Elementary Math - Recognizing number sense

While visiting a historic district during a field trip, the students were told the dates that four of the historic houses were built. The chart below list those dates. According to the list, which house was built last?

a. House no. (1)
b. House no. (2)
c. House no. (3)
d. House no. (4)

Answer : ( c ) – House no. (3)

In order to solve this problem, a student must recognize which number in the table represents the highest number. That number will indicate the year the last house was built. By examining the table, it is determined that 1853 is the highest number in the table. Therefore House no. (3) was built last.

The student can list the numbers in ascending or descending order to assist in identifying the highest number.

In order to list the numbers in ascending order, they can be written as follows:

1803 - 1819 - 1836 - 1853

In order to list the numbers in descending order, the can be written as follows:

1853 - 1836 - 1819 - 1803

Both of the above cases indicates that the highest number is 1853 and the lowest number is 1803.

©Copyright 2008.Najwa S. Hirn. All rights reserved.

College Level Math – Vertex, Axis of Symmetry and Maximum/Minimum points

Find the vertex, the axis of symmetry, the maximum or minimum value of the quadratic function.

The above equation is written in the standard quadratic equation form which is:

It is important to be able to calculate the vertex, line of symmetry, and the maximum or minimum points from a quadratic equation. These crucial points can assist in identifying the characteristics of the resulting parabola and thus being able to sketch the resulting graph.

It is important to note that the sign of the (a) term will determine the orientation of the parabola. A positive (a) term – an (a) term greater than zero - will indicate that the parabola will open upwards, while a negative (a) term – an (a) term less than zero – will indicate that the parabola will open downward.

A vertex is either the highest or lowest point of a parabola depending on its orientation. Once calculated, the vertex is written in an (x ,y) ordered pair.

Therefore, it is easy to determine the following characteristics about the parabola presented in this problem. Those characteristics are:

1) This parabola opens upwards since the (a) term is positive or greater than zero.

2) This parabola will have a vertex at its lowest point.

3) This parabola will have a minimum point.

4) The axis of symmetry will be the vertical line passing through the x-coordinate of the vertex and dividing the parabola into two equal parts.

The following steps detail how the vertex is calculated for this problem:

Vertex

Axis of Symmetry

The axis of symmetry is the line that splits the parabola into two identical halves. It passes through the vertex.
Since this quadratic opens upwards, that means that the axis of symmetry is the x co-ordinate of the vertex. Therefore, the axis of symmetry is the vertical line that passes through x = 5

Maximum or Minimum points

The vertex of a parabola is usually considered as either a maximum or a Minimum value.

We determine if the point is a maximum or minimum by looking at the a co-efficient of the a-term of the quadratic equation. Since the a-term in this quadratic equation is greater than zero, then the vertex is a minimum point because the parabola opens upwards.

Therefore for this quadratic the minimum value point occurs at (5,-28)
If a > 0, the vertex is a minimum point and the minimum value of the quadratic function f is equal to the y-co-ordinate of the vertex. That means the minimum point for this quadratic = -28
This minimum value occurs at x = = -b/2a which is the x-co-ordinate of the vertex. For this quadratic the minimum value = 5



© Copyright 2008.Najwa S. Hirn. All rights reserved.

Rate - High School Math

Math at Hand: A Mathematics Handbook
A patient receives her nutrition by a stomach tube with the use of a feeding pump. The rate of fluid pumped is usually measured in milliliters per hour (mL/hr). The feeding pump used to infuse the formula in the tube is set to pump at a rate of 300 mL/hr. The patient is fed for only 45 minutes. How many milliliters did this patient receive?

Answer: 225 milliliter

The term milliliter per hour means a certain volume of fluid is to be pumped into a patient for a length of one hour. Therefore, in this problem the patient’s pump is set to feed the patient 300-milliliter of formula for every hour the pump is running. This also means that the rate is set for 300 mL per 60 minutes.

