5.NF.A.2 Fractions--The Unusual Baker Problem

5.NF.A.2  Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers.

The more I work with the CORE standards, the more I appreciate the wisdom of slowing down and laying a solid foundation. I have been working all year in my 4th grade class on the concept of fractions, but my students still confuse ideas about "where is the whole"?





We had a lot of fun this week doing  The Unusual Baker problem from NCTM.

I asked the class to solve the cake problems individually, before sharing too many ideas with others.






Following this, the children worked in groups, divided the tasks, and heartily discussed the price and fractional size of each cake portion. We worked on this activity for 3 days and still need to review some of the key points.








The children could tell me that the cake was cut in half, and that the two equal parts on the right were fourths...but it didn't bother them that $4 did not equal $5! 

It took several questions to get them to focus on the numbers as money, and to realize that half of 5 dollars was $2.50.





Another group correctly made the four equal parts and proudly told me that $6 plus $4 was $10--so their answers were correct. (Ugh!) I asked them, "So which fourth would you like to buy, one for $2 or one for $3?" At least they were beginning to realize that their numbers were not reasonable.

 



  A third group showed me 4 equal parts on one side and 2 equal parts on the other. I asked them, "Where is the whole cake? ...Then does it make sense that these long strips are each half of that whole?" 

 
At that point, Cristal's face sparkled as she realized that the smaller portions were eighths. Whew!! We're beginning to make progress.

I showed them how to cut and glue new answers to repair their chart. Next, I will have to have them look again at Thursday's burger in the lower right-hand corner. (Interesting that they wanted to cut and serve portions of burgers instead of cake or pizza.)










Cesar's group understood the problem the best and felt the most confident when they presented their solutions to the class.  












Great work, Pizza group!!  You rock, Rigoberto!

 Great job with the mic, too.

Here is their finished poster. Well done, team.

For most of the class, I still need to reinforce the concept of always finding the whole, and then making sure you are equally dividing the entire area. A fourth of one cake, should match one-fourth of another cake of the same size.

Many of the students still do not have enough confidence to boldly declare that their answers are absolutely correct. So if another group member tells them a different answer, they simply erase without questioning it.

Here Martha knew that half a cake was $5, but she changed it when Angie told her that "It has to be six dollars, because 6 plus 4 is 10. And you can't have 3 plus 3 because that is too big." It never occurred to the group that you could use coins to make the money amounts correctly match the fractional sizes.

Way to go, Cesar!!  Good problem solving. Next time don't erase your work; rather, use it to prove your answers.














Even though Esmeralda still needs to finish, she has done an excellent job showing her work and proving her answers. Next, I need to have my students explain their thinking in writing--something they struggle with and need a great deal of practice doing.

Follow-up lesson to clarify fractional parts and cake prices...

Gabriela shows that these "halves" are actually two-fourths of the whole. She is then dividing the whole into eight equal parts to prove that the "fourths" are really eighths.

Angel explains why the two "half" sections on Jason's paper are actually eighths.

Next, I gave each child a circular protractor and a calculator. We talked about how the whole was the 360 degrees of the circle. Therefore, 1/4 of 360 degrees was 90 degrees and 1/4 of $10 was $2.50.

Rigoberto shows how he added $1.25 three times to get the value of 3/8 of the cake as being $3.75.

Jorge also explains where the 5/8 are on the circular "cake" and why the price is $6.25.

Stay tuned....we're not done with fractions yet, I'm sure!!  Why can't the U.S. just use the metric system like the rest of the world?!!

6.G.A.1 Finding Area on the Geoboard

Core Standard: 6.G.A.1  Find the area of right triangles by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.

The geoboard is the perfect manipulative for a student to demonstrate understanding of this standard. Esmeralda shows her octopus polygon and proves that the area is 11 square units.

 Galilea proves the area of her "X" by showing the four hidden right triangles inside each of the 4 "arms" of her polygon. This was a popular design, so several were able to compare  solutions to prove that the area is 10 square units.








Although Johnny made the same design, he proved the area by finding the outside area of 6 square units and subtracted it from 16 to get a polygon area of 10 square units.

Another fun activity I like to do with this is ask the children to find a shape that will trick the teacher. Thrilled with the challenge, several tried to create difficult shapes.  Soon, they become so good at solving the difficult puzzles that they don't need me to intervene.

Sometimes students are confused about the size of the square units. Whenever they see any square, they think it is a square unit, forgetting that all of the units must be the same size and measured with the same unit value. Ivette redoes her polygons to show 1 (square for sq. unit) + 2 sq. units + 2 sq. units + 1 sq. unit = 6 (square units).

Carl shows how he determined the area of the yellow quadrilateral by drawing a square around the top right-hand portion of his polygon.
He knows the area of the square is 4 square units, so he can subtract the 1 red square to get 3 square units. In addition, he can see that one-half of the top 2 squares is 1 square unit, so he can use this information to determine that the yellow area inside this square is also 1 square unit. Since his design is symmetrical, the area for the rest of the polygon is easy to figure out.


 Esmeralda shows how she combined half squares to count whole square units.

Rigoberto and Angel are both very proud of their shapes. I love the critical thinking skills involved here!

Although this is now a 6th grade standard, my 4th graders love this activity and everyone feels successful.

Carl and his group prove his solution to the class, giving everyone a chance to demonstrate his or her understanding.

I also taught the students how to prove the area by subtracting the outside or negative space from the total geoboard area. In this way, the children are able to prove their area in two different ways.

Galilea and Leslie present their solution for Leslie's fancy "Spiked Heel" design.

Don't have geoboards?

Here's a great virtual geoboard!

Can you help Cesar and Alejandra prove the area of this quadrilateral as 5 square units?

They prove the area by finding a hidden right triangle on the bottom right, and use the idea of a 4-unit square to determine the top portion of this shape.





Geoboards are an excellent way to explore fractional parts of the one-whole square on the geoboard. The children also enjoy creating quilt or garden designs.

Alejandra, a third grader, illustrated her geoboard garden. You can also see the Communication Guide I created to help my ELD students write down their mathematical ideas and strategies.

How many of these shapes can you solve and prove?

There are a number of great interactive websites where the children can create interesting shapes and click the button to compare their answers.








Great geoboard outfit, Andrew!
Interactive Geoboard

I am truly proud of these math geniuses!  :)  All of them!!

Angel and Gabriela explain how they solved Angel's "Chinese Fighter Fish" polygon.