A New Look in Math: A Visual Approach to Math Instruction
Visual Math 2 builds on the foundation of Visual Math 1. The images and concepts used in Visual Math 1 are applied to more advanced mathematical ideas. The first sixteen lessons will concentrate on fractions, decimals, and mastery of multiplication and division with single digit multipliers and divisors.
1. Goals: To present an overview of the study of Mathematics.
Objectives: The following questions are posed and discussed: What do numbers represent? How can a given quantity be represented? What can be done with numbers?The students practice various ways of representing a given quantity including visual representation, other bases, Egyptian and Arabic Numerals, expanded and standard form. Each of the mathematical operations are reviewed briefly and explained.
Materials: Pencil, workbook pages 2, 3, 4, and 5. Underlined pages are assigned as homework.
2. Goals: To introduce the concept of changing the appearance of a number without changing its value.
To begin intermediate multiplication.
Objectives: The students review lesson 1 to see the many ways they know how to change the appearance without changing the value of a quantity. The discussion includes regrouping, factoring, changing the order of factors (Commutative Principle), regrouping factors (Associative Principle), adding or subtracting zero, and multiplication by one. Several approaches to the operation of multiplication are discussed. Students use place value diagrams to locate multiplication answers.
Materials: Pencil, workbook pages 6, 7, 8, and 9.
3. Goals: Continue the study of intermediate multiplication.
Introduce decimals using Egyptian Numerals.
Objectives: Students review the Associative Principle in a short exercise, after which they practice solving multiplication problems using patterns and place value diagrams to locate the solutions. Homework in this is assigned. Egyptian Numeral symbols for tenths and hundredths are introduced. Students practice writing quantities including decimal values in both Arabic and Egyptian Numerals.
Materials: Pencil, workbook pages 10, 11, 12, 13, 14, 15, 16.
4. Goals: To present a fuller explanation of intermediate multiplication and to continue the study of decimal values.
Introduce addition and subtraction of decimals.
Objectives: The students continue to deepen their knowledge of multiplication by learning how the multiplication facts, the Associative and Distributive Principles are used in solving multiplication problems. Both diagrams and expanded notation calculations are used. A multiplication practice page is given, answers are found in the back of the workbook. An Egyptian Numeral symbol for thousandths is introduced. Students practice addition and subtraction of decimals.
Materials: Pencil, workbook pages 17, 18, 19 (practice page), and 20.
5. Goals: Introduce division, review addition, subtraction, multiplication, sequences, and rounding numbers.
Objectives: Students locate multiples on a number chart. They consider different aspects of division; i.e. division as a missing factor, division using Egyptian Numeral manipulatives, division using Egyptian Numeral symbols, and division with Arabic Numeral symbols. A review is presenting using the workbook pages.
Materials: Colored marker, pencil, workbook pages 21, 22, 23, and 24.
6. Goals: Continue building multiplication and division skills.
Continue addition and subtraction of decimals.
Review rounding numbers.
Objectives: The students observe the solution of sample problems. The Commutative, Associative, and Distributive Principles are explained and contrasted by using a reference page. Several different approaches to division are reviewed (see lesson 5). Strategies for rounding numbers are discussed.
Materials: Pencil, workbook pages 25, 26, 27, 28, and 29.
7. Goals: Begin fractions, compare fractions with decimals.
Objectives: Board drawings are used to illustrate fractional parts of a whole and to show the relationship of decimals and fractions. Students learn that the whole can be a single thing, a single group, or an arbitrary measure. Mention is made of the connection between fractions and division. Students solve addition and subtraction problems of fractions with common denominators. The diagram of multiplication and division of fractions is shown and discussed, some problems are demonstrated, a page of division is assigned for homework.
Materials: Blue and purple markers, pencil, workbook pages 30, 31, and 32.
8. Goals: Continue the study of fractions, review intermediate multiplication and division.
Objectives: Students learn to deal with mixed numbers by practicing changing a mixed number or whole number into an improper fraction. The relationship between this operation and subtraction of fractions is discussed. The diagram for addition and subtraction of fractions is shown. Several multiplication and division problems are solved as a review. Canceling as a means of simplifying problems is demonstrated.
Materials: Red and yellow markers, pencil, workbook pages 33 and 34.
9. Goals: Review fractions all operations.
Introduce multiplication of decimals by 1, 10, 100, and 0.1.
Review geometry vocabulary.
