Advanced Skip Counting: 2.5

• Indicator of Progress
• Teaching Strategies
• Further Resources

Supporting materials

• Related Progression Points
• Developmental Overview of Methods of Calculation (PDF - 38Kb)
• Professional Reading on Mental computation (PDF - 110Kb)

Indicator of Progress

At this level students can skip count fluently forwards and backwards by all single digit numbers starting at any number.

Before achieving this, they will be able to skip count forwards fluently by twos, fours, fives, tens and hundreds starting at any number and count backwards from multiples of these numbers.

With daily practice students will continue to expand the range of numbers with which they can skip count, gradually including 3, 6, 9, 8, 11, 12 and 7. They will also become able to skip count forwards and backwards from any number. Starting at zero results in the most familiar sequences of multiples (e.g. 0, 3, 6, 9 …) but at this level students should be able to continue sequences such as 1, 6, 11, 16 ...  and 4, 7, 10, 13, and 10, 18, 26, 34, 42, 50 …

Skip counting is important in the development of fluency in calculation, number sense and as the basis of multiplication and division. This transition to using fluent number facts is a key to success throughout school.

Illustration 1: Difficulties with counting backwards

Unless students can count fluently forwards they will not be able to count backwards. The general progression is that students will first learn to count forwards, then begin to count backwards by mentally counting forward, until they have sufficient practice to count backwards fluently. Crossing the tens and hundreds barriers is hardest.

Mia is an example of a student in the middle stage for skip counting by 3. She still has to count forwards to articulate the counting backwards sequence. When counting backwards by 3 from 21, Mia whispered “17, 18, 19, 20, 21“ then reversed this to “21, 20, (1), 19, (2), 18,  (3)“ before announcing 18 as the next required number in the sequence.

Teachers should take care to check whether students can count and skip count forwards and backwards, especially over the tens and hundreds ‘barriers’ e.g. 104, 102, 100, 98, 96 ... and 103, 101, 99, 97 ...

Illustration 2: Starting at any number

Once students have learnt to count fluently starting at zero, they need to learn to count confidently from any number. When asked to start at 13 and count by twos some students will accidentally slip into the more familiar sequence. For example, students might say 13, 15, 17, 19, 20, 22 …. The 2, 4, 6, 8 pattern of the even numbers is much more familiar than the odd numbers.

Similar comments are made when students are asked to count from 43 by tens. Some students count by ones and cannot attend to the patterns in the tens place because they are paying attention to all the numbers in between. They therefore cannot identify the pattern: 43, 53, 63, 73 … This relates to developing sound place value understanding.

Illustration 3: What is involved in skip counting?

(i) Skip counting can be performed by reciting a number sequence learned by heart, e.g. 2, 4, 6, 8 … Knowing a few sequences by heart forms a good basis on which other sequences can be built. Learning by heart does NOT mean learning without understanding.

(ii) Many skip counting sequences are built from known sequences. For example, if a student can skip count by 2, they can learn to skip count by 4 by subvocalising every second multiple of 2, i.e. (2), 4, (6), 8, (10), 12, (14), 16 … It is better that students do this than rely on counting by ones and subvocalising 3 numbers out of 4 i.e. “(1), (2), (3), 4, (5), (6), (7), 8, (9), (10), (11), 12, (13) …because important number patterns and relationships are more clearly evident. Efficient skip counting builds on fluency with number facts (in this case adding 2), not just on the ability to count by ones. In turn, practising skip counting builds fluency with number facts.

(iii) Skip counting at this level also requires the ability to quickly modify known sequences by addition or subtraction of a constant. Facility with basic number facts is essential. For example, to count by 8 starting at 12, Jack used the fact that adding 8 is the same as adding 10 and subtracting 2.  He explained his thinking as follows: “12, 20 (know 12 + 8), 28 (easy to add 8), 36 (added 10, subtracted 2), 44 (added 10, subtracted 2), 52 (added 10, subtracted 2), 60 (easy to add 8) …”

Illustration 4: Links to the Mathematics Online Interview

Examples of the types of tasks that would be illustrative of prior knowledge of skip counting at this level, aligned from the Mathematics Online Interview:

• Question 4 - Counting from 0 by 10s, 5s and 2s
• Question 5 (a)  - Counting from 'x' by 10s
• Question 6 - Counting from x by a single digit number

Teaching Strategies

It is appropriate that at every level students have experiences every day with oral counting and skip counting backwards and forwards. At more advanced levels students will be using negative numbers, fractions and decimals.

