**New Zealand Curriculum:** Level 3-4

**Learning Progression Frameworks: **Multiplicative thinking, Signpost 6 to Signpost 7

**Target students**

These activities are intended for students who have some previous experience with treating fractions as numbers. This should include a well-developed understanding of the meaning of the numerator and denominator. Students should also have well-developed knowledge of multiplication facts.

The following diagnostic questions indicate students’ understanding of, and ability to represent the part-whole relationships in one or more ratios as fractions. A connected cube model is used to gauge students’ ability to recognise the fractions created within ratios. 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, 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.

**Here is a mixture of blueberry and apple juice. The blue represent units of blueberry juice and the green represent units of apple juice. (Ratio 1:3)**

**What fraction of the mixture is blueberry?**

**What fraction is apple?**

__Signs of fluency and understanding:__

Identifies that there are four units in total, so each cylinder is equivalent to one quarter of the mixture. Gives the fraction of blueberry as one quarter and the fraction of apple as three quarters.

__What to notice if your student does not solve the problem fluently:__

Expresses the part-to-part relationship for blue to green as the fraction one third (i.e. one to three). This indicates that the student needs further experience identifying the whole and expressing each part as a fraction of that whole, in a range of contexts.

Notices that the number of blueberry units is one and the number of apple units is three but does not express the part-to-whole relationships as fractions. This may indicate that the student has difficulty naming a fraction of a set.

__Supporting activities:__

Identifying fractions in simple ratios

**What fraction of this mixture is blueberry?**

**What fraction is apple?**

**Can you give equivalent fractions for blueberry and apple?**

__Signs of fluency and understanding:__

Identifies that the mixture has ten units. Recognises that four units are blueberry, and six units are apple, gives identifies the fractions as four tenths blueberry and six tenths apple. Works out that these fractions can be simplified to the equivalent fractions two fifths blueberry and three fifths apple.

__What to notice if your student does not solve the problem fluently:__

Names the fractions as four tenths and six tenths but cannot name the equivalent fractions. This indicates that the student needs more experience working with equivalent fractions in sets contexts.

Names the part-to-part relationships as fractions, for example four sixths (as in four to six). This indicates that the student needs support with recognising and naming part-whole relationships.

Identifies the numbers of blueberry and apple units but cannot name the part-whole fractions. This indicates that the student needs further opportunities to identify fractions in sets contexts.

__Supporting activity:__

Equivalent fractions in simple ratios

**Which mixture of Blapple juice tastes most strongly of blueberry? (ratios 5:3 for Mixture A and 4:4 for Mixture B)**

**Explain how you know.**

__Signs of fluency and understanding:__

Recognises that mixture A has more blue units, although the total number of units in each whole is the same (8 units). Therefore, mixture A has the strongest blueberry flavour. Explains that the fraction of blueberry in mixture A is five eighths which is larger than the fraction of blueberry in mixture B (four eighths or one half).

__What to notice if your student does not solve the problem fluently:__

Believes that mixture A has a stronger blueberry flavour than mixture B based on the numbers of units that are blueberry, without making any reference to fractions. This may suggest that the student is relying on whole numbers. This may interfere with their comparison of ratios that are more complicated.

__Supporting activity:__

Comparing ratios with a common whole

**Which mixture of Blapple juice tastes most strongly of blueberry? (ratios 4:5 for Mixture A and 4:6 for Mixture B)**

**Explain how you know.**

__Signs of fluency and understanding:__

Recognises that the number of blueberry units is the same in both mixtures (4 units). Therefore, mixture A has the strongest blueberry flavour because there are fewer units of apple to ‘balance’ the blueberry. Explains that the fraction of blueberry in mixture A is four ninths which is a larger number than four tenths, the fraction of blueberry in mixture B.

__What to notice if your student does not solve the problem fluently:__

Believes that both mixtures have the same flavour of blueberry because they both have four units of blueberry. This may suggest that the student is relying on whole numbers and not thinking of the part-whole relations as fractions. The student may need practical experience with mixing ratios to support them in recognising that the attribute (strength of flavour) reflects the relationship between the parts and the whole.

Believes that mixture B is not as strong in blueberry flavour because it has six units of apple while mixture A has only five units of apple. This may also indicate that the student is relying on whole numbers.

__Supporting activity:__

Comparing ratios with a common part

**Which mixture of Blapple juice tastes most strongly of blueberry? (ratios 3:5 for Mixture A and 2:3 for Mixture B)**

**Explain how you know.**

__Signs of fluency and understanding:__

Uses fractions to measure the strength of blueberry flavour in both mixtures. Calculates these fractions as three eighths for mixture A and two fifths for mixture B. Compares the fractions as numbers and concludes that mixture B has a stronger blueberry taste because two fifths is a greater number than three eighths. The student may also use decimals or percentages to compare the fractions, e.g. 2/5 = 40% and 3/8 = 37.5%.

__What to notice if your student does not solve the problem fluently:__

Believes that mixture A, with three units of bluberry, has the strongest blueberry flavour since it has more units of blueberry compared to mixture B (with only two units). This indicates that the student relies on whole number thinking and compares ratios using only one of the measures. The student needs support in finding the part-whole fractions in ratios.

Attempts to copy the ratios until a common part or whole is found. For example, replicates 2:3 to create 4:6 then 6:9 (mixture B), and 3:5 to create 6:10 (mixture A). Concludes that mixture B is strongest since the number of parts of blueberry is the same, but mixture B has only nine parts apple, compared to ten parts for mixture A. This is a legitimate strategy but indicates that the student needs to develop a more efficient strategy that relies more on comparison of fractions.

__Supporting activity:__

Comparing ratios

#### Teaching activities