Study Guide
Grade 3 Math-a-thon
Add Decimals: Story Problems - A
Story problems, also called word problems, relate decimal numbers to actual situations. For example, if Sara
weighs 32.5 pounds and Kimberly weighs 31.2 pounds, how much do Sara and Kimberly weigh together? The
student must determine that addition is required to solve this problem. (Answer: 63.7 pounds)
Story problems are often very difficult for children to master. It may be beneficial for you to verify that
the student is comfortable with addition and subtraction skills, as well as with reading skills. Relate
word problems with everyday events.
Example 1: A recipe calls for 2.55 cups of flour and another recipe calls for 1.25 cups of flour. How
many cups of flour do you need to make both recipes?
Step 1: Rewrite the problem vertically. Always line up the decimal points.
Step 2: Add the numbers in the hundredths column (5 + 5 = 10). Write the 0 in the hundredths position.
Carry the 1 to the tenths column.
Step 3: Add the numbers in the tenths column, including the number carried over from the previous
column (1 + 5 + 2 = 8). Write the 8 in the tenths position.
Step 4: Bring the decimal point down.
Step 5: Add the numbers in the ones position (2 + 1 = 3). Write the 3 to the left of the decimal point.
Answer: You need 3.80 cups of flour for the two recipes.
Bar Graphs - A
A bar graph is a drawing used to show and compare information. A bar graph has rectangular bars at different
heights to show/compare data.
An interesting method for increasing the student's understanding of bar graphs is to help him or her
develop a graph of favorite colors based on a survey of family and friends. Limit the color choices to 3
or 4, then design and color a graph to match the results of the survey. The following is an example of a
bar graph:
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Example 1: Which color is the most popular?
Answer: According to the above graph, the most popular color is green because the bar above green is
the tallest (12).
Example 2: Which color is the least popular?
Answer: According to the above graph, the least popular color is blue because the bar above blue is the
shortest (4).
Symmetry - A
A symmetric figure is one that can be folded in half so that the two halves match exactly (or are the same size
and shape). The letter "A" is a symmetrical figure. It can be folded in half evenly. It does not change
appearance when held to a mirror.
One artistic way to help the student understand symmetry is for you to draw half of a simple shape (such
as a square, circle, triangle, or heart) on a piece of paper and have the student draw the other half. Fold
the paper on the halfway line and compare to see if both sides match.
In the following figures, the dotted lines are lines of symmetry.
Add Whole No: 2 Numbers with 1-3 Digits
Adding two numbers with more than one digit (columns of numbers) often requires regrouping (carrying,
trading, or renaming). Regrouping occurs when the total of the numbers in a column (i.e., ones position) is
equal to or greater than ten. Problems are presented in both vertical and horizontal formats.
It may be beneficial to verify that the student understands regrouping. The following is a step-by-step
example of a problem that requires regrouping.
Solve: 567 + 135 = ?
Step 1: Rewrite the problem vertically.
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Step 2: Add the numbers in the ones position (7 + 5 = 12). Write the 2 in the ones position and carry the
1 to the next column (tens).
Step 3: Add the numbers in the tens column, including the number carried over from the previous
column (1 + 6 + 3 = 10). Write the 0 in the tens position and carry the 1 to the next column (hundreds).
Step 4: Add the numbers in the hundreds position, including the number carried over from the previous
column (1 + 5 + 1 = 7). Write the 7 in the hundreds position.
Answer: 567 + 135 = 702.
Add Whole No: 2 Numbers with 4+ Digits
Adding two numbers with four or more digits (columns of numbers) often requires regrouping (carrying,
trading, or renaming). Regrouping occurs when the total of the numbers in a column (i.e., ones position) is
equal to or greater than ten. Problems are presented in both vertical and horizontal formats.
It may be beneficial to verify that the student understands regrouping. The following is a step-by-step
example of a problem that requires regrouping.
Example 1: 2,345 + 3,659 = ?
Step 1: Write the problem vertically.
Step 2: Add the ones column (5 + 9 = 14). Write the 4 in the ones place and carry the 1 to the tens
column above the 4.
Step 3: Add the tens column (1 + 4 + 5 = 10). Write the 0 in the tens place and carry the 1 to the
hundreds column above the 3.
Step 4: Add the hundreds column (1 + 3 + 6 = 10). Write the 0 in the tens place and carry the 1 to the
thousands column above the 2.
Step 5: Add the thousands column (1 + 2 + 3 = 6). Write the 6 in the thousands place. Write a comma
after the 6.
The answer is: 2,345 + 3,659 = 6,004.
Example 2: 13,286 + 11,491 = ?
