More Addition and Subtraction Formal algorithms in addition

More Addition and Subtraction Formal algorithms in addition and subtraction Objectives Day 1 Add 3 -digit numbers using expanded addition Subtract 3 -digit numbers using expanded decomposition Day 2 Add 3 -digit numbers using expanded addition Subtract 3 -digit numbers using expanded decomposition Day 3 Add 3 -digit numbers using expanded addition, estimate totals Subtract 3 -digit numbers using expanded decomposition Day 4 Subtract a 2 -digit number from a 3 -digit number using counting up (Frog) Subtracting using decomposition or Frog © hamilton-trust. org. uk 1 Year 3/4

More Addition and Subtraction Formal algorithms in addition and subtraction Short Mental Workouts Day 1 Add multiples of 10 p Day 2 Subtraction facts Day 3 Round 3 -digit numbers to the nearest 10 or 100 Day 4 Complements to 100 © hamilton-trust. org. uk 2 Year 3/4

More Addition and Subtraction Formal algorithms in addition and subtraction Short Mental Workout Add multiples of 10 p © hamilton-trust. org. uk 3 Year 3/4

More Addition and Subtraction Formal algorithms in addition and subtraction Short Mental Workout Subtraction facts © hamilton-trust. org. uk 4 Year 3/4

More Addition and Subtraction Formal algorithms in addition and subtraction Short Mental Workout Round 3 -digit numbers to the nearest 10 or 100 © hamilton-trust. org. uk 5 Year 3/4

More Addition and Subtraction Formal algorithms in addition and subtraction Short Mental Workout Complements to 100 © hamilton-trust. org. uk 6 Year 3/4

More Addition and Subtraction Formal algorithms in addition and subtraction Objectives Day 1 Add 3 -digit numbers using expanded addition Subtract 3 -digit numbers using expanded decomposition © hamilton-trust. org. uk 8 Year 3/4

Day 1: Add 3 -digit numbers using expanded addition; Subtract 3 -digit numbers using expanded decomposition. To add 243 + 725 first make the numbers with place value cards and lay them out like this. We can add 3 -digit numbers by partitioning. 2 0 0 4 0 3 Add the 1 s. . . 3 + 5 = ? 7 0 0 2 0 5 Then the 10 s. . . 40 + 20 = ? 8 Lastly the 100 s. . . 200 + 700 = ? 9 0 0 6 0 Finally recombine 900, 60 and 8…. © hamilton-trust. org. uk 9 06 008 9 Year 3/4

Day 1: Add 3 -digit numbers using expanded addition; Subtract 3 -digit numbers using expanded decomposition. 200 40 3 + 700 20 5 We can record it like this. Y 4 s, why do we leave a blank space under the second number? 900 60 8 900 + 60 + 8 = 968 Copy and complete it on your whiteboards. Add the 1 s, 10 s and 100 s… Finally recombine 900, 60 and 8… © hamilton-trust. org. uk Why were there no numbers on the waiting line? 10 Year 3/4

Day 1: Add 3 -digit numbers using expanded addition; Subtract 3 -digit numbers using expanded decomposition. Let’s try 427 + 345 without the place value cards. 400 20 7 + 300 40 5 10 The numbers are partitioned, lined up in 100 s, 10 s and 1 s and a blank space left under the second number. 700 70 2 700 + 70 + 2 = 772 Next add the 10 s. . . 20 + 40 + 10 = ? Add the 1 s. 7 + 5 = 12. The 1 s come to more than 10 so we write the 10 in the space under the 10 s and the 2 under the 1 s in the answer line. Lastly the 100 s. . . 400 + 300 = ? © hamilton-trust. org. uk Finally recombine 700, 70 and 2…. 11 Year 3/4

Day 1: Add 3 -digit numbers using expanded addition; Subtract 3 -digit numbers using expanded decomposition. Let’s try 528 + 333. 500 20 8 + 300 30 3 10 The numbers are partitioned, lined up in 100 s, 10 s and 1 s and a blank space left under the second number. 800 60 1 800 + 60 + 1 = 861 Add the 1 s. 8 + 3 = 11. The 1 s come to more than 10 so we write 10 in the waiting line under the 10 s and 1 under the 1 s in the answer line. Next add the 10 s. . . 20 + 30 + 10 = ? Lastly the 100 s. . . 500 + 300 = ? © hamilton-trust. org. uk Finally recombine 800, 60 and 1…. 12 Year 3/4

Challenge © hamilton-trust. org. uk 13 Year 3/4

Day 1: Subtract 3 -digit numbers using expanded decomposition. We could use Frog but today we are going to try another way. How could we work out 654 - 321? First we need to partition the two numbers. 6 05 004 6 0 0 5 0 4 3 02 001 3 0 0 2 0 1 © hamilton-trust. org. uk 14 Year 3/4

