Recognising and Using Patterns: 1.75

• Indicator of Progress
• Teaching Strategies
• Further Resources

Supporting materials

• Related Progression Points
• Developmental Overview of Working Mathematically (PDF - 31Kb)
• Developmental Overview of Methods of Calculation (PDF - 38Kb)

Indicator of Progress

Success depends on students recognising and using patterns and relationships. In some ways, all mathematics is about recognising and using patterns. Many calculations can be made simpler by using the appropriate patterns. For example, I can calculate 6 + 5 because I know that 5 + 5 is 10 and so 6 + 5 is 1 more.

Earlier, students may not have sufficient number knowledge to recognise number patterns, although they can recognise and continue other patterns. As their knowledge and skills develop, students will use known facts to derive other facts. If I know 6 + 4 = 10 then I can also work out 16 + 4, 6 + 14, 26 + 4, 6 + 24, 36 + 4.

 

Illustration 1: Learning from patterns

Students will learn a lot about numbers if they observe patterns. This is one of the reasons why patterns are important.

Initially this involves recognising how counting changes the units digit. For example, when counting by ones from 21 the units digit changes but the tens digit stays the same until I get to 29: 21, 22, 23, 24, 25...

This then leads to adding 2 more. For example, 21 + 2 = 23; 22 + 2 = 24...

Later students will see the pattern that adding ten only changes the tens digit, eg 32 + 10 = 42. Being able to add 10 in this way is quite different to the child who counts by ones ten times. As this counting process is very time consuming, there are many chances of making a mistake in the count.

 

Illustration 2

Examples of the types of tasks that would be illustrative of recognising and using patterns, aligned from the Mathematics Online Interview:

• Question 4 - Counting by 10s, 5s, 2s starting from 0
• Question 8 (a) - Read 2 digit numbers
• Question 9 (a) (b) - Use calculator to record and say numbers to 2 digits
• Question 10 (a) (b) - Order 1 and 2 digit sets of number cards
• Question 12 - Identify missing number in 100 chart

 

Teaching Strategies

These teaching activities aim to improve students' abilities in recognising and making number patterns by giving them plenty of experiences in pattern making and describing what they see.

Activity 1: Using the repeat function on the calculator provides opportunities for students to use the repeat function on the calculator to create patterns with numbers.
Activity 2: Sorting provides opportunities for students to sort numbers into groups and describe the groups they have chosen.
Activity 3: Patterns in a modified hundreds chart provides opportunities for students to use the hundreds chart to highlight and describe the patterns they found.
Activity 4: What is my counting pattern? emphasises the counting patterns that numbers belong to. Teachers find this a very rich task.

 

Activity 1: Using the repeat function on the calculator

The students press 5 + = = and continue pressing =. On most calculators, this produces the sequence 5, 10, 15, 20, 25... Students describe the pattern of numbers that they see. The teacher could record the numbers on the board, so that the pattern can be viewed while being discussed.

Supplementary activities

• Press 2 + = =
• Press 4 + = =
• Press 10203 + = =
• What are the similarities when they count by twos and fours?

It is important to record the pattern so that students can look back at it. However, it is also useful to do a lot of oral work supported by the calculator. For example, children count aloud by 2s supported by the calculator.

 

Activity 2: Sorting

Give the students a selection of cards each with a number on it (e.g., 2, 3, 4, 6, 9, 12, 15, 16). Ask the students to sort the numbers into groups and describe the groups they have made.

Discuss the variety of ways the students have sorted the cards. If all students have sorted into the same groups ask them to find other ways of sorting.

Then select a single number and ask the students where they would put this number, and why.

 

Activity 3: Patterns in a modified hundreds chart

A hundreds chart can be adapted to include numbers from 1 to 120. This addresses the problem where a significant number of students think numbers finish at 100. This design also caters for students who think that the number after 109 is 200.

Hundreds Chart (PDF - 14Kb) - Document with three different versions of a Hundreds Chart (1-120, 1-100, 0-99)

For example, give students a modified hundreds chart and ask them to shade all the numbers that have a 6 as one of the digits.

1	2	3	4	5	6	7	8	9	10
11	12	13	14	15	16	17	18	19	20
21	22	23	24	25	26	27	28	29	30
31	32	33	34	35	36	37	38	39	40
41	42	43	44	45	46	47	48	49	50
51	52	53	54	55	56	57	58	59	60
61	62	63	64	65	66	67	68	69	70
71	72	73	74	75	76	77	78	79	80
81	82	83	84	85	86	87	88	89	90
91	92	93	94	95	96	97	98	99	100
101	102	103	104	105	106	107	108	109	110
111	112	113	114	115	116	117	118	119	120

Ask students to discuss what they found. How can they check that they have all the numbers?

Other tasks

• "I am thinking of a 2 digit number. One of the digits is 7. What might the number be?" Ask students to describe the strategies they used to decide on the number. Do they think they have thought of all the possibilities?
• Shade in 2 numbers that have a difference of 10
• Shade all the numbers ending in 5 or 0

Students need to discuss the strategies they used to discover the appropriate sets of numbers. Teachers need to ask questions such as:

• Was the hundreds chart useful? How?
• Could you have done it another way?
• What other patterns do you see?

 

Activity 4: What is my counting pattern?

Students think of a counting pattern (eg 5, 10, 15, 20... or 2, 4, 6, 8...) and select a number from within the pattern. They take turns to come to the front of the group and say: "My number is 20 (for example). What might I have been counting by?"

Other students in the group suggest a counting sequence that 20 belongs to. For example, someone might suggest 2, 4, 6, 8... and someone else might suggest 5, 10, 15, 20... One of these counting patterns will be the one the child has thought of, but there is interesting mathematics to discuss in the other answers too.

This is a rich task and there are many opportunities for students to explain their strategies and calculations. For example, someone might say that they know 20 is an even number, someone else might say that it is in the counting by tens pattern. The teacher might note that the numbers in the counting by tens pattern are also in the counting by fives pattern.

Further Resources

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

Digilearn object *

Musical number patterns: music maker –students make some music by building up rhythms for four instruments. Students choose a starting point on a number line and build a counting rule. Students count in lots between 2 and 10 until they reach 36. They add their number several times on the number line to make a pattern.
(https://www.eduweb.vic.gov.au/dlr/_layouts/DLR/Details.aspx?ID=3333)

* Note that Digilearn is a secure site; DEECD login required.