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This article follows on from Journeys on the Gattegno Tens Chart, in which Alf describes the way a Year 3/4 class (7-9 year olds) used a Gattegno chart to gain an awareness of place value and the inverse relationship between multiplication and division. Here Alf outlines six activities using the Gattegno chart which begin by developing learners' awareness of reading, writing and ordering numbers, then progress to gaining awareness of multiplication facts, multiplication and division by 10, onto fractions as operations and, finally, to finding percentages of any number. These activities were created in the context of a project run through the charity 5x5x5=creativity, in collaboration with Penny Hay, Sara James, Kate Parsons, George Samios, Hannah Scott, Helen Shoubridge, Sam Thacker, Ellie Walsham, Carol Watts, Grace Webster.
Activity 1: Naming Numbers
Aim
When introducing the Gattegno chart to a group for the first time, students needs to see how numbers are named on the chart. Rather than concern about the meaning or place value of numbers, the focus of this activity is on how to say and write numbers and to gain awareness of how they are ordered.
A starting point
Tap on a number in the units row and get the class to chant back in unison the number name. Continue for other numbers of the units row and extend to numbers in the tens row.
Tap on “4” (class chants FOUR) and then “40” (class chants FOUR-TY); tap on “6” and then “60”; tap on “8” and then “80”. Focus attention on how the number name changes. All we do is add “-ty” to the end of the digit number. Practise this (you may want to allow students to say “five-ty” and “three-ty” for 50 and 30).
Having established these names, tap on “40” followed by “2” - students need to chant back “FOUR-TY-TWO”.
Keep tapping out two-digit numbers. When students seem confident in calling back these number names, invite a student to the board to write “forty-two” - or, if this is new to students, show them how to write it and invite someone to write 43, 44, etc.
Students could count up in 1s from different starting points, focusing on how the digit names keep repeating in sequence.
A task
Students, in pairs or small groups, could be challenged to create their own “100 square”, i.e., order the numbers 1 - 100 (or as an extension, 100-200). The Gattegno chart can be kept in view as support.
Activity 2: Multiples
Aim
Support students in exploring patterns in number, leading to awareness of multiplication table facts.
A starting point
With no introduction, say to the class: “One hundred”. Invite a response - someone will usually say “Two hundred”, you can then reply “Three hundred” and get the class to continue chanting multiples of 100. As they do this, begin tapping out the numbers on the Gattegno chart. When you have gone slightly higher than feels comfortable, invite the class to give you another starting number, or choose one for the class (e.g. 10 or 5 or 2).
Work on multiples of the new starting number, get the class to chant in unison - again, tapping out the numbers they say on the chart. Someone can write down the numbers as well. Invite the class to talk about any patterns they notice.
A task
Students pick their own starting number and write out multiples, going as high as they can. Students can be invited to pay attention to any patterns they notice. At some point you might want to gather the class on the carpet and invite individuals to describe the patterns they have seen. Other students can be asked if they can continue these patterns.
Activity 3: Addition and Subtraction Journeys
Aim
Students to develop an awareness of what happens at the 'transition' between tens and/or hundreds
A starting point
Tap on a number and get students to chant back the number one higher. Choose different numbers, particularly those ending in a “9”.
A task
Students choose a starting number and add one. They then keep on adding one, looking for patterns in their answers, writing, for example:
17+1=18
18+1=19
19+1=20
20+1=21
When they have gone as far as they want, they choose a new starting number.
The same process could be gone through with subtracting one, or going up/down in numbers other than one.
Activity 4: Multiplication and Division by Powers of 10
Aim
Students to gain an awareness of how to multiply and divide by powers of ten and awareness of the relations between these operations.
A starting point
Tap on a number, get the class to chant back the number that is 10 times bigger. Initially chose single-digit numbers, then progress to others, returning to single digit if the class lose confidence. At some point, invite someone to say how, on the chart, they get their answer. Focus attention on how you can get the answer simply by moving down one row on the chart. Try to steer the class away from mention of changing numbers of zeros as such methods do not extend to decimals. Return to chanting with the awareness of movement. Then point on a number and invite the class to chant back that number divided by 10. Repeat for multiplication and division by 100 (and possibly 1000).
A task
Students have to choose a number on the chart, then go on a 'journey' multiplying or dividing by 10, 100, etc. Their challenge is to return to the number they started with. Do one 'journey' with the class and then let them try out their own.
You may want to show some of the decimal rows of the chart in order to help students extend their journeys. Be prepared for some big numbers!
