Study Guide
Grade 2 Math-a-thon
Identify Number Sentences - A
A number sentence is a mathematical sentence that includes numbers and operational symbols (such as
+, -,
, ÷ , and =). Examples of number sentences:
3+5=8
7-2=5
This skill will focus on addition and subtraction number sentences. Students will be asked to determine
the number sentence that could be used to solve a real world addition or subtraction problem. The table
below shows common phrases associated with addition and subtraction.
Example 1: Kim has 8 paintbrushes. If she gets 10 more for her birthday, how would you figure out
how many paintbrushes she will have?
A.
B.
C.
D.
10 - 8 = 2 paintbrushes
8 + 10 = 18 paintbrushes
18 + 8 = 26 paintbrushes
18 - 8 = 10 paintbrushes
Step 1: Determine which operation to use. Since she already has 8 paintbrushes and she will get 10
more, this problem requires addition.
Step 2: Determine which numbers to use in the problem. Since she already has 8 paintbrushes and she
will get 10 more, the correct solution would be to add 8 and 10.
Step 3: Using the results of the first two steps, write a number sentence that could be used to solve the
problem: 8 + 10 = 18. Do not forget to label the answer since this is a word problem! So, 8 + 10 = 18
paintbrushes.
Step 4: Choose the correct answer from the answer choices that are given in the problem.
Answer: B. 8 + 10 = 18 paintbrushes
Example 2: Luis had 16 marbles. He lost 5 marbles when he was playing with them. How would you
figure out how many marbles Luis has now?
A. 16 + 5 = 21 marbles
B. 21 + 16 = 37 marbles
C. 21 - 5 = 16 marbles
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D. 16 - 5 = 11 marbles
Step 1: Determine which operation to use. Since he had 16 marbles and he lost 5 marbles, this
problem requires subtraction.
Step 2: Determine which numbers to use in the problem. Since he had 16 marbles and he lost 5
marbles, the correct solution would be to subtract 5 from 16.
Step 3: Using the results from the first two steps, write a number sentence that could be used to solve the
problem (remember to label the answer): 16 - 5 = 11 marbles.
Step 4: Choose the correct answer from the answer choices that are given in the problem.
Answer: D. 16 - 5 = 11 marbles
A creative way to help the student work on this skill is to make up a series of word problems and then
have the student write a number sentence that will solve each problem. To make a game, write each
word problem on an index card and the number sentence that will solve it on another index card, then
have the student match them up.
Measure Capacity
Capacity is the amount of liquid or solid a container can hold. In the U.S. customary system, capacity is
measured using cups, pints, quarts, or gallons.
When measuring capacity, read the mark on the measuring container that is at eye level to the surface of
the substance being measured. Reading the mark at eye level gives the most accurate reading,
particularly for liquids.
Example 1: How many cups of salt are in the measuring cup?
Step 1: Look for the eye level mark at the surface of the salt.
Step 2: Read the scale number at the eye level mark.
Answer: 3 cups
Example 2: How much juice is in the glass?
Step 1: Look for the eye level mark at the surface of the juice.
Step 2: Read the scale number at the eye level mark.
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Example 3: What is the greatest amount this mug can hold?
Solution: Read the scale on the mug. The largest amount on the scale is 8 ounces, so the greatest
amount the mug can hold is 8 ounces.
Answer: 8 ounces
The following activity can help reinforce the concept of measuring capacity. Allow students to explore
using various measuring containers. Have them fill the containers with a specified amount of a liquid or
solid. This process will give students practice filling containers to eye level with specified amounts.
Order Objects By Length
Ordering objects by length allows students to compare two objects, and decide which is shorter or longer.
When ordering three or four objects by length, this comparison occurs more than once.
Example 1: Which object is the longest? Which object is the shortest?
Step 1: Line all of the objects up with their left edges along a line -- the line could be a ruler, a piece of
paper, a table edge, etc.
Step 2: Compare the three items to find the one that extends the farthest to the right (the arrow). This
item is the longest. Then find the object that extends to the right the least (the diamond). This item is
the shortest.
Answer: The arrow is the longest object, and the diamond is the shortest object.
Example 2: Order the objects from longest to shortest.
Step 1: The star, rectangle, and octagon have already been placed on rulers with their left edges at the
zero mark. (Please note that the zero mark does not always occur at the "end" of a ruler.)
Step 2: Compare them to find the object that extends the farthest to the right. The rectangle extends to
the right the farthest, so it is the longest.
Step 3: Compare the octagon and the star to find the one that extends the farthest to the right. The star
extends the farthest to the right, so it is the second longest.
Step 4: The octagon is the only object left over, so it is the shortest.
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Answer:
Example 3: Order the objects from shortest to tallest.
Step 1: Line all of the objects up with their bottom edges along a line -- the line could be a ruler, a piece
of paper, a table edge, etc.
