Arithmetic Expressions

Context: In an expression having two terms, swapping them does not change the value: $\text{Term 1} + \text{Term 2} = \text{Term 2} + \text{Term 1}$

Q. Can you explain why this is happening using the Token Model of integers that we saw in the Class 6 textbook of mathematics?

Context: Let us consider the expression $(-7) + 10 + (-11)$ again. What happens when we change the order and add $-7$ and $-11$ first, and then add this sum to $10$? Will we get the same sum as before? We see that adding the terms of the expression $(-7) + 10 + (-11)$ in any order gives the same sum of $-8$.

Q. Does adding the terms of an expression in any order give the same value? Take some more expressions and check. Consider expressions with more than 3 terms also.

Context: Does adding the terms of an expression in any order give the same value? Take some more expressions and check. Consider expressions with more than $3$ terms also.

Q. Can you explain why this is happening using the Token Model of integers that we saw in the Class 6 textbook of mathematics?

What happens to the value of an expression if we increase or decrease the value of one of its terms?

Some expressions are given in following three columns. In each column, one or more terms are changed from the first expression. Go through the example (in the first column) and fill the blanks, doing as little computation as possible.

Expression Engineer!

Using three 3's along with the four operations (addition, subtraction, multiplication, and division) and brackets as needed we can create several expressions. For example, $(3 + 3)/3 = 2$, $3 + 3 - 3 = 3$, $3 \times 3 + 3 = 12$, and so on.

• Using four 4's, create expressions to get all values from 1 to 20.
• Using the numbers 1, 2, 3, 4, and 5 exactly once in any order get as many values as possible between $-10$ and $+10$.
• Using the numbers 0 to 9 exactly once in any order, make an expression with a value 100.
• What other similar interesting questions can you ask?

Fill in the blanks to make the expressions equal on both sides of the $=$ sign:

(a) $13 + 4 = \underline{\quad} + 6$

(b) $22 + \underline{\quad} = 6 \times 5$

(c) $8 \times \underline{\quad} = 64 \div 2$

(d) $34 - \underline{\quad} = 25$

Arrange the following expressions in ascending (increasing) order of their values.

(a) $67 - 19$

(b) $67 - 20$

(c) $35 + 25$

(d) $5 \times 11$

(e) $120 \div 3$

Find the values of the following expressions by writing the terms in each case.

(a) $28 - 7 + 8$

(b) $39 - 2 \times 6 + 11$

(c) $40 - 10 + 10 + 10$

(d) $48 - 10 \times 2 + 16 \div 2$

(e) $6 \times 3 - 4 \times 8 \times 5$

Write a story/situation for each of the following expressions and find their values.

(a) $89 + 21 - 10$

(b) $5 \times 12 - 6$

(c) $4 \times 9 + 2 \times 6$

For each of the following situations, write the expression describing the situation, identify its terms and find the value of the expression.

(a) Queen Alia gave 100 gold coins to Princess Elsa and 100 gold coins to Princess Anna last year. Princess Elsa used the coins to start a business and doubled her coins. Princess Anna bought jewellery and has only half of the coins left. Write an expression describing how many gold coins Princess Elsa and Princess Anna together have.

(b) A metro train ticket between two stations is ₹40 for an adult and ₹20 for a child. What is the total cost of tickets:

(i) for four adults and three children?
(ii) for two groups having three adults each?

(c) Find the total height of the window by writing an expression describing the relationship among the measurements shown in the picture.

Add brackets at appropriate places in the expressions such that they lead to the values indicated.

(a) $34 - 9 + 12 = 13$

(b) $56 - 14 - 8 = 34$

(c) $-22 - 12 + 10 + 22 = -22$

Using only reasoning of how terms change their values, fill the blanks to make the expressions on either side of the equality ($=$) equal.

(a) $423 + \underline{\quad\quad} = 419 + \underline{\quad\quad}$

(b) $207 - 68 = 210 - \underline{\quad\quad}$

Using the numbers 2, 3 and 5, and the operators '+' and '-', and brackets, as necessary, generate expressions to give as many different values as possible. For example, $2 - 3 + 5 = 4$ and $3 - (5 - 2) = 0$.

