Indicator of progress
At this level students appreciate that we can represent a number by a point on a number line, in contrast to representing numbers by sets of objects. This conceptual leap allows us to mark the number 0 as the point at the left end of a horizontal number line, or at the base of a vertical number line.
Identifying the length of 1 unit (the distance between the points labelled 0 and 1) allows us to mark points for all the whole numbers by repeatedly measuring this unit distance from left to right across a horizontal number line (or up a vertical number line), and also to mark fractions and decimals between the whole numbers.
Number lines arise from measuring, rather than counting. Before achieving this, students will be able to use a set of objects (such as labelled unifix blocks) organised in a ‘counting line’ to represent numbers and to assist in performing various operations (for example 9 + 5 = 14, 18 – 5 = 13). While such a ‘counting line’ shares some properties with a number line (specifically, the numbers are in order and increase to the right) it is important to note the differences. Critically, when the numbers are labelling objects (not points), there is no object labelled 0; nor will there be objects labelled with fractions or decimals.
Later, students will be able to move flexibly between representing numbers as points on a number line and as distances along a number line. This ability is required to fully model addition, subtraction, repeated addition and repeated subtraction.
Read more about the Importance of number lines as useful representations of numbers, as well as important mathematical representations in their own right: Number Lines with Whole Numbers (PDF - 157Kb).
Illustration 1: Counting produces ‘counting lines’ not number lines
In the diagram below, we can see a set of blocks, alternately coloured blue and green, with number labels. These have been placed in a ‘counting line’. Note that the first object is labelled with 1 which indicates that we are counting the blocks. We can use this ordered sequence of number labels to help with some calculations; for example, we can see 9 + 5 = 14. Starting on the blue 9 block, we can count 5 more to get to the green 14 block.
Calculations like 5 – 5 = 0 cannot be shown in a similar way as there is no 0 on this set of counted blocks. We cannot simply insert another block at the start and label it 0 because in counting, the first object is 1, not 0.
Additionally this model cannot show fractions or decimals, (or negative numbers). Where would we label 81/2 on these objects? If 8 refers to a complete object and 9 to the next, there is nothing that can be labelled 81/2
NOTE: The process of counting objects produces counting numbers. Measuring produces other types of numbers. Number lines come from measurement.
Illustration 2: Measuring produces number lines
The diagram below shows a (horizontal) number line with 0 at the starting point on the left. Here the points are labelled with their distance from the origin (0). The number 9 is represented by a point, and not an object or a region, between 8 and 10. As well as showing the same calculation as above (9 + 5 = 14) by starting at the point labelled 9 and moving a distance of 5 to the point labelled 14, a number line can also show clearly 5 – 5 = 0.
Until students need to deal with numbers that are not counting numbers, the ‘counting lines’ of objects are adequate. When they manipulate and compute with fractions, they need to use the more sophisticated ‘number lines’ based on measurement of distances in an agreed unit. The fractions shown in the number line below are positioned at points halfway, two thirds, or one quarter the distance between the whole numbers.
Illustration 3: Links to the Mathematics Online Interview
Examples of the types of tasks that would be illustrative of this understanding from the Mathematics Online Interview:
- Question 17 – Interpreting the Number Line
Class discussion needs to constantly highlight that we are labelling points on the number line rather than labelling objects. Measurement contexts provide excellent opportunities for using number lines marked with 0. As students meet new numbers (more fractions, decimals, negative integers, negative fractions and negative decimals) they need to revisit and extend their number lines to accommodate the new numbers.
These activities extend the ideas in the indicator of progress Early fraction ideas with models
Activity 1: Thermometers are vertical number lines and provide a familiar context to introduce number lines.
Activity 2: Create a number line on the white board is a class demonstration of how to draw a number line using the paper strips that were folded in Early fraction ideas with models
Activity 3: Create a number line on the wall is a variation of Activity 2 using a string pinned to the wall.
Activity 4: Consolidating links between representations with a think board extends an activity in Early fraction ideas with models to include number lines.
Activity 5: Fractions in cooking provides a context for representing multiples of 1/4 (cup) to a vertical number line, again emphasising the need for a measurement context.
Activity 1: Thermometers
Provide a large thermometer with a clearly labelled scale for students to examine. And draw a vertical number line on the board to illustrate how to read the thermometer.
Features to discuss with students:
- As the temperature increases (decreases), the mercury ‘goes up’ (‘goes down'). Vertical number lines match the common intuitive idea that larger numbers are higher than smaller numbers (more, up, higher) compared with horizontal number lines (which require a convention more is to the right).
- When the temperature is 20° the top of the mercury will be in line with the point on the side labelled 20 (focus on points on a line).
- The numbers on the thermometer are evenly spaced. The spacing between 10° and 20° will be the same as from 20° and 30°.
- The thermometer is an excellent context for the appearance of the number 0 and, later, for the study of negative numbers.
- Temperatures between labelled points can be determined by carefully reading the scale. For example, if there are 4 markers between 20° and 30°, this would indicate 5 intervals. Hence the marked points would be 20°, 22°, 24°, 26°, 28° and 30°. It is important to constantly remind students that we need to focus on the number of intervals not the number of markers.
- Small movements in the level of the mercury are related to small changes in temperature. The level of mercury moves slowly up and down in a smooth manner, and we can estimate the temperature when it is between markers.
- Ask students to draw their own thermometer and mark in various temperatures including numbers which are not whole numbers (i.e. the temperature is between 16° and 17°).
Ask students to draw a portion of a thermometer which does not include 0 (for example, from 20° to 40°) and ask them to mark in various temperatures.
Activity 2: Create a number line on the whiteboard
Draw a long horizontal line on the board and use the paper strips that were folded in Activity 3 of Early fraction ideas with models to measure and mark the line, as described in detail below. Involve students at each step.
A recommended variation is to draw a vertical number line.
Activity 3: Create a number line on the wall
Activity 4: Consolidating links between representations with a think board
In Activity 5 in Early fraction ideas with models a think board with three sections was used to link representations of fractions. Revisit this activity and include number lines in the Draw it sector. The following diagram shows how the point 3/4 can be shown on a number line.
Activity 5: Fractions in cooking
The diagram below shows a vertical number line starting at 0, which is based on the markings on the side of a measuring jug found in most kitchens. If we use a scoop to measure 1/2 cup of flour and pour it into the jug, the top of the flour will be in line with the point marked 1/2 cup (as shown in the first diagram). If we measure another 1/2 cup of flour and pour it into the jug the top of the flour will be in line with the point marked 1 cup, as in the second diagram. If we measure 1/4 cup of flour and pour this into the jug, and then repeat this a second and then third time, the top of the flour would then be in line with the point marked 13/4 cups (as in the third diagram).
Use this context to discuss the following important features:
- Emphasising the link between numbers (both whole cups as well as fractions of cups) to points on the number line
- If we remove flour from the jug, the top of the flour will drop until it reaches 0 (the bottom of the jug)
- We can add (or remove) small amounts of flour from the jug and the top of the flour will be slightly above or slightly below the previous mark. The flour in the jug can be more than 11/2 cups and less than 13/4 cups. This leads into the notion of the density of the number line: given any two numbers, no matter how close they appear to be, another number exists between the two.
- Addition and subtraction of fractions (being treated at Level 3) usually models adding ingredients in a measuring cup e.g. 1/4 cup flour + 1/2 cup flour = 3/4 cup flour but not for 1/2 cup nuts + 1/4 water (as the water fills the gaps between the nuts)
- It is important for mathematics (but not for cooking) to extend these models beyond halves and quarters to thirds and other fractions.