Appetizer
Sarah is an artist. She is painting a portrait of herself and is trying to create the perfect color for her eyes. Her eyes are a bluish green so she mixed blue and green paint. She made a mixture of 25 drops of blue paint and 15 drops of green paint. It was the perfect color that matched her eyes exactly. Before she started painting she realized her mixture would not be enough to paint both of her eyes. She added 9 more drops of green paint. How many more drops of blue paint must she add to create that perfect color again?
Solution to Appetizer
The answer is that she must add 15 more drops of blue paint to create the color. There are many ways to solve this problem. Here two possible ways:
Fraction strategy
The
24 came from adding 9 drops of green to 15 drops of green, giving us 24
green. The fractions must be
equivalent. To turn 15 into 24 we must
multiply by 1.6 because 15 * 1.6 = 24.
Therefore, we must also multiply 25 by 1.6 to keep the fractions
equivalent. Since 25 * 1.6 = 40, there
must be a total of 40 blue drops of paint.
Since we began with 25 drops of blue paint, and 40 is
15 more than 25, she must add 15 drops of blue.
Unit Rate Strategy
blue
drops per green drop. Therefore, if
there are a total of 24 drops of green now there must be 
drops of blue which
equals 40 drops of blue. Since 40 is 15 more than 25, she must add 15 more drops of blue.
Another Strategy
25:15 Reduce to a simpler ratio
5:3
For every 5 drops of blue, we need 3 drops of green. Since 9 is 3 x 3, we need 5 x 3 = 15 more drops of blue.
Amber went on a walk with her dog Rex. Amber let Rex run around without a leash on their walk because he was a good dog and always came to her side when she called for him. Which graph do you think represents Amber’s walk and which graph represent Rex’s walk? For both graphs, the scale is the same. Describe how you decided which graph belonged to Amber and Rex. Write a short story describing the events that happened on their walk.
Solution to Main
Course
While it is possible for Rex and Amber to be either graph it makes more sense for Amber’s walk to be represented in Graph 1 and Rex’s walk to be represented in Graph 2. Graph 2 shows a much higher speed and many more stops. Dogs often stop during walks to smell things and go to the bathroom. Graph 1 has more areas that are at a steady speed. During walks humans often walk steadily enjoying the scenery. While it makes more sense for Amber’s walk to be represented in Graph 1 and Rex’s walk to be represented in Graph 2, as long as the story backs up their decision, it could be right the other way also.
Explanations for
Graph 1 (Amber’s walk) should include:
1) Constant acceleration at the beginning
2) Constant velocity at the top of both maximum points
3) Explanation of blank areas (stops)
4) Explanation for curve in second high speed area
Here is an example of a story for Amber’s walk:
Amber started out slow and gained speed at a constant rate. She then walked at a stead pace for a while enjoying the scenery. She fairly abruptly when she noticed her shoe was untied. She took a little time to retie her shoe and continued on her way. She walked very slowly at first slowly gaining speed. She then got on the paved sidewalk and began to pick up speed faster for a little bit but then gained speed slower until she reached a constant speed. She called for Rex while still walking and slowed down steadily as she waited for him to come. When she stopped walking he was right there waiting for his treat.
Explanations for Graph
2 (Rex’s walk) should include:
1) Reasons for curved line in the first, third and fourth hill
2) Reasons for all three blank areas (stops)
3) Explanation of the slow constant speed section
4) Explanation of the abrupt stops
Here is an example of
a story for Rex’s walk:
Rex was so excited to be outside on a beautiful day. He immediately took off first gaining speed slowly and then gaining speed very quickly. He ran at a steady pace for a while until he smelled something good. He slowed down very quickly and came to a stop. He gained a little speed and walked at a very slow steady pace for a while, sniffing the ground for the source of the smell. He stopped almost abruptly when he had found what was making the smell. It was someone’s half eaten hot dog. He gobbled it up very quickly (yuck!) and looked up. He saw a squirrel running across the grass and took off after it very quickly. It ran up a tree so he stopped just as quickly as he had begun. Just then he heard Amber call for him, so he began running to her slowly picking up speed. He began to pick up more speed when he saw that she had a treat for him. He came to a quick stop in front of her.
Aaron’s mother gave him a list of errands to do one morning. He had to mail a letter at the post office, pick up some bread at the grocery store, and give his grandmother her birthday present. All of the places Aaron went that morning are on the map along with the possible roads he could have taken. He could have gone to his three stops in any order. How many different trips are possible?
The next day, Aaron’s mother asked him to make the same three trips along with a trip to the pet store to get some pet food. Create a new picture for this day and find out how many possible different trips there are if Aaron could go to his four stops in any order.
Solution to Dessert:
To solve this problem, we must create a list. This list could be organized or unorganized. An unorganized list would be made by picking
random trips and seeing how many you can find.
This makes it difficult to know if you have all of the possible
combinations, so an organized list is a better idea. In this organized list we
will let, GS = Grocery Store, GH = Grandma’s House and
Home – GS – GH –
Home – GS –
Home – GH – GS –
Home – GH –
Home –
Home –
We see that there are 6 possible trips.
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