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

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

New Jersey Student Learning Standards (Formerly Common Core) http://www.state.nj.us/education/cccs/2016/math/

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

Place Value and Number Sense: Grades 1-5 1,234 1,000 + 200 + 30 + 4 One thousand two hundred thirty four

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

Goal 2: Develop Decomposing/Composing Skills for Computation Success Gr 2-6 72 = 7 tens and 2 ones 70 + 2 6 tens and 12 ones 60 + 12 When adding: 56 5 tens + 6 ones + 27 2 tens + 7 ones 83 = 7 tens + 13 ones or 8 tens + 3 ones When subtracting: 72 7 tens + 2 ones OR 6 tens + 12 ones - 18 1 ten + 8 ones 54 5 tens + 4 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.

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?”

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

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

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

Bar Models Gr 1-8

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

Website for Bar Modeling http://www.mathplayground.com/thinkingblocks.html

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 10 + 0 0 + 10 10 – 0 = 10 2 + ? = 10 9 + 1 1 + 9 10 – 1 = 9 “what number, when added to 8 + 2 2 + 8 10 – 2 = 8 2, equals 10?” 7 + 3 3 + 7 10 – 3 = 7 6 + 4 4 + 6 10 – 4 = 6 3x + 5 = 17 5 + 5 5 + 5 10 – 5 = 5 “What number when added to 5, equals 17?” “What number, when multiplied by 3, equals 12?”

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

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

Reinforce the order/commutative property visually

Look for Patterns

Relate Multiplication Facts to Division Facts

Multi-Digit Multiplication with Modeling Gr 3-5

Make the connection between the model and the algorithm 39 X 15 195 390 585

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

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

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.

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 http://www.visualfractions.com/ Count the spaces in between the wholes not the tic marks

Comparing and Ordering Fractions Gr 3-4-5 Use benchmarks when possible What are your one half equivalents? Compare 1 5 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.

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)

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 1 + 1 could not equal 2 4 4 8

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

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

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?

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

Ratios and Unit Rate Gr 6-8

Adding/Subtracting Positive and Negative Integers Gr 6-8 Great Site to Use http://www.mathsisfun.com/positive-negative-integers.html