**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 understand multiplication and division of whole numbers and who know most, if not all, of the basic multiplication facts. It is also expected that students have an existing understanding of whole number place value, to six places at least.

The following diagnostic questions indicate students’ understanding of, and ability to apply, decimal place value. 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 and strengthen their fluency and understanding. Allow access to pencil and paper but calculator use is restricted to the questions that require it. (show diagnostic questions)

The questions should be presented orally and in a written form so that the student can refer to them. Ensure students can record on a piece of paper if they need to.

*Order these measurements by their length, shortest to longest.*

2.6 metres 2.59 metres 2. 487 metres

__Signs of fluency and understanding:__

Orders the measures as 2.487, 2.59, 2.6

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

Ordering as 2.6, 2.59, 2.487 may indicate the students applies whole number thinking to decimals, such as 59 is greater than 6 so 2.59 is greater than 2.6.

Ordering as 2.487, 2.59, 2.6 may indicate the student applies reciprocal or ‘longer is smaller’ thinking, such ‘decimal values get smaller, the more digits the number has.”

__Supporting activity:__

Ordering decimals

**Here is a number line that has zero and two marked on it.**

Mark where each decimal goes on the number line and write the decimal underneath the mark.

**0.4m 1.2m 0.75 1.95 0.333 **

__Signs of fluency and understanding:__

Locates the decimals correctly with reasonable attendance to scale. For example, 0.4 should be less than one half, preferably located at four tenths. 0.75 should be half-way between 0.7 and 0.8, so three quarters of the length between 0 and 1. 0.333 should be one third of that length.

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

Inability to use the divisions on the scale (tenths) indicate the student does not know that the o you lose first decimal place after the point is the tenths place and/or has insufficient experience working with scales.

Location of 0.75 away from three-quarters position and 0.333 away from one third position indicates that the students does not know common fraction to decimal facts.

Location of 1.95 close to one suggests that the student is not familiar with equally partitioning tenths into hundredths to form more precise locations on a scale.

__Supporting activity:__

Locating decimals on a scale

**Imagine you have two lengths of rope. One length is 2.8m long and the other is 1.65m long.**

You tie the two ropes together and make a new rope that is 3.95m long.

How much length do you lose by tying the ropes together?

Explain how you worked out your answer.

__Signs of fluency and understanding:__

Connects joining with addition and finds that the total length of the two pieces of rope is given by 2.8 + 1.65 = 4.45m. Finds the difference between 3.95 and 4.45 by adding on or subtraction, such as 4.45 – 3.95 = 0.5.

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

Recognises that addition is needed to work out the total lengths but incorrectly calculates 2.8 + 1.65 = 3.73 indicates the student treats decimals as whole numbers, e.g. 8 + 65 = 73. They might also calculate 3.95 – 3.73 = 0.22 correctly but not realise that the total length must exceed the joined length.

Incorrectly adds 2.8 + 1.65 = 3.73 which might indicate they do not understand that ten tenths make one which must be ‘carried’ across the decimal point. The student may see the decimal point as a separation of whole numbers from decimals.

Subtracts 4.45 – 3.95 = 1.5 indicates that the student subtracts higher digit from lower digit without considering whether the digit is in the total length or joined length.

Inability to turn the problem into operations may indicate that the student does not associate joining with addition and difference with subtraction in a decimals context or is unable to coordinate the multiple steps needed.

__Supporting activity:__

Adding and subtracting decimals

**Imagine you start a coin trail with ten $2 coins. Each coin measures 2.65 cm across.**

How long is the trail?

Show me with your hands how long that is.

__Signs of fluency and understanding:__

Anticipates the length correctly using multiplication, i.e. 10 x 2.65 = 26.5cm. Shows clearly that 26.5 cm is slightly less than the length of a desk ruler (30cm).

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

Inability to turn the problem into an action may indicate that the student has yet to transfer multiplication with whole numbers to similar contexts with decimals.

Incorrectly multiplying 10 x 2.65 = 2.650 or 10 x 2.65 = 20.650 may indicate that the student relies on rules like ‘add a zero’ to multiply by ten.

Repeated addition of 26.5 + 26.5 = 53.0, 53 + 26.5 = 79.5, etc. indicates that the student has yet to transfer multiplication as repeated addition to decimal situations.

__Supporting activity:__

Multiplying decimals by 10 or 100

#### Teaching activities