SOUTHERN LEHIGHSCHOOL DISTRICT

Mathematics Essentials

Grade K / Mathematics
Numbers, Number Systems and Number Relationships
Pennsylvania Academic Standards: 2.1.3
  1. Count using whole numbers (to 10,000) and by 2’s, 3’s, 5’s, 10’s, 25’s and 100’s.
  2. Use whole numbers and fractions to represent quantities.
  3. Represent equivalent forms of the same number through the use of concrete objects, drawings, word names and symbols.
  4. Use drawings, diagrams or models to show the concept of fraction as part of a whole.
  5. Count, compare and make change using a collection of coins and one-dollar bills.
  6. Apply number patterns (even and odd) and compare values of numbers on the hundred board.
  7. Use concrete objects to count, order and group.
  8. Demonstrate an understanding of one-to-one correspondence.
  9. Apply place-value concepts and numeration to counting, ordering and grouping.
  10. Estimate, approximate, round or use exact numbers as appropriate.
  11. Describe the inverse relationship between addition and subtraction.
  12. Demonstrate knowledge of basic facts in four basic operations.

Essential Understandings:
Place value is the foundation of our Hindu-Arabic numeration system
Our place value system is a decimal system (based on powers of 10)
Numbers can be compared, grouped and related by using common characteristics
An estimation is a reasonable judgment
Students will need to know and be able to do:
  • Match objects one to one
  • Make and identify equivalent sets of objects
  • Identify and make sets that have more items than other sets
  • Identify sets with fewer objects
  • Count, recognize, represent, and name examples of one and two
  • Count, recognize, represent and name examples of three, recognize and model three
  • Count, recognize, represent, and name examples of four, recognize and model four in different arrangements
  • Count recognize, represent and name examples of zero
  • Count, recognize and name examples of numbers 0-5
  • Recognize and represent fractional parts of a whole
  • Identify and model halves
  • Recognize 6, 7, 8, 9, 10, 11, and 12 in many different arrangements, count, recognize, represent and name 6-12.
  • Recognize and represent numbers 0-12 in many different arrangements and parts
  • Use more and fewer to compare sets of objects
  • Make and recognize reasonable estimates for numbers 12 and less
  • Model addition as combining sets and adding to sets.
  • Add one to numbers zero through nine.
  • Add two to numbers zero through five, using concrete objects.
  • Solve addition problems using pennies.
  • Model and record addition sentences.
  • Identify, model, and add doubles
  • Model subtraction as taking away from or separate sets of objects.
  • Model and record results of subtracting one from the numbers one through ten.
  • Model and record subtraction of two from the numbers two through five using concrete objects.
  • Solve subtraction problems, using pennies.
  • Model and record subtraction sentences.
  • Relate addition and subtraction as opposite operations.
  • Identify, represent, and write numbers 10-20.
  • Order numbers, 10-20 and identify numbers missing from an order.
  • Use dimes and pennies to represent amounts of money to 19 cents.
  • Count, identify and write numbers 21 to 25.
  • Count, identify and write numbers 26 to 30.
  • Identify and write numbers in order to 31.
  • Use numbers 1-31 in working with the calendar

Teaching and learning experiences (including differentiation and enrichments):
Lessons: 3.1, 3.2, 3.3, 3.4, 4.2, 4.3, 4.3, 4.5, 4.6, 7.1, 7.2, 7.3, 7.4, 7.5, 7.7, 7.8, 8.4, 8.5, 13.1. 13.2. 13.3, 13.4, 13.5, 13.6, 13.7, 14.1, 14.2, 14.3, 14.4, 14.5, 14.6, 15.1, 15.2, 15.3, 15.4, 15.5, 15.6, 15.7, 15.8, 16.1, 16.2, 16.3, 16.4, 16.5
Enrichments:
Chapter enrichments
DI enrichment tubs
Assessments:
Construction of a model
Student demonstration/construction using manipulatives/tools
Quizzes, tests
Teacher observations
Work samples
Student Self-Assessments: paired learning, small groups, daily review of daily classroom assignments
Materials and Resources:
Houghton Mifflin Math Textbook
Everyday Counts
Accommodations:
Follow all IEP and 504 Plan requirements, allow extra time, extra help outside of class, graphic organizers, guided notes, test accommodations, guided practice, hands-on activities, teacher provided study guides, small group instruction, peer study partners.
Time: 34lessons (some lessons overlap with other Standards)
Grade K / Mathematics
Computation and Estimation
Pennsylvania Academic Standards: 2.2.3
  1. Apply addition and subtraction in everyday situations using concrete objects.
  2. Solve single- and double-digit addition and subtraction problems with regrouping in vertical form.
  3. Demonstrate the concept of multiplication as repeated addition and arrays.
  4. Demonstrate the concept of division as repeated subtraction and as sharing.
  5. Use estimation skills to arrive at conclusions.
  6. Determine the reasonableness of calculated answers.
  7. Explain addition and subtraction algorithms with regrouping.

