Arithmetic of Variables (Group 1)
The Number Rods, Sand Paper Cyphers and Number Rods and Cards form the first half of the Maths Materials for Group 1, in which the ‘Addition of Variables’ takes place
The Number Rods
The Number Rods are similar to the Red Rods, only here they are rendered countable by blue stripes, so that each rod is clearly a multiple of the first rod, (the unit), each section of the subsequent number rods is equal in length to the first one. These rods materialise the quantities of the numbers from 1-10. Through the Number Rods ‘quantities’ can be seen as an unbroken multiple of a unit in which quantities are measured in terms of a unit; the ‘Arithmetic of Variables’. After completing these first activities of Group 1 the child is given quantities according to the ‘Arithmetic of Groups’, quantities formed by loose identical units (Spindles and Counters).
The advantages of using Number Rods to initially represent quantity are;
- Past experience with the Red Rods helps the child to use the Number Rods, sensorially appreciating the difference between them
- The child’s intelligence accepts, understands and associates different names with the differences she can see between the Rods.
- Each Rod differs by the same increment, which is equal to the first in the series (the unit)
- The Rods show the position of the succession of numbers 1 to 10
- They show the close relationship between Ordinal and Cardinal numbers
Material Description:
- Ten wooden rods, identical to the Red Rods, graded in length form 10 cms to 1 metre, each rod is coloured in alternate red and blue 10 cm long partitions. The first rod and the first partition in each succeeding rod is red
- Working Mat
Presentation:
- Invite the child, introduce the material and help her to bring the rods to a Working Mat, place them in sequence – pointing out that the red partition goes on the left
- Isolate the shortest three rods and give there names in a Three Period Lesson
First Period
- Begin with the 10 cm rod, touching it and saying, “This is one”
- Then take the 20 cm rod, touching each partition and saying, “This is two”
- Then take the 30 cm rod, touching each partition and saying, “This is three”
Second Period
- Mix up the rods and give various commands including asking the child to count the partitions. Continue to mix and give directions until the child can recognise them with ease
Third Period
- Take any rod and ask the child to name, saying, “what is this?”, mixing frequently
Note
- The other Number Rods are introduced in the same way, first check her understanding of rods introduced earlier and mix these in during the second period
Exercises:
- The rods are placed randomly on the mat, the director asks for a particular one and the child brings it. The child is asked to verify by touching each partition
- The rods are placed randomly on the mat, the directress picks a particular one and the child names it. The child is asked to verify by touching each partition
Criteria of Perfection (Control of Error):
- The partitions guide the child
Direct Aim:
- To teach the names 1 to 10 with the corresponding quantity
- To show that each number from 1 to 10 is a separate object
- To help the child memorise the sequence of numbers 1-10
Age at Presentation:
Four years onwards
Footnote
- Even if the child immediately recognises and names the rod it is still necessary to verify by touching and counting to fix the sequence of numbers.
The Sand Paper Cyphers
With the help of the Director, the children already know the names of the numbers 1-10 and associate them with their quantities and sequence, having worked with the Number Rods. The Cyphers are symbols for the quantities and different from the Sand Paper Letters (which the child will have already used) because they represent not just a sound, but a whole idea
Material Description:
The Numbers 1 to 9 and 0, cut rom sandpaper and mounted on separate boards.
Note
- It is important to check the child’s knowledge of the Number Rods before proceeding
- Do not introduce the 0 at this stage
Presentation:
- Ask the child to sensitise her fingers and select a group of three Sand Paper Cyphers in any order
- Invite the child, introduce the material and bring it to a chowki
- Trace each numeral with two fingers as with the Sand Paper Letters
- Give there names in a Three Period Lesson, asking the child to trace and name the Cypher
Games:
Two children can be show how to play these games together
- Show a Cypher, ask the child to name it and trace it
- Lay the Cyphers on a mat, say a number and ask one child to go to the mat, find it and return with it and trace it.
