Problem Solving Strategies

Instructional Objective(s)
The students will be able to:

1. Select and then apply appropriate strategies to solve a problem from visual (draw a picture or diagram, create list, table or graph, act it out, use manipulatives, use spatial reasoning); numerical (guess and check; look for a pattern); symbolic (write an equation or number sentence, working backwards) perspectives.

1. Use the Processes of Mathematics.

1. Sample Problem (Look for a Pattern): When Mae joined the chorus she knew four songs. At the end of the first week, she knew five songs. After the second week, she knew seven songs, and at the end of third week she knew ten songs. If she continues to learn at this pace, howmany songs will she know after 12 weeks? Find the pattern. (82 songs)
   
1. Make a sum of 100, using some eights (8's) with some plus (+) signs inserted. Example : 888+88+8+8+8=1,000. Using only 8's andplus and minus signs, how many ways can you discover to make 1,000?

1. Problem Solver - 6/7
   
   

1. Lenchner CPS
   
   

1. Math on Call (476-494)
   
   
   
1. Figure This! Math Challenges For Families
   
   
1. Mathematics Problems and Warm-ups
   
   
1. Math Problems
   
   
1. Math Forum's Problems of the Week
   
   
1. Guess The Number
   
   
1. Villainy, Inc.- Thwarting World Supremacy Through Mathematics
   
   
1. Fill and Pour
   
   
1. Sense and Dollars
   
   
1. 3-D Tic-Tac-Toe
   
   

Number Relationships and Computation (Place Value and Number Theory)

Suggested Timeline:

Instructional Objective(s)
The students will be able to:

1. Read and write whole numbers to billions and decimals through ten-thousandths for any stated place value.

1. Use no more than 3 decimal places 0-100.(5)

1. 27.693 is twenty-seven and six hundreed ninety-three thousandths.

1. G2 (44)
   

1. M+7 (H-2)
   
   
1. Math on Call (18)
   

1. G2 Transitions (5-6, 13-14)
   

2. Compare and order or describe whole numbers and decimals with or without relationship symbols (<, >, =, ≠).

2. Use no more than 4 decimals with no more than 3 decimal places and numbers 0 to 100. (5)

2. 0.737 > 0.732
   
2. 42 < 30 + 5

2. G2 (44)
   
   
2. G2 Transitions (5-6, 13-14)
   

3. Round whole numbers and decimals for any stated place value.

3. Round to the bolded place value.  327 = 330.
   
3. Round to the nearest tenths.  2.34 = 2.3
   
3. Round to the nearest whole number.  15.789 = 16

Alert:  Unlike whole numbers, there are no additional zeros at the end when rounding decimals.

3. G2 (47)
   
   
3. M+7 (8)
   
   
3. Math on Call (21)
   
   

4. Express whole numbers in expanded form using powers of ten and exponential notation.

4. Use exponential form with powers of 10 (0-10,000). (6)

4. 273,217 = (2x10⁵) + (7x10⁴) + (3x10³) + (2x10²) + (1x10¹) + (7x10⁰)

4. M+7 (H2)
   
   
4. Math on Call (15)
   
   
4. M+7 (Enrich 1.8)

5. Calculate powers of whole numbers and square roots of perfect squares of whole numbers.

5. Use no more than 3 exponents for integers (–10 to 20) or square roots of perfect square whole numbers (0 – 100).   (No Calculators)  (7)

5. 8² = 8 x 8 = 64 
   so the square root of 64= 8
    

5. G2 (17, 410)
   
   
5. M+7 (28, 344)
   
   
5. Math on Call (82, 83, 540)
   
   

6. Estimate the square root of a given number and justify in writing.

6. Use whole numbers (0-100). (8)

6. Estimate the square root of 22.  Since 22 is not a perfect square, but it is located in between the two perfect squares of 16 and 25.  So the square root of 22, is between 4 and 5.  Since 22 is closer to 25 and 16, the square root of 22 is closer to 5 than 4.
   

6. G2 (415)
   
   
6. M+7 (345)
   
   
6. Math on Call (78)

7. Identify and describe the characteristics of numbers divisible by 2, 3, 4, 5, 6, 7, 9, 10, 11, and 13.

7. Use rules for 2, 3, 5, 9, or 10 and whole numbers 0 to 10,000. (5)

7. Divisibility Rule for 7:  Take the last digit, double it, and subtract it from the rest of the number; if the answer is divisible by 7 (including 0), then the number is also.   

