Download Tuesday, July 7, pm

Content Session 4
July 7, 2009
Addition & Subtraction
M3N5. Students will understand the meaning of
decimal fractions and common fractions in simple cases
and apply them in problem-solving situations.
e. Understand the concept of addition and subtraction of
decimal fractions and common fractions with like
denominators.
f. Model addition and subtraction of decimal fractions and
common fractions with like denominators.
g. Use mental math and estimation strategies to add and
subtract decimal fractions and common fractions with
like denominators.
Addition & Subtraction
M4N5. Students will further develop their
understanding of the meaning of decimals and use
them in computations.
c. Add and subtract both one and two digit decimals.
M4N6. Students will further develop their
understanding of the meaning of decimal fractions and
common fractions and use them in computations.
b. Add and subtract fractions and mixed numbers with like
denominators. (Denominators should not exceed twelve.)
Addition & Subtraction
M5N4. Students will continue to develop their
understanding of the meaning of common
fractions and compute with them.
g. Add and subtract common fractions and mixed
numbers with unlike denominators.
Goals (Grades 3 & 4)
• Grade 3
– The meaning of addition and subtraction remains
the same even when numbers become decimal
numbers or fractions
– Decimal numbers and fractions are numbers, just
like whole numbers
• Grade 4
– Fluency with decimal addition and subtraction
– The sum of two fractions may exceed 1 (improper
fractions or mixed numbers), and the minuend
may also exceed 1.
Goals (Grades 5)
• Fluency with fraction addition and subtraction
Key Ideas
• Unitary perspective of numbers
– 0.3 is 3 0.1-units; 3/5 is 3 1/5-units; etc.
• Relative size of (decimal) numbers
– 0.32 is 32 0.01-units
• Addition/Subtraction can be performed only
when the two numbers are referring to the
same unit
How do these relate to 3 + 4?
– 30 + 40
– 300 + 400
– 3000 + 4000
– etc.
How about these?
– 300 + 40
– 30 + 4000
If you put a tape that is 0.3 meters long and
another tape that is 0.4 meters long together,
end to end, how long will it be?
• What math sentence will represent this
problem?
• What does 0.3 meters mean?
• What does 0.4 meters mean?
• How many 0.1 meter will there be altogether?
• What is the answer?
If you put a tape that is 3/8 meters long and
another tape that is 3/8 meters long together,
end to end, how long will it be?
• What math sentence will represent this
problem?
• What does 3/8 meters mean?
• What does 4/8 meters mean?
• How many 1/8 meter will there be altogether?
• What is the answer?
If you put a tape that is 0.6 meters long and
another tape that is 0.8 meters long together,
end to end, how long will it be?
• What math sentence will represent this
problem?
• What does 0.6 meters mean?
• What does 0.8 meters mean?
• How many 0.1 meter will there be altogether?
• What is the answer?
If you put a tape that is 0.07 meters long and
another tape that is 0.05 meters long together,
end to end, how long will it be?
• What math sentence will represent this
problem?
• What does 0.07 meters mean?
• What does 0.05 meters mean?
• How many 0.01 meter will there be altogether?
• What is the answer?
If you put a tape that is 3.6 meters long and
another tape that is 2.2 meters long together,
end to end, how long will it be?
• What math sentence will represent this
problem?
• What does 3.6 meters mean?
• What does 2.2 meters mean?
If you put a tape that is 3.6 meters long and
another tape that is 2.2 meters long together,
end to end, how long will it be?
• What math sentence will represent this
problem?
• What does 3.6 meters mean?
• What does 2.2 meters mean?
3 meters + 2 meters, and
6 0.1-meters + 2 0.1-meters
5 meters and 8 0.1-meters, or 5.8 meters
If you put a tape that is 3.6 meters long and
another tape that is 2.2 meters long together,
end to end, how long will it be?
• What math sentence will represent this
problem?
• What does 3.6 meters mean?
