1: Stage 4: Advanced counting
Welcome to Module Eight
This is one of seven modules describing effective teaching practices.
Once you have finished you will be able to select and implement learning activities appropriate for students working
at stage 4: advanced counting.
2: General principles for the teaching and learning activities presented:
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Activities are designed to strengthen students’ understandings within the ACTUAL STAGE that they are
linked to. Teachers wishing to move students into the next strategy stage need to select activities linked to
that NEXT STAGE.
Each of the teaching modules is structured under the following headings:
o Strategy stage review
o Knowledge being developed
o Activities to develop knowledge
o Strategy being developed
o Activities to develop strategy.
The knowledge component is described first as the knowledge outlined provides a foundation for the
development of strategy at each stage.
The activities assume that students understand the ideas from earlier stages so it is important that teachers
check that students have these understandings.
The activities do not necessarily require the purchase of specialised materials. In general they use
resources readily available within New Zealand schools or easily accessible from the project materials
section of the nzmaths website.
The activities presented here are provided as a starting point. Teachers are encouraged to develop other
activities using the methods presented here as the basis of an approach. Further ideas for activities can also
be found from the number section of nzmaths, from the Numeracy Development Project Books, and by
using the Numeracy Planning Assistant.
3: Strategy stage review
Stage 4: Advanced counting
The advanced counting stage is characterised by students who use counting on or counting back to solve simple
addition and subtraction problems. Students at this stage use a skip-counting strategy to solve multiplication
problems.
The videos on the following 2 pages illustrate students solving problems at this stage.
4: Strategy stage review
Stage 4: Advanced counting, addition and subtraction domain
Video Clip
5: Strategy stage review
Stage 4: Advanced counting, multiplication and division domain
Video Clip
Students at this stage will also be working in the proportional domain, finding fractions of shapes and
sets.
6: Knowledge being developed
At this stage it is important to extend students’ knowledge in the Number Sequence and Order domain, developing
their ability to:
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identify the number before and after a given number, to at least 100
count forwards and backwards from numbers other than one
count forwards and backwards to at least 100
skip count in twos, fives and tens to at least 100.
Students will need to be able to do these things in order to use the strategy of advanced counting.
7: Knowledge being developed
Students will also be developing knowledge in each of the other four knowledge domains. More specifically, students
need to develop the ability to:
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read and write numbers to at least 100 (Number Identification domain)
know groupings with ten / "ten and" facts. For example 10 and 2 is 12, 10 and 5 is 15 (Grouping and Place
Value / Basic Facts domain)
know groupings with 5, for example 5+3, 5+1 (Grouping and Place Value domain)
know doubles to 10, for example 3+3, 4+4 (Basic Facts domain)
record the results of addition and subtraction operations using number sentences such as 5+2=7 (Written
Recording domain)
8: Activities to develop knowledge: number sequence and order
Counting
Count together, both forwards and backwards, starting from numbers other than 1. Focus students’ attention on
numbers before and after "Let’s count from 35 to 79. What will be the first number we say after 35? What will be the
last number we say before 79?"
Flip strip
The Flip strip equipment animation shows how Flip strips can be used to help students develop the ability to identify
the number after any given number. Change the numbers in the strip, working with larger numbers as students
become more confident.
Flip strips are Material Master 4-2, (available from Material Masters).
Lucky Dip
Use a set of numeral cards. Have students take a "lucky dip" to get a card then ask "What’s the number before this?
What’s the number after this?" Identify the numbers in a hundreds board.
Numeral cards are Material Master 4-1 (available from Material Masters).
9: Activities to develop knowledge: number sequence and order
Skip Counting
Rote count with body percussion to help develop and learn skip counting sequences. For example students touch
their knees and quietly whisper "one" then clap and say "two", and so on.
Skip counting on the number line
Use a large number line and put pegs on every fifth number. Read the numbers with pegs on to learn the skip
counting sequence in fives. Gradually remove the pegs as the skip counting sequence is learned.
Repeat for other skip counting sequences, for example 2 and 10.
Number lines are Material Master 4-8, (available from Material Masters).
Bead Strings
The Bead strings equipment animation shows how bead strings can be used to develop and practice skip counting
sequences.
10: Activities to develop knowledge: number sequence and order
This video clip shows teaching activities using a number line, hundreds board and number flip strip.
11: Activities to develop knowledge: number sequence and order
This video clip shows teaching activities using a bead frame.
12: Activities to develop knowledge
Guess My Number (Number Identification)
Use a hundreds board and choose a "mystery number" between 1 and 100. Students ask questions, one at a time, to
find the mystery number. Students should be encouraged to ask open questions, for example, "Is it in the twenties?",
"Is it an even number?" or "Is it greater than 50?" Direct questions such as "Is it 24?" should be disallowed. As each
question eliminates possibilities for the mystery number, these are flipped over on the hundreds boards, so the
remaining possibilities can be clearly seen. Once the mystery number is identified, clearly identify it for all students.
Hundreds boards are Material Master 4-4, (available from Material Masters).
Teen numbers (Grouping / Place Value and Basic Facts)
Seat the students in pairs. One partner shows 10 fingers. The other partner shows any number of fingers from one to
nine, say six. The "ones" person says "six" and the other partner says "10", and together they say "is sixteen".
Writing Equations (Written Recording)
As students solve number problems ask them to record their work using number sentences such as 5+2=7 and 83=5. Provide students with number sentences and ask then to make up story problems to go with them.
13: Strategy being developed
At this stage students are learning to count on to solve number problems. To do this, students must develop
cardinality. Students have developed cardinally when they understand that:
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counting tells you how many things are in a set. The last word in the counting sequence tells you how many
items are in the set.
once the number of items in a set has been established, the numbers before and after this number give the
result of adding one or taking one away from the set.
the number of items in a set remains constant, no matter how the objects are rearranged.
numbers can be used to compare two or more sets without having to bring the sets together and match the
items one to one.
14: Strategy being developed
This video clip and the one on the next page show two activities you may want to try with students. Students’
responses will vary based on whether they have developed an understanding of cardinality.
Video Clip
15: Strategy being developed
Video Clip
16: Activities to develop strategy
The following video clips describe teaching activities that are appropriate for students at this stage and outline how
the teaching model is applied in these activities.
You can review the Teaching Model in Module 5.
17: Activities to develop strategy: addition and subtraction
Video Clip
18: Activities to develop strategy: multiplication and division
Video Clip
19: Activities to develop strategy
Number strips: counting back
The Number strip: counting on equipment animation shows how number strips can be used to develop the strategy of
counting back to solve subtraction problems.
Number strips: skip counting
The Number strip: multiplication by counting on equipment animation shows how number strips can be used to
develop the use of skip counting to solve multiplication problems.
Number strips are Material Master 5-1, (available from Material Masters).
20: End of Module Eight
This module has covered selecting and implementing learning activities appropriate for students working at stage 4:
advanced counting. It is intended as an introduction only. Further information and additional activity ideas can also be
found in:
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The teaching units and activities available from the number section of the nzmaths website. Select
appropriate activities from those listed under Level One.
The Numeracy Planning Assistant Find appropriate activities for this stage under "advanced counting."
The Numeracy Development Project Books (PDF versions available from Numeracy Development Project
Books). Relevant activities for this stage can be found in Book 4: Teaching Number Knowledge
Book 4: Teaching Number Knowledge
Book 5: Teaching Addition, Subtraction, and Place Value
Book 6: Teaching Multiplication and Division
Book 7: Teaching Fractions, Percentages and Decimals
21: End of Module Eight
Module 9 looks at teaching activities for students working at the first of the partitioning stages: stage 5, early additive.