Year+4+Numeracy

Year 4 || =Digit sum=
 * MATHEMATICS


 * Information ||

Objective
• To solve mathematical problems, recognise and explain patterns and relationships, generalise and predict. Suggest extensions by asking ‘What if …?’

Prior learning
To benefit from this lesson, children should: • have had experience of listing systematically all possibilities.

Vocabulary
digit, digit sum, systematic

Resources
• interactive whiteboard and/or data projector linked to a laptop • software which allows the generation of random numbers according to specific rules and then allows these numbers to be arranged in numerical order (in this Example, //Excel//) • a resource file with the appropriate number generation capacity (in this Example, Sheets 1, 2 and 3 of the spreadsheet, //Digit sum Excel// file)

ICT skills needed by teachers
To teach this unit, teachers need to know how to: • manipulate a spreadsheet file.

Preparation for this lesson
Before the lesson, set up the relevant software and resources on the shared area of the network or on the computers that the children are to use. Prepare the software on your own computer ready for display.


 * Lesson extract ||

Starter
Launch the appropriate resource file (in this Example, //Digit sum Excel file// sheet 1). Ask children to read aloud the four-digit number displayed. Q What is 10 more, 10 less than this number? Discuss answers and the impact on the digits of adding and subtracting 10. Click on the upwards arrow on the spinner to display another four-digit number and repeat. Q What is 100 more, 100 less than this number? Continue, discussing the impact on the digits. Incorporate the addition and subtraction of 1 and 1000. Use children’s answers to compare the impact on the digits with the impact on the whole number, e.g. subtracting one hundred from 4382 changes the hundreds digit 3 to a 2 and changes the number 4382 to 4282.

Main activity
Launch sheet 2, to display the three-digit number. Ask children to read this number aloud. Click on the spinner to increase or decrease the number and ask children to read this new number aloud. Repeat. Q What do you notice about all these three-digit numbers? Collect children’s observations. Generate new numbers to test their observations. Establish that the sum of the three digits in each number is 7. Say that we shall call the sum of the digits the ‘digit sum’. Ask children to work in pairs and record four three-digit numbers whose digit sum is 7. Q What numbers did you use for the hundreds digit? Collect answers and for each hundreds digit suggested, ask for a three-digit number whose digit sum is 7. Record these on the board. Q Could the hundreds numbers be an 8 or 9? Confirm that it could not and why not, so children understand that the largest possible hundreds digit is 7. Establish that 700 is the largest three-digit number as it is the only three-digit number in the 700s with digit sum 7. Q Could the hundreds digit be 0? Record 016 to establish that while its digit sum is 7, 016 is really 16, a two-digit number. Confirm that 1 is the smallest possible hundreds digit. Establish that 106 is the smallest three-digit number in the 100s with the digit sum 7. Say that you want the children to work in pairs and list all the three-digit numbers in the 100s with a digit sum 7. Launch sheet 3, and enter the number 106. Collect and discuss their answers, and record their numbers in the spreadsheet. Make sure 115 is recorded. Q Have we got all the 100s numbers? How can we check to see whether there are any we have missed? Discuss suggestions. Q How will putting these numbers in order help us to see which numbers are missing? Use the spreadsheet to put the hundreds numbers in order and ask children whether they can identify any that are missing. Establish that it is easier to identify missing numbers when the list is ordered. With the children, analyse the numbers in the list. Identify the first number as 106 and ask: Q What is the next biggest 100s number we have found? Establish that the number is 115. Emphasise the decreasing of the units digit from 6 to 5 and the increasing of the tens digit from 0 to 1. The digit sum has stayed at 7 because we have subtracted 1 and added 1, keeping the total at 7. Q What can we do to get the next biggest 100s number? Invite children to demonstrate the decrease in the units digit and the increase in the tens digit to get 124. Continue this process to generate the full list: 106, 115, 124, 133, 142, 151, 160 Explain that by being systematic we have made all the 100s numbers with a digit sum of 7. Q How many 100s numbers have a digit sum of 7? Agree that there are 7. Q If the three-digit numbers had a digit sum of 7, what would the smallest 200s number be this time? Agree it is 205 and enter 205 on the spreadsheet. Say that you want the children to work in pairs and list all the other three-digit numbers that have a digit sum of 7.