Since the patient received nutrition for only 45 minutes then the student must set up the problem as a ratio and proportion to be able to solve it. The steps are as follows:

[300 mL / 60 min] = [x mL / 45 min] = 225 milliliters

Note that the units for minutes disappear and the student is left with the milliliter units that are required to solve the problem.

©Copyright 2008.Najwa S. Hirn. All rights reserved.

Measurements - Middle School Math

A long piece of wood is 8.25 feet long. If this piece were to be cut into 5 equal pieces how long will each piece be?

a. 11.25 inches
b. 19.80 inches
c. 22.75 inches
d. 27.15 inches

Answer : 19.80 inches (answer b)

The student must notice that the dimension of the original piece of wood is in feet units while the answers are in inch units. This is an important observation since all units must match in order to solve the problem. Therefore, as a starting point, we will convert the 8.25 feet into inches. In order to do that, the student must realize that each 1 foot = 12 inches. Therefore:

8.25 feet x [12 inch/1 foot] = 99 inches
(note here that the units for feet are eliminated and we are left with only the inch units we need to continue solving the problem).

Since we need to cut this piece into 5 equal pieces then we must divide the 99 inches by 5 to reach the final answer:

99 inches / 5 = 19.8 inches

©Copyright 2008.Najwa S. Hirn. All rights reserved.

Subtraction - Elementary School Math

There are a total of 72 cars parked in the used parking lot 8 of them are blue. How many cars are not blue?

a. 58
b. 62
c. 64
d. 68

Answer: 64 (c )

This is a subtraction problem. The student must subtract the 8 blue cars from the total 72 cars in order to come up with what is remaining. The remaining number will be the number of cars that are not blue. Here is the step:

72 - 8 = 64

For additional interactive practice visit: Aplusmath.com

©Copyright 2008.Najwa S. Hirn. All rights reserved.

Functions - College Level Math

If

f(x) = (x-3)

and

g(x) = (2x + 4)

Evaluate:

(f + g) (2)

Answer : 7

The above two equations are examples of the use of the Algebra of functions.

A function is a relation that does not repeat any first coordinates.

The problem is asking the student to evaluate (f+g)(2). This is the same as saying take the sum of functions f(2) and g(2) respectively and evaluate the result.

The steps involved in solving this problem are as follows:

• Substitute the equation of f(x) for the (f) value of the function.
  
• Substitute the equation of g(x) for the (g) value of the function.
  
• Add the two equations.
  
• Take the function with respect to (2) Evaluate the answer.
  

Here is the algebraic solution for the problem.

f(2) = ( 2 - 3) = -1

g(2) = ( 2(2) + 4 ) = ( 4 + 4 ) = 8

(f + g) (2) = ( -1 + 8 ) = 7


© Copyright 2008.Najwa S. Hirn. All rights reserved.

Parallel Lines - High School Math

The line shown on the Cartesian coordinate graph below passes by points AB. Another line needs to be drawn parallel to line AB and passes by point C shown on the graph and point D.

What coordinates should point D have that will make line CD parallel to line AB?

a. (-3, 4)
b. (-5, 2)
c. (-6, 0)
d. (-9,-1)

Answer: The coordinates for point D are (-9,-1).

In order to solve this problem the student must recall that parallel lines have the same slope. Therefore, the slope that is calculated for line AB will be the same slope for the line parallel to it that passes through point C.

Therefore, step one is to calculate the slope for line AB.
The coordinates for point A are (6,1)
The coordinates for point B are (1,0)

The slope formula states that:

Let apply the formula to find the slope for line AB:

m = (0 - 4)/(1 – 6) = (- 4)/(- 5) = 4/5

An examination of point C on the graph reveals that the coordinates for this point are:
(- 4 , 3).

The student must now use this point and check each of the above coordinates provided as a possible answer to determine which pair resulted in a slope of 4/5.

Once the coordinate pair of (-9 , -1) is checked, it reveals the following:

m = (- 1 – 3)/ (- 9 – [- 4])
m = (- 1 – 3)/ (- 9 + 4) = (- 4)/(- 5) = 4/5

©Copyright 2008.Najwa S. Hirn. All rights reserved.