<p> Can You Combine Two To Make One?</p><p>Katrina Heit September 23, 2004 Math/Reasoning and Proof 4th Grade Standard(s)</p><p>Reasoning and Proof</p><p> Select and use various types of reasoning and proof.</p><p>Communication</p><p> Communicate their mathematical thinking coherently and clearly t peers, teachers, and others.  Analyze and evaluate the mathematical thinking and strategies of others.  Use the language of mathematics to express mathematical ideas precisely.</p><p>Outcomes/Objectives:</p><p> The Students will understand and become more familiar with multiplication and division pairs through arrays. (How to take 2 small arrays and make one big array by playing the small array/big array game with partners and independently).</p><p> a. The students will play small/array big array.</p><p> b. The students will come up with two smaller arrays that match up to make a big array.</p><p> c. The students will use their set of arrays to make as many 8x6 arrays as they can, using two or three arrays.</p><p> d. The students will share their combinations of arrays by describing the dimensions of the smaller arrays using the overhead so that the whole class and the teacher can tell what is going on.</p><p> e. The students will two smaller arrays into a bigger array independently.</p><p>Vocabulary: dimensions, array, shares, multiplication, division.</p><p>Materials: Array cards (set a and b one set per pair), array game directions, highlighted hundreds chart, overhead projector, transparencies of array cards, notebook paper, pencil. Procedure: (Ongoing assessment, multiple representations, and incorporated learning cycle) Students will work in pairs and independently. Concrete, Pictorial.</p><p>Engage: (the concept) Ask how can you make two smaller arrays into a bigger array? (Optional pre-assessment – have them draw a picture of two smaller arrays and how it can be added to form a larger array)</p><p>Explore: Playing the Small Array/Big Array game. Concrete.</p><p> a. One student from each pair will gather the materials needed for the game.</p><p> b. 10 array cards will be dealt out to each player with the dimensions side up. Spread out the cards in front of you.</p><p> c. Spread out 6 more cards, dimensions side up, in the center of the table. Place the remaining cards in the deck in the center of the table.</p><p> d. The goal is to make a “match” by covering a big array with two or three smaller arrays. Players take turns.</p><p> e. From the smaller arrays, choose one that matches one dimension of a big array in the center of the table. </p><p> f. If none of your array cards matches, you can choose a card that matches a dimension of the big array form the center of the table. Or you can pick the top card from the deck and play it if you can.</p><p> g. If you use an array from the center of the table to cover another array, you can either replace it with a card from the deck, or discard one of your array cards. There should always be 6 cards in the middle.</p><p> h. When you cover a big array, you can collect the “ match.” While playing, keep a list of dimensions of the large array and the smaller arrays: </p><p> 7x6 = 3x6 + 4x6  42 = 18 + 24  (Students will write their “matches” on a sheet of paper using mathematical sentences).</p><p> i. If you run out of cards take 4 cards form the deck. The game is over when there is no more card or no more matches can be made.</p><p> j. Possible teacher questions:  How are they figuring out the total # of squares? Are they counting each square or do they know multiplication parts?</p><p> As you played small array/big array, what clues helped you decide which two smaller arrays might match up to make a bigger array?A- because when you lay the small ones over the big one they match.</p><p> If you are trying to cover a 5x7 array and have a 2x7 array, what array would you need to complete the match? (As “matches are completed, ask students to express the match using mathematical notation and check it by comparing the totals of the smaller arrays with the bigger array. For example in the above situation, students would write: 5x7=2x7+3x7; this checks because =14+21).</p><p>Explain: (the idea/concept) </p><p> a. Have students come up to the overhead to show and explain their combination of matches to the class and teacher.</p><p> b. As a class discuss how they came up with their different combinations.</p><p>Expand (Extend/Apply) (the concept)</p><p> a. Take a look at the 8x6 array. (Teacher shows this array on the overhead).</p><p> b. What are some possible combinations of smaller arrays that would equal this array? (Show 1 example).</p><p> c. Use set of arrays to make as many 8x6 arrays as you can using either two or three arrays.</p><p> d. Every student or student pair builds 8x6 arrays on their desks. </p><p> e. When students have constructed a few examples, ask students to share their combinations by describing the dimensions of the smaller arrays. Record these on the board. (Tell students if they record combinations in an ordered list it will be easier to use the patterns to exhaust all possible combinations).</p><p>Evaluate (the concept) </p><p>Give students a new number and have them make the smaller arrays into a bigger array independently using array cards and then recording their answers using mathematical notation and check like in the explore stage. </p><p>Expand (2): Students will use arrays as a way of solving word problems using multiplication and division. Students will also learn problem-solving strategies to do this.</p><p>Adaptations and modifications: If you have lower-level students you can place them with a higher-level student for their pair. Monitor students to make sure everyone stays on task. Put directions on the board and in a handout. Verbally say the directions the first time through. Play music softly during this time for the musically inclined students.</p><p>References:</p><p>Economopoulos, K., Tierney, C., and Russell, J. S. (1995). Arrays and shares: multiplication and division. (Grade 4-5). Dale Seymour Publications: Massachusetts. Pp40-46, 121.</p><p>National Teachers of Mathematics, (2000). Principles and standards for school mathematics. The National Council of Teachers of Mathematics Inc, Reston, VA, pp. 188, & 194.</p><p>Reflection: I feel that this lesson is appropriate for fourth graders because they are familiar with multiplication, division, and the hundreds chart. This lesson meets the standards of the 3-5-grade reasoning and proof, and communication. Younger students may have a hard time doing this activity because of the prior knowledge that is required to do this activity. This activity has plenty of concrete experiences and explanations to improve student’s comprehension. I think that most students will grasp this concept right away but some may still be counting each square at a time or counting by 2’s, 3’s or 4’s. This could require lower level students be paired with higher-level students for clarification and reasoning. </p><p>Ideas for Integration: You could integrate science by using inquiry more and experimenting with larger numbers. They could write in a journal about what they learned and if they understood what was happening for a Language Arts integration. For Social Studies you can study different arrays found throughout the world. </p>