Plenary
Collect answers and ask the children for three-digit numbers whose digit sum is 7. Record these under 205 to generate the list: 205, 214, 223, 232, 241, 250 Q Have we listed all the 200s numbers? How can we check to see whether there are any we have missed? Discuss suggestions. Ask children to explain the strategies they used to list all the three-digit numbers with a digit sum of 7. Q Can we predict how many 100s numbers have a digit sum of 9? Collect responses. Enter 108. Ask children to use the strategies we have looked at during this lesson to list all the possible numbers. Take responses and enter these into the spreadsheet under the starting number of 108: 108, 117, 126, 135, 144, 153, 162, 171, 180 Q Was our prediction correct? Can we see a pattern in the numbers we have listed that will help us to predict? Discuss the strategy of starting with the smallest possible three-digit number and increasing one digit and decreasing another to generate the pattern. Say that this is an example of working systematically. Track the units digit, which decreases from 8 to 0. This covers a range of 9 digits so there are 9 100s numbers with a digit sum of 9. Q If the three-digit numbers had a digit sum of 8, how many 100s numbers are there? Collect answers and ask children to check this for homework.


 * Notes ||

Links to the Framework for teaching mathematics
The lesson links to units on properties of numbers and reasoning about numbers.

Context of this lesson
This lesson could be used in Unit 8 of the autumn, spring or summer terms from the sample medium-term plans.

Useful sources of resources
There are similar problems to this in //Mathematical challenges for able pupils in Key Stages 1 and 2//.

Why use ICT?
The advantages of using ICT are as follows. • ICT allows teachers to project enlarged visual images for whole-class demonstration and discussion. • The interactivity of the software is motivating and stimulating. • A spreadsheet can be used to order the numbers ascending or descending by using the sort facility, helping children to see a systematic way of ordering the solutions they have found. The workbook can be saved and used at a later date.

Year 4 || =Interpreting data=
 * MATHEMATICS


 * Information ||

Objective
• To solve a problem by collecting quickly, organising, representing and interpreting data in tables, charts, graphs and diagrams.

Prior learning
To benefit from this lesson, children should: • be able to interpret a bar chart; • know how to record results in a table.

Vocabulary
bar chart, justify, more likely, less likely, fair game

Resources
• interactive whiteboard and/or data projector linked to a laptop • software showing two spinners (in this Example, //Two spinners Excel// file) (Note: The game played using these two spinners is a fair game so results will tend to balance. If you want to present children with a game which is unfair, change the numbers in one of the spinners.) • a printout of the two spinners for each pair of children (in this Example, //Two Spinners Word// file) (Note: If you have changed the numbers on the spinners, remember to change the numbers on this resource) • a paperclip for each pair of children

ICT skills needed by teachers
To teach this unit, teachers need to know how to: • enter data into a spreadsheet; • use a spinner in //Excel//.

Preparation for this lesson
Become familiar with //Two spinners// //Excel// file. Reset the counters to zero on sheets 1 and 2 using the down arrow, and reset the values in columns 2 and 3 to zero on sheet 3.


 * Lesson extract ||

Starter
Load sheet 1 of //Two spinners// //Excel// file. Divide the class into four groups: A, B, C and D. Explain that: • group A is to sum the column of four numbers above A; • group B is to sum the column of four numbers above B; • group C is to sum the row of four numbers leading to C; • group D is to sum the row of four numbers leading to D. The four numbers are in the range 0 to 10. (Note: To change this range, enter a number that is 1 more than the range required into cell F5 in the bottom right-hand corner.) Say that each child in the group is to work out on their own the sum of their four numbers, and if they finish before you ask for the answer, they are to work out the other sum to check the other group’s answer. When you ask for a group’s answer, the whole group is to say the answer aloud. Use the upward arrow to generate a new set of numbers. After each group answers, ask the other children if they agree with that group’s answer. Invite explanations of their strategies and discuss these. Drag aside the letters to reveal the totals and confirm answers. Q What numbers were easy to add when you worked out your totals? Establish that 0, 1 and 10 are easy numbers to add in a list. Q What pairs of numbers should we look for when we add the four numbers? Remind children that looking for pairs totalling 10 is a helpful strategy. Repeat, collecting and discussing the children’s answers and explanations.

