Math 2 Album
February 14, 2017 | Author: Mersi Ta | Category: N/A
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montessori album...
Description
Math II General Outline I Powers of Numbers
1
II Negative Numbers
15
III Non-Decimal Bases
27
IV Word Problems
45
V Ratio & Proportion
65
VI Algebra
83
Copyright 2009 | Jonathan Feagle www.FreeMontessori.org
i
Contents
I Powers of Numbers
1
A. Powers of 2 2 Presentation:
Passage One: Introduction p.2 Extension I: Terminology Extension II: Exploration with Bases Other than Two Passage Two: Different Unit Size p.6 Passage Three: Hierarchical material p.8
B. Exponential Notation 10 Presentation: Passage One: Behavior of Exponents when Multiplying Number of the Same Base p.10 Passage Two: Behavior of Exponents when Dividing Numbers p.12
II Negative Numbers
A. Addition Using Negative Numbers 16 Presentation:
15
Passage One: The Snake Game with Negative Numbers and Negative Changing p.16 Passage Two: Writing p.18 Passage Three: Introduction to the Ten Bar p.18
B. Subtraction of Sign Numbers 20 Presentation: Deriving the Rule for Subtracting Sign Numbers p.22
C. Multiplication of Sign Numbers Presentation: D. Division of Sign Numbers 25 Presentation:
III Non-Decimal Bases
A. Numeration 29 Passage One: Numeration Part A: Counting on a strip p.29 Part B: Bases larger than 10 p.29
B. Operations in Bases
31
Part A: Addition p.31 Part B: Subtraction p.33 Part C: Multiplication p.35 Part D: Division p.36 Copyright 2009 | Jonathan Feagle www.FreeMontessori.org
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27
ii
C. Conversion from One Base to Another
37
Part A: To convert a number from any base to base 10 p.37 Part B: To convert from base 10 to another base p.39 Part C: Changing to bases larger than 10 p.41 Part D: Changing bases using the base chart p.42 Example I: Example II: Example III:
IV Word Problems
A. Introduction to Word Problems B. Distance, Velocity and Time 47 Presentation:
46
45
Passage One: Introduction p.47 Passage Two: Solving for Distance p.47 Level One Level Two Level Three Passage Three: Solving for Velocity p.49 Level One Level Two Level Three Passage Four: Solving for Time p.52 Level One Level Two Level Three
C. Principal, Interest, Rate and Time 53 Presentation: Passage One: Introduction p.53 Passage Two: Solving for Interest p.54 Level One Level Two Level Three Passage Three: Solving for Rate p.56 Level One Level Two Level Three
Passage Four: Solving for Principal Level One Copyright 2009 | Jonathan Feagle www.FreeMontessori.org
iii Level Two Passage Five: Solving for Time p.62 Level One Level Two Level Three
V Ratio & Proportion
A. Ratio 66 Presentation: Introduction Passage One: Introduction p.66 Passage Two: Introduction to the Language p.66 Passage Three p.68 Passage Four p.68 Passage Five: Exploring the Idea Arithmetically p.68 Passage Three: Passage Four: Passage Six: Ratios Written as Fractions p.70 Passage Seven: Stating the Ratio Algebraically p.71 Passage Eight: Word Problems p.72 Example A Example B Example B Algebraically Example C Example C Arithmetically Example C Algebraically
B. Proportion 77 Presentation: Introduction Exercise One: Determining if Something is in Proportion p.78 Exercise Two: Proportion Between Geometric Figures p.78 Exercise Three: With 3 Dimensional Figures p.79 Exercise Four: p.80
C. Calculations with Proportion 81 Exercise One: Arithmetically p.81 Exercise Two: Algebraically (for older children) p.81 Exercise Three: Applications p.82 Example I Example II
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VI Algebra
83
A. Introduction to Algebra 84 Exercise One: Balancing an Equation p.84 Exercise Two: Balancing an Equation When Something is Taken Away p.85 Exercise Three: Balancing an Equation When Something is Multiplied p.85 Exercise Four: Balancing an Equation When it is Divided p.85
B. Operations With Equations 86 Exercise One: Addition p.86 Exercise Two: Subtraction p.86 Exercise Three: Multiplication p.86 Exercise Four: Division p.86
C. Algebraic Word Problems
87
Example I: p.87 Example II: p.87 Example III: p.87 Example IV: p.88 Example V: p.88 Example VI: p.88 Example VI: p.89 Example IX: p.89 Example X: p.90 Example XI: p.90 Example XII: p.90
Copyright 2009 | Jonathan Feagle www.FreeMontessori.org
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I Powers of Numbers Contents A. Powers of 2 Presentation:
2
Passage One: Introduction p.2 Extension I: Terminology Extension II: Exploration with Bases Other than Two Passage Two: Different Unit Size p.6 Passage Three: Hierarchical material p.8
B. Exponential Notation Presentation: Passage One: Behavior of Exponents when Multiplying Number of the Same Base p.10 Passage Two: Behavior of Exponents when Dividing Numbers p.12
Copyright 2009 | Jonathan Feagle www.FreeMontessori.org
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A. Powers of 2 Introduction: This material is not presented to the child until after the exercises with squares and cubes, including notation and operations have been completed. This material is designed to present to the child the powers of numbers beyond three, and to present the hierarchical material in another way to reinforce in the child’s consciousness what is meant by the powers of ten. This lesson may be presented several times, perhaps a month or so apart, particularly if the child is having difficulties with it. Materials: The Power of Two’s Box, the cubing material, small tickets of paper and pencils
Presentation: Passage One: Introduction 1. Hold up a small red cube. State that it is one, a unit, and that it is powerless. Set it on the mat. 2. “Now we will make a group of two.” Move the first cube over, setting a second beside it. 3. Recognize this as your first group of two, call it two to the first power, and write and place a ticket under it. 4. “Now we will take 2 to the power of 1, two times.” Move the group of two over and add two to it, forming a square. 5. “It makes a square.” Write a ticket stating 22, place it under the square, stating that this is what it is. 6. “Now we will take 2 to the power of 2, two times.” Move the square over and add four cubes to it, forming a cube. 7. Replace the built cube with the cube of the same size from the cubing material. State that it is two to the power of three and label it as such. 8. Continue in the same way, each time taking to the previous power two times, double the number of cubes, exchange if possible, state its name and label it. 9. When two to the power of nine has been completed, reverse the procedure, dismantling the cube, exchanging as necessary, laying the pieces out at each level, and reading their names. 10. When you return to the cube, ask the child what you called it (unit). Remind her that you said it had no power. Label it as 20, and state its value as one. 11. The child may wish to write and place tickets stating the value (23, 2x2x2, 8) or the number of the previous power times two (23x2, 24x2, etc.).
Copyright 2009 | Jonathan Feagle www.FreeMontessori.org
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Copyright 2009 | Jonathan Feagle www.FreeMontessori.org
8 Extension I: Terminology 1. After the child has had some practice, tell her that the first number with which we make these powers of numbers, the larger one, is called the base. 2. Continue, stating that the second number, the smaller one written above the first, is called the exponent. 3. Ask the child what the exponent is (what does it tell you?) 4. Note that when you have multiplied the base by itself the number of times directed by the exponent, you have reached that power of the base.
Extension II: Exploration with Bases Other than Two 1. Present the unit; state that this time you would like to work with groups of three. Assemble one group of three cubes. 2. Ask the child how this should be written (31), have the child write and place a ticket. 3. “How do we get to the next power?” Determine that you would take three 3 times. Replace the three red cubes with a three square. Have the child write and place a label (32). 4. Continue, taking 32 three times, replace the square with the cube and label it accordingly (33). 5. Note that you do not have two more cubes of three. Add two stacks of three squares to the cube and label (34). 6. Note that for 35, you lack squares with which to construct the shape, ask the child what it would look like. 7. Lead the child to see that it would be the nine square. Lay it out. 8. You may continue to the power of nine, or the child may wish to reverse the procedure as you did for the power of two. 9. You may repeat with other bases. Also you may compare bases to each other, laying one base behind the other. Compare their sizes, shapes and other relationships.
Copyright 2009 | Jonathan Feagle www.FreeMontessori.org
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2
3
Extension I:
base
exponent what does the exponent tell you?
2 x 2 x 2 = power of the base
Extension II:
then 32 three times
then 31 three times
= 32
= 31
= 33
then 33 three times
= 34
=
= 3 taken three times = 3 4
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10 Passage Two: Different Unit Size 1. Take out the Power of Two’s Box, stating that you are going to look at it again. 2. “We will let the small cube (small yellow) be the unit, and we are in the base of two, what will we label this cube?” 3. Have the child write and place a label reading 20 below the unit cube. 4. Place the other cube next to it, label it 21, and replace the two cube with the prism, putting the cube back over its original label. 5. “Let’s take 21 two times to build to the next power.” Put the two cubes beside the prism, then exchange it for the square, replacing the cubes in their original locations. Have the child write a label. 6. Repeat, building the two square to a cube with the prism, and two cubes. Exchange, replace the other pieces and label the cube. 7. Repeat in the above fashion with 24, constructing the prism, then dismantling the other pieces back in their places. 8. Continue to 26, the limit of the material. 9. Summarize that the unit can be of any size.
