Mathematics, grade 8

Example

1. A campground charges $32 a night. Ask what four nights comes to, then hand over the receipt and ask whether it matches.
2. Ask how many packs of 8 water bottles it takes to get 40 bottles into the cooler.

Example

1. Give a two-step problem and ask what they would do first, before any arithmetic happens.
2. After an answer, ask why it is reasonable. Listen for a reason rather than a re-check of the steps.

Example

1. Ask for 5 x 60 and watch whether they reach for paper. Reaching for it at this size is the thing worth noticing.
2. Ask which is easier for 47 x 6, mental math or writing it down, and why.

Example

1. Ask them to show 6 x 7 two ways: with a drawing, and with a number sentence.
2. Ask them to explain a division out loud, without pointing at the page.

Example

1. Ask them to keep a tally of something on a drive and turn it into a table at the end.

Example

1. Ask what 6 x 4 has to do with 6 x 40.
2. Ask why knowing 5 x 8 helps with 6 x 8.

Example

1. Ask them to convince you that one third is bigger than one fourth, using the word "parts". Listen for whether the bigger bottom number pulls them the wrong way.

Example

1. Draw a Venn diagram of rational and irrational numbers inside the real numbers, and ask them to place 3, -1/2, √2, π, and 0.75 where each one belongs.
2. Give the set {4, -3, 2/3, √5} and ask which members are rational and which are only real.

Example

1. Ask them to approximate √50 to the nearest whole number, then mark that approximation on a number line.
2. Ask them to approximate π as 3.14, say which two whole numbers it sits between, and mark it on a number line.

Example

1. Ask them to write 4,500,000 in scientific notation, and 3.2 × 10⁴ back in standard decimal notation.
2. Ask them to write 0.00072 in scientific notation.

Example

1. Ask them to order -1.5, √4, π, and 2/3 from least to greatest.
2. Three trail markers read -12.5 feet, √100 feet, and 3π feet. Ask them to order the three elevations.

Example

1. Give a triangle and its dilation with every side doubled, and ask what the side ratios say about the two shapes.
2. Give a photo 4-by-6 enlarged to 6-by-9 and ask whether the corresponding sides stayed proportional.

Example

1. Dilate a rectangle by 2 on graph paper with the origin as the center and ask what changed about its position, its size, and its shape — and what did not.
2. Show a shape and its half-size dilation sitting in another quadrant and ask them to compare the coordinates, the side lengths, and the angles.

Example

1. Multiply every coordinate of (2, 3) by 3, with the origin as the center, and ask where the point lands and what the 3 did.
2. Apply scale factor 1/2 to a triangle at (4, 6), (8, 2), and (0, 2), and ask for the image coordinates.

Example

1. Draw two similar right triangles as slope triangles on one line and ask why the rise-over-run comes out the same.
2. Take the points (1, 2) and (4, 8) on a line and ask for the slope two ways: from the triangles and from (y2 - y1) / (x2 - x1).

Example

1. Graph y = 3x from a table and ask what the 3 means as a unit rate and as a slope.
2. A bike shop charges $15 per hour. Ask them to graph cost against hours and read the unit rate off the slope of the line.

Example

1. Give a table pairing 0, 2, and 4 with 5, 11, and 17, and ask for the rate of change and the starting value.
2. Show a line crossing (0, 10) with slope 2 and ask for the y-intercept and the rate of change.

Example

1. A lemonade stand earns $2 per cup. Ask for the table, the graph, and the equation.
2. Give the equation y = 5x and ask for a story, a table, and a graph that fit it.

Example

1. A phone plan costs $20 plus $0.10 a minute. Ask for the table, the graph, and the equation, and what the 20 does on the graph.
2. Give y = 4x + 7 and ask them to build the table and the graph, and say why it is not proportional.

Example

1. Show two scatterplots — one roughly straight, one clearly curved — and ask which suggests a linear relationship.
2. Show a scatterplot with points scattered everywhere and ask what it says about a linear relationship.

Example

1. Draw a trend line through a scatterplot of study hours versus scores and ask what it predicts for 6 hours.
2. Show arm-span versus height data with a trend line drawn and ask for the predicted height at a 65-inch arm span.

Example

1. y varies directly with x, and y = 12 when x = 3. Ask for y when x = 7.
2. A recipe scales directly: 2 cups of flour make 12 cookies. Ask how much flour makes 30 cookies.

Example

1. Show y = 3x and y = 3x + 2 as tables and ask which is proportional and how the table, the graph, and the equation each show it.
2. Show the graphs of y = 2x and y = 2x + 5 and ask which passes through the origin and why that decides it.

Example

1. Give {(1, 2), (2, 3), (1, 4)} and {(1, 2), (2, 3), (3, 4)} and ask which is a function and why.
2. Show a mapping with one input pointing to two outputs and ask whether it is a function.
3. Show a table pairing 1, 2, and 3 with 5, 6, and 5 and a graph with two points sharing an x-value, and ask which is a function and why.

Example

1. Ask them to name a proportional function from a road trip and a non-proportional one from a taxi meter.
2. Give d = 60t and C = 60t + 50 and ask which is proportional and which is not.

Example

1. A gym charges $30 to join plus $25 a month. Ask for the equation modeling cost against months, built from a table first.
2. A plant is 10 cm tall and grows 3 cm a week. Ask for the equation, then check it against a table and a graph.

