Fractions and Decimals 8_Guided and Independent Investigations
(Year 3) ACMNA058, NSW MA2 7NA
Multiplicative relationships between fractions while building a fraction wall.
GUIDED INVESTIGATION
Children learn how to investigate the concept by following teacher’s instructions until
they are ready to investigate the concept independently.
INDEPENDENT INVESTIGATION
Children investigate and explain independently over many lessons at their current level
of understanding informing both themselves and the teacher of their current level of
understanding
Resources: equal paper strips (at end of teaching plan, or children create using ruler), rulers, scissors, pencil, paper
What could we do?
What language could we use to
ask questions and explain?
What could we do?
What language could we use to
explain?
The guided and independent investigation in this teaching plan is contained within the Explicit Learning, as children question and answer one another as they create a
fraction wall.
Children investigate different ways that they can create fractions, developing their understanding of the multiplicative relationships between fractions.
For example:
 creating eighths by folding in quarters then halving, children develop understanding that an eighth is half of a quarter, and then creating eighths by folding in halves
and then quartering, children develop understanding that an eighth is quarter of a half.
 creating sixths by folding in thirds then halving, children develop understanding that a sixth is half of a third, and then creating sixths by folding in halves and then
thirding, children develop understanding that a sixth is third of a half.
 creating twelfths by folding in sixths then halving, children develop understanding that a twelfth is half of a sixth, then creating twelfths by folding in halves and then
sixthing, children develop understanding that a twelfth is sixth of a half, then creating twelfths by folding in thirds and then quartering, children develop
understanding that a twelfth is quarter of a third, then creating twelfths by folding in quarters and then thirding, children develop understanding that a twelfth is third
of a quarter.
 creating tenths by folding in fifths and then halving, children develop understanding that a tenth is half of a fifth, and then creating tenths by folding in halves and
then fifthing, children develop understanding that a tenth is fifth of a half.
The learning is in the investigative process of creating a fractions wall. The end product (creating perfectly-sized fractions) is not as important as this process. After children
have created their own fraction wall, they will use commercially available fraction walls to continue their investigations of fractions, as their creation will not be accurate
enough.
REFLECTION Before, during and after lessons,
children discuss then record responses to reflection
questions to inform themselves and the teacher of their
current level of understanding
What is a fraction?
What is the multiplicative relationship between:
• quarters and halves?
• eigthths and quarters?
• eighths and halves?
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• sixths and thirds?
• sixths and halves?
 twelfths and sixths?
• twelfths and thirds?
• twelfths and quarters?
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• twelfths and halves?
• tenths and fifths?
• tenths and halves?
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1
CONGRUENT INVESTIGATIONS
If you can’t explain it
simply, you don’t
understand it well enough.
These investigations allow children to investigate and explain the concept in new and varied situations, providing formative assessment
data for both the child and the teacher.
‘Doing’ mathematics is not enough and is not a good indicator of understanding.
Anyone who has
Investigation takes time as children develop both the capacity and meta-language to explain mathematical concepts
never made a
at their current level of understanding.
mistake has never
tried anything new.
As they investigate, allow children to experience confusion (problematic knowledge) and to make mistakes to develop
resilience and deep understanding,
Fraction wall.

In pairs, children have equal strips of paper. They create a fraction wall including halves, quarters, eighths, thirds, sixths,
twelfths, fifths and tenths, explaining the multiplicative relationships as they create each fraction in different ways (see
Guided and Independent Learning above).
Fraction circles.

In pairs, children have identical shapes of paper, for example, squares or circles. They divide the shapes into halves, quarters,
eighths, thirds, sixths, twelfths, fifths and tenths. They explain how they have created each fraction, when possible, in more
than one way. They explain the multiplicative relationships between the fractions.
Quartering.

In pairs, children create quarters of shapes and groups in 2 ways – by quartering 1, and by halving and halving again. They
explain the multiplicative relationship between 1, halves and quarters.
Eighthing.

In pairs, children create eighths of shapes and groups in 3 ways – by eighthing 1, by halving and quartering, and by quartering
and halving. They explain the multiplicative relationship between 1, halves, quarters and eighths.
Sixthing.

