Master Mathematical Thinking
Place Value 1
Learning Outcomes
Students will be able to: write the place value and expanded form for 4-digit numbers. show a 4-digit number using the abacus and place value blocks and write its number name. compare 2 or more 4-digit numbers and arrange them in ascending and descending order. form 4-digit numbers using the given 4 digits. round off numbers up to 3 digits to the nearest 10 using a number line.
Alignment to NCF
C-1.1: Represents numbers using the place value structure of the Indian number system, compares whole numbers, and knows and can read the names of very large numbers
Let’s Recall
Recap to check if students know how to read, write and show 2-digit and 3-digit numbers. Ask students to solve the questions given in the Let’s Warm-up section.
Vocabulary
place value: value of a digit based on its position in the number abacus: a set of rods, showing the place of digits in a number, with beads that slide along the rods expanded form: the sum of the values of all the digits in a number round off: to write the closest 10 or 100 number instead of the actual number
Teaching Aids
Place value chart showing 1000s, 100s, 10s and 1s; Place value blocks of 1000s, 100s, 10s and 1s; 4-rod abacus; Number line showing numbers from 300 to 400
Chapter: Place Value
Place Value and Expanded Form of 4-digit Numbers
Learning Outcomes
Students will be able to write the place value and expanded form for 4-digit numbers.
Teaching Aids
Place value chart showing 1000s, 100s, 10s and 1s; Place value blocks of 1000s, 100s, 10s and 1s
Activity
Distribute the place value charts and blocks to the students.
Instruct the students to show the 4-digit numbers 4319 and 6725 using place value blocks and on the place value chart by writing each digit in its correct place. This will help them identify the place value of each digit in a number. Then, in their notebooks, they will write the number of thousands, hundreds, tens and ones, and the numbers in their expanded form.
Ask questions like: How many thousands, hundreds, tens and ones are there in the number? How did you write the number using the place value and the expanded form?
Extension Idea
Ask: How will you write the expanded form for a number which has 2 hundreds more than 8623?
Say: 8623 has 6 hundreds. 2 hundreds more than 8623 will give us 8823. So, the expanded form of 8823 = 8 thousands + 8 hundreds + 2 tens + 3 ones = 8000 + 800 + 20 + 3.
Number Names; 4-digit Numbers on the Abacus
Learning Outcomes
Students will be able to show a 4-digit number using the abacus and place value blocks and write its number name.
Teaching Aids
Place value blocks of 1000s, 100s, 10s and 1s; 4-rod abacus
Activity
Discuss with the students how to show a number on an abacus and tell them that the number of beads on the rod of an abacus tells us the value of the digit.
Ask the students to take the place value blocks and the abacus and show the number 5691 using each. They will then write the number of thousands, hundreds, tens and ones; the number in its expanded form; and as a number name. Finally, they will compare to see if they got the same number of 1000s, 100s, 10s and 1s using both the place value blocks and the abacus.
Ask questions like: How is showing 1000s, 100s, 10s and 1s on the abacus different from showing them using place value blocks?
Learning Outcomes
Students will be able to compare 2 or more 4-digit numbers and arrange them in ascending and descending order.
Teaching Aids
Place value blocks of 1000s, 100s, 10s and 1s
Activity
Instruct the students to work in groups and use the place value blocks to show 2 numbers, 1234 and 1252, using the place value blocks. They will first compare the number of thousands blocks in both numbers. The number with the greater 1000s blocks is bigger. If the 1000s blocks are the same, tell them to compare the 100s blocks, then the 10s and finally the 1s blocks. Ask them to write the answer in their notebooks. Ask them to repeat with 2 more numbers and then write all the numbers in ascending order, in their notebooks.
Extension Idea
Ask: Which is a smaller number: 5000 + 300 + 20 or five thousand three hundred two?
Say: 5000 + 300 + 20 is 5320 and five thousand three hundred two is 5302. 5320 has two 10s and 5302 has zero 10s. So, 5302 is smaller or 5302 < 5320.
Forming 4-digit Numbers
Learning Outcomes
Students will be able to form 4-digit numbers using the given 4 digits.