Example

1. Give a cylinder with base area 20 and height 6. Ask them to say in words what V = Bh gives, then work it out.
2. Ask them to explain what the B and the h each stand for in the volume of a soup can.

Example

1. Fill a cone with rice and pour it into the matching cylinder with the same base and height. Ask how many pours it takes, and what that says about the two formulas.
2. Ask them to write both volume formulas side by side and say where the one third comes from.

Example

1. Build squares on all three sides of a 3-4-5 triangle and ask what the three areas say.
2. Cut out the two smaller squares and fit them onto the largest, then ask what that shows about the sides.

Example

1. A cylinder has radius 3 and height 10. Ask for its volume.
2. A cone has radius 6 and height 9. Ask for its volume.
3. A sphere has radius 3. Ask for its volume.

Example

1. A rectangular prism measures 5 by 3 by 2. Ask for the lateral and total surface area from its net.
2. A cylinder has radius 4 and height 6. Ask them to connect the rectangle in its net to the lateral area formula.
3. A triangular prism has two triangular faces of area 6 and three rectangular faces of areas 10, 12, and 14. Ask for the lateral and total surface area from its net.

Example

1. A right triangle has legs 6 and 8. Ask for the hypotenuse.
2. A 13-foot ladder leans with its foot 5 feet from a wall. Ask how high it reaches.
3. Ask whether sides 9, 12, and 15 form a right triangle, and how the converse decides it.

Example

1. Ask for the distance between (1, 2) and (4, 6) on a coordinate plane.
2. Ask for the distance between (-2, 1) and (1, 5).

Example

1. One phone plan costs $0.20 a minute with no fee; another costs $10 plus $0.10 a minute. Ask for the equation for the minutes where the two cost the same.
2. The left pan of a balance holds 3x + 2 and the right holds x + 10. Ask for the equation the balance shows.

Example

1. Give 4x + 5 = 2x + 15 and ask for a story it could tell.
2. Give 3n - 7 > n + 5 and ask for a real situation it fits.

Example

1. Model 5x - 3 = 2x + 9 with tiles and ask for x.
2. Model 2(x + 1) = x + 7 and ask for the solution.

Example

1. Tear the three corners off a paper triangle and fit them on a straight line. Ask what that shows about the angle sum.
2. Show two parallel lines cut by a transversal and ask which angles match, and what that says about the angles of similar triangles.
3. Fold one corner of a paper triangle outward to make an exterior angle and ask how it compares with the sum of the two far angles.

Example

1. Graph y = 2x + 1 and y = x + 4 and ask where the lines cross, then check the point in both equations.
2. Give the intersection (2, 5) of two graphed lines and ask them to verify it satisfies both equations.

Example

1. Rotate a paper flag 90° and ask what happened to its orientation versus its shape and size.
2. Reflect a triangle over the y-axis and ask what stayed the same and what flipped.
3. Slide a triangle 3 right and 2 up, then dilate a copy by 2, and ask what each move did to the orientation and whether the image still matches.

Example

1. Show a rotation and a dilation side by side and ask which one preserves congruence and why the other does not.
2. Ask them to sort translations, reflections, rotations, and dilations into keepers and non-keepers of congruence.

Example

1. Rotate a triangle 90° about the origin and ask for the image coordinates and the rule behind them.
2. Translate the point (3, 4) by 2 right and 1 down, then rotate it 180° about the origin, and ask where it lands each time and for the rule each move follows.
3. Reflect a shape over the x-axis and ask for the algebraic rule the reflection follows.

Example

1. Dilate a rectangle by 3 and ask what happens to its perimeter and its area.
2. Dilate a triangle by 1/2 and ask what happens to its side lengths and its area.

Example

1. Give arm-span versus height data and ask for a scatterplot and whether the association looks linear.
2. Plot shoe size versus test scores and ask what kind of association that is.

Example

1. Give the data 2, 4, 4, 4, 5, 5, 7, and 9, and ask for the mean absolute deviation.

Example

1. Simulate 20 samples of 10 marbles from a bag known to hold half red, and ask what share of the samples land near half red.

Example

1. Compare a 3-year and a 5-year car loan at the same rate and ask which costs more in interest and why.
2. Compare 12% and 18% on the same $1,000 balance and ask what the rate changes about the cost of credit.

Example

1. Run a $2,000 credit-card balance at 18% and an easy access loan at 10% through an online loan calculator over 1 year and over 2 years, and ask for the total repaid under each combination.

Example

1. Ask them to explain how $50 a month saved for college grows over 10 years, and what changes if it starts 5 years earlier.
2. Show two retirement savers, one starting at 20 and one at 30, and ask what the head start is worth at 60.

Example

1. Put $100 in an account at 5% simple interest and $100 at 5% compounded yearly. Ask them to compare the two balances after 3 years.
2. Ask which of the two earns more over 10 years, and by about how much.

Example

1. Ask them to explain when cash wins, when a debit card wins, and when a credit card wins — and what each one risks.
2. Compare paying by phone tap versus writing a check: what moves, and how fast.

Example

1. Show a plan to save 10% of every paycheck and ask whether it is financially responsible and what it buys over time.
2. Show maxing out a credit card for concert tickets and ask what the costs are beyond the tickets.

Example

1. Look up one year of in-state tuition nearby, estimate the two-year and four-year totals, and ask what saving the first year’s share monthly over 48 months takes.
2. Ask them to build the plan in three parts: family contribution, savings, and the remaining gap.