Year 3 Block E – Securing number facts, relationships and calculating

Unit 1

Learning overview

Children count on and back in regular steps of 1, 2, 3, 4, 5, 6, 10 and 100 using their knowledge of addition and subtraction facts to help them to count accurately. They find the difference between consecutive numbers to establish the step size to complete sequences such as:

1, 7, 13, 19, ☐, ☐; ☐, 26, 22, ☐, ☐, 10, 6, 2

Children identify patterns and relationships and use these to support their count. They investigate general statements such as: When you count in fives, the units digits form a pattern. Where they work in groups on a task, they ensure that all members try out examples and discuss what they have found.

Children know by heart the 2, 5 and 10 multiplication facts and use them to solve questions like:

They recognise questions that involve division, such as:

They count on and back from zero in steps of 2, 3, 4, 5, 6 and 10 to answer questions like:

Children research the question: What digits can multiples of 2 end in? What about multiples of 3, multiples of 4? They investigate by joining the last digits of each multiple in order on a digit wheel. For example, the last digits of the multiples of 2 ( 2 , 4 , 6 , 8 , 10 , 12 , 14 , 16 , 18 , 20 , 22 ) form this pentagon:

Children record the outcomes of this enquiry by recording in a table the number, the last digits of its multiples and the shape that they form on the digit wheel. They use their results to answer questions such as:

Children review multiplication as repeated addition and division as repeated subtraction by counting hops on a number line. For example, they find 6 fours by making 6 hops of 4.

Children divide a number of objects by using grouping . They understand that one way to find 30÷6 is to find how many sixes there are in 30 . Through practical experience, they understand that some division calculations have a remainder, for example 13 ÷ 4 = 3 R 1:

Children understand the relationship between multiplication and division . For example, they state two multiplication sentences and two division sentences that relate to a particular array, for example:

5 × 2 = 10, 2 × 5 = 10

10 ÷ 2 = 5, 10 ÷ 5 = 2

They use the image of an array to explain why, for example, 2 × 5 gives the same answer as 5 × 2. They also use the image to show how many fives make 10 and how many twos make 10.

Children derive quickly the doubles of all numbers 1 to 20 . They recognise that halving is the inverse of doubling. They understand that doubling is equivalent to multiplying by 2 and halving is equivalent to dividing by 2.

Children begin to use practical and informal methods to solve simple TU × U calculations. For example, to find 12 × 5 they appreciate that 10 fives are 50 and add on another 2 fives to make 60.

Children fold shapes in half and, where possible, repeat this to find ^₁/₂, ^₁/₄ or ^₁/₈ of a variety of shapes . By folding three identical rectangles into halves, quarters and eighths, they show and explain that ^₁/₂ is equivalent to ²/₄ and is also equivalent to ⁴/₈. They understand that a whole is, for example, two halves, four quarters or eight eighths.

Children recognise other unit fractions of shapes, realising, for example, that ¹/₅ of this shape is shaded because 1 piece out of 5 equal pieces is shaded.

Children find ^₁/₂, ^₁/₄ or ^₁/₈ of collections of objects by sharing or repeated halving. They recognise and use the notation of ^₁/₂,^₁/₄ or ^₁/₈, understanding that the numbers on the bottom of the fraction relate to sharing equally between 2, 4 or 8. They place ^₁/₂ and ^₁/₄ between 0 and 1 on a number line and half past, quarter past and quarter to on a 12–hour time line.