Teaching Aids
4-rod abacus
Activity
Discuss with the students how, in an abacus, when the number of beads are arranged in descending order, starting from the highest value to the lowest value, the digits are also arranged from the highest to the lowest, thus forming the largest number. Show them an abacus where the beads are arranged in descending order to form the number 7643.
Ask the students to form groups and take 1 abacus. Instruct them to show the smallest and then the largest 4-digit numbers using each of the digits, 5, 1, 8, 6, only once. Then, in their notebooks, they will then draw the abacus for each number and write the numbers formed.
Ask questions like: Which rod will have no beads when we want to form the smallest number using the digits 3, 1, 0 and 6?
Extension Idea
Ask: How many 3-digit numbers can be formed using the digits 1, 2 and 3 only once, such that the number can be divided by 2, leaving no remainder?
Say: The possible 3-digit numbers using each of the digits 1, 2 and 3 only once are: 123, 132, 231, 213, 312, and 321. Out of these, the numbers that can be divided by 2 with no remainder are 132 and 312.
Learning Outcomes
Students will be able to round off numbers up to 3 digits to the nearest 10 using a number line.
Teaching Aids
Number line showing numbers from 300 to 400
Activity
Discuss the terms ‘about’ and ‘estimate’ that show rounded off numbers.
Distribute number lines from 300 to 400 with 10 divisions shown between 350 and 360.
Ask the students to show the numbers 353 and 358 on the number line, find the nearest 10 and 100 for each number and write the rounded off numbers in their notebooks.
Extension Idea
Ask: A number, when rounded off to the nearest 10, reaches 80. The number was an even number less than 80 and more than 77. What is the number?
Say: The numbers that can be rounded off to 80 are: 75, 76, 77, 78, 79, 80, 81, 82, 83, 84. But the number is an even number so the number can be 76, 78, 80, 82 or 84. Since the number is less than 80 and more than 77, the number is 78.
1. Place Value and Expanded Form of 4-digit Numbers
Do It Together
Th H T O
4 3 1 9
Answers
4319 = 4000 + 300 + 10 + 9
Standard Form Expanded Form
9 ones or 9 × 1 = 9
1 tens or 1 × 10 = 10
3 hundreds or 3 × 100 = 300
4 thousands or 4 × 1000 = 4000
2. Number Names
Do It Together
3000 + 400 + 50 + 7
three thousand fifty seven four hundred
The number 3457 can be read as three thousand four hundred fifty-seven
3. 4-digit Numbers on the Abacus
Do It Together
Th H T O 5 2 4 1
The number 5241 is read as five thousand two hundred forty-one.
4. Comparing Numbers; Ordering Numbers
Do It Together
Start comparing from the thousands place.
1765 4372 6145 4538 6 > 4 > 1
Compare the hundreds place for 4372 and 4538.
4372 4538 5 > 3
The ascending order is: 1765 < 4372 < 4538 < 6145.
5. Forming 4-digit Numbers
Do It Together
Smallest number place the digits in ascending order: 3059
Th H T O 3 0 5 9
Greatest number place the digits in descending order: 9530
Th H T O 9 5 3 0
Therefore, 3059 is the smallest and 9530 is the greatest 4-digit number formed using the digits 3, 5, 9, 0.
6. Rounding off 3-digit Numbers
Do It Together
Closer to 290
Look for the nearest 10 of the number on the number line. 286 is between 280 and 290 but is closer to 290 Therefore, it is rounded off to 290.
Addition of 3-digit Numbers 2
Learning Outcomes
Students will be able to:
add using the order property and addition by 1 and 0 properties. mentally add 2-digit numbers by expanding numbers. add two 3-digit numbers without regrouping. add two 3-digit numbers with regrouping. add two 4-digit numbers without regrouping. round off two 2-digit numbers and estimate their sum. solve word problems on adding numbers with 3 or 4 digits.
Alignment to NCF
C-1.3: Understands and visualises arithmetic operations and the relationships among them, knows addition and multiplication tables at least up to 10 × 10 (pahade) and applies the four basic operations on whole numbers to solve daily life problems
C-4.3: Selects appropriate methods and tools for computing with whole numbers, such as mental computation, estimation, or paper-pencil calculation, in accordance with the context
Let’s Recall
Recap to check if students know how to add two numbers using pictures. Ask students to solve the questions given in the Let’s Warm-up section.
