Math // Mathematical Thinking

Finding efficient methods

Notice when a calculation or pattern repeats and use this to count more efficiently or predict results

Ages
5 to 6
Type
Thinking skill
Builds on
2 topics
Leads to
1 topic

Quick check

When the learner is counting or doing repeated additions, have they started to notice the pattern — like "5, 10, 15, 20…" — and used it to predict the next number without counting one by one?

Ask this to see whether the topic is secure.

Topic // Mastery

What mastery looks like

What a learner can do once this topic is secure.
  • Notice that skip counting by 2s follows a repeating odd/even pattern
  • Recognise that adding 1 to any number always gives the next counting number
  • Use a repeated pattern (e.g. +10 on a hundred chart always moves down one row) to predict answers

Topic // Map

Where this fits

What to learn first and what this topic opens up, with the reason for each link. Required links come first.
  • RequiredMust come first
  • HelpfulHelps, but not essential

Before this

2 topics
  • One More Each TimeHelpful

    Counting & Cardinality // Ages 4 to 6

    The +1 pattern in counting is the earliest repeated reasoning

  • Counting in 2sHelpful

    Counting & Cardinality // Ages 5 to 7

    Skip counting exercises recognising repeating calculation patterns

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Finding efficient methods

Ages 5 to 6 // Thinking skill

Leads to

1 topic
  • Generalising PatternsRequired

    Mathematical Thinking // Ages 6 to 7

    Age 6-7 generalising from repeated reasoning builds on age 5-6 noticing repeated patterns

Topic // Research-informed

Teaching notes

From the Marble Skill Taxonomy. These notes have not been validated in classrooms, so treat them as a starting point for your own judgement.

The approach

For ages 5-6, this is the seed of noticing that a calculation or move REPEATS and using that to count more efficiently or predict, rather than starting afresh each time. Children learn that adding 1 always gives the next counting number, that skip-counting in 2s from 2 lands on every second number (the even numbers, skipping the odd ones between), and that on a standard hundred square with 1 in the top row, +10 moves you straight down one row whenever there is a row below. The teaching job is to prompt the child to expect the repeat and test it ('will it happen again?'), not to hand over the rule. The most developed activity has the child predict the next repeat and then see it happen; recognition checks and highlighted references play supporting roles.

Common misconceptions

  • Counts in ones every time instead of using the repeated +10 move, treating each sum as brand new
  • Sees a pattern work once but does not expect it to continue, so won't use it to predict
  • Loses the skip-count structure partway (2, 4, 6, 8, 9) by reverting to counting in ones
  • Thinks 'the next number' and 'one more' are unrelated ideas rather than the same repeated +1

Useful representations

  • Hundred square (rows make +10 a repeated downward move)
  • Number track / number line for +1 and skip-count steps
  • Bead string or steps grouped in 2s, 5s, 10s
  • Ten-frames showing the steady 'one more' growth

Teaching practices

  • Use the hundred square so +10 is a physical, repeatable move (straight down a row) the child can predict before checking
  • Prompt prediction: 'what will happen next? will it keep happening?' so the child forms and tests an expectation rather than counting afresh
  • Skip-count with a visible rhythm (steps on a number track, claps) so the repeating structure is heard and seen, and highlight the odd/even alternation
  • Deliberately extend a pattern one more step together and confirm the rule held, building the habit of using a noticed repeat to predict

Activities to try

  1. 1

    Predict-the-jump hundred square

    Why it works: On this hundred square (1 in the top row), adding ten moves you straight down one row each time there is a row below, so once you notice the repeat you can predict the next result instead of counting ten squares each time.

    1. Point to 24 on a hundred square.
    2. Before moving anything, say out loud where you think 24 + 10 will land.
    3. Move a counter straight down one row to land on 34 and check your guess.
    4. Do it again: predict 34 + 10, then land on 44; then predict 44 + 10 and land on 54.
    5. After a few jumps, try to say where the next +10 will land without looking closely — notice you no longer need to recount.
  2. 2

    Skip-count rhythm track

    Why it works: Skip-counting in 2s follows a repeating, predictable rhythm with a steady even pattern, so you can anticipate the next number instead of counting in ones.

    1. Count in 2s along a number track or line of steps: 2, 4, 6, 8, ...
    2. Add a steady beat (clap, drum, or stepping) on each number so the repeating rhythm is heard as well as seen.
    3. After a few steps, try to call the next number before you reach it.
    4. Notice which numbers you land on and which you skip; talk about the odd/even alternation.
    5. Answer: 'Is 7 in your counting-by-2s? Why not?' and 'What comes after 2, 4, 6, 8 — how did you know without counting all of them?'
  3. 3

    Distinguish the +1 move from the +10 move

    Why it works: Different repeated moves have different fixed effects: +1 gives the next counting number, +10 drops down a row. 'One more' and 'the next number' are the same idea, not unrelated.

    1. On the hundred square, show where adding 1 takes you from a number — usually one place to the right, to the next counting number (at the end of a row it continues at the start of the next row).
    2. Then show where adding 10 takes you (straight down one row).
    3. Optionally, use a ten-frame: add one counter and see the total become the next number — the same steady 'one more' step.
    4. Say in your own words how the two moves are different.
    5. Connect 'the next number' and 'one more': confirm these are the same repeated +1 move.

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