Step 2: Compare the rectangle, pentagon, and surfboard to find the one that extends upward the least
(the pentagon). The pentagon is the shortest object.
Step 3: Compare the rectangle and the surfboard to find the one that is shorter. The rectangle is the
shorter of the two objects, so it is the second shortest of the original objects.
Step 4: The surfboard is the only object left over, so it is the longest.
Answer: pentagon, rectangle, surfboard
To reinforce the concept of ordering objects by length, give students 3 to 4 objects and ask them to
arrange the objects in order from shortest to longest, shortest to tallest, longest to shortest, or tallest to
shortest.
Elapsed Time
Approximating elapsed time periods and predicting future time periods are important skills. Students need to be
able to determine time periods that are appropriate for various activities, and to understand how to apply terms
related to time periods. These terms include "before," "after," "yesterday," "today," "tomorrow," "day," "night,"
"morning," "evening," "afternoon," and others that tell students when events occurred or will occur.
Example 1: When do people usually eat breakfast?
A. after lunch
B. in the morning
C. in the evening
D. in the afternoon
Solution: Think about the time period when people normally eat breakfast. Then reveiw each answer choice.
A. is not the correct answer because breakfast is usually eaten early in the day.
B. is the correct answer, since breakfast is usually one of the first things we do in the morning.
C. is not the correct answer because breakfast is usually eaten early in the day.
D. is not the correct answer because breakfast is usually eaten early in the day.
Answer: B. in the morning
Example 2: If Friday was yesterday, what day is today?
A. Monday
B. Thursday
C. Sunday
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D. Saturday
Solution: Starting with Friday, count forward one day in the week. The weekdays go in a particular order:
Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday. Since yesterday was Friday, today is
Saturday. Review each answer choice to determine the correct answer.
Answer: D. Saturday
Example 3: If today is November 25, what day was yesterday?
A. November 26
B. November 24
C. November 23
D. November 22
Solution: It may be helpful for the student to see an actual calendar when answering these types of questions.
Starting with November 25, count back one day on the calendar. Since the days on a calendar are arranged in
numerical order, yesterday would be November 24 if today is November 25. Review the answer choices to
determine the correct answer.
Answer: B. November 24
Reinforce the concepts of elapsed and future time by giving students scenarios, along with a limited number of
possible answers for each scenario. Then, ask the student to choose the most appropriate answer choice. For
example, "I brush my teeth ________(before, after) I go to bed." The correct answer is "before".
Add Whole No: No Regrouping
Addition of one to two digits without regrouping (carrying or trading) is an important step in the development
of strong math skills.
It may be helpful to use objects to explain addition to the student. For example, start with 4 baseball
cards and give him or her 2 more cards. The student should see that he or she now has 6 cards.
Continue this using different objects.
Example 1:
Step 1: Add the ones column (0 + 2 = 2). Write the 2 in the ones place below the line.
Step 2: Add the tens column (2 + 1 = 3). Write the 3 in the tens place below the line.
The answer is: 20 + 12 = 32.
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Add Whole No: Regrouping
Adding two numbers often requires regrouping (carrying, trading, or renaming). Regrouping occurs when the
total of the numbers in a column is equal to or greater than ten. Problems are presented in vertical format only.
It may be helpful to use counters to explain addition with regrouping to the student. Cut an end off an
empty egg carton so that you have a ten frame carton. Use 2 different colors of buttons or dried beans
for counters. Use one color of counter to represent the first digit in the problem by placing one button
per frame in the egg carton, the other color to show the second digit. The student should see that not all
of the counters will fit into the ten frame carton and need to be carried to the next place value.
Example 1: 32 + 28 = ?
Step 1: Write the problem vertically.
Step 2: Add the ones column (2 + 8 = 10). Write the 0 in the ones place below the line and "carry" the 1
to the tens column above the 3.
Step 3: Add the tens column (1 + 3 + 2 = 6). Write the 6 in the tens place below the line.
The answer is: 32 + 28 = 60.
Compare Whole Numbers - A
Comparing numbers involves determining which number is larger in value than the other. Knowledge of
numbers and the order symbols (<, >, and =) are needed for this skill.
It may be helpful to review ordering symbols with the student.
Develop a series of numbers and help the student insert the appropriate ordering symbols.
Example 1: Which one is correct?
A.
B.
C.
D.
24 < 19
12 > 15
5 < 10
34 < 30
Solution: Write each pair of numbers so that one is on top of the other one and compare each place
value working from left to right. When a number has a place value that is larger than the corresponding
place value in the number it is being compared to, that number is greater.
A. 24
19
B. 12
15
C. 5
10
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D. 34
30
A. 24 is greater than 19, so the statement is false.
B. 12 is less than 15, so the statement is false.
C. 5 is less than 10, so the statement is true.
D. 34 is greater than 30, so the statement is false.
Answer: C.