Objectives: Students review operations with fractions by analyzing and correcting three problems assigned for homework. Egyptian Numeral symbols are used to demonstrate multiplication of whole numbers and decimals by 1, 10, 100, and 0.1. Next the same quantities shown in Arabic Numerals are multiplied by 1, 10, 100, and 0.1. A page of decimal and fraction problems is assigned for homework and a similar page is completed in class. Students review geometry vocabulary by viewing the flash card pictures.
Materials: Pencil, workbook pages 35, 36, 37, and 38.
10. Goals: Continue to build skills in decimal calculations.
Relate decimals and fractions.
Introduce division of decimals.
Objectives: Students are taught to estimate products prior to calculating in sample multiplication problems. Placement of the decimal point is discussed. Addition, subtraction, and multiplication of money is accomplished. Line drawings are used to relate fractions with their decimal equivalents. These values are then obtained by division. The decimal equivalent of other fractions are found and the solution of several division problems is demonstrated. The page is to be completed for homework. Pages 43 and 44 are available for further practice in multiplication and division of decimals, answers are found in the back of the workbook.
Materials: Pencil, workbook pages 39, 40, 41, and 42.
11. Goals: Continue developing multiplication and division skills, especially multiplication and division of decimals.
Objectives: The students observe the change in the location of the decimal point on a place value diagram when a quantity is multiplied by 10, 100, and 0.1. The associative principle is reviewed. A method of two step multiplication when the multiplier is a decimal is demonstrated. Students practice using the workbook. An activity with manipulatives is shown which illustrates division when the quotient is a decimal.
Materials: Place value diagram and markers, pencil, workbook page 45, and 43. Optional, easily measured bulk material (beans or birdseed) and small paper cups.
12. Goals: To continue study of division of decimals, especially where the divisor is a decimal.
Circumference and perimeter.
Objectives: The students learn to image dividing into decimal groups by observing this done using a picture diagram. Several problems are solved in this manner. Circumference and perimeter are discussed, the formula for each is shown and word problems are solved. The word problems on page 48 are assigned for homework. The problems are analyzed and discussed in class.
Materials: Pencil, workbook pages 46, 47, and 48.
13. Goals: Continue perimeter and circumference.
Review fractions, especially mixed numbers, improper fractions, and decimal equivalents.
Introduce addition and subtraction of fractions without a common denominator.
Objectives: Students review perimeter and circumference by correcting and analyzing the homework problems. They study a reference page entitled “Changing the appearance of fractions”. Each of the four possibilities is discussed and demonstrated. Some examples of each are assigned for homework. Students examine a page of addition and subtraction where the fractions do not have common denominators. The addition and subtraction diagram is displayed. Students learn that a fraction is changed to an equivalent fraction by multiplying times one wearing a disguise. Possible common denominators are chosen for the problems on page 51.
Materials: Red, yellow, and blue markers, pencil, workbook pages 49, 50, 51, and 52.
14. Goals: Continue study of equivalent fractions, changing mixed numbers to fractions.
Introduce area of plane figures.
Objectives: The students re-enforce previously learned techniques for finding common denominators by determining equivalent fractions and solving addition and subtraction problems. A review page of equivalent fractions, addition, multiplication and division of decimals is analyzed and the solution of problems shown. The meaning of “area” is explained. Methods of finding the area of plane figures are discussed briefly.
Materials: Pencil, workbook pages 51, 53, 54.
15. Goals: To study alternate methods of finding the least common denominator.
Objectives: The students begin this lesson with a brief exercise in finding equivalent fractions and reducing fractions to lowest terms. They learn the terms “least common multiple” and “least common denominator”. Different ways of finding the least common multiple are explored, including dividing by prime numbers. Using the number chart on page 21, all of the prime numbers between 2 and 100 are determined. This is used in finding all of the prime factors of several quantities. Use of prime factors in canceling in multiplication is demonstrated. Students work on review pages.
Materials: Ruler, colored marker, workbook pages 21, 55, 56, 57.
16. Goals: Review fractions all operations, decimals all operations, equivalent fractions, circumference, perimeter, rounding, number sequence, etc.
Objectives: The students review these topics using the exercises provided in the workbook.
Materials: Pencil, workbook pages 58, 59, 60, and 61.
The second half of Visual Math 2 concentrates on multiplication and division using multi-digit multipliers and divisors. Other topics are included for completeness (fractions, decimals, percent, exponents, word problems, etc.) or as enrichment (base 5, negative numbers, Mayan Numerals, etc.).
