Enrichment Investigation
“Body Mass Mania”
Common Core State Standard(s):
6.EE.2 : Write, read, and evaluate expressions in which letters stand for numbers.
a. Write expressions that record operations with numbers and with letters standing for numbers. For example, express the calculation “Subtract y from 5” as 5 – y.
c. 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 = 6 s2 to find the volume and surface area of a cube with sides of length s = ½.
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. / Standard(s) for Mathematical Practice:
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
Materials Needed:
· Task Sheet – Body Mass Mania
· Calculator
· Graph paper
Instructions:
1. Student(s) should have background in working with formulas and substituting values for specific variables.
2. Student(s) read instructions and task(s) for investigation two.
3. Students create tables and charts as specified in task.
4. Estimated time for investigation: 1-2 class periods
Sources:
· Website: www.figurethis.org
· Website: “Tales of Statisticians” http://www.umass.edu/wsp/statistics/tales/quetelet.html
Investigation: Body Mass Mania
Generalization: Algebraic formulas using variables aid society in scientific research.
Background: The Body Mass Index(BMI) formula was developed by a Belgium mathematician named Adolphe Quetelet (1796-1874). Doctors today may use this formula as a health risk indicator of obesity. This is a simple formula where you only need to know height and weight.
The “Imperial BMI Formula” is W (703) ÷ (H)2 where W= weight in lbs. and H = height in inches
The metric “Imperial BMI Formula” is W ÷ (H)2 where W = weight in kilograms and H= height in meters.
According to the US Department of Health and Human Services, you can use the table above to determine a person’s weight status. From the above table, if the BMI is greater than 27 or less than 19, this may be an indicator of a health risk.
Part One: Harold has a height of 5 ft 2 in and a weight of 115 pounds. Your task is to first calculate Harold’s BMI using both formulas above. Show all of your work in an organized way. Be sure to label all equations and label all of your variables clearly. Explain if Harold’s BMI indicates a health risk.
* In order to calculate the Metric Imperial Formula you will need to investigate conversions from inches to meters and lbs to kilograms.
Part Two: Let’s investigate a little further. What would be a healthy weight range for Harold’s height? Explain your reasoning.
Part Three: According to BMI, determine the heights that a weight of 130 pounds would be safe. Show all of your calculations and explain your reasoning.
Further investigation of BMI in children: http://www.cdc.gov/healthyweight/assessing/bmi/childrens_bmi/about_childrens_bmi.html
Name ______Date ______Class ______
X in the Mix
Variables are used to describe the following recipe for Chocolate Brownies.
S = number of servings
B = cups of butter
C = cups of cocoa
X = cups of oil
G = cups of sugar
V = teaspoons of vanilla
E = number of eggs
F = cups of flour
N = cups of nuts
If the recipe requires cup of oil, use the formulas below to determine the remaining ingredients for the brownies.
B = X + C
C = or 3X
G = or 8X
V = X + C + B
E = G + V
F = E ÷ 3
N =
S = 2 x (E 3 ÷V 3)
Chocolate Brownie Recipe
This recipe makes ______servings.
_____ cups butter
_____ cups cocoa
cup oil
_____ cups sugar
_____ tsp. vanilla
_____ eggs
_____ cups self-rising flour
_____ cup nuts
Heat oven to 350°. Grease and lightly flour bottom of 8 or 9 inch square pan. In large saucepan, melt butter over low heat. Add cocoa and oil once butter is melted, stirring until completely blended.
Blend sugar and vanilla. Beat in eggs, one at a time. Lightly spoon flour into measuring cup; level off. Stir in flour and remaining ingredients. Spread into pan. Bake 20-25 minutes, or until set in center. Cool completely. Cut into bars.
Expressions
Directions: Read and answer each problem. Some problems may require more than one answer.
Remember – 1. Define the variables
2. Show your work
3. Draw a box around your answer(s)
1.) Joe’s weight is four times that of his baby sister. Write an expression for each of their weights.
2.) Francis solved 17 more problems than Beatrice. Write an expression for each person.
3.) Write an expression for the number of hours in an unknown number of days.
Problems may be credited to Edward Zaccaro’s book Real-World Algebra © 2001.
Expressions
Directions: Read and answer each problem. Some problems may require more than one answer.
Remember – 1. Define the variables
2. Show your work
3. Draw a box around your answer(s)
1.) Julie emptied her piggy bank and took all of the money with her to the store. She purchased a skateboard for $50 and then she found $11 on the sidewalk on her way home. Write an expression for the amount of money Julie has left.
2.) Write an expression for the value of a pile of quarters in dollars.
3.) Mark earns $420 more than one half of Jacob’s salary. Write an expression for Mark’s salary where ‘n’ stands for Jacob’s salary.
Problems may be credited to Edward Zaccaro’s book Real-World Algebra © 2001.
Expressions
Directions: Read and answer each problem. Some problems may require more than one answer.
Remember – 1. Define the variables
2. Show your work
3. Draw a box around your answer(s)
1.) Write an expression for the perimeter of a rectangle that has a length which is three times its width.
2.) A rancher has 25 horses and an unknown number of llamas. Write an expression for the number of legs on the ranch where ‘n’ is the number of llamas. (Don’t forget the rancher.)
3.) Write an expression for the average of four consecutive odd numbers where ‘c’ is the smallest number.