Main activity
Show the two spinners from //Two spinners// //Word// file: Explain that pairs of children play a game. One child has Spinner 1, the other Spinner 2. Together the two children each spin a paperclip around a pencil in the middle of their spinners and compare the numbers they get. The one with the larger number gets a point. After 20 games the player with the most points wins. Q If you played the game which spinners would you choose and why? Collect and discuss children’s motives and reasons. Record the most popular choice of spinner. Open Sheet 2. Explain that the two grids of numbers represent the two spinners. The number highlighted by the colour blue on spinner 1 and red on spinner 2 is a randomly selected number to be used to play the game. Interpret the case shown on the screen and decide whether the larger number has been selected from Spinner 1 or Spinner 2. Use the upwards arrow to play the game 20 times. Record the points on a table using tally marks. Q How many points did each player get? Ask the children to total the tally marks. Q Do these results support our choice of spinner? Refer back to the most popular choice and the children’s reasoning. Q Have we played the game enough to be convinced that our chosen spinner will win? Establish that more data will help us to decide. Explain that you want the children to work in pairs and play the game 20 times, using the resource sheet with the two spinners and a paper clip and pencil. They are to collect the points on a table using tally marks. When they have finished, they are to total up the points for each spinner and enter their results into the table. Divide the children into pairs. Remind them that one child will use Spinner 1, the other Spinner 2. It is important that the children make sure the numbers they get are not influenced by the way they spin their spinners as they play the game and they collect their data. They are not trying to win but to see what happens when they gather lots of data. Both children are to use their spinners at the same time, then compare their two numbers and record the results. Give the children the sheet with the two spinners and a paper clip they can spin around a pencil placed at the centre of the spinner. Open Sheet 3. Enter the results from the original ICT-based game to demonstrate how to enter the results and how the totals change along with the bars on the bar chart. Say that we will collect results from pairs of children. Q Would we expect these results to be the same as the first set of results already in the table? Q How will these results change the bar chart? Enter the results given by one pair of children into the table. Repeat with other results collected from pairs of children until the table is full.
 * Spinner 1 || / / / / ||
 * Spinner 2 || / / ||

Plenary
Remind children about the discussion they first had, their choice of spinner and their reasons for this choice. Look at the results of the pairs of 20 games in the table. Identify pairs whose points are imbalanced (e.g. 16, 4) and those whose points are better balanced (e.g. 9, 11). Q Why should we expect results like these? Agree that we cannot predict the results as we don’t know what numbers we will get on the spinners. With the children identify the number of pairs of players, and count up in 20s to work out how many games were played altogether. Q If the player with Spinner 1 and the player with Spinner 2 both won the same number of times, how many points would there be for each spinner in the totals boxes? Collect and discuss children’s answers and agree that there would be 150 in each total. Explain that when either player has the same chance of winning the game, the game is said to be a fair game. Compare the figures of 150 and 150 with the actual total results from the children’s games. Discuss the bars on the bar chart. Establish that the heights of the two bars correspond to the totals values in the table. Find the value 150 on the vertical axis and draw a horizontal line from this value across the bar chart. Q Do we think that using these two spinners, the game we played was a fair game? Collect and discuss the children’s responses. Relate their views back to the numbers on the spinners. Establish that as the two bars are about the same height, the game seems to be a fair game. Q Will the two bars always look the same as the two bars shown here? Establish that because we will not get exactly the same results on the spinners each time we play the game, the bars will not always look exactly the same as the two on the screen. Q Is the difference between the two bars very much when compared to the 150? Collect and discuss children’s views. Emphasise that a difference of 10 say would be big for one game for 20 points where the result might be 5 and 15, but for hundreds of games 10 is quite small. Q. Do you think it matters which spinner we choose when we play the game? Collect children’s views and see whether these have changed in the light of the data they have collected. Explain that as we think the game is fair, it does not matter which spinner we choose.


 * Notes ||

Links to Framework for teaching mathematics
The lesson links to: Units on Handling data

Why use ICT?
The advantages of using ICT are as follows. • ICT allows teachers to project enlarged visual images for whole-class demonstration and discussion. • The interactivity of the software is motivating and stimulating. • The software automatically constructs a bar chart as data is entered allowing children to see the effects of new information.

MATHEMATICS Year 4 || =Remainders=

Information ||

Objective
• To find remainders after division

Prior learning
To benefit from this lesson, children should: • know multiplication and division facts for the 3 and 4 times tables; • have met the idea of a remainder before.

Vocabulary
remainder, divided by, groups

Resources
• interactive whiteboard and/or data projector linked to a laptop and interactive whiteboard software (in this Example, //SMART notebook)// • ICT suite or set of laptops • software which enables the modelling of remainders in division (in this Example, the interactive teaching program (ITP) //Grouping)//

ICT skills needed by teachers
To teach this unit, teachers need to know how to: • use the chosen software (in this Example, the ITP //Grouping//); • set up the use of the chosen software in the computer suite.