Whenever Jasoda has to subtract 9 from a number, she subtracts 10 and adds 1 to it. For example, $36 - 9 = 26 + 1$.

(a) Do you think she always gets the correct answer? Why?

(b) Can you think of other similar strategies? Give some examples.

Consider the two expressions: a) $73 - 14 + 1$, b) $73 - 14 - 1$. For each of these expressions, identify the expressions from the following collection that are equal to it.

(a) $73 - (14 + 1)$

(b) $73 - (14 - 1)$

(c) $73 + (-14 + 1)$

(d) $73 + (-14 - 1)$

Figure it Out

1. Fill in the blanks with numbers, and boxes by signs, so that the expressions on both sides are equal.

(a) $3 \times (6 + 7) = 3 \times 6 + 3 \times 7$

(b) $(8 + 3) \times 4 = 8 \times 4 + 3 \times 4$

(c) $3 \times (5 + 8) = 3 \times 5 \ \Box\ 3 \times \underline{\ \ \ }$

(d) $(9 + 2) \times 4 = 9 \times 4 \ \Box\ 2 \times \underline{\ \ \ }$

(e) $3 \times (\underline{\ \ \ } + 4) = 3 \ \underline{\ \ \ } + \underline{\ \ \ }$

(f) $(\underline{\ \ \ } + 6) \times 4 = 13 \times 4 + \underline{\ \ \ }$

(g) $3 \times (\underline{\ \ \ } + \underline{\ \ \ }) = 3 \times 5 + 3 \times 2$

(h) $(\underline{\ \ \ } + \underline{\ \ \ }) \times \underline{\ \ \ } = 2 \times 4 + 3 \times 4$

(i) $5 \times (9 - 2) = 5 \times 9 - 5 \times \underline{\ \ \ }$

(j) $(5 - 2) \times 7 = 5 \times 7 - 2 \times \underline{\ \ \ }$

(k) $5 \times (8 - 3) = 5 \times 8 \ \Box\ 5 \times \underline{\ \ \ }$

(l) $(8 - 3) \times 7 = 8 \times 7 \ \Box\ 3 \times 7$

(m) $5 \times (12 - \underline{\ \ \ }) = \underline{\ \ \ } \ \Box\ 5 \times \underline{\ \ \ }$

(n) $(15 - \underline{\ \ \ }) \times 7 = \underline{\ \ \ } \ \Box\ 6 \times 7$

(o) $5 \times (\underline{\ \ \ } - \underline{\ \ \ }) = 5 \times 9 - 5 \times 4$

(p) $(\underline{\ \ \ } - \underline{\ \ \ }) \times \underline{\ \ \ } = 17 \times 7 - 9 \times 7$

In the boxes below, fill '<', '>' or '=' after analysing the expressions on the LHS and RHS. Use reasoning and understanding of terms and brackets to figure this out and not by evaluating the expressions.

(a) $(8 - 3) \times 29 \ \square\ (3 - 8) \times 29$

(b) $15 + 9 \times 18 \ \square\ (15 + 9) \times 18$

(c) $23 \times (17 - 9) \ \square\ 23 \times 17 + 23 \times 9$

(d) $(34 - 28) \times 42 \ \square\ 34 \times 42 - 28 \times 42$

Here is one way to make 14: $2 \times ( 1 + 6 ) = 14$. Are there other ways of getting 14? Fill them out below:

(a) $\underline{\quad} \times (\underline{\quad} + \underline{\quad}) = 14$

(b) $\underline{\quad} \times (\underline{\quad} + \underline{\quad}) = 14$

(c) $\underline{\quad} \times (\underline{\quad} + \underline{\quad}) = 14$

(d) $\underline{\quad} \times (\underline{\quad} + \underline{\quad}) = 14$

Find out the sum of the numbers given in each picture below in at least two different ways. Describe how you solved it through expressions.

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 from his orchard and Shyam supplies 11 kg of mangoes each day from his orchard to this market. Find the amount of mangoes supplied by them in a week to the local district market.

(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?

(c) During the daytime a snail climbs 3 cm up a post, and during the night while asleep, accidentally slips down by 2 cm. The post is 10 cm high, and a delicious treat is on its top. In how many days will the snail get the treat?