Copyright 2009 | Jonathan Feagle www.FreeMontessori.org
11 20
21
21
22
22
exchange
23
exchange
24
23
24
exchange
exchange
25
25
exchange
26 26
exchange
Copyright 2009 | Jonathan Feagle www.FreeMontessori.org
12 Passage Three: Hierarchical material Note: For this passage, you will need the hierarchical material. 1. Have the child lay out the hierarchical material in their families. 2. Have the child name each family. State that we know this is the decimal system, since each family is composed of ten of the previous (deci = ten). 3. Note that in the decimal system, the base is ten. Label the unit 100, ask the child what it means when you have a base (how much it takes to go from one power to the next). 4. “Is that what has been done to the ten? Is it ten to the first power?” The child may verify by counting. Ask the child what it is called (101). Have him write and place a label. 5. “Let’s go to the next power, what do we have to do?” (multiply the ten by ten) have the child write and place a label. 6. “For the next level, we take ten to what power?” (3rd) Have the child write and place the label by the cube. 7. Continue to the sixth power (106). The child may wish to place the powers of two alongside. Note which powers have the same shape. 8. The child may also wish to lay out the numerical values of the powers of ten. Note that the number of zeros equals the exponent number.
Copyright 2009 | Jonathan Feagle www.FreeMontessori.org
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101
102
10
100
21
Copyright 2009 | Jonathan Feagle www.FreeMontessori.org
103 1000
22
23
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B. Exponential Notation Introduction: This work allows the child to investigate and explore exponential numbers. It should not be presented until the child has completed the powers of two and ten. Materials: The cubing material, pencils and paper
Presentation: Passage One: Behavior of Exponents when Multiplying Number of the Same Base 1. Propose the problem: 33 x 32 =. Note that the child can probably do it, but that you are going to try a new way. 2. “We are going to take the three cube 32 times.” Lay out the three cube. 3. Ask what 32 is (9). State that you will take the three cube nine times. 4. Layout nine three cubes (assembled with three squares), forming a square. 5. “We can write this as a power of three, as three to the fifth power.” 6. “How did I know that?” Demonstrate by reconstructing the shape. Start with a three square (32), times 3 makes the three cube (33), times three yields a line of 3 three cubes (34), times three is 3 rows of 3 three cubes (35). 33 x 32 = 35 7. Propose another problem: 32 x 33 =. Note that the quantity is 32, and it is to be taken 33 times (27). 8. Set out 27 three squares. Take one, stating that it is 32. Add two more to form 33. Add six more to form a line of three cubes (34). Place the remaining 18 to form a square of 27 three squares (35). 9. Note that you will express the answer as a power of the base. In this case, the base is three, and youíve taken it to the power of five (35 is the answer). 10. After some work, ask the child if he can make an observation about the answers. Note the following rule: When multiplying exponential numbers of the same base, add the exponents and express the answer as the base with the sum of the exponents. 11. Later, propose the problem: 53 x 5 =. State that the problem says to take a five cube five times. 12. Add four groups of 5 five squares to the cube. Note that you get 54. 13. “What can we say about the five in this problem?” (It is five to the power of one and it gets added to the cube to make 54. Copyright 2009 | Jonathan Feagle www.FreeMontessori.org
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33
32
33 x 32 = x
(
=9
)
So,
x 9 =
= 35
Copyright 2009 | Jonathan Feagle www.FreeMontessori.org
16 Passage Two: Behavior of Exponents when Dividing Numbers 1. Propose the problem: 35 ÷ 33 =. Write in fraction form. Lay out three to the fifth, forming a large square. 2. Ask how many three cubes are contained in the square. [Lay the cubes in a line.] 3. Have the child count the cubes (9). Ask her if it is a number that can be written in the same base as the others (32). 4. Note the rule: When dividing exponential numbers of the same base, subtract the exponents and express the answer as the base with the difference of the exponents. 5. Propose the problem: 32 ÷ 32 =. Note that the answer would be 1, or 30 (expressing the unit as the base to the zero power).
Copyright 2009 | Jonathan Feagle www.FreeMontessori.org
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35 3
3
=
= 35
How many three cubes in the square?
9 cubes 9 = 32
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Copyright 2009 | Jonathan Feagle www.FreeMontessori.org
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II Negative Numbers Contents A. Addition Using Negative Numbers Presentation:
16
Passage One: The Snake Game with Negative Numbers and Negative Changing p.16 Passage Two: Writing p.18 Passage Three: Introduction to the Ten Bar p.18
B. Subtraction of Sign Numbers Presentation:
20
Deriving the Rule for Subtracting Sign Numbers p.22
C. Multiplication of Sign Numbers Presentation: D. Division of Sign Numbers Presentation:
Copyright 2009 | Jonathan Feagle www.FreeMontessori.org
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20
A. Addition Using Negative Numbers Note: It helps to give kids examples of negative numbers before starting: “If I have $7 and I owe John $8, I really have -$1.” Materials: A mat, boxes containing the colored bead bars from 1 - 10, the negative bead bars 1 - 10, the black and white bead stairs and the red and white bead stairs.
Presentation: Passage One: The Snake Game with Negative Numbers and Negative Changing 1. “Let’s call the colored bead bars positive, and the gray ones negative.” 2. Have the child form a long snake, dictating to him positive and negative numbers to be placed:
8 + +9 + -8 + +8 + -9 + +6 + +5 + -8 + +4 + +6
+
3. 4. 5. 6. 7.
Have the child lay out the black and white stair. Bring the first two bars down from the snake (+8 + +9). Exchange them for a ten bar and the seven from the black and white stair. Attach this to the snake. Place the used bars in a pile at the top. Bring down the black and white 7 and the - 8. When you add -8 and +7, you get a negative 1 (-1) | How are we going to do that without a -1 bar? Replace the bars with the -1 bar. Place the black and white 7 back into the b/w bead stair, and start a negative pile at the top. 8. Bring down the next two bars (-1 and +8). Continue as described above until the end of the snake. 9. Count to see what’s left on the snake. Ask the child how you are going to check it. Point out that positive and negative beads at the top represent the whole snake. 10. “We can add all the positives, and all the negatives, then find the difference between them.” Place bars in each pile in like groups and add their values on paper. 11. Compare the two answers, place a check if correct.
Copyright 2009 | Jonathan Feagle www.FreeMontessori.org
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Copyright 2009 | Jonathan Feagle www.FreeMontessori.org
22 Passage Two: Writing 1. Dictate a number (+8) and have the child take the bar out. Show him how to say and write it. 2. Continue dictating, saying and writing the rest of the numbers in the problem below, noting that the +/signs are placed close to the numbers in the problem. 3. Point out that the problem is a sum. Have the child work it out and place the answer at the end of the problem. +8 + -9 + +6 + -5 = Passage Three: Introduction to the Ten Bar Note: The child will most likely run across the following on her own, and you will therefore show her what to do then. It need not be a separate lesson. 1. Lay out and record the problem: 7 + +2 + +3 + -9 + -8 + +2 + -9 + -3 + +7 + +9 + +4 + +9 + -7 + -3 + -6 + +2 + +6 + +8 =
+
2. Have the child work out the problem as usual. When she encounters a negative sum over nine, introduce her to the negative ten bar, and have her place it in the snake, followed by the five from the red and white bead stair. 3. Continue in the manner described in Passage One through the snake. When the snake is complete, determine its value by canceling out the positive and negative ten bars. 4. Have the child check the answer. She may do so in a similar way, canceling the positive and negative bars to get the answer. Have her record the answer 7 + +2 + +3 + -9 + -8 + +2 + -9 + -3 + +7 + +9 + +4 + +9 + -7 + -3 + -6 + +2 + +6 + +8 = 14
+
5. Lead the child to the rule by having her complete the following statements: When you add numbers of different signs… (you subtract and the answer has the sign of the larger number in the problem). When adding numbers of the same sign….(you add them and the sign remains the same). 6. The child may discover that she can add all the same sign numbers and subtract the different sign totals and assign the sign of the larger number. 7. And/or, she may discover canceling. When doing this, she should cross out the canceled numbers, then add the same signs and subtract as described above. Copyright 2009 | Jonathan Feagle www.FreeMontessori.org
23 Passage Three:
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B. Subtraction of Sign Numbers Materials: A mat, boxes containing the colored bead bars from 1 to 10, the negative bead bars from 1 to 10, the black and white bead stairs and the red and white bead stairs.
Presentation: 1. Propose the problem: +7 + +3 + +4 + +3 + -4 =. Have the child build the snake, record the values, solve the problem and write the answer at the end (+13). 2. Recompose the snake and the problem by removing the +4 bar and hiding it in your hand. Have the child work the new problem and state the answer. 3. State that you will record the problem in a special way. Note that you took the +4 away and record this problem under the original: (+7 + +3 + +4 + +3 + -4) - +4 = +9. 4. Recompose the problem again by replacing the +4 and removing the -4. Have the child work the problem as above. 5. Write the problem, asking what you took away, and subtract that at the end: (+7 + +3 + +4 + +3 + -4) - -4 = +17. 6. Do a series of similar problems, recording them as you go, so that you may make observations at the end. 7. Ask the child what she notices. Lead her to understand that when a positive number was subtracted, the answer got smaller, while when a negative number was subtracted, the answer got larger. 8. The child may check her observations on other problems.
Copyright 2009 | Jonathan Feagle www.FreeMontessori.org
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7 + +3 + +4 + +3 + -4 =
+
Hide the four bar in hand. Recorded as: (+7 + +3 + +4 + +3 + -4) - +4 = +9
Replace +4 bar and remove the -4 bar then write the problem showing what you took away (-4).
Recorded as: (+7 + +3 + +4 + +3 + -4) - -4 = +9
Copyright 2009 | Jonathan Feagle www.FreeMontessori.org
26 Deriving the Rule for Subtracting Sign Numbers 1. Restate the rule arrived at above. Hold up the nine bar, and say that it is positive nine. Write it down. 2. “I’ll take +4 away from that.” Record this: +9 - +4, and cover four beads on the bar. 3. Ask the child what’s left (+5), record this.