Essential Understandings:
Numbers can be used to solve problem
Estimation skills can be used to determine a reasonable solution to a problem
The processes of addition and subtraction are related and can be used to determine exact answers
Students will develop foundational concepts of:
  • Putting objects together and taking them apart
  • Telling the difference between adding and subtracting
  • Solving simple addition and subtraction number sentences

Teaching and learning experiences (including differentiation and enrichments):
  • Integrated throughout the curriculum and daily activities

Assessments:
  • Teacher observations / Final Assessment

Materials and Resources:
  • Blocks, centers, other manipulatives, and real-life situations

Accommodations:
Follow all IEP and 504 Plan requirements, allow extra time, extra help outside of class, graphic organizers, guided notes, test accommodations, guided practice, hands-on activities, teacher provided study guides, small group instruction, peer study partners.
Time:
Grade K / Mathematics
Measurement and Estimation
Pennsylvania Academic Standards: 2.3.3
  1. Compare measurable characteristics of different objects on the same dimensions (e.g., time, temperature, area, length, weight, capacity, perimeter).
  2. Determine the measurement of objects with non-standard and standard units (e.g., US customary and metric).
  3. Determine and compare elapsed times.
  4. Tell time (analog and digital) to the minute.
  5. Determine the appropriate unit of measure.
  6. Use concrete objects to determine area and perimeter.
  7. Estimate and verify measurements.
  8. Demonstrate that a single object has different attributes that can be measured in different ways (e.g., length, mass, weight, time, area, temperature, capacity, perimeter).

Essential Understandings:
A variety of tools may be used for measuring quantities
An estimation is a reasonable judgment
A variety of measurement systems exist
Students will need to know and be able to do:
  • Identify activities that take place in the day/night, morning, afternoon, evening
  • Name and order the days of the week; understand that there are twelve months in a year
  • Use comparative words to describe differences in temperature
  • Identify which of two activities takes more or less time.
  • Order events with respect to time.
  • Use analog clocks to tell time to the hour.
  • Use digital clocks to tell time to the hour.
  • Match times on analog and digital clocks.
  • Identify a penny as one cent; count and compare a set of pennies.
  • Identify a nickel as five cents; relate 5 pennies to a nickel.
  • Identify a dime as ten cents; relate 10 pennies to a dime.
  • Identify a quarter; compare attributes of quarters, nickels, dimes and pennies
  • Compare the length of objects; identify and demonstrate taller, longer, and shorter.
  • Sort and order objects by length.
  • Measure lengths of objects using nonstandard units.
  • Estimate and measure length in nonstandard units.
  • Explore the concept of area.
  • Compare the weights of objects and determine which is heavier and which is lighter.
  • Order objects by weight.
  • Weigh objects using nonstandard units and a balance.
  • Estimate and measure weight in nonstandard units.
  • Compare the capacities of containers and determine which holds more and which holds less.
  • Put objects in order according to what holds least and what holds most.
  • Nonstandard units.
  • Estimate and measure capacity in nonstandard units.
  • Identify tools for measuring time, length, weight, capacity, and temperature.
  • Use benchmarks to estimate quantities of 20 and less.

Teaching and learning experiences (including differentiation and enrichments):
Lessons:
9.1, 9.2, 9.3, 9.5, 9.6, 9.7, 9.8, 9.9, 10.2, 10.3, 10.4, 10.5, 11.1, 11.2, 11.3, 11.4, 11.5, 12.1, 12.2, 12.3, 12.4, 12.5, 12.6, 12.8, 12.9
Enrichments:
Chapter enrichments
DI enrichment tubs
Assessments:
Construction of a model
Student demonstration/construction using manipulatives/tools
Quizzes, tests
Teacher observations
Work samples
Student Self-Assessments: paired learning, small groups, daily review of daily classroom assignments
Materials and Resources:
Houghton Mifflin Math Textbook
Everyday Counts
Accommodations:
Follow all IEP and 504 Plan requirements, allow extra time, extra help outside of class, graphic organizers, guided notes, test accommodations, guided practice, hands-on activities, teacher provided study guides, small group instruction, peer study partners.
Time: 27 lessons (some lessons overlap with other Standards)
Grade K / Mathematics
Mathematical Reasoning and Connections
Pennsylvania Academic Standards: 2.4.3
  1. Make, check and verify predictions about the quantity, size and shape of objects and groups of objects.
  2. Use measurements in everyday situations (e.g., determine the geography of the school building).