Step 1: Write the problem vertically.
Step 2: Add the ones column (6 + 1 = 7). Write the 7 in the ones place.
Step 3: Add the tens column (8 + 9 = 17). Write the 7 in the tens place and carry the 1 to the hundreds
column above the 2.
Step 4: Add the hundreds column (1 + 2 + 4 = 7). Write the 7 in the hundreds place.
Step 5: Add the thousands column (3 + 1 = 4). Write the 4 in the thousands place.
Step 6: Add the ten thousands column (1 + 1 = 2). Write the 2 in the ten thousands place. Write a comma
after the 4.
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The answer is: 13,286 + 11,491 = 24,777.
Add Whole No: Story Problems - B
Story problems, also called word problems, relate addition of whole numbers to actual situations. Operational
symbols, such as the addition (+) symbol, are replaced with text. For example, if Jill has 2 apples and Jack gave
her 2 more apples, how many apples would Jill have now? The student must determine that addition is required
to perform this problem.
Story problems are often very difficult for children to master. It may be beneficial for you to verify that
the student is comfortable with addition skills. Then, create humorous problems and help the student
determine the correct formulas.
Example: Joe picked 87 oranges on Saturday. On Sunday, he picked 65 more oranges. How many
oranges did he pick in all?
Step 1: This problem requires addition. Since we want to determine the total number of oranges Joe
picked, we need to add 87 and 65. Write a vertical equation.
Step 2: Add the numbers in the ones column (7 + 5 = 12). Write the 2 in the ones place and carry the 1 to
the tens column.
Step 3: Add the numbers in the tens column including the number carried over (1 + 8 + 6 = 15). Write
the 5 in the tens column and the 1 in the hundreds column.
Answer: Joe picked 152 oranges.
Money - B
Students at this level have been introduced to the concept of money. Problems deal with making change,
adding coins, etc.
Since currency is represented as decimal numbers (dollars are to the left of the decimal point and cents
are to the right of the decimal point), it may be helpful to review how to add and subtract decimal
numbers.
The difference between adding decimals with regrouping and adding whole numbers with regrouping is
the fact that the decimal points must be lined up before addition can occur. The following is a step-bystep example of adding decimals (written as money) with regrouping:
Example 1: Jasen had $2.40 and his brother gave him $1.68. How much money does Jasen have now?
(1)
$2.40
(2)
1
$2.40
(3)
1
$2.40
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+ $1.68
+ $1.68
.08
+ $1.68
$4.08
Step 1: Write the problem vertically. Make sure the decimal points are lined up.
Step 2: Add the hundredths column (0 + 8) to get 8. Write the 8 in the hundredths column. Add the
tenths column (4 + 6) to get 10. Write the 0 in the tenths column and carry the 1 to the ones column (left
of the decimal point). Bring the decimal point straight down (left of the 0).
Step 3: Add the ones column, including the 1 that was carried (1 + 2 + 1 = 4). Place the 4 to the left of
the decimal point to finish the problem. Bring down the dollar sign.
Answer: $2.40 + $1.68 = $4.08
The difference between subtracting decimals with regrouping and subtracting whole numbers with
regrouping is the fact that the decimal points must be lined up before subtraction can occur. The
following is a step-by-step example of subtracting decimals with regrouping.
Example 2: Solve: $5.30 - $2.90 = ?
Step 1: Rewrite the problem vertically. Always line up the decimal points.
Step 2: Begin by subtracting the hundredths (0 - 0). Subtraction follows the same format as subtraction
with whole numbers. Then subtract the tenths (13 - 9 = 4), after regrouping.
Step 3: Bring the decimal point straight down.
Step 4: Complete the problem by subtracting the ones (4 - 2 = 2). Bring down the dollar sign.
Answer: $5.30 - $2.90 = $2.40.
Once the student is comfortable with adding and subtracting currency, have him or her apply the skill to
real life situations. For example, give the student a handful of change and help him or her determine the
amount. Another technique that many children enjoy is "playing store." Help the student add the prices
of several items together and then make correct change.
Add Decimals: Hundredths
Adding two decimal numbers with more than one digit (columns of numbers) is very similar to adding whole
numbers. Like whole numbers, addition of decimals often requires regrouping (carrying, trading, renaming).
Regrouping occurs when the total of the numbers in a column (i.e., ones position) is equal to or greater than ten.
Problems are presented in both vertical and horizontal formats.
The following is a step-by-step example of a problem that requires regrouping:
Example 1: Solve. 8.97 + 5.36 = ?
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Step 1: Rewrite the problem vertically. Always line up the decimal points.
