Algebra. Suggested Materials: 40 Algebra

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1 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 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 : index cards counters dice pattern puzzles function tables grocery advertisements graph paper decks of playing cards attribute blocks 40 Algebra

2 Secret Number X 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. Representing variables as unknown quantities using a letter or symbol index cards envelopes counters math journals Pascal s Triangle 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 Pascal s Triangle Study the pattern shown in the triangular shape. Discover the pattern and fill in the missing numbers. 1 Algebra 1 Representing and analyzing patterns and functions Pascal s Triangle reproducible (page 45) math journals rulers calculators Algebra 41

3 Practicing operations such as addition, subtraction, multiplication, and division Expressing mathematical relationships using equations 6-sided dice 12-sided dice number cards paper 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. Representing and analyzing patterns and functions Using deductive reasoning Fit the Rule reproducible (page 46) Deck of playing cards 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. 42 Algebra

4 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. 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 graph paper yardstick or meter stick Slinky tape paper cup small candies ruler or measuring tape math journal 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. 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 Discovering Functions reproducible (page 47) graph paper Algebra 43

5 Understanding uses of multiplication and division Working with fractions and ratios Using equations to draw conclusions Recipes reproducible (page 48) Recipe Ratios reproducible (page 49) ingredients as listed on recipe math journals calculators (optional) 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. 44 Algebra

6 Name Date Pascal s Triangle Study the pattern shown in the triangular shape. Discover the pattern and fill in the missing numbers Hands-on Math Grades Creative Teaching Press Algebra 45

7 Name Fit the Rule Date Fits the Rule 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) Fits the Rule 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) Hands-on Math Grades Creative Teaching Press 46 Algebra

8 Name Date Discovering Functions Function Table 1 Function Table 2 Function Table 3 Function Table 4 Function Table 5 x (in) y (out) x (in) y (out) x (in) y (out) x (in) y (out) x (in) y (out) Rule Rule Rule Rule Rule Function Table 6 Function Table 7 Function Table 8 Function Table 9 Function Table 10 x (in) y (out) x (in) y (out) x (in) y (out) x (in) y (out) x (in) y (out) Rule Rule Rule Rule Rule Hands-on Math Grades Creative Teaching Press Algebra 47

9 Recipes Hands-on Math Grades Creative Teaching Press 48 Algebra

10 Name Recipe Ratios Date Recipe Recipe for 4 People Recipe for 6 People pint any flavor sorbet, slightly softened cup sweet granola sliced strawberries cup whipping cream Tbsp powdered sugar 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 Number of people served 1 4 = X 6 New amount of ingredient New number of people served Hands-on Math Grades Creative Teaching Press 4 x = 6 x = 6 4 x = 12 4 x = 1½ cross multiply get x by itself reduce Algebra 49