TEKSING TOWARD TAKS
Scope and Sequence Six Weeks #3
Grade 5 Mathematics
LESSON
TEKS/LESSON CONTENT
LESSON 1
5.2C/compare two fractional quantities in problem-solving situations using a variety of methods,
including common denominators
5.3B/use multiplication to solve problems involving whole numbers (no more than three digits times
two digits without technology)
5.4/use strategies, including rounding and compatible numbers to estimate solutions to
multiplication problems
5.3C/use division to solve problems involving whole numbers (no more than two-digit divisors and
three-digit dividends without technology)
5.4/use strategies, including rounding and compatible numbers to estimate solutions to division
problems
5.5B/identify prime and composite numbers using concrete objects, pictorial models and patterns in
factor pairs
5.8A/sketch the results of reflections on a Quadrant I coordinate grid
5.8B/identify the transformation that generates one figure from the other when given two congruent
figures on a Quadrant I coordinate grid
5.10B/connect models for volume with their respective formulas
5.10C/select and use appropriate units and formulas to measure volume
LESSON 2
LESSON 3
LESSON 4
LESSON 5
LESSON 6
LESSON 7
LESSON 8
LESSON 9
Assessment
5.12C/list all possible outcomes of a probability experiment such as tossing a coin
5.13A/use tables of related number pairs to make line graphs
5.13C/graph a given set of data using an appropriate graphical representation such as a line graph
5.14C/Select or develop an appropriate problem-solving plan or strategy, including drawing a
picture, looking for a pattern, systematic guessing and checking, acting it out, making a table,
working a simpler problem, or working backwards to solve a problem
Review lessons and Mini-Assessments to prepare for Six Weeks #1 Assessment
PARENT NOTES:
TEKSING TOWARD TAKS 2008
GRADE 5 MATHEMATICS
(5.2) Number, operation, and quantitative reasoning. The student uses fractions in problem-solving situations. The student is expected
to: (C) compare two fractional quantities in problem-solving situations using a variety of methods, including common denominators.
Parent Guide
Six Weeks 3 – Lesson 1
For this TEKS students should be able to use fractions in problem solving situations. The focus for
this lesson is comparing two fractional quantities in problem-solving situations using a variety of
methods, including common denominators.
MATH BACKGROUND
Understanding How to Compare and Order Fractions With Like Denominators
Sometimes pictures can be used to compare fractions with like denominators.
1
2
of a personal size pizza he ordered. Sally ate
of a personal size pizza
4
4
she ordered. Who ate more pizza, Ben or Sally?
EXAMPLE: Ben ate
Write:
2
1
>
4
4
Say: two fourths is greater than one fourth.
Sally ate more pizza than Ben.
NOTE: When two fractions are compared, the wholes must be congruent.
EXAMPLE: If Ben ordered a small pizza, and Sally ordered a large pizza, you could not
compare the fractions of the pizzas they ate because the wholes are not the same.
Comparing fractions with the same denominator is a lot like comparing whole numbers you are
comparing which is more of something and which is less of something.
TEKSING TOWARD TAKS
2008
Page 1
GRADE 5 MATHEMATICS
(5.2) Number, operation, and quantitative reasoning. The student uses fractions in problem-solving situations. The student is expected
to: (C) compare two fractional quantities in problem-solving situations using a variety of methods, including common denominators.
To compare fractions that have the same denominator, compare their numerators.
EXAMPLE: Compare the fractions
3
5
and .
6
6
Both fractions have the denominator 6.
Compare the numerators to find how many parts are in each fraction.
Since 3 < 5, then
3
5
< .
6
6
It is also true that 5 > 3, so you can also say that
5 3
> .
6 6
9
7
mile to school. Sheri walks
mile to school. Who walks a
10
10
greater distance to school?
EXAMPLE: Jorge walks
When fractions have the same denominator, compare the numerators.
Since 9 > 7, then
9
7
>
.
10 10
Jorge walks the greater distance to school.
Understanding How to Compare Fractions With Unlike Denominators
Sometimes pictures can be used to compare fractions with different denominators.
EXAMPLE: Which fraction is greater,
Write:
1
1
or ?
4
6
1
1
>
4 6
Say: one fourth is greater than one sixth.
TEKSING TOWARD TAKS
2008
Page 2
GRADE 5 MATHEMATICS
(5.2) Number, operation, and quantitative reasoning. The student uses fractions in problem-solving situations. The student is expected
to: (C) compare two fractional quantities in problem-solving situations using a variety of methods, including common denominators.
EXAMPLE: Which fraction is less,
Write:
2
2
or ?
8
4
2
2
<
8
4
Say: two eighths is less than two fourths.
Number lines can be used to compare fractions with denominators that are not the same.
Compare the fractions to benchmarks on the number lines.
EXAMPLE: Adrian runs
3
3
mile. Erin runs
mile. Who runs farther?
4
10
1
2
0
3
4
1
3 > 1 because
2
4
3 is more than half of 4.
3 < 1 because
10 2
3
10
0
1
2
1
3 is less than half of 10.
Adrian runs farther than Erin.
Equivalent fractions can be used to compare fractions with denominators that are not the same.
To compare fractions that have unlike denominators, rewrite them as equivalent fractions with a
common denominator. A common denominator is one that is the same in two or more
fractions.
To find a common denominator for two fractions, use the least common multiple of the two
different denominators. A multiple of a number is the product of the number and another whole
number.
EXAMPLE: 12 is a multiple of 3. The number 12 can be written as the product of the number 3
and the number 4: 12 = 3 4.
The least common multiple of two numbers is the smallest number that is in both lists of
multiples. To find the least common multiple of two numbers, list the multiples of both numbers.
Identify the smallest number found in both lists.
TEKSING TOWARD TAKS
2008
Page 3
GRADE 5 MATHEMATICS
(5.2) Number, operation, and quantitative reasoning. The student uses fractions in problem-solving situations. The student is expected
to: (C) compare two fractional quantities in problem-solving situations using a variety of methods, including common denominators.
EXAMPLE: Find the least common multiple of 3 and 5.
Multiples of 3 are: 3, 6, 9, 12, 15 , 18, …
Multiples of 5 are: 5, 10, 15 , 20, …
The smallest number that is in both lists of multiples is 15.
The least common multiple of 3 and 5 is 15.
Guidelines for Comparing Fractions That Have Different Denominators:
First find a common denominator for the two fractions.
Rewrite each fraction as an equivalent fraction with the common denominator.
Compare the numerators of the two rewritten fractions.
3
5
pound of peanuts. Beau has pound of peanuts. Who has the
4
6
greater amount of peanuts?
EXAMPLE: Jackson has
Find the least common multiple of 4 and 6.
Multiples of 4 are: 4, 8, 12, 16, …
Multiples of 6 are: 6, 12, 18, 24, …
The smallest number that is in both lists of multiples is 12.
The least common multiple of 4 and 6 is 12.
The least common denominator is 12.
Rewrite
3
5
and
to have 12 as their denominators.
4
6
3
?




4
12
4



3



12
3



3
9




4



3
12
3
9




4
12
5
?




6
12
6




2



12
5



2 10




6




2
12
5
10




6
12
Compare the numerators of the two rewritten fractions.
10 > 9
Since 10 > 9, then
5
3
> . Beau has the greater amount of peanuts.
6
4
TEKSING TOWARD TAKS
2008
Page 4
GRADE 5 MATHEMATICS
(5.2) Number, operation, and quantitative reasoning. The student uses fractions in problem-solving situations. The student is expected
to: (C) compare two fractional quantities in problem-solving situations using a variety of methods, including common denominators.
Confirm the comparison of the fractions using a model.
Shade and label a sketch below to represent the common denominators and compare the two
fractions.
Since
3
4
=
9
12
5
6
=
10
12
3
9
=
and
4
12
5
10
5
3
=
, then
>
.
6
12
6
4
.
The model confirms that Beau has the greater amount of peanuts.
TEKSING TOWARD TAKS
2008
Page 5
GRADE 5 MATHEMATICS
(5.3) Number, operation, and quantitative reasoning. The student adds, subtracts, multiplies, and divides to solve meaningful problems.
The student is expected to: (B) use multiplication to solve problems involving whole numbers (no more than three digits times two digits
without technology)
(5.4) Number, operation, and quantitative reasoning. The student estimates to determine reasonable results. The student is expected
to use strategies including rounding and compatible numbers to estimate solutions to addition, subtraction, multiplication, and division
problems.
Parent Guide
Six Weeks 3 – Lesson 2
For this TEKS students should be able to multiply to solve meaningful problems. The focus for this
lesson is using multiplication to solve problems involving whole numbers and using strategies
including rounding and compatible numbers to estimation solutions to multiplication problems.
MATH BACKGROUND
Understanding Multiplication
Multiplication is a shortcut for adding same-size groups. Two terms in multiplication are factor
and product. The factors are the numbers being multiplied. The product is the result of the
multiplication.
Use multiplication when you want to combine groups of equal size.
EXAMPLE: Your family is going to recycle the cans you have been collecting. There are 268
pounds of cans waiting to be turned in for 12¢ per pound. Adding 12¢ 268 times would take a very
long time, so using multiplication is a much faster process.
Understanding How to Multiply by a 1-Digit Number
If you know how to multiply 1-digit numbers such as 6 x 7, you can also multiply larger numbers
such as 6 x 777.
When one of the factors is larger than 10, and one of the factors I less than 10, there are two
procedures you can use to multiply:
List the partial products and then add.
Use what you know about regrouping.
EXAMPLE: A box of special order pencils with a school name and logo contains 777 pencils. How
many pencils are in 6 boxes?
To solve the problem, multiply 777 by 6.
TEKSING TOWARD TAKS
2008
Page 1
GRADE 5 MATHEMATICS
(5.3) Number, operation, and quantitative reasoning. The student adds, subtracts, multiplies, and divides to solve meaningful problems.
The student is expected to: (B) use multiplication to solve problems involving whole numbers (no more than three digits times two digits
without technology)
(5.4) Number, operation, and quantitative reasoning. The student estimates to determine reasonable results. The student is expected
to use strategies including rounding and compatible numbers to estimate solutions to addition, subtraction, multiplication, and division
problems.
Multiply by listing all the partial products, then adding the partial products.



