Authentic+Problems+for+6th+Grade

=Authentic problems for 6th grade, based on Ingham ISD power standards=

1. It's 20 degrees today. The temperature is supposed to reach -8 degrees tonight. How many degrees colder will it be tonight than it was today? Show this on a thermometer/number line. Variation: Chart temperatures for 7 days in the winter (or use Celcius temperatures to get more negatives). Compute the difference from one day to the next: "Today was 7 degrees colder than yesterday" or "The temperature went down by 10 degrees from Wednesday to Thursday." Eventually have students use this notation: "The difference in temperature from Wed to Thurs was -10 degrees" (for when it gets colder). Find the range of temperatures over the week and the mean temperature for the week.

2. An elevator in a tall building has buttons marked with integers. The integer represents how many floors you want to move: positive numbers make the car go up, negative numbers make the car go down. (This is an elevator in the math department building at the U of M.) Figure out which button to push for each of these situations, and write a number sentence to go with each one. (Also, use a vertical number line to show each problem): a) You're on the 25th floor and want to go to the 10th floor. b) You're on the 10th floor and want to go to the 3rd sub-basement (3 floors underground). c) You're on the ground floor and want to go to the 19th floor. d) You're on the 8th sub-basement (it's a big parking garage) and you want to go to the ground floor. e) You're on the 15th sub-basement (it's a really big parking garage) and you want to go to the 5th sub-basement, so you can then get some exercise by walking back to the ground floor. f) You don't know what floor you're on, so you push -7, and you wind up at the 5th floor. What floor did you start on? g) The lights have gone out in the building (but the power to the elevators still works). You're on the 34th floor, and you push some button (you can't see which one you pushed). When you get out of the elevator, you're three floors below where you parked your car on the 2nd sub-basement. What floor did you start on?

3. There's a party going on in your house. Some of the people at the party are happy, and the others are grumpy. The mood of the whole party depends on if there are more happy people or grumpy people. One grumpy person cancels out one happy person. We'll let a happy person be represented by a +1, and a grumpy person is represented by a -1. a) When the party starts, there are 8 happy people and 3 grumpy people. So the party is mostly happy: +8 + (-3) = +5 b) But then 7 grumpy people come in. What's the mood of the party now? Use integers to express the party's mood. Hint: +5 + (-7) = [Ans: -2] c) Now the party is too grumpy, so 4 of the happy people leave. Find the mood of the party now, using a number sentence. [Ans: -6] d) But then a limo full of happy people arrives. 12 happy people join the party. What's the mood now? [Ans: +12] e) Write a number sentence to express what happens to the mood of the party if 3 grumpy people leave. [Ans: 12 - (-3) = 15 Hence subtracting a negative is like adding a positive.] Multiplication: f) 3 limos with 8 happy people each arrive. What happens to the party's mood? 8(+3) = +24 --> the party gets much happier g) 4 limos arrive with 5 grumpy people each. What's the effect on the party's mood? 4(-5) = -20 --> the party gets much grumpier h) 2 limos arrive to remove 6 grumpy people each. What's the effect on the party's mood? -2(-6) = +12 --> the party also gets much happier

4. Another representation of integers that is similar to the grumpy/happy people representation, is from the Interactive Mathematics Program, called "Chef's Hot and Cold Cubes." You can imagine unmeltable ice cubes that represent -1, so when they're added to something they lower the temp by 1 degree, and unburnable hot charcoal cubes that represent +1, so when they're added to something they raise the temp by 1 degree. a) If a liquid is at 60 degrees and you add 3 hot cubes and 7 cold cubes, what's its new temperature? Write an expression to show what you would do. [Ans: 60 + 3 + (-7) which is 56 degrees] b) If you add 5 trays full of cold cubes (10 in each tray), what's the new temp? Write the expression. [Ans: 56 + 5(-10) which is 6 degrees] c) If you remove 3 trays of cold cubes (10 in each tray), what's the new temp? Write the expression. [Ans: 6 - 3(-10) or 6 + 30]

M6.2 Multiply and divide any two fractions fluently to solve applied problems.
1. A bakery has planned to make cakes today. They need the following ingredients for each cake. They want to bake 12 cakes. How much of each ingredient do they need for all 12 cakes? 3/4 cup of sugar 2 1/3 cups of flour 1/4 teaspoon of salt 2/3 tablespoon of baking powder

2. A pan of brownies was left out on the counter and 1/4 of the brownies were eaten. Then you came along and ate 2/3 of the brownies that were left. How much of the whole pan of brownies was eaten?