Example 2: Fill in the missing symbol.
34 _____ 43
A. >
B. <
C. =
Solution:
A. is not the correct answer because 34 is less than 43.
B. is the correct answer because 34 is less than 43.
C. is not the correct answer because 34 is not equal to 43.
Answer: B.
Use the following game to reinforce the skill of comparing whole numbers. Cut fifty index cards in half,
then write the numbers 1 - 100 on the cards. Next, write the <, >, and = symbols on three full-sized
index cards. Have the student draw two cards from the number cards and use the ordering symbols to
put the cards in order. If the student is struggling, write the words that each symbol stands for above the
symbol on the index cards. Once the student has mastered these numbers, remove 1 - 50 and make cards
for 101 - 150.
Length - A
The questions dealing with length assess a student's ability to measure objects using a standard ruler. An object
is shown next to a ruler, and the student is asked to determine the length of the object by reading the numbers
on the ruler.
To measure the length of an object, line the object up with the zero mark on the ruler. The length can be
determined by looking at the number where the object stops. In these examples, all of the rulers start at
zero, but in real life, not all rulers start at zero. When working with an actual ruler, the student will need
to determine where the zero mark is on the ruler.
Example 1: About how long is the wrench?
Solution: The left end of the wrench is lined up with the zero (or end) of the ruler and the right end of
the wrench stops at the 9, so the wrench is 9 units long. If the ruler had been marked with a specific unit
(such as inches), the wrench would be 9 inches long instead of 9 units long.
Answer: 9 units
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If an object is not lined up to the end of a ruler, the length can be determined by subtracting the number
where the object starts from the number where the object stops.
Example 2: About how long is the rope?
Solution: One end of a rope is lined up with 2 cm, and the other end is lined up with 10 cm. To find the
length of the rope, subtract 10 - 2 to get 8. The length of the rope is 8 cm.
Answer: 8 cm
If the object being measured is not straight or flat, but is oddly shaped, it may be necessary to use the
ruler to estimate the overall length or width.
Example 3: About how wide is the clock?
Solution: The widest part of the clock reaches from the 1 inch mark to the 3 inch mark, and 3 - 1 = 2, so
the clock is about 2 inches wide.
Answer: 2 inches
The best way to practice measuring length is to use a ruler to measure different objects around the
house. After the student has practiced measuring, challenge him or her to estimate length and find
objects that are close to an inch long, six inches long, twelve inches long, etc. Then have the student
measure the objects to check the estimated lengths. Remember to line up each object so that it starts at
zero (0) on the ruler.
Add Decimals: No Regrouping
Adding two decimal numbers with more than one digit is very similar to adding whole numbers. At this level,
problems are presented in vertical format with the decimal points lined up correctly. There is no regrouping,
carrying, borrowing, or trading involved in this skill.
A decimal number is a number that uses place value and a decimal point to show tenths, hundredths,
thousandths, etc. For example, the number 3.57 has a 5 in the tenths place and a 7 in the hundredths
place.
The difference between adding decimals and adding whole numbers is the fact that the decimal points
must be lined up before addition can occur. The following is a step-by-step example of adding
decimals.
Example: Solve.
2.3
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+ 1.4
Step 1: Add the tenths column (3 + 4 = 7). Write 7 in the tenths column (below the line).
Step 2: Bring down the decimal point.
Step 3: Add the ones column (2 + 1 = 3). Write 3 to the left of the decimal point to finish the problem.
The correct answer is 2.3 + 1.4 = 3.7.
Add Decimals: Regrouping
Adding two decimal numbers with regrouping is very similar to adding whole numbers with regrouping. These
problems are presented in a vertical format with the decimal points lined up correctly. These problems require
regrouping, carrying, or trading.
A decimal number is a number that uses place value and a decimal point to show tenths, hundredths,
thousandths, etc. For example, the number 3.57 has a 5 in the tenths place and a 7 in the hundredths
place.
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 with regrouping:
Example 1: Solve. 2.4 + 1.68 = ?
Step 1: Write the problem vertically. Make sure the decimal points are lined up.
Step 2: Add one zero to 2.4 so that the two numbers have the same number of digits.
Step 3: Add the hundredths column (0 + 8 = 8). Write the 8 in the hundredths column (below the line).
Step 4: Add the tenths column (4 + 6 = 10). Write the 0 in the tenths column (below the line) and carry
the 1 to the ones column (left of the decimal point).
Step 5: Bring the decimal point straight down.
Step 6: 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.
The correct answer is 2.4 + 1.68 = 4.08.
Example 2: Solve. 3.2 + 2.8 = ?
Step 1: Write the problem vertically. Make sure the decimal points are lined up.
Step 2: Add the tenths column (2 + 8 = 10). Write the 0 in the tenths column (below the line) and carry
the 1 to the ones column (left of the decimal point).
