ARITHMETIC EXPRESSION

Transcription

1 Section 1: Expression & Terms MATH LEVEL 2 LESSON PLAN 1 ARITHMETIC EXPRESSION 2017 Copyright Vinay Agarwala, Revised: 10/31/17 1. An arithmetic expression is made up of numbers joined by addition (+), subtraction ( ), multiplication (x) and division ( ). Here are examples of such expressions x An expression is made up of terms separated by plus (+) and minus ( ).The multiplications (x) and divisions ( ) are part of the term. In the following expressions the terms are underlined (3 terms) 4 x (4 terms) 3. Here are some examples of arithmetic expressions. Write these expressions down on a piece of paper. Circle the (+) and ( ), and then underline the terms separated by them. Verify the number of terms. 8 x (2 terms) 12 x x (5 terms) 12 x x 18 9 (2 terms) 12 x x 2 x 18 9 (1 term) Identify how many terms there are in each expression. (a) 6 x 16 x (d) x x (b) (e) 5 x (c) (f) Answer: (a) 1 term (b) 3 terms (c) 5 terms (d) 4 terms (e) 2 terms (f) 7 terms Section 2: Reducing Terms (x and only) 4. Complex terms are made up x and. Complex term: 6 x 2 3 In this term the x and apply to the number to their right. The sign applies to 3. The sign x applies to 2. The sign that applies to 6 is not shown but it is x, because 6 x 2 3 = 1 x 6 x 2 3

2 Determine the signs of the following numbers in the given term (a) 1 (b) 2 (c) 3 (d) 4 (e) 5 (f) 6 (g) 7 (h) 8 (i) 9 Term: 9 3 x x 8 4 x 7 x 2 Answer: (a) (b) x (c) (d) (e) x (f) (g) x (h) x (i) x 5. The sign explains the operation on the number to its right. The number may be moved around with its sign. This does not change the value of the term. 6 x 2 3 = 6 3 x 2 = 2 x 6 3 = 4 The sign explains the operation on the number to which it belongs. In the first example below 6 can only be used as a divisor to divide. 8 6 x 3 is NOT 8 18 because 6 is used to divide and not to multiply. 8 6 x 3 is 8 x 3 6 = 24 6 = 4 Find the value of the following terms. (a) 7 3 x 6 (b) 9 2 x 4 Answer: (a) 14 (b) Dividing by several divisors individually is same as dividing by the product of the divisors = 24 (2 x 3) = = 60 (5 x 2 x 2) = = 3 Divide the following by grouping the divisors. (a) (b) Answer: (a) = 2 (b) = 2 7. We may reduce a complex term from left to right as follows x 14 7 = 3 x 14 7 = 42 7 = 6 Sometimes it is not possible to reduce a term from left to right x So we move multipliers to the top and divisors to the bottom of a line as follows. We then divide the top numbers by the bottom numbers to reduce the expression.

3 8. We cancel out the same number above and below the line because they reduce to 1. Reduce the following complex terms (a) 6 x 16 x (d) 8 x 23 x (b) 21 8 x 2 21 x 8 (e) 17 8 x 5 17 x 8 (c) x 10 (f) x 32 Answer: (a) 2 (b) 2 (c) 1 (d) 3 (e) 5 (f) 2 Section 3: Reducing simple expressions (+ and only) 9. In a simple expression the terms are made up of single numbers. Simple Expression: (3 terms) In this expression the + and apply to the number to their right. The sign applies to 10. The sign + applies to 5. The sign that applies to 15 is not shown but it is +, because = Determine the signs of the following numbers in the given expression (a) 1 (b) 2 (c) 3 (d) 4 (e) 5 (f) 6 (g) 7 (h) 8 (i) 9 Expression: Answer: (a) (b) (c) (d) (e) + (f) (g) + (h) + (i) The sign explains the operation on the number to its right. The number may be moved around with its sign. This does not change the value of the expression. In the following example, the terms 10 is moved around. This does not change the value of the expression = = = 10 The sign explains the operation on the number to which it belongs. In the first example below 3 can only be subtracted. It cannot be added. Ex. 1: is NOT = is = 18