Step 2: Add the numbers in the hundredths position (7 + 6 = 13). Write the 3 in the hundredths position.
Carry the 1 to the next column (tenths).
Step 3: Add the numbers in the tenths column, including the number carried over from the previous
column (1 + 9 + 3 = 13). Write the 3 in the tenths position. Carry the 1 to the next column (ones).
Bring the decimal point down.
Step 4: Add the numbers in the ones position, including the number carried over from the previous
column (1 + 8 + 5 = 14). Write the 14 to the left of the decimal point.
Answer: 8.97 + 5.36 = 14.33
Example 2: Solve. $23.91 + $32.64 = ?
Step 1: Rewrite the problem vertically. Always line up the decimal points.
Step 2: Add the numbers in the hundredths position (1 + 4 = 5). Write the 5 in the hundredths position.
Step 3: Add the numbers in the tenths column (9 + 6 = 15). Write the 5 in the tenths position. Carry the
1 to the next column (ones).
Step 4: Bring the decimal point down.
Step 5: Add the numbers in the ones position, including the number carried over from the previous
column (1 + 3 + 2 = 6). Write the 6 in the ones position.
Step 6: Add the numbers in the tens position (2 + 3 = 5). Write the 5 in the tens position. Bring down the
dollar sign.
Answer: $23.91 + $32.64 = $56.55
Compare Decimals - A
Comparing decimal numbers involves the ordering of numbers by using ordering symbols, such as <, >, and =.
It may be helpful to review the ordering symbols with the student.
To improve the student's understanding of decimal placement, develop a series of decimal numbers and
help the student insert the appropriate ordering symbols. To compare 2 numbers, compare the
corresponding columns beginning with the highest place value.
Example 1: Which is greater, 5.9 or 5.8?
Answer: Since both decimals have a 5 in the ones column, move to the next column (to the right). Since
9 is larger than 8, 5.9 is greater than 5.8 (5.9 > 5.8).
Example 2: Which is greater, 3.45 or 3.46?
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Answer: Since both decimals have a 3 in the ones column and a 4 in the tenths column, move to the next
column (to the right). Since 5 is less than 6, 3.46 is the greater number (3.46 > 3.45).
Example 3:
Find the missing number.
1.3, _____, 1.8
A. 1.2
B. 2.0
C. 1.9
D. 1.6
Answer: Since the missing number falls between 1.3 and 1.8, it must be greater than 1.3 and less than
1.8.
Choice A, 1.2, is less than 1.3.
Choice B, 2.0, is greater than 1.8.
Choice C, 1.9, is greater than 1.8.
Choice D, 1.6, is greater than 1.3 and less than 1.8.
The answer is D, 1.6.
Ordering Numbers
In this skill, the student must place numbers in order by determining which number would come between two
other numbers.
Ordering Whole Numbers:
Example 1: Which number comes between the numbers below?
15, _____, 28
A.
B.
C.
D.
33
23
13
43
Step 1: Since the number must be between 15 and 28, the number must be greater than 15 but less than
28. Eliminate any answer choice that is less than or equal to 15. C. can be eliminated because 13 is less
than 15.
Step 2: Eliminate any answer choice that is greater than or equal to 28. A. and D. can be eliminated
because they are both greater than 28.
Step 3: The answer choice that is left is B. 23. Since 23 is larger than 15 and smaller than 28, it comes
between 15 and 28.
Answer: B.
Ordering Decimal Numbers:
Many students find it more difficult to order decimal numbers, so it may be important to review with the
student how to determine which decimal number is greater than or less than another. The chart below
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will help reveiw the decimal places with the students.
Example 2: Which is greater, 3.45 or 3.46?
Solution: Since both decimals have a 3 in the ones column and a 4 in the tenths column, move to the
next column to the right (the hundredths column). Since 6 is greater than 5, 3.46 is the greater number
(3.46 > 3.45).
Answer: 3.46
Example 3: Which of these numbers comes between the numbers below?
36.76, _____, 37.42
A. 36.67
B. 37.42
C. 37.43
D. 36.77
Step 1: Since the number must come between 36.76 and 37.42, the number must be greater than 36.76
but less than 37.42. Eliminate any answer choice that is less than or equal to 36.76. A. can be
eliminated because 36.67 is less than 36.76.
Step 2: Eliminate any answer choice that is greater than or equal to 37.42. B. and C. can be eliminated
because 37.42 is equal to 37.42 and 37.43 is greater than 37.42.
Step 3: The answer choice that is left is D. 36.77. Since 36.77 is larger than 36.76 and smaller than
37.42, it comes between 36.76 and 37.42.
