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TIPM3 November 16, 2010
What Children Should Know about 8
Minimal Skills• Count to 8• Count 8 objects and know that the last
number word tells how many• Write the numeral 8• Recognize and read the numeral 8
• More and less by 1 and 2• 8 is one more than 7, one less than 9, two more
than 6 and two less than 10
• Spatial patterns for eight
What Children Should Know about 8
• More and less by 1 and 2• 8 is one more than 7, one less than 9, two more than 6 and
two less than 10• Spatial patterns for eight • Anchors to 5 and 10:
• 8 is 3 more than 5 and 2 away from 10• Part-whole relationships:
• 8 is 5 and 3, 2 and 6, 7 and 1, and so on.• Doubles: double 4 is 8• Relationship to the real world
• My brother is 8 years old, my reading book is 8 inches ling
What Children Should Know about 8
Addition and Subtraction Problem Structures• Categories that the students do
not need to master• Teachers need to know as part of
Pedagogical content knowledge (PCK)
• Allows for proper sequencing to support student learning.
Main Ideas
• Two different approaches for developing operation meaning
• Word problems vs use of models• Cognitively Guided Instruction
Group and van De Walle support through word problems
• Parallel development of addition and subtraction
Important Ideas
• Avoid the phrase take away for subtraction.
• Instead use minus or subtract• Addition is not defined as join and
subtraction as remove• There are joining action problems
that can be classified as subtraction.• There are remove problems that
require addition.
Important Ideas4 + [ ] = 9 (join, change unknown)This is a subtraction situation9 – 4 = [ ]
For large numbers like4,376 + [ ] = 9,201Subtraction must be used to solve How do you get children to see the
equivalence of these equations?
More Important IdeasStrong tendency to use the equal
sign as a symbol to precede the answer.
The correct meaning of the equal sign is very difficult to develop
Develop the idea of the equal sign as « the same as »
Most ImportantlyView subtraction as finding the
missing part. Subtraction is used to name or find a
part of the whole when the whole and one of the parts is known.
When seeing 9 – 6, think of 6 and what number make 9?
Addition and Subtraction Problem Structures• Join Problems• Separate Problems• Part-Part-Whole Problems• Compare Problems
Each structure has three numbers. Any one of the three numbers can be the unknown in the story problem
Join Problems
Initial
Change
Result
Use pennies and nickels to model the stories.
Join: Result Unknown
Initial
Change
Result
Sandra had 8 pennies. George gave her 4 more. How many pennies does Sandra have altogether.
Join: Result Unknown
8
Change
Result
Sandra had 8 pennies. George gave her 4 more. How many pennies does Sandra have altogether.
Initial
4
12
Join: Change Unknown
Initial
Change
Result
Sandra had 8 pennies. George gave her some more. Now Sandra has 12 pennies. How many did George give her?
128
?
Join: Initial Unknown
Initial
Change
Result
Sandra had some pennies. George gave her 4 more. Now Sandra has 12 pennies. How many pennies did Sandra have to begin with?
?
4
12
Join Problems: On Your Own
Join, Result UnknownJoin, Change UnknownJoin, Initial Unknown
Write an equation to show the solution.
Separate Problems
Initial
Change
Result
• Any one of the three numbers can be the unknown in the story problem.
• In separate problems, the initial amount is the whole or the largest amount
Separate: Result Unknown
Initial
Change
Result
Sandra had 12 pennies. She gave 4 pennies to George. How many pennies does Sandra have now?
Separate: Change Unknown
Initial
Change
Result
Sandra had 12 pennies. She gave some pennies to George. Now she has 8 pennies. How many did she give to George?
Separate: Initial Unknown
Initial
Change
Result
Sandra had some pennies. She gave 4 pennies to George. Now Sandra has 8 pennies left. How many pennies did Sandra have to begin with?
Separate Problems: On Your Own
• Separate, Change Unknown• Separate, Result Unknown• Separate, Initial Unknown
• Write an equation to show the solution
Part-Part-Whole Problems
Initial
Change
Whole
Part Part
•There are two parts combined into one whole•The combining can be a physical action or a mental combination.•No meaningful distinction between the two parts in this part-part-whole situation.
Part-Part-Whole: Whole Unknown
Initial
Change
Whole
Part Part
1. George has 4 pennies and 8 nickels. How many coins does he have?
2. George has 4 pennies and Sandra had 8 pennies. They put their pennies into a piggy bank. How many pennies did they put into the bank?
Part-Part-Whole: Part Unknown
Initial
Change
Whole
Part Part
1. George has 12 coins. Eight of his coins are pennies. The rest are nickels. How many nickels does George have?
2. George and Sandra put their pennies into the piggy bank. George put in 4 pennies. How many pennies did Sandra put in?
Part-Part-Whole: On Your Own
• Whole Unknown• Part Unknown
• Write an equation to show the solution.
• Thinking Blocks• Model It
Compare Problems
Initial
Change
Small setLarge set
Difference
• Compare problems involve the comparison of two quantities.
• The third amount does not exist but is the difference between the two amounts.
Compare: Difference Unknown
Initial
Change
Small setLarge set
Difference
George has 12 pennies and Sandra has 8 pennies. How many more pennies does George have than Sandra?
12 8
Compare: Difference Unknown
Initial
Change
Small setLarge set
Difference
George has 12 pennies and Sandra has 8 pennies. How many fewer pennies does Sandra have than George?
12 8
Compare: Larger Unknown
Initial
Change
Small setLarge set
Difference
8
4
George has 4 more pennies than Sandra. Sandra has 8 pennies. How many pennies does George have?
Compare: Larger Unknown
Initial
Change
Small setLarge set
Difference
8
4
Sandra has 4 fewer pennies than George. Sandra has 8 pennies. How many pennies does George have?
Compare: Smaller Unknown
Initial
Change
Small setLarge set
Difference
George has 4 more pennies than Sandra. George has 12 pennies. How many pennies does Sandra have?
12
4
Compare: Smaller Unknown
Initial
Change
Small setLarge set
Difference
Sandra has 4 fewer pennies than George. George has 12 pennies. How many pennies does Sandra have?
12
4