After some time, gather students to discuss what they have done. They might have questions they want to explore further. This activity has continued with energy over four hours and more with some groups. Questions might emerge about how long they can make their journeys, or whether they can go on a journey and get back in one go, or whether they can get journeys that shift between columns (see Fraction Journey activity below).
Activity 5: Fraction Journeys
Aim
Let students develop an awareness of fractions as operations.
A starting point
This activity is best done following on from: 'Multiplication and Division by Powers of 10' (see above). In that activity, students may have noticed that their 'journeys' only move vertically (up or down). It may be that a student asks how they could do different journeys. This activity is best set up if this question has arisen in such a way.
The key to the starting point is to establish, through chanting on the chart, that to get from the first (1s) column to, say, the fifth (5s) column, we multiply by 5. To go in the reverse direction, we multiply by one fifth. To go from the 1s column to the 7s column, we multiply by 7. To go in the reverse direction we multiply by one seventh.
A task
Students have to pick two numbers from the chart: a starting number and a finishing number. Their task is to find a journey from one to the other.
e.g., From 200 to 70
200 x ½ = 100
100 ÷10 = 10
10 x 7 = 70
You may need to do a few such journeys all together. We have seen year 3 classes (7-8 year olds) confidently working with fractions in this way, having spent time beforehand working on multiplication and division by 10 on the Gattegno chart.
Activity 6: Percentages
Aim
Develop a method for finding a percentage of any number.
A starting point
Tap on a number, get the class to chant back 10% of that number. Repeat many times. As with multiplication and division by powers of ten, at some point focus attention on what you are doing visually (going up one row). Then repeat with finding 1% of a number (going up two rows).
A task
Students can set challenges for each other of finding percentages of a number. They should find their percentages initially by splitting it up into 10s and 1s.
e.g., 31% of 60
10% of 60 = 6
10% of 60 = 6
10% of 60 = 6
1% of 60 = 0.6
31% of 60 = 18.6
They may notice ways to get 20% or 5%, based on 10% and this can speed up their calculations.
A harder challenge is to choose two numbers from the chart and find, as close as possible, one as a percentage of the other.
e.g., choose numbers 16 and 60, what percentage of 60 gives us an answer as close as possible to 16?
10% of 60 = 6
10% of 60 = 6
5% of 60 = 3
So, 25% of 60 is 15
Can you get closer?
Activities on the Gattegno Chart
This article follows on from Journeys on the Gattegno Tens Chart, in which Alf describes the way a Year 3/4 class (7-9 year olds) used a Gattegno chart to gain an awareness of place value and the inverse relationship between multiplication and division. Here Alf outlines six activities using the Gattegno chart which begin by developing learners' awareness of reading, writing and ordering numbers, then progress to gaining awareness of multiplication facts, multiplication and division by 10, onto fractions as operations and, finally, to finding percentages of any number. These activities were created in the context of a project run through the charity 5x5x5=creativity, in collaboration with Penny Hay, Sara James, Kate Parsons, George Samios, Hannah Scott, Helen Shoubridge, Sam Thacker, Ellie Walsham, Carol Watts, Grace Webster.
An example of the Gattegno Chart
Activity 1: Naming Numbers
Aim
When introducing the Gattegno chart to a group for the first time, students needs to see how numbers are named on the chart. Rather than concern about the meaning or place value of numbers, the focus of this activity is on how to say and write numbers and to gain awareness of how they are ordered.
A starting point
Tap on a number in the units row and get the class to chant back in unison the number name. Continue for other numbers of the units row and extend to numbers in the tens row.
Tap on “4” (class chants FOUR) and then “40” (class chants FOUR-TY); tap on “6” and then “60”; tap on “8” and then “80”. Focus attention on how the number name changes. All we do is add “-ty” to the end of the digit number. Practise this (you may want to allow students to say “five-ty” and “three-ty” for 50 and 30).
Having established these names, tap on “40” followed by “2” - students need to chant back “FOUR-TY-TWO”.
Keep tapping out two-digit numbers. When students seem confident in calling back these number names, invite a student to the board to write “forty-two” - or, if this is new to students, show them how to write it and invite someone to write 43, 44, etc.
Students could count up in 1s from different starting points, focusing on how the digit names keep repeating in sequence.
A task
Students, in pairs or small groups, could be challenged to create their own “100 square”, i.e., order the numbers 1 - 100 (or as an extension, 100-200). The Gattegno chart can be kept in view as support.
Activity 2: Multiples
Aim
Support students in exploring patterns in number, leading to awareness of multiplication table facts.