Criteria of Perfection (Control of Error):
- The Director and the Sand Paper which guides the child’s fingertips
Direct Aim:
- To teach the written symbol or the quantities 1 to 10 which the child has learnt with the Number Rods
- To give the child a key to the World of Number
- Preparation for the writing of numerals
Age at Presentation:
Four years, after the Number Rods
The Number Rods and Cards
Material Description:
- Number Rods
- Set of wooden or cardboard cards on which are printed the numbers 1 to 10
- Working Mat
Presentation:
- Invite the child, introduce the material and help her bring it to a Working Mat, placing the Number Rods at random
- Ask the child to collect the Cards and name them, while placing them randomly on the mat
- Introduce the child to the card with the symbol for ten (ensure that the child is familiar with this before proceeding)
- Ask the child to find the ‘rod of ten’ and place the card on the last partition
- Show the child another card, at random and ask her to find it’s pair
- Continue to do this till all the rods are labeled
Exercises:
- Exercise 1 – as above
- Exercise 2 – the Two Mat Game – the Directress shows a card and the child brings the rod
- Exercise 3 – the Two Mat Game – the Directress points to a rod and the child brings the card
- Exercise 4 – Rods in Sequence
- Arrange the rods as for the Red Rods
- Place the cards randomly on a floor mat, point to any rod in the series and the child counts the partitions and places the card on the last partition
- Ask the child to read the numbers in ascending and descending sequence
Criteria of Perfection (Control of Error):
- The child counts the number of partitions
Direct Aim:
- To associate the written symbol with the corresponding quantity
- To introduce the sequence of the symbols
Age at Presentation:
Four to four and a half years, after the Sand Paper Numerals
Footnote
- This introduces the symbol for ’10’ and is the first time the child sees the role of ‘0’ to symbolise ’10’
Special Exercises:
These special exercises are done after the Card and Counters
Impressions of putting together (addition)
- Ask the child to build the Number Rods in sequence in the middle of a Working Mat and to place the cards on the last partition of each rod
- Ask the child to move the ‘rod of ten’ to the top of the mat
- Ask her to put the ‘rod of nine’ below it and then say, “Lets make the ‘rod of nine’ an long as the ‘rod of ten’
- The child places the ‘rod of one’ adjacent to the ‘rod of nine’ and puts their cards in place
- Continue until the ‘rods of six and four’ have been placed
- Move the ‘rod of five’ beneath the ‘rod of six’ and ask the child where we can get a rod to complete the length – possibly flip the ‘rod of five’, another ‘rod of five’ from a different set or use the corresponding Red Rod to give the impression that the ‘rod of five’ twice is equal to the ‘rod of ten’
- The same activity can be done with totals of the ‘rod of nine’ etc.
Variations:
The child can later make any total using three or more rods
Impressions of taking away (subtraction)
- Build the Number Rods and cards to the place where impressions of addition was completed, with the ‘rod of five’ in it’s original place, below the ‘rod of six and four’
- Remove the ‘rod of four’ and the child sees that the ‘rod of six’ remains
- Continue nil the rods are in their original Red Rod position
Direct Aim:
- To associate the written symbol with the corresponding quantity
- To introduce the sequence of the symbols
Indirect Aim:
- The building of tens is an indirect preparation for addition while the removal of rods gives a fleeting impression of subtraction
- The flipping of the ‘rod of five’ gives a fleeting impression of multiplication and division
‘Arithmetic of Groups’
The Spindle Boxes, where the symbols are fixed and Cards and Counters and Memory Game , where loose quantities are associated with loose symbols form the second half of the Maths Materials for Group 1, in which the ‘Arithmetic of Groups’ takes place
The Spindle Boxes
The Spindle Boxes are presented after the child has had sufficient experience with the ‘Arithmetic of Variable’ an is mentally prepared for the abstraction required for the ‘Arithmetic of Groups’
Material Description:
- Two boxes, each exactly the same, divided into five compartments or one box divided into ten compartments. At the interior back of each compartment is a painted symbol of the numbers 0 to 9
- 45 spindles in a basket or box
- Eight green coloured ribbons or bands
- Sand Paper Cypher for 0
- Working Mat
Note
The Spindle Boxes should be displayed empty
Presentation:
- Invite the child, introduce the material and help her to bring the Spindles and Spindle Boxes to a Working Mat
- Point to the symbols and ask the child to identify them
- Skipping zero, take one spindle with your right hand, place it in your left and aloud count “one”, place it in the corresponding compartment
- Then take two spindles, one at a time, counting them in her hand and then puts them on the mat and places a green band around them before putting them in the compartment. (This shows they have been grouped together counting them on to the mat in front of you and then transfer them as a group into the corresponding compartment).
