Mathnasium Training Exam

Addition With Half

"One half plus one half equals a whole" is the basic notion required for adding fractions., Then, "two and one half plus three and one half" can be thought of as: "two plus three is five" (add the whole numbers), "one half plus one half equals a whole" (add the fractions), and "five and one are six" (combine the results).

Dividing by half

"Three divided by one half" can be thought of as "How many halves are there in three wholes?" or "How many half sandwiches can you make out of three whole sandwiches?"

When they learn to count by ones ask

"What number is 1 bigger/smaller than five? 2 bigger/smaller than 9?", THEN they should be asked "name 2 #s that add up to 5" or "what and 3 add up to five?", THEN "5 and how much more make 7?" and "8 is how much more than 5?"

The meaning of the Law of SAMEness

"You can only add and subtract things of the same denomination, things that have the same name."

Ratio

"a comparison between two numbers by division"

Teach place value as

"the art of 'seeing' 10s.

The denominator

"the one that names." It is determined by the total number of equal parts into which the whole is divided.

The numerator

"the one that numbers." It tells us how many of the parts to use this time

arithmetic

"the process of transforming one number into another, using given rules."

Multiplying BY half

"three and one half, two times" means "three, two times, and one half, two times." Saying "3½, two times" does this; saying "2 times 3 ½" doesn't.

Teach the important properties of numbers

(Identity property, distributive, cumulative)

The Power of 10

10 is the base #, Stu must learn special properties of 10, "The given # and how much more make 10?", Point out that teen = digit + 10, THEN move on to doubles (1+1, 2+2, 3+3...), THEN 1X is how much more than 10?, THEN 1X is how much more than 10, 9, 8, 7..., Avoid the use of terms like "commutative property" and "associative property" until well after the student has mastered the number facts.

Improper fraction

A fraction whose absolute value is greater than one whole (1).

Proper fraction

A fraction whose absolute value is greater than or equal to zero (0) and is less than one whole

Unity

A fraction whose value is equal to one whole

Tier 1

Adding on to the larger # (adding 1 or 2 onto a larger #), 10 plus a single digit # (emphasizes that the 1 in the 10s place means 10, includes 10 and X more is 1X), decomposing single digit numbers (Do these numbers feel close together or far apart? The count FORWARD or BACK)

Tier 2

Addition facts Sums up to 10 (adding onto the larger number without finger counting), Complements of 10 (uses language from tier 1, puzzle piece helps visualize complements of 10), Doubles facts, subtraction facts Minuends up to 10 (builds upon up to and take away ideas to introduce subtraction, and find how much is left)

Tier 3

Addition facts over 10 (uses doubles, up to or over ten, and add/take away 10 and what method works for you), Subtraction Facts/MIssing Addends, Minuends Over 10 (scaffolding for up to and over 10 technique)

Direct Relationship

As one quantity gets larger, the other quantity gets larger by the same amount. As one quantity gets smaller, the other quantity gets smaller by the same amount.

Indirect (Inverse) Relationship

As one quantity gets larger, the other quantity gets smaller by the same amount. As one quantity gets smaller, the other quantity gets larger by the same amount.

Sections 1-5 of binder

Assessments and progress info, Prescriptives in works (grade specific), A+ splits active packs and new packs, Workout books in works (development dependent, simple and helps with mathematical thinking), Corrected prescriptives, Corrected workout books

Rules of team teaching

Assign team roles, Rotate and read the floor, Engage students appropriately, Interact with students effectively, Disengage and communicate, Communication and accountability for student progress, Understand team roles/accountability/communication, Gives students space to develop without instructor, Students hear multiple instructor voices, Instructors are productive and precise, Vibrant safe space

Numerical Fluency

Automaticity, learn by heart, effortless recall

Team leader

Center director or experienced instructor, Direct students to seats, Check students in, Direct instructors, Delegate, Acknowledge students in need, Assist struggling students

Checking out

Check workout plan goals and see if they've been accomplished, Make a note on issues if necessary

The Concept of Complements

Complements are separate things which, when taken together, form a whole. One way to think of complements is "the complement is the rest of it.", as one gets bigger, the other gets smaller by the same amount (see "Direct and Inverse Variation"), and • when one is zero, the other is the whole.

