Expressions and Equations Presenter: Janet Gately

When I am dissatisfied and would like to go back to my youth, I think of algebra. a retired lawyer

Every time I see a math word problem it looks like this: If I have 10 ice cubes and you have 11 apples, how many pancakes will fit on the roof? Answer: Purple, because aliens don’t wear hats.

“Students must learn mathematics with understanding, actively building new knowledge from experience and prior knowledge.” NCTM

The Math Things Mingle http://www.insidemathematics.org/index.php/classroom-video-visits/public-lesson-equations-inequalities-properties/243-equations-inequalities-a-properties-introduction-part-a?

Expressions—Vocabulary Look at the terms in each box. Write what you know about each one.

Labeling an Expression Cut out the vocabulary terms. Glue them next to the appropriate part of the expression. (You may need to draw arrows to get them to fit.) Check with your partner. Do you agree?

Vocabulary Match Match each term with the appropriate representation. Check with your partner. Do you agree?

Vocabulary Assessment Label each part of the expression with the correct vocabulary.

Common Core State Standards 6.EE.2b Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. For example, describe the expression 2(8 + 7) as a product of two factors; view (8 + 7) as both a single entity and a sum of two terms.

Students need ample opportunities to work on translating words and contexts into symbols.

Grounding early algebraic experiences in familiar contexts can help students to see the relevance of algebra to everyday life.

Expressions Numerical expression—a combination of numbers and operations 16 + 32 25 • 50 24/4 Algebraic expression—a combination of variables, numbers, and at least one operation 6x + 2 x² x + y

Writing Expressions with Bar Diagrams Kevin has 6 more baseball cards than Eli. Step 1: Eli has an unknown number of baseball cards c. Use a bar diagram to show Eli’s cards. Eli c cards Step 2: Kevin has 6 more baseball cards than Eli. Complete the bar diagram below to show how many baseball cards Kevin has. Kevin c cards 6 cards So, Kevin has c + 6 base ball cards.

Sam sent 10 fewer messages in July than in August. Step 1: Sam sent and unknown number of messages, m, in August. Label the bar diagram to represent the messages Sam sent in August. August m messages Step 2: Sam sent 10 fewer messages in July. Label the bar diagram to show the messages Sam sent in July. July m messages 10 fewer So, Sam sent m – 10 messages in July.

A bottlenose dolphin can swim d miles per hour A bottlenose dolphin can swim d miles per hour. Humans swim one third as fast as dolphins. Step 1: Dolphins can swim is an unknown number of miles per hour, d. Use a bar diagram to represent the speed a dolphin swims. dolphins d miles per hour Step 2: Humans swim one third as fast as dolphins. Divide and shade a second bar diagram to represent the speed humans can swim. dolphins d miles per hour humans So, humans can swim d ÷ 3 miles per hour.

Clark’s dog weighs 5 times as much as his cat. Step 1: The weight of Clark’s cat is an unknown number. Label the bar diagram to represent the weight of Clark’s cat. cat w cat Step 2: Clark’s dog weighs 5 times as much as his cat. Add four additional bars to represent the dog as five times the length of the cat. dog So, the weight of the dog is 5w.

With a partner, complete the table. Draw a bar diagram Write an algebraic expression for the situation

Common Core State Standards 6.EE.1 Write and evaluate numerical expressions involving whole-number exponents. 6.EE.2a Write expressions that record operations with numbers and the letters standing for numbers. For example, express the calculation “subtract y from 5” as 5 – y.

Common Core State Standards 6.EE.2c Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). For example, use the formulas V= s³ and A = 6s² to find the volume and surface area of a cube with sides of length s = ½.

Distributive Property Three friends are going to a concert at the fair. They each want admission to the fair, which is $6.00 and admission to the concert, which is $22.00. What is the total that the three friends will spend? Step 1: Write an expression to represent the amount spent in dollars. 3(6 + 22) friends fair admission concert

Step 2: Use area models to evaluate the expression. Method 1: Add the lengths. Then multiply. 6 22 3 3 (6 + 22) = 3(28) = 84 Method 2: Find each area. Then add. 6 22 3 3 (3 • 6) + (3 • 22) = 18 + 66

Since both expressions are equal to 84, they are equivalent. So, 3(6 + 22) = (3 ● 6) + (3 ● 22)

With a partner, complete the table. Rewrite the expression Evaluate the expression Check with your partner. Do you agree?

Prime Factorization The prime factorization of an algebraic expression contains both the prime factors and any variable factors. For example, the prime factorization of 6x is 2• 3 • x.

Factoring an Expression Write the prime factorization of 12 and 8. 12 = 2 • 2 • 3 8 = 2 • 2 • 2 The GCF of 12 and 8 is 2 • 2 or 4. Write each term as a product of the GCF and its remaining factor. Then use the Distributive Property to factor out the GCF. 12 + 8 = (4 ● 3) + (4 ● 2) = 4(3 + 2) So, 12 + 8 = 4(3 + 2)

= 3(x + 5) So, 3x + 15 = 3(x + 5) Factor 3x + 15 Write the prime factorization of 3x and 15. 3x = 3 • x 15 = 3 • 5 The GCF of 3x and 15 is 3. Write each term as a product of the GCF and its remaining factor. Then use the Distributive Property to factor out the GCF. 3x + 15 = (3 ● x) + (3 ● 5) = 3(x + 5) So, 3x + 15 = 3(x + 5)

With a partner, complete the table. Write the prime factorization Rewrite the term using the GCF Use the Distributive Property Check with your partner. Do you agree?

Common Core State Standards 6.EE.3 Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3(2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6(4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y.

6. EE. 4 Identify when two expressions are equivalent (i. e 6.EE.4 Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). For example, the expression y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for.

Equation An equation is a mathematical sentence showing two expressions are equal.

When you replace a variable with a value that results in a true sentence, you solve the equation. The value for the variable is the solution of the equation.

A solution to a problem in algebra might be 5x + 2, which students might interpret as a lack of closure because their previous experiences have taught them that an answer to a mathematics problem should consist of a single term or number.

Hands-on-Equations Hands-on-Equations is a concrete and pictorial model to help build Algebra understanding.

Common Core State Standards 6.EE.5 Understand solving an equation or inequality as a process of answering a question: Which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.

6.EE.6 Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.

6.EE.7 Solve real-world and mathematical problems by writing and solving equations of the form x + p = q for cases in which p, q, and x are all non-negative rational numbers.

Writing Inequalities You must be older than 13 to play in the basketball league. a > 13 l l l l l l l l l l l l 10 11 12 13 14 15 16 17 18 19 20 21

You must be at least 18 years old to vote. l l l l l l l l l l l l l l l l 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Jason has less than 65 pages of his book to read. l l l l l l l l l l l l l 58 59 60 61 62 63 64 65 66 67 68 69 70

The movie will be no more than 90 minutes in length. l l l l l l l l l l l l l l l 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98

Write an inequality and draw a number line diagram for each situation

Solving Inequalities Regina sent x text messages before lunch. She sent another 4 text messages after lunch. She sent less than 7 text messages today. x + 4 < 7 x + (4 – 4) < 7 – 4 x < 3 l l l l l l l l l l l 0 1 2 3 4 5 6 7 8 9 10

Write an inequality Solve it Draw a number line diagram

Common Core State Standards 6.EE.8 Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.

“Effective mathematics teaching requires understanding what students know and need to learn and then challenging and supporting them to learn it well.” NCTM