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Algebra
In grades 4–5, algebra involves higher-level problem-solving skills associated with patterns, functions, graphs,
and variables. In order to make these concepts more concrete, give students opportunity to play games and
complete problem-solving puzzles that will help develop students’ reasoning abilities.
NCTM Standard
• Understand patterns,
relations, and functions
• Represent and analyze
mathematical situations and
structures using algebraic
symbols
• Use mathematical models
to represent and understand
quantitative relationships
Skills
• Explore patterns, relations, and functions and represent them
• Identify commutative, associative, and distributive properties
• Represent variables as unknown quantities using a letter or a symbol
• Investigate mathematical relationships using equations
• Model problem solving using objects
• Draw conclusions with the help of graphs, equations, and other
representations
• Examine situations in which change in one variable relates to change in
a second variable
Suggested Materials:
• index cards
• counters
• dice
• pattern puzzles
• function tables
40
Algebra
• grocery advertisements
• graph paper
• decks of playing cards
• attribute blocks
Secret Number X
Skills
Give each pair of students six index cards on which each card features either
a large red X or +, –, ×, ÷, or = and 24 numeral cards in which each card
displays a different number less than 50. Place each set of numeral cards inside
an envelope labeled Secret Number X. Have one student pull a card from the
envelope and create an equation with the secret number as the variable. The
student should model the equation using counters and the set of symbol cards
and then invite his or her partner to solve it. For example, if the secret number
is 6, he or she could make a pile of 16 counters, and then place the minus sign,
the X, the equal sign, and 10 counters. The student solving the equation can
complete the equation in his or her math journal. Challenge students to make
the equations progressively more difficult.
Pascal’s Triangle
• Representing variables as
unknown quantities using a
letter or symbol
Materials
• index cards
• envelopes
• counters
• math journals
Skills
Present each student with a copy of the Pascal’s Triangle reproducible. Ask
students to fill in the numbers where the blanks are and make a list of every
pattern that they see in their math journal. Offer rulers so students can check
for patterns along different lines of the triangle (horizontal, diagonal, vertical).
Challenge students to check sums of different lines or extend Pascal’s Triangle
to more rows.
Name ________________________________________________________
Date _________________________
Algebra
1
Pascal’s Triangle
Study the pattern shown in the triangular shape. Discover the pattern and fill in the missing numbers.
• Representing and analyzing
patterns and functions
Materials
• Pascal’s Triangle reproducible
(page 45)
• math journals
• rulers
• calculators
1
1
1
1
1
1
1
1
_____
1
_____
7
8
9
_____
4
_____
120
6
35
4
_____
1
_____
21
_____
126
252
1
_____
_____
70
_____
210
1
_____
20
56
_____
1
3
_____
15
28
_____
45
3
5
6
1
2
7
_____
84
_____
_____
_____
_____
36
_____
1
_____
_____
_____
10
1
Algebra
41
Skills
• Practicing operations such
as addition, subtraction,
multiplication, and division
• Expressing mathematical
relationships using equations
Materials
• 6-sided dice
• 12-sided dice
• number cards
• paper
Skills
• Representing and analyzing
patterns and functions
• Using deductive reasoning
Materials
• Fit the Rule reproducible
(page 46)
• Deck of playing cards
42
Algebra
Dice Equations Game
Give each group of four to five students three 6-sided dice and one 12-sided
die. Have the first player roll one 6-sided die at a time and then the 12-sided
die, and write down the numbers in the order he or she rolled. Then the
player must add, subtract, multiply, and/or divide the numbers represented on
the six-sided dice to equal the number on the 12-sided die. If he or she cannot
make an equation that equals the number on the 12-sided die, he or she must
write the difference between the sum of the 3 rolls of the 6-sided dice and the
number on the 12-sided dice on the paper. Students total their differences
from all the rounds and the one with the smallest total at the end of the
10-round game wins.
Fit the Rule
Give each pair of students a deck of playing cards and the reproducible. Tell
students that this game requires each student to guess the rule maker’s rule
without talking. Have the first player, the rule maker, think of a rule such as
My rule will only include cards with single-digit prime numbers. Then the rule
maker demonstrates the rule by selecting one card that fits the rule to create
a top row and one card that does not fit the rule to place below that card to
create the bottom row. Player 2 now picks a card to place either in the “fits the
rule” row or the “does not fit the rule” row to help him or her guess what the
rule may be. The rule maker must nod or shake his or her head in response to
Player 2’s guess. Player 2 now has two more chances to guess the rule maker’s
rule using the same procedure, writing the card information on his or her
reproducible to keep track and eliminate possibilities. If Player 2 is ready to
guess the rule, he or she can write down the rule on the reproducible and
hand it to the rule maker. Each student keeps track of a correct or incorrect
guess and the number of points gained or lost for the round according to the
key on the reproducible. The first student to 50 points wins the game.
