Guide to the Final Exam
- Be sure you can create diagrams (area, chip, bar, number line) representing the following arithmetic operations, and that you are able to write a step-by-step paragraph-long grammatically accurate and cogent description of the entire algorithm used in the operation utilizing key words.
- Addition, subtraction, and multiplication of whole numbers
- Be sure you can distinguish and be able to create word problems according to the interpretations:
- Set Model (for addition and subtraction)
- Set Model: Part-Whole Interpretation (for subtraction)
- Set Model: Take-Way Interpretation (for subtraction)
- Set Model: Comparison Interpretation (for subtraction)
- Measurement Model (for addition and subtraction)
- Measurement: Part-Whole Interpretation (for subtraction)
- Measurement: Take-Way Interpretation (for subtraction)
- Measurement: Comparison Interpretation (for subtraction)
- Rectangular array model (for multiplication)
- Ordinal / Number Line Model (for addition and subtraction of integers)
- Addition and subtraction of abstract even and odd numbers
- Multiplication of whole numbers, and fractions
- Division (partitive and measurement) of whole numbers and fractions
- Be sure you can create a word problem representing a particular type of division, then provide a teacher's solution of this problem
- Be sure you can describe carefully and methodically using grammatical English a problem involving long division
- Addition and subtraction of decimal numbers (Sections 9.1)
- Be sure you can solve a problem using a chip diagram
- Addition, subtraction, and multiplication of Integers (Section 8.1, 8.2)
- Be sure you can depict and solve an addition and subtraction problem using the number line model, and black/white chips
- Be sure you can explain why the product of a negative number and a positive number is negative.
2. Be sure you understand place-value and can demonstrate this by:
- Adding, subtracting, and multiplying in base 5, 10 and 60
- Utilizing chip-diagrams to perform operations in base 10
- Utilizing the lattice method of adding in base 10
3. Be sure you can prove basic arithmetic theorems. See guide to Exam III.
4. Be sure you can provide a teacher's solution for a wide-variety of problems including:
- Basic one or two-step problems (Section 2.2)
- Percent problems using three methods of solution (See Section 7.3)
- Ratio problems (Section 7.2)
- Operations involving fractions (Sections 6.1-6.5)
5. Be sure you have understood the Introduction, and Ch. 1 of Liping Ma's book and can answer the basic question that appeared on previous guides and exams.
6. Be sure you have a solid grasp of basic arithmetic operations, including:
- Order of operations problems involving integers
- Complex arithmetic operations involving fractions
- Determining the LCM, GCF, and Prime Factorization of integers
- Divisibility criteria for 2, 3, 5, 9, and 11
- The procedure for determining whether a number is prime.
On Subitizing from Where Mathematics Comes From by George Lakoff
Introduction to Knowing and Teaching Elementary Mathematics by Liping Ma
Chapter 1 of Knowing and Teaching Elementary Mathematics by Liping Ma
Chapter 1 of Parker and Baldridge - Elementary Mathematics for Teachers
Solutions to Exercises in our Book on CSUN's website for the Course (FREE)
Detailed solutions to exercises in our book on Chegg ($14/month)
