NUMBER SENSE
1.0 Students understand the place value of whole numbers and decimals to two decimal places and how whole numbers and decimals relate to simple fractions. Students use the concepts of negative numbers:
KINDERGARTEN
1.0 Students understand the relationship between numbers and quantities (i.e., that a set of objects has the same number of objects in different situations regardless of its position or arrangement):
1.1 Compare two or more sets of objects (up to ten objects in each group) and identify which set is equal to, more than, or less than the other.
1.2 Count, recognize, represent, name, and order a number of objects (up to 30).
1.3 Know that the larger numbers describe sets with more objects in them than the smaller numbers have.
2.0 Students understand and describe simple additions and subtractions:
2.1 Use concrete objects to determine the answers to addition and subtraction problems (for two numbers that are each less than 10).
3.0 Students use estimation strategies in computation and problem solving that involve numbers that use the ones and tens places:
3.1 Recognize when an estimate is reasonable.
1.1 Read and write whole numbers in the millions.
Watch this animated math lesson.
Adding Roman numerals.
Subtracting Roman numerals.
Comparing Numbers
1.2 Order and compare whole numbers and decimals to two decimal places.
ASSESSMENT: Ordering and Compring Whole Numbers - Level 4
Rounding
1.3 Round whole numbers through the millions to the nearest ten, hundred, thousand, ten thousand, or hundred thousand.
Watch this animated math lesson on rounding using a number line.
1.4 Decide when a rounded solution is called for and explain why such a solution may be appropriate.
Students have direct instruction on how to estimate whole numbers and how it applies to the real world.
Students apply knowledge of estimating with whole numbers.
Fractions
1.5 Explain different interpretations of fractions, for example, parts of a whole, parts of a set, and division of whole numbers by whole numbers; explain equivalents of fractions (see Standard 4.0).
1.6 Write tenths and hundredths in decimal and fraction notations and know the fraction and decimal equivalents for halves and fourths (e.g., 1/2 = 0.5 or .50; 7/4 = 1 3/4 = 1.75).
Write tenths and hundredths in decimal and fraction notations
Draw lines to match each fraction with its equivalent decimal
1.7 Write the fraction represented by a drawing of parts of a figure; represent a given fraction by using drawings; and relate a fraction to a simple decimal on a number line.
Negative Numbers
1.8 Use concepts of negative numbers (e.g., on a number line, in counting, in temperature, in "owing").
Number Line
1.9 Identify on a number line the relative position of positive fractions, positive mixed numbers, and positive decimals to two decimal places.
2.0 Students extend their use and understanding of whole numbers to the addition and subtraction of simple decimals:
Decimal Numbers
2.1 Estimate and compute the sum or difference of whole numbers and positive decimals to two places.
2.2 Round two-place decimals to one decimal or the nearest whole number and judge the reasonableness of the rounded answer.
Rounding decimals to the nearest whole number.
Rounding decimal numbers
3.0 Students solve problems involving addition, subtraction, multiplication, and division of whole numbers and understand the relationships among the operations:
Computing Numbers
3.1 Demonstrate an understanding of, and the ability to use, standard algorithms for the addition and subtraction of multidigit numbers.
Subtract whole numbers through thousands using place-value models.
3.2 Demonstrate an understanding of, and the ability to use, standard algorithms for multiplying a multidigit number by a two-digit number and for dividing a multidigit number by a one-digit number; use relationships between them to simplify computations and to check results.
We don't multiply this way in "The States."
3.3 Solve problems involving multiplication of multidigit numbers by two-digit numbers.
3.4 Solve problems involving division of multidigit numbers by one-digit numbers.
4.0 Students know how to factor small whole numbers:
Factoring
4.1 Understand that many whole numbers break down in different ways (e.g., 12 = 4 x 3 = 2 x 6 = 2 x 2 x 3).
4.2 Know that numbers such as 2, 3, 5, 7, and 11 do not have any factors except 1 and themselves and that such numbers are called prime numbers.
Click on the apples with prime numbers as they fall from the tree.
ALGEBRA & FUNCTIONS
1.0 Students use and interpret variables, mathematical symbols, and properties to write and simplify expressions and sentences:
KINDERGARTEN
Algebra and Functions
1.0 Students sort and classify objects:
1.1 Identify, sort, and classify objects by attribute and identify objects that do not belong to a particular group (e.g., all these balls are green, those are red).
Variables
1.1 Use letters, boxes, or other symbols to stand for any number in simple expressions or equations (e.g., demonstrate an understanding and the use of the concept of a variable).
