Grade Level: 4, 5 Remediation Grade: 6, 7, 8

The first sixteen lessons in this series concentrate on laying the foundation for future math learning by strengthening and deepening student understanding of number concepts.

1. Goals: Discuss the use and value of visual symbols in math learning.

Introduce the Egyptian Numeral symbols for “one” and “ten”

Objectives: The students will hear about Egyptian writing and number system. They will learn to express oral and written integers and Arabic Numerals as Egyptian Numerals.

Materials: Crayons, pencil, workbook pages 1, 2, 3 and 4. Underlined pages are assigned as homework.

2. Goals: Introduce the idea of symmetry in connection with Egyptian numerals.

Discuss the meaning of a base ten number system.

Introduce Egyptian numeral for “hundred”.

Begin addition with Egyptian numerals using pipe cleaner manipulatives.

Objectives: The students will continue to use the Egyptian numerals to learn about base ten number systems. The study of symmetry will help them to see these numerals as picture symbols. The students will solve addition problems using Egyptian numerals.

Materials: Yellow crayon or marker, pencil, workbook pages 5, 6, 7, 8, and 9. (Optional, popsicle sticks, pipe cleaners)

3. Goals: Continue the discussion of symmetry using geometric shapes.

Introduce geometry vocabulary.

Introduce Egyptian numeral symbol for “thousand”.

Solve addition problems containing thousands.

Objectives: The students will learn to draw a line of symmetry and will hear about a plane of symmetry. They will learn the words triangle, rectangle, rhombus, cone, rectangular prism, cylinder, and cube. They will observe the use of manipulatives to show addition of Egyptian numerals and will solve addition problems themselves.

Materials: Pencil, ruler, yellow crayon or marker, workbook pages 10, 11, 12, and 13.

4. Goals: Introduce the concept of congruency using pictures and three dimensional forms.

Begin subtraction with Egyptian numerals.

Introduce the Egyptian numeral symbol for 10,000.

Review the < and > symbols.

Objectives: The students will compare the concepts of symmetry and congruency using diagrams and objects and learn to distinguish between the two. They will begin the process of subtraction with Egyptian numerals. They will learn to identify the symbol for 10,000 and to use the < and > symbols. An art activity is provided for practice with these symbols.

Materials: Pencil, red crayon or marker, workbook pages 14, 15, and 16.

5. Goals: To continue the study of number value using Egyptian numerals.

Introduce regrouping in addition.

Begin to teach number visualization using fixed configuration of spots.

Objectives: The students will identify and use Egyptian numerals greater than 10,000. They will practice regrouping using Egyptian numerals in addition problems. They will learn to recognize instantly the patterns for the integers 1-6 by using these patterns in an art activity.

Materials: Pencil, crayons, workbook pages 17, 18, 19, and 20.

6. Goals: Use the integer spot patterns to study common geometric shapes.

Review the patterns for 1 - 6.

Introduce the pattern for 9.

Continue subtraction using Egyptian numerals.

Continue addition with regrouping.

Objectives: The students will observe and be expected to identify numbers by seeing the pattern briefly. They will learn the pattern of the number 9. Geometric shapes will be learned in conjunction with the number patterns by identification of figures. The students will observe the use of manipulatives in solving addition and subtraction problems and will be expected to solve problems on their own.

Materials: Pencil, yellow and red crayons or markers, workbook pages 21, 22, 23, and 24. Optional: Pipe cleaners, etc. Note: The classroom teacher may wish to use the manipulatives or assign further addition problems after this lesson if they think this is necessary.

7. Goals: Introduce the number patterns for 7 and 8.

Review the other digit patterns.

Begin the use of regrouping in subtraction.

Objectives: The students will learn to identify the digit patterns for 7 and 8 through a discussion of their shape and configuration. They will practice recognizing the other patterns by seeing them briefly and writing the digit value. The students will observe and then solve subtraction problems requiring regrouping.

Materials: Pencil, red pen, crayon or marker, workbook pages, 25, 26, and 27.

8. Goals: Review.

Introduce number series.

Continue subtraction with regrouping.

Begin multiplication and division with Egyptian numerals.

