2. Arithmetic expressions

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Topics

2.1 Simple Expressions

• Explanation: Introduces mathematical phrases containing numbers and operations that evaluate to a specific value.

• Example: The expression $13+2$ evaluates to 15.

Figure it out

• Fill in the blanks
• Ascending order

Comparing Expressions

• Explanation: Demonstrates how to compare expressions using '<', '>', or '=' signs, often by reasoning through a scenario rather than fully computing the values.

• Example: To compare $113 - 25$ and $112 - 24$, you can reason that starting with 1 less marble but also losing 1 less marble results in the exact same amount, so $113 - 25 = 112 - 24$.

BODMAS rules consideration

BODMAS = PEMDAS = GEMDAS

2.2 Reading and Evaluating Complex Expressions

• Explanation: Explains that expressions with multiple operations need rules and tools to avoid ambiguity in how they are evaluated.

• Example: Without rules, $30 + 5 \times 4$ could be incorrectly evaluated as 140 (adding first) instead of the correct 50 (multiplying first).

Brackets in Expressions

• Explanation: Shows how brackets are used to explicitly state the order of operations; expressions inside brackets are evaluated first.

• Example: Calculating the change from ₹100 after spending ₹15 and ₹56 is written as $100 - (15 + 56)$.

Terms in Expressions

• Explanation: Defines "terms" as the parts of an expression separated by a '+' sign. To identify terms with subtraction, subtraction is treated as adding the inverse (a negative number).

• Example: The terms of the expression $83 - 14$ are 83 and -14, because it can be written as $(83) + (-14)$.

Swapping and Grouping

• Explanation: Explores the commutative property (swapping terms doesn't change the sum) and associative property (grouping terms doesn't change the sum) of addition.

• Example: $Term 1 + Term 2 = Term 2 + Term 1$. For grouping, $(-7) + 10 + (-11)$ yields the same result regardless of which two terms are added together first.

Removing Brackets-I & II

• Explanation:

  • Details how signs change when removing brackets.
  • If preceded by a negative sign, terms inside the bracket change signs.
  • It also covers the distributive property, showing that multiplying a number by a sum is the same as the sum of the individual multiples.

• Example (Sign Change):

  • $200 - (40 + 3)$ becomes $200 - 40 - 3$.
  • $500 - (250 - 100)$ becomes $500 - 250 + 100$.

• Example (Distributive): $(4 + 3) \times 5$ is equal to $4 \times 5 + 3 \times 5$.

Tinker the Terms I & II

• Explanation: Focuses on logically adjusting numbers within expressions or products to make mental computation easier.

• Example: To find $63 \times 18$, you can use a known value like $53 \times 18 = 954$, and write it as $(53 + 10) \times 18 = 53 \times 18 + 10 \times 18 = 954 + 180 = 1134$

Exercises

Page 2 - Figure it Out

• Question 1: Fill in the blanks to make the expressions equal on both sides of the = sign: (a) $13+4=\_+6$ (b) $22+\_=6\times5$ (c) $8\times\_=64\div2$ (d) $34-\_=25$

  • Answer: (a) 11 (b) 8 (c) 4 (d) 9.

• Question 2: Arrange the following expressions in ascending (increasing) order of their values: (a) $67-19$ (b) $67-20$ (c) $35+25$ (d) $5\times11$ (e) $120\div3$

  • Answer: (e) 40, (b) 47, (a) 48, (d) 55, (c) 60.

Page 11 - Figure it Out

• Question 1: Find the values of the following expressions by writing the terms in each case. (a) $28-7+8$ (b) $39-2\times6+11$ (c) $40-10+10+10$ (d) $18-10\times2+16\div2$ (e) $6\times3-4\times8\times5$

  • Answer:

    • (a) 29

    • (b) 38

    • (c) 50

    • (d) 6

    • (e) $18 - 160 = -142$.

• Question 2: Write a story/situation for each of the following expressions and find their values: (a) $89+21-10$ (b) $5\times12-6$ (c) $4\times9+2\times6$

  • Answer: (a) 100 (b) 54 (c) 48.

Page 14 & 15 - Figure it Out

• Question 1: Fill in the blanks with numbers, and boxes with operation signs such that the expressions on both sides are equal. (a) $24+(6-4)=24+6-\_$ (b) $38+(9-4)=38+9-4$ (c) $24-(6+4)=24\_6\_4$ (d) $24-6-4=24-(6\_4)$ (e) $27-(8+3)=27\_8\_3$ (f) $27-(8-3)=27-8+3$

  • Answer: (a) 4 (c) -, - (d) + (e) -, -.

• Question 2: Remove the brackets and write the expression having the same value. (a) $14+(12+10)$ (b) $14-(12+10)$ (c) $14+(12-10)$ (d) $14-(12-10)$ (e) $-14+(12-10)$ (f) $14-(-12-10)$

  • Answer: * (a) $14+12+10$

    • (b) $14-12-10$

    • (c) $14+12-10$

    • (d) $14-12+10$

    • (e) $-14+12-10$

    • (f) $14+12+10$.

Page 18 & 19 - Figure it Out

• Question 1: Fill in the blanks with numbers, and boxes by signs, so that the expressions on both sides are equal. (a) $3\times(6+7)=3\times6+3\times7$ (b) $(8+3)\times4=8\times4+3\times4$ (c) $3\times(5+8)=3\times5 + 3\times8$

  • Answer: (c) Add $+ 3\times8$.

Page 19, 20 & 21 - Figure it Out

• Question 1: Read the situations given below. Write appropriate expressions for each of them and find their values: (a) The district market in Begur operates on all seven days of a week. Rahim supplies 9 kg of mangoes each day… and Shyam supplies 11 kg of mangoes each day… Find the amount of mangoes supplied by them in a week…

  • Answer: Expression: $(9+11)\times7$. Value: 140 kg. (b) Binu earns ₹20,000 per month. She spends ₹5,000 on rent, ₹5,000 on food, and ₹2,000 on other expenses every month. What is the amount Binu will save by the end of a year?

  • Answer: Expression: $(20000 - (5000+5000+2000))\times12$. Value: 96,000. (c) During the daytime a snail climbs 3 cm up a post, and during the night… accidentally slips down by 2 cm. The post is 10 cm high… In how many days will the snail get the treat?

  • Answer: Expression: The snail makes a net progress of $(3-2) = 1$ cm per day, but on the 8th day it reaches the top ($7 + 3 = 10$). Value: 8 days.