Preparation for this lesson
The chosen software in this Example, the ITP //Grouping//) needs to be loaded onto the laptop connected to the interactive whiteboard and the children’s computers.

Lesson extract ||

Starter
Organise the class into four groups of children. The groups are to be of different sizes. Where possible, ensure that two groups contain even numbers of children and two contain odd numbers of children – a particularly good example would be to have four consecutive numbers, e.g. 6, 7, 8 and 9. Tell the children that within each group they are going to organise themselves into sub-groups and you want them to be ready to describe what they notice after they have formed the sub-groups. Ask children to organise themselves into groups of 2. Ask each group in turn: Q How many were in your group? Q What happened when you tried to get into these sub-groups? Collect and discuss children’s responses. Q Why was there one person left over in some groups but not in others? Ensure that children recognise that odd numbers, when divided by 2, have a remainder of 1. Now ask each group to divide themselves into sub-groups of 3. Repeat the questions and discuss when there were remainders and why. Ask the groups to collect information about the size of the sub-groups and the numbers left over and ask: Q For your group, when were there no people left over? Q For your group, when were there people left over and how many were left over? For each group, collect and record results. Q How can we complete the statement below? Record on the board: ‘There was no remainder when …’ Agree that the statement can be completed with ‘… the number in the group is a multiple of the number you were dividing by.’

Main activity
Write the calculation 12 ¸ 3 on the board and ask one of the children to read this calculation aloud. Q How does this relate to the activity we did at the start of the lesson? Encourage children to recognise that it could mean a group of 12 children are to be divided into groups of 3. Q What is the answer to this calculation? Collect and discuss answers and complete the number sentence: 12 ¸ 3 = 4. Q What number sentence would we write to complete the calculation 13 ¸ 3? Record the children’s answers on the board. Launch the chosen software (in this Example, the ITP //Grouping//) and enter a calculation, e.g. 13 ¸ 3, to confirm there is a remainder of 1. Draw attention to the number of groups and jumps on the number line and the way the answer is written on the screen to show there is a remainder of 1. Q What is the calculation if our answer is 3 remainder 1? Cover up the first group of 3 to help children to see that the calculation could be 10 ¸ 3. Q What other calculations would give a remainder of 1 when we divide by 3? Establish the pattern: 13 ¸ 3, 10 ¸ 3, 7 ¸ 3, 4 ¸ 3. Q What numbers larger than 13 can you suggest would give a remainder of 1 when divided by 3? Collect and record answers on the board to highlight the pattern. Q When we divide a number by 3 when would we get a remainder of 2? Collect answers and use the chosen software (in this Example, the ITP //Grouping//) to model the suggested divisions, e.g. 5 ¸ 3, drawing attention to the one group of 3 and the remainder of 2. Ask children to extend the pattern to other divisions by 3 that also give a remainder of 2. Q When we divide by 3, what remainders can we have? Establish that the possible remainders are 1 or 2, or the number is an exact multiple of 3 and there is no remainder. We can say that when dividing by 3 possible remainders are 0, 1 or 2. Tell the children that they are going to explore remainders when different numbers are divided by 4. You want them to find out what the possible remainders are, and then identify lists of numbers that, when divided by 4, give these remainders. Children are to work in pairs. They can use the ITP on laptops or in a computer suite or have access to appropriate resources such as counters. When they finish, ask them to go on to looking for the patterns when dividing by 5.

Plenary
Ask the children to remind you of the remainders when dividing by 3. Q What were the remainders when you divided by 4? Q What were the remainders when you divided by 5? In each case, list the remainders: 0, 1 and 2; 0, 1, 2 and 3; 0, 1, 2, 3 and 4. Q What will the remainders be when you divide by 6, 7, …? Draw out the fact that the number of possible remainders is the same as the number you are dividing by, and the remainders go from 0 to 1 less than the number you are dividing by. Q Can you tell me a number that will leave a remainder of 3 when we divide by 4? Use the ITP to confirm children’s responses. As before, use the representation on the number line to draw out the pattern of numbers: 7, 11, 15, 19, … Repeat with other remainders when dividing by 4 or 5.

Notes ||

Links to the Framework for teaching mathematics
The lesson links to units on understanding multiplication and division.

Context of this lesson
This lesson could be used in Units 9 and 10 of the autumn, spring or summer terms from the sample medium-term plans.

Why use ICT?
The advantages of using ICT are as follows. • ICT allows teachers to project enlarged visual images for whole-class demonstration and discussion. • The interactivity of the software is motivating and stimulating. • The interactivity of the ITP allows the teacher to control the model in flexible ways.