Starter WWK Product notation – a way of simplifying repeated sums, e.g., a + a  2 ‘lots’ of a  2 x a  2a Variable – a symbol that represents an unknown number Expression – algebraic form consisting of numbers, variables, and operation signs Equation – algebraic form which contains an = sign Term – algebraic forms which are separated by + or – signs, eg., 3x + 2y - 8 (3 terms) Distributive property– 4(x + 2)  4x + 8

Chapter 4: Algebraic Operations Essential Question How do we use algebraic expressions, properties of numbers, and the distributive property to simplify and evaluate? Title Page Number

I can: Use product notation to simplify algebraic expressions. Algebraic Notation I can: Use product notation to simplify algebraic expressions.

Algebraic Notation In algebra, we: Usually leave out the multiplication sign (x). Write numbers first in any product. Where products contain 2 or more letters, we write them in alphabetical order. For example: write in product notation 4 × a = 4a 1 × b = b We don’t need to write a 1 in front of the letter. Ask pupils why they think we try not to use the multiplication symbol ×. One reason is that it is easily confused with the letter, x Another reason is that when we use algebra we try to write thing as simply as possible, only writing what is absolutely necessary. It’s simpler to write 2n than 2 x n. It is also unnecessary to write a 1 in front of a letter to multiply it by 1. Multiplying by 1 has no effect so we can leave it out altogether. b × 5 = 5b We don’t write b5. 3 × d × c = 3cd We write letters in alphabetical order.

Distributive property Multiplying terms together Write in product notation. Example 1: f x 15e Example 2: 6 x m + 11 x d Example 3: 3 x (a + b) Distributive property

Distributive property Sometimes we need to multiply out brackets and then simplify. Example 4: 2(5 – x) We need to multiply the bracket by 2. 10 – 2x Show pupils that it is not necessary to construct a grid to multiply out a bracket. If required pupils can use lines, as shown here in orange, to make sure that every term inside the bracket is multiplied by the term outside the bracket.

Example 5: r + r + r + s + s Example 6: a + a – (b + b + b + b + b) Writing sums as products Simplify. Example 5: r + r + r + s + s Example 6: a + a – (b + b + b + b + b) Example 7: d + d + d - (c + c)

Example 8: Example 9: 9 x d x d x c x c x c k + k - 2 x t x t x t Using index notation Simplify: x + x + x + x + x = 5x x to the power of 5 Simplify: x × x × x × x × x = x5 Example 8: 9 x d x d x c x c x c Example 9: k + k - 2 x t x t x t Start by asking pupils to simplify x + x + x + x + x. This is 5 lots of x, which have seen is written as 5x. Now, ask pupils how they might simplify x × x × x × x × x. Impress upon pupils the difference between this, and the previous expression, as they are often confused. If x is equal to 2, for example, x + x + x + x + x equals 10, and x × x × x × x × x equals 32. Some pupils may suggest writing xxxxx. This is not strictly incorrect, however, it should be discouraged in favour of using index notation. When we write a number or term to the power of another number it is called index notation. The power, or index, is the raised number, in this case 5. The plural of index is indices. The number or letter that we are multiplying successive times, in this case, x, is called the base. x2 is read as ‘x squared’ or ‘x to the power of 2’. x3 is read as ‘x cubed’ or ‘x to the power of 3’. x4 is read as ‘x to the power of 4’.

P77 - 78  #1 - 5 and Distributive property puzzle Independent Practice P77 - 78  #1 - 5 and Distributive property puzzle

Starter 1. 3p × 4p 2. q × q × q × 4 x q × q WWK Coefficient – the number factor of an algebraic term, eg. 4xy or 12z Like terms – terms with exactly the same variable form, eg. 2x + 7x or 11xy – 3xy Constant terms – a term which does not contain a variable, eg. 6 or -43 1. 3p × 4p 2. q × q × q × 4 x q × q 3. f + f + f + f – (h x h x h) 4. 2t × 2t

Top 3 HW HW Quiz

Homework Quiz Simplify using product notation. Please complete 7 of the 8 questions. p x q x (r - 2) a x 5 x b s – (t + t) s – t + t 5. (w – x) x 8 6. p x h x d 7. Expand: (5a)2 8. Write in simplest form: a x a + 2 x b x b – a x b x b

Homework Quiz Simplify using product notation. Please complete 7 of the 8 questions. 12 – r x s x 6 m x 4n a + a + a + a + 7 s – t + t 5. 3 x (x + y) 6. 3 + z + y + y 7. Expand: 7f2g3 8. Write in simplest form: 2 x p x q x q + 6 x r x r x s

4B – The Language of Mathematics I can: Identify coefficients, constant terms and like terms. Understand the difference between an expression and an equation.