Day 1: Subtract 3 -digit numbers using expanded decomposition. Set out 654 – 321 like this with the numbers partitioned. 600 50 4 - 300 20 1 300 30 3 Subtract the 1 s 4– 1=? Finally recombine and write the answer: 300 + 3 = 333 Subtract the 10 s 50 – 20 = ? Subtract the 100 s 600 – 300 = ? © hamilton-trust. org. uk 15 Year 3/4

Day 1: Subtract 3 -digit numbers using expanded decomposition. 40 600 50 - 300 20 How could we work out 651 – 324? Partition the numbers and set it out. 11 1 4 7 651 – 324 = 327 4 is bigger than 1! Now subtract the 1 s, 10 s and 100 s. We take 10 from 50 and add it to the 1 s. What do you notice about 50 + 1 and 40 + 11? This is called decomposition. © hamilton-trust. org. uk 16 Year 3/4

Day 1: Subtract 3 -digit numbers using expanded decomposition. How could we work out 742 – 328? 30 700 40 - 300 20 400 10 Set it out like before. 12 2 8 4 Look how we change the numbers. 12 – 8 = ? 8 is bigger than 2. We take 10 from 40 and add it to the 1 s. 30 – 20 = ? 700 – 300 = ? So, 742 – 328 = 414. © hamilton-trust. org. uk 17 Year 3/4

Day 1: Subtract 3 -digit numbers using expanded decomposition. 30 40 - 300 30 100 0 Now on your whiteboards try 445 – 339. 15 5 9 6 Set it out like this. 15 – 9 = ? Let’s check through. 30 – 30 = ? 400 – 300 = ? 9 is bigger than 5. We take 10 from 40 and add it to the 1 s. © hamilton-trust. org. uk So, 445 – 339 = 106. 18 Year 3/4

Challenge © hamilton-trust. org. uk 19 Year 3/4

More Addition and Subtraction Formal algorithms in addition and subtraction Objectives Day 2 Add 3 -digit numbers using expanded addition Subtract 3 -digit numbers using expanded decomposition © hamilton-trust. org. uk 20 Year 3/4

Day 2: Add 3 -digit numbers using expanded addition; Subtract 3 -digit numbers using expanded decomposition. Let’s add 346 and 437. 300 40 6 + 400 30 7 10 Partition the numbers and line them up neatly. Don’t forget a blank space under the second number. 700 80 3 700 + 80 + 3 = 783 Add the 1 s. 6 + 7 = 13. The 1 s come to more than 10 so we write 10 under the 10 s in the waiting line and 3 under the 1 s in the answer line. Next add the 10 s. . . 40 + 30 + 10 = ? Lastly the 100 s. . . 300 + 400 = ? © hamilton-trust. org. uk Finally recombine 700, 80 and 3. 21 Year 3/4

Day 2: Add 3 -digit numbers using expanded addition; Subtract 3 -digit numbers using expanded decomposition. Now let’s try 473 + 257. 400 70 3 + 200 50 7 100 10 Set it out neatly! Will there be an extra 10 or 100 this time? How do you know? 700 30 0 700 + 30 + 0 = 730 Add the 1 s, 10 s and 100 s and recombine. This time the 1 s add to more than 10 AND the 10 s add to more than 100. © hamilton-trust. org. uk 22 Year 3/4

Day 2: Add 3 -digit numbers using expanded addition; Subtract 3 -digit numbers using expanded decomposition. Work with a partner to generate other additions where 100 and or 10 will need to be written on the waiting line. Let’s check out some of your ideas… © hamilton-trust. org. uk 23 Year 3/4

Challenge © hamilton-trust. org. uk 24 Year 3/4

Day 2: Subtract 3 -digit numbers using expanded decomposition. 600 50 7 - 400 30 2 200 20 5 How could we work out 657 – 432? Partition the numbers and set it out like this. Now subtract the 1 s, 10 s and 100 s. Don’t forget to recombine at the end: 200 + 20 + 5 = 225. © hamilton-trust. org. uk 25 Year 3/4

Day 2: Subtract 3 -digit numbers using expanded decomposition. 500 600 - 400 100 Now let’s try 657 – 472. First subtract the 1 s. 7– 2=? 150 50 7 70 2 80 5 70 is bigger than 50… What might we do about this? What do you notice about the 10 s? Now we can subtract the 10 s. 150 – 70 = ? © hamilton-trust. org. uk 657 – 472 = 185 …so take 100 from 600 and add it to the 10 s. Finally subtract the 100 s. 500 – 400 = ? 26 Year 3/4

