MrSmith Posted July 11, 2014 Share Posted July 11, 2014 Please explain how to solve this problem using the bar model method. I looked at the answer and it seems so simple but I just can't understand where it comes from. I even read the Process Skills (Book 5!) topic on this, but the explanation is so terse. The question comes from Challenging Word Problems Level 4, Section 5 (Fractions of a Set), Challenging Problem #7 (page 59 in my book): "Rod P is 6 cm longer than rod Q. 3/5 of the length of rod Q is equal to half the length of rod P. What is the length of rod Q?" Using algebra I can find the answer, but I need to see how the bar model is used to solve this. Thanks! Quote Link to comment Share on other sites More sharing options...
EKS Posted July 11, 2014 Share Posted July 11, 2014 Draw rod P a bit longer than rod Q and show that the extra length is 6 cm. Then divide Q into five pieces. Show that 3/5 of Q is equal to 1/2 of P. It then becomes obvious that each fifth of rod Q is equal to 6 cm. Quote Link to comment Share on other sites More sharing options...
crazyforlatin Posted July 11, 2014 Share Posted July 11, 2014 It helps to convert 1/2 to 3/6 since these problems are easier to see if both numerators are the same. So when you compare 3/6 and 3/5, you can see that each section is 6cm. So for P, draw out a bar with 6 sections, and for Q, draw out a bar with 5 sections. Have the P bar right above the Q bar. Quote Link to comment Share on other sites More sharing options...
MrSmith Posted July 11, 2014 Author Share Posted July 11, 2014 The part I don't follow is how you know that each square is 6cm? As Q is 5 squares and P is Q plus 6cm all I know for sure is that each of Q's squares is the same value, but cannot claim that those squares are 6cm. Quote Link to comment Share on other sites More sharing options...
Alte Veste Academy Posted July 11, 2014 Share Posted July 11, 2014 Because P=Q+6cm, and half of P=3/5Q, the other half of P=2/5Q+1/5Q. We know 6 cm must equal 1/5Q because that 6cm constitutes the remainder of that half of P. Wow. That is a horrible explanation. LOL Hopefully someone with bar diagram typing skills will come along! Quote Link to comment Share on other sites More sharing options...
wendyroo Posted July 11, 2014 Share Posted July 11, 2014 Well, you know that 3/5 Q = 1/2 P. And since P = Q + 6cm then the other half of P = the other 2/5 Q + 6 cm [ 1/2 P ][ 1/2 P ] [ ][ ][ ][ ][ ] So 3/5 Q = 1/2 P = 2/5 Q + 6cm 3/5 Q = 2/5 Q + 6 cm 1/5 Q = 6 cm Q = 30 cm Edited to add... To visualize it you could stack like this: [ 1/2 P ] [ ][ ][ ] [ ][ ]6cm Then it becomes obvious visually that 1/5 Q must equal 6 cm. Quote Link to comment Share on other sites More sharing options...
marlowefamily Posted July 12, 2014 Share Posted July 12, 2014 From 3rd to 5th grade, the use of bar diagrams to solve problems can be confusing for both the child and adults. The intent is to get the child thinking and visualizing somewhat in a pre-algebra type manner. There is a good instructors book that explains how adults should understand and explain the bar diagram problems for kids...I ended up purchasing it and that was the only way that I could help my two sons with getting through SM. Very easy once you get the hang of it. Quote Link to comment Share on other sites More sharing options...
Guest Posted July 12, 2014 Share Posted July 12, 2014 From 3rd to 5th grade, the use of bar diagrams to solve problems can be confusing for both the child and adults. The intent is to get the child thinking and visualizing somewhat in a pre-algebra type manner. There is a good instructors book that explains how adults should understand and explain the bar diagram problems for kids...I ended up purchasing it and that was the only way that I could help my two sons with getting through SM. Very easy once you get the hang of it. Which book is this? My son really doesn't care for doing the bar diagrams but then he gets to a tough problem and because he hasn't been practicing the easier problems with bar diagrams, he can't set up the harder problem with a bar diagram. It is a little frustrating right now for us, to say the least. Quote Link to comment Share on other sites More sharing options...
marlowefamily Posted July 12, 2014 Share Posted July 12, 2014 I think it was: Step by Step Model Drawing: Solving Word Problems the Singapore WayYou can find it on Amazon. Quote Link to comment Share on other sites More sharing options...
kiwik Posted July 12, 2014 Share Posted July 12, 2014 OK I drew it to a scale. Each fifth of Q is 1cm. Therefore 3 cm on the drawing is half P making P six cm long. So the extra cm on the drawing is the 6 in the equation making q=30 and P=36. Would have sounded better if I had graph paper. Quote Link to comment Share on other sites More sharing options...
madteaparty Posted July 12, 2014 Share Posted July 12, 2014 And this is why we don't do graphs. This is an easy Algebra problem that my DS, not gifted in math, can set up and solve. I'm too old to draw to scale! Quote Link to comment Share on other sites More sharing options...