777







6





42 Multiply the ones 6 x 7 ones = 42



420 Multiply the tens 6 x 7 tens = 420
4,200 Multiply the hundreds 6 x 7 hundreds = 4,200
4,662 Add the partial products. 42 + 420 + 4,200 = 4,668
There are 4,662 special order pencils in 6 boxes.
Multiply without listing the partial products. Use regrouping.
44





777








6

4,662
Multiply the ones 6 x 7 ones = 42 ones
Regroup: 42 ones = 4 tens + 2 ones
Write 2 in the ones place. Write 4 above the tens place.
Multiply the tens 6 x 7 tens = 42 tens
Regroup: 42 tens = 420 = 4 hundreds + 2 tens
Add the 2 tens to the 4 tens you already have 2 tens + 4 tens = 6 tens
Write 6 in the tens place. Write 4 above the hundreds place.
Multiply the hundreds 6 x 7 hundreds = 42 hundreds
Regroup: 42 hundreds = 4,200 = 4 thousands + 2 hundreds
Add the 2 hundreds to the 4 hundreds you already have 2 hundreds + 4 hundreds = 6 hundreds
Write 6 in the hundreds place. Write 4 in the thousands place.
There are 4,662 special order pencils in 6 boxes.
Using either procedure for multiplying by a 1-digit number, there are 4,662 special order pencils in
6 boxes.
Understanding How to Multiply Two-Digit Numbers
When both of the factors are larger than 10, there are two procedures you can use to multiply:
Multiply the value of each digit in one factor by the value of each digit in the other factor.
List the partial products and then add.
Use what you know about regrouping.
Break apart one of the factors before multiplying.
TEKSING TOWARD TAKS
2008
Page 2
GRADE 5 MATHEMATICS
(5.3) Number, operation, and quantitative reasoning. The student adds, subtracts, multiplies, and divides to solve meaningful problems.
The student is expected to: (B) use multiplication to solve problems involving whole numbers (no more than three digits times two digits
without technology)
(5.4) Number, operation, and quantitative reasoning. The student estimates to determine reasonable results. The student is expected
to use strategies including rounding and compatible numbers to estimate solutions to addition, subtraction, multiplication, and division
problems.
EXAMPLE: The school auditorium has 14 rows. Each row has 28 seats. How many seats are in
the auditorium?
To solve the problem, multiply 14 by 28.
Multiply the value of each digit in one factor by the value of each digit in the other factor.
List the partial products and then add.
Tens
TO




14

Ones



28 Multiply by the ones.
8 x 4 ones = 32



32



80
8 x 10 ones = 80



80

200
Multiply the tens.
20 x 4 tens = 80
20 x 10 tens = 200

392
Add the partial products 32 + 80 + 80 + 200 = 392
There are 392 seats in the auditorium.
Multiply without listing every partial product. Use what you know about regrouping.
TO
3




14



28
112

Multiply by the ones 8 x 14 ones = ?
8 x 4 ones = 32
2 ones with 3 tens to regroup
8 x 10 ones = 80
8 tens + 3 tens = 11 tens
So, 8 x 14 = 112
3




14


28
112

280
Multiply by the tens 20 x 14 tens = ?
20 x 4 ones = 80
8 tens + 0 ones
20 x 10 ones = 200
2 hundreds
So, 20 x 14 = 280.
392
Add the partial products. 112 + 280 = 392
There are 392 seats in the auditorium.
TEKSING TOWARD TAKS
2008
Page 3
GRADE 5 MATHEMATICS
(5.3) Number, operation, and quantitative reasoning. The student adds, subtracts, multiplies, and divides to solve meaningful problems.
The student is expected to: (B) use multiplication to solve problems involving whole numbers (no more than three digits times two digits
without technology)
(5.4) Number, operation, and quantitative reasoning. The student estimates to determine reasonable results. The student is expected
to use strategies including rounding and compatible numbers to estimate solutions to addition, subtraction, multiplication, and division
problems.
Break apart one of the factors before multiplying.
Break apart one factor into numbers that are easy to multiply.
14 x 28 = (10 + 4) x 28
Multiply.
10 x 28 = 280
4 x 28 = 112



112

280
Add the two products.



392
There are 392 seats in the auditorium
Using any of the 3 multiplication procedures for multiplying two-digit numbers, there are 392 seats
in the auditorium.
NOTE: Zeros may seem like “nothing” in a factor or product, but they are very important.
EXAMPLE: The website http://users/htcomp.net receives an average of 305 visits per week. At
this rate, about how many visits would the website receive in 4 weeks?
To find the answer, multiply 305 by 4.
HTO
2




305
4 x 5 = 20
2 tens + 0 ones
There are no tens in 305, but that does not mean we
can forget about the tens.








4


1,220

4 x 0 = 0 tens
0 tens + 2 tens = 2 tens
4 x 300 = 1,200 tens
1 thousand + 2 hundreds
At this rate, the website would receive about 1,220 visits in 4 weeks.
Understanding How to Check Multiplication
Following are 2 different methods that can be used to check multiplication:
Reverse the factors.
Use the lattice method.
TEKSING TOWARD TAKS
2008
Page 4
GRADE 5 MATHEMATICS
(5.3) Number, operation, and quantitative reasoning. The student adds, subtracts, multiplies, and divides to solve meaningful problems.
The student is expected to: (B) use multiplication to solve problems involving whole numbers (no more than three digits times two digits
without technology)
(5.4) Number, operation, and quantitative reasoning. The student estimates to determine reasonable results. The student is expected
to use strategies including rounding and compatible numbers to estimate solutions to addition, subtraction, multiplication, and division
problems.
Reverse the factors.
\
EXAMPLE:
1
3
1
3




24




38



38



24


192


152


720


760


912


912
1
If reversing the factors gives the same product, the multiplication
is correct.
1
If reversing the factors does not give the same product, on of the
products is not correct.
Use the lattice method.
2
EXAMPLE:
4
STEP 1: Draw a grid.
Write one factor on top.
Write the other factor on the right.
3
8
STEP 2: In each square, write a product.
Multiply the digit at the top of the column by the digit to the right of the row.
Use a diagonal line to separate the digits in each product.
2
0
If the product is a single digit, write the product as 0 __ .
Examples: Write 2 x 3 as
0
6
. Write 4 x 3 as
4
1
6
1
1
2
2
STEP 3: Add along the diagonals.
2
1
6
0
1
2
3
6
1
3
8
14
0
9
STEP 4: Read the product.
3
6
.
Begin at the lower right.
For 2-digit sums, add the tens digit to the digits in the next diagonal.
2
2
3
8
2
Begin at the top left and end at the bottom right
24 x 38 = 912
Using either method to check the multiplication, the product is correct.
TEKSING TOWARD TAKS
2008
Page 5
GRADE 5 MATHEMATICS
(5.3) Number, operation, and quantitative reasoning. The student adds, subtracts, multiplies, and divides to solve meaningful problems.
The student is expected to: (B) use multiplication to solve problems involving whole numbers (no more than three digits times two digits
without technology)
(5.4) Number, operation, and quantitative reasoning. The student estimates to determine reasonable results. The student is expected
to use strategies including rounding and compatible numbers to estimate solutions to addition, subtraction, multiplication, and division
problems.
Understanding When to Estimate an Answer
When an exact answer is not needed for a problem, the numbers in the problem can be rounded
before you do the arithmetic. When you do this, you are making an estimate of the answer. For
example, some problems ask about how many or approximately how much. Use estimation when
solving such problems.
You can estimate the answer to any problem before you find the exact answer. The estimate tells
you about how big or small the exact answer should be. If you estimate first, you will know whether
your exact answer is reasonable. For example, some problems ask you whether a certain number
is a reasonable answer to a problem. Use estimation to answer such questions.
One way to estimate is to round numbers in a problem before working it out. To round to a number
to a specific place value, look at the digit in the place value to the right of the digit you are
rounding.
To round to the nearest one, look at the digit in the tenths place.
If the digit in the tenths place is 0, 1, 2, 3, or 4, leave the digit in the ones place the same.
If the digit in the tenths place is 5, 6, 7, 8, or 9, round the digit in the ones place to the next-highest
value.
Replace the digit to the right of the ones place with a 0.
To round to the nearest ten, look at the digit in the ones place.
If the digit in the ones place is 0, 1, 2, 3, or 4, leave the digit in the tens place the same.
If the digit in the ones place is 5, 6, 7, 8, or 9, round the digit in the tens place to the next-highest
value.
Replace the digits to the right of the tens place with a 0.
To round to the nearest hundred, look at the digit in the tens place.
If the digit in the tens place is 0, 1, 2, 3, or 4, leave the digit in the hundreds place the same.
If the digit in the tens place is 5, 6, 7, 8, or 9, round the digit in the hundreds place to the nexthighest value.
Replace the digits to the right of the hundreds place with a 0.
EXAMPLE 1: Alfred can bicycle around the lake bicycle trail in 37.3 to 41.8 minutes. What is a
reasonable total number of minutes it will take him to bicycle around the trail 8 times to the nearest
10 minutes?
Estimate when an exact answer to a problem is not needed.
One way to estimate is to round numbers in a problem before working it out. For this problem,
estimate the least amount of time and the most amount of time Alfred bicycles around the lake
trail to the nearest 10 minutes.
Round 37.3 to 40. Round 41.8 to 40.
It takes Alfred about 40 minutes to bicycle around the lake trail one time.
TEKSING TOWARD TAKS
2008
Page 6
GRADE 5 MATHEMATICS
(5.3) Number, operation, and quantitative reasoning. The student adds, subtracts, multiplies, and divides to solve meaningful problems.
The student is expected to: (B) use multiplication to solve problems involving whole numbers (no more than three digits times two digits
without technology)
(5.4) Number, operation, and quantitative reasoning. The student estimates to determine reasonable results. The student is expected
to use strategies including rounding and compatible numbers to estimate solutions to addition, subtraction, multiplication, and division
problems.
Use the operation of __________________ to find the estimated time it takes Alfred to bike
around the lake 8 times.
40 x 8 = 320
A reasonable amount of time it takes Alfred to bicycle around the lake bicycle trail 8 times is 320
minutes.
EXAMPLE 2: Trisha spends between $18.75 and $22.25 each week on lunch in the school
cafeteria. What is a good estimate of the total amount of money, to the nearest dollar, she will
spend during a semester that lasts 18 weeks?
Estimate when an exact answer to a problem is not needed.
One way to estimate is to round numbers in a problem before working it out. For this problem,
estimate the amount of money Trisha spends in 1 week and the number of weeks in a semester.
Round 18.75 to 19. Round 22.25 to 22.
Round 18 to 20.
Trisha spends between $19 and $22 each week on lunch in the school cafeteria. A semester
lasts about 20 weeks.
Use the operation of __________________ to find the estimated least amount of money she will
spend and the estimated most amount of money she will spend during a semester.
The least amount of money she spends each week is about $19. A semester lasts about 20
weeks.
19 x 20 = 380
A good estimate of the least amount of money she will spend in a semester is $380.
The most amount of money she spends each week is about $22. A semester lasts about 20
weeks.
22 x 20 = 440
A good estimate of the most amount of money she will spend in a semester is $440.
A good estimate of the amount of money Trisha will spend during a semester is between $380
and $440.
TEKSING TOWARD TAKS
2008
Page 7
GRADE 5 MATHEMATICS
(5.3) Number, operation, and quantitative reasoning. The student adds, subtracts, multiplies, and divides to solve meaningful problems.
The student is expected to: (C) use division to solve problems involving whole numbers (no more than two-digit divisors and three-digit
dividends without technology)
(5.4) Number, operation, and quantitative reasoning. The student estimates to determine reasonable results. The student is expected
to use strategies including rounding and compatible numbers to estimate solutions to addition, subtraction, multiplication, and division
problems.
Parent Guide
Six Weeks 3 – Lesson 3
For this TEKS students should be able to divide to solve meaningful problems. The focus for this
lesson is using division to solve problems involving whole numbers and using strategies including
rounding and compatible numbers to estimation solutions to division problems.
MATH BACKGROUND
Understanding Division
Division involves equal groups. Three terms in division are dividend, divisor, and quotient. The
dividend is the number being divided. The divisor is the number by which another number is
being divided. The quotient is the result of the division.
divisor
7
2 14
quotient
dividend
Use division when you want to separate a whole into groups of equal size.
There are two reasons to divide:
You know the original amount and the number of shares, so divide to find the size of one share.
You know the original amount and the size of one share, divide to find the number of shares.
EXAMPLE: An extra large pizza is cut into 12 pieces. Six people want equal shares. How many
pieces will each person get?
You know
the original amount (12)
the number of shares (6)
You need to know
the size of 1 share
(12 6 = 2)
Each person will get 2 slices of pizza.
EXAMPLE: A peanut butter sandwich uses 2 slices of bread. How many sandwiches can you
make with 18 slices of bread?
You know
the original amount (18)
the size of one share (2)
You need to know
how many shares
(18 2 = 9)
You can make 9 sandwiches.
TEKSING TOWARD TAKS
2008
Page 1
GRADE 5 MATHEMATICS
(5.3) Number, operation, and quantitative reasoning. The student adds, subtracts, multiplies, and divides to solve meaningful problems.
The student is expected to: (C) use division to solve problems involving whole numbers (no more than two-digit divisors and three-digit
dividends without technology)
(5.4) Number, operation, and quantitative reasoning. The student estimates to determine reasonable results. The student is expected
to use strategies including rounding and compatible numbers to estimate solutions to addition, subtraction, multiplication, and division
problems.
Understanding How to Relate Multiplication and Division
Multiplication is the opposite, or inverse of division. Division is the inverse, of multiplication.
To multiply with the table below, choose one number from the beginning of a row and one number
from the top of a column. The product will be where the row and column meet. 4 x 3 = 12
x
1
2
3
4
5
1
1
2
3
4
5
2
2
4
6
8
10
3
3
6
9
12
15
4
4
8
12
16
20
5
5
10
15
20
25
To divide, choose a number in the table. The quotient of that number and the row number is the
column number. 12 4 = 3
x
1
2
3
4
5
1
1
2
3
4
5
2
2
4
6
8
10
3
3
6
9
12
15
4
4
8
12
16
20
5
5
10
15
20
25
Or, to divide choose a number in the table. The quotient of that number and the column numbers is
the row number. 12 3 = 4
x
1
2
3
4
5
TEKSING TOWARD TAKS
2008
1
1
2
3
4
5
2
2
4
6
8
10
3
3
6
9
12
15
4
4
8
12
16
20
5
5
10
15
20
25
Page 2
GRADE 5 MATHEMATICS
(5.3) Number, operation, and quantitative reasoning. The student adds, subtracts, multiplies, and divides to solve meaningful problems.
The student is expected to: (C) use division to solve problems involving whole numbers (no more than two-digit divisors and three-digit
dividends without technology)
(5.4) Number, operation, and quantitative reasoning. The student estimates to determine reasonable results. The student is expected
to use strategies including rounding and compatible numbers to estimate solutions to addition, subtraction, multiplication, and division
problems.
Understanding How to Divide Whole Numbers Without Remainders
Use multiplication and subtraction to help with division.
EXAMPLE:
2
8 16
16
0
2