Essential Understandings:
Reasoning is an essential component in understanding mathematical relationships, making reasonable judgments, drawing conclusions, and applying appropriate mathematical strategies to problems
Students will develop foundational concepts of:
  • Measuring objects
  • Comparing quantities and sizes

Teaching and learning experiences (including differentiation and enrichments):
  • Integrated throughout the curriculum and daily activities

Assessments:
  • Teacher observations / Final Assessment

Materials and Resources:
  • Blocks, centers, other manipulatives, and real-life situations

Accommodations:
Follow all IEP and 504 Plan requirements, allow extra time, extra help outside of class, graphic organizers, guided notes, test accommodations, guided practice, hands-on activities, teacher provided study guides, small group instruction, peer study partners.
Time:
Grade K / Mathematics
Mathematical Problem Solving and Communication
Pennsylvania Academic Standards: 2.5.3
  1. Use appropriate problem-solving strategies (e.g., guess and check, working backwards).
  2. Determine when sufficient information is present to solve a problem and explain how to solve a problem.
  3. Select and use an appropriate method, materials and strategy to solve problems, including mental mathematics, paper and pencil and concrete objects.

Essential Understandings:
  • Strategies for solving problems differ
Reasoning skills must be used for selecting and using appropriate problem solving strategies and tools.
Justification for using precise language for selection of strategies, tools, and final answers is necessary in problem solving
Students will need to know and be able to do:
  • Use logical reasoning to solve problems with numbers
  • Use pictures to solve real word problems
  • Use logical thinking to solve sorting problems
  • Model, describe and extend growing patterns
  • Gather and interpret data to answer questions
  • Use ordinal numbers first to fifth to show positions
  • Predict and record outcomes of activities dependent on chance
  • Count, compare and graph solid shapes
  • Identify and extend and solve problems with simple number patterns
  • Solve a problem by acting it out to make and identify equal groups
  • Find and use information from a calendar to answer questions.
  • Using pictures to represent and solve addition problems.
  • Decide whether to add or subtract in problem situations

Teaching and learning experiences (including differentiation and enrichments):
Lessons:
1.7, 2.9, 3.8, 4.7, 5.8, 6.6, 7.6, 8.6, 9.4, 10.6, 11.6, 12.10, 13.8, 14.8, 15.9, 16.6
Enrichments:
Chapter enrichments
DI enrichment tubs
Assessments:
Construction of a model
Student demonstration/construction using manipulatives/tools
Quizzes, tests
Teacher observations
Work samples
Student Self-Assessments: paired learning, small groups, daily review of daily classroom assignments
Materials and Resources:
Houghton Mifflin Math Textbook
Everyday Counts
Accommodations:
Follow all IEP and 504 Plan requirements, allow extra time, extra help outside of class, graphic organizers, guided notes, test accommodations, guided practice, hands-on activities, teacher provided study guides, small group instruction, peer study partners.
Time: 13 lessons (some lessons overlap with other Standards)
Grade K / Mathematics
Statistics and Data Analysis
Pennsylvania Academic Standards: 2.6.3
  1. Gather, organize and display data using pictures, tallies, charts, bar graphs and pictographs.
  2. Formulate and answer questions based on data shown on graphs.
  3. Predict the likely number of times a condition will occur based on analyzed data.
  4. Form and justify an opinion on whether a given statement is reasonable based on a comparison to data.

Essential Understandings:
Statistical data can be displayed in a variety of formats
Statistical displays assist in the understanding of the given data
Reasoning skills must be used to interpret and extend statistical data
Students will need to know and be able to do:
  • Collect and sort data to create graphs using concrete objects
  • Sort objects by one attribute to make real graphs
  • Sort objects by various attributes and make pictographs of the results
  • Identify and discuss the likelihood of a given situation based on known facts and chance