Homework Due 3/25
1. Multiply the following numbers in expanded form:
(a) $25\times 37$
(b) $752\times 248$
2. Multiply the following base-5 numbers using the algorithm (no need to use expanded form).
(a) $(34)_5 \times (24)_5$
(b) $(143)_5 \times (234)_5$
3. Multiply the following base-60 numbers using the multiplication algorithm:
(a) $1,7 \times 2, 9 $
(b) $3, 9 \times 4, 10$
(c) $25,23,49 \times 35$
4. Divide the following numbers in expanded form:
Example: $593 \div 3 =(5\times 100+9\times10+3)\div 3 =(3\times100+2\times100+9\times10+3)\div 3=$
$(3\times100+20\times10+9\times 10+3)\div 3=(3\times 100+29\times10+3)\div 3=$
$(3\times100+27\times10+2\times10+3)\div 3=(3\times100+27\times10+23)\div 3=$
$(3\times100+27\times10+21+2)\div 3=\frac{3}{3}\times100+\frac{27}{3}\times 10+\frac{21}{3}+\frac{2}{3}=$
$1\times100+9\times10+7 R2=197 R2$
(a) $262 \div 2$
(b) $74 \div 2$
(c) $778 \div 3$
5. Complete reading Chapter 1 of Liping Ma's Knowing and Teaching Elementary Mathematics
6. Complete the following homework sets:
Homework Set 10: Question 1-5
Homework Set 11: Questions 1-9
Homework Set 12: Questions 1-8
Homework Set 13: Questions 1-3
7. Read Sections 3.5, 3.6 in our textbook
Homework Due April 8
Page. 76 #4 through 7.
Homework Set 14: #1, 3, and 6
Homework Set 15: #1a,b,c,d,e,f., #3, 4
Homework Set 16: #3, 4, 5, 6, 7, 9, 11
Homework Set 17: #3, 4, 7, 9, 10, 11.
Homework Due April 29
1. Use a (a) diagram proof, and (b) an algebra proof Prove that the sum of an even number and an odd number is an odd number
2. Prove using Algebra only that the product of two odd numbers is again an odd number.
3. Prove that if the ones digit of a number n is either 0 or 5 then 5|n. You may assume that n is a 3-digit number.
4. Prove that if the last two digits of a number (ones and tens place) is divisible by 4, then the number is divisible by 4.
5. Prove the divisibility test for 9 for 4 digits numbers. On other words, prove that if the sum of the digits of a number
is divisible by 9, then the number is divisible by 9.
6. Prove that a number n has an even number of factors unless it is the square of a whole number. (Hint: Each factor a
has a "partner" b=n/a and a and b are different unless... unless what?)
7. Prove that the sum of any three consecutive numbers is divisible by 3. Hint: Call the first number x.
Homework Set #20 (p. 117)
#1, 2
Homework Set 21 (p. 121)
#1, #2,
Read about the Sieve of Eratosthenes on pages 118 and 119 and answer question 7 in homework set 21
Homework Set 22 (p. 124)
#1, 2, 3 (important), 6.
Now read the proof of Theorem 4.5, and answer questions 7, and 8.
Read section 5.5, and as much of Ch. 6 as you can.
Homework Due May 6
Homework Set 23 (p. 130): #1-6 (all), 11
Homework Set 24 (p. 137): #1, 3, 6
Homework Set 25 (p. 142): #1, 3, 4, 9
Homework Due May 20
Homework Set 26 (p. 149): #10
Homework Set 27 (p. 154): #2, 3, 4, 5, 6
Homework Set 28 (p. 159): #5
Homework Set 30 (p. 159): #1, 2
Homework Set 31 (p. 159): #5
Guide to Exam II
- Be able to explain the algorithmic procedure involved in subtracting, say by utilizing mathematical terms in a grammatically, and logically correct manner.
Avoid using such algorithmic and conceptually impoverished phrases as “borrow a 1 from the tens place”, “the 3 becomes 13”, “cross out the 2 and write 1”
Be sure to use the following key terms:
digit, number, minuend, subtrahend, difference, decompose/decomposition, higher-value unit, tens place, ones place.
Sample Answer: To subtract 18 from 23 we first notice that since we must subtract 8 ones from 23, and that we only have 3 ones, we must first decompose the higher-value unit from the tens place of the minuend. We take a unit from the digit 2 in the tens place of 23, and decompose it into 10 ones, therefore converting 23 into 13 ones and 1 ten. We now subtract the 8 ones of the subtrahend from 13 ones of the minuend, obtaining 5 ones. We write this digit 5 in the answer box in the ones place. We now have 1 ten in the minuend, and 1 ten in the subtrahend. Subtracting these ones we obtain 0 for the tens place of the difference. Therefore, the difference of 23 and 18 is 5.