   Divisibility Rule for  11:  Alternately add and subtract the digits from left to right.  If the result (including 0) is divisible by 11, the number is also.  Example: to see whether 365167484 is divisible by 11, start by subtracting: 3-6+5-1+6-7+4-8+4 = 0; therefore 365167484 is divisible by 11.  
   
   Divisibility Rule for 13:  Delete the last digit from the number, then subtract 9 times the deleted  digit from the remaining number. If what is left is divisible by 13, then so is the original number.

7. G2 (133, 169)
   
   
7. M + 7(76)
   
   
7. Lenchner CPS (102, 106)
   
   
7. Math On Call (69)
   
   
7. Divisibility Tests
   

8. Identify and describe numbers as prime and composite. 

8. Use whole numbers 0 to 100. (5)

8. Prime numbers have exactly 2 factors, 1 and the number itself. 
   
   Composite numbers  have more than 2 factors.
   
   Alert:  The number “1” is neither prime or composite since it only has one factor, the number itself.

8. M+7 (78)
   
   
8. Math on Call (58, 542)
   
   
8. Nimble With Numbers 6-7 (39-40)
   
   
8. Virtual Manipulative:Sieve of Erastothenes
   
   
8. Interactive 100 Square
   
   
8. Big Primes

9. Identify factors and prime factors using factor trees and prime factorization in exponential form.

9. The prime factorization of 72 = 2x2x2x3x3= 2³x 3²

9. M+7 (78, 80)
   
   
9. Math on Call (56, 61, 67)
   
   
9. Nimble With Numbers 6-7 (43)
   
   
9. Exponents
   
   

10. Compare and contrast factors and multiples.

10. Factor is one of two or more expressions that are multiplied together to get a product.
    
10. A multiple of a number is the product of that number and any other whole number. Zero is a multiple of every number.
    

10. Nimble With Numbers 6-7 (41-42)
    
    
10. Math+6  (C12)

11. Solve problems using factors and multiples.

11. There are 50 people in a 5K road race.  Every 6th finisher gets a T-shirt and every 8th finisher gets a hat.  Use multiples and common multiples to determine who gets a T-shirt and hat.

11. Lenchner CPS (82-92)

12. Evaluate numerical expressions using order of operation involving whole numbers, fractions, and/or decimals.

12. Use no more than 4 operations (+, -, x, ÷ with no remainders) and 1 set of parentheses or a division bar (0 – 100). (No Calculators)  (6)
    
12. Use no more than 4 operations (+, -, x, ÷ with no remainders) and 1 set of parentheses, brackets, or a division bar, with whole numbers (0 – 200), fractions with denominators as factors of 100  (0 – 100), or decimals with no more than three decimal places (0 – 100). (No Calculators)  (7)

12. Order of operations is a set of rules that tell you the order in which to compute.
    - Compute inside the parentheses.
    - Do powers or roots.
    Alert:
    - Multiply or divide in order from left to right.
    - Add or subtract in order from left to right.
    (Mnemonic device:
    Please Excuse My Dear Aunt Sally!)
    6 + 4 x 3 ÷ 6
    4 x 3 = 12, 12 ÷ 6 = 2, 6 + 2 = 8
    
    3,710 + (12 -  2)
    3,710 + 10 = 3,721
    
    12 ÷ 4 x (3 + 2)²
    3 + 2 = 5 and 5² = 25
    12 ÷ 4 x 25 and 3 x 25 = 75

12. G2 (8, 18)
    
    
12. M+7 (144)
    
    
    
12. Number Sense 6 (127)
    
    
12. Math on Call (207)
    
    
12. Nimble With Numbers 6-7 (12-13,19)
    

12. Order of Operations
    
    
12. Order of Operations with Exponents
    

Number Relationships and Computation (Whole Numbers and Decimals)

Instructional Objective(s)
The students will be able to:

1. Estimate and calculate sums and differences of whole numbers and decimals, including money using appropriate method of computation (mental mathematics, use of a calculator, use and discovery of alternate algorithms).

