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Level Three > Number and Algebra

# Number Relations

Achievement Objectives:

Achievement Objective: NA3-1: Use a range of additive and simple multiplicative strategies with whole numbers, fractions, decimals, and percentages.
AO elaboration and other teaching resources
Achievement Objective: NA3-6: Record and interpret additive and simple multiplicative strategies, using words, diagrams, and symbols, with an understanding of equality.
AO elaboration and other teaching resources

Purpose:

This unit of work is based on the Māori medium unit Te Whakaaro Tūhonohono, and investigates how relational thinking is applied to number operations.

Specific Learning Outcomes:

Students will learn

• to use relational thinking as a basis for a range of number strategies
• to explain number strategies using materials and diagrams
Description of mathematics:

The equal sign signifies a relationship between the numbers and operations on either side.

If the numbers on one side of the equal sign are changed, then relational changes must be made to the other side to maintain equality.

Understanding the relationship between numbers on either side of the equal sign is fundamental in a range of operational strategies.

Required Resource Materials:
counters
pūkeko array (Copymaster 1)
Activity:

Session 1

This session looks at aspects of relational thinking that relate to common addition strategies.

Write the equation 48 + 17 = 50 + 15 on the board and get students to think about how they know that the equation is correct. They could also think about a number story for the equation. Some ideas to discuss:

• Completing the addition on either side of the equal sign is an inefficient way of showing equality.
• 2 is added to 48 and 2 is taken away from 17 to get 50 + 15. Adding 2 and taking away 2 means the total doesn’t change so they must be equal.
• 2 is taken from 17 and added to 48. This means the numbers are re-arranged but the total doesn’t change.

Get students to show the equality using place value equipment.

Show students the following diagrams and get them to explain how they show the equality 48 + 17 = 50 + 15.

Give similar equations for students to decide whether or not they are true. Get them also to show the equality (or inequality) using materials, and in a picture:

 39 + 18 = 40 + 17 56 + 23 = 59 + 20 38 + 17 = 40 + 19 67 + 24 = 57 + 34 236 + 38 = 238 + 40 52 + 227 = 50 + 229

Get students to go back and look at the equations and think about which side of the equal sign would be the easiest to work out and why. Use this discussion as a basis for the ‘tidy number’ strategy which could be used to solve the following equations:

 47 + 18 34 + 69 26 + 37 438 + 48

The following additions require students to think about the additive relationship between the numbers on either side of the equal sign. It is important to get student to explain and discuss their strategies.

 87 + 36 = 77 + 342 + 678 = + 478 29 + 74 = 31 + 488 + 396 = 427 +

To wind up this session get students to think about what numbers could be put in the square and circle to make these equations true. Ask students to explain the relationship between the square number and the circle number, and how this relationship comes about:

 15 + = 14 + + 78 = + 68 + 362 = 261 + 278 + = + 268

Session 2

This session focusses on relational thinking and subtraction.

Get students to think about why you don’t need to work out the subtractions on both sides of the equal sign to decide that this equation is true: 43 – 28 = 63 – 48. The key idea is that if you add the same amount to both numbers, the difference between them stays the same.

To consolidate, students could look at equations like the following and decide whether or not they are true without working out the subtractions on either side of the equal sign:

 23 – 9 = 24 – 10 267 – 46 = 261 – 40 468 – 239 = 478 – 249 63 – 38 = 61 – 40 2,548 – 1,376 = 2,648 – 1,476 456 – 78 – 450 – 70

Students should also be able to explain their reasoning using diagrams and equipment. For example:

Introduce students to the term ‘equal addition’ as a strategy to make the subtraction easier to work out. This could also be ‘equal subtraction’, the main concept being that adding or subtracting the same amount to or from both of the numbers does not change the difference between them. Talk about the following two strategies used to make this subtraction easier to solve, and why it becomes easier:

Consolidate relational thinking with regard to subtraction by getting students to use the ‘equal addition’ or ‘equal subtraction’ strategy to solve the following.

 35 – 18 = 64 – 39 = 263 – 98 = 632 – 148 =

To wind up this session get students to think about what numbers could be put in the square and circle to make these equations true. Ask students to explain the relationship between the square number and the circle number, and how this relationship comes about:

 65 – = 63 - – 32 = – 30 – 18 = 74 –

Session 3

This session explores how relational thinking applies to multiplication.

Discuss why this multiplication equation is true, and what the relationship is between the numbers on the left of the equal sign and the right: 14 x 5 = 7 x 10

Re-arranging a 14 x 5 array to become 7 x 10 will help students to understand that nothing has been added or taken away to the array, so there will be the same number of items (Copymaster 1 is the pūkeko array):

Relate the re-arrangement of the array with the more abstract equation representation such as:

Give students some multiplication equations like the following for them to decide whether or not they are true, without working out either side of the equal sign:

 34 x 5 = 17 x 10 34 x 4 = 17 x 8 ½ x 20 = 2 x 5 20 x 4.5 = 10 x 9 68 x 2 = 39 x 3 1½ x 4 = 3 x 2

The following multiplications require students to think about the additive relationship between the numbers on either side of the equal sign. It is important to get students to explain and discuss their strategies.

 18 x 4 = x 8 124 x 48 = x 24 60 x 15 = x 45 72 x = 36 x 150

To wind up this session get students to think about what numbers could be put in the square and circle to make these equations true. Ask students to explain the relationship between the square number and the circle number, and how this relationship comes about:

 56 x  = 28 x x 7 = x 21 66 x = x 22 x 15 = x 5 15 + = 14 + + 78 = + 68 + 362 = 261 + 278 + = + 268

Session 4

In this final session students are challenged to come up with their own ‘relational thinking’ with regard to solving division problems. They should be able to demonstrate and explain using materials and equation diagrams.

Measurement is a good context for showing how relational thinking applies to division problems. For example, dividing 24 litres of juice into 6 litre buckets will give you the same number of buckets as dividing 12 litres of juice in to 3 litre buckets:

24 ÷ 6 = 12 ÷ 3

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Number_Relations_CM.pdf427.8 KB