Lesson 32 · Tables and Equations · Osaka Budo Dojo
Tournament Scoreboard
Complete a table from an equation, and write an equation from a table or from words.
The Osaka Budo Dojo is hosting the Kansai Junior Open. Behind the mats, a scoreboard operator tracks every fighter's total points. Each fighter starts with a starting bonus, then points are added (or taken away) at a steady rate.
The same story can be shown in two ways: a table of pairs, or one equation. Today you travel in both directions, and you must always know which direction you are going.
Equation → table
You are given the rule. For each $x$ in the table, substitute it into the rule to find $y$.
Table or words → equation
You are given evidence. Check the rate is constant, find the rate $m$ and the start $b$, then write $y=mx+b$.
The scoreboard machine
Blue = the input $x$ (matches won). Rose = the output $y$ (total points). Green = the rate (the steady $y$-step for each $+1$ in $x$). Amber = the start (the $y$ at $x=0$). The dashed lines show that the machine and the table describe the same pair, $(3,14)$.
Worked example · From a table to an equation
The scoreboard shows this table for a junior fighter, where $x$ is matches won and $y$ is total points.
$x$ (matches)
0
1
2
3
$y$ (points)
5
8
11
14
1 · Check the steps. $x$ rises by $1$ each time; $y$ rises by $3$, $3$, $3$. The change is the same every time, so the rate is constant and a line equation can fit.
2 · Find the rate $m$. $m=\dfrac{\text{change in }y}{\text{change in }x}=\dfrac{3}{1}=\color{#15803d}{3}$ points per match. If the $x$-steps are uneven, divide anyway: from $x=1$ to $x=3$, $y$ goes $8\to14$, so $\dfrac{6}{2}=3$.
3 · Find the start $b$. Look at the row where $x=0$: $y=\color{#b45309}{5}$ points. (If there is no $x=0$ row, step backwards: subtract one rate-step for every $x$ you go back.)
4 · Write the equation. $$y=\color{#15803d}{3}x+\color{#b45309}{5}$$
Check with a different row: $x=3$ gives $3(3)+5=14$, which matches the table.
Practice
Scaffolded · the equation or structure is shown
1 · The opening round.EQUATION → TABLE At the dojo, a fighter's total points follow $y=4x+6$, where $x$ is matches won and $y$ is total points. The row for $x=1$ is done for you.
$x$ (matches won)
0
1
2
3
$y$ (points)
?
10
?
?
(a) Complete all missing table entries, showing one substitution as evidence. (b) Describe the constant change in points for each additional match won.
substitute with parentheses → write y beside each x → read the rate from the equation
2 · Cautions cost points.EQUATION → TABLE Every fighter starts a bout with $20$ conduct points, and each caution removes $2$. The rule is $y=-2x+20$, where $x$ is cautions received and $y$ is conduct points left. Notice that the $x$-values are not evenly spaced.
$x$ (cautions)
0
1
3
6
$y$ (points left)
20
?
?
?
(a) Complete all missing entries. (b) Verify the signed rate between $x=3$ and $x=6$ using the actual coordinate changes, and explain why counting table columns would be wrong.
substitute → keep the negative rate → signed change in y ÷ change in x → units
3 · Read the scoreboard backwards.TABLE → EQUATION Now the direction flips: nobody tells you the rule. Pool B's scoreboard shows $x$ = matches won and $y$ = total points.
$x$ (matches)
0
1
2
3
$y$ (points)
7
12
17
22
Derive the equation $y=mx+b$ from the table. Support it by identifying the constant rate and starting value, then verify it with a row you did not use to calculate the rate.
Before you divide, check that all three $y$-steps are equal.
check steps equal → rate → start → fill the structure
From here, write the equation yourself. No structure is shown.
4 · The speed round.TABLE → EQUATION In the speed round, the scoreboard gives a starting bonus and then awards points for each clean strike. $x$ is clean strikes and $y$ is total points. The strikes were logged at uneven moments.