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Step 3: Bring the decimal point straight down.
Step 4: Add the ones column, including the 1 that was carried (1 + 3 + 2 = 6). Place the 6 to the left of
the decimal point to finish the problem.
The correct answer is 3.2 + 2.8 = 6.0.
Counting
Counting requires the student to complete a given number pattern. For example:
What are the next three numbers? 45, 46, 47, ___, ___, ___.
The student must determine which set of numbers given in the answer choices will fit correctly in the blanks
(48, 49, 50).
Use 3 x 5 cards to make a stack of cards numbered 1-100. Shuffle the cards and turn them face down in a pile.
Take turns picking the top card and then counting to the next three places.
Example: If the student draws a card with the number 23 on it, he or she must say "23, 24, 25, 26."
As the student shows improvement with this skill, change the game to count backwards from the value of the
card picked. (Answer: 23, 22, 21, 20)
Tables - A
A table or chart is a set of data organized into columns and/or rows. Tables and charts are often used to display
or record information in an organized way. They often have labels, called headings, that explain the type of
information included. Common examples include calendars, schedules, and menus. Students need to be able to
read tables and charts and use the information to make comparisons or determine quantities.
The table below shows information on the growth of a plant over several weeks. Each week, the plant's
height was recorded.
To determine the height of the plant in week 1, for example, find the "1" in the "Week" column, then
look to the right to see the corresponding height. The chart shows that the plant was two inches tall the
first week, three inches tall the second week, five inches tall the third week, and seven inches tall the
fourth week.
Example 1: How many push-ups can Ali complete?
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Solution: To determine the answer, find the name "Ali" in the top row and the activity "Push-Ups" in the
left column. Follow the "Ali" column down, and the "Push-Ups" column across until the two meet (at 8)
Answer: 8 push-ups
Example 2: Who completed the most sit-ups?
Solution: To determine who completed the most sit-ups, look across the "Sit-Ups" row to find the
largest number, which is 8. Then, look above the 8 in the same column to find Syd's name.
Answer: Syd
Example 3: The chart below shows the rainy days in June.
Which day of the week had the most rainy days in June?
Answer: Friday. According to the calendar, there were three rainy Fridays: the ninth, the sixteenth, and
the thirtieth.
A creative method for improving the student's data interpretation skills is to use actual tables and charts
from magazines or books. Help the student understand the information provided in these tables by
asking him or her specific questions about it. Have the student find who or what has the highest or
lowest amounts.
Missing Elements - A
Equations are presented with either a number or operational symbol omitted. Students must solve the equation
by determining what is missing.
It may be useful to develop a series of mathematical equations with either a number or operational
symbol missing. Help the student replace the missing elements.
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Example 1: 6 ? 4 = 10
The missing element is the plus symbol (+), we know this because 6 + 4 = 10.
Example 2:
Solution: Since 5 + 2 = 7, the missing digit is 2.
Example 3:
Solution: Since 7 - 5 = 2, the missing digit is 7.
Operational Symbols - A
Operational symbols are the symbols used to indicate the type of equation. Addition (plus, +) and subtraction
(minus, -) are examples of operational symbols. In this skill, the student must determine which symbol is
needed to make the problem correct.
It may be beneficial to verify that the student can identify the operational symbols commonly used in this grade
level.
Symbol
+
-
Definition
addition
subtraction
Help the student identify operational symbols in various equations.
Example: Replace the ? with a plus (+) or minus (-) sign.
10 ? 3 = 7
A. +
B. 10 + 3 = 13. 10 - 3 = 7. Therefore, minus (-) is the correct symbol for this equation.
Answer: B
Before/After/Between
This skill involves ordering numbers using the terms "before", "after", and "between."
Use numbered cards (1-100) made from 3 x 5 cards. Shuffle and take turns drawing the top card. During the
first round, you must say the number that comes immediately AFTER the drawn card. During round two, say
the number that comes immediately BEFORE the drawn card. To practice "between," ask the student to tell
you what number comes between two numbers, such as 47 and 49.
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It may be helpful to create questions like those illustrated below:
Example 1: Which number comes right BEFORE 37?
A.
B.
C.
D.
38
36
40
39
Answer: B
Example 2: What number comes BETWEEN 57 and 59?
A.
B.
C.
D.
56
57
58
59
Answer: C
Example 3: What number comes AFTER 29?
A. 28
B. 31
C. 210
D. 30
Answer: D
Geometric Patterns - A
A geometric pattern is a series of geometric shapes that follow a specific rule (such as "square, circle, triangle,
oval, square, circle, triangle, oval"). Geometric patterns can consist of a series of repeating geometric shapes,
they can consist of a series in which the same shape is oriented in different directions, or they can consist of
geometric shapes filled with a series of repeating colors or designs inside the shapes. Patterns can follow more
than one of these rules at a time.