17. Goals: Review division, present several strategies for solving division problems.
Objectives: Students listen to a discussion of division, the meaning of division, division as finding a missing factor, visualizing division using Egyptian Numerals, etc. Several problems are solved as a demonstration. A visual memory aid consisting of a three color diagram is shown and explained. Students observe problem solving following the color diagram.
Materials: Red, blue, and purple markers, pencil, workbook pages 62, 63.
18. Goals: Continue division.
Introduce division with a 2 digit divisor.
Discuss area, volume, and use of exponents.
Objectives: Students check and correct the assigned homework. The use of a division practice page (64) is recommended so that students can determine their skill level in division. Division with a two digit divisor is introduced using patterns and a place value diagram. Area and volume are demonstrated using drawings. The word “exponent” is introduced, as is “square” and “cube” when referring to exponents.
Materials: Red, blue, and purple color markers, pencil, workbook pages 63, 65, 66, 67, and 68.
19. Goals: Continue division.
Review fractions, area, perimeter, and symmetry.
Objectives: Students continue to solve division problems having 2 digit divisors. They learn the advantage of picturing the divisor as the number in each group, rather than the number of groups. The assigned homework on page 68 is corrected. A three color memory aid diagram of division with two digit divisors is shown and explained. Students are taught to use this color aid when solving problems. Problems on pages 69 and 70 are explained. Students are shown how to use a file card with 20, 30, ....times tables to estimate quotients. Fraction problems are analyzed.
Materials: Blue, red, and purple markers, 3X5 card, pencil, workbook pages 68, 69, and 70.
20. Goals: Review fractions, least common multiple, prime factors, exponents, and base 5 numbers.
Continue the study of division by two digit divisors.
Objectives: The students review fractions, common multiples, prime factors, and exponents by the analysis and solution of word problems on page 71 and 72. Multiplication, subtraction, and addition problems are reviewed. The color code for operation is used with division problems as in the diagram. Manipulatives, box diagram and coins, are used to review base five numbers.
Materials: Red, blue, and purple markers, pencil, (optional: a cardboard base five diagram) workbook pages 71, 72, and 73.
21. Goals: Continue division.
Review fractions, base 5 numbers, and exponents.
Objectives: Students continue to build division skills by checking the homework pages 70 and 72 to ensure accuracy. The “times table file card” is used to estimate quotients. The word problems on page 70 are analyzed and possible errors discussed. Multiplication of base five numbers is introduced. Base five times tables are derived. Students learn to regroup in base five by solving addition and subtraction problems needing regrouping. Two rules for exponents are derived by simplifying the expressions on page 74.
Materials: Pencil, workbook pages 70, 72, and 74.
22. Goals: Continue division.
Consider the relationship among fractions, division problems, decimals, and decimal fractions.
Objectives: Students continue to solve division problems with assistance using color codes for operation and times table card to estimate quotients. Division word problems are analyzed, and the significance of the word “each” is mentioned. Page 75 is assigned for homework. The connection between a fraction and a decimal is reviewed. Students study a diagram showing decimal - fraction relationships, and practice writing fractions as division problems. A chart of equivalent decimal - fraction values is completed.
Materials: Color markers, times table card, workbook pages 75, 76, 77, and 78.
23. Goals: Continue division.
Review fractions and decimals, introduce percent.
Objectives: The division homework is corrected, several of the problems are analyzed. The relationship between fractions and decimals is reviewed. Students are reminded that both are measures of some one thing or one group. Percent is introduced and discussed in this context. Students add % to the diagram on page 77. Percentage problems are solved. Several division problems are assigned for homework.
Materials: Color markers, pencil, workbook pages 75, 77, 79, and 80.
24. Goals: Review division.
Begin multiplication by tens.
Review rounding numbers, multiplication of fractions, and word problems.
Objectives: Students correct the division homework, page 80. A practice page of division examples is given for skill assessment. Multiplication by a one digit multiplier is reviewed with a single example showing the Associative Principle. The effect of multiplication by ten is demonstrated using manipulatives and diagrams. Two digit multiplication examples are solved using diagrams and “patterns”. Workbook page 85 is used to review rounding numbers, multiplication of fractions with canceling and word problems with f fractions.
Materials: Color markers, pencil, workbook pages 80, 81, 82, 83, 84 and 85.
25. Goals: Continue multiplication, review multiplication and division of decimals, percent.
Objectives: Students will review the word problems on page 85 and the solution of #7 is explained. Multiplication using diagrams and drawing the patterns for the partial products is explained and demonstrated. Two optional pages are provided (87, 89) for student practice if necessary. A review page is discussed and assigned for homework, problems include percent and multiplication and division of decimals. The subject of ratio is introduced and explained briefly.
