**New Zealand Curriculum:** Level 4

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

**Target students**

These activities are intended for students who understand simple fractions, know most basic multiplication and division facts, and apply multiplicative thinking to whole numbers. By the end of Level 3 students are expected to be applying multiplication and division to different contexts across all strands of mathematics and statistics.

The following diagnostic questions indicate students’ understanding of, and ability to use, percentages as a representation of part whole relationships. The questions are 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 or strengthen fluency and understanding. Allow access to pencil and paper but not to a calculator unless it is stated in the question. (show diagnostic questions)

The questions should be presented orally and in a written form so that the student can refer to them. The questions have been posed using a basketball context but can be changed to other contexts that are engaging to your students.

**You take eight basketball shots at goal. You get four shots in. What is your shooting percentage?**

__Signs of fluency and understanding:__

Recognises that four out of eight equals one half. Knows that the percentage for one half equals 50%.

__What to notice if they don’t solve the problem fluently:__

Recognises that four out of eight equals one half but does not know that one half equals 50%. This indicates a need for the student to build up their knowledge of percentages that are equivalent to simple fractions.

Unable to rename four out of eight as one half. This indicates that student needs to work on equivalent fractions.

__Supporting activity:__

Simple fractions as percentages

*You take 12 basketball shots at goal. You get nine shots in. What is your shooting percentage?*

__Signs of fluency and understanding:__

Recognises that nine out of 12 equals three-quarters. Knows that the percentage for three quarters equals 75% or works 75% out using knowledge of ½ = 50 or ¼ = 25%.

__What to notice if they don’t solve the problem fluently:__

Attempts to scale 12 up to 100 but unable to find the multiplier. This indicates the students understands the multiplicative relationship in proportions but needs to work on scaling. There is an intervention about multiplicative scaling. (Link?)

Recognises that nine out of 12 equals three quarters but does not know that three quarters equals 75%. This indicates a need for the student to build up their knowledge of percentages that are equivalent to simple fractions.

Unable to rename nine out of 12 as three quarters. This indicates that the student needs to work on equivalent fractions.

__Supporting activity:__

More complex fractions as percentages

*You take 20 basketball shots at goal. You get 14 shots in. What is your shooting percentage?*

__Signs of fluency and understanding:__

Recognises that 20 can be scaled up to 100 using five as the multiplier. Calculates 5 x 14 = 70% probably with support from written recording.

Renames 14 out of 20 as equivalent to 7 out of ten. Scales ten to 100 by multiplying by ten, then calculates 10 x 7 = 70% probably with support from written recording.

__What to notice if they don’t solve the problem fluently:__

Attempts to scale 20 up to 100 but unable to find the multiplier. This indicates the students understands the multiplicative relationship in proportions but needs to work on scaling. There is an intervention about multiplicative scaling. (Link?)

Attempts to change 14 out of 20 to a trusted equivalent fraction but either cannot rename 14/20 or does not recognise 7/10 as a familiar percentage. The student may need to work on equivalent fractions or expressing common fractions as percentages.

__Supporting activity:__

Scaling up to 100%

**You take 44 basketball shots and score with 29 shots.**

*Is your shooting percentage closer to 46%, 56%, 66% or 76%?*

*Use a calculator to check if you want.*

*Explain your answer.*

__Signs of fluency and understanding:__

Recognises that 44 can be scaled to 100 by multiplying by a bit more than two. Doubles 29 to get 58 and choses 66% since the percentage must be greater than 56%.

Rounds 44 to 45 and 29 to 30 and creates the fraction 30/45 which they recognise as equivalent to 2/3. Knows that two thirds equal 66.66…%.

Uses the calculator to find 29 ÷ 44 = 0.659 which they know equals 65.9%. Chooses 66%.

__What to notice if they don’t solve the problem fluently:__

Recognises that 29 out of 44 is greater than one half so eliminates 46%. Speculates that the percentage could be 56%, 66% or 76% but with no certainty about why. The indicates the students has one half as a benchmark percentage but needs to learn more fraction to percentage knowledge.

Tries various calculations to see if the result is reasonable without a clear idea of what the numbers mean. For example, they might find 29 ÷ 44 = 0.659 with a calculator but not be able to relate 0.659 to a percentage. This indicates that the student needs support to structure the problem and to develop a sensible calculation.

Reverses the calculations to find the answer by trial and error. Uses the calculator to find 46% x 44 = 20.24, 56% x 44 = 24.64, 66% x 44 = 28.64, and 76% x 44 = 32.64. Concludes that 66% is closest. This indicates the student has a valid way to choose from the options available but needs support to develop estimation strategies.

__Supporting activity:__

Rounding to estimate percentages

*You take 25 basketball shots at goal. 48% of your shots are goals.*

**How many shots are goals?**

__Signs of fluency and understanding:__

Recognises that 25 is one quarter of 100 so divides 48 by four to get 12 shots. Checks to see that 12 out of 25 equals 48% by scaling back to 100.

Estimates that 48% is slightly less than one half so anticipates that 12 shots are goals, since 12 is slightly less than one half of 25. Checks to see that 12 out of 25 equals 48% by multiplying both 12 and 25 by four. May use supportive recording such as 12/25 = 48/100 = 48%.

__What to notice if they don’t solve the problem fluently:__

Attempts various numbers of goals to find a number that works. May scale the fraction to a base of 100 using multiplication by four. The process involves trial and error. This indicates the student understands that percentages are equivalent fractions and needs support in renaming equivalent fractions.

Confuses the numbers in the problem and may try to work out 25 out of 48 as a percentage or perform other operations with the numbers, unaware of the impossibility of their answer. The student needs support in recognizing the part whole fraction involved and/or identifying the unknown in the situation, i.e. □/25 = 48/100.

__Supporting activity:__

Finding the missing part

*You score with 70% of your basketball shots. In total 28 shots go in.*

*How many shots do you take?*

*Use a calculator if you need to.*

__Signs of fluency and understanding:__

Recognises that if 70% of the shots equals 28 then four equates to 10% (dividing by seven). Scales 10% to 100% and four to 40 by multiplying by ten. May use written recording or the calculator to manage the solution.

Divides 28 by 70 using the calculator to get 0.4. Knows that this is the unit rate of 0.4 shots: 1%. Multiplies 0.4 by 100 to get 40 shots.

Use a proportional reasoning algorithm to find 70 x s = 28 x 100, where s is the unknown number of shots. Applies algebra to find s = 28 x 100 ÷ 70 = 40 shots. Question the student about the meaning of numbers in their algorithm to check they know what the procedure is doing.

__What to notice if they don’t solve the problem fluently:__

Attempts various numbers of shots to find a number that works, using the calculator to manage the operations. May use sensible estimation such as recognise that the number must be about 40 since 70% is more than 50% and 28 is more than 20. The student needs to work on efficient methods using the numbers provided.

Tries various calculations to see if the result is reasonable. For example, the student might multiply 70 x 28 = 1960 with a calculator and realise that 1960 shots is an unrealistic answer, then try 70 ÷ 28 = 2.5 or 28 ÷ 70 = 0.4 but not know what the output means. This indicates that the student needs support to structure the problem situation and to create a workable algorithm.

__Supporting activity:__

Finding the whole