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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 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. 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. 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