9 - +4 = +5
+
4. “It’s also possible to have -9 - -4. Help the child to show you this on the negative nine bead bar. Determine that the answer is negative five and write: 9 - -4 = -5
-
5. “You can also have -9 - +4 =. But there isn’t a positive four on the negative nine bar.” Place both a positive and a negative four bar beside the negative nine bar. Affirm with the child that you have now added zero to the negative nine bar. 6. Take the positive four bar away, determining that -13 is the answer. Record this: 9 - +4 = -13
-
7. Propose the problem: +9 - -4 =. Have the child take the positive nine bar and add zero to it by placing a positive and a negative four bar beside it. Remove the negative four to arrive at +13. 9 - -4 = +13
+
8. Have the child look at the first problem (+9 - +4 = +5). Ask if there is another way to get a +5. Determine that you can change the sign of the subtrahend and add. 9. Write out the rule and try it on other problems: When subtracting sign numbers, change the sign of the subtrahend and add it to the minuend.
Copyright 2009 | Jonathan Feagle www.FreeMontessori.org
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-
9 - -4 =
cover four
cover four
-
9 - +4 =
add zero
take away the +4 = -13
9 - -4 =
+
add zero
take away the -4 = 13 Copyright 2009 | Jonathan Feagle www.FreeMontessori.org
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C. Multiplication of Sign Numbers Materials: A mat, boxes containing the colored bead bars from 1 - 10, the negative bead bars 1 - 10, the black and white bead stairs and the red and white bead stairs.
Presentation: 1. Roll out the mat, and ask the child what is on it (nothing). “I would like you to give me +5 three times.” 2. Place the bars on the mat and ask the child what is there now (+15). “We took +5, +3 times and we got + 15.” Have him write: 5 x +3 = +15
+
3. “Now, I would like you to give me -5 three times.” Lay them out after the child hands them to you, and have him restate the problem with the answer: 5 x +3 = -15
-
4. Lay out three positive and three negative five bars, and ask the child what is there (zero). 5. Take away the three positive five bars and ask what is left (-15). (Take +5 a -3 times.) Record: 5 x -3 = -15
+
5 x -3 =
+
What is here? Zero!
Take +5 negative three times.
What is here now?
-15. Copyright 2009 | Jonathan Feagle www.FreeMontessori.org
29 6. Again, lay out three positive and three negative five bars, and state that you want to take negative five negative three times. 7. Remove three negative five bars, and ask the child what’s left (+15). Record: 5 x -3 = +15
-
8. After some practice, help the child to observe the rule: When you multiply numbers of the same sign, the answer is positive. When you multiply numbers of different signs, the answer is negative.
D. Division of Sign Numbers Materials: In addition to the materials mentioned above, division cups and skittles will be necessary.
Presentation: 1. Place two skittles on the mat. Gather 4 positive seven bars into your hand. Show them to the child, ask her what they are (four +7 bars), and what their value is (+28). 2. “I’m sharing these bars between the skittles.” Have the child record: 28 ÷ +2 = +14
+
3. Note that this matches the multiplication rule (+ x + = +). 4. Return the positive sevens and gather four negative seven bars into your hand. Ask the child what they are (four -7 bars), and what their value is (-28). 5. Distribute the bars to the skittles, ask the child what each skittle got (-14), and have her write the problem: 28 ÷ +2 = -14
-
6. Gather the negative seven bars again. Ask the child how many groups of -14 you could make (+2). 7. Ask how to write this as a division problem and record: Copyright 2009 | Jonathan Feagle www.FreeMontessori.org
28 ÷ +2 = 14
+
30 28 ÷ -14 = +2
-
-
28 ÷ +2 = -14
8. Propose that 2 people are debtors, they owe money, and each owes +14. Place two +7 bars each in two cups. Ask the child what is there (+28). 9. Ask her how to write this as a division problem. “There are +28 beads in the cups (write +28), and two debtors. The debtors are negative because they took the +28 away (write -2). 10. State that the debtors took the positive 28 away, do so, and ask the child what each skittle now gets (-14). Record the problem and answer: 28 ÷ -2 = -14
+
11. Determine that the rule is the same as that for multiplication: When you divide numbers of the same sign, the answer is positive. When you divide numbers of different signs, the answer is negative.
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III Non-Decimal Bases Contents A. Numeration Passage One: Numeration
29
Part A: Counting on a strip p.29 Part B: Bases larger than 10 p.29
B. Operations in Bases
31
Part A: Addition p.31 Part B: Subtraction p.33 Part C: Multiplication p.35 Part D: Division p.36
C. Conversion from One Base to Another Part A: To convert a number from any base to base 10 p.37 Part B: To convert from base 10 to another base p.39 Part C: Changing to bases larger than 10 p.41 Part D: Changing bases using the base chart p.42 Example I: Example II: Example III:
Copyright 2009 | Jonathan Feagle www.FreeMontessori.org
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32
Introduction: Children should have worked extensively with the decimal system. They must understand that in our system there are only nine symbols (well, okay, ten (i.e. 0)) to represent digital quantity. As soon as ten is reached, it is necessary to move to the next place and employ the zero as a marker in the units category. At ten, we no longer have a group of units, we have a single unit of the next higher order: we have a ten. The child must recognize that ours is a place value system in which zero is a necessity. The child must have worked with powers of number between 2 and 10 and exponential notation. They are also aware of the geometric shapes of the various powers, i.e.: the power of zero is a point, and the powers of 2 and 5 create lines. The child also must have studied the history of numbers and understand that other cultures have used different-base number systems than ours, and had entirely distinct concepts of number (the Egyptians used no place value) from ours. Materials: Use as appropriate: the colored bead bars, the cubing material, the golden unit beads, a roll of adding machine tape, number bases board, made of felt and marked into four categories as below, a chart of numeration in four or more bases with base 10 in red, blank paper tickets, and pencils
0
1
2
3
4
5
6
7
8
9
A
B
C
D
E
F
Fifteen
0
1
2
3
4
5
6
7
8
9
A
B
C
D
E
10
Fourteen
0
1
2
3
4
5
6
7
8
9
A
B
C
D
10
11
Thirteen
0
1
2
3
4
5
6
7
8
9
A
B
C
10
11
12
Twelve
0
1
2
3
4
5
6
7
8
9
A
B
10
11
12
13
Eleven
0
1
2
3
4
5
6
7
8
9
A
10
11
12
13
14
Ten
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Nine
0
1
2
3
4
5
6
7
8
10
11
12
13
14
15
16
Eight
0
1
2
3
4
5
6
7
10
11
12
13
14
15
16
17
Seven
0
1
2
3
4
5
6
10
11
12
13
14
15
16
20
21
Six
0
1
2
3
4
5
10
11
12
13
14
15
20
21
22
23
Five
0
1
2
3
4
10
11
12
13
14
20
21
22
23
24
30
Four
0
1
2
3
10
11
12
13
20
21
22
23
30
31
32
33
Three
0
1
2
10
11
12
20
21
22 100 101 102 110 111 112 120
Two
0
1
10
11 100 101 110 111 1000 1001 1010 10111100 1101 1110 1111
The Relations among a few base systems.
Sixteen
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A. Numeration Passage One: Numeration Part A: Counting on a strip 1. Introduce the number bases board, perhaps relating it to a question that the child may have had regarding numbers in other bases. 2. “Let’s work in the base of five and see how to count and write in it.” Write “5” on a ticket and place it into the box at the top of the board. 3. Cut a long strip from the roll of adding machine tape and write “base 5” at the top. Place a unit bead in the units column of the board and write 1 on the strip. 4. Place another bead on the board and write “2” under the “1”. Continue until five beads are laid in the unit column. 5. Exchange the beads for a five bar and record “10”(read “one, zero”) on the strip. 6. Place a gold bead in the unit column and record “11”. Place a second and record “12”. 7. Continue until there are five beads in the unit column. Exchange for a five bar and record “20” on the strip. 8. Continue further until there are four unit beads and four five bars on the board. Exchange the beads for a bar, and exchange the bars for a square, recording “100” on the strip. Continue in the same manner, writing “101” for the next unit. 9. Encourage the child to work in different bases. You may wish to give two children different bases and run a race to see who can get to “1000” first.
Part B: Bases larger than 10 1. Note that you can only write a single digit in a particular place. 2. Place a ticket reading “12”into the box at the top of the base board. 3. Place beads to 9 on the board, recording them as in Part A. At ten, note that a single digit symbol is necessary for this quantity, since “10” has a different meaning in this base. Suggest “t” for ten and “e” for eleven. Ask what twelve will be (10). 4. Note that the last bead invites an exchange to the next level. Ask the child what we would normally exchange for (a bar). Acknowledge that there is no bar for twelve, invite the child to find an appropriate solution.
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34
Cubes
Squares
5
Bars
Units
use strip of paper base 5 1 2 3 4 5 10 1 2 3 4 5 20
Cubes
Squares
5
Bars
Units 10 20 30 40 50 100 200 300 400 500
1000
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35
B. Operations in Bases Part A: Addition 1. Review that when the child was in the primary classroom, learning to add, he used an addition chart. Show the child the base ten addition chart. 2. State that we can do the same thing in other bases. Suggest that they make their own so they can refer to it when adding in other bases. 3. Discuss with the child how to set the chart up, referring to the base ten chart. 4. First lay out the top and left sides of the chart. Then have the child fill in the chart by adding the numbers on the axes. Start with 1 + 1, write the answer (2) in the correct space. 5. Continue as above, picking numbers to add at random. 6. The complete board should look like the example on the page to the right: 7. Note that now you can add with any number in this base. 8. Point out that when working in a non-decimal base, first you must know what base you are working in. 9. You may either write the base at the top of the page if all the work is in the same base. Or you can write it to the lower right of the number - 12five 10. Propose the problem:
12five 13five 4five +3five
11. Begin by adding 4five and 3five to get 12five (from the chart). Add 3five to this for 20five. Add 2five to complete the column and get 22five. Have the child record 2five as the first digit of the answer and carry in their heads. 12. Add the mentally-carried-2 to the two 1’s in the second column for an answer of 42five.