Answer: D
The following is an activity to help reinforce the skill of ordering numbers. Write the numbers 1 - 100
on index cards. Shuffle the cards and place them face down on the table in a pile. Have the student
choose two cards and then name two numbers that would come between the numbers on the cards.
Once the student has mastered whole numbers, make index cards with decimal numbers on them and
repeat the game.
Number Lines - A
A number line is a line with equally spaced points marked by numbers. Problems include calculating the sum
and difference of points, as well as determining the value of a specific point.
An interesting method for improving the student's understanding of number lines is to develop a series
of number lines. Help the student plot specific points on the number lines.
Example 1: Point A is at which number on this number line?
A. 20
B. 24
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C. 15
D. 10
Solution: Start at the number 12 on the number line and count over 3 to point A. Point A is at 15 on the
number line.
Answer: C
Example 3: Add 3 to Point B. At what point are you on the number line?
A.
B.
C.
D.
27
30
21
24
Solution: Count over from 21, B is at the number 24 on the number line. When we add 3, we get 27.
Answer: A
Line Graphs - A
A graph is a drawing used to show and compare information. A line graph is typically used to show how data
changes over a period of time.
The following is an example of a line graph. A company is interested in how their product sold over a
three month period. The left side of the graph represents the number of products sold. The bottom of
the graph represents the months.
Example 1: During which month did this company sell the most products?
Answer: According to the above graph, the most products were sold in August (40).
Function/Pattern - A
Students must identify the equation that creates a specific pattern or function.
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It may be helpful to develop a series of number patterns and help the student identify the pattern and
determine the correct equation.
Example: The numbers in Column A have been changed to the numbers in Column B by using a
specific rule.
Which number sentence shows that rule?
A.
B.
C.
D.
Ax3=B
A-3=B
A+3=B
A÷ 3=B
Solution: Since the numbers in Column B are less than Column A, we can conclude that Column B is
either being subtracted or divided from Column A. Starting with the first row of numbers, 3 - 3 = 0, so
subtracting will not work. If we divide (3 ÷ 3 = 1) it fits our pattern. Continue down the row to see if
dividing works for all numbers in the pattern:
9÷ 3=3
12 ÷ 3 = 4
15 ÷ 3 = 5
Answer: D
Example 2: Use the given rule to complete the table.
(1) 8 x 2 = 16
(2) 16 + 5 = 21
Step 1: Multiply 8 by 2 to get 16.
Step 2: Add 5 to 16.
Answer: 21
Multiple-step Story Problems - A
Students are presented with problems that assess their ability to interpret data from a written word problem.
Answers are found by solving equations using multiple operations.
It may be helpful to develop a series of multiple-step word problems that relate to the student's activities,
such as allowance or sports. The following is a step-by-step example of a multiple step story problem.
Solve: On Saturday, Stella earned $3.50 for each hour she worked. She earned $3.25 for each hour of
work on Sunday. She worked 5 hours each day. How much money did she earn for both days?
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Step 1: Develop 2 separate equations. One to find the earnings on Saturday, and one to find the earnings
on Sunday.
Step 2: Find the products of the two equations.
Step 3: Add the two products together.
Answer: Stella earned $33.75.
Irrelevant Information - A
Students are assessed on their ability to read a word problem and answer questions using only the information
pertinent to the question asked.
An interesting method for improving the student's awareness of information in a story problem is to develop an
entertaining story filled with obscure details. Ask a question related to only part of the story.
Example: Arnold found 3 green bugs, 4 green lizards, and 8 yellow bugs. How many bugs did Arnold find?
The correct answer is 11 (8 yellow and 3 green). The irrelevant information is the 4 green lizards.
Lines
A line is a straight path extending in both directions with no endpoints. Students must identify sets of lines,
such as parallel lines.
It is important for the student to understand the different types of lines.
Intersecting lines are lines that meet or cross. Intersecting lines have only one point in common.
A diagonal is a line segment that joins two vertices of a polygon, but is not a side of the polygon. For
example, each of the four corners of a rectangle are labeled (ABCD).
A diagonal goes through the middle of the rectangle, connecting either BD or AC.
Parallel lines are lines in the same plane that do not intersect. Parallel lines have no points in common.
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Perpendicular lines are two lines that intersect and form right angles. Perpendicular lines have only one
point in common.
Horizontal lines run parallel to the horizon (left-right).
Vertical lines run perpendicular to the horizon (up-down).
Figures - B
Students must identify various geometric figures.