Graphing Candy
Have each student make a graph with the x-axis labeled Number of Candies
and the y-axis Distance from the Floor. Arrange two tables about a foot
apart and lay a yardstick across the gap. Attach one end of the Slinky to the
yardstick with a paper clip by weaving the end of the Slinky in and out of
the paper clip and taping down the end to the yardstick. Attach the cup to
the Slinky’s other end and secure it with tape. Ask a student to measure the
distance between the bottom of the paper cup and the floor. Have the other
students plot the distance on the graph paper. Then ask another student to
place one of the candies in the paper cup and measure the distance from
the floor, and have the other students plot these coordinates on the graph.
Continue the activity until the students can see the shape created on the
graph. Is it straight or is it curved? Invite students to write in their math
journals about why they think this shape appears.
Discovering Functions
Explain that a function is a rule that you apply to an input number to produce
an output number. Ask students to fill in the blanks on the function tables
by finding a pattern and using guess and check to discover the function
equation. Then give each student a piece of graph paper to graph the x and
y coordinates listed on the completed function tables. Challenge students
to write the rule at the bottom of each function table by asking themselves,
“What do I do to x in order to get the y?” Ask students to notice how the
rule dictates the shape of the graph. Finally, give students the opportunity to
create their own function equation, fill in the blank table partially, and ask a
classmate to discover the rule.
Skills
• Using representations such
as graphs and tables to draw
conclusions
• Investigating how a change in
one variable relates to a change
in a second variable
Materials
• graph paper
• yardstick or meter stick
• Slinky
• tape
• paper cup
• small candies
• ruler or measuring tape
• math journal
Skills
• Representing and analyzing
patterns and functions
• Investigating how a change in
one variable relates to a change
in a second variable
• Using representations such
as graphs and tables to draw
conclusions
Materials
• Discovering Functions
reproducible (page 47)
• graph paper
Algebra
43
Skills
• Understanding uses of
multiplication and division
• Working with fractions
and ratios
• Using equations to draw
conclusions
Materials
• Recipes reproducible (page 48)
• Recipe Ratios reproducible
(page 49)
• ingredients as listed on recipe
• math journals
• calculators (optional)
44
Algebra
Recipe Ratios
Place students into groups of four, and give each group a copy of the Sweet
Parfaits recipe. Tell the class that each recipe makes enough for four people,
but their job is to find the correct amount of each ingredient if they were
making the recipe for six people. Each group decides how much of the
other ingredients they need by finding the ratio of the ingredient listed to
the amount needed. Challenge the students to fill in the ratio chart. Check
the students’ answers, and allow them to make the recipe according to their
figures when they have completed the chart correctly.
1
_____
_____
9
1
1
8
45
1
7
_____
1
6
28
120
1
5
_____
_____
1
4
15
56
210
1
3
_____
35
_____
1
2
6
Date _________________________
20
70
252
1
3
_____
_____
126
1
4
_____
_____
_____
1
_____
21
84
1
_____
_____
_____
1
7
36
_____
_____
_____
_____
_____
Study the pattern shown in the triangular shape. Discover the pattern and fill in the missing numbers.
Pascal’s Triangle
Name ________________________________________________________
_____
1
10
_____
1
Hands-on Math • Grades 4–5 © 2009 Creative Teaching Press
45
Algebra
Name ________________________________________________________
Fit the Rule
Fits the Rule
Date _________________________
Does Not Fit the Rule
Rule (1st Guess: Correct +10, Incorrect –2) _____________________________________
Rule (2nd Guess: Correct +5, Incorrect –5) _____________________________________
Rule (3rd Guess: Correct +2, Incorrect –10) _____________________________________
Does Not Fit the Rule
Rule (1st Guess: Correct +10, Incorrect –2) _____________________________________
Rule (2nd Guess: Correct +5, Incorrect –5) _____________________________________
Rule (3rd Guess: Correct +2, Incorrect –10) _____________________________________
46
Algebra
Hands-on Math • Grades 4–5 © 2009 Creative Teaching Press
Fits the Rule
Name ________________________________________________________
Date _________________________
Discovering Functions
y (out)
0
x (in)
y (out)
Function Table 5
x (in)
34
Function Table 4
y (out)
17
Function Table 3
x (in)
13
Function Table 2
y (out)
0
Function Table 1
x (in)
12
y (out)
7
x (in)
0
4
0
19
6
15
1
2
8
13
2
8
42
4
3
8
17
21
14
46
1
6
9
10
16
16
3
11
2
4
5
Rule ________________
10
Rule ________________
12
Rule ________________
Function Table 10
40
Rule ________________
Function Table 9
5
Rule ________________
Function Table 8
x (in)
14
y (out)
3
x (in)
8
y (out)
Function Table 7
y (out)
4
24
Rule ________________
x (in)
Function Table 6
x (in)
19
5
Rule ________________
168
120
80
48
y (out)
4
26
y (out)
7
8
x (in)
0
24
74
62
38
5
16
24
Rule ________________
11
11
6
7
39
2
3
15
8
4
5
9
Rule ________________
6
Rule ________________
Hands-on Math • Grades 4–5 © 2009 Creative Teaching Press
47
Algebra
Recipes
Sweet Parfaits
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1 pint any flavor sorbet, slightly softened
½ cup sweet granola
2 sliced strawberries
(or another fruit of your choice)
½ cup whipping cream
2 Tbsp powdered sugar
Preparation: In your cups, layer sorbet and granola.