12
Base 5 Addition Chart
1 2 3 4 10
Copyright 2009 | Jonathan Feagle www.FreeMontessori.org
1
2
3
4
10
2
3
4
10
11
3 4
4 10
10 11
11 12
12 13
10
11
12
13
14
11
12
13
14
20
36 base 10 addition chart 1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
11
2
3
4
5
6
7
8
9
10
11
12
3
4
5
6
7
8
9
10
11
12
13
4
5
6
7
8
9
10
11
12
13
14
5
6
7
8
9
10
11
12
13
14
15
6
7
8
9
10
11
12
13
14
15
16
7
8
9
10
11
12
13
14
15
16
17
8
9
10
11
12
13
14
15
16
17
18
9
10
11
12
13
14
15
16
17
18
19
10
11
12
13
14
15
16
17
18
19
20
Copyright 2009 | Jonathan Feagle www.FreeMontessori.org
37 Part B: Subtraction 1. Display the base ten subtraction chart, and note how the chart is set up. Set the numbers up in a similar fashion for a base five chart. 2. Have the child choose the numbers she wants to subtract. The child may want to lay the beads on the base board to solve the problem. Make sure she exchanges as necessary. 3. Continue working until the chart is completed. 4. Note that they are now prepared to do any subtraction problem in base five. Suggest the following:
24five -13five 11five
21five -3five 13five
123134five -23412five 44222five
Base 5 Subtraction Chart Minuend
1
Subtrahend
2 3 4 10
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1
2
3
4
10
0
1
2
3
4
0
1
2
3
0
1
2
0
1 0
38
minuend
Base 10 Subtraction Chart
Subtrahend
1
2
3
4
5
6
7
8
9
10
1
0
1
2
3
4
5
6
7
8
9
2
-1
0
1
2
3
4
5
6
7
8
3
-2
-1
0
1
2
3
4
5
6
7
4
-3
-2
-1
0
1
2
3
4
5
6
5
-4
-3
-2
-1
0
1
2
3
4
5
6
-5
-4
-3
-2
-1
0
1
2
3
4
7
-6
-5
-4
-3
-2
-1
0
1
2
3
8
-7
-6
-5
-4
-3
-2
-1
0
1
2
9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
Copyright 2009 | Jonathan Feagle www.FreeMontessori.org
39 Part C: Multiplication 1. Set up a chart for base five multiplication in a manner similar to the preceding. Randomly fill in the chart by working out the problems on the base board. 2. Have the child work out problems using the chart. She should carry in her head, not on paper. Suggest:
123five x 3five 424five
Base 5 Multiplication Chart Multiplicand
0
1
2
3
4
10
Multiplier
1
1
2
3
4
10
2
2
4
11
13
20
3
3
11
14
22
30
4
4
13
22
31
40
10
10
20 30 40 100
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40 Part D: Division 1. Look at the base ten division chart as a guide to making a base five division chart. 2. Note that you will need the products from the multiplication chart to start the division chart. 3. Point out that the product of 9 x 9 (81) is the largest digit on the base ten chart, and that the corresponding number in base 5 would be the product of 4five x 4five (31five). Start with this at the top of the chart. 4. Note that the products (quotients?) on the base ten chart descend to zero in order and write them as such on the base five chart. 5. Also, note that not all the squares are filled in - Only whole quotients are used. 6. Fill the chart randomly, checking your work against the multiplication chart. 7. Use the chart to solve problems in the same way as in base ten. Check by multiplying and adding the remainder. Suggest: (20344five ÷ 3five) 3244r2five 3244five 3five 20344five x 3five -14 20342five 13 + 2five -11 20344five 24 -22 2
base 5 division c hart 31
1
31
30
22
10 10
3
10
14
30 22 20 14
2
4
20
4
4
11
13
11
4
3
3
3 30
13
10
10
4 2
2 1
2 2
4
1 Copyright 2009 | Jonathan Feagle www.FreeMontessori.org
41
C. Conversion from One Base to Another Note: Children must understand that in different bases, the categories represent different powers. Part A: To convert a number from any base to base 10 1. “Suppose I wanted to know what a number in base five equaled in base ten. How could I figure that out?” 2. Propose the problem: 1432five = ____ten. Have the child lay the beads, bars, squares and cubes onto the base chart in the appropriate places. Note that unit beads are the same in either base. 3. Direct the child to the cube (53). State that it is 1 times 53, or 125. Record: 1 x 53 = 125 4. Direct the child to the squares. Ask what one is worth (52 or 25), and how many there are (4). 1 x 53 = 125 4 x 52 = 100 5. Direct the child to the bars. Determine that each bar represents 51 and that there are 3. Record 3 x 51 below the others. 6. Direct the child to the units. Ask how they may be expressed as a power of five. Record 2x50=2 below the others and add them together, putting the answer in the original equation: 1 x 53 = 125 4 x 52 = 100 3 x 51 = 15 2 x 50 = + 2 242
1432five = 242ten
7. Note that you wrote the number in expanded notation in order to make the transition from base five to base ten.
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42
Cubes
53 125
+
Squares
4 x 52 100
15
3 x 51
Bars
1432five = _______ten
5
+
+
Units
2 x 50 2
=
10
20
30
1 2
use strip of paper
400
300
200
100
base 5
1000
242ten
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43 Part B: To convert from base 10 to another base 1. Suppose I wanted to know what a number in base ten equalled in base four. How could I figure that out?” 2. Propose the problem: 54ten = ____four. Have the child place 5 ten bars and 4 unit beads on the base board. 3. Have the child exchange as many of the beads as he can into 4-bars (don’t have him exchange to squares or cubes yet). 4. “Let’s record what we did.” Write: 54 ÷ 4 = 13r2. 5. Have the child exchange the bars for squares and record what he did: 13 ÷ 4 = 3r1. 6. “Let’s see if we can exchange any more. OOPS, 3 ÷ 4 doesn’t work.”; state that you’ll take it one step further to show the fact that you can’t change any more. Write: 3 ÷ 4 = 0r3 Note that what you did was take out multiples of four, the remainders stayed behind. 7. Point out that you continue because what’s left over becomes the quotient (work to zero); the remainders are the digits of the preceding columns. The first remainder (2) is the number in the digits column, and the second remainder is in the second (41) column. 54ten = 312four 54 ÷ 4 = 13 r 2 13 ÷ 4 = 3 r 1 3 ÷ 4 = 0 r 3
Copyright 2009 | Jonathan Feagle www.FreeMontessori.org
(40) (41) (42)
44
54ten= _______four Cubes
Squares
Bars
Units
Exchange for 4 bars (54 ÷ 4 = 13r2) Cubes
Squares
Bars
Units
Exchange for 4-squares (13 ÷ 4 = 3r1) Cubes
Squares
Bars
Units
Can we exchange any more? What happens if we divide 3 by 4? = 0r3. 312four 54ten= _______ Copyright 2009 | Jonathan Feagle www.FreeMontessori.org
45 8. Have the child repeat the same procedure for: 6821ten = 100100122three 6821 ÷ 3 = 2273 r 2 2273 ÷ 3 = 757 r 2 757 ÷ 3 = 252 r 1 252 ÷ 3 = 84 r 0 84 ÷ 3 = 28 r 0 8 ÷ 3 = 9 r 1 9 ÷ 3 = 3 r 0 3 ÷ 3 = 1 r 0 1 ÷ 3 = 0 r 1
(30) (31) (32) (33) (34) (35) (36) (37) (38)
9. The child may check his answer by expanding it, as was done when converting from a given base to base ten: 1 x 38 = 6561 0 x 37 = 0 0 x 36 = 0 etc. Part C: Changing to bases larger than 10 1. Propose the problem: 6821ten = ____twelve. 2. Ask the child if he expects the base 12 number to be larger or smaller than the base 10 (smaller). 3. Work out the problem in the manner described above, substituting “t” and “e” for “10” and “11” as necessary.
6821ten = 3b45twelve
6821 ÷ 12 = 568 r 5 (120)
568 ÷ 12 = 47 r 4
(121)
47 ÷ 12 =
3 r b
(122)
3 ÷ 12 =
0 r 3
(123)
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46 Part D: Changing bases using the base chart Example I: 1. Propose the problem: 1432five = ____ten. 2. “Let’s do this one using the rule. It’s in base five right now, let’s change it to base ten.” 3. “We need to know how many groups of 10 there are in this base five number. We’ll find that out by dividing by ten. This chart will tell us what number ten is in base five.” Find the number you’re converting to in the base ten column and slide across to the base five column to see what its equivalent is (20five). 4. Divide 1432five by 20five, noting that the answer is in groups of ten. 20five
44 r 2 1432five -130 132 - 130 2
5. Continue to divide out the answers as demonstrated above:
1432five = 242ten
1432five ÷ 20five = 44 r 2 44five ÷ 20five = 2 r 4 2five ÷ 20five = 0 r 2
(100) (101) (102)
Example II: 1. Propose the problem: 1424five = ____four. Complete in the same manner described above.