- Be sure you can tell the difference between a digit and a number. A number is composed of the digits 0 through 9. There are ten digits, but infinitely many numbers. All ten of these digits are also numbers, but obviously the rest of the numbers are not digits. When you are referring to the digits 0 through 9 when talking about other numbers, please do not call them numbers, but digits.
- Be able to perform the three mathematical operations of addition, multiplication, and subtraction, using the chip model diagrams.
- Be sure you can perform all arithmetic operations (except division) in base 5, and in base 60.
- Be able to perform basic arithmetic operations and correctly use the “=” sign in doing the steps. Be sure to perform only one operation between each equal sign. For example:
$2-3+6\div2-1=-1+6\div2-1=-1+3-1=2-1=1$
$(12-2·5+3)\div5=(12-10+3)\div5=(2+3)\div5=5\div5=1$
- Be able to explain the division algorithm in dividing a 3-digit number by a 2-digit number and utilizing the following terms: divisor, dividend, quotient, estimate, remainder, digit, decompose, difference/subtract, product, ones, tens
- Be able to provide a teacher’s solution for the exercises in Primary Math 5A Problems 28-32 on page 63-64, and Problems 9, 16-18 for pages 89-90. See exercises 4, and 5 in Homework Set 9.
- Be able to perform addition, and multiplication using the lattice algorithm. See sections 3.1, and 3.3
- Be sure to be able to provide a teacher’s solution for Problems 8 and 9 of Practice 2B of Primary Math 4A (See exercise 6 of Homework Set 12 for the sample solution). Note that you cannot use algebra in providing the teacher’s solution.
- Be able to provide a range estimate in a word problem. See examples 5.4, and 5.5 in section 3.5.
- Be able to distinguish between an algebraic variable being used as:
(1) metavariable as part of an algebraic identity such as $x^2-y^2=(x+y)(x-y)$
(2) as part of a geometric, or physical formula such as area, parameter, or interest
(3) as representing a specific but unknown quantity such as when we assign x to represent an unknown quantity in a word problem.
- Be sure you can identify the various Arithmetic Properties (on p. 96) of section 4.2, and be able to justify them using a geometric, or set diagram.
- Any previous homework problems may show up on the exam.
- Be sure you have read the first chapter of Liping Ma, and understand the various issues discussed, including:
- Multiple Ways of Regrouping as discussed by several Chinese teachers. Keep in mind that the subtraction algorithm classically requires only one kind of regrouping such as when subtracting 53-26 we regroup 53 into 40+13 so that we can then do 40-20 and 13-6. However, as Liping Ma discussed, Chinese teachers also consider others forms of regrouping.
- The notion of decomposing a unit of higher value. This is the notion that I would like you to use instead of “borrowing” from now on.
- The notion of “rate of composing a higher value unit”. This is the idea that we can group 5 ones into one 5 (as in base 5) or sixty 1s into a single unit representing 60 (as in base 60). However, in the Hindu-Arabic number system, this rate of composing is simply 10. Be sure to read the section of the book on this topic.
- Be sure you understanding the three levels of understanding subtraction as discussed on p. 15:
Level 1: Understanding subtraction where the minuend is between 10 and 20.
Level 2: Problems with minuends between 19 and 100, where a higher value unit must be decomposed where there already are several such higher value units
Level 3: Problems with larger minuends, especially when decomposition must occur “across 0” such as when subtracting 205-19. In this problem we must decompose a unit which represents 100 into ten 10s first, then decompose one of those 10s into ten 1s to be able to then subtract 15-9. Be sure you can explain this procedure!
- Be sure you can distinguish between procedural understanding vs. conceptual understanding of subtraction as discussed on pp. 22-23.
Guide to Exam III
- Be sure you can provide (A) Algebraic, and (B) Diagrammatic proofs of the following facts:
- Sum of an even number and an odd number is an odd number
- The sum of two even numbers is even
- The sum of two odd numbers is even
- Be sure you can provide an algebra-based proof of the following theorems:
- The product of an even number and an odd number is an even number
- The product of two odd numbers is odd.