A creative method for improving this skill is to help the student draw the geometric figures commonly
studied in this grade. After he or she has drawn all the figures, develop a series of flash cards. On one
side of the card, draw the figure. On the other side, write the name of the figure. The following are
figure definitions to help you get started:
Cylinder - a solid with two bases that are congruent circles
Sphere - a solid with all points at a fixed distance from the center
Cone - a solid with one circular face and one vertex
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Pyramid - a solid with one face that is a polygon and three (or more) faces that are triangles with a
common vertex
Cube - a rectangular prism with six congruent square faces
Example 1: How many triangles are there in this figure?
Step 1: Count the obvious triangles - the outside of the figure is one triangle and there are 6 obvious
triangles inside the figure.
Step 2: Count the triangles in the figure that are not obvious. (see diagrams below)
Step 3: Count the total number of triangles.
1 + 6 + 2 + 2 + 2 = 13
Answer: There are 13 triangles in the figure.
Number Patterns - A
A number pattern is a series of numbers that follow a specific rule (such as "add 2" or "subtract 10"). Number
patterns can be created by adding, subtracting, multiplying, or dividing numbers based on the rule of the pattern.
This study guide will focus on patterns that follow addition and subtraction rules.
Determining Rules for Patterns:
An element of a pattern is an individual number that is part of the pattern. Since patterns are read from
left to right, the starting element is the element farthest to the left.
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In order to determine the rule for a pattern, one must look at the individual parts of the pattern. If the
numbers in a pattern increase as the pattern is read, the rule for the pattern most likely involves adding
specific numbers to each element in the pattern (it could also involve multiplying, but this study guide
focuses on addition and subtraction patterns). If the numbers in a pattern decrease, the rule for the
pattern most likely involves subtracting specific numbers from each element in the pattern (it could also
involve dividing, but this study guide focuses on addition and subtraction patterns).
Look at the pattern: 6, 8, 10, 12, 14. As the pattern is read from left to right, the numbers increase, so
this pattern follows an addition rule. The next step is to determine which number or numbers are being
added to each element of the pattern. Since 6 + 2 = 8, the rule for this pattern could be "add 2." Try
adding two to each element in the pattern to see if this is the correct rule. The diagram below shows that
"add 2" is the correct rule for this pattern.
Example 1:
What is the rule for the pattern below?
17, 15, 13, 11, 9
Answer: As the pattern is read from left to right, the elements decrease by two each time, so the rule for
this pattern is "subtract 2." The diagram below shows how the pattern rule works.
All elements of a pattern must fit the determined rule in order for that rule to govern the pattern. If any
element of a pattern does not fit the rule developed by the student, the rule needs to be revised.
Continuing a Number Pattern:
Once a student is able to determine the rule for a pattern, he or she will be able to determine which
number would come next in a pattern.
Example 2:
Complete the number pattern.
6, 10, 14, 18, _____
Answer: 22. The elements in the pattern increase by four each time, so the rule for the pattern is "add
4." Since 18 + 4 = 22, 22 is the number that will complete the pattern.
Example 3:
Complete the number pattern.
36, 33, 30, 27, _____
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Answer: 24. The elements in the pattern decrease by three each time, so the rule for the pattern is
"subtract 3." Since 27 - 3 = 24, 24 will complete the pattern.
One way to discover more about patterns is to have the student create number patterns using his or her
own rules. Write numbers (start with 1-30) on pieces of paper so that the numbers can be moved into
many different patterns. This activity can easily be turned into a game by having one person make a
pattern and timing how long it takes the second person to either determine the rule for the pattern or find
the next element in the pattern.
Similar Figures - A
Similar figures are figures that have the same shape, but not necessarily the same size.
Imagine that you have reduced or enlarged a figure in a photocopy machine - the figure has the same
shape, but not the same size. The figures below are similar figures (they have exactly the same shape,
but they are different sizes).
The figures below are NOT similar because they are not exactly the same shape.
The following are ways to determine that figures are NOT similar:
• The figures are the same height, but different widths.
• The figures are the same width, but different heights.
• One figure is wide and short, while the other figure is tall and skinny (as in the case above).
Example 1:
Which two shapes are similar?
Solution:
A. is not the correct answer. The first triangle is short and wide, while the second triangle is tall and
skinny.
B. is the correct answer. The shape of the triangles is the same and they are different sizes.
C. is not the correct answer. The triangles appear to be the same width, but they are different heights, so
the triangles are not exactly the same shape.
D. is not the correct answer. The triangles appear to be close to the same height, but one triangle is
much skinnier than the other. Therefore, the triangles are not the same shape.
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Example 2:
Choose the pair of similar shapes.