Watch this animated math lesson on finding the value of a variable.
Order of Operations
1.2 Interpret and evaluate mathematical expressions that now use parentheses.
1.3 Use parentheses to indicate which operation to perform first when writing expressions containing more than two terms and different operations.
Formulas
1.4 Use and interpret formulas (e.g., area = length x width or A = lw) to answer questions about quantities and their relationships.
1.5 Understand that an equation such as y = 3 x + 5 is a prescription for determining a second number when a first number is given.
2.0 Students know how to manipulate equations:
2.1 Know and understand that equals added to equals are equal.
2.2 Know and understand that equals multiplied by equals are equal.
MEASUREMENT & GEOMETRY
1.0 Students understand perimeter and area:
Perimeter and Area (Watch the video)
KINDERGARTEN
Measurement and Geometry
1.0 Students understand the concept of time and units to measure it; they understand that objects have properties, such as length, weight, and capacity, and that comparisons may be made by referring to those properties:
Students use non-standard units to estimate and measure.
1.1 Compare the length, weight, and capacity of objects by making direct comparisons with reference objects (e.g., note which object is shorter, longer, taller, lighter, heavier, or holds more).
1.2 Demonstrate an understanding of concepts of time (e.g., morning, afternoon, evening, today, yesterday, tomorrow, week, year) and tools that measure time (e.g., clock, calendar).
1.3 Name the days of the week.
1.4 Identify the time (to the nearest hour) of everyday events (e.g., lunch time is 12 o'clock; bedtime is 8 o'clock at night).
2.0 Students identify common objects in their environment and describe the geometric features:
2.1 Identify and describe common geometric objects (e.g., circle, triangle, square, rectangle, cube, sphere, cone).
2.2 Compare familiar plane and solid objects by common attributes (e.g., position, shape, size, roundness, number of corners).
Area & Perimeter
1.1 Measure the area of rectangular shapes by using appropriate units, such as square centimeter (cm2), square meter (m2), square kilometer (km2), square inch (in2), square yard (yd2), or square mile (mi2).
Finding the area of plane figures using grids and formulas.
1.2 Recognize that rectangles that have the same area can have different perimeters.
1.3 Understand that rectangles that have the same perimeter can have different areas.
1.4 Understand and use formulas to solve problems involving perimeters and areas of rectangles and squares. Use those formulas to find the areas of more complex figures by dividing the figures into basic shapes.
2.0 Students use two-dimensional coordinate grids to represent points and graph lines and simple figures:
The Coordinate Grid
2.1 Draw the points corresponding to linear relationships on graph paper (e.g., draw 10 points on the graph of the equation y = 3 x and connect them by using a straight line).
2.2 Understand that the length of a horizontal line segment equals the difference of the x- coordinates.
2.3 Understand that the length of a vertical line segment equals the difference of the y- coordinates.
3.0 Students demonstrate an understanding of plane and solid geometric objects and use this knowledge to show relationships and solve problems:
3.1 Identify lines that are parallel and perpendicular.
3.2 Identify the radius and diameter of a circle.
3.3 Identify congruent figures.
Symmetry
3.4 Identify figures that have bilateral and rotational symmetry.
Angles
3.5 Know the definitions of a right angle, an acute angle, and an obtuse angle. Understand that 90°, 180°, 270°, and 360° are associated, respectively, with 1/4, 1/2, 3/4, and full turns.
Using a protractor to measure angles.
Shapes
Identifying speres, cylindars, cones and cubes.
Visualize, describe, and make models of geometric solids (e.g., prisms, pyramids) in terms of the number and shape of faces, edges, and vertices.
Use nets to recognize the relationship between planes and solids.
Interpret two-dimensional representations of three-dimensional objects; and draw patterns (of faces) for a solid that, when cut and folded, will make a model of the solid.
Click on the solid figure created when the net is folded.
Know the definitions of different triangles (e.g., equilateral, isosceles, scalene) and identify their attributes.
Know the definition of different quadrilaterals (e.g., rhombus, square, rectangle, parallelogram, trapezoid).
Rotate the building until you get the right side view (see the same shape)..
STATISTICS, DATA ANALYSIS, & PROBABILITY
1.0 Students organize, represent, and interpret numerical and categorical data and clearly communicate their findings:
KINDERGARTEN
Statistics, Data Analysis, and Probability
1.0 Students collect information about objects and events in their environment:
1.1 Pose information questions; collect data; and record the results using objects, pictures, and picture graphs.
1.2 Identify, describe, and extend simple patterns (such as circles or triangles) by referring to their shapes, sizes, or colors.