Objectives Children's learning outcomes are emphasised	Assessment for learning
Follow a line of enquiry by deciding what information is important; make and use lists, tables and graphs to organise and interpret the information I can make a table to record my results	What information will you find? How will you record it? What did you find out? Show me what in your results helped you to draw this conclusion. Why did you choose to record your results in a table? What number should go in the shaded square? What multiplication fact did you use?
Identify patterns and relationships involving numbers or shapes, and use these to solve problems I can describe the pattern when I count in fives	What are the missing numbers in this pattern? How did you find them? 83, 78,☐, 68, 63, 58,☐ Find 1/2 of 16. Find 1/4 of 16. Find 1/8 of 16. What do you notice? Sam says: 'When you count from zero in fours, every number is even.' Is he right? How do you know?
Derive and recall all addition and subtraction facts for each number to 20, sums and differences of multiples of 10 and number pairs that total 100 I know addition and subtraction facts for number to 20 I can add and subtract multiples of 10	What is the missing number in this pattern? 4, 7, 10, 13, ☐, 19 What facts did you use to work this out? What other fact could you use? Three numbers add up to 100. Two of the numbers are 50 and 20. What is the third number? Put + or – symbols in the circles to make the answer correct: 9◯7◯3◯5=8
Derive and recall multiplication facts for the 2, 3, 4, 5, 6 and 10 times–tables and the corresponding division facts; recognise multiples of 2, 5 or 10 up to 1000 I know the 2, 5 and 10 times–tables I can use multiplication facts to answer division questions	How many fives make the same number as three tens? What multiplication and division facts does this array show? Complete this division fact in as many ways as you can: 12 ÷ ☐ = ☐ Is 113 a multiple of 5? How do you know? How many multiples of 2 are there between 175 and 183?
Use practical and informal written methods to multiply and divide two–digit numbers (e.g. 13 × 3, 50 ÷ 4); round remainders up or down, depending on the context I can multiply a 'teen' number by 2, 3, 4, 5 or 6	Paul buys 12 lollies that cost 5p each. Work out how much this will cost altogether. How did you find the answer? Did you jot anything down? You are given that 10 × 3 = 30 and 3 × 3 = 9. How many threes are there in 39? How many teams of 3 people can be made from 10 people? Draw a picture that shows that your answer is correct.
Find unit fractions of numbers and quantities (e.g. 1/2, 1/3, 1/4 and 1/6 of 12 litres) I can find fractions of numbers by using division	Which is heavier: 1/2 of 18 kg or 1/4 of 32 kg? What calculation would you do to work out 1/8 of 32? Mary says that 1/4 of the numbers on a 100–square are bigger than 60. Is she right? How do you know?
Sustain conversation, explaining or giving reasons for their views or choices I listen to the views of everyone in my group and make sure that everyone has a turn to talk	You are going to solve this problem as a group. Start by agreeing what everyone in the group is going to do. How can you make sure that the discussion involves everyone and that everyone has a chance to express their point of view?

Resource links to existing published material

Mathematical challenges for able pupils Key Stages 1 and 2

Activities	Resources
Activity 29 – Spaceship	Puzzles and problems for Year 3 and 4

Intervention programmes

Objectives for Springboard intervention unit	Springboard unit
Know by heart doubles of numbers to 10; doubles of multiples of ten up to 50 Identify near doubles using doubles already known Halve even numbers from 20 to 2 Measure and compare lengths using a standard measure	Springboard 3 Unit 4: Doubling and halving
Understand the operation of multiplication as describing an array Know by heart the facts of the two–, five– and ten–times tables	Springboard 3 Unit 7: Understanding multiplication

Supporting children with gaps in their mathematical understanding (Wave 3)

Diagnostic focus	Resources
Confuses numbers when counting in twos; has difficulty understanding that a pair consists of two objects	1 YR×/÷ Wave 3 (1 YR ×/÷) Teaching activities to help children understand pairs and counting in twos
Makes unequal groups and is unable to compare groups	3 YR×/÷ Wave 3 (3 YR ×/÷) Teaching activities to help children make and compare equal groups
Has difficulty with counting reliably in tens from a multiple of ten	5 YR×/÷ Wave 3 (5 YR ×/÷) Teaching activities to help children count reliably in tens from a multiple of ten
When halving, makes two unequal groups or splits a single object unequally	6 YR×/÷ Wave 3 (6 YR ×/÷) Teaching activities to help children understand halving
Still counts in ones to find how many there are in a collection of equal groups; does not understand vocabulary, for example, 'groups of', 'multiplied by'	1 Y2×/÷ Wave 3 (1 Y2 ×/÷) Teaching activities to help children count in equal groups
Does not link counting up in equal steps to the operation of multiplication; does not use the vocabulary associated with multiplication	2 Y2×/÷ Wave 3 (2 Y2 ×/÷) Teaching activities to help children understand multiplication
Does not focus on 'rows of' or ' columns of', but only sees an array as a collection of ones	3 Y2×/÷ Wave 3 (3 Y2 ×/÷) Teaching activities to help children understand rows and columns in arrays
Has difficulty relating multiplying by two to known facts about doubles; records double 4 as 4 + 4	4a Y2×/÷ Wave 3 (4a Y2 ×/÷) Teaching activities to help children relate multiplying by two to doubling
Does not use knowledge of doubles to find half of a number	5 Y2×/÷ Wave 3 (5 Y2 ×/÷) Teaching activities to help children find half of a number
Does not understand that 'sets of' or 'groups of' need to be subtracted to solve the problem	7 Y2×/÷ Wave 3 (7 Y2 ×/÷) Teaching activities to help children understand division as repeated subtraction
Is not confident in recalling multiplication facts	1 Y4×/÷ Wave 3 (1 Y4 ×/÷) Teaching activities to help children recall multiplication facts
Is muddled about the correspondence between multiplication and division facts	2 Y4×/÷ Wave 3 (2 Y4 ×/÷) Teaching activities to help children understand the correspondence between multiplication and division facts