In pairs, children create sixths of shapes and groups in 2 ways – by sixthing 1, and by thirding and halving, and by halving and
thirding. They explain the multiplicative relationship between 1, thirds and sixths.
Twelfthing.

In pairs, children create twelfths of shapes and groups in 5 ways – by twelfthing 1, by halving and sixthing, by sixthing and
halving, by thirding and quartering, and by quartering and thirding. They explain the multiplicative relationship between 1,
twelfths, thirds, sixths, thirds and quarters.

When dividing by 2 and by 4 (and 3, 5, 6, 8, 9 and 7), children record their number sentence as a division and as a fraction,
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for example, 24 ÷ 4 = 6 and 4 of 24 = 6 (see Multiplication and Division 10 – 17).
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Investigating Multiplicative Relationships between Fractions while building a
Fraction Wall
FRACTIONS AND DECIMALS 8 Multiplicative relationships between fractions while building a fraction wall
Halves, Quarters and Eighths
Have equal strips of paper.
Keep 1 whole and place it at the top of the fraction wall as 1.
Divide one strip into halves by halving.
Explain the multiplicative relationship between halves and 1 (a half is half of 1).
Place the halves under the 1 to continue the fraction wall.
Divide one strip into quarters by halving a half.
Explain the multiplicative relationship between quarters and 1 (a quarter is a quarter of 1) and
between quarters and halves (a quarter is half of a half).
Place the quarters under the halves to continue the fraction wall.
Divide one strip into eighths by halving a quarter.
Explaining the multiplicative relationship between eighths and 1 (an eighth is an eighth of 1),
between eighths and quarters (an eight is half of a quarter) and between eighths and halves (an
eighth is a quarter of a half).
Place the eighths under the quarters to continue the fraction wall.
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Investigating Multiplicative Relationships between Fractions while building a
Fraction Wall
FRACTIONS AND DECIMALS 8 Multiplicative relationships between fractions while building a fraction wall
Third and Sixths
Have equal strips of paper.
Keep 1 whole and place it at the top of the fraction wall as 1.
Divide one strip into thirds by estimating a third, then estimating the remaining 2 thirds.
Explain the multiplicative relationship between thirds and 1 (a third is a third of 1).
Place the thirds under the 1 to continue the fraction wall.
Divide one strip into sixths by halving a third, or by thirding a half.
Explain the multiplicative relationship between sixths and 1 (a sixth is a sixth of 1), between sixths
and thirds (a sixth is half of a third) and between sixths and halves (a sixth is third of a half).
Place the sixths under the thirds to continue the fraction wall.
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Investigating Multiplicative Relationships between Fractions while building a
Fraction Wall
FRACTIONS AND DECIMALS 8 Multiplicative relationships between fractions while building a fraction wall
Thirds, Sixths and Twelfths
Have equal strips of paper.
Keep 1 whole and place it at the top of the fraction wall as 1.
Divide one strip into thirds by estimating a third, then estimating the remaining 2 thirds.
Explain the multiplicative relationship between thirds and 1 (a third is a third of 1).
Place the thirds under the 1 to continue the fraction wall.
Divide one strip into sixths by halving a third, or by thirding a half.
Explain the multiplicative relationship between sixths and 1 (a sixth is a sixth of 1), between sixths
and thirds (a sixth is half of a third) and between sixths and halves (a sixth is third of a half).
Place the sixths under the thirds to continue the fraction wall.
Divide one strip into twelfths by halving a sixth or by sixthing a half or by thirding a quarter or by
quartering a third.
Explain the multiplicative relationship between twelfths and 1 (a twelfth is a twelfth of 1),
between twelfths and sixths (a twelfth is half of a sixth), between twelfths and halves (a twelfth is
a sixth of a half), between twelfths and thirds (a twelfth is a quarter of a third) and between
twelfths and quarters (a twelfth is a third of a quarter).
Place the twelfths under the sixths to continue the fraction wall.
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Investigating Multiplicative Relationships between Fractions while building a
Fraction Wall
FRACTIONS AND DECIMALS 8 Multiplicative relationships between fractions while building a fraction wall
Fifths and Tenths
Have equal strips of paper.
Keep 1 whole and place it at the top of the fraction wall as 1.
Divide one strip into fifths by estimating a fifth at each end, then estimating the remaining 3 fifths.
Explain the multiplicative relationship between fifths and 1 (a fifth is a fifth of 1).
Place the fifths under the 1 to continue the fraction wall.
Divide one strip into tenths by halving a fifth, or by fifthing a half.
Explain the multiplicative relationship between tenths and 1 (a tenth is a tenth of 1), between
tenths and fifths (a tenth is half of a fifth) and between tenths and halves (a tenth is fifth of a half).
Place the tenths under the fifths to continue the fraction wall.
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Investigating Multiplicative Relationships between Fractions while building a
Fraction Wall
FRACTIONS AND DECIMALS 8 Multiplicative relationships between fractions while building a fraction wall
Once you have constructed your own fraction wall, explaining multiplicative relationships between
the fractions as you build it, you may use a commercially available fraction wall to continue to
investigate the multiplicative relationships.
Build the fraction wall by:


Starting with 1, identifying that all of the fractions are fractions of this sized 1.
Placing a fraction on the wall that has a multiplicative relationship to the fractions already
on the fraction wall.
For example, you may select any fraction as you first fraction
because every fraction has a multiplicative relationship to 1.
Let’s select thirds.
Explain the multiplicative relationships on the fraction wall, for
example a third is a third of 1.
Our next fraction must have a multiplicative relationship to
thirds. Let’s select sixths.
Explain the multiplicative relationships on the fraction wall, for
example, a sixth is a sixth of 1, a sixth is half of a third.
Our next fraction must have a multiplicative relationship to
thirds and / or sixths. Let’s select twelfths.
Explain the multiplicative relationships on the fraction wall, for
example, a twelfth is a twelfth of 1, a twelfth is half of a sixth, a
twelfth is quarter of a third.
Continue building the fraction wall placing fractions on the wall that have a multiplicative
relationship to fraction/s already on the wall, explaining all multiplicative relationships.
http://www.alearningplace.com.au
Investigating Multiplicative Relationships between Fractions while building a
Fraction Wall
FRACTIONS AND DECIMALS 8 Multiplicative relationships between fractions while building a fraction wall
Have identical shapes of paper, for example, squares or circles.
Divide the shapes into halves, quarters, eighths, thirds, sixths, twelfths, fifths and tenths.
Explain how you have created each fraction.
Explain the multiplicative relationships between the fractions.
http://www.alearningplace.com.au
Investigating Multiplicative Relationships between Fractions while building a
Fraction Wall
FRACTIONS AND DECIMALS 8 Multiplicative relationships between fractions while building a fraction wall
Create quarters of shapes and groups in 2 ways – by quartering 1, and by halving and halving
again.
Explain the multiplicative relationship between 1, halves and quarters.
http://www.alearningplace.com.au
Investigating Multiplicative Relationships between Fractions while building a
Fraction Wall
FRACTIONS AND DECIMALS 8 Multiplicative relationships between fractions while building a fraction wall
Create eighths of shapes and groups in 3 ways – by eighthing 1, by halving and quartering, and by
quartering and halving.
Explain the multiplicative relationship between 1, halves, quarters and eighths.
http://www.alearningplace.com.au
Investigating Multiplicative Relationships between Fractions while building a
Fraction Wall
FRACTIONS AND DECIMALS 8 Multiplicative relationships between fractions while building a fraction wall
Create sixths of shapes and groups in 2 ways – by sixthing 1, and by thirding and halving, and by
halving and thirding.
Explain the multiplicative relationship between 1, thirds and sixths.
http://www.alearningplace.com.au
Investigating Multiplicative Relationships between Fractions while building a
Fraction Wall
FRACTIONS AND DECIMALS 8 Multiplicative relationships between fractions while building a fraction wall
Create twelfths of shapes and groups in 5 ways – by twelfthing 1, by halving and sixthing, by
sixthing and halving, by thirding and quartering, and by quartering and thirding.
Explain the multiplicative relationship between 1, twelfths, thirds, sixths, thirds and quarters.
http://www.alearningplace.com.au