4 Ex. 2: is = = 6 Find the value of the following expressions. (a) (b) Answer: (a) 10 (b) Taking away the negative numbers individually is the same as taking them away as a group (sum) = 5 (2 + 2) = 5 4 = = 25 ( ) = = 5 Subtract the following by grouping the negative numbers (a) (b) Answer: (a) = 5 (b) = We may reduce a simple expression from left to right as follows = = 13 4 = 9 Sometimes it is not possible to reduce expression from left to right as in the following example So we move positive numbers to the left and negative numbers to the right, and then reduce the expression = = (9 + 20) (17+ 4) = = We can cancel out the same number in the + and groups because a number subtracted from itself gives 0 remainder. Reduce the following simple expressions (a) (d) (g) (b) (e) (h) (c) (f) (i) Answer: (a) 5 (b) 4 (c) 3 (d) 3 (e) 5 (f) 7 (g) 0 (h) 8 (i) 4

5 Section 4: Reducing Complex Expressions 14. Complex terms are reduced first and then the whole expression. We identify the terms by underlining them. Here is an example: Expression: 13 2 x x = 13 2 x (10 x 24 16) + 5 = = The expression within parentheses represents a single number: Therefore, parenthesis are reduced first to a single number. Expression: (16 + 6) = (16 + 6) = = = 7 Expression: 4 x 7 9 x = 4 x 7 9 x = = 5 Reduce the following expressions (a) x 4 (d) (3 + 8) x 5 (g) 4 x x 5 (b) 8 x (e) x 3 (h) x (c) x 5 (f) 6 x (5 + 3) (i) x (2 + 5) Answer: (a) 16 (b) 20 (c) 43 (d) 55 (e) 21 (f) 48 (g) 22 (h) 15 (i) In general lower order operations are grouped together into higher order operations. The most basic operation is COUNTING Order 0: COUNTING Repeated Counting is the grouping called ADDITION. The reverse of addition is called SUBTRACTION. We may resolve addition and subtraction together. Order 1: ADDITION & SUBTRACTION Repeated Addition is the grouping called MULTIPLICATION. The reverse of multiplication is called DIVISION. We may resolve multiplication and division together. Order 2: MULTIPLICATION & DIVISION

6 Expressions in parentheses are treated as a single number. Therefore, such expressions are resolved within the isolation of parentheses. 17. Schools teach the order of operations as PEMDAS (Please Excuse My Dear Aunt Sally) This order results in error when multiplication is carried out before division, and addition is carried out before subtraction without understanding the signs of numbers. 6 2 x 3 = 9 and NOT 1 as per PEMDAS = 6 and NOT 0 as per PEMDAS 18. Higher operations of multiplication (x) and division ( ) are part of terms. Lower operations of addition (+) and subtraction ( ) separate the terms of expression. The order of operations is automatically taken into account when we reduce the terms first and then reduce the expression. Reduce the following expressions to a number (a) 6 x 6 5 x (d) x x (b) (e) x (c) 8 4 x 3 4 x x 5 (f) 13 2 x x Answer: (a) 4 (b) 3 (c) 0 (d) 20 (e) 19 (f) 36 ADDITIONAL EXERCISE Reduce the following expressions to a number (a) 8 x 3 12 (b) (c) 20 x (d) 3 x (e) (f) 2 x 6 x 7 x 1 14 (g) 21 x (h) 5 x x 8 20 (i) 5 6 x 5 x (j) 8 x x (k) 21 x 35 x (l) (m) 56 x 54 6 x (n) 6 x 6 6 x (o) x x 21 9 (p) 8 6 x x 2 2 x 6 4 Answer: (a) 2 (b) 11 (c) 2 (d) 15 (e) 1 (f) 6 (g) 6 (h) 5 (i) 2 (j) 80 (k) 2 (l) 1 (m) 35 (n) 4 (o) 15 (p) 9 Section 5: Word Problems 19. Solve the word problems from Article 49 of DUBB S ARITHMETICAL PROBLEMS (see the link on Level 2 page). Check your answer against the answers given.

7 L2 Lesson Plan 1: Check your Understanding 1. Explain term and expression. 2. Isolate the terms containing x and in parentheses. (a) 6 x 6 5 x (b) 8 4 x 3 4 x x 5 3. Reduce the following expressions to a single number? (a) x (2 + 5) (b) 13 2 x x Check your answers against the answers given below. Answer 1) Numbers joined together by addition, subtraction, multiplication and division generate an arithmetic expression. In an expression, addition (+) and subtraction ( ) separate the terms. Within terms we have multiplication (x) and division ( ). A term can simply be a number. 2) (a) (6 x 6) (5 x 6) + (3 3) + (3 3) 4 (b) (8 4 x 3) (4 x 4 2) + (6 15 x 5) 3) (a) 25 (b) 36