1424five = 3233four
1424 ÷ 4 = 214 r 3 214 ÷ 4 = 24 r 3 24 ÷ 4 = 3 r 2 3 ÷ 4 = 0 r 3
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47 Example III: 1. Propose the problem: 1424five = ____seven. Complete in the same manner described above, noting that sometimes when changing to a larger base, the remainder may need to be changed. (7 = 12 in base five[10 (5) + 2])
1424five = 461seven
424five ÷ 12five = 114 r 1 114five ÷ 12five = 4 r 3 4five ÷ 12five = 0 r 4
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48 notes:
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49
IV Word Problems Contents A. Introduction to Word Problems B. Distance, Velocity and Time Presentation:
46 47
Passage One: Introduction p.47 Passage Two: Solving for Distance p.47 Level One Level Two Level Three Passage Three: Solving for Velocity p.49 Level One Level Two Level Three Passage Four: Solving for Time p.52 Level One Level Two Level Three
C. Principal, Interest, Rate and Time Presentation: Passage One: Introduction p.53 Passage Two: Solving for Interest p.54 Level One Level Two Level Three Passage Three: Solving for Rate p.56 Level One Level Two Level Three
Passage Four: Solving for Principal Level One Level Two Passage Five: Solving for Time p.62 Level One Level Two Level Three
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53
50
A. Introduction to Word Problems Word problems are an important aspect of the cosmic approach to mathematics. Just as the other mathematical areas are considered to have abstract and practical applications, so do word problems. It is best to introduce word problems through situations that arise in class. If, however, problems do not arise, some may be made up. The scope and variety of vocabulary used to describe operations in our language should be employed when doing word problems in class. It is in this way that the child comes to understand how all of these words describe the operations. Word problems should not be a constant feature in the classroom. They should be brought in on occasion. The first set of word problems should be coded as to what operation is required to complete it. After the children have explored these and are comfortable, these coded problems may be exchanged for uncoded problems. Later, problems that are uncoded and which involve mixed operations may be introduced. Further, when these problems are mastered, it is time to introduce more complex problems involving decimals, fractions and mixed operations. When teaching word problems, remember to solve them in a step-by-step fashion, ensuring that the child understands the method. First, read the entire problem, making sure the child knows all the words in it. Then, help the child to determine “what you know” by listing to the facts presented in the problem, “what do you want to know” by evaluating the request of the problem, and how to solve it (what operations on what numbers, etc.). The child should then carry out the work of the problem and check her work by asking herself if the answer seems right, given what is known, and what is to be discovered. The following section covers two types of word problems, those involving distance, time and velocity, and those involving interest rate, principal and time, leading to endeavors with formulas. Each type is presented in three levels. The first level is introductory, sensorial and may be presented around seven years of age. The second level leads to abstraction, and more precise identification of the problem’s request. Level two is presented between seven and eight years. The third level is abstract and presents the rule for the type of problem’s solution. It is presented around eight to nine years of age.
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51
B. Distance, Velocity and Time Materials: The golden bead material, a box of tickets containing one each for velocity, distance, time and their abbreviations (v, d, t), what is known?, what is wanted?, operation signs, blanks, two fraction bars, and pencils
Presentation: Passage One: Introduction 1. Set up a race: measure a straight course, set out start and finish lines, and have the children run. Record each child’s time and the distance of the race. 2. Make a chart of this information. As you do, mention that the straight course is a distance and that the children’s different foot speed affected the times. 3. Ensure the children understand the relationships between time, distance and speed, then introduce the terms, using “velocity” for “speed”. 4. Before beginning the problems, ask the children to remind you what the name of what they ran was (distance). Introduce the distance card. 5. Ask the children what the stopwatch recorded (how long, time). Introduce the time card. 6. Ask the children why some finished before others (they were faster). You could introduce velocity as the measurement of “fastness”. Introduce the velocity card, and all the abbreviation cards. Lay the abbreviations alongside the terms.
Note: Sometimes problems will arise from this discussion. Solve them first. Passage Two: Solving for Distance Level One 1. Propose the problem: “If a plane travels 500 miles per hour, how far will it travel in 3 hours?” 2. “What information does the problem give us?” Lay out the ticket reading “What is known?” 3. “We know the plane was going 500 miles per hour; what is that (velocity)?” Place the velocity card, then its abbreviation to the right of the “What is known?” card. 4. Write 500 on a blank ticket and place it to the right of the cards. 5. Determine what else is known (time) and place that below the velocity material. Finish the layout by placing the “What is wanted?” card below “What is known?” and place the distance card to the right with a question mark at the end:
Copyright 2009 | Jonathan Feagle www.FreeMontessori.org
52 Layout the cards thus:
6. Get out 5 hundred squares, noting that this is the velocity. State that the plane went that fast for 3 hours, and put out 3 skittles. 6. Place 5 hundred squares beneath each skittle. Add them together for the answer (1500 miles). 7. Write 1500 on a ticket, replacing the “?” with it. 8. Continue with other similar problems before advancing to Level Two.
Level Two 1. Have the child reconstruct the final layout from Level One. 2. Ask the child what operations were used on what numbers (multiplication). Lay these tickets out with operation cards.
3. Propose other problems and solve them in the same manner. Soon, the child won’t need the material at all.
Copyright 2009 | Jonathan Feagle www.FreeMontessori.org
53 Level Three 1. Propose the problem from Level One again. This time, ask the child how we might express the problem using the abbreviations. 2. “We want to know what the distance was.” Place out the “d” abbreviation card with an equal sign written on a ticket D
=
3. Ask what you did (multiplied velocity by time). Lay out these abbreviations:
4. Have the child solve the problem using the formula. Continue practicing on other problems Passage Three: Solving for Velocity Level One 1. Propose the problem:
Layout the cards thus:
“A plane travels 1500 miles in 3 hours. At what speed is it traveling?” 2. Talk through laying out the cards for what is known and what is wanted. 3. Place out 15 hundred-squares, stating that they are the distance. Placing out three skittles, state that this is the time. 4. Distribute the squares to the skittles. Note that the plane travels 500 miles per hour. Write a ticket to reflect this, and replace the “?” with it. 5. Note that you’ve solved for velocity.
Copyright 2009 | Jonathan Feagle www.FreeMontessori.org
54
Write ticket for the answer 500
Copyright 2009 | Jonathan Feagle www.FreeMontessori.org
55 Level Two 1. Have the child lay the cards, skittles and squares out in the manner above. 2. Ask the child what operation he used to perform the lay out (division). 3. Determine that you divided 1500 miles by 3 hours lay this out with the tickets: Level Three
1. Propose the problem from Level One again. Ask the child how we might express the problem using the abbreviations. 2. “We want to know what the velocity was.” Place out the “v” abbreviation card with an equal sign written on a ticket 3. Ask what you did (divided distance by time). Lay out these abbreviations:
4. Work out the problem on paper, plugging the facts into the formula.
Copyright 2009 | Jonathan Feagle www.FreeMontessori.org
56 Passage Four: Solving for Time Level One 1. Propose the problem: “A plane traveled 1500 miles at 500 miles per hour. How many hours did the plane fly?” 2. Lay out the cards and information as below: Lay out the cards thus -
3. Place out a stack of 15 hundred-squares. Count from this stack groups of five for each hour traveled. 4. Place a skittle on top of each stack. Count the skittles for the answer.
Level Two 1. Ask the child what you did. Note that you found out how many groups of 500 there were in 1500. 2. Note that you divided, and lay out the tickets as follows:
Copyright 2009 | Jonathan Feagle www.FreeMontessori.org
57 Level Three 1. Determine the formula, noting that you were searching for time. Lay out
Note that you divided the distance by the velocity: 2. Plug in the number and solve using the formula.
C. Principal, Interest, Rate and Time Materials: A box labeled “Interest” containing cards marked “What is known?”, “What is wanted?”, principal, rate, interest, time, P, T, R, I, 100, years, “?”, several marked “$”, and division bars.
Presentation: Note: This should be presented after the distance, time and velocity problems, and after the child understands fractions. Passage One: Introduction 1. Give an oral introduction stemming from an article read or a savings passbook. 2. Note that the money in the account is called the principle, and the bank uses it to loan to other people. They are using your money, so they pay you a certain amount. This is called interest. They charge the borrower interest on the loan they made as well. 3. To make the calculations easier, the bank pays you a certain amount based on every hundred dollars in the account for a specific period of time. This is called the rate. 4. Because the amount is paid for every hundred dollars, it is called a percent (per = for, and cent = hundred). 5. “One way to pay interest is every year. The interest is paid at a certain rate for every year the bank has your money. The amount of interest you money earns depends on the number of years it is left in the account.” 6. The term for which the money is left is called the time.
Copyright 2009 | Jonathan Feagle www.FreeMontessori.org
58 Passage Two: Solving for Interest Level One 1. Propose the Problem: “I left $1600 in the bank for 3 years. Each year the bank paid me $2 for every $100. How much did they pay me?” 2. 3. 4. 5. 6.
Lay out what is known and wanted as follows: Place 16 hundred-squares in a square on the rug to represent the principle. Place 3 skittles beside them to represent the years. Ask the child how much was paid for each $100 ($2). Place two beads on each of the hundred squares. Remind the child that $2 per $100 was paid for each year, but the money was in the bank for three years. State that what is laid out represents one year’s earnings. 7. Collect the beads from the hundred squares, exchanging as necessary. State that this is what you got in Layout the cards thus:
one year, but the money was there for three years. Place the 32 beads under one skittle. 8. As you place 32 beads in front of each of the other skittles, note that now you are putting out the total interest earnings. 9. Collect the beads and bars together and count them for the answer. 10. Replace the “?”card with a ticket reading $96.
Copyright 2009 | Jonathan Feagle www.FreeMontessori.org
59 Level Two 1. Ask the child what you did. Note that you took the principal times the rate. Lay it out:
2. “This told us how much I earned per year. Now we have to multiply by the number of years.” Lay it out:
3. Do the arithmetic for an answer of $96.
Level Three 1. Work out the formula, noting that you were solving for interest, and place out:
2. Ask the child what you did (took the principal times the rate then times time). Lay it out, noting that you donít write multiplication symbols between the letter in a formula:
Solve similar problems on paper using the formula.
