Mathematics, grade 5

Example

1. A gallon of milk costs $3.79 and a loaf of bread costs $2.85. Ask for an estimate of the total, then an estimate of how much more the milk costs, before working either one out exactly.
2. Ask them to estimate 289 divided by 6 using compatible numbers before dividing it out.
3. Before multiplying 68 by 21, ask them to round each number first and say roughly what to expect.

Example

1. Ask for one half plus one third, using fraction strips cut into sixths to find a common size of piece.
2. Ask for three fourths minus one sixth, and have them find a common denominator with paper strips before subtracting.

Example

1. Ask for two fifths plus one fourth, done fluently without drawing a picture first.
2. Ask for 5.6 minus 2.75, done fluently using the standard algorithm.

Example

1. Draw three circles and shade two thirds of each, then ask for the total shaded amount as 3 times two thirds.
2. Draw an area model for 4 times three fifths using four rows of a fifths strip.

Example

1. Cut a strip standing for one third into 4 equal pieces and ask what fraction of the whole strip each small piece is — that is one third divided by 4.
2. Ask how many one-third cups it takes to fill 5 whole cups, then draw the pictorial model that shows five wholes each split into thirds.

Example

1. Ask for 6 divided by one third, and then one third divided by 6, and notice the two answers are not the same operation reversed.
2. Ask how many one-fourth servings come out of 3 whole pies.

Example

1. Count how many campsites had a fire ring, a picnic table, both, or neither, and ask them to build a frequency table, then a bar graph from it.
2. Record five hikes with distances that include halves and quarters of a mile, and ask them to build a dot plot marked in quarters.
3. Record a week of gas-tank fill amounts in gallons, several of them decimals, and ask them to build a stem-and-leaf plot.

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. Give the rule y = x + 3 and ask them to build a table for x = 1 through 5, then plot the pairs.
2. Give the rule y = 2x and ask them to build the table, graph the points, and say what shape the points make.

Example

1. Show two tables side by side — one where y is always x plus 4, one where y is always x times 4 — and ask which pattern is additive and which is multiplicative.
2. From a graph of points climbing by the same amount at every step, ask whether getting from one point to the next means adding or multiplying.

Example

1. Draw a graphic organizer with "quadrilaterals" at the top and ask them to nest "square," "rectangle," and "rhombus" inside it to show which shapes are also which other shapes.
2. Ask whether every square is a rectangle, and whether every rectangle is a square, then have them sort a mixed stack of quadrilaterals into a diagram that shows the difference.

Example

1. Ask them to explain, in their own words, the steps for plotting the point (3, 5) starting from the origin.
2. Give the pair (4, 2) and ask which direction to move first and which second, and why the order matters.

Example

1. Give an input-output table where the output is always the input plus 2, and ask them to plot each pair on the coordinate plane.
2. Record the miles hiked next to the day number for a five-day stretch, and ask them to graph the pairs in the first quadrant.

Example

1. Record the number of pages read and the number of days it took for five different books, and ask them to plot the pairs on a scatterplot.
2. Record hours driven and miles covered for a week of travel days, and ask them to plot the pairs on a scatterplot.

Example

1. From a bar graph of tent-site counts at three campgrounds, ask how many more sites two of them have combined than the third, then ask the same question off the frequency table the graph was built from.
2. From a dot plot of hike distances in quarter-miles, ask for the total distance of the three longest hikes, then have them find those same three on a stem-and-leaf plot of the week’s mileage.
3. From a scatterplot of hours driven against miles covered, ask how many more miles the longest driving day covered than the shortest.

Example

1. Hand over a single unit cube and ask what one cubic unit means, then have them build a shape from 12 of them with no gaps or overlaps and count the volume.
2. Build an L-shaped figure from unit cubes and ask them to count the cubes for the volume, checking there are no gaps or overlaps.

Example

1. Ask how many quarts are in 3 gallons, given that 1 gallon equals 4 quarts.
2. Ask how many centimeters are in 2.5 meters, given that 1 meter equals 100 centimeters.
3. A recipe calls for 24 ounces of trail mix. Ask how many pounds that is, given that 1 pound equals 16 ounces.

Example

1. A camp store sells trail mix at $2.35 a bag. Ask for the cost of 4 bags, worked out by multiplying.
2. Ask for 0.6 times 0.4, then for 6 times 4, and ask what they notice about the two answers and the decimal point.

Example

1. Ask them to explain the difference between a sales tax added at a register and a property tax billed once a year.
2. Ask what payroll tax is taken out of a paycheck for, and how income tax differs from it.

Example

1. Give a pay stub showing $600 earned before deductions and $478 after, and ask which number is gross income and which is net.
2. Ask why the amount that lands in a bank account is usually smaller than the amount a job pays before deductions.

Example

1. Give a budget where $340 comes in and $380 goes out, and ask what two things could change to bring it back into balance.
2. Ask what happens to a budget if the price of gas jumps for a month, and what could be cut to cover it.

Example

1. Ask for 428 times 36 using the standard algorithm.
2. Ask for 512 times 47, and check the answer by estimating first — is it close to 500 times 50?

Example

1. Ask for 4,896 divided by 24 using the standard algorithm.
2. Ask for 7,308 divided by 18, then check the answer by multiplying back.

Example

1. Draw a 10-by-10 grid and shade to show 0.4 times 0.6, then ask them to count the double-shaded squares for the product.
2. Draw an area model for 1.2 times 0.3 on a grid split into tenths, and ask them to find the product from the shaded region.

Example

1. Draw an area model for 84.96 divided by 24 and ask them to fill in the missing side.
2. Draw a pictorial model for 15.75 divided by 15 and ask them to compare the strategy to whole-number division.

Example

1. Ask for 74.52 divided by 12 using the standard algorithm.
2. Ask for 61.25 divided by 25, then check the answer by multiplying back.

Example

1. Ask what happens to money right away with a debit card versus a credit card, and which one can leave a bill due later.
2. Ask why a campground might prefer an electronic payment over a check, and what a check offers that the others do not.

Example

1. Write 0.617 and ask them to expand it as 0.6 + 0.01 + 0.007, matching each part to the digit it came from.
2. Write 5.203 and ask what the 2 is worth versus what the 3 is worth — two tenths against three thousandths.

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. A three-stop drive covers legs of 212, 96, and 158 miles, split evenly between two drivers. Ask for an equation with a letter standing for one driver’s share, then ask them to solve it.
2. You start with $140, buy 4 tents at the same price, and have $12 left. Ask for an equation with a letter standing for one tent’s price, then ask them to solve it.

Example

1. A rectangular tent pad measures 10 feet by 8 feet. Ask for the perimeter of the edging and the area of the ground cloth it needs.
2. A cooler measures 2 feet long, 1 foot wide, and 1 foot tall. Ask for its volume in cubic feet.