x

8 16
Multiplication and subtraction can also be used to help with long division.
EXAMPLE:
13
5 65
50
15
15
0
10

x

5 50
3
x

5 5
Understanding How to Check Division by Multiplying
Checking an answer is important. Checking helps make sure each step in the long division was
correctly recorded. One way to check division is to use multiplication. Multiplication can be used
to check division because multiplication is the opposite, or inverse, of division.
EXAMPLE: Is 28 a correct answer for 140 5?
28
5 140
10
40
40
0
quotient
divisor
28



5



140
dividend
140 = 140, so the division is correct.
TEKSING TOWARD TAKS
2008
Page 3
GRADE 5 MATHEMATICS
(5.3) Number, operation, and quantitative reasoning. The student adds, subtracts, multiplies, and divides to solve meaningful problems.
The student is expected to: (C) use division to solve problems involving whole numbers (no more than two-digit divisors and three-digit
dividends without technology)
(5.4) Number, operation, and quantitative reasoning. The student estimates to determine reasonable results. The student is expected
to use strategies including rounding and compatible numbers to estimate solutions to addition, subtraction, multiplication, and division
problems.
Understanding Division Without Remainders
Following are ways to divide by a 1-digit number.
Use models to show the division.
Think about multiplication to help you with division.
EXAMPLE: Jason has 75 baseball cards. He wants to divide them equally among 5 pages in his
collection book. How many cards will he put on each page?
Use models to show division to find the number of cards Jason will put on each page.
Use models to show 75.
Put the 7 tens into 5 equal groups.
There is 1 ten in each group.
There are 2 tens and 5 ones left over.
2 tens = 20 ones.
Add the 20 ones to the 5 ones you
already have.
Put the 25 ones into 5 equal groups.
There are 5 ones in each group.
Each group has 1 ten and 5 ones.
The model shows that Jason will put 15 baseball cards on each of the 5 pages in his collection
book.
TEKSING TOWARD TAKS
2008
Page 4
GRADE 5 MATHEMATICS
(5.3) Number, operation, and quantitative reasoning. The student adds, subtracts, multiplies, and divides to solve meaningful problems.
The student is expected to: (C) use division to solve problems involving whole numbers (no more than two-digit divisors and three-digit
dividends without technology)
(5.4) Number, operation, and quantitative reasoning. The student estimates to determine reasonable results. The student is expected
to use strategies including rounding and compatible numbers to estimate solutions to addition, subtraction, multiplication, and division
problems.
Think about multiplication to help you divide to find the number of pages Jason needs.
Divide the tens.
70 5 =
Multiply to estimate.
5 x 1 ten = 50
5x
tens = 70
5 x 2 tens = 100
Use 5 x 1 ten.
(2 tens is too much)
Write 1 in the tens place.
Write 50 below 75.
Subtract and compare.
75 50 = 25
1
5 75
50
25
5 x 1 ten
75 50 = 25
There are 25 ones remaining.
25 > 5
There is enough to keep dividing.
Divide the ones.
25 5 =
Multiply to estimate.
5x
ones = 25
5 x 5 ones = 25
Write 5 in the ones place.
Write 25 below 25.
Subtract and compare.
25 25 = 0
There is no remainder.
The division is finished.
15
5 75
50
25
25
0
5 x 5 ones
25 25 = 0
Using multiplication to help you divide shows Jason will put 15 baseball cards on each of the 5
pages in his collection book.
Either way, Jason will put 15 baseball cards on each of the 5 pages in his collection book.
TEKSING TOWARD TAKS
2008
Page 5
GRADE 5 MATHEMATICS
(5.3) Number, operation, and quantitative reasoning. The student adds, subtracts, multiplies, and divides to solve meaningful problems.
The student is expected to: (C) use division to solve problems involving whole numbers (no more than two-digit divisors and three-digit
dividends without technology)
(5.4) Number, operation, and quantitative reasoning. The student estimates to determine reasonable results. The student is expected
to use strategies including rounding and compatible numbers to estimate solutions to addition, subtraction, multiplication, and division
problems.
Understanding Division With Remainders
In division there are times when equal groups cannot be made without having some extra or left
over. The part left over is called a remainder.
Understanding How to Divide Whole Numbers With Remainders
Use multiplication and subtraction to help with division.
EXAMPLE:
5 R2
4 22
20
5

x

4 20
2
Multiplication and subtraction can also be used to help with long division.
EXAMPLE:
124 R3
6 747

600
100

x

6 600
147
120
20

x

6 120
27
24
4

x

6 24
3
Understanding How to Check Division by Multiplying
Checking an answer is important. Checking helps make sure each step in the long division was
correctly recorded. One way to check division is to use multiplication. Multiplication can be used
to check division because multiplication is the opposite, or inverse, of division.
EXAMPLE: Is 28 R2 a correction answer for 142 5?
whole number
part of quotient
divisor
28



