Post on 22-Sep-2020
MY204: Using manipulatives to enhance
understanding in KS3 maths
Monday 17th October 2016
09:30 – 15:00
Michael Andersonm.anderson@stem.org.uk
@STEMLearning_MA
• develop methods that use manipulatives to enhance
mathematical understanding
• explore the wealth of resources which can be used in
conjunction with manipulatives that are freely
available in the STEM Learning online resource
collection
• see how resource packages, which 'add value' and
pedagogical support, can be linked to schemes of
work
Aims of the Day
Format of the DayTime Session Title
09:30 to 10:15 1 Introduction and welcome:
What are Manipulatives?
10:15 to 11:30 2 The magical world of interlocking cubes
Refreshments available from 11:15
11:30 to 12:30 3 I’ve never seen a dice like that before:
Dice, Dominoes and Pentominoes
12:30 to 13:15 LUNCH
13:15 to 14:30 4 Manipulatives circus
14:30 to 15:00 5 Action planning and feedback
Stairwell
Stairwell
Impact Toolkit and
Learning and Evaluation Tool
• Why are they used?
• Where are they used?
• Why should we use them?
• What does the research say?
• How can we use them in the classroom?
What are manipulatives?
I hear and I forget
I see and I remember
I do and I understand
Building Understanding and Clarifying Concepts
Manipulatives help students develop conceptual
understanding of mathematical ideas by representing
the ideas in multiple ways.
Using mathematics manipulatives and models offers many benefits.
Just as a picture can be worth a thousand words, manipulatives can
provide visual representations of ideas, helping students to know and
to understand mathematics. Manipulatives enhance the abilities of
students at all levels to reason and communicate.
Working with manipulatives deepens understanding of concepts and
relationships, makes skills practice meaningful, and leads to retention
and application of information in new problem-solving situations. In
turn, the valuable time spent on manipulative - and model - based
lessons has the sustained, long-term effect of building student
confidence and deepening mathematics understanding.
Indeed, it is time well spent!
Many teachers that I have worked with over the course of
the last year have been confused about what
manipulatives to use in specific contexts and I have seen
quite a bit of practice where one specific manipulative
might appear to teach a given concept and then disappear
never to be seen again in the classroom. For children this
can make the practical apparatus rather mystifying and
encourage them to think that each manipulative has a
specific function in relation to a specific task
The teachers involved gave various reasons for using manipulatives. One of
these was that using them was more enjoyable than doing mathematics that
was solely abstract and symbolic. This was substantiated by the
researcher’s observations that students were active, engaged and
interested in lessons when manipulatives were used. The enjoyment
experienced by teachers and learners in using manipulatives meant that
teachers tended to use them as a reward for good behaviour rather than
solely when they would be a useful adjunct to learning. Some of the
teachers used the manipulatives only at the end of the week, the end of the
year or when they had time. They didn’t seem to view their use as intrinsic
to the substance of the core of the curriculum but rather an addition that
enhanced enjoyment.
The magical world of interlocking cubes
Using the grid
How did you approach the task?
What have you found?• One is the only number which only has one factor• Most numbers have an even number of factors• Square numbers are the only numbers which have an
odd number of factors• Prime numbers have exactly two factors
Questions that can be asked• Is one a prime number?• Why do square numbers have odd factors?• Do the prime numbers form a pattern?
Strategies, observations and patterns
Find a formula for the nth triangular number
Explore the result when two consecutive triangle
numbers are added together
Describe how you see this shape growing.
Make the shapes using blocks
Is it possible to rearrange the blocks in each shape to form a different
picture?
What do you find?
Notation
T1 = first triangle number S1 = first square number
Tn = nth triangle number Sn = nth square
Can the nth triangle number be expressed in
terms of square and triangle numbers?
Squares on a square
How many nxn squares will fit on an mxm
square?
Squares from a square
Explore how many cubes are left when a smaller
square is subtracted from a larger square
Can this be made into a square?
Introducing Cuisenaire
Can this be made into a square?
Completing the square
Can this be made into a rectangle?