Vocabulary
successor: the number that comes after a number regrouping: rearranging numbers into groups according to their place values estimating: finding the approximate value of a number or a
Teaching Aids
Small boxes; Green and blue marbles; Chits with numbers, along with the method (add mentally or add by expanding numbers); Place value blocks of 100s, 10s and 1s; Straws in red, blue, yellow and white; Sheets with blank number lines drawn on them; Word problem written on a sheet of paper with spaces given for each element of the CUBES strategy
Chapter: Addition of 3-digit Numbers
Properties of Addition
Learning Outcomes
Students will be able to add using the order property and addition by 1 and 0 properties.
Teaching Aids
Small boxes; Green and blue marbles
Activity
Instruct the students to work in groups. Distribute 2 boxes of marbles to each group with 4 green and 5 blue marbles in one box and 5 green and 4 blue marbles in another box.
Ask the students to count the green and blue marbles, and the total number of marbles in the first box. Help them to write the addition sentence for it in their notebooks. Then, ask them to count the green and blue marbles in the second box, along with the total, and write the addition sentence.
Discuss with them how the order of addition doesn’t change the final result.
Next, ask them to put 4 green marbles in one box. Ask them to add 1 more green marble and tell the sum. Explain that when we add 1 to any number, we get its successor. Now, ask the students to put 5 blue marbles in a box and to not add anything to it. Ask them to show this with the help of an addition sentence. Discuss the fact that when 0 is added to any number, the number remains unchanged.
Extension Idea
Ask: Which is bigger: 67 + 1 or 67 + 0?
Say: When we add 1 to any number, the sum is the successor of that number, so 67 + 1 = 68. Similarly, if we add 0 to any number, the sum remains the same, so 67 + 0 = 67. Therefore, 67 + 1 is bigger.
Addition by Expanding Numbers; Adding by Counting Forward
Learning Outcomes
Students will be able to mentally add 2-digit numbers by expanding numbers.
Teaching Aids
Imagine Maths Page 19
Chits with numbers, along with the method (add mentally or add by expanding numbers)
Activity
Instruct the students to work in groups. Distribute the chits with numbers among the groups. Write a 2-digit addition problem on the board, like 23 + 12. Discuss both the the methods of adding numbers –adding mentally by counting forward and adding by expanding smaller numbers. Instruct them to read the addition sums on the chits carefully and find the sum using the given method. The group that solves all the sums first will win a point. Make them practice more such problems.
Addition
without Regrouping
Learning Outcomes
Students will be able to add two 3-digit numbers without regrouping.
Teaching Aids
Place value blocks of 100s, 10s and 1s
Activity
Instruct the students to work in groups. Distribute the place value blocks among the groups. Display an addition problem, like 435 + 242, on the board. Introduce the column method by creating 3 columns (hundreds, tens and ones) and writing the digits of 435 and 242 in their corresponding columns.
Instruct the students to copy the table and place the same number of 100s, 10s and 1s blocks as given in both numbers. Ask them to count the place value blocks of each type and record the total below their corresponding columns. Give them 1 more problem to solve.
Addition with Regrouping Imagine Maths Page 24
Learning Outcomes
Students will be able to add two 3-digit numbers with regrouping.
Teaching Aids
Place value blocks of 100s, 10s and 1s
Activity
Instruct the students to work in groups. Distribute the place value blocks among the groups.
Instruct them to show 545 and 456 using the blocks by placing them side by side. Ask them to add the 1s blocks. Discuss how ten 1s blocks become one 10s block. Ask them to now add the 10s blocks. Discuss how ten 10s blocks become one 100s block. Ask them to now add the 100s blocks. Ask how many blocks they have after adding the 100s blocks. Discuss how ten 100s blocks become one 1000s block. Then, in their notebooks, they will add the numbers using the vertical method and verify their answer. Ask questions like: How will you show thirty 100s blocks in 1000s blocks?