Assign a Task

Concise but as long as necessary, Assign a task as you disengage, 5-10 PK/Workout/HW exercises, Reassure that they can ask for help

The Student Binder

Contains material from student learning plan, assembled from assessment results

Correcting student work

Correct right away in front of stu, Stu should not proceed until you review their page, Strengthen stu's understanding, Ask stu to explain their answer - correct or incorrect, Circle the exercise # of errors, Give page back for stu to correct, Box circled exercises to indicate a correction was made, Star punch when completely correct

Subtraction tips

Counting how far apart and how much is left: The process of removing a part from a whole., Sub asks how much is left and how far apart are 2 #s, Use "how much is left?" when the numbers are fairly far apart, and count down. (~10), Use "how far apart are the two numbers?" when the numbers are fairly close to each other, and count up (<<10)

Mastery checks

Demonstrates mastery and independence, can only be reviewed once completed independently, Review and return for stu to correct, Date and initial, Notes to center director if unable to complete MC

Variation

Direct and Inverse

Division Tips

Division is repeated subtraction, and answers the question, "How many of these are there in that?", How many groups of 3 are there in 12? How many times can 3 be taken out of 12? How many times can 3 be subtracted from 12 before the answer is less than 3?, The written process of division should not be started until the student can reliably, orally answer questions like these: How many 10s are there in 30? How many stacks of 5 can you make out of 20 pennies? How many quarters are there in $3.00? ...in $4.50? How many 20s are there in 100? How many 125s are there in 1,000? How many 15s are there in 60?, Say "'15 divided by 5' means 'How many 5s are there in 15?'."

Workout plan

For director use, not for parents

If a student is reluctant to work on own

Give reassurance that you can come back, Positive reinforcement of skills and effort, Offer extra punches, Offer a game later

Problem Solving Strategy

If the whole is unknown, then the task is to build it up from its known parts, If the parts are equal, we multiply; If the parts are not equal, we add., The whole is equal to the sum (total) of its parts.

Multiplication Tips

Instead of asking, "What is 5 times 4?" which calls for a memorized response, a much better question form is: "How much is 5, four times?"

How to rotate and read the floor

Look around see if students have lost focus/talking/staring into space/asking for help/stuck on one page for a long time, Use peripheral vision to see if students are raising their hands while engaged with a student

Documenting session progress

Mathnasium work, Footer #s of PK, Tally pages, Homework spot check, Learning reflection

Types of students interactions

Normal instruction, Responsive or proactive

How to disengage and communicate

Once they have it figured out, let them try it on their own and help someone else, Always leave them with a task

PER CENT

Percent is an appropriate topic for first and second grade students if it is presented as an outgrowth of Counting, Wholes and Parts, and Proportional Thinking., Reverse questions involving percent take the form, "10% of what number is 40?"

How to appropriately engage students

See how they're doing, Be inviting and non threatening, Students will be self conscious to ease them in, See what they know to adjust your vocabulary, Scaffolding by working on easier stuff to work on what they have an issue with

Instruction Teams

Separate areas or student needs, Higher level students may be sat separately from lower level, Ratio should not exceed 4:1 students to instructor

Guidelines for instruction delivery

Sit across from students, Supporting and encouraging voice, Be concise

Tips for teaching addition

Start at x and count up to y, Doubles +1, Doubles - 1, Breaking down #s (6 + = 8), How far apart are x and y?, How far is it from x up to y?, Combinations that make 10, 10 plus a number, 10 plus what number, Putting it all together

Routine

Students program is in their binder, Students pull binder off shelf, Scan in, Pick up where left off, Work on workout plan

Team roles

TT works best when the team is coordinated to support students

Blueprint for Change

Teach number facts as "the process of counting in groups." • addition is "counting-up the total." • subtraction is "counting 'how far apart' two numbers are." • multiplication is "a fast way to count equal groups." • division is "counting the number of this group that are inside of that group."

Number Sense

The ability to appreciate the size and scale of numbers, in the context of the question at hand

Student types

Unruly, easily distracted or unwilling students, Overloaded fatigued students, Quiet or shy students

10 rules of engagement

Use mathnasium teaching constructs, try a new explanation if it isn't working, fall back on old knowledge when a student is struggling and expand knowledge with student is succeeding, praise and criticize constructively when appropriate, socratic vs direct teaching, use mathnasium vocabulary, use tools for visual learning, use metacognition, use mental math, master team teaching

Session Wheel

Use to see student's productivity and communicate for max productivity

Subtraction with half

WRONG: "Seven take away two and one half" is often answered by students as "five and one half" because they subtract the whole numbers and just "bring down" the half, RIGHT: Hold up seven fingers, five on the left hand and two on the right, and say, "Here are seven." • Next, put down the two fingers on the right hand and say, "I've taken away two, so how many are left?" (Five.) • Then say, "I have five left. Now I'll take away the half, so how much is left?" Now, fold down the first joint of the thumb of the left hand. (Four and one half.)