Copyright 2009 | Jonathan Feagle www.FreeMontessori.org
60 Passage Three: Solving for Rate Level One 1. Propose the Problem: “I left $1600 in the bank for three years. The total interest I earned was $96. What was the rate of interest?” 2. Lay out the cards, talking through what is known and wanted:
Layout the cards thus:
3. Lay 16 hundred-squares into a large square. Set three skittles beside them: 4. Bring out 96 beads. State that for the rate, you must find out how much was earned each year. Distribute the beads evenly to each (32). 5. Take the beads from one skittle, exchange them, and distribute them one at a time to each of the 16 hundred squares. When finished, note that the rate is 2 for each 100, and that this can also be read as 2%. Replace the “?” card with a ticket reading 2.
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61
Passage Three
Then distribute the beads across the 16 hundred-squares (2 each).
Copyright 2009 | Jonathan Feagle www.FreeMontessori.org
62 Level Two 1. 2. 3. 4.
Ask the child what you did. You found how much you got for each year by dividing $96 by 3 years. Then, you divided that by the amount of money (96/3 ÷ 1600). “Let’s do the arithmetic. 96 divided by 3 is 32. Now 32 divided by 1600; let’s make this a fraction.” Set it up as a fraction and reduce terms to something over 100. 32/1600 = 2/100
Level Three 1. State that you needed to find the rate, and place:
2. Ask the child what you did in the problem (first, I ÷ t, then I/t ÷ p). Lay this out in cards:
3. Note that in formulas, you donít use division signs. You can invert the divisor and multiply:
4. Then, multiply the fractions to get:
Copyright 2009 | Jonathan Feagle www.FreeMontessori.org
63
Passage Four: Solving for Principal Level One 1. Propose the Problem: “I received $96 for money I left in the bank for 3 years at a rate of $2 per hundred. How much did I put in the bank?” 2. Lay out what is known and wanted as follows: Layout the cards thus:
3. “First, let’s figure out how much we got each year.” Take out 96 beads and distribute them to 3 skittles. Note that each skittle got 32 beads. 4. Saying “We only need to worry about one year’s interest to find the principal.” Put two skittles and their beads away. 5. “The rate is 2 per hundred or 2%. Let’s see how many groups of two we can make.” Make 16 pairs of beads, exchanging as necessary. 6. Match the pairs of beads to hundred squares, since each two beads represents a hundred deposited. 7. Count the hundred squares to get the answer (1600); Replace the “?” card with a ticket reading 1600.
Copyright 2009 | Jonathan Feagle www.FreeMontessori.org
64 Level Two 1. Ask the child what you did. Note that you took the interest ($96) and divided it by the number of years (3) to discover how much interest was earned each year. Lay this out in tickets:
2. Note that you took the yearly interest and divided by the rate (2/100) to learn the principal. Lay this out in tickets:
Do the arithmetic, invert and multiply, then divide for the principal (1600):
Copyright 2009 | Jonathan Feagle www.FreeMontessori.org
65 Level Three 1. “Let’s work out a formula for solving for principal.” Lay out the cards and signs, continue, “To get the principal, we first divided the interest by time, then we divided by the rate.”
2. “In a formula, we can’t have a division sign, so we’ll make the rate a fraction, invert it and multiply.”:
3. Multiply and alphabetize the terms to get the following. Use this formula to solve other problems for principal.
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66 Passage Five: Solving for Time Level One 1. Propose the Problem: “I left $1600 in the bank at a rate of $2 per hundred for 3 years, and the bank paid me $96. How long was my money in the bank?” 2. Lay out what is known and wanted as follows: Layout the cards thus:
3. Lay out 16 hundred-squares in a square to represent the principal. Noting that it is the interest - place out 96 beads. 4. State that for each $100 in the bank, you were paid a certain amount. Share out the interest beads until all are out. There should be 6 on each hundred square. 5. “We know that each $100 received $2 per year.” Put out a skittle, lay 2 beads beside it. 6. Continue with a second then a third skittle, noting that each represents a year of time. 7. Determine that there are 3 skittles representing 3 years. Replace the “?” card with a ticket reading 3.
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67 Level Two 1. Ask the child what you did. Determine that you divided 96 by 1600. Lay this out in tickets:
2. Note that you got an answer, then divided that by the rate (2/100 annually). Lay out:
3. “Let’s invert and multiply to find out how long the money was in the bank.”
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68 Level Three 1. “We were solving for time.” Lay out:
2. “We divided the interest by the principal, then by the rate.” Lay out:
3. Remove the ÷ by inverting and multiplying:
4. Multiply and alphabetize for:
5. Have the child use this formula to work out some problems on paper.
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V Ratio & Proportion Contents A. Ratio
66 Presentation: Introduction
Passage One: Introduction p.66 Passage Two: Introduction to the Language p.66 Passage Three p.68 Passage Four p.68 Passage Five: Exploring the Idea Arithmetically p.68 Passage Three: Passage Four: Passage Six: Ratios Written as Fractions p.70 Passage Seven: Stating the Ratio Algebraically p.71 Passage Eight: Word Problems p.72 Example A Example B Example B Algebraically Example C Example C Arithmetically Example C Algebraically
B. Proportion Presentation: Introduction
77
Exercise One: Determining if Something is in Proportion p.78 Exercise Two: Proportion Between Geometric Figures p.78 Exercise Three: With 3 Dimensional Figures p.79 Exercise Four: p.80
C. Calculations with Proportion Exercise One: Arithmetically p.81 Exercise Two: Algebraically (for older children) p.81 Exercise Three: Applications p.82 Example I Example II
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A. Ratio Introduction: length = 5 inches
Ratios compare objects. This comparison is one of division. Because of this, it is vital that the divisor and dividend be identified. For example, the ratio of length to width of a 3x5-card is 5 to 3, while its length to width ratio is3 to 5. One yields an answer greater than one, the other, an answer less than one. The child will have worked with ratios before this, comparing the unknown length of an object to a fixed length, width = 3 inches and investigating the relationship of pi to the radius and circumference of a circle. Materials: The geography stamps, various objects from the environment, the peg board and pegs, paper and pencils
Presentation: Introduction Passage One: Introduction 1. Place 2 green pegs across from 3 red pegs on the pegboard (bead bars may also be employed here). 2. “We have some pegs here. Let’s compare them. One way I can do this is to say that there are 2 pegs here to 3 pegs there.” 3. “There is another way of saying this. We could say the ratio of green pegs to red pegs is 2 to 3.” 4. Using the geography stamps, make illustrations of ratios. For example, a ratio of corn to wheat of 5 to 3. 5. Show the child how to write the ratio as such: 5 : 3. The child may wish to find and express ratios of objects in the environment. Passage Two: Introduction to the Language 1. After the child has had some experience, point out that the order in which the objects are stated is important. 2. Introduce the term antecedent (ante- meaning before) to describe the first term in the ratio. 3. Also introduce consequence to describe the second term in the ratio. 4. Note that the antecedent is the number to which the consequence is compared. Ratios are always stated antecedent to consequence. 5. Write 2 : 3. State that the green pegs are two thirds of the red ones, and this is what is meant when saying that the antecedent is compared to the consequence. 6. Switch the terms, and read the ratios of red to green as 3 : 2. Note that the red pegs are 1 ‡ times the green ones.
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72 Passage Three 1. “Let’s look at some other ratios.” Place two red pegs across from three green pegs. 2. State that you are going to do something else. Add two rows of each below the first. 3. Have the child examine the second group. Lead him to see that the ratio is still 2 : 3 even though there are more beads. 4. Have the child make another ratio using the whole quantity of pegs (6 : 9). 5. The child may notice that these are multiples and can be reduced to the first quantities. 6. Ask how the child would figure out if 23 : 36 had the same ratio as the ones you’d been working with. Lead the Note that if you multiply or divide both terms by the same quantity, the ratio will remain the same. Passage Four 1. “Let’s say that we have two numbers in a ratio of 2 : 4, and the smaller of the terms is 8, how can we find out what the other term is?” Write tickets for 2 and 4. 2. Have the child lay out 2 red pegs to 4 green pegs. 3. “We know the smaller number will be eight, which of these will be the smaller number (the red)?” 4. Lay out red pegs in groups of 2 beneath the first, as the child lays out corresponding groups of four green pegs. 5. Note that each line of the red pegs contains 2 while each line of green is four. Stop laying out pegs when 8 red is reached. 6. Count the pegs in each group, noting that 8 : 16 is the same ratio as 2 : 4. 7. Work through other examples, giving the ratio and one term and having the child find the second. 8. Later, the children may pose problems to each other. Passage Five: Exploring the Idea Arithmetically 1. Lay out 2 groups of pegs in the ratio 2 : 4 (perhaps 8 : 16). 2. “To reach 8 what did I do?” Lead the child to understand the smaller number was multiplied by something. 3. “What number did we multiply 2 by to get 8 (4)?” 4. “To keep the same ratio, we have to multiply the other side by 4 as well.” Write 2 :4 down the work you’ve done so far: 5. Repeat with other examples, working them out on the board and recording them.
2x4 : 4x4
8 : 16
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Passage Three:
Passage Four:
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74 Passage Six: Ratios Written as Fractions 1. 2. 3. 4.
