# Addition inquiry

# The prompt

Mathematical inquiry processes: Generate examples; list permutations; reason. Conceptual field of inquiry: Addition of two- and three-digit numbers.

Sanchita Dalal, a primary teacher at Kunskapsskolan Gurgaon in Gurugram (India), contacted Inquiry Maths to request more prompts for grades 1 to 3.

In particular, she wanted ideas for topics that were coming up in the curriculum. The prompt on this page is the fourth of a sequence of five designed for the topics of number sense and addition and subtraction (see below).

Question, notice, wonder

The following questions and statements can help teachers guide novice inquirers.

Questions: Is the equation true? How do we know the two sides are equal?

Notice: The digits are consecutive. The digits are switched round.

Wonder: Would the two sides be equal if the digits are non-consecutive? Would they be equal if we changed the position of the digits (for example, 14 + 32 = 41 + 23)?

Teacher's aim

In the inquiry the teacher's aim is to encourage pupils to create their own examples with consecutive and non-consecutive digits and show their equations are true or not.

In the initial stage the teacher should ensure pupils can find the sum of two two-digit numbers - perhaps by using an appropriate representation (see 'Lines of inquiry' below). She might also create examples to structure a period of whole-class exploration.

Intrigue

The addition prompt is intriguing with lots to discover. If you reverse the digits in the two two-digit numbers and the digits are consecutive, the two sides of the equation are equal if the largest and smallest digits are in the same column - that is, both tens or both units. For example, 65 + 34 works because the six (largest) and three (smallest) are both in the tens column.

However, non-consecutive digits will not always work if you follow the rule. For example, 97 + 14 does not work, but 97 + 13 does.

Pupils might be able to generalise from a set of examples to find a rule that applies to all sets of four digits. If the two numbers on the left-hand side are 10a + b and 10c + d, the relationship between the digits has to be a + c = b + d. For example, in 74 + 25, a 7, b = 4, c = 2, d = 5, and 7 + 2 = 4 + 5.

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Sanchita would love to connect with teachers who use the prompt or other prompts from the sequence. Send feedback on your classroom inquiry to Contact Inquiry Maths.

July 2024

# Lines of inquiry

1. Represent the equation

The equation can be represented in different ways: with base 10 tiles (diagram below), place value diagrams, and number lines among others.

(See the PowerPoint for more information about all the lines of inquiry.)

2. Generate examples

Pupils generate more examples by searching for all the permutations with the digits from one to four. How many are repeats when the digits are reversed? How many are true?

After pupils choose their own four digits. They could test other consecutive digits or four non-consecutive digits.

3. Proof

A proof is likely to be beyond the majority of pupils, but could be used to stretch the most talented mathematicians in grade 3. The proof uses the variables a, b, c, and d, which are defined as different digits from one to 10. 10a + b, for example, is an expression for the first two-digit number.

4. Three-digit numbers

Another line of inquiry involves exploring three-digit numbers. Pupils choose six different digits, arrange them in two numbers and then reverse the digits. Are the two sides of the equation equal? Under what conditions are they equal?

The example below, which can generate discussion and conjectures, is intriguing because the three equations seem to follow the same approach in placing the digits. Yet, the first two are equal and third is not.

# A sequence of inquiry prompts

Sanchita requested prompts about these topics:

Number sense - an introduction to two-digit numbers and different ways to represent them;

Addition and subtraction of two- and three-digit numbers with and without regrouping.

We linked the concepts in a sequence of five inquiry prompts, of which the addition prompt on this page is the fourth.

We show the other prompts below. Next to each one we suggest contributions pupils might make when asked to question, notice, and wonder. Even though novice inquirers are unlikely to contribute the ideas, the teacher can use them to scaffold the inquiry process.

We also include teachers' notes that might facilitate further inquiry in the classroom.

Prompt 1: Representations

Question

What does the arrow symbol mean?

Notice

The digits are switched round in the numbers.

In the first diagram, the bars represent tens and the squares represent units.

The diagrams and numbers show the same inequality.

Wonder

Does the inequality always work when you switch the digits?

Is there another type of diagram to represent the inequality?

Teachers' notes

The prompt contains more information than a normal Inquiry Maths prompt. The teacher might choose to use only one representation at a time and support its use with concrete resources, such as counters or base ten tiles (pictured). Pupils could be invited to compare the four representations, giving advantages and disadvantages for each one.

The main activity might see pupils represent their own inequalities (or ones that the teacher gives them) in the four ways.

The inequality does not always work when you switch the digits. When the larger digit is in the tens column of the first number, the inequality sign should be reversed - for example, 74 > 47.

Prompt 2: Doubling inequalities

Question

Why have the arrows turned round compared to the first prompt?

Notice

The digits have been switched round in the two numbers.

The numbers in the second inequality are double those in the first one.

Wonder

If the first inequality is true and we double the numbers, is the second inequality always true?

Teachers' notes

The prompt follows on from the first one. The teacher's aim is to encourage pupils to use diagrams to explain that the second inequality must be true if the first one is true.

Prompt 3: Switching tens

Question

Is the equation true?

How do we know the two sides are equal?

Notice

The numbers are all even.

The digits in the tens place have been switched round.

Wonder

Would it always work if you switch the tens?

What would happen if we switch the digits in the units place?

Can we use all odd digits or a combination of odd and even?

Teachers' notes

The teacher's aim is to encourage pupils to create their own examples with digits that are either even, odd, or odd and even. In the initial stage of the inquiry the teacher would ensure pupils can add the two numbers on either side of the equation (using diagrams and number lines as appropriate). She might also create some examples to test if switching the tens always works.

Pupils should be aware (or the teacher should make them aware) of the properties of the examples they generate. By thinking about and defining the type of digits they use (odd, even, or both), pupils learn to test particular cases and, more broadly, to regulate their mathematical activity.

Prompt 5: Subtraction

Question

What does the symbol in the second equation mean?

Are the equations true? How do we know?

Notice

The digits are consecutive and switched round.

The numbers are the same in both equations, but the symbols are different.

Wonder

Is it possible to make the two equation (addition and subtraction) equal using the same numbers?

Teachers' notes

The prompt uses the same equation as the main prompt (prompt 4) on this page and, therefore, could be used as an extension to that inquiry. By generating more examples, pupils realise that it is not possible to make both sides of the two equations equal using the same numbers.

An alternative prompt that focuses attention on the commutative property of addition is:

43 + 12 = 12 + 43

43 - 12 ≠ 12 - 43

However, the prompt leads the class into negative numbers, which might not be appropriate.