Working towards the Level 3 standard, skip counting builds on the simpler cases of counting by 2, 10, 5, and 4. A good order to continue is to first learn multiples of 3 (so counting by 3 and then using that knowledge for 6), numbers close to ten (9, 11, 8 and 12) and then the difficult case of 7. This sequence capitalises on relationships between numbers (e.g. 6 is double 3, 9 is one less than 10). Skip counting by 20, 50 etc also draws on known sequences and should also be included.

Skip counting requires, and in turn develops, fluency in mental computation.

The activities below are flexible in level, time, and group size and build on activities in the Indicator of Progress Skip Counting 2.0.

Activity 1: Using the Hundreds Grid for counting is a very useful scaffold for all counting activities.

Activity 2: Advanced Whisper Count is an activity that helps students who are still counting by ones to hear and learn the skip counting with varied starting points, and helps all other students to use their knowledge of simpler skip counting sequences to work out related difficult sequences.

Activity 3: How Far Can You Go? is an activity that builds on students’ ability to record their knowledge of the skip counting sequences.

Activity 4: Using Counting Grids allows students to read the counting sequence.

Activity 5: Counting Games is a set of group counting games, which are easily adjusted to different levels.

Activity 1: Using the Hundreds Grid for counting

To develop fluency in skip counting students should place transparent counters on a Hundreds Grid to mark the counting sequence. Transparent counters are used so the students can still see the numbers. Once the counters are in place students whisper the numbers that are not part of the counting sequence and touch the counters as they say these numbers out-loud. Once students have confident recall with the counting sequences or can calculate the next number readily, they will not need to use the Hundreds Grid.

Note that students should always be encouraged to move on from ‘counting by ones’ strategies. They may learn to skip count by 3 whispering all numbers, as above, but they should not use such a strategy for skip counting by 9, which they should see as adding 10 and subtracting 1.

On the Hundreds Grid below, a student has started to place red counters on the counting pattern starting at 53 and counting by threes. When students have said and recorded their counting sequences the teacher can ask questions such as:

• Where would the next counter go on your grid?
• How far can you continue counting by threes?
• How would this help you count by sixes?
• What patterns can you see in where the counters are? (e.g. are they in rows?)

Activity 2: Advanced Whisper Count

In the earlier Indicator of Progress Skip Counting 2.0 the emphasis was on counting by numbers starting at zero. For example, if counting by twos the students would whisper the odd numbers and say loudly the even numbers (in bold): 1 2 3 4 5 6 7 8 9 10.

For this activity the focus is on starting with non-multiples of the counting sequence. It builds on Activity 1, although some students may be able to do this directly without the preliminary visual support of the hundreds chart. When counting from a particular starting number to the finishing number the students still whisper the numbers that are not part of the count and say loudly the numbers that are part of the count. In the following example the students are using Whisper Count to count by sixes starting at 2, based on their good knowledge of counting by twos.

“2  4  6  8 10 12  14  16  18  20  22  24  26  28  30  32  34 . …

Starting, finishing and counting sequences can be changed according to the needs of the students. Stress the relationships between sequences – a student can use their knowledge of one sequence (e.g. counting by threes) to work out other skip counting sequences.

Activity 3: How far can you go?

Students are given a strip of coloured paper or a roll of paper. Students record a counting sequence as far as they can. Teachers may assign the counting sequence and starting number or may allow students to choose their own. Teachers can place a time limit on this activity or allow students to explore the pattern as far as they are able. A hundreds grid or a calculator can assist students. Writing the numbers vertically assists the students to focus on any patterns that are formed. This is also an excellent activity to take home.