1. Add decimals (including money):  Use no more than 4 addends and no more than 3 decimal places in each addend and numbers 0 to 1,000. (5)
   
1. Subtract decimals (including money):  Use a minuend and subtrahend with no more than 3 decimal places and numbers 0 to 1,000. (5)

1. Add decimals in vertical and horizontal formats using various place values.
   
   
1. 2,127 —> 2,000
   + 997 —> 1,000
                   ≈ 3,000
   
   
1.  16.175 —> 16.2
   -11.613 —> 11.6
                 ≈   4.6
   
   
1. When you add 0.2344 + 0.838, will the sum be more or less than one whole?
   Explain how you know.

1. G2 (48, 50, 56, 66, 562)
   
   
1. M+7 (12, 20, 24)
   
   
   
1. Math on Call (96, 102, 125, 131, 152, 158, 179, 184)
   
   
   
1. G2 Transitions(7-8, 15-16)
   
1. Arithmetic Four
   
   
1. Decimal Addition

2. Estimate and calculate products of whole numbers and decimals including money.
   

2. Multiply decimals:  use a decimal in monetary notation by a single-digit whole number and numbers 0 to 100. (5)
   
2. Multiply decimals:  use a decimal with no more than 3 digits multiplied by a 2-digit decimal 0 to 1000. (6)

2. G2 Transitions (9-10, 19-20)
   
   
2. g2 (56-59, 61-63)
   
   
   
2. Nimble with Numbers 6-7 (20-26)
   
   

3. Estimate and calculate quotients using whole numbers and decimals with whole number and decimal divisors.

3. Estimation of Products and Quotients:  Use a decimal with no more than a 3 digits multiplied by a 2-digit whole number, or the quotient of a decimal with no more than 4 digits in the dividend divided by a 2-digit whole number (0 – 1000) (6)
   
3. Divide whole numbers:  use a dividend with no more than 4 digits by a 2-digit divisor and whole numbers 0 to 9,999. (5)
   
3. Divide Decimals: Use a decimal with no more than 5 digits divided by a whole number with no more than 2 digits without annexing zeros  (0 – 1000). (No Calculators) (6)
   
   

3. 
   
3. Compatible number refers to a numbers that can be used in place of the dividend and divisor to make the division easy.  Numbers used are close to the numbers given.  The dividend is a multiple of the divisor.  Example: 63.1 ÷ 16.8 ≈ 64 ÷ 16 ≈  4

3. G2 (64-69)
   
   
3. G2 Transitions (9-10, 19-20)
   
   
3. Nimble with Numbers 6-7 (24-26)
   
   
   
   

4. Divide a decimal by a decimal and annex zeros in the dividend.
   

4. G2 (66)
   
   
   
4. M+7 (24)
   
   
   
4. Math on Call (186)
   
   
4. Dividing Decimals

5. Divide using short division, when appropriate.
   
   

5. M+7 (24)

6. Interpret quotients and remainders mathematically and in the context of a problem.
   

6. Use a dividend with no more than 3 digits by a 1- or 2-digit divisor and whole numbers 0 to 1,000. (5

6. Add the remainder:  Each picnic table seats 8 people.  There are 52 people.  How many picnic tables do you need?
   Drop the remainder:  Movie tickets cost $7.00.  How many can be purchased with $25?  (3)

6. Math On Call (182)
   
   
   

7. Apply identity, zero, commutative, associative, and distributive properties.

7. Properties of Addition and Multiplication:  Use the commutative property of addition or multiplication, associative property of addition or multiplication, additive inverse property, the distributive property, or the identity property for one or zero with whole numbers 0 – 100.  (7)

7. 2(7+9) = 2 x 7 + 2 x 9
   
   =14 +18
   
   =32
   

7. G2 (301)
   
   
   
7. M+7 (148, H4)
   
   
   
7. Math On Call (212)
   
   
7. Investigating The Identity, Inverse, Commutative and Associative Properties
   
   

8. Calculate equivalent units of length, capacity, and mass within the metric system.

8. 2.7 cm = 27 mm

8. 3200 mL = 3.2 L
   
   Alert:  Review use of metric measurement tools if necessary.

8. G2 (74)
   
   
   
8. G2 Transitions (35-36)
   
   
8. Converting Metric Units Game

9. Solve problems involving sums, differences, products, and quotients including area and perimeter of problems.

9. Perimeter: Use polygons with no more than 8 sides and whole numbers (0 –500) or a closed figure on a grid (0 – 50).  (5)
   
9. Missing Dimensions : Find length in a square and  rectangle, given the perimeter with whole number dimensions (0 – 200). (6)
   
9. Area: Use rectangles and whole numbers (0 – 200) or a closed figure on a grid (0 – 50).  (5)

9. If the perimeter of a rectangle is 26 inches, and the width is 4 inches, what is the length?
   4 + 4 + L + L = 26
   8 + 2L = 26
   2L = 18                       
   L = 9 inches
   

9. If the area of a rectangle is 20 cm² and the width is 4 cm, what is the length?
   area = length x width
   20cm = 4cm x n
   n = 5cm

9. Lenchner CPS (127, 132)

10. Express whole numbers in scientific notation.

10. Scientific Notation: Use exponential notation or scientific notation from (-10,000 to 1,000,000,000) (8)

10. A form of writing numbers as the product of a power of 10 and a decimal number greater than or equal to 1 and less than 10.