Solution:
A. is not the correct answer. The hexagons appear to be close to the same height, but one is much
skinnier than the other. Therefore, the hexagons are not exactly the same shape.
B. is not the correct answer. One of the flowers is fatter and the other is longer. Therefore, the flowers
are not exactly the same shape.
C. is the correct answer. The shape of the ovals is the same, and they are different sizes.
D. is not the correct answer. The rectangles appear to be the same width, but one rectangle is much
taller than the other. Therefore, the rectangles are not exactly the same shape.
Try the following activity to help reinforce this skill. Cut out different shapes, some that are similar and
others that are not. Mix up the shapes and have the student choose the similar figures. This could be
made into a game in which each person gets a turn to choose a pair of similar figures. Another activity
might involve taking a picture of a cartoon character, drawings of miscellaneous shapes, or other images
to a local copy store. Use the reduction or enlargement feature to change the size of each image in
several different ways. The results will be figures that are similar to the originals.
Identify Shapes in Real World Figures
Plane figures are two-dimensional geometric figures. Students need to be able to identify squares, rectangles,
triangles, circles, ovals, and diamonds by sight and to identify the word names of the figures. In this skill,
students will be asked to find geometric shapes in real world figures. The examples below will help the student
recognize the different ways in which these shapes can be displayed.
Rectangles:
A rectangle is a closed four-sided figure in which the opposite sides are parallel (they will never
intersect or cross) and congruent (exactly the same length), and all four angles are congruent (exactly the
same measure).
Examples of Rectangles:
Squares:
A square is a closed four-sided figure in which the opposite sides are parallel and all four sides are
congruent. Although a square is technically a type of rectangle, students at this level typically do not
consider squares to be rectangles.
Examples of Squares:
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Triangles:
A triangle is a closed three-sided figure.
Examples of Triangles:
Ovals:
An oval is an oblong curved figure that is similar to the shape of an egg.
Examples of Ovals:
Circles:
A circle is a curve in which every point is equally distant from a fixed point, the center.
Examples of Circles:
Diamonds:
A diamond is a closed four-sided figure in which two inner angles are obtuse (larger than 90º ) and the
other two inner angles are acute (less than 90º ).
Examples of Diamonds:
Students will be required to identify these shapes in real world diagrams for this skill. There are three
types of problems. The first involves several real world figures and asks the student to identify the
figure that contains one of the shapes listed above. The second involves matching a real world diagram
to its simplified geometric shape. The third involves naming the geometric shape that makes some part
of the diagram. Examples of these three types of problems are provided below.
Example 1:
In which picture can you find a triangle?
Solution:
The envelope, diagram D, is the only diagram with a triangle. Diagrams A and B contain rectangles and
diagram C contains circles, but none of these have any triangles.
Answer: Diagram D contains a triangle.
Example 2:
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The package is most like which shape?
Answer: The package is most like the prism in answer (B).
Example 3:
The door on the microwave is _______________ .
A. a rectangle
B. an oval
C. a triangle
D. a circle
Answer: A. The door on the microwave is a rectangle.
An activity that will help students learn this skill is to draw circles, squares, rectangles, triangles, and
diamonds on index cards. Then have the student go through a magazine or book, or look around the
surroundings to try to identify each of the shapes in at least one real world setting. For example, the
student could see a circle in the face of a clock in the house, a rectangle in the shapes of the drawers in
the kitchen, and an oval in the face of a favorite cartoon character.
Make Change: Coins/Bills
The ability to make change correctly is a skill that the student will use extensively in real life. There are two
methods for making change. An example of each method follows.
The first method for finding the amount of change that is due a person is to subtract the amount of the
purchase from the amount of money that was given to the cashier.
Example 1:
Kato spent $8.70 to play miniature golf. How much change should he get back if he gives the cashier
$20.00?
Step 1: Set up the problem. Since Kato gave the cashier $20.00 and only spent $8.70 of the $20.00, this
is a subtraction problem. Be sure to line up the decimal points.
Step 2: Subtract the numbers in the hundredths column (0 - 0 = 0).
Step 3: It is now time to subtract the numbers in the tenths column. Since 7 cannot be taken from 0,
trading (borrowing) must occur. There is a zero in the ones place of the top number, so the student must
first borrow from the 2 in the tens column. Cross off the 2 and make it a 1. Add 10 to the amount in the
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ones place of the top number (0 + 10 = 10).
Step 4: It is still not possible to subtract the numbers in the tenths column, but it is now possible to trade
using the ones column. Cross out the 10 and make it a 9. Add 10 to the amount in the tenths column of
the top number (0 + 10 = 10). Now it is possible to subtract the numbers in the tenths column.