Determining Rules for Patterns:
An element of a geometric pattern is an individual geometric shape that is part of the pattern. Since
patterns are read from left to right, the starting element is the element farthest to the left.
In order to determine the rule for a pattern, one must look at the individual parts of the pattern. First,
determine whether the pattern is made up of one shape or multiple shapes. If the pattern is made up of
multiple shapes, write out the names of the shapes to see if the shapes follow a rule.
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A pattern with multiple shapes can consist of more than one rule. The pattern below follows two rules:
there are three shapes that repeat and there is a design pattern that repeats inside the shapes.
If the pattern is made up of one shape, then the pattern could follow a rule that is based upon the
orientation of the shape or the pattern could follow a rule that is based upon the color or design inside
the shape.
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, then the rule needs to be revised.
Extending Geometric Patterns:
Once a student is able to determine the rule for a geometric pattern, he or she will be able to extend (or
continue) a pattern.
Example 1:
Which shape comes next?
Answer: (C) is the correct answer (circle). The pattern rule is: triangle, circle, square, oval, repeat.
Since the last element shown in the pattern is a triangle and circle follows triangle, circle is the next
element in the pattern.
Example 2:
Which shape comes next?
Answer: (A) is the correct answer (white). The pattern rule is: square with side-to-side stripes, black
square, white square, square with up-and-down stripes, repeat. Since the last element shown in the
pattern is black square and white square follows it, white square is the next element in the pattern.
One way to discover more about patterns is to have the student create geometric patterns using his or her
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own rules. Cut geometric shapes out of pieces of paper so that the shapes can be moved into many
different patterns. Draw designs on some of the shapes to make patterns with gradients. This activity
can 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 missing element in the pattern. To make
the activity easier, visit a craft store and buy shapes that are already cut out of foam or wood.
Convert Time
This skill requires that students convert time between weeks and days, years and months, hours and minutes, or
days and hours. Students will need to know the basic time conversions below.
1 hour
1 day
1 week
1 year
=
=
=
=
60 minutes
24 hours
7 days
12 months
First, ask the student the following questions to make sure he or she understands the basic conversions.
• How many days are in one week? (Answer: 7)
• How many hours are in one day? (Answer: 24)
• How many minutes are in one hour? (Answer: 60)
• How many months are in one year? (Answer: 12)
Once the student knows the basic conversions, the student should be able to do more difficult
conversions like the ones below.
Example 1:
2 weeks = _____ days
Answer: 14. The student could use 7 days + 7 days = 14 days to determine the answer.
Example 2:
24 months = _____ year(s)
Answer: 2. The student could use repeated subtraction to determine the answer. There are 12 months in
one year, so subtract 12 from 24 (24 - 12 = 12). There are months (12) left over from the first
subtraction, so subtract 12 from the number of months left over (12 - 12 = 0). Since 12 can be
subtracted from 24 twice, it would mean that 24 months equals 2 years).
An active way to work on time conversions is to write the conversions on index cards. Then, cut the
cards in half and have the student match the parts back up again. The pieces could either be cut exactly
in half or they could be cut to look more like puzzle pieces. See the diagram below.
Combined Shapes
Combining shapes to make other figures relies heavily on the student's skill level with spatial relationships.
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Spatial relationships deal with the ability to mentally flip, slide, and turn shapes to match them to another
drawing or to fit them together to make another figure.
An activity that will help the student explore how shapes can be combined is to cut circles, squares,
rectangles, triangles, diamonds, pentagons (five-sided shapes), hexagons (six-sided shapes), and
octagons (eight-sided shapes) out of construction paper. Put two of the construction paper shapes
together on top of a plain piece of paper and trace the outline of the new figure. Remove the
construction paper shapes and trace more figures using different shapes. Have the student use the
shapes that were cut out of construction paper to decide which shapes were used to make up each figure.
When the student seems to have mastered the concept that simple shapes can be used to make more
advanced figures, try using three shapes to make figures.
Example 1:
Which two shapes can be put together to make the following figure?
Hint: Look for shapes in the original figure that are easily recognizable, such as squares, rectangles,
triangles, circles, etc.
(A) is not the correct answer. While the original figure is made up of a triangle and a rectangle, the
triangle in this answer choice is not wide enough to match the triangular part of the original figure.
(B) is not the correct answer. While the original figure is made up of a triangle and a rectangle, the
rectangle in this answer choice is too wide to match the rectangular part of the original figure.
(C) is the correct answer. The triangle needs to slide to the right, but the triangle and the rectangle
would fit exactly over the triangular and rectangular parts of the original figure.
(D) is not the correct answer. The original figure is made up of a triangle and a rectangle. This answer
choice shows a triangle and a square.
Example 2:
Which new figure can be made using the shapes below?
(A) is the correct answer. If the lower triangle is turned so that the side on the right sits on the bottom,
the two triangles will fit together to make a square.
