Math // Mathematical Thinking

Shape patterns

Look for and use mathematical structure: apply properties of operations, place-value patterns, and relationships between shapes to solve problems efficiently

Ages
6 to 7
Type
Thinking skill
Builds on
3 topics
Leads to
1 topic

Quick check

When the learner notices a pattern in their maths work — like how multiplying by 10 always adds a zero — do they use that pattern to solve similar problems more quickly?

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.
  • Use the commutative property deliberately (e.g. reorder 3 + 9 as 9 + 3 to count on from the larger number)
  • Use place-value structure to add or subtract tens (e.g. 47 + 10 = 57 because only the tens digit changes)
  • Recognise structural similarities between shapes (e.g. rectangles and squares both have 4 sides and 4 right angles)

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

3 topics
  • Spotting mathematical patternsRequired

    Mathematical Thinking // Ages 5 to 6

    Age 6-7 using structure deliberately builds on age 5-6 noticing simple patterns and structure

  • Addition in any orderHelpful

    Addition & Subtraction // Ages 6 to 7

    Deliberately applying commutativity to make addition easier exercises using structure

  • The two digits of a two-digit numberHelpful

    Number Representation & Place Value // Ages 6 to 7

    Using place-value structure (tens and ones) to solve problems efficiently

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Shape patterns

Ages 6 to 7 // Thinking skill

Leads to

1 topic

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 6-7 year-olds, using structure shifts from noticing a pattern to deliberately exploiting it: reordering 3+9 as 9+3 to count on from the larger addend, using place value so 47+10 changes only the tens digit, and seeing a square as a special rectangle because both have four sides and four right angles. The core teaching move is to make the efficient and inefficient approaches visible side by side so the child feels why structure pays off — fewer counts, a digit that predictably changes, a shared property — rather than being told a rule, and then justifies why the shortcut is allowed.

Common misconceptions

  • Always counts on from the first number written, not the larger, because order is treated as fixed
  • Adds 10 by counting in ones or by changing the ones digit (47+10 = 48) rather than seeing the tens digit change
  • Believes a square is not a rectangle because they look different / have different names (prototype and mutually-exclusive categories)
  • Applies commutativity to subtraction (34-10 handled like 10-34)

Useful representations

  • Number line for count-on-from-larger (comparing hop counts)
  • Place-value / arrow cards and a tens-and-ones column machine
  • Venn-style overlapping shape hoops (four sides; four right angles)
  • Base-ten blocks showing +10 as one added ten-rod

Teaching practices

  • Contrast counting on from the smaller vs the larger addend on the same number line so the child sees three hops beats nine hops and connects that to 'order doesn't change the total'
  • Use a place-value machine / arrow cards so adding a ten visibly ticks only the tens column, and ask the child to predict which digit will change BEFORE it does
  • Sort and overlap shape sets (all four-sided, all with right angles) so squares fall inside rectangles, surfacing the structural relationship rather than treating them as separate names
  • Ask 'is there a quicker way? why does it work?' so the structural strategy is justified, not just performed

Activities to try

  1. 1

    Choose your start: count on from the larger addend

    Why it works: Starting the count-on from the larger addend reaches the same total with far less work, and this is allowed because addition can be done in any order (commutativity).

    1. For 3 + 9, first count on starting from 3 — draw or hop all nine moves across a number line and watch it crawl from 3 to 12.
    2. Now start from 9 instead and count on three — land on 12 in just three hops.
    3. Say what is the same (both reach 12) and what is different (far fewer hops when you start from the larger number).
    4. Answer: why is swapping the order allowed here? Connect to 'the order you add in doesn't change the total.'
    5. Try the same choice for 2 + 8 and 8 + 2: decide which number to start from and explain your choice before you count.
  2. 2

    Adding ten: predict which digit moves

    Why it works: For two-digit numbers that do not cross into the hundreds, adding ten changes only the tens digit (it goes up by one) because ten is one whole ten; place value, not counting in ones, tells you which column moves.

    1. Before adding, predict: when I add 10 to 47, which digit will change and which will stay the same?
    2. Add one ten-rod to a base-ten representation of 47, or move the tens on a place-value / arrow-card setup, and check: the tens change (4→5) and the ones stay the same, giving 57.
    3. Repeat the predict-then-check for 63 + 10, saying which digit changes and why before you do it.
    4. Explain in your own words why adding ten leaves the ones digit alone.
  3. 3

    Is a square a kind of rectangle?

    Why it works: A square is a special rectangle because it has the defining properties of a rectangle (four sides, four right angles); classification should be based on properties, not names or appearance.

    1. Sort a mixed set of shapes into groups by property: all shapes with four sides, and all shapes with four right angles.
    2. Overlap the two groups (e.g. two overlapping hoops): the overlap holds every shape with four sides and four right angles, which is the rectangle set. See that squares land there alongside the other rectangles, because they have both properties.
    3. Answer: what is the same about a square and a rectangle, and what is different?
    4. State why a square counts as a special rectangle, pointing to the shared properties rather than the name or how it looks.

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