EKS Posted July 12, 2014 Share Posted July 12, 2014 Sometimes it helps to draw these things backwards. You want 3/5 of rod Q to equal 1/2 of rod P. So show three equal pieces of rod Q equal to one piece of rod P. Then draw the second half of rod P (which also equals 3/5 of rod Q). Then draw two more fifths of rod Q. The part missing from rod Q (the sixth fifth) is the 6 cm difference. Therefore, each fifth of rod Q is 6 cm. There are 5 fifths in rod Q, so 6 x 5 is 30, so rod Q is 30 cm. Quote Link to comment Share on other sites More sharing options...
MrSmith Posted July 12, 2014 Author Share Posted July 12, 2014 Thanks everyone who offered their solutions. It helped me to expand my understanding of bar model. My intent with this question is to try and understand the concept and rationale behind the usage of the bar model in general, and for these types of questions in particular. I'm not trying to teach DS any algebra with these questions, although we may circle back sometime and work this problem that way. In particular, I want to highlight the Process Skills method for this concept (which they call Comparison Unit Model Concept). Here is the question from Process Skills Level 5 (Section 2.3 example 2): '3/4 of Tony's savings was equal to 2/5 of Rizal's savings. If Tony saved $350 less than Rizal, how much did they save altogether?' The solution offered by the book: 'Before drawing the model, convert 3/4 and 2/5 to fractions with same numerator, such that 6 units of Tony's savings was equal to 6 units of Rizal's savings. 3/4 = 6/8; 2/5 = 6/15. 15 - 8 = 7 units. 7 units -> $350. 1 unit = $350 / 7 = $50. (15+8) units -> 23 x $50 = $1150 Total'. In their solution, they highlight that yes 6 of Tony's is equal to 6 of Rizal's, and on Tony's bar there are 8 total sections (2 unshaded), and on Rizal's bar there are 15 total sections (9 unshaded). Also in the picture is that $350 represents the last 7 of Rizal's sections, which extend past Tony's 8 sections. Please explain how they arrived at this methodology. I cannot follow it, even though I was able to solve this problem with the a bar model (but I solved for Rizal's money first then circled back to Tony's money then added them up). I can't follow what the numerator and denominator represent anymore in their solution. In fact no part of the solution makes sense to me. TIA again :laugh: Quote Link to comment Share on other sites More sharing options...
kiwik Posted July 12, 2014 Share Posted July 12, 2014 Thanks everyone who offered their solutions. It helped me to expand my understanding of bar model. My intent with this question is to try and understand the concept and rationale behind the usage of the bar model in general, and for these types of questions in particular. I'm not trying to teach DS any algebra with these questions, although we may circle back sometime and work this problem that way. In particular, I want to highlight the Process Skills method for this concept (which they call Comparison Unit Model Concept). Here is the question from Process Skills Level 5 (Section 2.3 example 2): '3/4 of Tony's savings was equal to 2/5 of Rizal's savings. If Tony saved $350 less than Rizal, how much did they save altogether?' The solution offered by the book: 'Before drawing the model, convert 3/4 and 2/5 to fractions with same numerator, such that 6 units of Tony's savings was equal to 6 units of Rizal's savings. 3/4 = 6/8; 2/5 = 6/15. 15 - 8 = 7 units. 7 units -> $350. 1 unit = $350 / 7 = $50. (15+8) units -> 23 x $50 = $1150 Total'. In their solution, they highlight that yes 6 of Tony's is equal to 6 of Rizal's, and on Tony's bar there are 8 total sections (2 unshaded), and on Rizal's bar there are 15 total sections (9 unshaded). Also in the picture is that $350 represents the last 7 of Rizal's sections, which extend past Tony's 8 sections. Please explain how they arrived at this methodology. I cannot follow it, even though I was able to solve this problem with the a bar model (but I solved for Rizal's money first then circled back to Tony's money then added them up). I can't follow what the numerator and denominator represent anymore in their solution. In fact no part of the solution makes sense to me. TIA again :laugh: I can kind of see how that works but it makes my head hurt and I couldn't justify it. I will bump you and leave it to the experts. Quote Link to comment Share on other sites More sharing options...