Day 2: Subtract 3 -digit numbers using expanded decomposition. Now on your whiteboards try these 465 – 283, 625 – 362 and 936 – 571. Set them out neatly and think carefully about when you need to do any regrouping. Let’s check 465 – 283. 300 400 - 200 100 80 is bigger than 60. We take 100 from 400 and add it to the 10 s. 160 60 5 80 3 80 2 5– 3=? 160 – 80 = ? 300 – 200 = ? 465 – 283 = 182 © hamilton-trust. org. uk 27 Year 3/4

Day 2: Subtract 3 -digit numbers using expanded decomposition. 60 is bigger than 20. We take 100 from 600 and add it to the 10 s. Let’s check 625 – 362. 500 600 - 300 200 120 20 5 60 2 60 3 5– 2=? 120 – 60 = ? 625 – 362 = 263 © hamilton-trust. org. uk 28 500 – 300 = ? Year 3/4

Day 2: Subtract 3 -digit numbers using expanded decomposition. 70 is bigger than 30. We take 100 from 900 and add it to the 10 s. Let’s check 936 – 571. 800 900 - 500 300 130 30 6 70 1 60 5 6– 1=? 130 – 70 = ? 936 – 571 = 365 © hamilton-trust. org. uk 29 800 – 500 = ? Year 3/4

Challenge © hamilton-trust. org. uk 30 Year 3/4

More Addition and Subtraction Formal algorithms in addition and subtraction Objectives Day 3 Add 3 -digit numbers using expanded addition, estimate totals Subtract 3 -digit numbers using expanded decomposition © hamilton-trust. org. uk 31 Year 3/4

Day 3: Add 3 -digit numbers using expanded addition, estimate totals; Subtract 3 digit numbers using expanded decomposition. In the first we moved 10 s to 1 s, in the second Y 4 s – who can explain we moved 100 s to 10 s. expanded decomposition to Y 3? Who can talk us through this one? 50 400 60 - 200 30 20 10 0 8 2 600 700 - 300 734 – 351 = 383 460 – 238 = 222 © hamilton-trust. org. uk 130 30 4 50 1 80 3 32 Year 3/4

Day 3: Add 3 -digit numbers using expanded addition, estimate totals; Subtract 3 digit numbers using expanded decomposition. Y 4 are going to look at reversing digits and then subtracting, e. g. 753 – 735 and investigate the digit sum of the answers…. 40 700 50 - 700 30 0 10 © hamilton-trust. org. uk 753 – 735 = 18. So the digit sum is 1 + 8 = 9. 13 3 5 8 33 Year 3/4

Challenge © hamilton-trust. org. uk 34 Year 3/4

Day 3: Add 3 -digit numbers using expanded addition, estimate totals. What would be a good estimate for 418 + 235? We can round each number to the nearest 100. A good estimate would be 600. Let’s go through that using expanded addition. 400 10 8 + 200 30 5 10 600 50 3 600 + 50 + 3 = 653 © hamilton-trust. org. uk 35 Year 3/4

Day 3: Add 3 -digit numbers using expanded addition, estimate totals. What would be a good estimate for 492 + 233? A good estimate would be 700. Who can explain why? Let’s check using expanded addition. 400 90 2 + 200 30 3 100 700 20 5 700 + 20 + 5 = 725 © hamilton-trust. org. uk 36 Year 3/4

Day 3: Add 3 -digit numbers using expanded addition, estimate totals. 227 + 109, 582 + 321 and 273 + 164. In pairs agree an estimate for each of these. Ask each third of the class to calculate one answer and compare with the estimates. © hamilton-trust. org. uk 37 Year 3/4

More Addition and Subtraction Formal algorithms in addition and subtraction Objectives Day 4 Subtract a 2 -digit number from a 3 -digit number using counting up (Frog) Subtracting using decomposition or Frog © hamilton-trust. org. uk 39 Year 3/4

Day 4: Subtract a 2 -digit number from a 3 -digit number using counting up (Frog); Subtracting using decomposition or Frog helps us to find the difference between two numbers (the size of the gap between them on the number line). Frog starts at the smaller number, hops to the next ‘friendly number’ (the next 10 or 100) until landing on the larger number. Then we add the hops to find the difference. © hamilton-trust. org. uk 40 Year 3/4

Day 4: Subtract a 2 -digit number from a 3 -digit number using counting up (Frog); Subtracting using decomposition or Frog. Add the jumps. Let’s try 136 – 87. 36 + 10 + 3 = ? Frog jumps 3 to 90… … and then 10 to 100. . . … then 36 to 136. 10 3 9 87 0 36 100 © hamilton-trust. org. uk 41 Year 3/4