Step 5: Subtract the numbers in the tenths column (10 - 7 = 3). Bring the decimal point straight down.
Subtract the numbers in the ones column (9 - 8 = 1). Subtract the numbers in the tens column (1 - 0 =
1). Finally, bring down the dollar symbol.
Answer: Kato will get $11.30 in change.
The second method for determining the amount of change someone should receive is to count back.
Counting back is the process of starting with the amount of money that was paid and counting back to
the amount of money that was given to the cashier. Counting back is often done by cashiers when they
give change back from a purchase.
Example 2:
Sunee decided to buy a smoothie and a chocolate covered frozen banana. How much change should she
get back if she gives the cashier $10.00?
(1)
(2)
(3)
(4)
$5.25 + $4.15 = $9.40
$9.40 + $0.10 = $9.50
$9.50 + $0.50 = $10.00
$0.10 + $0.50 = $0.60
Step 1: Add the cost of a smoothie and the cost of a chocolate covered banana to determine the total
amount Sunee spent.
Step 2: Since Sunee spent $9.40 and she gave the cashier $10.00, start counting back from $9.40 and
stop when $10.00 is reached. The first step is to add an amount that will get the total to the nearest
twenty-five, fifty, or seventy-five cents. In this case, it is easiest to get the total to the nearest fifty cents
($9.40 + $0.10 = 9.50).
Step 3: Once the total is to the nearest twenty-five, fifty, or seventy-five cents, add an amount that will
bring the total up to the nearest whole dollar. In this case, add $0.50 to $9.50 to get $10.00. (If the total
had come to less than the amount given to the cashier, it would have been necessary to add whole dollar
amounts until the counting up total equaled the amount given to the cashier.)
Step 4: Add together the amounts that were added to $9.40 in the process of counting back ($0.10 +
$0.50 = $0.60). This amount is the change that Sunee will receive.
Answer: $0.60
To help the student become proficient at making change, use play or real money to create scenarios in
which the student must determine the amount of change someone is to receive. Also, the next time he or
she buys something at the store, have the student try to determine how much change he or she should
receive.
Most Likely Outcomes
This skill will help students learn to identify the most likely outcome in a probability situation. A probability
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situation is a situation in which outcomes have a chance of occurring (for example, choosing a particular card
from a deck of cards). An outcome is one of the possible events in a probability situation. The most likely
outcome is the outcome that will occur most of the time if it is chosen randomly (as when a person chooses the
card with closed eyes).
To determine which outcome is most likely, determine the possible outcomes and the number of times
each outcome can occur. The outcome with the highest number of chances of occurring is the most
likely outcome.
Example 1: Lilo is trying to get money out of her piggy bank. The coins below are in the bank. If she
shakes one coin out without looking, which type of coin is it most likely to be?
Step 1: Determine all of the types of coins that are in Lilo's piggy bank: quarters, dimes, nickels, and
pennies.
Step 2: Determine the number of each type of coin that is in her piggy bank.
Step 3: Determine which type of coin occurs the most in
the piggy bank. Since there are three nickels in the piggy bank, the nickel is the most likely outcome.
Answer: B.
Example 2: If Hector spins the spinner, which section is the arrow most likely to land on?
Solution: The spinner has four sections. Since the question is asking for the section that the arrow is
most likely to point to, look for the section that is the largest. The gray section is the largest, so it is the
section that the arrow is most likely to point to.
Answer: C.
An activity that will help reinforce this skill involves cutting shapes out of construction paper. Cut out
five identical copies of each shape (for a shortcut, buy pre-cut foam shapes at a craft store). Use the
shapes to create probability situations for the student and have the student determine the most likely
outcome for each situation. Then, have the student create probability situations for someone else to
answer.
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Tally Chart
A tally chart is a type of table that uses marks to help count the results. Common uses of tally charts include
keeping track of points during a game and keeping track of the number of people who like a specific item. The
tally chart below shows the favorite types of sandwiches of a group of students.
On the tally chart, each vertical mark stands for 1 student and each group of marks with a slash through
them stands for 5 students. According to the chart, 17 students chose peanut butter & jelly as their
favorite sandwich.
Example 1: The Garrison family decided to keep track of the number of chores each child did for one
week. The tally chart shows the results.
How many chores did Magna do?
Solution: Count the tallies.
Answer: 8 chores
Example 2: The reading groups in Mr. Silva's class recorded the number of books they read for a week.
The tally chart shows the results.
How many more books did the Blue Jays read than the Purple Marlins?
(1) Blue Jays = 19 books
(2) Purple Marlins = 17 books
(3) 19 - 17 = 2 books
Step 1: Determine the number of books that the Blue Jays read (19).