Problems may be credited to Edward Zaccaro’s book Real-World Algebra © 2001.
.
Properties with Whole Numbers Enrichment
Name ______Date ______Class ______
Mental Math Using Properties
Simplify each problem mentally (no calculator or paper/pencil). Record each answer and your solution strategy (be specific – write down each thought that you went through to arrive at your solution). After you’ve done this, determine the mathematical property that you used to help you mentally simplify the problem. If you do not think that you used any property to simplify the problem, determine a property that could have been used to simplify the problem mentally.
1. + + + 10. 19 (21)
2. 15 • • • 16 11. 4 • • • 25 • 6
3. ( + ) 24 12. 3x + 2y + 4x + 9y
4. () • 5942 13. 36 (x + 2y)
5. 24 + 17 + 95 + 13 + 26 + 5 14. 5x + 2x2 + 3x + 5x2
6. 25 • 22 • 3 • 4 15. ( 4 + 2y)
7. (42 + 54)
8. 24 • 599 • 0 • 278 • 92
9. 29 • 74 + 29 • 26
Name ______Date ______Class ______
Alien Math
x / $ / * / # / @ / !$ / @ / ! / # / * / $
* / ! / @ / # / $ / *
# / # / # / # / # / #
@ / * / $ / # / ! / @
! / $ / * / # / @ / !
+ / $ / * / # / @ / !
$ / @ / # / $ / ! / *
* / # / ! / * / $ / @
# / $ / * / # / @ / !
@ / ! / $ / @ / * / #
! / * / @ / ! / # / $
Wow! A family of aliens just had lunch in your backyard, and one of their kids left his homework for you to find. It seems they use the same symbols for addition and multiplication that we do, but different symbols for the numbers. Use the addition and multiplication tables above to discover the following about the alien math.
- Is there an additive identity? If so, what is it? How do you know?
- Is there a multiplicative identity? If so, what is it? How do you know?
- Is addition commutative? How do you know?
- Is multiplication commutative? How do you know?
- Make up some alien addition and multiplication problems to see whether the associative properties hold for this number system.
- Make up some problems to see whether the distributive property works in this system.
- What is the additive inverse of each number?
- Do all the numbers have a multiplicative inverse? What is the multiplicative inverse of each number?
- Can you figure out the Earth number that matches each symbol? How do these aliens do math?
Name ______Date ______Class ______
Rectangle Algebra
Online at http://illuminations.nctm.org/Lesson.aspx?id=1820, you will find a rectangle with dotted lines. Carefully cut along the dotted lines of the rectangle. Keep the rest of the page intact. When you are done cutting, you should have two pieces that look like this:
Shape Sorter Rectangle
There are four different moves that you will perform with the rectangle. If you start with the rectangle placed in the shape sorter, the rectangle will completely fill the hole after any of these moves is performed.
N...... do nothing
...... one-half turn in a clockwise direction
V...... reflection across the vertical center
H ...... reflection across the horizontal center
Place an N at the top front of the rectangle. This represents a “do nothing” move. When performing any move in this activity, begin with the N at the top front. This is the “original position.”
The other labels mark the top front after a move has been made from the original position. Therefore, appears at the end opposite the N, indicating the top front after a 180° rotation about the center of the rectangle; the V is at the same end as the N but on the other side, indicating a reflection across the vertical center of the rectangle; and, the H is at the end opposite the V, indicating a reflection across the horizontal center of the rectangle. With all labels in place, your rectangle should look like this:
front back
Start with the rectangle in original position — that is, in the shape sorter with the N at the top front.
The symbol # means “followed by,” so the expression # V means to do a first move of a one-half turn followed by a second move of a reflection across the vertical center of the rectangle. What is the result; that is, what letter is at the top front after these two moves? In the chart below, indicate the result in the appropriate square.
Now, perform all possible combinations of first moves followed by second moves to complete the entire chart. Each time, begin with the rectangle in the starting position.
2ND MOVE1ST MOVE / # / / V / H / N
V
H
N
1. What is # V? 2. What is H # H? 3. What is (H # V) # ( # N)?
4. What happens when the second move is the same as the first move?
5. What happens if either move is N?
6. Is there an identity move in the system? Explain.
7. Does every move have an inverse? Explain.
8. Is the # operation commutative? Explain.
9. Is the # operation associative? Explain.
Name ______Date ______Class ______
Everything Balances Out in the End
Step 1: Complete the Who is Correct? handout.
Based on your work in step 1, you probably determined that Andy could benefit from using a step-by-step process to simplify numeric expressions.
Step 2: You will now need to connect to the Internet. Go to http://illuminations.nctm.org. Click Activities. Then click 6-8. Find the activity called Pan Balance – Numbers. Select Pan Balance – Numbers.
1. Type 25 into the red box. What happens to the balance?
2. Type 25 into the blue box. What happens to the balance?
3. What can you conclude about the way the balance works?
Click Reset Balance.
Step 3: Use the Pan Balance – Numbers to complete the Balancing Expressions handout. Be sure to follow the directions and record each step you are following in the order of operations as you work.
Step 4: Use the Pan Balance – Numbers to complete the Balancing Exponents handout. Be sure to read all of the directions and follow them. In the last paragraph before the problems, instructions are given on how to enter exponents into the computer.