5



140
140



2



142
remainder
142 = 142, so the division is correct.
dividend
Understanding Ways to Divide With Remainders
Use models to show the division.
Think about multiplication to help you with division.
TEKSING TOWARD TAKS
2008
Page 6
GRADE 5 MATHEMATICS
(5.3) Number, operation, and quantitative reasoning. The student adds, subtracts, multiplies, and divides to solve meaningful problems.
The student is expected to: (C) use division to solve problems involving whole numbers (no more than two-digit divisors and three-digit
dividends without technology)
(5.4) Number, operation, and quantitative reasoning. The student estimates to determine reasonable results. The student is expected
to use strategies including rounding and compatible numbers to estimate solutions to addition, subtraction, multiplication, and division
problems.
EXAMPLE: Mrs. Jerrard has 74 pieces of chocolate fudge she has made for holiday gift boxes.
She wants to put the 6 pieces of fudge in each gift box. How many pieces of fudge will she put in
each box?
Use models to show division to find the number of pieces of fudge Mrs. Jerrard will put in each
box.
Use models to show 74.
Put the 7 tens into 6 equal groups.
There is 1 ten in each group.
There are 1 ten and 4 ones left over.
1 ten = 10 ones.
Add the 10 ones to the 4 ones you
already have.
Put the 14 ones into 6 equal groups.
There are 2 ones in each group.
Each group has 1 ten and 2 ones.
There are 2 ones left over.
The model shows Mrs. Jerrard can make 6 gift boxes with 12 pieces of fudge in each box, and
she will have 2 pieces of fudge left over.
TEKSING TOWARD TAKS
2008
Page 7
GRADE 5 MATHEMATICS
(5.3) Number, operation, and quantitative reasoning. The student adds, subtracts, multiplies, and divides to solve meaningful problems.
The student is expected to: (C) use division to solve problems involving whole numbers (no more than two-digit divisors and three-digit
dividends without technology)
(5.4) Number, operation, and quantitative reasoning. The student estimates to determine reasonable results. The student is expected
to use strategies including rounding and compatible numbers to estimate solutions to addition, subtraction, multiplication, and division
problems.
Think about multiplication to help you divide to find the number of pieces of fudge Mrs. Jerrard
can put each gift box.
Divide the tens.
70 6 =
Multiply to estimate.
6 x 1 ten = 60
6x
tens = 70
6 x 2 tens = 120 (2 tens is too much)
Use 6 x 1 ten.
Write 1 in the tens place.
Write 60 below 74.
Subtract and compare.
74 60 = 14
There are 14 ones remaining.
1
6 74
60
14
6 x 1 ten
74 60 = 14
ten
14 > 6
There is enough to keep dividing.
Divide the ones.
14 6 =
Multiply to estimate.
6 x 2 ones = 12
6x
ones = 14
6 x 3 ones = 18 (3 ones is too much)
Write 2 in the ones place.
Write 12 below 14.
Subtract and compare.
14 12 = 2
2<6
You are finished dividing.
The remainder is 2.
12 R2
6 74
60
14
12
2
6 x 2 ones
14 12 = 2
Using multiplication to help you divide shows Mrs. Jessup can put 12 pieces of fudge in each
box, with 2 pieces of fudge left over.
Either way, Mrs. Jessup can put 12 pieces of fudge in each box, with 2 pieces of fudge left over.
TEKSING TOWARD TAKS
2008
Page 8
GRADE 5 MATHEMATICS
(5.3) Number, operation, and quantitative reasoning. The student adds, subtracts, multiplies, and divides to solve meaningful problems.
The student is expected to: (C) use division to solve problems involving whole numbers (no more than two-digit divisors and three-digit
dividends without technology)
(5.4) Number, operation, and quantitative reasoning. The student estimates to determine reasonable results. The student is expected
to use strategies including rounding and compatible numbers to estimate solutions to addition, subtraction, multiplication, and division
problems.
Understanding How to Interpret Quotients and Remainders
In whole-number division you sometimes get a remainder. When you have divided as far as you
can without using decimals, what has not been divided yet is the remainder. The problem helps
you decide what the remainder means and how to interpret the quotient and remainder.
Following are four way of thinking about remainders.
Ignore the remainder.
The answer is the next greater number.
Use the remainder in the answer.
Write the remainder as a fraction.
EXAMPLE: Ms. Petri has 52 ounces of pecan halves. How many 8-ounce bags of can she fill?
6 R4
8 52

48
4
PECANS
PECANS
PECANS
PECANS
8 oz
8 oz
8 oz
8 oz
PECANS
8 oz
PECANS
PECANS
8 oz
8 oz
The quotient shows that you can fill 6 bags. The remainder, 4, tells the weight of the pecans in
the bag that is not full.
Ignore the remainder.
The remainder is not needed for the answer. You only need to know how many 8-ounce bags
Ms. Petri can fill.
EXAMPLE: In the art classroom each table seats 6 students. Mr. Sinz has thirty-five students in
his last class. How many tables will he need for his classroom?
5 R5
6 35
30
5
The quotient, 5 R5, shows that you can not fit all the students at 5 tables.
The answer is the next greater number.
The answer is the next whole number, 6.
TEKSING TOWARD TAKS
2008
Page 9
GRADE 5 MATHEMATICS
(5.3) Number, operation, and quantitative reasoning. The student adds, subtracts, multiplies, and divides to solve meaningful problems.
The student is expected to: (C) use division to solve problems involving whole numbers (no more than two-digit divisors and three-digit
dividends without technology)
(5.4) Number, operation, and quantitative reasoning. The student estimates to determine reasonable results. The student is expected
to use strategies including rounding and compatible numbers to estimate solutions to addition, subtraction, multiplication, and division
problems.
EXAMPLE: Magee has 218 Super Hero trading cards that he does not need for his collection. He
decides to share the cards equally among 5 friends. He gives them as many cards as he can.
How many cards does he have left?
Super
Super
Super
Hero
Hero
Hero
43 R3
5 218
200
18
15
Trading
Trading
Trading
Cards
Cards
Cards
Super
Super
Super
Hero
Hero
Hero
Trading
Trading
Trading
Cards
Cards
Cards
Super
Hero
Trading
Cards
3
Super
Super
Super
Hero
Hero
Hero
Trading
Trading
Trading
Cards
Cards
Cards
Super
Hero
Trading
Cards
Super
Super
Super
Hero
Hero
Hero
Trading
Trading
Trading
Cards
Cards
Cards
Super
Super
Super
Hero
Hero
Hero
Trading
Trading
Trading
Cards
Cards
Cards
Super
Hero
Trading
Cards
The quotient, 43 R3, shows that Magee can give 43 cards to each of his 5 friends and he will
have 3 cards left over.
The remainder is the answer.
The remainder, 3, shows the number of cards Magee will have left.
EXAMPLE: Editors were needed to edit the fifth grade yearbook. The entire editing job took 61
hours. If 4 editors shared the job equally, how long will each one work on the book?
15 R1
4 61

4
21

20
1
The quotient, 15 R1, shows that each editor worked 15 hours and they shared 1 extra hour.
Write the remainder as a fraction.
The remainder, 1, shows the extra hour they shared.
To write the remainder as a fraction, write the remainder as the numerator and the divisor as the
denominator.
remainder
1
divisor
4
1
Each of the editors worked 15 hours.
4
TEKSING TOWARD TAKS
2008
Page 10
GRADE 5 MATHEMATICS
(5.3) Number, operation, and quantitative reasoning. The student adds, subtracts, multiplies, and divides to solve meaningful problems.
The student is expected to: (C) use division to solve problems involving whole numbers (no more than two-digit divisors and three-digit
dividends without technology)
(5.4) Number, operation, and quantitative reasoning. The student estimates to determine reasonable results. The student is expected
to use strategies including rounding and compatible numbers to estimate solutions to addition, subtraction, multiplication, and division
problems.
Understanding Zeros in a Quotient
Sometimes a zero is written in the quotient to show there is nothing in that place value.
Pay attention to place value and estimates and an important zero will not be forgotten.
EXAMPLE: The area of a fifth grade classroom is 981 square feet. How many square yards is the
area of the classroom?
From the Grade 5 Mathematics Chart: 1 yard = 3 feet.
1 yd
1 ft
1 ft
1 yd
1 sq yd
9 sq ft
There are 9 square feet in 1 square yard.
To solve the problem, divide 981 by 9.
First, estimate the answer.
981 9 = 900 9 = 100
The answer should be close to 100 square yards.
Now use long division to find the answer.
Divide the hundreds.
900 9 =
Multiply to estimate.
9x
hundreds = 900
9 x 1 hundred = 900
Write 1 in the hundreds place.
Write 900 below 981.
Subtract and compare.
981 900 = 81
81 > 9
Keep dividing.
TEKSING TOWARD TAKS
2008
1
9 981
900
81
9 x 1 hundred
976 900 = 76
ten
Page 11
GRADE 5 MATHEMATICS
(5.3) Number, operation, and quantitative reasoning. The student adds, subtracts, multiplies, and divides to solve meaningful problems.
The student is expected to: (C) use division to solve problems involving whole numbers (no more than two-digit divisors and three-digit
dividends without technology)
(5.4) Number, operation, and quantitative reasoning. The student estimates to determine reasonable results. The student is expected
to use strategies including rounding and compatible numbers to estimate solutions to addition, subtraction, multiplication, and division
problems.
Divide the tens.
80 9 =
Multiply to estimate.
9 x 0 tens = 0
9x
tens = 80
9 x 1 ten = 90 (1 ten is too much)
Use 9 x 0 tens
Write 0 in the tens place.
10
9 981
900
81
When 8 tens are divided
into 9 equal groups, there
are 0 tens in each group.
You must write the zero in
the quotient in the tens place.
Divide the ones.
81 9 =
Multiply to estimate.
9x
ones = 81
9 x 9 ones =
Write 9 in the ones place.
Write 81 below 81.
Subtract and compare.
109
9 981
900
81
81
0
When you are dividing ones,
think of this as 81 ones.
81 81 = 0
There is no remainder.
You are finished dividing.
Using multiplication to help you divide shows the area of the fifth grade classroom is 109 square
yards.
The answer is close to the estimate, 100 square yards.
TEKSING TOWARD TAKS
2008
Page 12
GRADE 5 MATHEMATICS
(5.3) Number, operation, and quantitative reasoning. The student adds, subtracts, multiplies, and divides to solve meaningful problems.
The student is expected to: (C) use division to solve problems involving whole numbers (no more than two-digit divisors and three-digit
dividends without technology)
(5.4) Number, operation, and quantitative reasoning. The student estimates to determine reasonable results. The student is expected
to use strategies including rounding and compatible numbers to estimate solutions to addition, subtraction, multiplication, and division
problems.
Understanding How to Use Estimation as a Strategy to Divide
With this strategy for dividing, different estimates can be made, but the answer will always be the
same.
EXAMPLE: Divide 720 by 30.
Estimate the quotient.
30 720 20
Multiply your estimate by the divisor.
30 720 20
600
120
If the product is less than the dividend, subtract.
If the product is greater than the dividend, try a lower estimate.
Keep estimating, multiplying, and subtracting until the difference
is less than the divisor.
Add your estimates to get the whole number part of the quotient.
The remainder is the final difference.
30 720 20
600
120 2
60
60 2
60
0
The quotient is 20 + 2 + 2 = 24. There is no remainder
TEKSING TOWARD TAKS
2008
Page 13
GRADE 5 MATHEMATICS
(5.5) Patterns, relationships, and algebraic thinking. The student makes generalizations based on observed patterns and relationships.
The student is expected to: (B identify prime and composite numbers using concrete objects, pictorials models, and patterns in factor pairs.
Parent Guide
Six Weeks 3 – Lesson 4
For this TEKS students should be able to make generalizations based on observed patterns and
relationships. The focus for this lesson is identifying prime and composite numbers using concrete
objects, pictorial models, and patterns in factor pairs.
MATH BACKGROUND
Understanding Prime Numbers
A prime number is a whole number greater than zero that has exactly two different factors, 1 and
itself.
EXAMPLE: 1 is a whole number, but 1 is not prime because 1 does not have two different factors
EXAMPLE: 11 is a prime number because the only factors of 11 are 1 and 11.
EXAMPLE: Find all the prime numbers from 2 to 10.
Number
1
2
3
4
5
6
7
8
9
10
Factors
1, 1
1, 2
1, 3
1, 2, 4
1, 5
1, 2, 3, 6
1, 7
1, 2, 4, 8
1, 3, 9
1, 2, 5, 10
Prime?
No (does not have exactly 2 different factors
Yes (only factors are 1 and 2)
Yes (only factors are 1 and 3)
No (more than 2 different factors)
Yes (only factors are 1 and 5)
No (more than 2 different factors)
Yes (only factors are 1 and 6)
No (more than 2 different factors)
No (more than 2 different factors)
No (more than 2 different factors)
The numbers 2, 3, 5, and 7 are all the prime numbers from 1 to 10.
FUN FACT: Every even number greater than 2 can be written as the sum of two prime numbers.
This fact is called Goldbach’s conjecture, named after the mathematician Christian Goldbach.
EXAMPLES:
4 = 2 +2
26 = 13 + 13
10 = 3 + 7
48 = 11 + 37
FUN FACT: Every number greater than zero, except 1, is either a composite number or a prime
number.
TEKSING TOWARD TAKS
2008
Page 1
GRADE 5 MATHEMATICS
(5.5) Patterns, relationships, and algebraic thinking. The student makes generalizations based on observed patterns and relationships.
The student is expected to: (B identify prime and composite numbers using concrete objects, pictorials models, and patterns in factor pairs.
Understanding Composite Numbers
A composite number is a number that has factors other than 1 and itself. For example, in the
table below, 4, 6, and 8, 9 and 10 are composite numbers.
Number
1
2
3
4
5
6
7
8
9
10
Factors
1, 1
1, 2
1, 3
1, 2, 4
1, 5
1, 2, 3, 6
1, 7
1, 2, 4, 8
1, 3, 9
1, 2, 5, 10
No
No
No
Yes
No
Yes
No
Yes
Yes
Yes
Composite?
(does not have more than two factors)
(only factors are 1 and 2)
(only factors are 1 and 3)
(factors other than 1 and itself)
(only factors are 1 and 5)
(factors other than 1 and itself)
(only factors are 1 and 6)
(factors other than 1 and itself)
(factors other than 1 and itself)
(factors other than 1 and itself)
Understanding Prime Factorization
Every composite number can be written as a product of its prime factors. A prime factor of a
number is a factor that is also a prime number. Writing a composite number as a product of prime
numbers is called prime factorization.
For some problems you must find all the prime factors of a number.
One way to find all the prime factors of a number is to use a factor tree.
Following are guidelines to make a factor tree:
First, write the number you are factoring at the top of the tree.
Next, find any pair of factors.
Then find pairs of factors for the factors.
Keep this up until you can’t do it any more.
EXAMPLE: Use a factor tree to find the prime factors of 18.
Write the number you are factoring at the top of the tree.
18
2
2