Objectives: A review lesson will be presented after which the students will be expected to solve addition and subtraction problems, write numbers in both Egyptian and Arabic numerals. They will complete several number sequences with help. The students will be introduced to the operations of multiplication and division using Egyptian numerals.

Materials: Pencil, red and yellow marker or crayon, workbook pages 28 and 29.

9. Goals: To discuss number sequences and strategies for dealing with them.

Continue the use of patterns and review the symbols < and >.

Introduce multiplication times 10 and 100 using Egyptian numerals.

Objectives: Using a visual aid the students will learn strategies for completing number series correctly. They will review the digit patterns and do an exercise page putting the appropriate sign, < <, >, or = > between each number pair. The students will learn to multiply an Egyptian numeral times 10 and 100.

Materials: Pencil, workbook pages 30, 31, 32, and 33.

10. Goals: Review the concept of estimating with Egyptian numerals.

Present problems containing Egyptian numeral values of ten millions.

Continue multiplication and division with Egyptian numerals.

Begin discussing composition of numbers using the digit patterns.

Objectives: The students will estimate the value of Egyptian numerals in ordering numbers from least to greatest. They will solve addition, subtraction, and multiplication problems involving quantities of ten millions. The students will observe the solution of a division problem. The students will complete an activity page involving estimating with Egyptian numerals and number patterns. They will observe a demonstration of number composition using discs and a magnetic board.

Materials: Pencil, red and yellow markers, workbook pages 34, 35, 36, and 37.

11. Goals: To practice using Egyptian numerals and estimating when possible.

Teach number series strategies using Arabic numerals, Egyptian numerals, and patterns.

Review the < and > symbols.

Continue discussing composition of numbers.

Objectives: The students will complete with help an addition, subtraction, multiplication, and division problem involving quantities of ten millions. They will solve several problems on their own. The students will be assigned an activity using the symbols < and >. Number series using Arabic numerals, Egyptian numerals and patterns will be completed by the student.

Materials: Pencil, yellow marker, workbook pages 38, 39, 40, 41, and 42.

12. Goals: Simple addition and subtraction facts will be taught using digit patterns. Number combinations which make ten will be discussed.

The use of patterns to solve subtraction problems will be shown.

Patterns will be used to represent multi-digit numbers using a place value diagram and manipulatives.

Objectives: The students will learn to visualize simple math facts using the number patterns by seeing the pattern combinations demonstrated. They will practice using these patterns to solve subtraction problems. They will continue to work with number series. They will begin learning to visualize multi-digit numbers by seeing these numbers displayed on a place value diagram.

Materials: Pencil, red marker, workbook pages 43, 44, 45 and 46.

Optional: It would be very helpful to some students to have nine disc shaped objects to make these addition and subtraction facts for themselves. The classroom teacher may wish to make a place value diagram similar to that used in this lesson.

13. Goals: Continue demonstration of the use of patterns for multi-digit numbers.

Show the use of patterns in solving simple addition problems.

Review the tens facts and sue this knowledge to solve addition problems.

Objectives: The students will observe the use of patterns on a magnetic board to indicate multi-digit numbers. We will review composition of numbers and the students will use the digit patterns to solve addition problems. They will complete activity pages to review the tens facts and sue that knowledge in solving addition problems.

Materials: Pencil, yellow marker if necessary still, workbook pages 47, 48, 49, 50, and 51.

14. Goals: Teach how to show an Egyptian numeral as a number pattern.

Review subtraction with Egyptian numerals and with patterns.

Review Geometry especially quadrilaterals.

Demonstrate a method of subtracting from ten.

Objectives: The students will learn to express Egyptian numerals as number patterns by drawing the patterns. They will review geometry vocabulary by drawing a square, a rectangle, a trapezoid, and a parallelogram. They will learn an alternate method of subtraction from ten when regrouping is necessary.

Materials: Pencil, red marker, workbook pages 52, 53, 54, 55, and 56.

15. Goals: Demonstrate transferring a number pattern to both Egyptian and Arabic numerals.

Use a ‘place value machine’ (wooden beads on dowels) to explain expanded notation.