- Continue up till nine
- Point to 0 and explain to the child, “zero means not anything”
- Show the Sand Paper Cypher for 0 and ask the child to trace it and associate it with the symbol on the first compartment
Criteria of Perfection (Control of Error):
- The number of spindles
Direct Aim:
- While the Number Rods were each a quantity in themselves, the spindles show a number which is composed of separate objects
- The concept of zero is introduced as representing no quantity when the first compartment is left empty
Indirect Aim:
- Indirect preparation for the fact that there are no other symbols but 0 to 9 in the decimal system
Age at Presentation:
Four to four and a half years, after number rod and cards
Footnote
- The child may participate in the presentation after compartment 5
- Alternatively the child places the spindles directly into the boxes and completes the activity, later she removes the contents from one compartment at a time, counting them aloud and places the band around them before replacing them in the compartment to show they have been grouped together The child removes the contents from one compartment at a time, counting them aloud and places the band around them before replacing them in the compartment to show they have been grouped together.
Games to reinforce the concept of Zero
With either one child or a small group ask children to perform actions a number of times, when you ask for zero times pause expectantly and see if the child can explain why she hasn’t performed an action
Cards and Counters
Material Description:
- Small Number Cards 1 to 10
- 55 identical counters in a box
Presentation:
- Invite the child, introduce the material and help her to bring the Cards and Counters to a Working Mat
- Place the Number Cards at random on the Working Mat
- Ask the child to place them in sequence horizontally leaving a gap between them the width of her palm
- Place the correct number of counters in pairs under each symbol if the number is odd put the remaining counter centrally beneath the last pair
Note
When the child understands how to arrange the counters let her take over
Exercise:
- As above
- Much later build the sequence and run your index finger from the base of the Card between the counters, saying the name of the numeral, stopping abruptly at the blocking counter if the number is odd, or continuing, lengthening the word if the number is even
Criteria of Perfection (Control of Error):
- The number of counters
Direct Aim:
- To reinforce the knowledge that each number is made up of separate quantities
- To verify whether the child has mastered;
- the sequence of numbers
- how many separate units form each number
Indirect Aim:
- To sensorially indicate the odd and even numbers
- To give a visual impression of odd and even
- Indirect preparation for the divisibility of numbers as well as multiples and sub-multiples (e.g. even numbers divide by two)
Age at Presentation:
Five and a half years, after the Spindle Boxes
Footnote:
Give the words “odd” and “even” only after the decimal system has been presented
Memory Game of Numbers
Can be played with up to ten children, if less children, use less cards but ensure that 0 is always in circulation
Material Description:
- Folded Cards, with a number from 0 to 10 written
- A basket for these cards
- 55 identical counters
- A mat or chowki for each child
Presentation:
- With a group of children take the cards and explain the game
- Each child chooses a card, reads the number silently, memorises it, refolds it and puts it on her mat or chowki
- One at a time each child is asked to collect the amount of counters corresponding to the number on her card
- The child with zero will not move however much the Director entices her
- When all of the children have their objects they count them to verify to themselves and the Director hat they have understood
Variation:
- Later an older child can take the place of the Director
- Later use objects other than the counters
Criteria of Perfection (Control of Error):
- The Director
Direct Aim:
- To train the child’s memory to keep the numerical symbol in her mind
- To relate the understanding of quantities 0 to 10 to any object in the environment
Age at Presentation:
Around four and a half years,
Footnote:
This exercise also develops character and will as the children have to restrain themselves from taking more than their card allows, especially difficult for the child with zero, inhibiting their actions.