New Zealand Curriculum: Level 2
Learning Progressions Framework: Multiplicative Thinking: Signpost 4 to Signpost 5
Target students
These activities are intended for students with some previous experience with reflection symmetry, who are now ready to build on their knowledge of additive strategies to solve multiplication and division problems. They may have some simple multiplication fact knowledge and be able to skip count in twos, fives and tens.
The following diagnostic questions indicate students’ ability to order simple fractions by size and represent unit fractions, and simple non-unit fractions, on a number line between zero and one. A number line is used to demonstrate students’ understanding of the size of fractions in relation to other numbers. The questions are given in order of complexity. If the student answers a question confidently and with understanding proceed to the next question. If not, then use the supporting activities to build and strengthen their fluency and understanding. Allow access to pencil and paper. (show diagnostic questions)
The questions should be presented orally and in a written form so that the student can refer to them. The questions and lessons presented below are context-neutral and illustrate the mathematical ideas without the ‘noise’ of context. To increase motivation you might frame the learning situations in contexts that appeal to the interests of your students. Fractions apply to situations where a total amount is partitioned into equal smaller amounts. Students might relate to forming equal parts, and naming those parts, in contexts like completed parts of a linear journey, like a bicycle ride or tramp, or other linear contexts like sharing fruit loops, sharing lengths of fibre, or fabric, or sharing eels among seals or worms among birds.
- Point to the strip that has one quarter shaded. Explain why you chose that strip.
Signs of fluency and understanding:
Chooses the third strip down and explains that there are four parts, so the strip is in quarters.
If provoked by the question, “The second strip has four parts. Is one quarter shaded in that strip?” the student identifies that the parts in the second strip are unequal in size and explains that the parts must be of equal length.
What to notice if your student does not solve the problem fluently:
Chooses the second strip and third strip because they both have four parts. This may indicate that the student does not understand that the parts must be equal.
Chooses one of the strips that has three or five parts. This may indicate that the student does not know that quarters are one of four equal parts.
Supporting activity:
Equal parts
- Match these cards. Each fraction has number, word, and picture cards.
Provide the student with cards made from the last page of the copymaster.
Read the word cards to the student if necessary.
Signs of fluency and understanding:
Fluently matches words, pictures, and symbols. Explains the matching of each trio.
What to notice if your student does not solve the problem fluently:
Inconsistent placement of cards indicates that the student needs experience with naming fractions with words and symbols.
Words and symbols are matched correctly but the picture cards are inconsistently placed. This may indicate that the student is unaware of the meaning of ‘th’ words, especially halves, thirds, quarters, and fifths that have special names.
Correct placement of words and pictures but not symbols. This may indicate that the student does not understand the meaning of the numerator and denominator in a fraction symbol.
Supporting activity:
Matching unit-fraction words, symbols, and amounts
- Here are three strips of paper. (Provide the student with three strips of equal length.)
Fold one strip into quarters.
Fold the second strip into eighths.
Fold the last strip into thirds.
Signs of fluency and understanding:
Uses repeated halving to create four and eight equal parts.
Uses trial and improvement to make thirds that are equal. This involves visualising an estimate of the length of one third and ‘iterating’ that unit three times to see it fits the strip.
What to notice if your student does not solve the problem fluently:
Folds the strips into unequal parts without any use of symmetry or iteration. This indicates the student needs more experience with equi-partitioning of lengths and other shapes, especially using symmetrical partitioning.
Folds the strips into the correct number of equal parts but removes the ‘extra’ at the end of the strip. This suggests that the student does not understand that the whole strip must be exhausted (completely used).
Supporting activity:
Folding fractions
- Which of these strips has three fifths shaded? Explain why you chose that strip.
Signs of fluency and understanding:
Chooses the correct strip and explains that the strip is divided into five equal parts called fifths. Three of the equal parts are shaded.
What to notice if your student does not solve the problem fluently:
Chooses the 5/8 strip, attending to the connection to three and five, in three fifths or chooses the 2/3 strip, attending to the three in three fifths. In both instances the student needs more opportunities to make sense of the numerator and denominator in fractions, using both written symbolic forms and words.
Chooses the strip with unequal parts.
The student correctly identifies three parts out of five but does not recognise that the parts are equal. This may indicate misunderstanding of the concept that fractions involve equi-partitioning.
Supporting activity:
Non-unit fractions
- Here is a number line showing zero, one, and one half.
Write these fractions in the correct places. (Say the fractions to the student one at a time)
One quarter, Three quarters, One third, Seven eighths
Explain why you put the fractions in the places you did.
Signs of fluency and understanding:
Locates all fractions in approximately the correct position in relation to zero, one, and one half.
Uses halving to locate one quarter and three quarters correctly.
Uses measurement (iteration) to locate one third in a place where three units make one.
Locates the size of one eighth by halving one quarter and measures seven spaces to locate seven eighths, or adds one eighth onto three quarters, or removes one eighth from one.
What to notice if your student does not solve the problem fluently:
Locates equivalent fractions with reference to numbers in the fractions without the use of both the numerator and the denominator (for example, places one third as less than one quarter). This may indicate that the student needs experience with relating fraction symbols to quantities. The student needs to learn about the use of both numerator and denominator in determining the size of a fraction.
Unable to locate the non-unit fractions, three quarters and seven eighths. This may indicate that the student does not understand that non-unit fractions are composed of unit fractions. For example, three quarters is composed of one quarter plus one quarter plus one quarter, 3/4 = 1/4 + 1/4 + 1/4.
Locates one third in a place where three measures of it do not make one. They may place it to be greater than one half (attending to the denominator of three). This may indicate the student needs experience with locating unit fractions on a number line and with understanding the meaning of the denominator, to recognise that the larger the denominator is, the smaller the equal parts.
Supporting activity:
Fractions on a number line
Teaching activities