Day 4: Subtract a 2 -digit number from a 3 -digit number using counting up (Frog); Subtracting using decomposition or Frog. Add the jumps. Let’s try 143 – 75. 43 + 20 + 5 = ? Frog jumps 5 to 80… … and then 20 to 100. . . … then 43 to 143. 20 5 75 80 43 100 © hamilton-trust. org. uk 42 Year 3/4

Day 4: Subtract a 2 -digit number from a 3 -digit number using counting up (Frog); Subtracting using decomposition or Frog. Using Frog we found 143 – 75 = 68. How can we check this? Try adding 68 and 75 on you whiteboards. It should equal 143. Were we right? © hamilton-trust. org. uk 43 Year 3/4

Day 4: Subtract a 2 -digit number from a 3 -digit number using counting up (Frog); Subtracting using decomposition or Frog. Let’s try 515 – 475. Y 3 s use Frog, Y 4 s decomposition. Now decomposition. 400 500 - 400 0 110 10 5 70 5 40 0 515 – 475 = 40 Let’s check with Frog. 20 + 15 + 5 = 40 20 5 475 15 48 0 © hamilton-trust. org. uk 500 44 515 Year 3/4

Day 4: Subtract a 2 -digit number from a 3 -digit number using counting up (Frog); Subtracting using decomposition or Frog. Which do you prefer? Today we can choose a method. Try 807 - 766. What do you think will be the best method to use? © hamilton-trust. org. uk 45 Year 3/4

Challenge © hamilton-trust. org. uk 46 Year 3/4

Challenge © hamilton-trust. org. uk 48 Year 3/4

More Addition and Subtraction Formal algorithms in addition and subtraction Well Done! You’ve completed this unit. Objectives Day 1 Add 3 -digit numbers using expanded addition Subtract 3 -digit numbers using expanded decomposition Day 2 Add 3 -digit numbers using expanded addition Subtract 3 -digit numbers using expanded decomposition Day 3 Add 3 -digit numbers using expanded addition, estimate totals Subtract 3 -digit numbers using expanded decomposition Day 4 Subtract a 2 -digit number from a 3 -digit number using counting up (Frog) Subtracting using decomposition or Frog © hamilton-trust. org. uk 49 Year 3/4

Problem solving and reasoning questions Year 3 Add four hundred and fifty-six to three hundred and seventy-two. What is the total of two hundred and sixty-eight and two hundred and eighty-six? Write the missing numbers 643 + 174 = ☐ 327 + 258 = ☐ 416 + ☐ = 888 Amit uses 346 Lego pieces in building his model X-wing, and his sister uses 287 in building her Millennium Falcon. How many Lego pieces have they used altogether? Fill the gaps in this subtraction: 81☐ – 4☐ 7 = ☐ 46 © hamilton-trust. org. uk 50 Year 3/4

Problem solving and reasoning answers Year 3 Add four hundred and fifty-six to three hundred and seventy-two. 828. Have children estimated? In this and subsequent addition questions, have children set out the expanded addition correctly? What is the total of two hundred and sixty-eight and two hundred and eighty-six? 554 Write the missing numbers 643 + 174 = 817 327 + 258 = 585 416 + 472 = 888 (Solve by subtracting 416 from 888). Amit uses 346 Lego pieces in building his model X-wing, and his sister uses 287 in building her Millennium Falcon. How many Lego pieces have they used altogether? 633 Fill the gaps in this subtraction: 813 – 467 = 346 © hamilton-trust. org. uk 51 Year 3/4

Problem solving and reasoning questions Year 4 Complete each diagram: 912 604 868 287 1003 578 Here is Ali’s homework. a. 403 – 265 b. 648 – 356 c. 752 – 199 d. 812 – 754 Advise him on the most efficient or error-proof subtraction strategy to calculate each one, explaining why you are suggesting that method. Calculate the answers. Complete this subtraction calculation that uses decomposition: 700 30 4 300 60 5 © hamilton-trust. org. uk 52 Year 3/4

Problem solving and reasoning answers Year 4 Complete each diagram: 912 604 287 317 44 1003 868 578 425 Here is Ali’s homework. a. 403 – 265 = 138 Best solved by Frog (counting up) since 403 is just larger than a multiple of 100; the zero makes decomposition tricky. b. 648 – 356 = 292 Best solved by expanded decomposition as the gap is large. c. 752 – 199 = 553 Best solved by subtracting 200 and adjusting by adding back 1. d. 812 – 754 = 58 Best solved by Frog (counting up) since the numbers are close and either side of a multiple of 100. Complete this subtraction calculation that uses decomposition: © hamilton-trust. org. uk 53 Year 3/4