Representing Data
1.1 Formulate survey questions; systematically collect and represent data on a number line; and coordinate graphs, tables, and charts.
How to read a line graph.
Circle graphs.
1.2 Identify the mode(s) for sets of categorical data and the mode(s), median, and any apparent outliers for numerical data sets.
Mean, median, and mode
1.3 Interpret one-and two-variable data graphs to answer questions about a situation.
PROBABILITY
2.0 Students make predictions for simple probability situations:
2.1 Represent all possible outcomes for a simple probability situation in an organized way (e.g., tables, grids, tree diagrams).
Determining the probability of an event and showing it as a fraction.
2.2 Express outcomes of experimental probability situations verbally and numerically (e.g., 3 out of 4; 3 /4).
MATHEMATICAL REASONING
1.0 Students make decisions about how to approach problems:
KINDERGARTEN
Mathematical Reasoning
1.0 Students make decisions about how to set up a problem:
1.1 Determine the approach, materials, and strategies to be used.
1.2 Use tools and strategies, such as manipulatives or sketches, to model problems.
2.0 Students solve problems in reasonable ways and justify their reasoning:
2.1 Explain the reasoning used with concrete objects and/ or pictorial representations.
2.2 Make precise calculations and check the validity of the results in the context of the problem.
Solving Word Problems
1.1 Analyze problems by identifying relationships, distinguishing relevant from irrelevant information, sequencing and prioritizing information, and observing patterns.
1.2 Determine when and how to break a problem into simpler parts.
2.0 Students use strategies, skills, and concepts in finding solutions:
2.1 Use estimation to verify the reasonableness of calculated results.
2.2 Apply strategies and results from simpler problems to more complex problems.
2.3 Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and models, to explain mathematical reasoning.
2.4 Express the solution clearly and logically by using the appropriate mathematical notation and terms and clear language; support solutions with evidence in both verbal and symbolic work.
2.5 Indicate the relative advantages of exact and approximate solutions to problems and give answers to a specified degree of accuracy.
2.6 Make precise calculations and check the validity of the results from the context of the problem.
3.0 Students move beyond a particular problem by generalizing to other situations:
3.1 Evaluate the reasonableness of the solution in the context of the original situation.
3.2 Note the method of deriving the solution and demonstrate a conceptual understanding of the derivation by solving similar problems.
3.3 Develop generalizations of the results obtained and apply them in other circumstances.
Levels in Math?
When we use a mastery approach to instruction, students tend to learn at an accelerated pace.
By Will Duckworth
TrinityTutors.com Lead Tutor
Are you familiar with a phenomenon called “summer learning loss,” a loss of learning over the summer that can mean an academic setback that, for some children, takes months to remedy in the fall?
All young people are at risk of losing ground academically over the summer months, regardless of where they are in the socioeconomic spectrum.
Teachers typically spend weeks at the beginning of each school year re-teaching material that students have forgotten over the summer. That’s because the material was never truly mastered in the first place. At TrintyTutors.com, we regard such re-teaching as a waste of precious time and take steps to eliminate its necessity.
While forgetting things is something that all humans do, most individuals from my generation can go literally years without reciting the “Pledge of Allegiance,” and yet, rattle it off on demand without a second thought. That’s because it constitutes material that was truly mastered in childhood.
At TrinityTutors.com we know that certain learning conditions can make for dramatic progress – especially that of individual tutoring. Moreover, we recognize that the home environment is the key factor in influencing academic success, espceially at the elementary level.
Consequently, we insure that critical skills and concepts are learned as well by our participants as kids once learned “The Pledge of Allegiance” by embeding the practice of critical skills and the use of related concepts throughout the curriculum, and by adopting a mastery approach to education and instruction in which you, the parent-teacher, take on the role of “private tutor.”
The core idea of mastery learning is: Everyone can learn given the right circumstances.
Because we encourage a mastery approach, TrinityTutors.com does not divide curriculum by grade level.
On the contrary, a mastery approach to instruction is characterized by the following:
- Main objectives representing the purposes of the unit or lesson are what define mastery of the material.
- The topic is divided into relatively small learning units, each of which has its own objectives and assessments.
- Included within learning materials and instructional strategies are teaching, modeling, practicing, formative evaluation, reteaching, reinforcement, and summative evaluation.
- Each unit is preceded by a brief diagnostic test.