• What does 2.2 meters mean?
36 0.1-meters + 22 0.1-meters
58 0.1-meters, or 5.8 meters
If you put a tape that is 3.73 meters long and
another tape that is 2.2 meters long together,
end to end, how long will it be?
• What math sentence will represent this
problem?
• What does 3.73 meters mean?
• What does 2.2 meters mean?
3. 7 3
+ 2. 2
3. 7 3
+ 2. 2 0
If you put a tape that is 5/8 meters long and
another tape that is 7/8 meters long together,
end to end, how long will it be?
• What math sentence will represent this
problem?
• What does 5/8 meters mean?
• What does 7/8 meters mean?
• How many 1/8 meter will there be altogether?
• What is the answer?
If you put a tape that is 5/8 meters long and
another tape that is 3/4 meters long together,
end to end, how long will it be?
• What math sentence will represent this
problem?
• What is different about this problem?
• What does 5/8 meters mean?
• What does 3/4 meters mean?
• What can we do?
If you put a tape that is 5/8 meters long and
another tape that is 3/4 meters long together,
end to end, how long will it be?
• Let’s make these fractions refer to the same
unit.
• We can change 3/4 into 6/8 [or we can
change both to 10/16 and 12/16, or some
other equivalent fraction pairs]
• Now, we know how to add those fractions.
Do we need the least common denominator?
• No – we just need a common denominator
(common unit) in order to add.
Multiplying & Dividing Decimals
M4N5.
e)Multiply and divide both one and two digit decimals by
whole numbers.
M5N3.
b)Explain the process of multiplication and division,
including situations in which the multiplier and divisor are
both whole numbers and decimals.
Multiplying & Dividing Decimals
M4N5.
e)Multiply and divide both one and two digit decimals by
whole numbers.
M5N3.
b)Explain the process of multiplication and division,
including situations in which the multiplier and divisor are
both whole numbers and decimals.
Which problem can we use our whole
number multiplication knowledge to solve?
• 1m of wire weighs 1.4 lb. How much will 6m
of the same wire weigh?
• 1m of wire weighs 6 grams. How much will
1.4 m of the sa,e wore weigh?
What does 1.4 x 6 mean?
Double Number Line
0
0
1.4
1
?
Weight (in grams)
Length (in meters)
6
Decimal Unit Approach
0
0
•
•
•
•
14
1
?
Weight (in 0.1 grams)
Length (in meters)
6
We have 6 groups of 14 0.1 grams.
14 x 6 = 84; Altogether, we have 84 0.1 grams.
84 0.1 grams  8.4 grams
1.4 x 6 = 8.4
Which problem can we use our whole
number division knowledge to solve?
• 4 m of iron pipe weighs 3.6 kg. How much will
1m of the same pipe weigh?
• 3.6 m of iron pipe weighs 4 kg. How much will
1m of the same pipe weigh?
What does 3.6 ÷ 4 mean?
Double Number Line
0
0
?
3.6
Weight (in kg)
Length (in meters)
1
4
Decimal Unit Approach
0
0
•
•
•
•
?
36
Weight (in 0.1 kg)
Length (in meters)
1
4
Divide 36 0.1-kg to make 4 equal groups.
36 ÷ 4 = 9; Each group will have 9 0.1-kg.
9 0.1-kg = 0.9 kg.
3.6 ÷ 4 = 0.9
What if we had 3.7 ÷ 4 mean?
• 37 0.1-kg: make 4 equal groups
• BUT,
37 ÷ 4 = 9 rem. 1
• Each group will get 9 0.1-kg and there will
be 1 0.1-kg left over.
• 3.7 ÷ 4 = 0.9 rem. 0.1
Dividing on: 3.7 ÷ 4
• Model 3.7 ÷ 4 using base-10 blocks – use a
flat as 1.
• What will be left over?
• Can we trade it? With what?
What is 8 ÷ 5?