1. Mathematics October 27, 2016 L Voza, Ed.D.
NHS Tutor Training
Mathematics
October 27, 2016
L Voza, Ed.D.

3. Important Site to Become Familiar With
Think Central Site for Math in Focus Books
K-2
4-8

4. New Jersey Student Learning Standards (Formerly Common Core)

5. Goal 1: Develop and Improve Number Sense
Reflection: You may have a keen number sense and not understand those who struggle with it.

6. Place Value and Number Sense: Grades 1-5
1,234 1,
One thousand two hundred thirty four

7. Activity: Administer a verbal “number writing activity”
Check grade level for appropriate place value limits
Verbally state numbers and have students write them.
Say the number at least three times
First time: student decides how many digits (can put spaces)
and begins to write the number.
Second time: student writes the number
Third time: student checks the number for appropriate commas
Be sure to include numbers with zeros and discuss
For struggling students, tell them how many digits prior and have them write the blanks:
____ _____ , _____ ______ ______ or _____ ______ . _____ _____
Don’t say “and” for whole numbers, say it for decimals

8. Goal 2: Develop Decomposing/Composing Skills
for Computation Success Gr 2-6
72 = 7 tens and 2 ones
6 tens and 12 ones
When adding: tens + 6 ones
tens + 7 ones
= 7 tens + 13 ones or 8 tens + 3 ones
When subtracting: tens + 2 ones OR 6 tens + 12 ones
ten ones
tens ones
Reinforce the concept that the value remains the same even if the number is regrouped in a different way. This works with decimals also.

9. Goal 3: Utilize Visual Tools Gr K-1
Five and Ten Frames
“Subitizing” as a skill
“What can you add to these
to make ten?”

10. Number Paths for Counting, Adding, Subtracting
Grade K-2

11. Hundreds Chart Gr 3-6 Look for patterns when adding
9 + 3, 19+3, 29+3, etc.
Rounding to closest ten
Skip counting and multiplication
Dividing with remainders
Prime and Composite Numbers
Common Multiples/Factors
GCF, LCM

12. Fraction Charts and Number Lines Gr 3-6
How are they alike? How are they different?

13. Bar Models Gr 1-8

14. There are a total of 40 gumballs
There are a total of 40 gumballs. 24 of them are cherry, the rest are grape. How many gumballs are grape?
Paige has 25 cents. Justin has 3 times as much as Paige. Justin has how many cents?
This can also be used to model a 1:3 ratio

15. Website for Bar Modeling

16. Goal 4: Improve Fact Fluency/Computation Skills,
but with Reasoning Not Rote Gr K- 8
There are 4 operations that can be used with numbers.
Addition/Subtraction: joining/separating, combining/taking from
comparing values/location on number line
Addition/Subtraction Facts: Strategies for Reinforcement:
Facts that make 10: Relationship Add/Subtract: Missing Value
– 0 = ? = 10
– 1 = “what number, when added to
– 2 = , equals 10?”
– 3 = 7
– 4 = x + 5 = 17
– 5 = 5
“What number when added to 5, equals 17?”
“What number, when multiplied by 3, equals 12?”

17. Multi-Digit Addition/Subtraction Gr 1-6
Use graph paper for students who struggle with alignment of numbers

18. Multiplication Gr 3-5 Repeated addition with equal groups
Use skip-counting, area grids, arrays for conceptual understanding
Look for patterns for reasoning skills

19. Reinforce the order/commutative property visually

20. Look for Patterns

21. Relate Multiplication Facts to Division Facts

22. Multi-Digit Multiplication with Modeling Gr 3-5

23. Make the connection between the model and the algorithm39
X 15
195
390
585

24. Use estimation to assess reasonableness: 2 x 4 = 8
Answer should be between 8 and 10

25. Division Gr 3-4-5
Repeated subtraction with equal groups, may have some left over

26. Long Division Gr 4-5
Avoid the tricks for sequencing
Relate to the inverse of multiplication: How many groups of _ are in _ ?
For long division algorithms, keep it visual
For more complex numbers, consider multiplying on the side to help.

27. Goal 5: Develop an Understanding of Fractions-Proportion
Fractions are numbers, part of whole numbers, an extension of previous understanding of whole numbers
Use visual models and benchmarks
Relate to decimals when possible
Avoid needless conversions, okay to leave it
Refer to standards for limits (denominators, etc.)
Stress the use of a number line over a pizza
Excellent Site for Visual Models
Count the spaces in between the wholes not the tic marks

28. Comparing and Ordering Fractions Gr 3-4-5
Use benchmarks when possible
What are your one half equivalents?
Compare
4 to 6
If one fourth is < one half
And five sixths is > one half
Then it is less than (<)
Also check out patterns as you get closer to 0 and closer to 1 whole.

29. AVOID!!!! Mixed to Improper and Improper to Mixed Much better to use
Ask: How many ninths in one whole? (9) How many ninths in two wholes? (18) Plus 8 more ninths? (26)

30. Adding and Subtracting Fractions Gr 3-5
Extend previous understanding of adding/subtracting whole numbers
One plus One equals Two
One fourth plus One fourth equals Two fourths
Use reasoning skills to understand that the could not equal 2

31. Adding/Subtracting Unlike Fractions Gr 4-5
Suggest making the algorithm vertical
Check if you have to rename one or both fractions
Use reasoning/estimation to check
Half + Half = Whole
2. >Half + <Half = ? Whole
3. <Half + Half = Between Half & Whole
4. <Half + <Half = <Whole

32. Multiplying Fractions Gr 4-6
Extend previous understanding of whole numbers
Multiplication as Repeated Addition Gr 4
Multiplication as Resizing Gr 5-6
x 1 = same value
x >1 = increased value
x <1= decreased value

33. Dividing Fractions Gr 5 Fraction and Whole Number
Extend previous understanding of Division
Understand Division as the Inverse of Multiplication
Use Modeling
One half of the whole is divided into 4 parts; what fraction of the whole is each of the 4 parts?
How many one halves
are in 8 wholes?

34. Dividing Fractions Gr 6 Fraction and Fraction
How many one sixths are in two thirds?

35. Ratios and Unit Rate Gr 6-8

36. Adding/Subtracting Positive and Negative Integers Gr 6-8
Great Site to Use