Objectives: The students will write a given pattern value in both Egyptian and Arabic numerals. They will observe a demonstration of the use of the ‘place value machine’ to indicate quantity, and will learn to express these quantities in expanded notation. Egyptian numerals will also be used to practice writing quantities in expanded form.

Materials: Pencil, workbook pages 57, 58, 59, 60 and 61.

16. Goals: Review expanded notation, tens facts, use of the symbols <, >, and =, place value, addition, patterns of multi-digit numbers, and geometry vocabulary.

Objectives: Except expanded notation and the geometry material, the students will review these topics by using the exercises provided. The geometry material will be studied using flash card diagrams.

Materials: Pencil, workbook pages 62, 63, 64, 65, and 66.

Lessons 17 through 32 will continue to deepen understanding in the area of number concepts and develop problem solving skills. Topics to be discussed will include number systems other than base ten, working with fractions, estimating, rounding numbers, number lines, graphs, simple multiplication and division, and special properties of zero and one.

17. Goals: To continue building addition and subtraction skills.

Introduce the base five number system.

Objectives: The students will observe the use of Egyptian numeral manipulatives to solve subtraction problems. After an explanation of the relationship to Arabic numeral solutions and to the pattern diagrams, students will solve addition and subtraction problems using different strategies and working with both Arabic numerals and patterns. Base five numbers will be discussed and some examples will be shown.

Materials: pencil, yellow and red markers, workbook pages 67, 68, 69, 70.

18. Goals: To continue building addition and subtraction skills.

Review number sequence.

Continue discussion of base five numbers.

Objectives: The students will solve addition and subtraction problems involving very large quantities. The place value “machine” will be used to compare base ten numbers to base five numbers. The students will complete an exercise page on base five numbers. A number sequence exercise having Egyptian numerals, Arabic numerals, and patterns will be assigned.

Materials: Pencil, yellow and red markers, workbook pages 71, 72, 73, and 75.

19. Goals: Continue working with base five numbers.

Demonstrate an alternate method of solving subtraction problems with zeros.

Introduce fractions.

Objectives: A place value diagram will be used in conjunction with an exercise page to demonstrate base five numbers. The students will listen to an explanation of a subtraction method and observe the solution of several problems. The lesson will end with a general discussion of fractions and observe a comparison of different fraction sizes.

Materials: Pencil, workbook pages 74, 77, 78, 152, 153.

20. Goals: Review patterns, 10 combinations, addition, and subtraction.

Continue with a general discussion of fractions.

Demonstrate methods of solving addition and subtraction problems containing fractions.

Objectives: The students will observe and write the value of a series of digit patterns. They will perform a similar exercise to show the number combinations which equal ten. They will listen to a discussion of addition and subtraction of fractions. They will draw a color diagram as a memory aid to solving fraction problems. They will analyze and solve problems.

Materials: Pencil, paper, red and yellow marker, workbook pages 79, 80, 81, 82, and 83.

21. Goals: Review symmetry, base five numbers.

Continue subtraction of fractions.

Begin multiplication.

Objectives: After a review discussion the students will complete a base five exercise page. We will work together on solving subtraction problems with fractions. The students will begin the study of multiplication using pattern manipulatives and then they will draw the patterns. They will compare addition and multiplication patterns.

Materials: Pencil, small disks (optional), workbook pages 84, 85, 86, 87, and 88.

22. Goals: Review congruency.

Continue addition and subtraction of fractions.

Review estimating.

Continue multiplication.

Objectives: The students will analyze and solve addition and subtraction problems involving fractions. They will estimate using both Egyptian and Arabic numerals. They will continue making or drawing multiplication patterns and comparing them with addition patterns.

Materials: Pencil, small disks (optional), workbook pages 89, 90, 91, and 92.

23. Goals: Continue estimating.

Introduce changing mixed numbers into fractions.

Continue multiplication and addition patterns.

Review.

Objectives: The students will analyze problems and estimate answers. They will learn to change mixed numbers into fractions. They will make and draw examples of addition and multiplication patterns using drawings. A review page will be assigned for homework.

Materials: Pencil, small disks (optional), workbook pages 93, 94, 95, 96, and 97.