Materials: Pencil, workbook pages 85, 86, 88, and 90.
26. Goals: Continue multiplication.
Review addition, subtraction and multiplication of base five numbers.
Review exponents, multiplication of fractions and canceling.
Objectives: Students check homework and review the placement of decimals in multiplication and division problems. Students continue to deepen their understanding of multiplication by using color coded answer boxes in solving multiplication problems. These are compared with color in division problems. A color diagram for multiplication is displayed. Regrouping with base five numbers is shown in addition to subtraction and multiplication problems. The use of exponents with unknown quantities is demonstrated. Division of decimals assigned for homework.
Materials: Blue, red and yellow markers, pencil, workbook pages 90, 91, and 92.
27. Goals: Continue multiplication.
Review division of decimals, changing percent to decimals, fractions.
Introduce negative numbers.
Objectives: Students are reminded why the decimal point can be moved when dividing by decimals. Multiplication problems are solved using color answer boxes and a practice page (94) of multiplication is shown. Both fractions and decimals are reviewed with workbook examples. The meaning and use of negative numbers is shown by practical examples. Students see a method of visualizing negative numbers using the magnetic board and patterns. Some examples of addition with negative numbers are shown.
Materials: Blue and yellow markers, pencil, workbook pages 92, 93, 95, and 96.
28. Goals: Continue negative numbers.
Discuss ratio in greater depth.
Review multiplication, division, exponents.
Objectives: Students listen to a discussion of ratio and proportion. They learn to solve ratio problems using several simple word problems. Manipulatives are used in demonstrating addition and subtraction of negative numbers. Students solve problems using the pattern drawings. Multiplication and division are reviewed with an emphasis on location of the decimal point in division of decimals. The rules of exponents are reviewed using both known and unknown quantities. Students are instructed to draw the patterns when adding and subtracting negative numbers assigned for homework.
Materials: Blue and yellow marker, pencil, workbook pages 95, 96, and 97.
29. Goals: Introduce problems with three digit multipliers and three digit divisors.
Practice addition and subtraction of negative numbers, explain multiplication of negative numbers.
Introduce Roman and Mayan Numerals.
Objectives: Students are instructed to draw color coded answer boxes and solve several multiplication problems. Division problems with three digit divisors are demonstrated. A division page page (99) is explained. Arabic, Egyptian, Roman and Mayan Numerals are compared. Students transfer examples from Arabic to Roman or Mayan Numerals. The homework assignment includes practice in multiplication and working with negative numbers.
Materials: Blue and yellow marker, pencil, workbook pages 98, 100, and 101.
30. Goals: Review multiplication, addition and subtraction of negative numbers, Roman and Mayan Numerals, division, area, and word problems.
Objectives: Multiplication homework is corrected; one problem is analyzed carefully. Problems involving addition and subtraction of negative numbers are corrected as well. The Mayan number system is explained and compared with a base ten number system. Students complete a chart of Arabic, Roman, and Mayan Numerals. Division by three digit divisors is reviewed solving several problems. The formulas for area are given. The origin of the formula for area of a triangle is shown visually with two identical triangles. Students are advised to try a similar activity for themselves. “Area” word problems are analyzed and solved. The word problems assigned for homework are discussed.
Materials: Pencil, paper and scissors, workbook pages 101, 102, 103, and 105.
31. Goals: Review.
Introduce and explain Algebraic Notation using Egyptian Numerals.
Objectives: This lesson provides an extensive review. The workbook pages are used as the basis for discussion. Topics include percentage, number sequence, geometry, negative numbers, fractions, Egyptian, Arabic, Roman and Mayan Numerals. Standard and expanded notation are used to introduce Algebraic notation in which both Arabic and Egyptian Numerals indicate a given quantity. Examples of Algebraic notation are presented. An example of multiplication with a four digit multiplier is given, two problems are assigned for homework. Ratio and proportion are mentioned briefly.
Materials: Pencil, workbook pages 104, 105, 106, 107, and 108.
32. Goals: Continue ratio and proportion.
Review exponents, explain square root.
Review general picture of mathematics.
Objectives: An activity suitable for students is described and suggested to study ration and proportion. Optional material on finding square root is presented. A general discussion follows.
Materials: Pencil, workbook pages 108, 109, and 110.