Teaching and learning experiences (including differentiation and enrichments):
Lessons:
3.5, 3.6, 3.7, 5.9
Enrichments:
Chapter enrichments
DI enrichment tubs
Assessments:
Construction of a model
Student demonstration/construction using manipulatives/tools
Quizzes, tests
Teacher observations
Work samples
Student Self-Assessments: paired learning, small groups, daily review of daily classroom assignments
Materials and Resources:
Houghton Mifflin Math Textbook
Everyday Counts
Accommodations:
Follow all IEP and 504 Plan requirements, allow extra time, extra help outside of class, graphic organizers, guided notes, test accommodations, guided practice, hands-on activities, teacher provided study guides, small group instruction, peer study partners.
Time: 4 lessons (some lessons overlap with other Standards)
Grade K / Mathematics
Probability and Predictions
Pennsylvania Academic Standards: 2.7.3
  1. Predict and measure the likelihood of events and recognize that the results of an experiment may not match predicted outcomes.
  2. Design a fair and an unfair spinner.
  3. List or graph the possible results of an experiment.
  4. Analyze data using the concepts of largest, smallest, most often, least often and middle.

Essential Understandings:
Probability is the likelihood of the occurrence of an event
Probability can be represented mathematically
Reasoning skills must be used to interpret, evaluate, and extend probability data
Students will need to know and be able to do:
  • Collect and sort data to create graphs using concrete objects
  • Sort objects by one attribute to make real graphs
  • Sort objects by various attributes and make pictographs of the results
  • Identify and discuss the likelihood of a given situation based on known facts and chance

Teaching and learning experiences (including differentiation and enrichments):
Lessons:
3.5, 3.6, 3.7, 5.9
Enrichments:
Chapter enrichments
DI enrichment tubs
Assessments:
Construction of a model
Student demonstration/construction using manipulatives/tools
Quizzes, tests
Teacher observations
Work samples
Student Self-Assessments: paired learning, small groups, daily review of daily classroom assignments
Materials and Resources:
Houghton Mifflin Math Textbook
Everyday Counts
Accommodations:
Follow all IEP and 504 Plan requirements, allow extra time, extra help outside of class, graphic organizers, guided notes, test accommodations, guided practice, hands-on activities, teacher provided study guides, small group instruction, peer study partners.
Time: 4 lessons (some lessons overlap with other Standards)
Grade K / Mathematics
Algebra and Functions
Pennsylvania Academic Standards: 2.8.3
  1. Recognize, describe, extend, create and replicate a variety of patterns including attribute, activity, number and geometric patterns.
  2. Use concrete objects and trial and error to solve number sentences and check if solutions are sensible and accurate.
  3. Substitute a missing addend in a number sentence.
  4. Create a story to match a given combination of symbols and numbers.
  5. Use concrete objects and symbols to model the concepts of variables, expressions, equations and inequalities.
  6. Explain the meaning of solutions and symbols.
  7. Use a table or a chart to display information.
  8. Describe and interpret the data shown in tables and charts.
  9. Demonstrate simple function rules.
  10. Analyze simple functions and relationships and locate points on a simple grid.

Essential Understandings:
Algebra is a system for identifying an unknown value
Patterns, equations, inequalities, and graphs are common components of algebra
Student will need to know and be able to do:
  • Compare objects and describe how they are alike or different
  • Identify, sort, and classify objects by color
  • Identify, sort, and classify objects by size
  • Identify, sort, and classify objects by shape
  • Identify, sort and classify objects by kind
  • Develop and explain own rules for sorting
  • Identify and describe spatial relationships: top, middle, and bottom
  • Model, describe and extend rhythmic patterns
  • Identify and extend patterns
  • Create and translate patterns
  • Identify and extend and solve problems with simple number patterns
  • Recognize examples of 0-12 and sort and classify by number
  • Sort and graph pennies, nickels, dimes and quarters by attributes.
  • Solve a problem by acting it out
  • Model and record subtracting two from the numbers six through ten.
  • Orally count by two’s, five’s, and tens; recognize and describe numbers patterns

