Mathematics Developmental Continuum P-10

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Complements To Ten: 1.5

Supporting materials

 

Indicator of Progress

Success depends on students being able to give two numbers that total ten fluently. For example, given the number seven, students can say that three more are needed to make ten.

Before achieving this, students may need to count physical objects by ones to find totals that make ten. They may be able to find complements to ten (e.g. “I have 8 – think 8, 9 10 – so 2 more make ten”) but they do not know them fluently. Students may not recognise the commutativity of addition (e.g. they may not recognise that knowing 6 + 4 =10 means they know 4 + 6 =10).

Being able to fluently identify complements to ten is an important building block for computation, and helps establish ten as a unit.

 

Illustration 1: Stages in finding the total of two collections

Students who count by ones usually keep track of the count by touching the objects being counted, nodding their heads or tapping the objects to be counted as they track with their eyes. Learning to use simple strategies like this is important for them to maintain one-to-one correspondence between objects and number names.

Initially, given two separate collections of counters, and asked to find the number in each group as well as the total number, students will use three steps:

  • the first count of the top six dots (below) will be 1, 2, 3, 4, 5, 6
  • then the four dots will be counted: 1, 2, 3, and 4
  • the third count will be of the combined groups 1, 2, … , 9, 10.

Later, students replace the third count above by ‘counting on’ using the number from the earlier count (e.g. sequentially from one to ten: 1, 2, 3, 4, 5, 6 (top row of dots) 7, 8, 9, 10 (bottom row of dots).

Progressively, when asked to find 6 + 4, students who use the ‘count on’ strategy will start from the six and count on four saying: “6, 7, 8, 9, 10”.  Watch for the student who has not appreciated how ‘count on’ works and counts on four by saying only “6, 7, 8, 9”.

6 blue counters and 4 red counters

Fluency in number facts including complements to ten builds on these foundations.

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Illustration 2: Moving on from counting by ones

Tens Frames are useful to demonstrate the addition of two groups and assist students to develop fluency in number combinations. The most effective use of Tens Frames is to determine the missing addend, as the number of counters to be added is clearly defined by the number of blanks on the Tens Frame.

For example, in the Tens Frame below the student can see the six green circles and four blank spaces. A student would say: “I have six and I need four more, so six and four make ten”.

2 x 5 array with 6 counters.

 

 

Illustration 3: Complements to five

Learning complements to ten is based on knowledge of complements to 5. Many materials can be used, but fingers are excellent. They provide a visual representation and a way of checking that is always available. Moreover, students can see links between the facts (e.g. “ If I move one finger across, then instead of showing 1 + 4 = 5, I can show 2 + 3 = 5”). Soon, however, these number facts should be known without looking at the hand.

Set of hands counting to 5 using 5 fingers

Teaching Strategies

For Complements to Ten teaching strategies, see: Part 2

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