Refreeze until firm. When ready to serve, beat cream
and sugar together until stiff, and dollop a bit on each
parfait. Top with fruit. Serves 4.
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Preparation: In your cups, layer sorbet and granola.
Refreeze until firm. When ready to serve, beat cream
and sugar together until stiff, and dollop a bit on each
parfait. Top with fruit. Serves 4.
Sweet Parfaits
ö
ö
ö
ö
ö
1 pint any flavor sorbet, slightly softened
½ cup sweet granola
2 sliced strawberries
(or another fruit of your choice)
½ cup whipping cream
2 Tbsp powdered sugar
Preparation: In your cups, layer sorbet and granola.
Refreeze until firm. When ready to serve, beat cream
and sugar together until stiff, and dollop a bit on each
parfait. Top with fruit. Serves 4.
Sweet Parfaits
ö
ö
ö
ö
ö
ö
ö
ö
ö
1 pint any flavor sorbet, slightly softened
½ cup sweet granola
2 sliced strawberries
(or another fruit of your choice)
½ cup whipping cream
2 Tbsp powdered sugar
Preparation: In your cups, layer sorbet and granola.
Refreeze until firm. When ready to serve, beat cream
and sugar together until stiff, and dollop a bit on each
parfait. Top with fruit. Serves 4.
48
Algebra
1 pint any flavor sorbet, slightly softened
½ cup sweet granola
2 sliced strawberries
(or another fruit of your choice)
½ cup whipping cream
2 Tbsp powdered sugar
Preparation: In your cups, layer sorbet and granola.
Refreeze until firm. When ready to serve, beat cream
and sugar together until stiff, and dollop a bit on each
parfait. Top with fruit. Serves 4.
Sweet Parfaits
ö
1 pint any flavor sorbet, slightly softened
½ cup sweet granola
2 sliced strawberries
(or another fruit of your choice)
½ cup whipping cream
2 Tbsp powdered sugar
Sweet Parfaits
ö
ö
ö
ö
ö
1 pint any flavor sorbet, slightly softened
½ cup sweet granola
2 sliced strawberries
(or another fruit of your choice)
½ cup whipping cream
2 Tbsp powdered sugar
Preparation: In your cups, layer sorbet and granola.
Refreeze until firm. When ready to serve, beat cream
and sugar together until stiff, and dollop a bit on each
parfait. Top with fruit. Serves 4.
Hands-on Math • Grades 4–5 © 2009 Creative Teaching Press
ö
Sweet Parfaits
Name ________________________________________________________
Date _________________________
Recipe Ratios
Sweet Parfaits Recipe
Recipe for 4 People
ö
ö
ö
ö
ö
Recipe for 6 People
1 pint any flavor sorbet, slightly softened
½ cup sweet granola
2 sliced strawberries
(or another fruit of your choice)
½ cup whipping cream
2 Tbsp powdered sugar
ö
ö
ö
ö
ö
Preparation: In your cups, layer sorbet and granola.
Refreeze until firm. When ready to serve, beat cream
and sugar together until stiff, and dollop a bit on each
parfait. Top with fruit. Serves 4.
____ pint any flavor sorbet, slightly softened
____ cup sweet granola
____ sliced strawberries
(or another fruit of your choice)
____ cup whipping cream
____ Tbsp powdered sugar
Preparation: In your cups, layer sorbet and granola.
Refreeze until firm. When ready to serve, beat cream
and sugar together until stiff, and dollop a bit on each
parfait. Top with fruit. Serves 6.
Ratio 4 : 6
Recipe for 4 people:
Recipe for 6 people:
1 pint any flavor sorbet,
slightly softened
____ pints any flavor sorbet,
slightly softened
Solve:
Original amount
of ingredient
1
Number of
people served
4
=
X
New amount
of ingredient
6
New number
of people served
Hands-on Math • Grades 4–5 © 2009 Creative Teaching Press
4x=6
x = 6⁄4
x = 12⁄4
x = 1½
cross multiply
get x by itself
reduce
Algebra
49