Step 2: Determine the number of books that the Purple Marlins read (17).
Step 3: Subtract the number of books the Purple Marlins read from the number of books the Blue Jays
read.
Answer: 2 books
An activity to help the student read and interpret information from tally charts is to find tally charts in
magazines, newspapers, and on the Internet. Using the tally charts that were found, ask the student
questions similar to those in the examples given above. To further the student's knowledge of tally
charts, have him or her collect data from family members and create an original tally chart using the
data.
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Least Likely Outcomes
This skill will help students learn to identify the least likely outcome in a probability situation. A probability
situation is a situation in which outcomes have a chance of occurring (such as choosing a specific card from a
deck of cards). An outcome is one of the possible events in a probability situation. The least likely outcome is
the outcome that will occur the least number of times if it is chosen randomly (such as with the eyes closed).
To determine which outcome is least likely, determine the possible outcomes and the number of times
each outcome can occur. The outcome with the lowest number of chances of occurring is the least likely
outcome.
Example 1: Lien collected these bugs from her backyard and put them in a jar. If one bug gets out
while Lien is putting food in the jar, which type of bug is least likely to get out?
Solution: Determine the number of each type of bug that is in the jar.
The striped bug is the outcome with the lowest number of chances of occurring because there is only
one. Therefore, it is the least likely outcome.
Answer: C.
An activity to reinforce this skill involves cutting shapes out of construction paper. Cut five identical
copies of each shape (or use pre-cut foam shapes from a craft store). Use the shapes to create
probability situations for the student and have the student determine the least likely outcome for each
situation. Then have the student create probability situations for someone else to answer.
Inverse Operations - A
Operations are actions, such as addition, subtraction, multiplication, and division, that are performed on
numbers. Certain pairs of operations are considered inverse operations because they have opposite effects on
numbers. In other words, they "undo" each other. For example, start with 2 and add 3 to it; the result is 5 (2 + 3
= 5). Then start with 5 and subtract 3 from it; the result is 2 (5 - 3 = 2). Since adding 3 to 2 makes 5 and
subtracting 3 from 5 makes 2, addition and subtraction are inverse operations. The other pair of inverse
operations at this level is multiplication and division, but the focus of this study guide will be addition and
subtraction. The student may have been taught to call these pairs of operations opposite operations. To avoid
confusion, use the terminology the student has already been taught.
Inverse operations can be used to check whether computation has been completed correctly. To do this, use
addition to check subtraction and subtraction to check addition. In order for a number sentence to check the
correctness of a computation, both number sentences must contain identical numbers, use inverse operations,
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and be true.
Example 1: Which number sentence can be used to check that 5 - 1 = 4?
A.
B.
C.
D.
5+1=6
5-2=3
4+1=5
4-1=3
Solution: Since the operation that was used in the question was subtraction (5 - 1 = 4), the operation that can be
used to check the correctness of the problem is addition. Answer choices B. and D. can be eliminated because
they do not use addition to check the subtraction. Answer choice A. can be eliminated because it does not
contain the same numbers as 5 - 1 = 4. The answer choice that is left is C. 4 + 1 = 5. This is the correct answer
because it uses addition to check subtraction, it contains all the same numbers as the original number sentence,
and it is true.
Answer: C.
Example 2: Which number sentence would you use to check that 10 + 5 = 15?
A.
B.
C.
D.
15 - 5 = 10
15 + 5 = 20
10 - 5 = 5
10 + 15 = 25
Solution: Since the operation that was used in the question was addition, the operation that can be used to check
the correctness of the problem is subtraction. Answer choices B. and D. can be eliminated because they do not
use subtraction. Answer choice C. can be eliminated because it does not contain the same numbers as 10 + 5 =
15. The answer choice that is left is A. 15 - 5 = 10. This is the correct answer because it uses subtraction to
check addition, it contains all the same numbers as the original number sentence, and it is true.
Answer: A.
An activity that will help reinforce this skill is to create an inverse operations game. Write an addition or
subtraction number sentence on an index card that has been cut in half. On the other half, write a number
sentence that could be used to check it. Do this for about ten number sentences. Shuffle the cards and lay them
face up on a table. Challenge the student to match the number sentences that will check each other. For more
of a challenge, lay the cards face down on a table and have the student turn over two cards to see if they will
check each other. If they will check each other, the student gets to pick up the two matching cards. If they will
not check each other, the student must turn over the two cards and try again. This can easily be made into a
game for two players - when a match is not made, the next player gets a turn; if a match is made, the player that
made the match gets to try again.
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