Give two numbers in a ratio of 2 : 3 (6 : 9). Lay them out in number cards and pegs. “This ratio compares this number, the antecedent, to this number, the consequence.” “We can say something about the 6 in relation to the 9. We can say the 6 is 2/3 of the 9.” “Because it is a ratio, we can write it as a fraction.” Show the child how to write it:
5. Point out that the colon between the terms is like a division sign. 6. “Now we can write ratios in two ways. Let’s try some of each.” Have the child experiment with other ratios. 7. Demonstrate how to connect 2 : 3 to 6 : ?. Set it up as follows:
8. Have the child work it out:
9. Remind the child that she did the same thing when working with fractions.
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75 Passage Seven: Stating the Ratio Algebraically Note: Present this lesson after the child has been introduced to algebra. 1. Propose the same problem: two numbers are in a ratio of 2 : 3 and the smaller number is 6. 2. Set the problem up and solve it as follows: 3. Check as follows:
4. “What if the ratio was 3 : 2?” Set up 9 : 6 pegs on the board. 5. Ask the child what this ratio would be in a fraction (9/6). Ask what the 9 is compared to (6). 6. “If we were given this ratio, and we knew only that the larger number is 9, we could set up the problem in the same way.” Set it up: 7. Work it out algebraically:
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76 Passage Eight: Word Problems Example A 1. Propose the problem: “A man left some money when he died. It was to be distributed to his wife and only son in a ratio of 2:3. The wife received $2400, how much did the son get?” 2. The child can work it out with pegs and calculate:
3. Or she may attempt it algebraically:
4. Continue with other problems. Work up to more complicated ones.
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77 Example B 1. Propose the problem: “I’m giving you 2 numbers in a ratio of 4 : 7. The sum of the numbers is 55. What are the numbers?” 2. Lay out the ratio with pegs. Count them up to get 11, and place a ticket stating this to the right of the pegs. 3. Continue with a second row and a ticket reading 22. 4. Repeat until there are five rows which total fifty-five. 5. Count each group of pegs to arrive at 20 and 35. 11 6. “Let’s analyze arithmetically 22 what we did.” Record the 33 44 ratio as a fraction: 55
7. Note that you added 4 and 7 to get 11, and each time you placed a row, you used up 11 pegs. You continued until you reached fifty-five (5 rows). 8. “Is there another way we could have gotten to the answer?” Lead the child to see that you could divide 55 by 11 to get 5 and then multiply the terms of the fraction by 5 to get the answer: Repeat with other problems. Think them through arithmetically, then check with the peg board.
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78 Example B Algebraically 1. State that if the ratio is 4 : 7, we know that 4/7ths of the larger number is the smaller number. 2. Record:
3. “We also know that y + x = 55.” Work the problem out as follows:
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79 Example C
1. Propose the problem below and lay it out in pegs: “Here are 2 numbers in a ratio of 3 : 5. If I say the difference of the two terms is 10, what are the terms?” 2. Determine that the difference between 3 and 5 is 2. Continue to 6 : 10, noting that the difference is 4, closer, but not quite what we’re looking for. 3. Continue until 15 : 25 is laid out. Note that the difference between them is 10, and these are the numbers you are looking for. 4. Repeat with other problems.
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80 Example C Arithmetically
1. Discuss what was done arithmetically as you write the following: 2. Note that the difference between the ratio was 2, while the difference between the terms was 10. To discover what to multiply by, you divide 10 by 2.
Example C Algebraically 1. Set the problem up and work it out as follows:
2. To solve: (one way to do it)
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B. Proportion Introduction: Proportion is a statement in which 2 ratios are equal. If there are more than 2 ratios, the proportion is continued. The children have worked with both ratios and proportion; now it is brought to their consciousness. Prerequisite: Children should have studied ratio and be able to balance equations. Materials: Objects of the environment.
Presentation: Introduction 1. Have children make a ratio with the geography stamps e.g. 2 black sheep to 1 white sheep. 2. Ask children to do another which is in the same ratio e.g. 4 black sheep to 2 white. State, “Let’s write this.”
3. Note that writing these ratios as fractions, we can say they are equivalent to each other; ask why. 4. Talk through why pointing out when 2 ratios are equal this statement is said to be proportion and the numbers are said to be proportional or in proportion. 5. Ask what this means; note that it means the relationship is the same. Point out how this worked with the sheep - for every 2 black sheep there is one white sheep. 6. Children can investigate other proportions.
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82 Exercise One: Determining if Something is in Proportion 1. Place a one bar and a 5 bar on a mat; state that the beads are in a relation of 1 to 5. 2. Lay out next to this one-5 bar and five-5 bars; note that there is another relationship. 3. Ask if these are in proportion; write them out: 1 5 5 25 4. Ask if they are equal; note that they are because for every one on one side there are 5 on the other; they are in proportion.
1 5 5 25
Exercise Two: Proportion Between Geometric Figures 1. Present 2 geometric figures which are equal in proportion; state that you’ll compare these. 2. Note that these are 2 things to compare, the width and length of both. 3. Compare the widths first, measuring one against the other; note that the width of the small one fits in the width of the large one twice. 4. Note that you can write this as: 1 2 5. Do the same with the lengths noting that 2 of the smaller figure’s lengths fit in the length of the larger figure. 6. Record this next to the other ratio noting that they are the same and are equal therefore they are in proportion. 1 1 2 = 2 7. Repeat with two which are not in proportion; note that these were not similar explaining that to be similar they need to be in proportion. 8. Children can investigate with other metal inset pieces; introduce more than 2 as equal as continual proportion.
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83 Excercise Two
Exercise Three: With 3 Dimensional Figures 1. Lay out the pieces of the power of 2 cube; state that you want to find which ones are in proportion. 2. Note that the small yellow cubes are because they are the same; note how many dimensions it has (length, width and height). 3. Compare the cube to the next larger cube; write out the ratio of the length, width and height: L = 1:2 W = 1:2 H = 1:1 4. Note that the height is not the same ratio as the length and width so it is not in proportion. 5. Continue comparing the other pieces; build the whole cube and ask if it is in proportion to anything; note that all the cubes are proportional.
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84 Exercise Four: Materials: Peg board and pegs, white decimal cards 0 to 9. 1. 2. 3. 4. 5.
State that you have 2 ratios; 2:3 and 4:6; lay out the cards for these at the top of the peg board. Note that you have 5 pegs in a ratio of 2 to 3; place these under the 2:3 number cards in ratio. State that you have 10 pegs in a ratio of 4 to 6; lay these under the 4 to 6 number cards in that ratio. Ask if these are proportional; determine that they are. Ask if they can express both groups as the same ratio; show this by changing the 10 pegs laid out in a line of 4 and 6 to 2 lines of 2 and 3: 6. Note that if all the pegs look like they’re in the same ratio, they’re proportional.
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C. Calculations with Proportion Introduction: This section works with proportion arithmetically and algebraically. Exercise One: Arithmetically 1. State when you say 1/2 and 3/6 are in proportion, what it means is that 1/2 is equal to 3/6. 2. Record this; ask if it is true and if they can prove it to you. 3. Note that you can also determine if they are in proportion if the fractions are divided out and their quotients equal: 1 ÷ 2 = 0.5 3 ÷ 6 = 0.5 0.5 = 0.5 4. State that you want to look at these 2 numbers in a different way; to do so you’ll multiply the first side by 2. 5. Note if you do one side you have to do the other: 1/2 x 2/1 = 3/6 x 2/1 Work out:
1x2 = 3x2 2x1 6x1
6. State that you want to look at something; point out the 3x2/6x1, have the children look at the original problem; note where the numbers are (diagonal). 7. Note that the opposite numerators and denominators are multiplied. 8. Ask why we care if this happens; state that suppose we wrote this down: 6/9 = x/12 9. Note that to solve for x we can cross multiply: work out: 9x = 72, x = 8 10. Show how to check this: 6/9 = 8/12 6 x 12 = 9 x 8 72=72 ck 11. Note that because this is a cross pattern, it’s called cross multiplication. Exercise Two: Algebraically (for older children) 1. 2. 3. 4.
State that the pattern of cross multiplication works for those numbers, but what about others. Set up ratios: a/c = c/d Multiply this out: a/b x b = c/d x b, a = bc/d, multiply each side by d to get: ad = bc Note that you have the same pattern.
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86 Exercise Three: Applications Example I 1. Give the problem: “There’s a painting 4 feet by 6 feet high. This painting appears in a photograph 2 inches by 3 inches. The eye in the picture is 5 mm long; how long is the eye in the painting?” 2. Set up the ratio; one of the width of the picture to the width of the painting (2/48) as equal to the width of the eye in the picture to the width of the eye in the painting (5/x):
2/48 = 5/x (48 inches = 4 feet)
3. Work out as follows: 2x = 240, x = 120 4. Children can work on their own to make scale drawings Example II 1. Give the problem: “These is a recipe which calls for 9 teaspoons of pineapple juice mixed with 6 teaspoons of cranberry. It’ll taste like a drink made with 12 cups of pineapple and 8 cups of cranberry. Is this true? 2. Determine that it is by cross multiplying and arriving at 72 for each. 3. Continue that you only have 10 cups of pineapple juice; how much cranberry juice will be needed for the same taste? 4. Set up as: 9/6 = 10/x, 60 = 9x, x = 6 2/3 c. 5. Suppose you only needed 10 cups? Set up and work out:
x/y = 3/2
3/2y + y = 10 3y + 2y = 20 5y = 20 y=4 x=6
x + y = 10
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VI Algebra Contents A. Introduction to Algebra
84
Exercise One: Balancing an Equation p.84 Exercise Two: Balancing an Equation When Something is Taken Away p.85 Exercise Three: Balancing an Equation When Something is Multiplied p.85 Exercise Four: Balancing an Equation When it is Divided p.85
B. Operations With Equations
86
Exercise One: Addition p.86 Exercise Two: Subtraction p.86 Exercise Three: Multiplication p.86 Exercise Four: Division p.86
C. Algebraic Word Problems Example I: p.87 Example II: p.87 Example III: p.87 Example IV: p.88 Example V: p.88 Example VI: p.88 Example VI: p.89 Example IX: p.89 Example X: p.90 Example XI: p.90 Example XII: p.90
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A. Introduction to Algebra Introduction: This is a study of statements of equality; how to manipulate to get what you want from it. Children have worked with this, e.g. 2 + 2 = 4. Prerequisites: Children should have done word problems to the point where they can write formulas; they should have a good grasp on the operations of fractions and negative numbers. Materials: Felt mat, colored bead bars, box of white decimal cards, operation signs, brackets, division bars, pencil and paper. Exercise One: Balancing an Equation 1. Lay out the beads and operation signs as follows and have the child read what’s there: (5 + 3 = 8) ••••• + ••• = •••••••• 2. State that this is a statement of equality; read what it says again and ask if it’s true (yes). 3. Note that you can call this statement an “equation” (equa meaning equal) 4. Have the child watch, state that suppose I do this, add a plus sign and a 2 bead bar to the left side of the equation. 5. Note that it is not an equation any more, but watch; add a plus sign and 2 bar to the right side of the equation. 6. Ask what it says now (5 + 3 + 2 = 8 + 2) and if it’s true noting that now again it’s a statement of equality. 7. State that if you have something that is true and you do something to change the left side of the equation, you don’t have an equation unless we do the same on the right side. (restore the balance) 8. Children can work with these, one laying out an equation, another changing it then the first or a third child deciding what to do to balance it.