9

3

Choose any pair of factors as branches, if either of these is not prime, factor again.
3
Keep factoring until you have a row of prime numbers.
In this factor tree the factors of 18 are 2 and 9. The number 2 is prime, and the number 9 is
composite.
The factors of 9 are 3 and 3. The number 3 is prime.
The prime factorization of 18 is 2 3 3.
TEKSING TOWARD TAKS
2008
Page 2
GRADE 5 MATHEMATICS
(5.5) Patterns, relationships, and algebraic thinking. The student makes generalizations based on observed patterns and relationships.
The student is expected to: (B identify prime and composite numbers using concrete objects, pictorials models, and patterns in factor pairs.
Use a factor tree to find the prime factors of 18 in a different way.
Write the number you are factoring at the top of the tree.
18
3
3

6

2

Choose any pair of factors as branches, if either of these is not prime, factor again.
3
Keep factoring until you have a row of prime numbers.
In this factor tree, the factors of 18 are 3 and 6. The number 3 is prime, and the number 6 is
composite.
The factors of 6 are 2 and 3. Both 2 and 3 are prime.
In both factor trees, the prime factors of 18 are 2 and 3.
The prime factorization of 18 is 2 3 3.
Division can also be used to find the prime factorization of a composite number.
Following are guidelines for using division to find the prime factorization of a composite number:
First, divide the composite number by a prime number.
Then keep dividing each quotient by a prime number until the quotient is 1.
NOTE: If you try a prime number and you get a remainder, that means your dividend is not
divisible by that prime. Try a different prime.
EXAMPLE: Find the prime factorization of 112.
Then keep dividing each quotient by a prime number until the quotient is 1.
1
77
2 14
2 28
2 56
First, divide the composite number by a prime number.
2 112
The prime factorization of 112 is the product of the divisors, 2 2 2 2 7.
NOTE: Remember, 1 is not included in the prime factorization because 1 is not prime.
TEKSING TOWARD TAKS
2008
Page 3
GRADE 5 MATHEMATICS
(5.8) Geometry and Spatial Reasoning. The student models transformations. The student is expected to: (A) sketch the results of
translations, rotations, and reflections on a Quadrant I coordinate grid; (B) identify the transformation that generates one figure from the
other when given two congruent figures on a Quadrant I coordinate grid.
Parent Guide
Six Weeks 3 – Lesson 5
For this TEKS students should be able to model transformations. The focus of this lesson is
sketching the results of reflections on a Quadrant I coordinate grid and identifying a reflection as
the transformation that generated one figure from the other when given two congruent figures on a
Quadrant I coordinate grid.
MATH BACKGROUND
Understanding Reflections
In geometry a figure can be moved from one place to another without changing its shape or size.
Transformations are ways of moving a figure in a plane. Three kinds of transformations are
translations, reflections, and rotations.
In geometry a reflection is a transformation in which a figure is flipped over a line. The line acts as
a mirror to reflect the figure. This line is called the line of reflection.
Each point in a reflected image is the same distance from the line as the corresponding point in the
original figure.
When you reflect a figure, it does not change the size or shape of the figure. The new figure is
congruent to the original figure.
EXAMPLE: Look at the figures on the grid below.
Original Figure
This is the line of reflection.
It acts as a mirror to reflect
a figure.
Reflected Figure
Check points on the reflected figure to make sure they are the same distance from the line as the
corresponding points of the original figure.
TEKSING TOWARD TAKS
2008
Page 1
GRADE 5 MATHEMATICS
(5.8) Geometry and Spatial Reasoning. The student models transformations. The student is expected to: (A) sketch the results of
translations, rotations, and reflections on a Quadrant I coordinate grid; (B) identify the transformation that generates one figure from the
other when given two congruent figures on a Quadrant I coordinate grid.
Check the first set of corresponding points:
Original Figure
Reflected Figure
The point on the reflected figure is 5 units below the line of reflection and is straight across from
the point on the original figure.
The corresponding point on the original figure is 5 units above the line of reflection and is straight
across from the point on the reflected figure.
The corresponding points are straight across from each other and are 5 units from the line of
reflection.
Check a second set of corresponding points:
Original Figure
Reflected Figure
The point on the reflected figure is 1 unit below the line of reflection and is straight across from
the point on the original figure.
The corresponding point on the original figure is 1 unit above the line of reflection and is straight
across from the point on the reflected figure.
The corresponding points are straight across from each other and are 1 unit from the line of
reflection.
TEKSING TOWARD TAKS
2008
Page 2
GRADE 5 MATHEMATICS
(5.8) Geometry and Spatial Reasoning. The student models transformations. The student is expected to: (A) sketch the results of
translations, rotations, and reflections on a Quadrant I coordinate grid; (B) identify the transformation that generates one figure from the
other when given two congruent figures on a Quadrant I coordinate grid.
Check a third set of corresponding points:
Original Figure
Reflected Figure
The point on the reflected figure is 4 units below the line of reflection and is straight across from
the point on the original figure.
The corresponding point on the original figure is 4 units above the line of reflection and is straight
across from the point on the reflected figure.
The corresponding points are straight across from each other and are 4 units from the line of
reflection.
EXAMPLE: Look at the figures on the grid below.
This is the line of reflection.
It acts as a mirror to reflect a figure.
Reflected Figure
Original Figure
Check points on the reflected figure to make sure they are the same distance from the line as the
corresponding points of the original figure.
TEKSING TOWARD TAKS
2008
Page 3
GRADE 5 MATHEMATICS
(5.8) Geometry and Spatial Reasoning. The student models transformations. The student is expected to: (A) sketch the results of
translations, rotations, and reflections on a Quadrant I coordinate grid; (B) identify the transformation that generates one figure from the
other when given two congruent figures on a Quadrant I coordinate grid.
Check the first set of corresponding points:
Reflected Figure
Original Figure
The point on the reflected figure is 5 units to the left of the line of reflection and is straight across
from the point on the original figure.
The corresponding point on the original figure is 5 units to the right of the line of reflection and is
straight across from the point on the reflected figure.
The corresponding points are straight across from each other and are 5 units from the line of
reflection.
Check a second set of corresponding points:
Reflected Figure
Original Figure
The point on the reflected figure is 7 units to the left of the line of reflection and is straight across
from the point on the original figure.
The corresponding point on the original figure is 7 units to the right of the line of reflection and is
straight across from the point on the reflected figure.
The corresponding points are straight across from each other and are 7 units from the line of
reflection.
TEKSING TOWARD TAKS
2008
Page 4
GRADE 5 MATHEMATICS
(5.8) Geometry and Spatial Reasoning. The student models transformations. The student is expected to: (A) sketch the results of
translations, rotations, and reflections on a Quadrant I coordinate grid; (B) identify the transformation that generates one figure from the
other when given two congruent figures on a Quadrant I coordinate grid.
Check a third set of corresponding points:
Reflected Figure
Original Figure
The point on the reflected figure is 4 units to the left of the line of reflection and is straight across
from the point on the original figure.
The corresponding point on the original figure is 4 units to the right of the line of reflection and is
straight across from the point on the reflected figure.
The corresponding points are straight across from each other and are 4 units from the line of
reflection.
TEKSING TOWARD TAKS
2008
Page 5
GRADE 5 MATHEMATICS
(5.10) Measurement. The student applies measurement concepts involving length (including perim eter), area, capacity/volume, and
weight/mass to solve problems. The student is expected to: (B) connect models for perimeter, area, and volume with their respective