The aim is that students notice patterns. Whilst students are completing their strips teachers might ask questions such as:

• What pattern do you notice when counting by fives starting at 23?
• When counting by fives starting at 23, will the number 70 be part of your counting pattern? How do you know?
• If you were counting by fives and started at 133, what would the next few numbers be?
• If you were counting backwards by fives from 98, what would the next few numbers be?

For students at a higher level, this activity can use fraction or decimal skips. As an example, start at 9.1 and skip count by 0.2, to get the sequence 9.1, 9.3, 9.5, 9.7, 9.9, 10.1, 10.3, 10.5, 10.7 …

Activity 4: Using Counting Grids

Counting Grids can be developed for different counting patterns. The first two examples below are of a Four Counting Grid and an Eight Counting Grid. The Four Counting Grid has four columns and usually ten rows (six rows are complete in the diagram and the seventh has been started) but the number of rows can be varied depending on the needs of the students. A Three Counting Grid would have three columns.

Teacher asks questions such as:

• How are the numbers within a column related to each other? (e.g. they go up by 8)
• What patterns do you see in the numbers in a column? (specify a column and expect patterns such as ‘all odd’, ‘numbers end in 1,9,7,5,3,1,9, …’ etc)
• How can you use the Eight Counting Grid to count forwards by eights from 25?
• Show how you can use your Eight Counting Grid to count backwards by eight from 46.
• What is the same and what is different about your Eight Counting Grid and your Four Counting Grid? Can you explain why this happens?  (Answer may be that 2 rows of 4 make one row of 8).

Students can use their counting grids as an aid to building up computational fluency. They should also understand the link between their counting grid and the multiplication tables.

Activity 5: Counting Games

These games were also used in Indicator of Progress Skip Counting 2.0 but have been varied to suit the needs of more advanced students.

Advanced Magic Number

This is a game that can be varied depending on the needs of the students. Choose starting, finishing and magic numbers that consolidate and extend students’ skills. (This same game format can be used for any counting sequence including decimals, fractions and negative numbers for students at higher levels.)

Students stand in a circle. The teacher announces the number the students are counting by, the starting and finishing numbers and the ‘magic’ numbers. Students count forwards and backwards between the starting and finishing numbers. If a student says one of the predetermined ‘magic’ numbers they must say the number then sit down and are eliminated from the game. The ‘magic’ numbers introduce an element of chance of being eliminated, unlike the similar game of Buzz where students are eliminated when they are incorrect. As students become more familiar with the counting sequences they can also be eliminated from Magic Number if they say an incorrect number.

For example, in the following game students are counting by nines, starting at the number 3 and finishing at 102 and with magic numbers of 39 and 75. Students would start counting: 3, 12, 21, 30, the next student says 39 but sits down, since 39 is a designated ‘magic’ number. The count continues so the next students would say 48, 57, 66 then the next student would say 75 and sit down. The count continues: 84, 93, 102. As 102 is the finishing number then students would count backwards starting at 93, the next student says 84, next says 75 and sits. The game continues in this way forwards and backwards between 3 and 102 until one child remains standing.

Advanced Buzz

Teacher announces starting and finishing numbers and the numbers on which students say “buzz”. For example, we might start at 31, finish at 91, and buzz on numbers that are the numbers that would be part of the count if we were counting by three starting at 31. Students stand in a circle to count in turn by ones, but they say “buzz” instead of the specified numbers.

For example, a correct sequence would be 31, 32, 33, buzz, 35, 36, buzz, 38, 39, buzz, 41, 42, buzz …

If a student forgets to buzz, they are out of the game.

Vary rules as required (e.g. give several chances before a student is out).

Fizz Buzz is another useful variation.

Further Resources

The following resource contains sections that may be useful when designing learning experiences:

Digilearn object *

Number trains: skip counting (https://www.eduweb.vic.gov.au/dlr/_layouts/dlr/Details.aspx?ID=4448) - Students sequence whole numbers from 1 to 120. They apply skip counting, forwards and backwards by twos, fives and tens.

* Note that Digilearn is a secure site; SMART::tests login required.