Moving on to rectangles
Task Maths
Learning and Evaluation Tool
• Dice
• Dominoes
• Pentominoes
• Geoboards
• Masking Tape and Cups
• Tangrams
• Cuisenaire Rods
More Manipulatives
One Roll Yahtzee
Numberphile
One Roll Yahtzee
• Draw the net of a dice
• Mark the sides such that, when folded, it will make a dice. Are all dice
the same.
• What are the different symmetries of each side.
Dice Activities
An activity taken from the “Wild Maths” site.
http://wild.maths.org/roll-over-dice
Dice Activities: Roll Over Dice
Dice Games For Number
Domino Activities
Cuisenaire Rods
Singapore Bar method
The crocodile
A crocodile’s head is 10cm shorter than it’s body.
It’s body is 16cm shorter than it’s tail.
The crocodile’s tail is the same length as it’s head and body combined.
Do you have enough information to find the total length of the crocodile?
If not, what more information do you need?
50
I share some sultanas between Alice and
Bob in the ratio 3:5. Alice gets 28g fewer
sultanas than Bob.
How many grams of sultanas does Bob get?
Robert Wilne, NCETMhttps://www.ncetm.org.uk/resources/46711
I share some sultanas between Alice and Bob in the ratio 3:5. Alice gets 28g
fewer sultanas than Bob.
How many grams of sultanas does Bob get?
I share some sultanas between Alice and Bob in the ratio 3:5. Alice gets 28g
fewer sultanas than Bob.
How many grams of sultanas does Bob get?
I share some sultanas between Alice and Bob in the ratio 3:5. Alice gets 28g
fewer sultanas than Bob.
How many grams of sultanas does Bob get?
Further questions:
What is Bob’s share as a fraction of the total?
Write Alice's share as a fraction of Bob’s share
“Alice’s share is Bob’s share reduced by ___%”
What is the scale factor from Alice’s share to Bob’s?
Bar Modelling
Bar Modelling
I think of three different numbers. When I add the
numbers up two at a time, I get 49, 57 and 64.
What are the three numbers?
The ratio of Sam’s money to Mary’s money was
4:1. After Sam had spent £26, Sam had £2 less
than Mary.
How much money did Sam have at first?
I think of three different numbers. When I add the
numbers up two at a time, I get 49, 57 and 64.
What are the three numbers?
49
113
64
? 57
113
The ratio of Sam’s money to Mary’s money was 4:1. After
Sam had spent £26, Sam had £2 less than Mary.
How much money did Sam have at first?
Before
Sam
Mary
After
Mary
Sam
£26
£2
Solve 3x + 7 = 31
7x
Visual Algebra
x x
31
Solve 3x = 24
x x x
24
Solve 2x - 3 = 19
x
Visual Algebra
19
2x - 3
3
x x 2x
22
Online bar models
62
NRICH:
Virtual Bar Modelling
Math Playground
Thinking Blocks
Geoboards
A game for two players
One player needs a geoboard.
The other player needs a white board and pen.
Geoboards
Geoboards
How can you find the area of this triangle?
GeoboardsHow can you find the area of this triangle?
GeoboardsIs the overlap a square?
Find the area of the overlap.
Masking Tape and Paper Cups
Tangram Algebra
Tangram Areas
Manipulatives Circus
Resource Collection
https://www.stem.org.uk/lx3efk
Community Group
https://www.stem.org.uk/community/groups/37580
/my204-using-manipulatives-to-enhance-
understanding-in-the-ks3-mathematics
Action Planning and Feedback
Secondary Mathematics Community Group
https://www.stem.org.uk/community/groups/37016/resources-secondary-
mathematics
Courses
One day courses
• Resourcing the mathematics curriculum
• Using resources to develop problem solving skills
• Using manipulatives to enhance understanding ay key stage 3
Residential courses
• New and Aspiring Leaders of Mathematics
• New to Teaching A-level Summer School
• Developing Mathematical Understanding Through Reasoning and
Problem Solving
• Teaching the New GCSE Content with Understanding
• Building Confidence as a Newly Qualified Teacher
• Effective Feedback and Assessing Progress in Mathematics Without
Levels
MY204: Using manipulatives to enhance
understanding in KS3 maths
(and beyond!)Monday 17th October 2016
09:30 – 15:00
Michael Andersonm.anderson@stem.org.uk
@STEMLearning_MA