24. Goals: Begin graphs.

Begin division using manipulatives, prime numbers.

Continue base five numbers.

Review.

Objectives: Egyptian numerals will be used to consider the idea of graphs as representing quantity. The students will study a pictograph and draw a similar graph for themselves. Two optional bar graph activities will be explained. The idea of division will be introduced using patterns. A base five exercise will be done as review and a review page will be assigned as a test or as homework. Several multiplication patterns will be demonstrated.

Materials: Pencil, ruler, (various fruit will be needed for one of the optional graphs), workbook pages 98, (99 and 100 optional) 101, 102, and 103 (test or homework).

25. Goals: Review fractions.

Practice estimating and reasoning skills.

Continue pictographs and bar graphs.

Introduce the idea of a number line.

Review Egyptian numerals and place value.

Objectives: The students will complete with help a worksheet of addition and subtraction of fractions. They will be expected to change mixed numbers into fractions and fractions into mixed numbers. An exercise page will be used for analysis. A pictograph and a bar graph will be studied and compared. A number line will be presented as a kind of graph and several questions will be answered. Two review pages are included in this lesson which could be used as a quiz or homework.

Materials: Pencil, workbook pages 104, 105, 106, 107, 108, and 109.

26. Goals: Consider the special properties of one and zero.

Continue working with number lines.

Continue division using patterns and Egyptian numerals.

Review number sequence, subtraction, and multiplication.

Objectives: The special properties of one and zero will be discussed. The students will use the given examples to practice these properties. Various kinds of number lines will be studied and information will be obtained from them. Review pages are included to review multiplication, division, number sequence and subtraction as needed.

Materials: Pencil, workbook pages 110, 111, 112, 113, 114, 115, and 116.

27. Goals: Estimation using number lines, Arabic numerals, and Egyptian numerals.

Review rounding numbers and number sequences.

Continue multiplication and division.

Objectives: The students will study various kinds of number lines to determine the information which can be obtained from each. They will also complete exercises on place value, number sequence and rounding numbers. They will continue to use patterns to learn about multiplication and division.

Materials: Pencil, small disks, workbook pages 117, 118, 119, 120, 121 and 122.

28. Goals: Review addition and subtraction of fractions.

Begin multiplication of fractions.

Review all operations.

Review rounding numbers.

Objectives: The students will review the color diagram memory aids for addition and subtraction of fractions. They will draw the diagram for multiplication and discuss. They will solve and listen to an analysis of fraction problems. Review material will be found on workbook pages.

Materials: Pencil, yellow, red and blue markers, workbook pages 123, 124, 125, 126, 127 and 128.

29. Goals: Review: Symbol value vs. place value, special properties of one and zero, multiplication, multiplication times ten, geometry.

Discuss the distributive principle.

Discuss prime numbers.

Objectives: The students will work on several exercise pages to practice the review material. They will listen to a discussion of the solution of problems involving addition and multiplication. They will be given the assignment to write a rule about grouping numbers. After a discussion of prime numbers, the students will be given a homework assignment on this topic.

Materials: Pencil, scrap paper, small disks, workbook pages 129, 130, 131, 132, 133 and 134.

30. Goals: Review multiplication, base five numbers, measurement, rounding numbers.

Continue commutative and distributive principles.

Objectives: The students will use the workbook pages for an overview of number concepts with special emphasis on commutative and distributive properties. They will observe the base five diagrams and study these in conjunction with the workbook page to review this material. They will chart distance on a number line.

Materials: Pencil, ruler, workbook pages 135, 136, 137, 138 and 139.

31. Goals: Review all operations, geometry (solid figures), fractions.

Introduce multiplication of mixed numbers.

Objectives: The students will review all operation using the workbook pages as a guide. Forming a mental image of the correct pattern will be emphasized. The students will be shown the color diagrams to review the solution of fraction problems. Several multiplication problems having mixed numbers will be explained.

Materials: Pencil, fraction diagrams, workbook pages 140, 141, 142, 143, and 144.

32. Goals: Review.

Objectives: The workbook pages will be used as a general review.

Materials: Pencil, workbook pages 145 and following.

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