Extension Idea
Ask: How many 100s should we add to 625 to regroup the 100s as 1000s?
Say: There are 6 hundreds in 625. Let us count forward to make it 1000 as 700, 800, 900, 1000. We need to add 4 hundreds to 625 to regroup the 100s as 1000s.
Learning Outcomes
Students will be able to add two 4-digit numbers without regrouping.
Teaching Aids
Straws in red, blue, yellow and white
Activity
Instruct the students to work in groups. Distribute the straws among the groups.
Display an addition problem, like 2541 + 7354, on the board. Introduce the column method by creating four columns (thousands, hundreds, tens and ones) and writing the digits of 2541 and 7354 in their corresponding columns. Specify that the red straws represent thousands, the blue straws represent hundreds, the yellow straws represent tens and the white straws represent ones.
Instruct the students to copy the table and organise the straws based on the numbers: 2541 with 2 red, 5 blue, 4 yellow and 1 white straw; and 7354 with 7 red, 3 blue, 5 yellow and 4 white straws. Ask them to count the straws of each colour and record the total below their corresponding columns.
Estimating
the Sum
Learning Outcomes
Students will be able to round off two 2-digit numbers and estimate their sum.
Teaching Aids
Sheets with blank number lines drawn on them
Activity
Instruct the students to work in pairs.
Distribute the sheets with number lines drawn to the students.
Give them two 2-digit numbers, say 48 and 64, and ask the students to first write numbers on the number lines that have these numbers on them, and then find the nearest 10 for each number. Ask them to add the rounded-off numbers in their notebooks. Explain that the sum they got is the estimated sum. Ask them to find the actual sum and compare it with the estimated sum. Give more problems for practice.
Extension Idea
Ask: Find the estimated sum of 10 + 23. Is it >, = or < than the estimated sum of 17 + 12?
Say: 10 + 23 on rounding off will be 10 + 20 = 30. 17 + 12 on rounding off will be 20 + 10 = 30. So, the estimated sums of both the problems are equal.
Learning Outcomes
Students will be able to solve word problems on adding numbers with 3 or 4 digits.
Teaching Aids
Word problem written on a sheet of paper with space given for each element of the CUBES strategy
Activity
Instruct the students to work in groups. Distribute the word problem sheets among the groups. Instruct them to circle the numbers, underline the question and box the key words. Discuss with the students what needs to be found. Then, ask them to solve and write the answer using the vertical method.
Ask questions like: How many 1000s, 100s, 10s and 1s blocks will you need to show the total number of t-shirts?
Extension Idea
Instruct: Create your own word problem where you need to add 1542 and 3435.
Say: There can be many such word problems. One can be: A toy factory produced 1542 toys in the month of January and 3435 toys in the month of February. What is the total number of toys produced in these 2 months?
A cloth store sold 526 shirts and 745 t-shirts in a month. How many shirts and t-shirts were sold altogether?
C Circle the numbers.
U Underline the question. B Box the key words.
E Evaluate/draw S Solve and check. 3
1. Properties of Addition
Do It Together
1 6 + 9 = 9 + 6 2 13 + 1 = 14
3 8 + 0 = 8 4 18 + 1 = 19
2. Addition by Expanding Numbers
Do It Together
1 15 + 53 15 → 10 + 5 53 → 50 + 3 1 5 → 10 + 5 5 3 → 50 + 3 6 8 ← 60 + 8
Thus, 15 + 53 = 68.
3. Adding by Counting Forward
Think and Tell
The smallest 2-digit number is 10.
The largest 2-digit number = 99
So, the sum is 109.
Do It Together
Keep the number 32 in mind.
Skip count by 5 tens.
Count forward by 3 ones.
32 + 53 = 85
4. Addition without Regrouping
Do It Together H T O 6 5 4 + 3 2 1 9 7 5 654 + 321 = 975
Answers
5. Addition with Regrouping Do It Together Do It Together Th
5 876 + 129 = 1005
6. Adding 4-digit Numbers Do It Together Th H T O 1 4 5 4 + 2 3 2 3 3 7 7 7 1454 + 2323 = 3777
7. Estimating the Sum Do It Together
24 rounded off to the nearest 10 is 20 78 rounded off to the nearest 10 is 80.
