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Key Strategies for Teaching Elementary Mathematics Where Are the Cookies? Mrs. James left a tray of cookies on the counter early one morning. Larry walked by before lunch and decided to take 1/3 of the cookies on the tray. Later that afternoon Barry came in and ate 1/4 of the remaining cookies. After supper Terry saw the tray of cookies and ate 1/2 of the cookies remaining at that time. The next morning Mrs. James found the tray with only 6 cookies left. How many cookies were on the tray when Mrs. James first left it on the counter? Georgia Formative Assessment tasks Visual representations are critical for learning. Mrs. James found the tray with only 6 cookies left. 6 Terry ate 1/2 of the cookies remaining. 6 Terry’s 6 6 remaining Barry ate 1/4 of the remaining cookies. 12 Barry remaining Larry took 1/3 of the cookies. 16 Larry starting 4 4 4 4 There are 338 players on a soccer team. 186 are girls and the rest are boys. How many boys are on the soccer team? 186 ? 338 Misha has 34 dollars. How many dollars does she have to earn to have 47 dollars? Visual representations for multiplication Visual representations translate to symbolic (CRA) Estimate: 1.4 x 1.3 is somewhere between 1 and 4 Distributive Property: 1.4 x 1.3 0.12 0.3 0.4 1 1.82 Knowing the reasons behind a procedure is as important as being fluent with it. Reason? 2 13 x8 104 13 x8 24 +80 104 10 8 8∙3 + 8∙10 = 8(3+10) a∙b + a∙c = a(b+c) 3 Reason? 1 1 1 × = 3 4 12 What problem does this illustrate? Students should work things out with their hands. Division by Partitioning 354 photos to share among 3 children Work with manipulatives also translates to procedures 354 ÷ 3 (300 + 50 + 4) ÷ 3 = 100 + 10 + 1 r 21 100 + 10 + 1 + 7 Try this with 251 ÷ 8. Partition base 10 blocks, then write a corresponding algorithm. Problem-solving is enhanced by knowing the structures of word problems. Part-whole where a part is unknown There are 23 players on a soccer team. 18 are girls and the rest are boys. How many boys are on the soccer team? 18 ? 23 Joining (adding to) where the change is unknown Misha has 34 dollars. How many dollars does she have to earn to have 47 dollars? Equal groups Each person on a relay race team runs 5/8 of a mile. There are 4 people on the team. How long is the total race? 5 8 5 8 5 8 5 8 + + + = 4×5 8 Multiplicative comparison The tree is 40 times taller than the person. If the person is 5 feet tall, how tall is the tree? Measurement division Our class baked 225 cookies for a bake sale. We want to put them in bags with 6 in each bag. How many bags can we make? 225 – 60 = 165 165 – 60 = 105 105 – 60 = 55 45 – 30 = 15 15 – 12 = 3 10 bags 10 bags 10 bags 5 bags 2 bags 37 bags with 3 cookies left over Try this with 251 ÷ 8. What’s a corresponding word problem? A recipe that makes 20 cookies needs ¾ cup of sugar. How much sugar is needed to make 100 cookies? Measurement division How many groups of 20 are in 100? (repeated subtraction, or 20 x what = 100?) Equal groups How much is 5 groups of 3 fourths? (Skip count on a number line) If a serving size for Cheerios is 3/4 cup, how many servings are in a box that has 12 1/2 cups? While division by a fraction is in the 6th grade curriculum, this example shows how important it is to understand the underlying structure of word problems. Alternative ways of solving a problem should be celebrated. T-shirts with the school logo cost $8 wholesale. The Pep Club has saved $496. How many t-shirts can they buy for their fund-raiser? Mental Math is Important Number Talks 32 x 15 14 64 4 -38 26 Practice and drill are essential for developing fluency Do these multiplication problems as quickly as you can using the partial product method. 25 x 164 562 x 36 285 x 102 Practice makes perfect: Frequent cumulative review is the most underused method for ensuring long-term retention.