Melvin reads a two-page story every day except on Tuesdays and Saturdays. How many stories would he complete reading in 8 weeks? Which of the expressions below describes this scenario?

(a) 5 × 2 × 8

(b) (7 - 2) × 8

(c) 8 × 7

(d) 7 × 2 × 8

(e) 7 × 5 - 2

(f) (7 + 2) × 8

(g) 7 × 8 - 2 × 8

(h) (7 - 5) × 8

Find different ways of evaluating the following expressions:

(a) 1 - 2 + 3 - 4 + 5 - 6 + 7 - 8 + 9 - 10

(b) 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1

Compare the following pairs of expressions using '<', '>' or '=' or by reasoning.

(a) $49 - 7 + 8 \quad \square \quad 49 - 7 + 8$

(b) $83 \times 42 - 18 \quad \square \quad 83 \times 40 - 18$

(c) $145 - 17 \times 8 \quad \square \quad 145 - 17 \times 6$

(d) $23 \times 48 - 35 \quad \square \quad 23 \times (48 - 35)$

(e) $(16 - 11) \times 12 \quad \square \quad -11 \times 12 + 16 \times 12$

(f) $(76 - 53) \times 88 \quad \square \quad 88 \times (53 - 76)$

(g) $25 \times (42 + 16) \quad \square \quad 25 \times (43 + 15)$

(h) $36 \times (28 - 16) \quad \square \quad 35 \times (27 - 15)$

Identify which of the following expressions are equal to the given expression without computation. You may rewrite the expressions using terms or removing brackets. There can be more than one expression which is equal to the given expression.

(a) $83 - 37 - 12$

(i) $84 - 38 - 12$

(ii) $84 - (37 + 12)$

(iii) $83 - 38 - 13$

(iv) $- 37 + 83 - 12$

(b) $93 + 37 \times 44 + 76$

(i) $37 + 93 \times 44 + 76$

(ii) $93 + 37 \times 76 + 44$

(iii) $(93 + 37) \times (44 + 76)$

(iv) $37 \times 44 + 93 + 76$

Choose a number and create ten different expressions having that value.

Choose your favourite number and write as many expressions as you can having that value.

Use '>' or '<' or '=' in each of the following expressions to compare them. Can you do it without complicated calculations? Explain your thinking in each case.

(a) $245 + 289 \ \square \ 246 + 285$

(b) $273 - 145 \ \square \ 272 - 144$

(c) $364 + 587 \ \square \ 363 + 589$

(d) $124 + 245 \ \square \ 129 + 245$

(e) $213 - 77 \ \square \ 214 - 76$

Context: Subtracting a number is the same as adding the inverse of the number. Recall that the inverse of a given number has the sign opposite to it. For example, the inverse of $14$ is $-14$, and the inverse of $-14$ is $14$. Thus, subtracting $14$ from $83$ is the same as adding $-14$ to $83$. That is, $83 - 14 = 83 + (-14)$ Thus, the terms of the expression $83 - 14$ are $83$ and $-14$.

Q. Check if replacing subtraction by addition in this way does not change the value of the expression, by taking different examples.

Can you explain why subtracting a number is the same as adding its inverse, using the Token Model of integers that we saw in the Class 6 textbook of mathematics?

In the following table, some expressions are given. Complete the table.

Does changing the order in which the terms are added give different values?

Context: We already know that swapping the terms does not change the sum when both the terms are positive numbers.

Q. Will this also hold when there are terms having negative numbers as well? Take some more expressions and check.

Context: While adding positive numbers, grouping them in different ways gives the same sum: $(\text{Term 1} + \text{Term 2}) + \text{Term 3} = \text{Term 1} + (\text{Term 2} + \text{Term 3})$

Q. Will this also hold when there are terms having negative numbers as well? Take some more expressions and check.

Manasa is adding a long list of numbers. It took her five minutes to add them all and she got the answer 11749. Then she realised that she had forgotten to include the fourth number 9055. Does she have to start all over again?

Manasa is going outside to play. Her mother says, "Wear your hat and shoes!" Which one should she wear first? She can wear her hat first and then her shoes. Or she can wear her shoes first and then her hat.