(B) is not the correct answer. Turning the figures to make one triangle out of the two that are given will
create a much wider triangle than the figure in answer choice B.
(C) is not the correct answer. If the lower triangle is turned so that the side on the right sits on the
bottom, the two triangles will fit together to make a square, not a rectangle.
(D) is not the correct answer. The figure in answer choice D has six sides and six corners. There is no
way to flip, slide, or turn the two original triangles to make a figure that has six sides or six corners.
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Sort/Classify Objects
Sorting and classifying are methods that are used to place objects into groups. Students will be asked to sort the
items that are in groups and use classification to determine which item belongs in a group. Objects can be
sorted by one or more characteristics such as shape, size, color, or function.
Example 1: Which shape belongs in the box?
Solution: Analyze the items in the box. Look for common shapes, colors, sizes, etc. The only
characteristic that all of the shapes in the box have in common is color. All of the shapes in the box are
solid black. It is safe to assume that the shape that would go in the box would also be black. A, B, and
D are not correct answer choices because they are not solid black.
Answer: C. The shape that would go in the box is the black shape.
Example 2: Which shape in the box does NOT belong?
Solution: Analyze the items in the box. Do the items in the box have anything in common? Three of
the items in the box are the same shape, so the characteristic that the items in the box have in common is
shape (color is not the common characteristic because only two of the four items are the same color).
The item that is not the same shape as the others does NOT belong in the box. Three of the four items in
the box are shapes with five sides and the fourth item has six sides. The item with six sides does NOT
belong.
Answer: C. The item that does NOT belong in the box is the six-sided figure.
Example 3: Which object belongs in the box?
Solution: Analyze the items in the box. Do the items in the box have anything in common? The items
in the box are all types of food. This is the only common characteristic, so the item that would fit in the
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box would also have to be a food item. B, C, and D are not correct answer choices because they are
types of drink, not types of food.
Answer: A. The item that would go in the box is the food item.
An activity to help the student master sorting and classifying is to have the student make a collage using
old magazines. Have the student cut out items that go together in some way, such as items that are all
yellow or animals that are all pets. The student should then glue the items that are similar onto a piece
of construction paper, labeling what the items have in common. An alternate method would be to cut
items out for the student so that he or she could sort and classify them.
Fractions: Estimation - A
Estimating fractions requires an understanding of fractional parts. Fractional parts are the portions that when
put together make a whole. At this level, students generally deal with estimating fractions to the half and to the
whole.
One way to help a student visualize fractions is to cut circles out of construction paper. Explain to the
student that each circle is one whole. Then cut the circle exactly in half and explain to the student that
each of the two pieces of the circle is one-half. Put the two pieces back together again to show the
student that the two halves of the circle make one whole. Do this with different-sized circles to show the
student that one-half can be different sizes depending upon the size of the whole. Also show the student
that the one-half pieces from two circles that were different sizes to begin with will not fit together to
make one whole.
Estimating One-Half:
To help the student learn to estimate one-half, cut a wedge out of a circle. Ask the student if the wedge
is more than half or less than half of the original circle. Do this a few times with circles and rectangles
to help the student master the concept of half.
Once the student has mastered the concept of half, help the student estimate one-half when the object is
broken into equal-sized pieces that are smaller than one-half. Draw a rectangle on paper, then draw
lines on it to break it into pieces of equal size (see the diagram below). Color in several of the pieces
and ask the student whether more than half or less than half of the pieces are shaded.
Example 1:
How much of the circle is shaded gray?
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A. more than half
B. less than half
Answer: B. less than half. The circle is cut into ten equal parts; therefore, five of the parts would equal
one-half. Since only four of the ten parts are shaded, this represents less than half.
Example 2:
Pick the picture that shows more than half of the parts are shaded gray.
A. is not the correct answer. There are eight squares, so four of them would be half. Since there are
four squares shaded gray, exactly half are shaded gray.
B. is not the correct answer. Since only three of the eight squares are shaded, this would be less than
half.
C. is the correct answer. Five of the eight squares are shaded. Since five is larger than four (which
would be exactly half), more than half of the squares are shaded gray.
D. is not the correct answer. Since only two of the eight squares are shaded, this would be less than half.
Example 3:
How much cake has been eaten?
A. more than half
B. less than half
Answer: B. less than half. Since more than half of the cake is left, less than half of the cake has been
eaten.
Estimating One Whole:
When estimating one whole, it is important to realize that more than half of the parts need to be shaded
in order to estimate up to the whole (see the diagram below).
To help the student understand the concept of estimating to the whole, draw a few different rectangles
and circles and shade in a different number of parts on each shape. Have the student decide whether or
not the parts that are shaded would estimate to one whole.
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Example 4:
Pick the picture that shows almost one whole is shaded gray.
A. is not the correct answer. Two of the four parts are shaded gray and that is exactly half, so it would
not estimate to one whole.
