Mathematics Developmental Continuum
Skip Counting: 2.0
Supporting materials
Indicator of Progress
At this level students can count fluently forwards and backwards by twos, fours, fives, tens and hundreds starting at any number.
Before achieving this, they will be able to skip count forwards fluently, but may experience difficulty counting backwards.
Skip counting is a gradually developing skill as students continue to expand the range of numbers with which they can skip count. They will also become able to skip count from any number, not just starting at zero, which results in the most familiar sequences (e.g. 0, 2, 4, 6 … instead of 1, 3, 5, 7 ...).
Skip counting is important in the development of fluency in calculation, number sense and as the basis of multiplication and division. It is also important to help students move from calculating by counting by ones to using number facts. For example, instead of working out 12 + 4 by counting 12, 13, 14, 15, 16, students can immediately add 4, or possibly add 2 twice. This transition to using fluent number facts is a key to success throughout school.
Illustration 1: Crossing the place value barriers
Bridging across barriers such as 100 causes difficulties in skip counting. For example, when asked to count backwards from 120 by tens, some students will say: 120, 110, 190, 180, 170 … and stop part way through the sequence as they realise they have already said some of the numbers before. Alternatively students may omit 100, saying 120, 110, 90, 80 … 10.
At an earlier stage, students will have difficulty crossing tens barriers when counting backwards by 2s (e.g. 24, 22, 20, unsure ...)
Illustration 2: Starting at any number
Once students have learnt to count fluently starting with the multiples of the number they are counting by, they need to skip count from a non-multiple of the number. For example, when asked to start at 13 and count by twos students will say “I can’t” or slip into a 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.
Illustration 3
Examples of the types of tasks that would be illustrative of skip counting fluency, aligned from the Mathematics Online Interview:
- Question 2 - Counting forwards and backwards by 1s
- Question 4 - Counting from 0 by 10s, 5s and 2s
- Question 5 (a) - Counting from 'x' by 10s
Teaching Strategies
Students should count or skip count every day. This may be an oral count, a count of physical objects to determine the number of objects in the set or a written record of skip counting. It is appropriate at every level, even in a more advanced context using negative numbers, fractions and decimals.
These activities are flexible in level, time and group size.
Activity 1: Counting Games is a set of group counting games, which are easily adjusted to different levels.
Activity 2: Whisper Count is an activity that helps students who are counting by ones to hear and learn the skip counting sequence.
Activity 3: How Far Can You Go? allows students to record their knowledge of the skip counting sequences.
Activity 4: Using the hundreds grid for counting shows a very useful scaffold for all counting activities.
Activity 1: Counting Games
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 a 'magic' number 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 fives, with the starting number zero and the finishing number of 50 and with magic numbers of 20 and 40. Students would start counting: 0, 5, 10, 15, the next student says 20 but sits down. The count continues so the next students would say 25, 30, 35 then the next student would say 40 and sit down. The count continues: 45, 50. As 50 is the finishing number then students would count backwards starting at 45, the next student says 40 and sits. The game continues in this way forwards and backwards between 0 and 50 until one child remains standing.
Buzz
Teacher announces starting and finishing numbers and which numbers will be ‘buzz’. For example, we might start at 1, finish at 100, and buzz on numbers that are multiples of 5. 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 “1, 2, 3, 4, buzz, 6, 7, 8, 9, buzz, 11 ….”
If a student forgets to buzz, they are out of the game. Vary rules as required (eg. give several chances before they are out).
Fizz Buzz
This is a variation of Buzz for students at Level 3 and beyond. The rules are the same as for Buzz, but we ‘buzz’ on some numbers, ‘fizz’ on others and ‘fizzbuzz’ when both are required.
For example if we are buzzing on multiples of 3 and fizzing on multiples of 10 the correct sequence is “1, 2, buzz, 4, 5, buzz, 7, 8, buzz, fizz, 11, buzz, 13, 14, buzz, 16, 17, buzz ... and fizzbuzz” is said instead of 30.
Activity 2: Whisper Count
This is an activity which assists students to move from orally counting by ones to skip counting. When counting from the starting number to the finishing number the students whisper the numbers that are not part of the count and say loudly the numbers that are part of the count. 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.
In the following example the students are using Whisper Count to count by twos starting at 2 and stopping at 20.
Initially the teacher records all the numbers from 2 to 20 on the board.
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Children take it in turns to circle the numbers that are part of counting by twos starting with two.
Starting at one the students whisper the odd numbers and say loudly the even numbers forwards from 2 to 20 and then count backwards from 20 to 2. Gradually increase the range.
Starting, finishing and counting sequences can be changed according to the needs of the students.
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 restrict the amount of time spent on this activity or allow students to explore the pattern as far as they are able. This is an excellent activity to take home. A hundreds grid or a calculator can assist students. Writing the numbers vertically assists the students to focus on any patterns that are formed.
After students have completed their strips teachers might ask questions such as:
- What pattern did you notice when counting by fives?
- When counting by fives would the number 70 be part of your counting pattern? How do you know?
- What if you were counting by fives and started at 135 what would the next few numbers be?
- What if you were counting backwards by fives from 90 what would the next few numbers be?
Activity 4: Using the Hundreds Grid for counting
Initially students should place clear counters on a paper Hundreds Grid to mark the counting sequence. Clear 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 loudly the numbers that are part of the counting sequence. Once students are confident with the counting sequences they will not need to use the Hundreds Grid.
On the Hundreds Grid below a student has started to place red counters on the fives counting pattern which will assist them to count fluently by fives. When students have said and recorded their counting sequences the teacher can ask questions such as:
- What number will come after 100 in your pattern?
- Can you continue counting by fives from 100?
- How would this help you count by tens?
A Hundreds Grid on a frame with rotatable numbers can readily display patterns and scaffold counting for the students. The image below shows the pattern for counting by fours. Number tiles can be rotated so that some numbers are not visible, to gradually reduce support.
Further Resources
The following resource contains sections that may be useful when designing learning experiences:
Digilearn object *
Number trains – students arrange train carriages according to numbers on their sides. The numbers are represented in a range of formats such as words, numerals, dice dots or counting frames. Students identify the numbers that come before and after starting numbers.
(https://www.eduweb.vic.gov.au/dlr/_layouts/dlr/Details.aspx?ID=4453)
* Note that Digilearn is a secure site; SMART::tests login required.