Manasa will look exactly the same in both cases. Imagine a different situation: Manasa's mother says "Wear your socks and shoes!" Now the order matters. She should wear socks and then shoes. If she wears shoes and then socks, Manasa will feel very uncomfortable and look very different.

Context: Amu, Charan, Madhu, and John went to a hotel and ordered four dosas. Each dosa cost ₹23, and they wish to thank the waiter by tipping ₹5.

Q. If the total number of friends goes up to 7 and the tip remains the same, how much will they have to pay? Write an expression for this situation and identify its terms.

Context: Children in a class are playing "Fire in the mountain, run, run, run!". Whenever the teacher calls out a number, students are supposed to arrange themselves in groups of that number. Whoever is not part of the announced group size, is out. Ruby wanted to rest and sat on one side. The other 33 students were playing the game in the class. The teacher called out '5'. Once children settled, Ruby wrote $6 \times 5 + 3$ (understood as 3 more than $6 \times 5$).

Q. Think and discuss why she wrote this.

For each of the cases below, write the expression and identify its terms:

(a) If the teacher had called out '4', Ruby would write

(b) If the teacher had called out '7', Ruby would write

Write expressions like the above for your class size.

Context: We have already come across simple expressions like $x + 3$, $y - 5$, $4x + 5$, $10y - 5$ and so on. For example, to form $4x + 5$, we first form $4x$ and then add $5$ to it. To form $10y - 5$, we first form $10y$ and then subtract $5$ from it.

Q. Identify the terms in the two expressions above.

Context: Dinu paid Kiran ₹432 using four ₹100 notes, three ₹10 notes, and two ₹1 coins.

Q. Can you think of some more ways of giving ₹432 to someone?

Context: Together they have to pay $2 \times (43 + 24)$. This is also the same as paying for two vegetable cutlets and two rasgullas: $2 \times 43 + 2 \times 24$

Therefore, $2 \times (43 + 24) = 2 \times 43 + 2 \times 24$

Q. If another friend, Sangmu, joins them and orders the same items, what will be the expression for the total amount to be paid?

5 × 4 + 3 ≠ 5 × (4 + 3). Can you explain why?

Is 5 × (4 + 3) = 5 × (3 + 4) = (3 + 4) × 5?

Use this method to find the following products:

(a) $95 \times 8$

(b) $104 \times 15$

(c) $49 \times 50$

Is this quicker than the multiplication procedure you use generally?

Context: Use this method to find the following products: (a) $95 \times 8$ (b) $104 \times 15$ (c) $49 \times 50$

Is this quicker than the multiplication procedure you use generally?

Q. Which other products might be quicker to find like the ones above?

Figure it Out

1. Fill in the blanks with numbers, and boxes with operation signs such that the expressions on both sides are equal.

   (a) $24 + (6 - 4) = 24 + 6 \ \square \ \underline{\quad}$

   (b) $38 + (\underline{\quad} \ \square \ \underline{\quad}) = 38 + 9 - 4$

   (c) $24 - (6 + 4) = 24 \ \square \ 6 - 4$

   (d) $24 - 6 - 4 = 24 - 6 \ \square \ \underline{\quad}$

   (e) $27 - (8 + 3) = 27 \ \square \ 8 \ \square \ 3$

   (f) $27 - (\underline{\quad} \ \square \ \underline{\quad}) = 27 - 8 + 3$

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) $14 - (-12 - 10)$

3. Find the values of the following expressions. For each pair, first try to guess whether they have the same value. When are the two expressions equal?

(a) $(6 + 10) - 2$ and $6 + (10 - 2)$

(b) $16 - (8 - 3)$ and $(16 - 8) - 3$

(c) $27 - (18 + 4)$ and $27 + (-18 - 4)$

4. In each of the sets of expressions below, identify those that have the same value. Do not evaluate them, but rather use your understanding of terms.

(a) $319 + 537$, $319 - 537$, $-537 + 319$, $537 - 319$

(b) $87 + 46 - 109$, $87 + 46 - 109$, $87 + 46 - 109$, $87 - 46 + 109$, $87 - (46 + 109)$, $(87 - 46) + 109$