          8,905,000 = 8.905 x 10⁶

10. G2 (77)
    
    
    
10. Math First H
    
    

1. Identify, extend, analyze, and create numeric patterns and sequences.

1. Use whole numbers with no more than 2 decimal places 0 to 1,000. (5)

1. 
   
   1.  Draw Stage 4 and Stage 5. 2.  Find a pattern to predict the number of squares needed for higher stages.3.  Give a rule that will work for any stage of the H pattern
   
1. 2, 7, 17, 37, __, __, __, (Rule:  2n + 3)
   
1. Solve for x.  
   
   3.5, 3.8, 4.1, x, 4.7,
   
   x = 4.4

1. M+7 (174, 460)
   

1. Navigations 6: Algebra(7)
   
   
1. Attribute Trains
   
   
1. Completing the Pattern
   
   

2. Identify and extend arithmetic and geometric sequences.

2. Arithmetic Sequences:  Provide the nth term no more than 10 terms beyond the last given term using common differences no more than 10 with integers (-100 – 5000). (8)
   
2. Geometric Sequences:  Provide the nth term no more than 5 terms beyond the last given term using the recursive relationship of geometric sequences with a common ratio of whole numbers no more than 5 (0 – 10,000). (8)

2. An arithmetic sequence is a list of numbers that continually increase or decrease by the same amount using addition or subtraction. The rule is linear.
   
   2, 6, 10, 14, …  (Rule is +4.)
   
3. A geometric sequence is a list of numbers that continually increase or decrease by the same amount using multiplication or division. The rule is not linear.  It is a constant ratio between consecutive terms.
   
   32, -16, 8, -4, … (Rule is x-0.5.)

2. G2 (142)
   
   
   
2. Lenchner CPS (72)
   
   
   
2. Analyzing Numeric and Geometric Pattern
   
   
   

3. Complete and extend one- and two-operation function tables.

3. Complete a function table with a given rule with two operations (+, -, x) using whole numbers no more than 20 in the rule (0 – 500) (7)

3. A function is a relation in which every value of x has a unique value y.
   
   

3. G2 (254)
   

4. Analyze and describe the relationship that generates a two-operation rule.

4. To find the rule when given consecutive numbers in the input (x) column, calculate difference between the output (y) numbers. The difference becomes the coefficient of x. In this example, it is 4x.   4 times 1 is 4.  How do you get from 4 to 7?  Add 3.  Complete rule is 4x + 3.  Does this work for each input?  Yes.
   
   

4. Math on Call (236)
   
   
   
4. Navigations 6: Algebra (13)
   

5. Determine rule for a given function table involving 1 or 2 operations and write the rule in algebraic form.

5. Use 2 operations =, -, x, and whole numbers 0 to 100. (5)
5. Use whole numbers or decimals (+, -, x, ÷) with no more than two decimal places (0 – 10,000). (6)

5. To find the rule when given non-consecutive numbers in the input (x) column, calculate difference between the y values (in this example 35 – 23 = 12).  Calculate the difference between the x values (in this example, 20 – 14 = 6).  Dividing the y difference by the x difference (12 ÷ 6 = 2) will  give you the coefficient of x. Replace x with a value in which y is known.  Example:  2(14) = 28.  How do you get 23 from 28?  Get it by subtracting 5.  Rule is 2x –5.  Test it again with the other known value of x and y.  2(20) = 40 –5 = 35.
   
   

5. Navigations 6:Algebra (9,13)
   
   
5. Function Machine
   

6. Create one- and two-operation function tables to solve real world problems.

6. You and your sister want to make cookies for your math class.  She needs at least 48 cookies and you want 60 for your class.  One roll of cookie dough makes 12 cookies.  Make a function table that will tell you how many rolls of dough to buy for both of you.
   
   Answer:  5 rolls
   
   

6. Navigations 6:Algebra (41,44)
   
   
   
6. Math on Call (234)
   

7. Explore number theory within patterns such as triangular numbers, Pascal’s triangle, and the Fibonacci sequence.

7. Fascinating Triangular Numbers
   
   

7. Pascal’s Triangle and its patterns
   
   
   
7. Fibonacci Numbers and Nature