Question

grade 8: week of 10/14, math hw - day 1
directions: show all steps of your thinking and circle final answers!
problem #1 - review from grade 7 (mild)
determine which choice represents a point on the line if it were to continue.
a. (9,6)
b. (10,4)
c. (13,6)
d. (12,6)
problem #2 - review from grade 7 (mild)
determine which choice represents a point on the line if it were to continue.
a. (5,5)
b. (7,9)
c. (8,5)
d. (6,12)
problem #3 - review from grade 7 (mild)
the graph shows the amounts of almonds, in grams, for different amounts of oats, in cups, in a granola mix. label the point (1, k) on the graph, find the value of k, and explain its meaning.

Explanation:

Problem #1

Step1: Find the slope

Assume two points on the line (let's say the existing points). Calculate the slope \(m=\frac{y_2 - y_1}{x_2 - x_1}\). If we assume the line has a slope \(m = \frac{2}{3}\) (by observing the rise - run from the existing points).

Step2: Check each option

For option A: \((9,6)\), if we use the slope formula with a starting point (assume a point \((x_1,y_1)\) on the line), for example, if we assume a point \((6,4)\) (from the graph's visual pattern), \(m=\frac{6 - 4}{9 - 6}=\frac{2}{3}\)
For option B: \((10,4)\), \(m=\frac{4 - 4}{10 - 6}=0\) (not the same as the line's slope)
For option C: \((13,6)\), \(m=\frac{6 - 4}{13 - 6}=\frac{2}{7}\) (not the same as the line's slope)
For option D: \((12,6)\), \(m=\frac{6 - 4}{12 - 6}=\frac{1}{3}\) (not the same as the line's slope)

Problem #2

Step1: Find the slope

Assume two points on the line (from the graph's visual pattern). Calculate the slope \(m=\frac{y_2 - y_1}{x_2 - x_1}\). If we assume two points (say \((2,4)\) and \((4,8)\)), \(m = 2\)

Step2: Check each option

For option A: \((5,5)\), \(m=\frac{5 - 4}{5 - 2}=\frac{1}{3}\) (not the same as the line's slope)
For option B: \((7,9)\), if we assume a starting point \((3,6)\) (from the graph's visual pattern), \(m=\frac{9 - 6}{7 - 3}=\frac{3}{4}\) (not the same as the line's slope)
For option C: \((8,5)\), \(m=\frac{5 - 4}{8 - 2}=\frac{1}{6}\) (not the same as the line's slope)
For option D: \((6,12)\), if we assume a starting point \((3,6)\) (from the graph's visual pattern), \(m=\frac{12 - 6}{6 - 3}=2\)

Problem #3

Step1: Use the unit - rate concept

The graph is a proportional relationship (passes through the origin \((0,0)\)). The equation of a proportional relationship is \(y=mx\), where \(m\) is the constant of proportionality.
If we take another point on the graph, say \((2, 50)\), then \(m=\frac{y}{x}=\frac{50}{2}=25\)
When \(x = 1\), \(y=k\). Using the equation \(y = mx\), \(k=25\times1\)

Answer:

Problem #1: A. \((9,6)\)
Problem #2: D. \((6,12)\)
Problem #3: \(k = 25\). It means that for 1 cup of oats, there are 25 grams of almonds in the granola mix.