of the wire she needs to make the remaining stars?

9. 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