$x$ (strikes)
0
2
3
6
$y$ (points)
4
7
8.5
13
(a) Calculate the rate using the rows $x=2$ and $x=6$ (include units). (b) Name the starting value, with units. (c) Write the equation. (d) Calculate $y$ for $x=3$ using your equation, and compare it with the table.
uneven jump → divide the changes → read the start → write y = mx + b → test a different row
5 · Round-robin pool: does a line fit?TABLE → EQUATION In a round-robin pool, every fighter fights every other fighter once. $x$ is the number of fighters in the pool and $y$ is the number of matches the dojo must schedule.
$x$ (fighters)
2
3
4
6
$y$ (matches)
1
3
6
15
(a) Calculate the rate from $x=2$ to $x=3$ (matches per extra fighter). (b) Calculate the rate from $x=4$ to $x=6$ (matches per extra fighter). (c) Decide whether a single equation $y=mx+b$ can describe this table. Frame: "___, because the rate from $2$ to $3$ is ___ but the rate from $4$ to $6$ is ___."
always check constancy first → two intervals → compare → decide before writing any equation
6 · The final, in words only.WORDS → EQUATION Haruka walks into the final with $15$ points on the scoreboard. Every warning from the referee removes $2.5$ points from her total. (a) Name what $x$ and $y$ stand for, including units. (b) Write the equation for Haruka's total points. (c) Calculate her total after $4$ warnings, using your equation.
name the variables → which number is the start, which is the rate (sign?) → write → substitute
Review · Skills from earlier lessons
7 · Valencia club uniforms. Our club in Valencia compares two shops for gi uniforms. $x$ is the number of uniforms and $y$ is the total cost in euros (€).
Shop A
$x$ (uniforms)
2
5
9
$y$ (€)
30
75
135
Shop B
$x$ (uniforms)
2
5
9
$y$ (€)
28
70
120
Decide which shop has a constant cost per uniform. Support the decision with the two interval rates for each shop, taking account of the unequal $x$-steps.
uneven steps → change in € ÷ change in uniforms → compare intervals → decide
8 · The kick bonus. The Osaka judges award $2$ points for every $3$ kicks landed, plus $5$ starting points: $y=\frac{2}{3}x+5$, where $x$ is kicks landed and $y$ is total points. A fighter finishes with $y=17$ points. (a) Write the equation with $y=17$ substituted. (b) Solve for $x$ using balance notation. Dividing by $\frac{2}{3}$ is the same as multiplying by $\frac{3}{2}$. (c) Check your $x$ by substituting it back into $y=\frac{2}{3}x+5$.
replace y, not x → subtract the start → undo the fraction with its reciprocal → check
9 · Prize money in Valencia. The club deposits €$800$ of prize money in an account that pays $3.5\%$ simple interest per year, using $I=Prt$ ($I$ is interest in euros, $P$ is the principal in euros, $r$ is the yearly rate as a decimal, $t$ is time in years). (a) Convert $3.5\%$ to a decimal. (b) Calculate the interest $I$ after $4$ years. (c) Write an equation for the total money $y$ in the account after $x$ years (principal plus interest).
percent ÷ 100 → P × r × t → interest per year is the rate, the deposit is the start
Retention check
10 · The timing drill. In a timing drill the scoreboard shows two readings: $(2,18)$ and $(6,6)$, where $x$ is minutes into the drill and $y$ is points left. Calculate and interpret the signed slope in points per minute, showing the coordinate changes you used.
second minus first → signed changes → divide → units → rise or fall
Show answers
Problem 1a
$y=4(0)+6=6$ points. Pair $(0,6)$.
Problem 1b
$y=4(2)+6=14$ points. Pair $(2,14)$.
Problem 1c
$y=4(3)+6=18$ points. Pair $(3,18)$.
Problem 1d
"Each match won adds $4$ points." (the number multiplying $x$)
Problem 2a
$y=-2(1)+20=18$ points left.
Problem 2b
$y=-2(3)+20=14$ points left.