B. is not the correct answer. Two of the four parts are shaded gray and that is exactly half, so it would
not estimate to one whole.
C. is the correct answer. Three of the four parts are shaded gray and that is greater than half, so it would
estimate to one whole.
D. is not the correct answer. One of the four parts is shaded gray, so it would not estimate to one whole.
Identify Equal Parts
A fraction is a portion of a whole. One of the first parts of learning about fractions is understanding equal parts.
When an object is divided into pieces that are the same size and shape, it is divided into equal parts. Each one
of the equal parts represents a fraction. Students need to be able to identify equal parts in order to completely
understand the concept of fractions.
Example 1: Which picture shows 2 equal parts?
Solution:
A. is the correct answer. The triangle is divided into 2 pieces and the pieces are the same size and shape.
To demonstrate this, if the pieces were folded on the cut line, they would match.
B. is not the correct answer. The square is divided into 2 rectangles, but the rectangles are not the same
size.
C. is not the correct answer. The triangle is divided into 2 pieces, but the pieces are not the same size or
shape.
D. is not the correct answer. The square is divided into 2 pieces, but the pieces are not the same size or
shape.
Example 2: Which picture shows 8 equal parts?
Solution:
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A. is not the correct answer. The figure is divided into 8 pieces, but the pieces are not the same shape or
size.
B. is not the correct answer. The figure is divided into 8 rectangles, but the rectangles are not all the
same size.
C. is the correct answer. The figure is divided into 8 equal pieces.
D. is not the correct answer. The figure is divided into 8 pieces, but the pieces are not the same shape or
size.
To help the student gain further confidence in recognizing equal parts, have the student name items in
his or her surroundings that have equal parts (such as a pizza that has been cut into equal pieces, a
checkerboard, or a quilt). Then have the student draw pictures of those things (or cut them out of
magazines) to make a poster.
Relate Fractions to One Whole Unit
A fraction is a portion of a whole. In this skill, students will be asked to identify a whole that is divided into a
specific number of parts, a number of which are shaded.
Example 1: Which shows 1 out of 4 parts shaded?
Solution: Since the question is asking for 1 out of 4 shaded parts, 1 part should be shaded and the figure
should be divided into 4 parts (fractions).
A. is not the correct answer because it is divided into 5 parts.
B. is not the correct answer because it is divided into 6 parts.
C. is the correct answer. It is divided into 4 parts and 1 part is shaded.
D. is not the correct answer because it is divided into 3 parts.
Answer: C.
Example 2: Pick the picture that shows 5 out of 8 parts are shaded.
Solution: Since the question is asking for 5 out of 8 shaded parts, 5 parts should be shaded and the
figure should be divided into 8 parts.
A. is not the correct answer because it is divided into 5 parts.
B. is the correct answer. The figure is divided into 8 parts and 5 of the parts are shaded.
C. is not the correct answer because it is divided into 16 parts.
D. is not the correct answer because it is divided into 10 parts.
Answer: B.
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To practice this skill with the student, begin by cutting shapes out of construction paper. Then draw
lines on each shape to divide it into parts. Shade a specific number of the parts and have the student
determine that "____ out of ____ parts are shaded."
Equally Likely Outcomes - A
This skill will help students understand that outcomes in a probability situation can be equally likely to occur.
A probability 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.
Equally likely outcomes are outcomes that have the same chance of occurring if they are chosen randomly (as
when a person chooses the card with closed eyes).
To determine whether or not outcomes are equally likely, determine all possible outcomes and the
number of times each outcome can occur. If all possible outcomes have the same number of chances of
occurring, they are equally likely.
Example 1: Elata has these stickers in her bag. If she picks one out of the bag without looking, which
would be true?
Solution:
A. is not the correct answer because in order for one sticker to be more likely, there would have to be
more of that sticker than the other stickers.
B. is not the correct answer because in order for one sticker to be more likely, there would have to be
more of that sticker than the other stickers.
C. is not the correct answer because in order for one sticker to be most likely, there would have to be
more of that sticker than any other type of sticker.
D. is the correct answer because Jenna only has one of each type of sticker, so each sticker is equally
likely to be picked.
Answer: D.
Example 2: Daniel has the toy cars below in a box. If he picks one without looking, which would be
true?
Solution:
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A. is not the correct answer because in order for one car to be most likely, there would have to be more
of those cars than any other cars.
B. is the correct answer because there are two of each car, so the three cars are equally likely to be
picked.
C. is not the correct answer because in order for one car to be least likely, there would have to be fewer
of those cars than any other type of car.
D. is not the correct answer because in order for one car to be more likely, there would have to be more
of that type of car than there are of the other type of car.
Answer: B.
