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

 
     
 
     
 
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6th
Problem Solving
Show Your Work and Describe It in Words
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.
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.
 

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:

Extra Help Problems

(1)

On the night before a big game, Saul sleeps twice as long as usual. Write an expression to represent the situation, if s represents the amount of time he usually sleeps. Show your steps to solve the problem, if Saul usually sleeps for 5-1/2 hours.

(The expression is 2s. Since I know that s is 5-1/2, I can insert this where the s is in the expression. I know that 2s is the mathematical notation for two times s. So, I will rewrite the problem as 2 x 5-1/2. On the night before a big game, Saul sleeps for 11 hours. )

(2)

Matt has 16 video games less, Dwayne’s total videos. Use the variable v to represent Dwayne’s total videos and write an equation to relate these to Matt’s videos (m). Explain the steps to use this equation to find Matt’s videos if Dwayne has 20.

(Use the equation m = v-16 to show that Matt’s videos are 16 less than Dwayne’s. Next, substitute v with 20 to show how many videos Dwayne has; m = 20 -16. Solve the equation, m = 4.)

(3)

Yasmin has a large photo album that holds 150 photos. It is full. She also has 3 smaller albums that are full of pictures. Write an expression to show the total pictures that Yasmin has. Then use it to show the steps to solve for the small photo albums holding 60 pictures each.

(The number of pictures that Yasmin has can be expressed as 150 + 3n. If the small albums hold 60 pictures and they are full, then n – 60. So the problem can be written as the numerical expression; 150 + 3 x 60. 3 x 60 must be solved first because of the order of operations; 150 + 90 = 240. Yasmin has 240 pictures.)

(4)

The Balenos are renting a van for their family vacation. The cost for the van is $30 per day plus a $100 rental fee. Write an expression to show the amount the Balenos will owe for x days. Then use it to solve for a 7 day rental.

(30x + 100 can be used to show the price, $30 for x days, plus the 100 dollar rental fee. Any number of days will be able to be substituted in for x. 30x means 30 times x. So to find the total for 7 days you multiply 30 x 7. The Balenos will pay 210 plus the 100 rental fee. The total they will pay is $310.00 )

(5)

Macy is running in the jog-a-thon. Her dad pledged to pay $3 for every lap she runs. Her grandparents gave her a donation of $25. Write an expression to show the total money Macy will raise if she runs y laps. Then show the steps to solve for 15 laps.

(3y + 25; First, replace y with 15 because y = laps; 3 x 15 + 25. Next, solve the problem using the order of operations. 45 + 25 = 70. Macy will earn $70 if she runs 15 laps.)

(6)

There are two similar rectangles. The first with a width of W and a length of L. The second triangle is ½ the size of 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 = 2 ft. and L = 5 ft.

(The algebraic expression is: (2W + 2L)2. This expression stands for two times the width plus two times the length, divided in half. Since I know that W is 2 and L is 5, I must replace these variables with the numbers in order to solve. The numerical expression is (2 x 2 + 2 x 5) 2. The multiplication problems inside the parenthesis must be solved first: (4 + 10)2. Next, the addition problem inside the parenthesis will be solved;14. Lastly, I will divide the sum by 2 to get the final answer of 7 feet.)

(7)

There are two similar triangles. The first with a base of b and a height of h. The second triangle is ½ the size of the first. Write an expression to show how to find the area of the first triangle. Then show the steps to solve for b = 12 in. and h = 6 in..

(A= b x h 2 can be used to find the area of a triangle. First, substitute 12 for b and 6 for h; A = 12 x 6 2. 12 x 6 = 72 2 = 36. The first triangle has an area of 36 sq. inches.

(8)

There are two similar triangles. The first with a base of b and a height of h. The second triangle is ½ the size of the first. Write an expression to show how to find the area of the second triangle. Then show the steps to solve for b = 12 in. and h = 6 in..

(A= (b x h 2) 2 can be used to find the area of a triangle. First, substitute 12 for b and 6 for h; A = 12 x 6 2. 12 x 6 = 72 2 = 36 2 = 18. The second triangle has an area of 18 sq. inches.

(9)

A Frisbee has a diameter of 4 in. Express algebraically the formula that should be used to solve for circumference. Express algebraically the formula that should be used to solve for circumference. Then clearly show the appropriate steps to solve.

(C d is the formula used to find the circumference of a circle. The symbol has an approximate value of 3.15. This equation can be written numerically as C = 3.14 x 4. So, to find the circumference, you must multiply the diameter by 3.14. C = 12.56 in. )

(10)

A Frisbee has a radius of 4 in. Express algebraically the formula that should be used to solve for circumference when you only know radius. Then, clearly explain the steps needed to find C.

(The equation C (r x 2 ) must be used, because we don’t know the diameter. The diameter is always twice the size of the radius. We can use this to solve for d. So, C (4 x 2 ); C (8 ). . The symbol has an approximate value of 3.15. The final step is to multiply the diameter by 3.14. C = 25.12 in.)

(11)

Mr. Ruiz drives at an average speed of 65 mph. How long will it take him to drive 455 miles? Explain your steps to solve this problem.

(Mr. Ruiz can drive 65 miles in one hour. To solve this, you can use the formula d = rt or distance = rate x time. In this problem 65 mph is the rate (r) and 455 is the distance (d). The numbers can be inserted into the formula a 455 = 65t. To find t, I will isolate the variable b by moving 65 to the other side of the equal sign and using the inverse operation. The inverse operation of multiplication is division. So, 455/65 = 7. It will take Mr. Ruiz 7 hours to drive 455 miles.)

(12)

A dolphin swims at an average rate of 15 mph. At this speed, how long will it take the dolphin to travel 10 miles? Write the symbolic formula needed to solve this problem, then use it to show how to solve the problem.

(Use the formula d = rt; d = 10, r=15. So, 10 = 15t. The t needs to be isolated before the value can be found. To do this the 15 must be moved to the other side of the equal sign and the inverse operation performed. The opposite of multiply is divide, so 10/15 = .67. It will take about .67 hours to travel 10 miles.)

(13)

The measures of the four angles of a rhombus have a total value of 360. The angles are labeled, a, b, c and d. Show the steps to solve for the value of a, if b = 80, c = 100 and d = 80.

(The equation a + b + c + d = 360 can be used to solve this problem. The variables can be replaced as follows; a + 80 + 100 + 80. Next, add up all known angles and rewrite with the variable a; a + 260 = 360. To find the value of a, subtract 260 from 360, a = 100.)

(14)

Two complementary angles are labeled m and n. Write an expression to find the value of m, if n = 75. Solve and explain.

(Complementary angles have a total measure of 90. So, m + n = 90. Since n = 75, the equation can be written m + 75 = 90. To isolate m, 75 can be subtracted from 90. Thus, m = 15.)

(15)

Show how to find the value of t, if it is supplementary to y, which is 34.

(t + y = 180, t + 34 = 180, t = 180 – 34, t = 146.)

 

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