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89 Exercise Two: Balancing an Equation When Something is Taken Away 1. 2. 3. 4.
Lay out the original equation, 5 + 3 = 8; state let’s look at this again; note that it is equal and is an equation. Have the children watch, take the +3 away; note that it says 5 = 8, ask if this is true. Ask what you could do to balance it noting you could take away from the other side as well. Do the same on the right by writing a ticket for 3 and placing it with a minus sign to the right of the 8.
••••• = •••••••• - 3
5. State, Let’s do it; replace the 8 bar with a 5 bar; note that it is again true, the equation is balanced.
••••• = •••••
6. Children can practice with this; have pick an equation, so something to it to change it then restore it. Exercise Three: Balancing an Equation When Something is Multiplied 1. Note that there are other things that can happen to change the equation; can also multiply, 2. Set up the equation in the same way as the original adding a multiplication sign and ticket with a 2 on it after the 5 plus 3 which can be put in brackets:
(••••• + •••) x 2 = ••••••••
3. Ask if it is an equation (no); ask what can be done to balance it; child knows to do the same to the quantity. 4. Lay out a x2 in tickets after the 8 bar; have look again to see if it is really true; note it is equivalent, the equation is balanced. 5. Let children each have a turn and practice in the same way as the other exercises. Exercise Four: Balancing an Equation When it is Divided 1. 2. 3. 4.
Ask what else can be done (divide it); do so by placing a division bar under the 5+3 bead bars and a 2 under it. Note these are not equal any more; ask what to do; children should know to divide the 8 bar in the same way. Check to see if it is balanced. Two things should come to the children’s awareness: 1. what is to the left of the equal sign must be the same value as what is to the right of the equal sign to be an equation 2. if some operation is performed on one side to change it, it can be read as true by changing the other side in the same way.
••••• + •••/2 = •••••••• ( ••••••••/2)
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B. Operations With Equations Materials: Same. Exercise One: Addition 1. Place a problem out in beads as follows: ••••• + x = •••••••• 2. Have the children write this down on paper; note that you are saying it is an equation but you don’t have all the numbers. 3. State that you want to find out what the mission number is; ask the child if she knows. 4. If the child does, state that let’s try to find it out if we didn’t know what it was. 5. Note that if we took the +5 away we could do it, but that wouldn’t be right; ask the child what could be done. 6. Lead the child to see if you take 5 away from the other side it would work; do so to get x = 3. 7. Note that we know that but were able to find out by balancing. 8. Have check it on paper; substitute 3 for the value of x and do the arithmetic to check it.
1. 2. 3. 4. 5.
Exercise Two: Subtraction Lay the problem out: x - 6 = •••• Have the child write this on paper; ask the child what can be done. Lead the child to see that + 6 can be added to both sides so that it cancels out on the left side and is added to the 4 bead bar on the right: x = •••• + •••••• Add the bead bars to get x = 10; substitute 10 for x on paper and work out. Of child knows her negative numbers well, she can take a minus 6 from each side with -6 becoming a plus 6.
Exercise Three: Multiplication 1. 1. Lay out the problem and have the child write it: 2 x = •••••• 2. Ask what should be done; lead child to see that both sides need to be divided by 2; set out: 2x/2 = ••••••/2 3. Work out so that x is equal to a 3 bar; have the child plug in 3 for x on paper and work out to confirm.
1. 2. 3. 4. 5.
Exercise Four: Division Note that you could also have a division problem; set up as follows: x/2 = •••• Ask what could be done; note that both sides can be multiplied by 2. Lay out tickets for x 2 on each side of the equation: x/2 x 2 = •••• x 2 Work out plugging in 8 for x on paper. Children can switch to all tickets early on; they can set up problems for each other. Copyright 2009 | Jonathan Feagle www.FreeMontessori.org
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C. Algebraic Word Problems Example I: 1. Give the problem: “Find the number 1/5th of which is equal to 15” 2. Note that you represent the number not known with a letter e.g. “x”. 3. Form an equation with what is known about “x”: 1/5x = 15 4. Note that his can be written: x/5 = 15 5. Work out by multiplying both sides by 5 to find that “x” is equal to 75. 6. Divide 75 by 5 to check; it should be equal to 15. Example II: 1. Give the problem: “What is the width of a rectangle that has an area of 1200 square feet and a length of 40 feet?” 2. Bring out that the child knows A = b x h; note and write out in this case it is A = lw. 3. Plug in what is written to have the problem read: 1200 = 40w ; record this under A = lw. 4. Note that you have to find w, can do this by dividing by 40: work out and check as follows:
A = lw 1200 = 40w 1200/40 = 40w/40 30 = w
1200 = 40 x30 1200 = 1200
Example III: 1. State problem: “Find the principal when the interest for one year amounts to $20 at 5 %.” 2. Set up the formula I = Prt; plug in what is known, work out and check as follows: I = Prt 20 = P x 0.05 x 1 20 = 0.05 P Check: Copyright 2009 | Jonathan Feagle www.FreeMontessori.org
20 = 400 x 0.05 x 1 20 = 20 ck.
20/0.05 = P, $400 = P
92 Example IV: 1. State the problem: “If 15 is added to twice a certain number the result will be equal to the number increased by 23. What is the number?” 2. Help the child to set up and solve the problem as follows:
15 + 2x = x + 23 (-15) + 15 + 2x = x + 23 + (-15) (-x) 2x = x + 8 (-x) x=8
3. Plug in the answer to see if it works; should come out to 31 = 31. Example V: 1. Give the problem: “The area of a rectangle is 200 square feet, the length is twice as much as the width. Find the dimensions.” 2. Again help the child set up and work out:
A = lw 200 = lw 200 = (2w) w 200 = 2w2 100 = w2 10 = w
(1 = 2w)
(1 = 20)
Example VI: 1. Present the problem: “Find the 3 dimensions of a rectangle prism with the volume of 500 cubic feet, the length and height each being twice the width.” 2. Set up and work out:
500 = lxh 500 = 2w + 2w + w 500 = 4w3 125 = w3 (w = 5 l = 10 h = 10)
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93 Example VI: 1. Give a similar problem: “The same man and boy work together for 2 days then the man leaves the boy to do the remaining part. How long does it take?” 2. Talk through that the boy does one section a day and the man does 3 so 4 are completed in a day. 3. Note that they work together for 2 days so 8 sections are completed; subtract this from how many all together to find 4 more need to be done. 4. Because there are 4 more to be done and the boy can only do one section a day, it will take 4 more days to complete. Example IX: 1. State the problem: “A garden is divided into 12 parts. A boy and a man dig it and the man works twice as fast as the boy. If each works for 2 days, the work is finished. See how many each digs in a day.” 2. Write out the know information, let: b = # of parts boy digs m = # of parts man digs m = 2b 3. Note that the man digs twice as much as the boy and they dig 12 sections in all; set up and work out: 2m + 2b = 12 2(2b) + 2b = 12 6b = 12 b = 2 parts m = 4 parts 4. Plug in to check: (2 x 4) = 2(2) = 12 8 + 4 = 12 12 = 12ck.
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94 Example X: 1. Relate the problem: “There is a water tank and it’s filled by 2 pipes from the top (a + b). When it is needed, there is a drainage pipe c. If the tank starts off empty, if “a” opens, it takes 8 hours to fill, “b” takes 12 hours to fill, “c” can drain the tank in 6 hours. If a,b, and c are open, how long does it take to fill?” 2. Set the problem up so that x = the number of hours to fill the tank; set up and work out as follows: x/8 + x/12 - x/6 = 1 3x/24 + 2x/24 - 4x/24 = 1 x/24 = 1 3. Determine that x = 24 hours to fill the tank. Example XI: 1. Give the problem: “A laborer does a job in 15 days, another does it in 10 days. How long does it take if they work together?” 2. Figure as follows:
x/15 + x/10 = 1 x = 6 days 2x/30 + 3x/30 = 1
5x/30 = 1 5x = 30
Example XII: 1. Give this problem: “A women bought some prized live turkeys and ducks. She spent $31 and each turkey cost $5, each duck $2. How many of each did she buy?” 2. Let t = # of turkeys, d = # of ducks; set up so that 5t + 2d = 31. 3. Try different values of t to find three possible solutions: t = 1 t = 3 t=5 5 + 2d = 31 15 +2d = 31 25 + 2d = 31 2d = 26 2d = 16 2d = 6 d = 13 d = 8 d=3 4. Note the three possibilities: 1 turkey, 13 ducks 3 turkeys, 8 ducks 5 turkeys, 3 ducks Copyright 2009 | Jonathan Feagle www.FreeMontessori.org
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