So, the estimated sum of 24 and 78 is 20 + 80 = 100.
100 is close to 102. So, the estimated sum is close to the actual sum.
8. Word Problems Do It Together
Number of girls = 480
Number of boys = 435
Total number of students = 480 + 435
Thus, the total number of students in the school is 915
Subtraction of 3-digit Numbers 3
Learning Outcomes
Students will be able to:
subtract using the property of subtraction by 1, 0 or the same number. mentally subtract a 2-digit number from a 2-digit number. subtract a 3-digit number from a 3-digit number without regrouping. subtract a 3-digit number from a 3-digit number with regrouping. subtract a 4-digit number from a 4-digit number without regrouping. round off two 2-digit numbers and estimate their difference. solve word problems on subtracting numbers with 3 or 4 digits.
Alignment to NCF
C-1.3: Understands and visualises arithmetic operations and the relationships among them
C-4.3: Selects appropriate methods and tools for computing with whole numbers, such as mental computation, estimation, or paper pencil calculation, in accordance with the context
Let’s Recall
Recap to check if students know how to subtract 1-digit numbers. Ask students to solve the questions given in the Let’s Warm-up section.
Vocabulary
regrouping: rearranging numbers into groups according to their place values subtraction: finding the difference of two numbers
Teaching Aids
Counters; Paper cups; Number grid (1 to 100); Place value blocks; Place value chart; Paper slips with numbers up to 50; Bowl; Word problem sheets with spaces for each element of the CUBES strategy
Chapter: Subtraction of 3-digit Numbers
Learning Outcomes
Students will be able to subtract using the property of subtraction by 1, 0 or the same number.
Teaching Aids
Counters; Paper cups
Activity
Distribute a few counters of different colours and a paper cup to each pair. Instruct them to put a few counters in the cup, and then take away 1 counter and give it to their partner. Then, ask them the number of counters they are left with. Ask them to write the subtraction sentence in their notebooks. Repeat the process for taking away 0 counters and for taking away all the counters that they have.
Discuss the properties of subtraction: subtracting 0, subtracting 1 and subtracting the same number. Bring out the fact that, on subtracting 1, the number we get is nothing but the predecessor of the given number.
Extension Idea
Ask: What is the predecessor of one hundred forty?
Say: One hundred forty is 140. So, 140 – 1 is 139. Thus, 139 is the predecessor of 140.
By Counting Forward on a Number Line; By Expanding the Smaller Number
Learning Outcomes
Students will be able to mentally subtract a 2-digit number from a 2-digit number.
Teaching Aids
Number grid (1 to 100); Counters
Activity
Distribute a number grid with numbers up to 100 and some counters to each group. Instruct them to use the counters to subtract 2-digit numbers on the number grid. Write a subtraction sentence on the board, say 45 – 24 = ?
Ask: What number would you start from to find the difference? In which direction would you move your counter to subtract?
Instruct them to start from the bigger number and move the counters up by jumping to the previous ten to subtract the tens first. Then, tell them to move to the left to subtract the ones. Explain that the number that they land on is the difference of the 2 numbers.
Ask the students to find the difference of 84 and 31.
Subtraction without Regrouping
Learning Outcomes
Students will be able to subtract a 3-digit number from a 3-digit number without regrouping.
Teaching Aids
Place value blocks
Activity
Instruct the students to form groups. Distribute the place value blocks to each group. Write a subtraction sentence on the board, say 245 – 124 = .
Ask them to use the place value blocks to subtract the 2 numbers.
Instruct the students to show the subtraction using the place value blocks by first showing the bigger number using the place value blocks and then removing the place value blocks equivalent to the smaller number. Now, ask them the value of the remaining place value blocks. Then, instruct them to solve the subtraction problem by writing the numbers one below the other in their notebooks. Finally, ask them to compare the answers they got using the 2 methods.
Extension Idea
Ask: From what number should 730 be subtracted from, to get 250?
Say: To find the answer, we will add 730 and 250. So, 730 + 250 is 980.