3. You have 6 donuts and you want to give 2/3 of them to a friend and keep 1/3 for yourself. How many donuts would your friend get? That is, how much is 2/3 of 6?

4. A baker is making cakes for a big party. She uses 1/4 cup of oil for each cake. How many cakes can she make if she has a bottle of oil that has 6 cups in it?

5. The serving size for the granola that Ted likes to eat for breakfast is 3/4 cup. How many servings are there in a box that holds 13 cups?

6. Mrs. Murphy's class is making pillow cases. Each pillow case uses 3/4 of a yard of fabric. How many pillow cases can they make out of 12 1/2 yards of fabric? Will any fabric be left over? If so, how much?

7. A book shelf is 3 1/2 feet long. Each book on the shelf is 5/8 inches wide. How many books will fit on the shelf?

M6.3 Solve applied problems that use the four operations with appropriate decimal numbers.
1. The circumference of a circle is found by multiplying the diameter of the circle times pi (approximately 3.14). The trunk of a tree is more or less round. You want to figure out the diameter of a tree without cutting it down. So you measure the distance around the tree using a tape measure. You find that the distance around the tree trunk (its circumference) is 7 feet. Calculate the diameter of the tree. You know that the tree is 45 1/2 years old. What is the average yearly rate of growth of the tree's trunk? That is, on average, how many inches bigger did the tree get every year? (decimals, fractions, converting units of measure, manipulating formula for circumference)

2. Suppose it costs $20 per day and $0.08 per mile to rent a car. What is the total bill if the car is rented for 2 days and is driven 120 miles?

M6.4 Use ratios and rates in problem situations.
1. 5 boys and 3 girls hold a car wash. They get a donation of $4 for every car. They want to split the money fairly between the boys and the girls, in the ratio of 5 to 3. How much money do the boys get, and how much money do the girls get, for each car they wash?

2. To make donuts, a baker uses 3 pounds of flour for every 2 pound of sugar. This will make about 10 dozen donuts. The baker needs to order supplies for the week. She knows that the bakery makes about 30 dozen donuts a day, every day of the week including the weekends. How much flour and sugar should she order for the week?

3. Using string (or yarn) and cans of different sizes (especially different diameters), wrap the string around the can, mark the place where the string starts and ends, then lay it flat and measure it. This is the circumference of the can. Also measure the diameter of the can (the distance across the widest part of the can) with a ruler (using the same measuring units). Record the circumference and diameter of each can in a table. Find the ratio of the circumference to the diameter. Do you notice any patterns?

M6.5 Represent situations using algebraic expressions and solve simple equations.
1a. Every student in a school is given 3 pens at the beginning of the year. If there are 127 students in the school, how many pens are needed for all the students? 1b. Write an algebraic expression to represent how many pens are needed if there are an unknown number of students in the school. Let x equal the unknown number. 1c. A school buys 723 pens for their students. All the pens are given to students, 3 to each student. How many students are in the school. Write an equation to show this.

2a. Make up a situation that would be represented by this algebraic expression: 25t 2b. If 25t = $200, what does t equal?

3. You took a long bike ride with your friends. You biked 10 miles every hour. At the end of the day, the map showed that you biked 75 miles. Write an equation to represent this situation. Then solve it to find how many hours you biked.

4. Alice is 5 years younger than twice her cousin's age. Alice is 21. How old is her cousin? Write an equation to represent this, and show how you would solve it.