Problem 2c
$y=-2(6)+20=8$ points left.
Problem 2d
$\dfrac{8-14}{6-3}=\dfrac{-6}{3}=-2$. "$y$ changes by $-6$ points while $x$ changes by $3$ cautions, so the rate is $-2$ points per caution." This matches the $-2$ in the equation.
Problem 3a
$y$-steps are $5,5,5$ with $x$-steps of $1$, so $m=\dfrac{5}{1}=5$ points per match.
Problem 3b
At $x=0$, $y=7$, so the start is $7$ points.
Problem 3c
$y=5x+7$. Check: $5(3)+7=22$.
Problem 4a
$m=\dfrac{13-7}{6-2}=\dfrac{6}{4}=1.5$ points per clean strike. (Rows $2\to3$: $\frac{1.5}{1}=1.5$ and $0\to2$: $\frac{3}{2}=1.5$ agree.)
Problem 4b
At $x=0$, $y=4$: the start is $4$ points.
Problem 4c
$y=1.5x+4$.
Problem 4d
$y=1.5(3)+4=4.5+4=8.5$ points, which matches the table.
Problem 5a
$\dfrac{3-1}{3-2}=\dfrac{2}{1}=2$ matches per extra fighter.
Problem 5b
$\dfrac{15-6}{6-4}=\dfrac{9}{2}=4.5$ matches per extra fighter.
Problem 5c
No, because the rate from $2$ to $3$ is $2$ but the rate from $4$ to $6$ is $4.5$. The rate is not constant, so no single $y=mx+b$ fits; the graph would curve, not be a straight line. (Rate from $3$ to $4$ is $3$, so the rate keeps growing.)
Problem 6a · x
$x=$ number of warnings (warnings).
Problem 6a · y
$y=$ Haruka's total points (points).
Problem 6b
$y=-2.5x+15$ (start $15$, rate $-2.5$ points per warning).
Problem 6c
$y=-2.5(4)+15=-10+15=5$ points.
Problem 7a
Shop A: $\dfrac{75-30}{5-2}=\dfrac{45}{3}=15$ € per uniform.
Problem 7b
Shop A: $\dfrac{135-75}{9-5}=\dfrac{60}{4}=15$ € per uniform.
Problem 7c
Shop B: $\dfrac{70-28}{5-2}=\dfrac{42}{3}=14$ € per uniform.
Problem 7d
Shop B: $\dfrac{120-70}{9-5}=\dfrac{50}{4}=12.5$ € per uniform.
Problem 7e
Shop A. Its rate is $15$ € per uniform both times; Shop B's rate changes from $14$ to $12.5$. (Shop A's rule would be $y=15x$.)
Problem 8a
$17=\frac{2}{3}x+5$.
Problem 8b
$$17=\tfrac{2}{3}x+5\quad\mid-5$$$$12=\tfrac{2}{3}x\quad\mid\div\tfrac{2}{3}\ \text{(same as }\times\tfrac{3}{2}\text{)}$$$$18=x$$The fighter landed $18$ kicks.
Problem 8c
$\frac{2}{3}(18)+5=12+5=17$. It checks.
Problem 9a
$3.5\%=\dfrac{3.5}{100}=0.035$.
Problem 9b
$I=Prt=800\times0.035\times4=€112$.
Problem 9c
Interest per year is $800\times0.035=€28$, so $y=28x+800$ ($x$ = years, $y$ = total in €). Check: $28(4)+800=912=800+112$.
Problem 10a
$\Delta y=6-18=-12$ points.
Problem 10b
$\Delta x=6-2=4$ minutes.
Problem 10c
$m=\dfrac{-12}{4}=-3$ points per minute. The points fall (negative slope).
Next · Lesson 33: Graph the scoreboardEvery table you completed today is a set of ordered pairs. In Lesson 33 you will plot those pairs on a coordinate grid and see that a constant rate makes a straight line, with the start where the line crosses the $y$-axis. The round-robin table from Problem 5 will show you what a table that is not linear looks like when it is drawn.