formulas, and (C) select and use appropriate units and formulas to measure length, perimeter, area, and volume.
Parent Guide
Six Weeks 3 – Lesson 6
For this TEKS students should be able to apply measurement concepts involving length (including
perimeter), area, capacity/volume, and weight/mass to solve problems. The focus for this lesson is
connecting models for volume with their respective formulas, and using appropriate units and
formulas to measure volume.
MATH BACKGROUND
Understanding How to Connect Models for Volume to Formulas for Volume
Finding the Volume of a Figure
The volume of a prism tells you how many cubes of a given size it takes to fill the prism. The
volume of a prism is the number of cubic units it takes to fill the prism.
Volume is measured in cubic units. The unit used to measure volume is the cube of the unit
used to measure the lengths of the edges.
The units used to measure volume are based on the units used to measure length. Following are
some units for measuring volume:
Customary Units of Volume
Metric Units of Volume
cubic inch
cubic centimeter
cubic foot
cubic meter
EXAMPLE: The cube below has edges that measure exactly 1 unit long on each edge. It has a
volume of 1 cubic unit.
1 unit
1 unit
1 unit
If each edge of the cube is 1 inch long, then the volume is 1 cubic inch.
If each edge of the cube is 1 centimeter long, then the volume is 1 cubic centimeter.
If each edge of the cube is 1 foot long, then the volume is 1 cubic foot.
If each edge of the cube is 1 meter long, then the volume is 1 cubic meter.
TEKSING TOWARD TAKS
2008
Page 1
GRADE 5 MATHEMATICS
(5.10) Measurement. The student applies measurement concepts involving length (including perim eter), area, capacity/volume, and
weight/mass to solve problems. The student is expected to: (B) connect models for perimeter, area, and volume with their respective
formulas, and (C) select and use appropriate units and formulas to measure length, perimeter, area, and volume.
Finding the Volume of a Rectangular Prism
The volume of a prism is the amount of space inside the prism. Volume is measured in cubic
units.
One way to find the volume of a rectangular prism is to count the number of cubes it takes to fill it.
Another way to find the volume of a rectangular prism is to find the number of cubes in the length
of the base, the number of cubes in the width of the base, and the number of cubes in the height
of the prism and use a formula that represents this relationship.
EXAMPLE: The rectangular prism shown below is made of 1-centimeter cubes.
What is its volume?
One way to find its volume is to count the cubic units. The prism has three layers.
Top layer
There are 6 cubic units in the top layer.
Middle layer
There are 6 cubic units in the middle layer, even though you cannot see all of them in the original
figure.
Bottom layer
There are also 6 cubic units in the bottom layer, even though you cannot see all of them in the
original figure.
top layer + middle layer + bottom layer = 6 + 6 + 6 = 18
There are 18 cubic units in all.
Since each cubic unit has a volume of 1 cubic centimeter, the volume of the rectangular prism is 18
cubic centimeters.
TEKSING TOWARD TAKS
2008
Page 2
GRADE 5 MATHEMATICS
(5.10) Measurement. The student applies measurement concepts involving length (including perim eter), area, capacity/volume, and
weight/mass to solve problems. The student is expected to: (B) connect models for perimeter, area, and volume with their respective
formulas, and (C) select and use appropriate units and formulas to measure length, perimeter, area, and volume.
EXAMPLE: The rectangular prism shown below is made of 1-centimeter cubes. Find the volume
using a formula.
The volume of a prism is the relationship between the number of cubes that fit on the base and
the number of layers of cubes. This relationship is always true.
height = 3 cubes
width = 3 cubes
length = 2 cubes
Find the number of cubes that fit on the base.
3 cubes
2 cubes
Base
The length of the base is 2 cubes. The width of the base is 3 cubes.
The number of cubes that fit on the base is the length times the width of the base.
2 3 = 6
The number of cubes that fit on the base is 6.
The volume of the prism is the total number of cubes in the prism. The volume is the number of
cubes that fit on the base times the number of layers of cubes. The prism has 3 layers.
6 3 = 18
The prism has a total of 18 cubes. The volume of the prism is 18 cubic centimeters.
Volume (Rectangular Prism) = length width height
V = l w h
V = 2 3 3
V = 6 3
V = 18
The volume of the prism is 18 cubic centimeters.
TEKSING TOWARD TAKS
2008
Page 3
GRADE 5 MATHEMATICS
(5.10) Measurement. The student applies measurement concepts involving length (including perim eter), area, capacity/volume, and
weight/mass to solve problems. The student is expected to: (B) connect models for perimeter, area, and volume with their respective
formulas, and (C) select and use appropriate units and formulas to measure length, perimeter, area, and volume.
Finding the Volume of a Cube
A cube is a special rectangular prism. Its length, width, and height are all the same.
The volume of a cube is the amount of space inside the cube. Volume is measured in cubic units.
One way to find the volume of a cube is to count the number of cubes it takes to fill it.
Another way to find the volume of a cube is to find the number of cubes in the length of the side
of the base, the number of cubes in the width of the side of the base, and the number of cubes in
the height of the side of the prism and use a formula that represents this relationship.
EXAMPLE: The cube shown below is made of 1-inch cubes.
What is its volume?
One way to find its volume is to count the cubic units. The prism has three layers.
Top layer
There are 9 cubic units in the top layer.
Middle layer
There are 9 cubic units in the middle layer, even though you cannot see all of them in the original
figure.
Bottom layer
There are also 9 cubic units in the bottom layer, even though you cannot see all of them in the
original figure.
top layer + middle layer + bottom layer = 9 + 9 + 9 = 27
There are 27 cubic units in all.
Since each cubic unit has a volume of 1 cubic inch, the volume of the rectangular prism is 27 cubic
inches.
EXAMPLE: The cube shown below is made of 1-inch cubes. Find the volume using a formula.
The volume of a prism is the relationship between the number of cubes that fit on the base and
the number of layers of cubes. This relationship is always true.
height = 3 cubes
width = 3 cubes
length = 3 cubes
TEKSING TOWARD TAKS
2008
Page 4
GRADE 5 MATHEMATICS
(5.10) Measurement. The student applies measurement concepts involving length (including perim eter), area, capacity/volume, and
weight/mass to solve problems. The student is expected to: (B) connect models for perimeter, area, and volume with their respective
formulas, and (C) select and use appropriate units and formulas to measure length, perimeter, area, and volume.
Find the number of cubes that fit on the base.
3 cubes
3 cubes
Base
In a cube, the length of the side of the base and the width of the side of the base are the same.
The length and width of the base are both 3 cubes.
The number of cubes that fit on the base is the length of the side times the width of the side of
the base.
3 3 = 9
The number of cubes that fit on the base is 9.
The volume of the cube is the total number of cubes in the prism. The volume is the number of
cubes that fit on the base times the number of layers of cubes in the side of the cube. The cube
has 3 layers. In a cube the length of the side of the base, width of the side of the base, and
height of the side of the cube are the same measure.
9 3 = 27
The cube has a total of 27 cubes. The volume of the cube is 27 cubic inches.
Volume (cube) = length of side of base width of side of base height of side of cube
V = s s s
V = 3 3 3
V = 9 3
V = 27
The volume of the cube is 27 cubic inches.
NOTE: Multiplying the same number three times is called finding the cube of the number.
Formulas for Volume
The formulas listed below are volume of a cube and volume of a rectangular prism.
Volume Formulas
cube
V s 




s



s
rectangular prism
V 
l



w


h
Students are NOT expected to memorize the formulas in the chart. Students ARE expected to be
able to utilize the formulas in the chart.
TEKSING TOWARD TAKS
2008
Page 5
GRADE 5 MATHEMATICS
(5.10) Measurement. The student applies measurement concepts involving length (including perim eter), area, capacity/volume, and
weight/mass to solve problems. The student is expected to: (B) connect models for perimeter, area, and volume with their respective
formulas, and (C) select and use appropriate units and formulas to measure length, perimeter, area, and volume.
Perimeter
square
P = 4 s
rectangle
P = (2 l) + (2 w)
Area
square
A = s s
rectangle
A = l w
Volume
cube
V s




