# 6th Grade - Show Your Work And Describe It In Words

 Grade Level: 6th Skill: Problem Solving Topic: Show Your Work and Describe It in Words Goal: Express the solution clearly and logically by using the appropriate mathematical notation and terms and clear language; support solutions with evidence in both verbal and symbolic work. Skill Description: Use clear language and academic vocabulary to clearly and logically express a solution. Express solutions using the correct notations which will use mathematical symbols and expressions. Understand that symbols include all numbers, operation signs, and symbols that represent a mathematical idea (i.e. greater than, pi). Understand that an expression is a sequence of symbols that can be used to find a solution (i.e. 2 + 3 or 4a). Support solutions by showing evidence with words and step-by-step symbolic work.

## Building Blocks/Prerequisites

### Sample Problems

 (1) On the night before a big test, Jen studies twice as long as usual. Write an expression to represent the situation, if h represents the amount of time she usually studies. Show your steps to solve the problem, if Jen usually studies for 1 hour. (The expression is 2h. Since I know that h is 1, I can insert the one where the h is in the expression. I know that 2h is the mathematical notation for two times h. So, I will rewrite the problem as 2 x 1. Two times 1 is 2. On the night before tests, Jen studies for 2 hours. ) (2) Haley had five less than 3 times an amount of dollars, d. Write an expression to represent the situation. Then express your solution logically, showing how you solved for \$10. (The expression is3d – 5. First, it must be understood that the expression means 3 times d minus five. Next, insert \$10 where d is because d is equal to 10. The algebraic expression will be re-written as the numeric expression: 3 x 10 -5. Lastly, the order of operations must be used to find the solution. In the order of operations, multiplication always comes first, so 30 – 5 = 25. Haley had \$25.) (3) There are two similar rectangles. The first with a width of W and a length of L. The second triangle is 4 times larger than the first. Write an expression to show how to find the perimeter of the second triangle. Then show the steps to solve for a W = 5 ft. and L = 10 ft. (The algebraic expression is: 4 (2W + 2L). This expression stands for two times the width plus two times the length times four. Since I know that W is 5 and L is 10, I must replace these variables with the numbers in order to solve. The numerical expression is 4 x (2 x 5 + 2 x 10). The multiplication problems inside the parenthesis must be solved first: 4 x (10 + 20). Next, the addition problem inside the parenthesis will be solved; 4 x 30. Lastly, I will multiply 4 x 30 to get the final answer of 120 feet.) (4) A triangle has three angles; a, b, and c. Angle a measures 40 degrees and angle b measures 70 degrees. Show how you would find the measure of angle c. (a + b + c = 180, which can be rewritten as 40 + 70 + c = 180. This expression must first be simplified by adding the numbers of the known angles together 40 + 70 = 110. To find the measure of angle c, you simply need to subtract the sum of angles a and b from 180. This is because the sum of all three angles of a triangle is 180. So, 180 – 110 = 70. Angle c measures 70.) (5) Matt is 8 years younger than his sister, Regina. The sum of their ages is 22 years. What equation could be used to find Matt’s age? (The equation m + (m + 8) = 22 could be used to find the solution. The variable m is worth the same value, so the same number would need to replace m in both parts of the sentence to make the equation true. 7 is the only number that will make the equation true. So, Matt is 7 years old.)

### Learning Tips

 (1) Expressions are mathematical phrases or sentences. Expressions are shown using numbers or variables. A variable is a letter or symbol that is used to take the place of a value in an expression. Expressions must use different mathematical symbols for different operations. A numerical expression uses all numbers and symbols. For example, 45 x 5 is a numerical expression. An algebraic expression uses a combination of numbers, mathematical symbols and variable, (i.e. 4 x c). (2) Mathematical notations can be made using the correct numerical or algebraic expression. Use key words to determine the operation when converting a sentence or word problem into an algebraic expression. Here is a list of some of the words that may be used to determine which operation to use. Addition: added to, more than, increased Subtraction: decreased by, less than, subtracted from, taken away Multiplication: times, multiplied by, product of, times as many Division: quotient of, divided by (3) In order to be successful in writing algebraic expressions from word form, you child will need to use semantics. In other words, he/she needs to think of the words in a real-world situation to make sense of how they are related. This is especially true for subtraction and division. Though the variable and numbers can switch places to get he same result for addition and multiplication problems, this is untrue for subtraction problems. For example, 5 -3 = 2, but 3 – 5 = -2. For this reason, the placement of the variable in a subtraction or division problem is important. While working with the sentence “10 less than a number x”, many children will write 10 – x. This is incorrect. In this sentence, 10 is being taken away from x and the correct expression is x – 10. However, if the sentence say, “9 decreased by a number t” the problem would be written 9 – t because t is decreasing or taking away from the 9. For division expressions such as, “23 divided by a number z”, the expression would be written 23  r. Relatively, “the quotient m and 17”, would be p  17. (4) For a visual and kinesthetic approach to solving equations use sticky notes and a white board or paper. Write out an expression, such as 120 + t, for t = 5. On a small sticky note write 5. Now, have your child place the sticky note right on top of the t in the original problem. After doing this he/she will see a numerical expression: 120 – 5. Repeating this activity several times will alleviate any anxieties caused by seeing numbers mixed with letters. It allows children to see that they’re still working with numbers and simply need to substitute them into the problem. (5) Make sure that your child knows the most commonly used mathematical symbols. Here is a list of the most commonly used mathematical symbols in 6th grade. Addition: + Subtraction: - Multiplication: x, a(b ), ab,  Division: , / Square root:  Percent:  Greater Than:  Greater Than or Equal To:  Less Than:  Less Than or Equal To:  Equal to:  Approximately Equal to:  Not equal to:  Pi:  Parallel lines: || Perpendicular lines:  Angle: 

### Online Resources

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