About Placeholders: 3.25

Students need to be able to use a 'placeholder' in a number sentence. Various symbols can be used for the unknown number(s), for example a box (□), a question mark (?) or a letter such as m.

It is common to use a box to hold the place of the missing number for various reasons:

• it looks like a card is being held in front of a number, so it is hiding the number written behind
• the box can then be 'filled in' with an appropriate number more easily than some other shapes, such as triangles and stars
• a box is quicker to draw than many other shapes, such as a star.

There are some difficulties, however, with using a box as a placeholder in a number sentence:

• machine-marked tests often require exactly one digit to be written in a box, so a two – digit number like 26 requires two boxes, as does a fraction like 3/5, and 158 requires three boxes (this is in stark contrast to the idea that the box represents a number, not a digit)
• over-generalised concept: How many answers are there for this number sentence? □ + □ = 10. If the box stands for a particular number, then there is only one possible answer (i.e. 5+5=10), while if the boxes are considered to be merely 'covering up' any number behind, then there are many possible answers: 0 + 10 = 10, 1 + 9 = 10, 2 + 8 = 10, 2.5 + 7.5 = 10, etc. When students encounter algebra later, a letter in a number sentence must stand for only one number. For example, there is only one answer (m = 5) to the equation m + m = 10.
• over-specialised concept: How many answers are there for this number sentence? □ + ∆ = 10. There are many possible answers (as above, 0 + 1 0 = 10, 1 + 9 = 10, 2 + 8 = 10, 2.5 + 7.5 = 10, etc). Some students will reject 5 + 5 = 10 as they believe that the different placeholders indicate that the numbers must be different.

 

Recommendation

(a) If open shapes are used for placeholders, then teachers need to make clear to children what conventions are being used:

• whether the placeholders are to hold digits or complete numbers (e.g. whether 46 goes in one or two boxes)
• whether placeholders of the same shape must contain the same number
• whether placeholders of different shapes must contain different numbers.

We recommend that placeholders hold numbers (not just digits), that placeholders of the same shape MUST hold the same number, and that place holders of different shapes need not hold different numbers. Consequently:

• □ + ∆ = 100 has many solutions including 31 + 69 = 100 and 50 + 50 = 100
• □ + □ = 100 has only one solution 50 + 50 = 100

(b) If letters are used for place holders for numbers, then you should use the normal algebra conventions: in an equation or missing number sentence, a letter stands for only one number in an equation and different letters might stand for the same number.

(c) Question marks can be used in number sentences with only one missing number.