wendyroo Posted July 12, 2014 Share Posted July 12, 2014 '3/4 of Tony's savings was equal to 2/5 of Rizal's savings. If Tony saved $350 less than Rizal, how much did they save altogether?' The solution offered by the book: 'Before drawing the model, convert 3/4 and 2/5 to fractions with same numerator, such that 6 units of Tony's savings was equal to 6 units of Rizal's savings. 3/4 = 6/8; 2/5 = 6/15. 15 - 8 = 7 units. 7 units -> $350. 1 unit = $350 / 7 = $50. (15+8) units -> 23 x $50 = $1150 Total'. In their solution, they highlight that yes 6 of Tony's is equal to 6 of Rizal's, and on Tony's bar there are 8 total sections (2 unshaded), and on Rizal's bar there are 15 total sections (9 unshaded). Also in the picture is that $350 represents the last 7 of Rizal's sections, which extend past Tony's 8 sections. Okay, I wasn't thinking of it in these terms, but that makes sense. Let's look at the Q and P problem first since the numbers are smaller. 3/5 Q = 1/2 P Make like numerators: 3/5 Q = 3/6 P [1/6P][1/6P][1/6P ][1/6P][1/6P][1/6P] [1/5Q][1/5Q][1/5Q][1/5Q][1/5Q][6cm] (Pretend all my bars line up nicely) So 6cm is equal to 1 "unit" because it makes up the difference between the length of 5 Q pieces and 6 P pieces. We made sure the pieces of P and Q were the same size because we knew 3/5 Q = 1/2 P and we wrote that as 3/5 Q = 3/6 P so since 3 pieces = 3 pieces they all must be the same length, so 6 pieces - 5 pieces = 1 pieces = 6cm Now the new problem... '3/4 of Tony's savings was equal to 2/5 of Rizal's savings. If Tony saved $350 less than Rizal, how much did they save altogether?' [ 1/8T ][ 1/8T ][ 1/8T ][ 1/8T ][ 1/8T ][ 1/8T ][ 1/8T ][ 1/8T ][ <---------------$350-------------------------------------------->] [1/15R][1/15R][1/15R][1/15R][1/15R][1/15R][1/15R][1/15R][1/15R][1/15R][1/15R][1/15R][1/15R][1/15R][1/15R] So we know all the "units" are the same length because we made them that way on purpose by converting 3/4 and 2/5 into fractions with like numerators. We know 1/8 T = 1/15 R because we know that 6/8 T = 6/15 R because we know that 3/4 T = 2/5 R So we know that $350 = 7 units (15 units - 8 units) 1 unit = $50 and all 23 unit together = $1150 I can probably explain it a different way if you don't see it. Let me know. Wendy Quote Link to comment Share on other sites More sharing options...
MrSmith Posted July 12, 2014 Author Share Posted July 12, 2014 Thank you. I understand this now. Despite the section being titled 'Fractions of a Set' it didn't occur to me that the size of each square ('unit') could be a fraction of each person's total. For a while there I was afraid the concept of 'Fraction' became unglued :D After reading your bar model, I went back and re-did the algebra with T/8 = R/15. Growing up I was always taught to eliminate the fractions to make manipulations easier (No exceptions under penalty of death!). Of course it still worked, but (7/15)R = 350 is more intuitive once I got over breaking that rule. Okay, I wasn't thinking of it in these terms, but that makes sense. Let's look at the Q and P problem first since the numbers are smaller. 3/5 Q = 1/2 P Make like numerators: 3/5 Q = 3/6 P [1/6P][1/6P][1/6P ][1/6P][1/6P][1/6P] [1/5Q][1/5Q][1/5Q][1/5Q][1/5Q][6cm] (Pretend all my bars line up nicely) So 6cm is equal to 1 "unit" because it makes up the difference between the length of 5 Q pieces and 6 P pieces. We made sure the pieces of P and Q were the same size because we knew 3/5 Q = 1/2 P and we wrote that as 3/5 Q = 3/6 P so since 3 pieces = 3 pieces they all must be the same length, so 6 pieces - 5 pieces = 1 pieces = 6cm Now the new problem... '3/4 of Tony's savings was equal to 2/5 of Rizal's savings. If Tony saved $350 less than Rizal, how much did they save altogether?' [ 1/8T ][ 1/8T ][ 1/8T ][ 1/8T ][ 1/8T ][ 1/8T ][ 1/8T ][ 1/8T ][ <---------------$350-------------------------------------------->] [1/15R][1/15R][1/15R][1/15R][1/15R][1/15R][1/15R][1/15R][1/15R][1/15R][1/15R][1/15R][1/15R][1/15R][1/15R] So we know all the "units" are the same length because we made them that way on purpose by converting 3/4 and 2/5 into fractions with like numerators. We know 1/8 T = 1/15 R because we know that 6/8 T = 6/15 R because we know that 3/4 T = 2/5 R So we know that $350 = 7 units (15 units - 8 units) 1 unit = $50 and all 23 unit together = $1150 I can probably explain it a different way if you don't see it. Let me know. Wendy Quote Link to comment Share on other sites More sharing options...
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