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Teaching Math Conceptually • Students who cannot easily memorize rules tend to not do as well in math. • Students who can memorize rules tend to do better in math, but do not always understand HOW or Why the math works. • What are some rules that we teach students? • Intergers – add the opposite, two negative make a positive (but not always) • Dividing Fractions – Keep Change Flip • Proportions – Cross Multiplying • Converting Mixed Numbers and Improper Fractions How and Why •Eventually, they will probably use the “rule” because it is faster, but they will make fewer mistakes if they truly understand how the math is working. •Turn to your neighbor and discuss how you usually teach adding, subtracting, multiplying and dividing integers. • When I start teaching integers, I always tell my students that after we start learning this there will be some students that miss 3+5 and 9-7 because they are so concerned about the rules instead of THINKING about how the numbers make SENSE. They think they will never miss such easy questions, but it happens every year!! Integers with a Number Line or Thermometer • + number walk forward (to the right) • - numbers walk backward (to the left) • “Plus” go in same direction • “minus” turn around One Way to Add Integers Is With a Number Line When the number is positive, count to the right. When the number is negative, count to the left. + -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 Examples We can use a number line to help us add positive and negative integers. –2 + 5 == 3 -2 3 Remember: To add a positive integer we move forwards up the number line. Examples -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 Examples –3 + (–4) = –7 -7 -3 Remember: To add a negative integer we move backwards down the number line. is the same as Examples -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 •This works with Subtraction also. Students know that – means opposite. •Examples •-5-3 = 8-10= 3-(-3)= Integers Using Algebra tiles • This represents a negative one. • This represents a positive one. • A positive one and a negative one cancel each other and make a zero pair. 2 + 4=6 Positives plus some MORE positives equal lots of POSITIVES. -2 + -4=-6 Negatives plus some MORE Negatives equal lots of Negatives. 2 + -4 = 2 Cross out the zero pairs and whatever left is the answer. Students quickly figure out that if you are adding a positive and a negative it is actually a SUBTRACTION problem Multiplying and Dividing 2 x 3……….This problem means “two groups of three” =6 • • 2 x - 3……..This problem means “two groups of negative 3” = -6 • • Multiplying and Dividing - 2 x 3……..This problem means “the opposite of two groups of three” = .) - 2 x - 3…….This problem means “the opposite of two groups of negative three” Subtracting •Subtracting with the tiles is a little more difficult, but it is a good way for kids to visually see why subtracting involves adding sometimes. •We will be saying “take away” for the subtraction sign and will be using zero pairs. 5–2=3 • Positive 5 take away positive 2 • That leaves us with positive 3 -5 – (- 2) = -3 • Negative 5 take away negative 2 • That leaves us with negative 3 5 – (-2) = 7??? I don’t have any negatives to take away. I can add as many zeros as I want without changing the number. Now I have two negatives to take away and I am left with 7 positives. -5 – 2 = -7 I don’t have any positives to take away. I can add as many zeros as I want without changing the number. Now I have two positives to take away and I am left with 7 negatives. Multiplying Fractions In Primary school students associate multiplying with numbers getting larger and dividing with numbers getting smaller. Multiplying proper fractions is probably the first time the answer to a multiplying question is LESS than what they started with. Why bother with pictures? • • Because we won’t help make mathematics work for a much larger proportion of the student body until and unless we recognize that abstractions like 3⁄4 or rules for dividing fractions work fine for some students but must be grounded in pictures and models for others. • (Leinwand, 2009. pp. 20-21) 1/2 x 1/3 •How would you model this? 1/2 x 1/3 •You had a birthday party and had ½ of a cake left over. 1/2 x 1/3 •Your brother ate 1/3 of the left over cake. 1/2 x 1/3 • How much of the whole cake did your brother eat? • 1/6 of the whole cake is shaded twice. 3/5 x 3/4 3/5 x 3/4 •9 out of 20 are double colored •9/20 Try it #1 Try it #2 How would you model this? Try it #1 Use the blocks (or draw) to show Try it #2 Try it #3 Use the blocks (or draw) to show Try it #4 How would you model this? This is a student’s example Try it #1 Try it #2 Converting Mixed Numbers and Improper Fractions What does this mean? • The denominator is 4 which means our object (candy bars) is cut into fourths. We have 2 whole candy bars and ¼ of another. How many total fourths do we have? • The denominator is 5 which means our object (candy bars) is cut into fifths. We have 1 whole candy bar and 2/5 of another. How many total fifths do we have? • • • How many Wholes and how many thirds will this • make? 2 whole candy bars and 2/3 of • another. • • The site to create models • This Glencoe site allows the user to explain their thinking using manipulatives.