Teaching and learning experiences (including differentiation and enrichments):
Lessons:
1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 2.1, 2.2, 2.6, 2.7, 2.8, 4.6, 8.3, 10.1, 14.4, 16.5
Enrichments:
Chapter enrichments
DI enrichment tubs
Assessments:
Construction of a model
Student demonstration/construction using manipulatives/tools
Quizzes, tests
Teacher observations
Work samples
Student Self-Assessments: paired learning, small groups, daily review of daily classroom assignments
Materials and Resources:
Houghton Mifflin Math Textbook
Everyday Counts
Accommodations:
Follow all IEP and 504 Plan requirements, allow extra time, extra help outside of class, graphic organizers, guided notes, test accommodations, guided practice, hands-on activities, teacher provided study guides, small group instruction, peer study partners.
Time: 16 lessons (some lessons overlap with other Standards)
Grade K / Mathematics
Geometry
Pennsylvania Academic Standards: 2.9.3
A.Name and label geometric shapes in two and three dimensions (e.g., circle/sphere, square/cube, triangle/pyramid, rectangle/prism).
B.Build geometric shapes using concrete objects (e.g., manipulatives).
C.Draw two- and three-dimensional geometric shapes and construct rectangles, squares and triangles on the geoboard and on graph paper satisfying specific criteria.
D.Find and describe geometric figures in real life.
E.Identify and draw lines of symmetry in geometric figures.
F.Identify symmetry in nature.
G.Fold paper to demonstrate the reflections about a line.
H.Show relationships between and among figures using reflections.
I.Predict how shapes can be changed by combining or dividing them.
Essential Understandings:
Geometry is the exploration of shapes and the relationships among these shapes
Students will need to know and be able to do:
  • Identify and describe spatial relationships: top, middle, and bottom
  • Identify and describe spatial relationships: before, after, between,
  • Compare and identify positions left and right
  • Compare and identify positions of objects using inside and outside
  • Identify and describe spatial relationships: over, under, on, up, down, on top of
  • Identify and describe circles and rectangles
  • Identify and describe squares and triangles
  • Identify and extend patterns with plane shapes, identify shapes that comprise larger shapes
  • Create symmetrical pictures and representations, identify symmetry
  • Sort three dimensional shapes
  • Identify and describe geometric shapes
  • Explore ways to build three-dimensional shapes
  • Match the surfaces of three-dimensional shapes to plane shapes
  • Combine and separate parts used in making a whole object

Teaching and learning experiences (including differentiation and enrichments):
Lessons:
2.1, 2.2, 2.3, 2.4, 2.5, 5.1, 5.2, 5.4, 5.5, 6.1, 6.2, 6.3, 6.4, 6.5
Enrichments:
Chapter enrichments
DI enrichment tubs
Assessments:
Construction of a model
Student demonstration/construction using manipulatives/tools
Quizzes, tests
Teacher observations
Work samples
Student Self-Assessments: paired learning, small groups, daily review of daily classroom assignments
Materials and Resources:
Houghton Mifflin Math Textbook
Everyday Count
Accommodations:
Follow all IEP and 504 Plan requirements, allow extra time, extra help outside of class, graphic organizers, guided notes, test accommodations, guided practice, hands-on activities, teacher provided study guides, small group instruction, peer study partners.
Time: 14 lessons (some lessons overlap with other Standards)
Grade K / Mathematics
Trigonometry
Pennsylvania Academic Standards: 2.10.3
  1. Identify right angles in the environment.
  2. Model right angles and right triangles using concrete objects.

Essential Understandings:
The relationships among the parts of a right triangle form the foundation for trigonometry
Students will develop foundational concepts of:
  • Identifying and naming triangles as shapes
  • Identifying triangles of equal size

Teaching and learning experiences (including differentiation and enrichments):
  • Integrated throughout the curriculum and daily activities

Assessments:
  • Teacher observations / Final Assessment

Materials and Resources:
  • Blocks, centers, other manipulatives, and real-life situations

Accommodations:
Follow all IEP and 504 Plan requirements, allow extra time, extra help outside of class, graphic organizers, guided notes, test accommodations, guided practice, hands-on activities, teacher provided study guides, small group instruction, peer study partners.
Time:
Grade K / Mathematics
Concepts of Calculus
Pennsylvania Academic Standards: 2.11.3
  1. Identify whole number quantities and measurements from least to most and greatest value.
  2. Identify least and greatest values represented in bar graphs and pictographs.
  3. Categorize rates of change as faster and slower.
  4. Continue a pattern of numbers or objects that could be extended infinitely.

Essential Understandings:
The basics of calculus include the identification and comparison of minimum and maximum values
The relationship between rates of change and time can be compared
Relationships among units of area and volume can be estimated
Students will develop foundational concepts of:
  • Ordering numbers/amounts from greatest to least and least to greatest
  • Identifying which group/number is greatest or least
  • Constructing a pictograph/bar graph to compare amounts

Teaching and learning experiences (including differentiation and enrichments):
  • Integrated throughout the curriculum and daily activities

Assessments:
  • Teacher observations / Final Assessment

Materials and Resources:
  • Vinyl graph mats, other manipulatives, and real-life situations

Accommodations:
Follow all IEP and 504 Plan requirements, allow extra time, extra help outside of class, graphic organizers, guided notes, test accommodations, guided practice, hands-on activities, teacher provided study guides, small group instruction, peer study partners.
Time:

1