The Problem with Memorization

When they forget the answer, they have no way of getting around it because they lack understanding and strategies, Without strategies, stu uses inefficient techniques or wild guess

Place Value

Whole numbers are formed by counting by 1s, decimal point, a marker that separates the whole numbers (on the left) from the fractions (on the right)

Factor

a number quantity that when multiplied by and number equals a product

Number facts

addition, sub, mult, div resulting in single digit and some double digit number answers, These are things students must know in order to learn more advanced math

Area

amount of space in a 2-dimensional figure, measured in square units

Volume

amount of space in a 3-dimensional figure, measured in cubic units.

Workout Books (WOBs)

based on student number sense evaluation, part of NSD series and Intervention series

When finding half of an uneven fraction

breaking a number into easier parts, • finding half of those parts, and • then putting the parts back together again

Fractions

created by dividing (cutting, breaking, "fracturing") a whole thing into equal parts.

Mathnasium is all about

custom and personalized learning, One on one instruction, Independent students

General approach

demonstrate exercise; observe, comment correct, ask Qs; disengage

PKs (prescriptives)

designed like typical math practice pages with increasing difficulty ranging in length (2-22 pgs) (Don't tell them the PK level)

Size vs scale

establishes magnitude, vs establishes relationship

Focus Ons (FOs)

extra practice, not grade related, general topics, adds depth, advanced students, code at bottom right of each page

Zero

first number in our counting system

Half and half model

half HW, half mathnasium activity

A multiplication fact

has 2 factors and 1 product

We can only add and subtract things that...

have the same name, things that are of the same denomination.

GENE Teaching

help a student for a while, allow them to be independent, help someone else, then come back

Instructional pages

help instructors refresh on techniques

Team teaching

instructors rotate on the floor, helping students in strategic moments, students benefit from multiple instructors, Each instructor helps students individually

Arithmetic

is "the Art of Counting." ari- (to join) + the + metic (measure)

Quantity

is the amount of that something; the number of things

Individual instruction in a group environment

many classes of one

Proportion

means "according to amount." pro- (according to) + portion (amount)

Let's talk about it icon

means have a conversation with the students

Mastery check

mini-assessment, should be done independently

ORDER

more than half - less than half

A whole number

number without a fractional part—no common or decimal fraction.

Negative numbers

numbers that are less than 0

Direct variation

occurs for changing quantities x and y when y x = k, for a constant value k

A fraction

part of a whole.

Distance

shortest amount of space between two points, and is measured in linear unit

Determine teaching method

socratic (ask questions), direct teaching, or combination or both

Rate

something happens for each or in each ("per") occurrence of something else

Student incentive

star punches, red level 1, red 2, black 3, yellow 4, grey 5

Number Sense Includes

the ability to count from any number to any number by any number, forward and backward, the ability to compute (mentally and on paper) using single- digit, and some double- and triple-digit numbers, understanding how 10s work, understanding the nature of fractions, decimal, and percents, understanding and applying measurement, being able to compare numbers by subtraction (difference), and by division (ratio), developing a "sense of proportion," a "sense of scale," and being able to explain in words, pictures, and symbols one's mathematical thinking.

When things are in proportion

the relationship between the parts stays the same even though one or more of the parts changes in value; as the amount of one thing changes, both the relationship between the object and the whole, and the relationship between the object and the other parts, stay the same

Counting

transforms one number into another

Wholes and Parts, and the relationship between them

unite various aspects of math.

Mathnasium model

warmup, warm down, mathnasium activities, HW

The Laws of Equality (the Laws of Transformation)

• "Equals operated on by equals remain equal.", • "Equals may anywhere be substituted for equals.", and • "Things equal to the same thing are equal to each other."

Teach the Laws of Wholes and Parts

• "The whole is equal to the sum of its parts." • "Any one part is equal to the whole minus all of the other parts."

Every whole number and every fraction can be written three ways:

• as a common fraction, • as a decimal fraction, • as a percent