- If the ones digit of a number n is either 0 or 5 then 5|n. You may assume that n is a 3-digit number.
- If the last two digits of a number (ones and tens place) is divisible by 4, then the number is divisible by 4.
- If the sum of the digits of a number is divisible by 9, then the number is divisible by 9. (You may assume that the number has only 4 digits)
- Prove that a number n has an even number of factors unless it is the square of a whole number. (Hint: Each factor a has a "partner" b=n/a and a and b are different unless... unless what?)
- There are infinitely many primes.
- Be sure you can determine whether a specific number is:
- Divisible by 2, or 3, or 5, or 9, or 11.
- Is Prime
- Be sure you understand the fundamental theorem of Arithmetic, and can write it in your own words.
- Be sure you can prime factorize large numbers such as 234235 using a calculator.
- Be sure you can find the GCF, and LCM of given two numbers utilizing the following methods:
- Listing the multiples, or factors, then using the least or greatest common
- Using the column method of listing the prime factorization
- In the case of GCF using Euclid’s Algorithm and in the case of LCM utilizing the formula on p. 128.
- Be sure you can provide a teacher’s solution along with a picture or diagram of problem 6 on p. 137 of our textbook.
- Be sure you can identify whether a problem is using measurement division or partitive division such as in problem #11 on p. 149.
- Be sure you can explain the diagrammatic meaning behind finding the least common multiple of two denominators when adding two numbers with unlike denominators.
- Be sure you can illustrate using diagrams the meaning behind multiplication by fractions when multiplying a whole number by a fraction, or multiplying a fraction by a fraction.
- Be sure you can illustrate using a bar diagram several division problems involving fractions.See question 2 on p. 154 of our book as a guide.
Guide to Exam IV
- Be sure you understand partitive and measurement division, and that you can:
- Understand all cases of division, including Whole$\div$Whole; Whole$\div$Fraction and Fraction$\div$Whole and Fraction$\div$Fraction
- Depict a division problem using a diagram.
- Create a word problem illustrating the division.
- Solve the word problem using a teacher’s solution.
EXAMPLES
Problem Type 1: Whole$\div$Whole (Divisor is smaller than Dividend)
Partitive Division: 6$\div$2
Here the question is asking: If 6 units is divided into 2 parts, how much is in each part?
Word Problem: Juana wants to split her 6 apples between herself and her friend. How many apples would each of them get?
Answer: 6$\div2=3$ apples
Teacher's Solution:
The number of units in half of 6 units is 3 units.
Measurement Division: 6$\div$2
Here the question is asking: If 6 units are separated into parts
so that there are 2 units in each part, how many parts do we have?
Word Problem: Juana wants do make small candy boxes with 2 candies in each box. How many boxes can she make if she has only 6 candies?
Answer: 6$\div2=3$ boxes
Teacher's Solution:
The number of 2-unit parts within a 6 unit whole is 3.
Problem Type 2: Whole$\div$Whole (Divisor is larger than Dividend)
Partitive Division: 3$\div$4
Here the question is asking: If three units are divided into four parts, what is the size of each part?
Word Problem: Suppose three large slices of cake are to be divided among four people. What is the size of the slice that each would get?
Answer: Each would get 3/4 of a size of a slice, or equivalently, if the slices were split into four pieces each, obtaining 12 smaller pieces. Each person would get 3 pieces.
Teacher's Solution:
If each unit is divided into four smaller units, there will be a total of
12 small units. Dividing these 12 small units into four equal parts
gives us 3 small units. So, each person would get three-fourths of a
size of a slice, or equivalently, each person would get 3 small slices.
Measurement Division: 3$\div$4
Here the question is asking: What fractional portion of 4 is 3?
