Ir-Rational Number Institute
Bureau of Indian Education
Santa Fe Community College
Problem Set for October 4, 2014
Big Ideas that We’ll Explore in This Weekend’s Institute:
1. Examine the proportionality of two quantities using mathematical models and use
proportional relationships to solve problems.
2. Understand that solutions to a system of two linear equations in two variables
correspond to points of intersection of their graphs, because points of intersection
satisfy both equations simultaneously.
3. Solve systems of two linear equations in two variables algebraically, and estimate
solutions by graphing the equations
4. Estimation of measures and the development of benchmarks for frequently used
units of measure help students increase their familiarity with units, preventing errors
and aiding in the meaningful use of measurement.
I. Let’s do a few warm-up problems just to get the juices flowing!
1. You decided to check the accuracy of the speedometer in your car by timing your
travel between mile markers on the highway. If you found that it was 50 seconds
between markers, what would you know?
2. When I drive to the Big Ole’ Happenin’ Mall on city streets at 40 mph, it takes me 20
minutes to get there. When I return the same distance at 50 mph on the highway.
How long does the return trip take?
II. Okay, now that you’re good to go, let’s solve a few more problems and think
about how graphing calculators may help our students visualize mathematical
ideas.
CCSS.Math.Content.8.EE.C.8.A: Understand that solutions to a system of two linear
equations in two variables correspond to points of intersection of their graphs, because
points of intersection satisfy both equations simultaneously.
3. Marian opened a video store. Her customers must pay a one-time fee of $10 to join
her video club and $3.50 for each video rented. Kathy also opened a video store, just
down the street from Marian. The cost to join Kathy’s video club is a one-time fee of
$20, but it only costs $3.20 for each video rented from her store. Create a table and a
graph that examines the cost of renting a video from Marian and from Kathy, and
then compare and contrast the advantages to joining each video club.
4. What is the slope of each line created in problem #1 and the y-intercept of each?
Interpret the slope and y-intercept for each line.
5. Larry and Leroy are in a race. Larry runs at an average speed of 7 meters per second
and is not given a head start. Leroy runs at an average speed of 5 meters per second
and is given a head start of 7 meters. Will they ever meet? If so, when?
CCSS.Math.Content.7.RPA.2.A: Decide whether two quantities are in a proportional
relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate
plane and observing whether the graph is a straight line through the origin.
6. Debra runs a 10 km race in 55 minutes and Catherine runs a 25 km race in 140
minutes. Create a graph on the coordinate grid to determine if Debra and Catherine’s
rates are proportional.
III. CCSS.Math.Content.8.EE.C.8.B: Solve systems of two linear equations in two
variables algebraically, and estimate solutions by graphing the equations. Solve simple
cases by inspection.
7. Use mathematical modeling to estimate solutions for each of the following systems
of linear equations and then solve algebraically. For simple cases, solve by inspection.
a. 4x - 3y = 11 and -5x + y = -12
b. 2y + 3x = 5 and 3x + 2y = 6
c. 5x – 2y = 6 and 10x – 4y = -4
IV. Additional Challenge Modeling Problems (Source: Bar modeling: A problem-solving
tool by Yeap Ban Har).
Use mathematical modeling to solve the following [CCSS.Math.Content.7.NSA.3:
Solve real-world and mathematical problems involving the four operations with
rational numbers]:
8. ½ of a group are children. 1/3 of the adults are men. There are 36 women. How
many people are in the group? (p. 134)
9. In a game, Zack sank 3 times as many baskets as Xin. Yani sank half as many baskets
as Zack. They sank a total of 44 baskets. Find the number of baskets that Zack sank.
(p. 135)
10. At first, the ratio of the number of apples to the number of oranges at a grocer’s was
5 : 3. After an equal number of apples and oranges were sold, the ratio became 7 : 1.
How many oranges were there at first if 32 apples were sold? (p. 136)
11. 5/8 of a group of girls chose the tiger as their favorite animal. 5/6 of the rest chose
the monkey. The remaining 18 girls chose the giraffe. How many girls were there in
the group? (p. 138)
12. 3/4 of the lions and 1/3 of the tigers in a wild cat enclosure are females. 3/5 of the
females are lions. What fraction of the animals in this enclosure are females? (p. 143)
V. Estimation of Measures and the Development of Benchmarks…
13. Measuring Body Parts 1: Humans have long used different body parts to measure
common objects. Some of these measures included the thumb, hand span, foot, yard,
pace, and fathom (i.e., “wingspan”). Measure the length of your table using all of these
measures. List your results on the class chart paper. What are advantages and
disadvantages are associated with using body parts to make measurements?
14. Measuring Body Parts 2: What do you think the relationship is between shoe size
and foot length? Make a table that lists the foot length (in centimeters) and
corresponding shoe size for each member of your group. Graph the results. Put foot
length in centimeters on the horizontal axis and shoe size on the vertical axis. What is
the relationship between foot length and shoe size?
15. Measuring Body Parts 3: In your group, use a measuring tape to measure each
person’s “wingspan” in centimeters. Your wingspan or “fathom” is the distance from
the middle fingertip of one hand to the middle fingertip of the other hand when your
arms are fully extended. Compare the measure of each person’s wingspan to their
height. What do you notice?
16. Surface of a Hand: Trace your hand on grid paper. Estimate the area of your
handprint in square centimeters. One method for estimating body surface area is to
multiply 100 times the surface area of your hand. Another is to find 3/5 of the area of
the square that you “fit into” when your arms are fully extended. Calculate the surface
area of your body using both of these methods. How accurate do these methods appear
to be?
17. Crooked Paths: Make some crooked or curvy paths on the floor with masking tape
or chalk. The task is to determine which path is longest, next longest, and so on.
Students should suggest ways to measure the crooked paths so that they can be
compared easily.