The Learning Principle...

Ms. Ashley Ward

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

The 411 on the Learning Principle•Learning and understanding mean being able to

make applications•Learning involves taking what is known about one situation and applying it properly to a new one

•“Learning with understanding also makes subsequent learning easier.”

•Learned concepts are more easily recalled

•“…Learning with understanding is essential to enable students to use what they learn to solve the new kinds of problems they will inevitably face in the future.”

Basically, this follows Skemp’s ideas about understanding relationally… This has many implications for teachers in the classroom:

•The mathematical mis-match

•Frustration

•Time constraints

A Lesson on Percentages Grades 6-

8

The Meaning of Percent Grid 1 Grid 2 Grid 3

We can represent each of these fractions and decimals as a percent using the symbol %.

96/100 = 96% 9/100 = 9% 77/100 = 77%

Comparing Shaded Boxes to Total BoxesGrid Ratio Fraction Decimal

1 96 to 100 96/100 .962 9 to 100 9/100 .093 77 to 100 77/100 .77

Definition: A percent is a ratio whose second term is 100. Percent means

parts per hundred. In mathematics, we use the symbol % for percent.

Applied Percents

30% of 60 is the same as 30% x 60.

Both quantities are represented as the decimal form of 30% multiplied by the number 60.

.30 x 60 = 18

To find a 15% increase in 5, there are two ways to solve this.

1: Find 15% of 5, then add that quantity to the original quantity (5), or

2: Find 1.15% (1+.15) of 5.

By finding 1.15% of 5, we are doing both of the steps from the first method at one time.

Applied Percents

Applied Percents

The same works for a decrease in percent, only the amount is subtracted.

To find a 75% decrease in 80:

1. Find 75% of 80 and subtract that quantity from 80, or

2. Find 25% (1-.75) of 80.

Activity: an Experiment in PercentsDirections: Given two scenarios, determine in which case you will be able to afford your purchase.

Scenario 1:

You have $20.

You wish to purchase a sweater that is $24.99, at %25 discount.

The sales tax is 7.0%.

Scenario 2:

You have $20.

You wish to purchase shoes for $17.99.

The sales tax is 7.5%

The Answer

Scenario 1:

25% of $24.99: $24.99 - (0.25 x $24.99) or 0.75 x $24.99 = $18.74.

Adding 7.0% sales tax: (0.07 x $18.74) + $18.74 or 1.07 x $18.74 = $20.05

No, you cannot afford this purchase.

The Answer

Scenario 2:

Including the 7.5% sales tax to $17.99:

(0.075 x $17.99) + $17.99, or

1.075 x $17.99 = $19.34

Yes, you can afford this purchase.

Percents for High SchoolScience Class:

20% Test 1

20% Test 2

20% Test 3

25% Term Paper

10% Quizzes

5% Attendance

Grades:

Test 1: 87

Test 2: 91

Test 3: 84

Paper: 88

Quizzes: 95

Attendance: 90

Directions:Determine your grade in the class.

Answer

Your grade:

88.4% B+