Subtraction with Regrouping;
Learning Outcomes
Students will be able to subtract a 3-digit number from a 3-digit number with regrouping.
Teaching Aids
Place value blocks
Activity
Instruct the students to form groups. Distribute the place value blocks to each group. Write a subtraction sentence on the board, say 245 – 156 = .
Instruct the students to show the subtraction using the place value blocks. Take as many place value blocks as the bigger number. Start subtracting the ones by taking away as many blocks as the smaller number. If the number of ones are not enough, remove 1 tens blocks with 10 ones blocks, and then take away. Now, repeat the process for tens and hundreds. The remaining blocks show the answer. Then, instruct them to solve the subtraction problem by writing the numbers one below the other. Ask them to write the answers in their notebooks and to compare the answers they got using the two methods. Ask questions like: How will you subtract in the ones place if you have fewer ones to subtract from?
Subtracting 4-digit Numbers
Learning Outcomes
Students will be able to subtract a 4-digit number from a 4-digit number without regrouping.
Teaching Aids
Place value chart; Counters
Activity
Instruct the students to form groups.
Write a subtraction sentence on the board, say 9687 – 7564 =
Instruct the students to show the subtraction using the place value chart.
Instruct the students to show the bigger number on the place value chart using counters and remove as many counters equivalent to the smaller number. Ask them the value of the remaining counters on the place value chart and say that number is the answer.
Then, instruct them to solve the subtraction problem by writing the numbers one below the other, in their notebooks. Finally, ask them to compare the answers they got using the 2 methods.
Extension Idea
Ask: What is two thousand five hundred sixty-seven less than 8977?
Say: Two thousand five hundred sixty-seven is 2567. So, 8977 – 2567 is 6410.
Estimating the Difference
Learning Outcomes
Students will be able to round off two 2-digit numbers and estimate their difference.
Teaching Aids
Paper slips with numbers up to 50; Bowl; Place value blocks
Activity
Divide the class into groups.
Call a student from each group to pick 2 numbers from the paper slips in the bowl kept on the table.
Instruct them to to show the numbers using place value blocks and check if the number of ones blocks is less than 5. They will then remove the ones blocks to get the rounded off number. If the number of ones is 5 or more, they will remove the ones blocks and add a tens block to get the rounded off number. Finally, they will subtract the 2 numbers to get the estimated difference. Encourage them to write the subtraction sentence in their notebooks. Repeat the activity once more if time permits.
Learning Outcomes
Students will be able to solve word problems on subtracting numbers with 3 or 4 digits.
Teaching Aids
Word problem sheets with spaces for each element of the CUBES strategy
Activity
Instruct the students to form groups. Distribute the word problem sheets to each group. Instruct them to circle the numbers, underline the question and box the keywords. Discuss with the students what needs to be found.
Ask them to then solve and write the answer using the vertical method, in their notebooks.
Extension Idea
C Circle the numbers.
Anna has 541 stamps. Susan has 167 stamps. How many stamps do they have in all?
U Underline the question. B Box the key words. E Evaluate/draw S Solve and check. 3
Instruct: Create and solve a word problem using the numbers 118 and 145. Say: There can be multiple word problems. One such problem can be: There are 118 Christmas bells in a box. Jane bought some more bells and added to the box. How many bells did Jane buy if the total number of bells in the box increases to 145?
Answers
1. Properties of Subtraction Do
2. By Counting Forward on a Number Line
3. By Expanding the Smaller Number Do It Together
35 – 12 = 23 The difference between 35 and 12 is 23
– 10 = 35; 35 – 6 = 29 The difference of 45 and 16 is 29.
4. Subtraction without Regrouping Do It
5. Subtraction with Regrouping
Do It Together
1.
640 – 386 = 254
–
6. Adding and Subtracting Together
Do It Together
8. Estimating the
Estimated difference = 30 – 20 = 10
9. Word Problems Do It Together
Manish has 423 seashells. He has 345 more seashells than Babita. How many seashells does Babita have?
Solve:
Evaluate: 423 – 345 = ?
7. Subtracting 4-digit Numbers
Do It
So, 3789 – 1569 = 2220.
Babita has 78 seashells.