Language of Mathematics Example 1: Consider 3y2 – 4x - 6xy – 8 a) Expression or equation? b) How many terms does it contain? Write them. c) State the coefficient of xy. d) State the constant term Start by asking pupils to simplify x + x + x + x + x. This is 5 lots of x, which have seen is written as 5x. Now, ask pupils how they might simplify x × x × x × x × x. Impress upon pupils the difference between this, and the previous expression, as they are often confused. If x is equal to 2, for example, x + x + x + x + x equals 10, and x × x × x × x × x equals 32. Some pupils may suggest writing xxxxx. This is not strictly incorrect, however, it should be discouraged in favour of using index notation. When we write a number or term to the power of another number it is called index notation. The power, or index, is the raised number, in this case 5. The plural of index is indices. The number or letter that we are multiplying successive times, in this case, x, is called the base. x2 is read as ‘x squared’ or ‘x to the power of 2’. x3 is read as ‘x cubed’ or ‘x to the power of 3’. x4 is read as ‘x to the power of 4’.

Identifying Like Terms Example 2: Identify like terms in the expression. 2x + 4y3 - 8x + 5z - 5y3 - 8z Look for like variables with like powers. Like terms: 2x and -8x, 4y3 and -5y3, 5z and -8z

Language of Mathematics Example 3: Complete the table by identifying parts of each expression. Coefficients Like Terms Constant Terms 5x + 7y - 7 5, 7 none -7 5a + 2b - 3a + 4 4st - 5s + 3st + 6 Start by asking pupils to simplify x + x + x + x + x. This is 5 lots of x, which have seen is written as 5x. Now, ask pupils how they might simplify x × x × x × x × x. Impress upon pupils the difference between this, and the previous expression, as they are often confused. If x is equal to 2, for example, x + x + x + x + x equals 10, and x × x × x × x × x equals 32. Some pupils may suggest writing xxxxx. This is not strictly incorrect, however, it should be discouraged in favour of using index notation. When we write a number or term to the power of another number it is called index notation. The power, or index, is the raised number, in this case 5. The plural of index is indices. The number or letter that we are multiplying successive times, in this case, x, is called the base. x2 is read as ‘x squared’ or ‘x to the power of 2’. x3 is read as ‘x cubed’ or ‘x to the power of 3’. x4 is read as ‘x to the power of 4’.

Activity – “Math Mad Libs” There are 9 cards around the room. You may work in groups of two. You can go in any order, but please write the card number, answer, and the word beside your answer. You will use these words to create a paragraph afterwards. State the coefficient of xy in the expression: -y2 + 6xy – 12y. 6 12 1 -12 -6 Ms. Store Mr. Roy Ms. Dahlia Mr. Yenshaw Mrs. Sue Ellen

Complete “math libs” paragraph Independent Practice: p79  #1 – 6 and Exit card (write an expression and identify the variables, like terms, constant terms, and coefficients)

Complete “math libs” paragraph flying a kite Ms. Store Taylor Swift Halloween the zoo a hot air balloon pajamas juggling to get some exercise Independent Practice: p79  #1 – 6 and Exit card (write an expression and identify the variables, like terms, constant terms, and coefficients)

Starter For each expression below, write the number of terms, and identify all coefficients. If there is a constant term, identify it as well. 1. 2. 3. 4. 5.

4C – Collecting Like Terms I can: Identify like terms. Collect like terms and simplify algebraic expressions.

Collecting like terms When we add or subtract like terms in an expression we say we are simplifying an expression by collecting like terms. An expression can contain different like terms. For example, 3a + 2b + 4a + 6b = 3a + 4a + 2b + 6b Explain the meanings of each key word and phrase. In the example 3a + 2b + 4a + 6b, explain that it is helpful to write like terms next to each other. (Remember that we can add terms in any order.) Stress that we cannot simplify 7a + 8b any further. We can’t combine a’s and b’s. This says 7 times one number plus eight times another number. = 7a + 8b This expression cannot be simplified any further.

Collecting like terms Example 1) a + a + a + a + a = 5a Simplify these expressions by collecting together like terms. Example 1) a + a + a + a + a = 5a Example 2) 5b – 4b = b Example 3) 4c + 3d + 3 – 2c + 6 – d = 4c – 2c + 3d – d + 3 + 6 = 2c + 2d + 9 Whenever possible make comparisons to arithmetic by substituting actual values for the letters. If it is true using numbers then it is true using letters. For example, in 1) we could say that 7 + 7 + 7 + 7 + 7 is equivalent to 5 × 7. For example 1) and example 2), stress again that in algebra we don’t need to write the number 1 before a letter to multiply it by 1. 1a is just written as a and 1b is just written as b. For example 3), explain that when there are lots of terms we can write like terms next to each other so that they are easier to collect together. The numbers without any letters are added together separately. In example 4) stress that n2 is different from n. They cannot be collected together. 4n – 3n is n and n2 stays as it is. If we can’t collect together any like terms, as in example 5), we write ‘cannot be simplified’. Example 4) 4n + n2 – 3n = 4n – 3n + n2 = n + n2 Example 5) 4r + 6s – t Cannot be simplified

Simplifying Expressions – Algebra Pyramids Example 7d + 6e 2d + 3e 3e + 5d 3e 5d 2d -2d + 10e 2x + 8y 2d + 5e 5e – 4d 3x + 4y 4y - x 2d 5e -4d 3x 4y - x

p80 - 81  #1 - 4 and Codebreaker puzzle Independent Practice p80 - 81  #1 - 4 and Codebreaker puzzle