- The results of diagnostic tests are used to guide instruction by identifying strengths and weaknesses, clarifying which skills and concepts the instructor and learner should focus on the most.
- Time to learn is adjusted to fit aptitude, and most importantly . . .
- NO STUDENT IS ALLOWED TO PROCEED TO NEW MATERIAL UNTIL BASIC PREREQUISITE MATERIAL IS MASTERED!
Moreover, it is my experience that students who are able to answer short-answer questions are often befuddled by “thought questions” and may experience difficulty when it comes to problem solving. Given questions like, “Give the reasons which would have influenced a typical Virginia tobacco farmer to support the ratification of the Constitution of 1789, and the reasons which would have influenced him to oppose the ratification.” students who can respond readily to true/false, multiple choice questions, are often stumped or at a loss.
Our program therefore makes a point of requireing students to “think out loud” during lessons and when responding to assessment prompts.
A home school is the perfect setting in which to remedy this sort of deficiency. Our model requires parent-teachers to, at the end of an instructional unit, [about every two weeks], give a formative [not used for grading] test to find out what has been learned or not learned in order to determine corrective instruction by reteaching – perhaps in a different way/style – and test again on the same items using altered questions.
Grading for mastery is not on a curve, so every student can get an “A” using this approach, which we fully expect every student to do!
Through mastery learning, the average student can pass at the 95th percentile of traditionally-taught classes, and by establishing a solid foundation of knowledge in the beginning, the stage is set for the academic progress of most students to eventually “take off,” accelerating exponentially as time goes on.
I also suggest that you . . .
- Give a pre-test and review at the beginning of a semester on those basic and essential facts, skills, and concepts that are necessary for later success.
- Give two chances to succeed on each quiz/final exam, reteaching the areas missed in the first test and using a different explanation/example/demonstration than the first time or a different style of instruction. Then use a different form of the test.
- Repeat the above process every 2 or 3 weeks.
- Establish regular times for eating, studying, sleeping, working, playing.
- Require that school work and reading come before play – even before other work.
- Give praise for good school work – doing so occasionally in front of others.
- Provide a quiet place to study.
- Discuss what is being studied and have the student verbalize skills and concepts, explaining what he or she is doing, steps that are involved, processes undertaken, the definitions of key terms, materials used, etc., when appropriate.
- Visit libraries, zoos, museums, etc., as a family.
- Encourage good speech habits, even during the couse of family exchanges such as dinner time or other times when everyone is engaged in conversation.
- Provide special help when needed.
- Talk about plans for the future, preparation for college, possible vocations, etc. (There is a very high correlation between home environment/attitude toward education and school success.)
5 Keys to success:
- Mastery approach to learning.
- Pre-requisite enhancement.
- Make reading automatic, beyond decoding.
- Emphasis on creativity, higher mental processes, critical thinking, etc.
The Basic Practice Model of Teaching
- Provide the objective of the lesson and level of performance required for proficienty.
- Describe the content of the lesson and its relationship to prior knowledge/experience.
- Discuss the procedures of the lesson – the different parts of the lesson and the learner’s responsibility during each activity.
Direct Instruction: Explain new concept(s) or skill(s), demonstrate and provide examples – orally and visually. For concepts, include their attributes (characteristics), pertinent rules and definitions, along with several examples. For skills, identify the necessary steps with examples of each. It is important that pupils have a visual representation of the task (VRT) in the early stages of learning!
Structured Practice: Lead students through practice examples working in a lock-step fashion through each step of the task as it appears in the VRT. [e.g., use an overhead projector doing practice examples on a transparency so that students can see the generation of each step. Then provide a visual instructional plan (VIP) – in which each step is detailed – for pupils to use when they get stuck in individual practice or independent practice.] Refer to the VRT while working practice examples as a group.
Guided Practice: Circulate while the learners complete their seat work [e.g., “praise, prompt, and leave”]. Monitor students’ work, providing corrective feedback as necessary, and assess performance in determining whether a learner is ready for the next instruction. Allow additional time for those whose aptitude calls for a longer learning period while giving “extra credit” assignments, supplementary activities, etc. to those that are ready to move on.
Independent Practice: [additional class time or homework] begins when students have achieved an 85 to 90% accuracy level. To insure retention and develop fluency, students practice on their own without assistance and with delayed feedback [e.g., comments on graded papers]. Five or more brief practice activities distributed over a month or more may be required to “fix” the new concept/skill with periodic re-assessments in the months that follow all the way to the end of the year.