An activity to reinforce the concept of equally likely is as follows. Begin by cutting pairs of identical
shapes out of construction paper. Place three pairs of the shapes on the table in front of the student and
ask the student which shape is most likely to be picked or least likely to be picked. The student should
see that since there are exactly two of each type of shape, none of the shapes will be more or less likely
to be picked. Have the student use the shapes to create probability situations in which all outcomes have
an equally likely chance of being picked.
Fact Families - Addition and Subtraction
A fact family is a set of related number sentences. The example below is a fact family for the numbers 2, 3, and
5.
2+3=5
3+2=5
5-3=2
5-2=3
A number sentence is a mathematical sentence that includes numbers and operational symbols (such as
+, -,
, ÷ , and =). Learning about fact families is one of the first steps toward learning that operations
can "undo" one another. In order for number sentences to be considered as being in the same fact
family, they must contain exactly the same numbers, use inverse operations (inverse operations are
operations that "undo" each other such as addition/subtraction or multiplication/division), and they must
be true. This study guide will focus on addition/subtraction fact families.
Each addition/subtraction fact family contains four number sentences. Two of the number sentences use
addition and two of the number sentences use subtraction. An exception to this rule occurs when two of
the numbers in the fact family are the same. For example, the fact family that contains the numbers 4, 4,
and 8 has only two number sentences: 4 + 4 = 8 and 8 - 4 = 4.
Example 1: Choose the number sentences that are members of the addition/subtraction fact family for
4, 5, and 9 from the number sentences below.
Solution: It is important to remember that in order for number sentences to be considered members of a
fact family they need to meet three conditions:
• They must contain exactly the same numbers (4, 5, and 9 in this case).
• They must use inverse operations (addition and subtraction in this case).
• They must be true.
Cross out any number sentences that do not meet these conditions:
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The following number sentences can be crossed off because they do not contain the correct numbers.
9 + 5 = 14
9 + 4 = 13
4+1=5
5-4=1
14 - 5 = 9
The number sentence 9 + 5 = 4 can be crossed off because it is not true.
The four number sentences that are left are the number sentences in the fact family 4, 5, and 9.
Answer: 4 + 5 = 9, 5 + 4 = 9, 9 - 5 = 4, and 9 - 4 = 5
The questions in this skill will ask the student to choose only one number sentence that belongs to a
particular fact family from four possible number sentences.
Example 2: Which number sentence belongs in the fact family for 8, 5, and 13?
A.
B.
C.
D.
8-5=3
13 - 5 = 8
13 + 5 = 18
8 + 13 = 21
Solution:
A. is not the correct answer because the number sentence does not contain the exact numbers.
B. is the correct answer. It contains all three numbers, it uses inverse operations, and it is true.
C. is not the correct answer because the number sentence does not contain the exact numbers.
D. is not the correct answer because the number sentence does not contain the exact numbers.
Answer: B.
An activity that may help reinforce this skill is to make a fact family concentration game. Write the
number sentences that are members of four or five fact families on index cards (one number sentence
per card). Shuffle the cards and place them face down on a flat surface in rows and columns. Each
player takes turns turning over two cards at a time. If the two cards are members of the same fact
family, the player gets to keep them and play again. The player with the most matches when all of the
cards are gone is the winner.
Recognize Fraction Illustrations
A fraction is a portion of a whole. In this skill, students will be asked to identify whether a shape or an object is
cut in half, thirds, fourths, or fifths.
A figure is said to be divided in half when it is cut into two equal pieces.
A figure is said to be divided in thirds when it is cut into three equal pieces.
A figure is said to be divided in fourths when it is cut into four equal pieces.
A figure is said to be divided in fifths when it is cut into five equal pieces.
Once a student can recognize pictures of objects that are cut in half, thirds, fourths, and fifths, he or she
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will be able to determine whether specific figures are cut into those fractions.
Example 1: The heart is cut in _______________ .
A. thirds
B. fifths
C. half
D. fourths
Solution: Since the heart is cut into two equal pieces, it is cut in half.
Answer: C. half
Example 2: Pick the rectangle that is cut in fourths.
A. is not the correct answer because it is cut into two equal pieces, so it is cut in half.
B. is not the correct answer because it is cut into three equal pieces, so it is cut in thirds.
C. is the correct answer. It is cut into four equal pieces, so it is cut in fourths.
D. is not the correct answer because it is cut into five equal pieces, so it is cut in fifths.
Answer: C.
An activity to help the student master this concept is to bake (or buy) either cupcakes or soft cookies and
help the student cut the cupcakes or cookies in half, in thirds, in fourths, and in fifths. Once the student
has mastered these fractions, move to more complicated fractions such as sixths, sevenths, eighths,
ninths, and tenths. Baking is a great way to show students how fractions are used, because the
measurements are usually in fractions of cups, teaspoons, or tablespoons. If sweets are not an option,
use some modeling clay, dough, or construction paper to make circles, squares, and other shapes to cut
into fractions.
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