A starting point
With no introduction, say to the class: “One hundred”. Invite a response - someone will usually say “Two hundred”, you can then reply “Three hundred” and get the class to continue chanting multiples of 100. As they do this, begin tapping out the numbers on the Gattegno chart. When you have gone slightly higher than feels comfortable, invite the class to give you another starting number, or choose one for the class (e.g. 10 or 5 or 2).
Work on multiples of the new starting number, get the class to chant in unison - again, tapping out the numbers they say on the chart. Someone can write down the numbers as well. Invite the class to talk about any patterns they notice.
A task
Students pick their own starting number and write out multiples, going as high as they can. Students can be invited to pay attention to any patterns they notice. At some point you might want to gather the class on the carpet and invite individuals to describe the patterns they have seen. Other students can be asked if they can continue these patterns.
Activity 3: Addition and Subtraction Journeys
Aim
Students to develop an awareness of what happens at the 'transition' between tens and/or hundreds
A starting point
Tap on a number and get students to chant back the number one higher. Choose different numbers, particularly those ending in a “9”.
A task
Students choose a starting number and add one. They then keep on adding one, looking for patterns in their answers, writing, for example:
17+1=18
18+1=19
19+1=20
20+1=21
When they have gone as far as they want, they choose a new starting number.
The same process could be gone through with subtracting one, or going up/down in numbers other than one.
Activity 4: Multiplication and Division by Powers of 10
Aim
Students to gain an awareness of how to multiply and divide by powers of ten and awareness of the relations between these operations.
A starting point
Tap on a number, get the class to chant back the number that is 10 times bigger. Initially chose single-digit numbers, then progress to others, returning to single digit if the class lose confidence. At some point, invite someone to say how, on the chart, they get their answer. Focus attention on how you can get the answer simply by moving down one row on the chart. Try to steer the class away from mention of changing numbers of zeros as such methods do not extend to decimals. Return to chanting with the awareness of movement. Then point on a number and invite the class to chant back that number divided by 10. Repeat for multiplication and division by 100 (and possibly 1000).
A task
Students have to choose a number on the chart, then go on a 'journey' multiplying or dividing by 10, 100, etc. Their challenge is to return to the number they started with. Do one 'journey' with the class and then let them try out their own.
You may want to show some of the decimal rows of the chart in order to help students extend their journeys. Be prepared for some big numbers!
After some time, gather students to discuss what they have done. They might have questions they want to explore further. This activity has continued with energy over four hours and more with some groups. Questions might emerge about how long they can make their journeys, or whether they can go on a journey and get back in one go, or whether they can get journeys that shift between columns (see Fraction Journey activity below).
Activity 5: Fraction Journeys
Aim
Let students develop an awareness of fractions as operations.
A starting point
This activity is best done following on from: 'Multiplication and Division by Powers of 10' (see above). In that activity, students may have noticed that their 'journeys' only move vertically (up or down). It may be that a student asks how they could do different journeys. This activity is best set up if this question has arisen in such a way.
The key to the starting point is to establish, through chanting on the chart, that to get from the first (1s) column to, say, the fifth (5s) column, we multiply by 5. To go in the reverse direction, we multiply by one fifth. To go from the 1s column to the 7s column, we multiply by 7. To go in the reverse direction we multiply by one seventh.
A task
Students have to pick two numbers from the chart: a starting number and a finishing number. Their task is to find a journey from one to the other.
e.g., From 200 to 70
200 x ½ = 100
100 ÷10 = 10
10 x 7 = 70
You may need to do a few such journeys all together. We have seen year 3 classes (7-8 year olds) confidently working with fractions in this way, having spent time beforehand working on multiplication and division by 10 on the Gattegno chart.
Activity 6: Percentages
Aim
Develop a method for finding a percentage of any number.
A starting point
Tap on a number, get the class to chant back 10% of that number. Repeat many times. As with multiplication and division by powers of ten, at some point focus attention on what you are doing visually (going up one row). Then repeat with finding 1% of a number (going up two rows).
A task
Students can set challenges for each other of finding percentages of a number. They should find their percentages initially by splitting it up into 10s and 1s.
e.g., 31% of 60
10% of 60 = 6
10% of 60 = 6
10% of 60 = 6
1% of 60 = 0.6
31% of 60 = 18.6
They may notice ways to get 20% or 5%, based on 10% and this can speed up their calculations.
A harder challenge is to choose two numbers from the chart and find, as close as possible, one as a percentage of the other.
e.g., choose numbers 16 and 60, what percentage of 60 gives us an answer as close as possible to 16?
10% of 60 = 6
10% of 60 = 6
5% of 60 = 3
So, 25% of 60 is 15
Can you get closer?