Word Problem: If Juan cut 3 feet of ribbon from an original roll of
4 feet of ribbon, which fraction of the total length of the ribbon did
Juan cut?
Answer: Juan cut three-fourths of the original ribbon.
Teacher's Solution:
Problem Type 3: Whole$\div$Fraction (Divisor is smaller than Dividend)
Partitive Division: 6$\div\frac{1}{3}$
Here the question is asking: one-third of which number is 6? Or
equivalently, if 6 is the fractional (one-third) portion of a number,
what is this number?
Word Problem: Jenny used one-third of her long ribbon to make a large bow. If she used 6 feet of ribbon to make the bow, how long was her original ribbon?
Answer: 6$\div\frac{1}{3}=18 feet$
Teacher's Solution:
1 unit = 6 feet
3 units = 18 feet
Measurement Division: 6$\div\frac{1}{3}$
Here the question is asking: How many one-third size units are there within
a 6 unit piece?
Word Problem: Helena had 6 cakes, and she wanted to prepare desert plates,
each having a slice the size of one-third of a cake. How many such plates could
she prepare?
Answer: 6$\div\frac{1}{3}=6\times3=18$ plates
Teacher's Solution:
If 6 cakes are divided into 3 slices each, how many 1/3-size slices are
there in total?
Answer: 18
Problem Type 4: Fraction$\div$Fraction (Divisor is smaller than Dividend)
Partitive Division: $\frac{3}{5}\div\frac{1}{3}$
Here the question is asking: $\frac{1}{3}$ of what amount is $\frac{3}{5}$?
Since the divisor is supposed to represent a fractional part, we can think of this
problem as $\frac{3}{5}=\frac{1}{3}\times x$ where $x$ is the quotient.
Word Problem: Jose cut out one-third of a long wooden plank. After measuring
he realized that the piece he cut was three-fifths of a yard. How long was the
original wooden plank?
Measurement Division: $\frac{3}{5}\div\frac{1}{3}$
Here the question is asking: How many $\frac{1}{3}$-size units are there within
$\frac{3}{5}$-size unit. In other words, how many one-third yards are there
in three-fifths of a yard? This question differs considerably from the previous one
since within the measurement division interpretation the divisor represents the
magnitude of the part, and not the fractional part of a larger magnitude.
Word Problem: Juana had $\frac{3}{5}$ quart of punch concentrate. If the recipe
requires $\frac{3}{5}$ quart of punch per 10 people, how many people could Juana
entertain with the punch?
Answer: There are $\frac{3}{5}\div\frac{1}{3}$
$\frac{3}{5}\times\frac{3}{1}$
$=\frac{9}{5}=1 \frac{4}{5}$
So, Juana can make punch for $(1+\frac{4}{5})\times10 = 10+8=18$ people.
2. Be sure you can solve all five types of percent word problems in three different ways.
Types of Percent Problems:
Type 1: Part is Missing
Example: What is 25% of $120?
Type 2: Whole is Missing
Example: 35% of what number is 4?
Type 3: Percent is Missing
Example: What percent of 60 is 10?
Type 4: New is Missing
Example: Jose's average grade was 70%. His grade this semester is 25% higher. What is his current grade?
Type 5: Old is Missing
Example: Helena currently earns $10/hour. This is 10% higher than her previous salary. What was her previous salary?
Methods of Solving Problems:
Method 1: Proportion
Example: If Jose paid $27 tax for an iPad, and the tax rate it 9%. What was the sale price of the shirt?
Proportion Method Solution:
$$\frac{9}{100}=\frac{27}{x} \longrightarrow 100\times27=9x \longrightarrow x=\frac{2700}{9}=$300$$
Method 2: Translating into Linear Equations
Example: 35% of what number is 72?
Equation Solution: $0.35x=72 \longrightarrow x=\frac{72}{0.35}= 205.7$
Method 3: Diagram and Unitary Method
Example: 7% of what number is 28?
7% -> 28
1% -> 4
100% -> 400