s




s
rectangular prism V 
l


w 

h
When using a formula to solve a problem, follow these steps:
Identify what is being asked in the problem.
Identify the type of figure in the problem.
Identify the quantities in the problem.
Identify the formula needed to solve the problem
Substitute the variables in the formula with the quantities from
the problem.
Solve the problem.
TEKSING TOWARD TAKS
2008
Page 6
GRADE 5 MATHEMATICS
(5.12) Probability and statistics. The student describes and predicts the results of a probability experiment. The student is expected to:
(C) list all possible outcomes of a probability experiment such as tossing a coin.
Parent Guide
Six Weeks 3 – Lesson 7
For this TEKS students should be able to describe and predict the results of a probability experiment.
The focus for this lesson is listing all possible outcomes of a probability experiment, such as tossing
a coin.
MATH BACKGROUND
Understanding Probability
Probability is a way of describing how likely it is that a particular outcome will occur.
To calculate probability, you need to know all the different things that can happen, or all of the
possible outcomes.
EXAMPLE: Suppose you toss a coin. One of only two things could happen. The coin could land
on heads, or it could land on tails. Since only two things can happen, there are only two possible
outcomes: heads or tails.
EXAMPLE: Look at the bag of marbles below. The bag contains 1 black marble, 1 gray marble,
and 1 white marble.
Suppose you are going to pick a marble from the bag without looking and then check its color.
What are the possible outcomes?
Think of all the possibilities. You could pick a black marble, a gray marble, or a white marble.
The possible outcomes are as follows:
black marble
gray marble
white marble
Sometimes more than one item is selected at random from a group of items.
TEKSING TOWARD TAKS
2008
Page 1
GRADE 5 MATHEMATICS
(5.12) Probability and statistics. The student describes and predicts the results of a probability experiment. The student is expected to:
(C) list all possible outcomes of a probability experiment such as tossing a coin.
EXAMPLE: A bag contains different color tiles. Of the 20 tiles in the bag, 1 is red, 13 are yellow,
and 6 are blue. If you pick 2 color tiles at the same time without looking, what are the possible
outcomes? NOTE: the outcome red and yellow is the same as the outcome yellow and red.
The possible colors that could be chosen if two color tiles are picked at one time are as follows:
Red and yellow
Red and blue
Yellow and Yellow
Yellow and blue
Blue and blue
Is it possible for both color tiles to be blue?
There is only 1 blue color tile. It is not possible for 2 color tiles to be blue.
Understanding a Sample Space
A sample space is a list of all the possible outcomes of an event.
A tree diagram or an organized list can be used to show all the possible outcomes in a sample
space.
EXAMPLE: Pollo’s Pizza Parlor gives you a choice of two crusts – regular or extra crispy – and 5
different toppings – cheese, pepperoni, sausage, peppers, or mushrooms. How many different
pizzas does Pollo’s Pizza make?
regular
cheese
sausage
pepperoni
peppers
mushrooms
extra crispy
cheese
sausage
pepperoni
peppers
mushrooms
An organized list of the possible combinations for ordering a pizza at Pollo’s Pizza Parlor follows:
regular cheese
regular sausage
regular pepperoni
regular peppers
regular mushrooms
extra crispy cheese
extra crispy sausage
extra crispy pepperoni
extra crispy peppers
extra crispy mushrooms
The sample space contains 10 ;possible outcomes. Pollo’s Pizza Parlor makes 10 different pizzas.
TEKSING TOWARD TAKS
2008
Page 2
GRADE 5 MATHEMATICS
(5.13) Probability and Statistics. The student solves problems by collecting, organizing, displaying, and interpreting sets of data.
The student is expected to (A) use tables of related number pairs to make line graphs; and (C) graph a given set of data using an
appropriate graphical representation such as a picture or line graph.
Parent Guide
Six Weeks 3 – Lesson 8
For this TEKS students should be able to solve problems by collecting, organizing, displaying, and
interpreting sets of data. The focus for this lesson is using tables of related number pairs to make
line graphs, and graphing a given set of data using an appropriate graphical representation such
as a line graph.
MATH BACKGROUND
Understanding How to Represent Sets of Data
When data are organized and displayed, it is easier to see relationships among the data. The
relationships among data can be helpful in problem solving. Tables, line graphs, bar graphs, and
pictographs can be used to organize and display data.
A table organizes data in rows and columns so that the data are easier to read.
EXAMPLE: Airports organize fight data in a table to show which flights are leaving and at what
time. The airport displays the table to help passengers identify flight information.
Time
7:32 A.M.
7:58 A.M.
8:05 A.M.
8:35 A.M.
9:22 A.M.
Airline Departures
Airline
Flight Number
Southwest
1234
American
961
Delta
2567
Southwest
504
American
618
Destination
Chicago
Austin
Miami
Houston
Hawaii
Understanding How to Represent Pairs of Related Numbers
Numbers can be related in different ways.
EXAMPLE 1: A table of related number pairs is shown below.
Point
R
S
T
x
3
6
9
y
4
8
7
EXAMPLE 2: A set of related numbers might include distances and the amounts of time it takes to
travel those distances.
Miles
120
180
300
TEKSING TOWARD TAKS
2008
Travel Time (in hours)
2
3
5
Page 1
GRADE 5 MATHEMATICS
(5.13) Probability and Statistics. The student solves problems by collecting, organizing, displaying, and interpreting sets of data.
The student is expected to (A) use tables of related number pairs to make line graphs; and (C) graph a given set of data using an
appropriate graphical representation such as a picture or line graph.
Understanding How to Represent Pairs of Related Numbers in a Line Graph
A coordinate plane is a grid used to locate points. A point is located using an ordered pair of
numbers. The two numbers that form the ordered pair are called the points coordinates (x, y).
The data in a table can be used to graph points on a coordinate grid. A line graph consists of
points graphed on a coordinate grid, with line segments drawn from one data point to the next.
EXAMPLE 1: The table below shows the coordinates of 3 points.
Point
A
B
C
x
1
4
7
y
5
8
6
The line graph below shows the line containing the points in the table.
y
9
B
8
7
6
5
C
A
4
3
2
1
0
1
2
3
4
5
6
7
8
9
x
Point A is 1 unit to the right of the origin. Its x-coordinate is 1.
Point A is 5 units above the origin. Its y-coordinate is 5.
Point A has the coordinates (1, 5).
Point B is 4 units to the right of the origin. Its x-coordinate is 4.
Point B is 8 units above the origin. Its y-coordinate is 8.
Point B has the coordinates (4, 8).
Point C is 7 units to the right of the origin. Its x-coordinate is 7.
Point C is 6 units above the origin. Its y-coordinate is 6.
Point C has the coordinates (7, 6).
EXAMPLE 2: The able shows the scores Jaime made on his math quizzes this six weeks.
Quiz Number
1
2
3
4
5
6
TEKSING TOWARD TAKS
2008
Quiz Score
60
80
70
70
90
85
Page 2
GRADE 5 MATHEMATICS
(5.13) Probability and Statistics. The student solves problems by collecting, organizing, displaying, and interpreting sets of data.
The student is expected to (A) use tables of related number pairs to make line graphs; and (C) graph a given set of data using an
appropriate graphical representation such as a picture or line graph.
The data in the table can be used to create a graph on a coordinate grid.
Write “Quiz Number” as the title of the x-axis.
Write “Quiz Score” as the title of the y-axis.
Place the quiz number on the x-axis. Mark lines for 1, 2, 2, 4, 5, and 6.
Place the numbers for quiz score on the y-axis. Mark lines for each 10 from 0 to 100.
Graph the points (1, 60), (2, 80), (3, 70), (4, 70), (5, 90), and (6, 85).
Connect the data points.
The line graph below shows the data in the table.
Quiz Score
Jaime’s Quiz Scores
100
90
80
70
60
50
40
30
20
10
0
1
2
3
4
Quiz Number
5
6
Statements about the data can be supported by the line graph.
Jaime made his highest quiz score on quiz number _____.
Jaime made his lowest quiz score on quiz number _____..
Jaime made at least ______ on all six quizzes.
Jaime made 80 or better on _____ of the quizzes.
TEKSING TOWARD TAKS
2008
Page 3
Grade 5 Mathematics
Process Skills
5.14A
5.14B
5.14C
5.15B
5.16A
TEKS 5.14A/5.14B/5.14C/5.15B/5.16A
identify mathematics in everyday situations
solve problems that incorporate understanding the problem, making a plan, carrying out the plan, and evaluating the solution
select or develop an appropriate problem-solving plan or strategy
relate informal language to mathematical language and symbols
make generalizations from patterns or sets of examples and nonexamples
Parent Guide
Six Weeks 3 – Lesson 9
For these TEKS students should be able to apply mathematics to everyday problem situations.
The focus for this lesson is selecting or developing an appropriate problem-solving plan or strategy.
MATH BACKGROUND
Understanding Problem-Solving Strategies
A problem-solving strategy is a plan for solving a problem.
The strategy you choose depends on the type of problem you are solving. Sometimes you can use
more than one strategy to solve a problem. Some problem-solving strategies include:
drawing a picture or a diagram;
looking for a pattern;
guessing and checking;
acting it out;
making a table;
working a simpler problem;
working backwards.
Understanding How to Draw a Picture or a Diagram to Solve a Problem
One way to solve a problem is to draw a picture or a diagram.
STEP 1: Organize the information given in the problem.
STEP 2: Draw a picture of a diagram using the information.
STEP 3: Use the picture or the diagram to help you solve the problem.
EXAMPLE: Sue lives 10 blocks directly west of Glenn. Carlos lives 4 blocks directly east of
Glenn. Lea lives directly west of Glenn, halfway between Sue and Carlos. How many blocks does
Leesa live from Gamon?
STEP 1: Organize the information given in the problem.
The following information is given in the problem:
Callie lives 12 blocks directly west of Gamon.
Sean lives 6 blocks directly east of Gamon.
Leesa lives directly west of Gamon, halfway between Callie and Sean.
TEKSING TOWARD TAKS
2008
Page 1
Grade 5 Mathematics
Process Skills
5.14A
5.14B
5.14C
5.15B
5.16A
TEKS 5.14A/5.14B/5.14C/5.15B/5.16A
identify mathematics in everyday situations
solve problems that incorporate understanding the problem, making a plan, carrying out the plan, and evaluating the solution
select or develop an appropriate problem-solving plan or strategy
relate informal language to mathematical language and symbols
make generalizations from patterns or sets of examples and nonexamples
STEP 2: Draw a picture of a diagram using the information.
Draw a horizontal line and place a mark for Callie 12 units to the left of a mark for Gamon.
Place a mark for Sean 6 units to the right of the mark for Gamon.
Place a mark for Leesa halfway between the marks for Callie and Sean.
12 blocks
Callie
