Mathematics By Design
Original Lesson Samples
Look at other Lesson Samples under Designs for Educators, Math Can Take You Places from PBS Affiliate KERA.
Look at other Lesson Samples under Designs for Educators, Math Can Take You Places from PBS Affiliate KERA.
A Method to clarify solving two-step equations:
Solve for the value of x,
that makes this equation true: 2x + 14 = 32.
Use the "Cover Up" Method to solve the equation.
Cover the variable term 2x:
2x + 14 = 32
Solve for the value of x,
that makes this equation true: 2x + 14 = 32.
Use the "Cover Up" Method to solve the equation.
Cover the variable term 2x:
2x + 14 = 32
- This means that some number x, times 2 plus 14 is 32 or another way of stating this is that some number x, times 2 is the same (value) as 32;
- Therefore, the 2x term is taking the place of 18. How do we know this? Because [18] plus 14 is 32.
- Two times some number (2x) is equal to or the same as 18.
- We know that 2 times 9 is equal to or is the same as 18.
- Therefore the solution to this equation is 9 (the one value that makes it both sides of the equation the same value).
Check your solution:
2 times 9 ( our solution) is 18.
18 plus 14 is equal to or "is the same as" 32.
(2 x 9) + 14 = 32.
Note: Always, clear the procedure inside of the parenthesis first. Parentheses are one componet of a set of mathematics symbols known as grouping symbols.
You have checked for reasonableness of the solution and the value 9 makes this equation true!
Using Multiplication Tables in the Secondary Classroom
Multiplication tables are basically graphic organizers, which hold
an orderly arrangement of multiplication fact families. The green
diagonal splits the table in half showing the multiplication is
commutative. In other words,you can exchange the order of the
factors and get the same product. The commutative property
also applies to addition: 3 + 5 = 5 + 3, but does not apply to
subtraction or division.
The technical term for the first factor is called the multiplicand.
The technical term of the second factor is called the multiplier.
What is unique about square numbers, is that the multiplicand
and the multiplier are the same number. Mathematics has
language of its own and be challenging for any learner. We could
all be considered ELL or Second Language Learners when it
comes to mathematics. See Glencoe-McGraw/Hill's eGlossary
of mathematics terms and vocabulary.
The numbers that we call perfect squares are along the green
diagonal. A square number is the product you get when you
multiply a whole number by itself, such as 5 x 5 = 25. When we
undo square numbers, we call that taking the square root. So,
positive square root of 25, [written symbolically below the chart]
is 5, because 5 x 5 = 25.
Numbers such as the square root of 32 [written symbolically
below the chart] are not perfect squares because there is not
a whole number that when multiplied by itself produces a product
of 32. Observe that the product 32 is not along the green
diagonal on the multiplication table.
However, the square root of 32, is what we call an
irrational number, because it produces a decimal that does
not terminate and does not repeat. We can see that along the green diagonal, the square root of 32 would lie between 25 and 36. Therefore, the square root would fall between 5 and 6, a decimal closer to the number 6, since 32 is closer to 36.
An estimate of the square root(32) = 5.65685425....Remember, imperfect squares are irrational numbers, the decimals keep on going and going and going and going...
Multiplication tables are basically graphic organizers, which hold
an orderly arrangement of multiplication fact families. The green
diagonal splits the table in half showing the multiplication is
commutative. In other words,you can exchange the order of the
factors and get the same product. The commutative property
also applies to addition: 3 + 5 = 5 + 3, but does not apply to
subtraction or division.
The technical term for the first factor is called the multiplicand.
The technical term of the second factor is called the multiplier.
What is unique about square numbers, is that the multiplicand
and the multiplier are the same number. Mathematics has
language of its own and be challenging for any learner. We could
all be considered ELL or Second Language Learners when it
comes to mathematics. See Glencoe-McGraw/Hill's eGlossary
of mathematics terms and vocabulary.
The numbers that we call perfect squares are along the green
diagonal. A square number is the product you get when you
multiply a whole number by itself, such as 5 x 5 = 25. When we
undo square numbers, we call that taking the square root. So,
positive square root of 25, [written symbolically below the chart]
is 5, because 5 x 5 = 25.
Numbers such as the square root of 32 [written symbolically
below the chart] are not perfect squares because there is not
a whole number that when multiplied by itself produces a product
of 32. Observe that the product 32 is not along the green
diagonal on the multiplication table.
However, the square root of 32, is what we call an
irrational number, because it produces a decimal that does
not terminate and does not repeat. We can see that along the green diagonal, the square root of 32 would lie between 25 and 36. Therefore, the square root would fall between 5 and 6, a decimal closer to the number 6, since 32 is closer to 36.
An estimate of the square root(32) = 5.65685425....Remember, imperfect squares are irrational numbers, the decimals keep on going and going and going and going...
