Decimals With A Tape

Task 204 ... Years 4 - 8

Summary

In an environment made non-threatening by encouraging estimation, provision of a measuring tape as a number line model and a calculator for checking, students explore how the four basic operations work with decimals. There is a strong emphasis on conversation and explanation.

Materials

• One 150cm measuring tape for each student
• One calculator

Content

• arithmetic, multiplication / division
• decimals, calculations
• decimals, number line
• estimating number
• fractions, whole & parts
• measurement, length
• mental arithmetic
• number line
• place value
• recording mathematics

Iceberg

A task is the tip of a learning iceberg. There is always more to a task than is recorded on the card.

The task takes the arithmetic of decimals out of the text book and into the sort of practical context that students may have already experienced in helping an adult with carpentry or sewing. The calculator and the tape help to make the experience non-threatening and the question Can you check it another way? places the experience firmly in the domain of working like a mathematician. However, there are subtleties to be discussed and explained and that is the main purpose of the task. It is not about doing lots of decimals exercises. It is about exploring and explaining. The students might like to do this one operation at a time, revisiting the task over time.

In an attempt to 'hook' students, the first part of the card asks them to estimate (not calculate as in the text book) the answers to four decimal exercises, one for each operation. Estimating then checking either confirms that the student's reasoning processes are 'on the track' or, perhaps more importantly, makes them question their reasoning. Through the story they are then immediately offered the tape as a tool to 'check it another way'.

• Note: Some versions of this card, such as the one in the photo, have a misprint in the division question. It should be 0·72 ÷ 0·03, not 0·73 ÷ 0·03.
However the tape brings its own challenge. It is numbered by the centimetres and appears as a number line of units with 'little marks' between. How do we reconcile this with the diagram showing the decimals 0·30 and 0·06? The hints are there to help and as part of the CRTRA process for exploring tasks, students have been encouraged to ask others, including the teacher. The intention is to engineer a 'point of need' teaching opportunity (if necessary).

So, the answers to this task will be the answers your students develop and record in their journal. There was a time in history when mathematicians had to work out how arithmetic operations applied to decimal numbers, so, however far the students get with this task, they are working like mathematicians. You might like to compare and contrast the teaching craft involved in this discovery approach with the text book approach of giving a rule and then practising many exercises.

When you discover interesting explanations in student journals, we would be happy to display them here for others to consider.

Note
Teachers in the Calculating Changes network can read of the success of this discovery approach used in a Year 2/3 developing their own subtraction processes with whole numbers. The story is recorded here. (Your school will need to be a Calculating Changes member.)
Note: This investigation has been included in Maths At Home. In this form it has fresh context and purpose and, in some cases, additional resources. Maths At Home activity plans encourage independent investigation through guided 'homework', or, for the teacher, can be an outline of a class investigation.
• For this specific activity click the Learners link and on that page use Ctrl F (Cmd F on Mac) to search the task name.

Whole Class Investigation

Tasks are an invitation for two students to work like a mathematician. Tasks can also be modified to become whole class investigations which model how a mathematician works.

Assuming the students have developed a concept of decimals (see Maths300 lesson below), calculators and a class set of tapes (one between two would be enough), the task suggests a structure for the whole class investigation. However, given the time it may have taken mathematicians to develop decimal calculation rules, this investigation will take more than one lesson.

Begin with estimation and spend time discussing student strategies. Check with the calculators. An important component in working with decimals is 'reality checking' (Does my answer make sense in the context of the starting numbers?) and this discussion is an important opportunity to share and develop strategies.

The calculator gives us the exact answer. It does that because the person who programmed it knew a rule (s)he could teach the machine. The purpose of our investigation is to try to work out what that rule might be.

Introduce the tape and suggest that if we think of 1m as the whole, then the markings tell us parts of the whole. Discuss and record the decimal representations of the parts, then challenge the students to use the tape to work out an addition as on the card. It doesn't really matter if they use the calculator to find out what they are working towards. Develop and continue a process of trial, check, discuss, record, improve until the class is confident that they can work out any decimal addition they choose.

Over time, continue this way with the other operations.

At this stage, Decimals With A Tape does not have a matching lesson on Maths300, but Lesson 182, Fractions to Decimals (on a rope!) uses a practical number line context to develop the concept of a decimal.

Is it in Maths With Attitude?

Maths With Attitude is a set of hands-on learning kits available from Years 3-10 which structure the use of tasks and whole class investigations into a week by week planner.

The Decimals With A Tape task is an integral part of:

• MWA Chance & Measurement Years 3 & 4
It can also be used to enhance the Years 7 & 8 Number & Computation kit.