6 blocks
Leesa
Gamon
9 blocks
Sean
9 blocks
Callie and Sean are 12 + 6, or 18, blocks apart.
One-half of 18 blocks is 9 blocks.
Leesa must live 9 blocks from both Callie and Sean.
Since 9 units - 6 units = 3 units, 9 units to the left of Sean is the same as 3 units to the left of
Gamon.
Leesa lives 3 blocks west of Gamon.
Understanding How to Find a Pattern to Solve a Problem
Organizing data in a table can help find a pattern that can be used to solve a problem.
EXAMPLE: Sarah is buying matching plates and napkins for a party. The plates are sold in
packages of 6 and the napkins are sold in packages of 15. What is the least number of packages
Sarah can buy to have the same number of plates and napkins?
The number of plates Sarah buys depends on the number of packages she buys. Make a table
to see the pattern.
Number of Packages
1
2
3
Number of Plates
6
12
18
The number of plates Sarah buys will be multiples of 6.
TEKSING TOWARD TAKS
2008
Page 2
Grade 5 Mathematics
Process Skills
5.14A
5.14B
5.14C
5.15B
5.16A
TEKS 5.14A/5.14B/5.14C/5.15B/5.16A
identify mathematics in everyday situations
solve problems that incorporate understanding the problem, making a plan, carrying out the plan, and evaluating the solution
select or develop an appropriate problem-solving plan or strategy
relate informal language to mathematical language and symbols
make generalizations from patterns or sets of examples and nonexamples
The number of napkins Sarah buys also depends on the number of packages she buys. Make a
table to see the pattern.
Number of Packages
1
2
3
Number of Napkins
15
30
45
The number of napkins Sarah buys will be multiples of 15.
To find an equal number of plates and napkins, look for a common multiple of 6 and 15.
Multiples of 6: 6, 12, 18, 24, 30, …
Multiples of 15: 15, 30, 45, 90, …
The least common multiple of 6 and 15 is 30.
If Sarah buys 5 packages of plates, she will have 6 5 = 30 plates.
If Sarah buys 2 packages of napkins, she will have 15 2 = 30 napkins.
The least numbers of packages Sarah can buy to have an equal number of matching plates and
napkins are 5 packages of plates and 2 packages of napkins.
Understanding How to Guess-and-Check to Solve a Problem
Another way to solve a problem is to use the guess-and-check strategy. Try numbers that make
sense until you find the correct number. Each number you guess should help you make a better
guess the next time.
EXAMPLE: Remie has $1.00 in nickels and dimes. She has 12 coins in all. How many nickels
does Remie have?
1st guess: She has 5 dimes.
If Remie has 12 coins in all, then she must have 12 5 7 nickels.
Check:
The value of 5 dimes is $0.50
The value of 7 nickels is $0.35
The value of the nickels and dimes $0.85
Remie has $1.00 in coins, so $0.85 is too small. This guess is close, but it is incorrect.
Guess again: Use your first guess to help you decide which numbers to try next.
Since $0.85 < $1.00, try a combination of coins with more value. Try one with more dimes and
fewer nickels.
TEKSING TOWARD TAKS
2008
Page 3
Grade 5 Mathematics
Process Skills
5.14A
5.14B
5.14C
5.15B
5.16A
TEKS 5.14A/5.14B/5.14C/5.15B/5.16A
identify mathematics in everyday situations
solve problems that incorporate understanding the problem, making a plan, carrying out the plan, and evaluating the solution
select or develop an appropriate problem-solving plan or strategy
relate informal language to mathematical language and symbols
make generalizations from patterns or sets of examples and nonexamples
2nd guess: She has 7 dimes.
If Remie has 12 coins in all, then she must have 12 7 5 nickels.
Check:
The value of 7 dimes is $0.70
The value of 5 nickels is $0.25
The value of the nickels and dimes $0.95
Remie has $1.00 in coins, so $0.95 is still too small. This guess is closer, but it is still incorrect.
Guess again: Use your first and second guesses to help you decide which numbers to try next.
Since $0.95 < $1.00, try a combination of coins with more value. Try a guess with more dimes
and less nickels.
3rd guess: She has 8 dimes.
If Remie has 12 coins in all, then she must have 12 8 4 nickels.
Check:
The value of 8 dimes is $0.80
The value of 4 nickels is $0.20
The value of the nickels and dimes $1.00
Remie has $1.00 in coins. This guess is correct. Remie has 4 nickels.
Understanding How to Act it Out to Solve a Problem
Another way to solve a problem is to act it out. Acting a problem out means modeling the actions
in the problem in a real-life way.
EXAMPLE: Maggie is stacking boxes of printer paper in rows. Each row is 4 boxes wide. The
stack of boxes is 3 rows high. Each box has two labels, one on the top and one on the front side.
How many labels can Maggie see?
Printer Paper
Printer Paper
Printer Paper
Printer Paper
Printer
Paper
Printer Paper
Printer
PaperPaper
Printer
Printer
Paper
Printer Paper
Printer
PaperPaper
Printer
Printer
Paper
Printer Paper
Printer
Paper
Printer Paper
Printer
Paper
Printer Paper
Printer
Paper
Printer Paper
Printer
Paper
Printer
Paper
Printer
Paper
Printer
Paper
Act out the problem by using linking cubes to model the boxes of printer paper. Count the
number of tops and front sides you can see on the linking cubes. This is the same as the
number of labels Maggie can see on the boxes of printer paper.
TEKSING TOWARD TAKS
2008
Page 4
Grade 5 Mathematics
Process Skills
5.14A
5.14B
5.14C
5.15B
5.16A
TEKS 5.14A/5.14B/5.14C/5.15B/5.16A
identify mathematics in everyday situations
solve problems that incorporate understanding the problem, making a plan, carrying out the plan, and evaluating the solution
select or develop an appropriate problem-solving plan or strategy
relate informal language to mathematical language and symbols
make generalizations from patterns or sets of examples and nonexamples
Count the tops and front sides you can see on the unit cubes starting at the bottom.
1 2 3 4
1 2 3 4
1
2
3
4
1
2
3
4
Row one: 4 fronts show
Row two: 4 fronts show
Row three: 4 fronts show and 4 tops show.
Add the number of tops and front sides that can be seen in each row.
4 + 4 + 4 + 4 = 16
You can see 16 tops and front sides on the linking cubes.
That means Maggie can see 16 labels on the printer paper boxes.
Understanding How to Use a Table to Solve a Problem
Sometimes the information in a table can be used to solve a problem.
EXAMPLE: The students at Garcia Elementary School collected cans of food for a local food
pantry after a recent tornado. The table below shows the number of cans Mr. Bazan’s’s class
collected.
Mr. Bazan’s Class Collection Record
Day
Number of Cans
Monday
17
Tuesday
16
Wednesday
15
Thursday
14
Friday
15
Ms. Marsh’s class collected twice as many cans as Mr. Bazan’s class. Mrs. Wagner’s class
collected 24 cans more than Ms. Marsh’s class collected. What is the total number of cans
collected by the three classes?
TEKSING TOWARD TAKS
2008
Page 5
Grade 5 Mathematics
Process Skills
5.14A
5.14B
5.14C
5.15B
5.16A
TEKS 5.14A/5.14B/5.14C/5.15B/5.16A
identify mathematics in everyday situations
solve problems that incorporate understanding the problem, making a plan, carrying out the plan, and evaluating the solution
select or develop an appropriate problem-solving plan or strategy
relate informal language to mathematical language and symbols
make generalizations from patterns or sets of examples and nonexamples
First add the numbers in the table to find the number of cans Mr. Bazan’s class collected.
17 + 16 + 15 + 14 + 15 = 77
Mr. Bazan’s class collected 77 cans.
Ms. Marsh’s class collected twice as many cans as Mr. Bazan’s class did. Multiply by 2 to find a
number of cans that is twice as large.
2 77 = 154
Ms. Marsh’s class collected 154 cans.
Mrs. Wagner’s class collected 24 more cans than Ms. Marsh’s class did. Add 24 to the number
of cans collected by Ms. Marsh’s class.
154 + 24 = 178
Ms. Warren’s class collected 178 cans.
Add to find the total number of cans collected by the three classes.
77 + 154 + 178 = 409
The classes collected 409 cans of food in all.
Understanding How to Work a Simpler Problem to Solve a Problem
Sometimes the numbers in a problem can make it seem difficult. You can change the numbers in
the problem so that you can solve a simpler problem. Then use the same steps to solve the
original problem.
EXAMPLE: A box of cereal weighs 1 pound 14 ounces and includes 6 ounces of raisins and nuts.
What fractional part of the cereal box’s weight is raisins and nuts?
The numbers in this problem make it complicated to solve.
Change the numbers so they are easier to work with.
Suppose the box weighed 10 ounces and the raisins and nuts weighed 6 ounces.
Ask : What fractional part of 10 is 6?
The number 4 is the part and is the numerator of the fraction.
The number 10 is the whole and is the denominator of the fraction.
The answer for this easier problem would be
6
3
which simplifies to .
10
5
Use that answer to help you find the answer to the real problem.
TEKSING TOWARD TAKS
2008
Page 6
Grade 5 Mathematics
Process Skills
5.14A
5.14B
5.14C
5.15B
5.16A
TEKS 5.14A/5.14B/5.14C/5.15B/5.16A
identify mathematics in everyday situations
solve problems that incorporate understanding the problem, making a plan, carrying out the plan, and evaluating the solution
select or develop an appropriate problem-solving plan or strategy
relate informal language to mathematical language and symbols
make generalizations from patterns or sets of examples and nonexamples
The box of cereal really weighs 1 pound 14 ounces.
You need the weight of the box of cereal in ounces if you are going to compare it to the weight of
the raisins and nuts.
Use the Mass and Weight section of the Mathematics Chart.
1 pound = 16 ounces
1 pound 14 ounces = 16 ounces + 14 ounces = 30 ounces
The cereal box weighs 30 ounces.
The raisins and nuts weigh 6 ounces.
What fractional part of 30 is 6?
The number 6 is the part and is the numerator of the fraction.
The number 30 is the whole and is the denominator of the fraction.
The fraction
6
1
simplifies to .
30
5
The weight of the raisins and nuts is
1
the weight of the cereal box.
5
Understanding How to Work Backwards to Solve a Problem
Another way to solve a problem is to work backwards. When you work backwards, you start with
the last information given in the problem.
EXAMPLE: The fifth-grade students collected donations for the Red Cross after a recent
hurricane. They collected donations at the city park one morning for 3 hours. Then they took a 30minute lunch break. After lunch they collected donations for 2 hours 25 minutes. They left the city
park at 3:00 P.M. At what time did the students arrive at the city park?
Work backwards. Begin with the last information given.
The students left the city park at 3:00 P.M.
They collected donations for 2 hours 25 minutes after lunch.
Since 2 hour 25 minutes before 3:00 P.M. is 12:35 P.M., their lunch break ended at 12:35 P.M.
They had a 30-minute lunch. Since 30 minutes before 12:35 P.M. is 12:05 P.M., their lunch break
started at 12:05 P.M.
The students collected donations for 3 hours before lunch.
Three hours before 12:05 P.M. is 9:05 A.M.
The students arrived at the city park at 9:05 A.M.
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