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Errors with Fractions and Decimals Draw a representation for each task. 1) In my sock drawer I have 4 blue socks, 6 white socks and 2 black socks. What fraction represents the number of black socks that I have? 2) I have a piece of string that is 3 feet long. I need to cut the piece into three equal pieces. How long is each piece? Fractions Part of a group – A group of items in which the objects have different characteristics Part of a whole – A whole or “unit” that has broken into pieces Most errors are misconceptions related to part of a whole Fractions- Representational Errors Working with shaded parts – Gretchen (p. 138) What’s the error? Gretchen Interventions- Start with the Whole 4 2 1) Read “2 out of 3 parts” 2) Draw a rectangle 9 3 3) Break it into 3 equal parts 4) Shade in two parts 5) This is “2 out of 3 parts” which is also called two-thirds Gretchen (#2) Start with an unshaded rectangle 4 2 9 3 Gretchen (#3) Still, in traditional curricula students are given: “Write the fraction that shows the number of regions that are shaded.” What options do we have?? Fractions- Representational Working with shaded parts – Carlos (p. 139) What’s the error? Carlos Equal area – Draw a number of squares on your paper Divide each square into fourths in a different way How many ways can you think of? Fractions- Symbolic Equivalent Fractions – Jill (p. 140) 4 2 3 1 3 1 4 2 9 3 9 3 8 4 8 4 Jill makes every… Why does 4 into a ____ 8 into a ____ she make 3 into a ____ 9 into a ____ this error? Fractions- Symbolic Equivalent Fractions – Jill (p. 140) Help by 1) Using fractional parts of regions 2) Build an array with fractional parts of a set 3) Look for a pattern in a list 4) Make sets of equivalent fractions Equivalent Fractions Concrete Modification – Cover your hexagon with six 4 2 2 equal pieces – Can you cover only four- 6 2 3 sixths of your hexagon? – How many triangles do you have on your hexagon? – Can you cover those triangles with another shape? Equivalent Fractions Representational Modification Using fractional parts of regions – Draw six parts (rectangle model) – For 4/6, shade four of the parts – “Can we use larger parts to cover exactly what we have?” – When students use thirds, the two-thirds will cover 4/6ths – Bring in mathematical rule 4 2 2 after students have experienced the representation 6 2 3 Equivalent Fractions 2/3 2/3 4/6 6/9 Look for a pattern in a list 10 5 12 6 Pattern– dividing 8 4 both the numerator and the denominator 10 5 by two 6 3 8 4 4 6 Make sets of equivalent fractions *1 *2 *3 *4 *5 *6 *7 *8 Numerator 1 2 3 4 5 6 7 8 Denominator 3 6 9 12 15 18 21 24 Decimals- Symbolic Tonya (p. 144) – What’s the error? Tonya Multiple representations – Hundreds chart Tonya Multiple representations – Ordering decimals Put three in a row (least-to-greatest, greatest-to- least) 2nd- 0.3560 Small- 0.3500 3rd- 0.3562 Tonya Multiple representations – Number line 0 1 Tonya Multiple representations – Models to decimals with base-10 blocks Decimals- Concrete Use of base-10 blocks – Students learn Common Decimal use Cube = 1 Cube = 0.01 Rod = 10 Rod = 0.1 Flat = 100 Flat = 1 Block = 1,000 – Students constantly mix them up Decimals- Concrete Use of base-10 blocks – Place value when the task is given orally “Show 2.04” – 2 flats, 4 cubes – Place value when reading the task “2 and 4 hundredths” – Modify with… Place value chart with words and pictures as headings Practice using base-10s to show decimals Operations with Fractions and Decimals Adding Fractions Covering the hexagon – Using your pattern blocks… Cover the hexagon in as many possible ways Fill out the chart for each solution that you found Pieces Picture Fraction Pieces Fraction 6 triangles 6 * (1/6) 2 trapezoids 2 * (1/2) 3 rhombi 3 * (1/3) 1 trap, 3 tri (1/2) + 3* (1/6) 2 rhombi, 2 tri 2*(1/3) + 2*(1/6) 1 trap, 1 tri, 1 rhom (1/6)+(1/2)+(1/3) Pieces Fraction 4 tri, 1 rhom 4*(1/6) + 1/3 Adding Fractions Clock Model 1 1 4 6 If students learn equivalent fractions, then this method is effective. Adding Fractions- Errors – Robbie (p. 151) Verticalorientation Use unit regions – Fraction strips – Fraction tiles Counting pieces – Only if the denominator is the same Estimate answers Adding Fractions- Symbolic Modifications for fractions with unlike denominators (5th grade) – Estimate answers before adding – Emphasize adding requires having like denominators – Use a number line to focus on equivalent fractions Subtracting Fractions Johnny says that… So, he claims that… 1 1 1 and (3)(4) 12 3 4 12 1 1 1 1 1 1 a b a *b and ( 4)(5) 20 4 5 20 1 1 1 Will his method and (5)(6) 30 always work? 5 6 30 Why or why not? Adding Fractions- Symbolic Errors adding Incorrect answers fractions 4 1 What – Steve runs 2 ½ 2 4 5 did miles in the morning 4 and 2 ¾ miles at 2 these 5 1 three night. How many 3 1 2 4 4 students miles did he run 4 do? total? 4 2 – What is the correct 4 4 ? 6 3 answer? Subtracting Fractions 1 Draw two rectangles 1 4 Shade 1 ¼ of them lightly 3 Cross out ¾ of a rectangle 4 What’s remaining? Adding Decimals- Concrete Use of base-10 blocks – At the track and field meet, triple jumpers’ scores are the sum of their two jumps. Monique jumped 3.4 and 2.7 yards. What was her total score? – Students use base-10 blocks and record the following answers: Use base-10 blocks. What 6.1 were there errors? 5.11 51.1 16.1 16 Adding Decimals- Symbolic On paper – Harold (p. 167) What’s the error? Adding Decimals- Symbolic On paper – Harold (p. 158)- Modifications Use base-10 blocks or c-rods – Exchange – “use as few blocks as possible!!” Use a number line Adding Decimals- Symbolic On paper – Harold (p. 167)- Modifications Use vertically lined paper (similar to division) Use meter sticks Adding Decimals Math Applet National Library of Virtual Manipulatives

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