Unit 1: Multiplication and division
Usually taught in August to October
- Equal groups and arrays4 shelves of 6 books as groups, then as an array: 4 × 6 = 24.
- Groups or how many in each?Fixes a mistake4 bags of 6 is not 6 bags of 4 in the story. The first number counts groups.
- Two kinds of division12 ÷ 3 as 3 groups, how many in each, and as groups of 3, how many groups.
- Equal jumps on a number line8 jumps of 6 land on 48.
- Turn the array4 × 6 turned is 6 × 4. Same dots, same product.
- Facts from facts you know6 × 9 as 4 × 9 + 2 × 9, one array split in two.
- Add the partsFixes a mistake(4 × 70) × (4 × 2) multiplies the parts. Add them: 280 + 8.
- One digit times two digits3 × 24 = 3 × (20 + 4) = 60 + 12.
- Multiply both partsFixes a mistake4 × (11 + 6) is not (4 × 11) + 6. The 4 multiplies the 6 too.
- Three factors, any order2 × 7 × 5 regrouped as 7 × (2 × 5) = 70.
- Times tens and hundreds6 × 7 = 42, so 6 × 7 tens is 42 tens, 420, and 6 × 700 is 4,200.
- Counting zeros failsFixes a mistake5 × 80 is 40 tens, 400, not 40. The fact already ends in 0.
- One array, four facts3 rows of 4: 3 × 4, 4 × 3, 12 ÷ 3, 12 ÷ 4.
- True or false?3 × 8 = 4 × 6 is true. 24 ÷ 4 = 12 ÷ 3 is false. Both sides same value.
- The answer can go firstFixes a mistake24 = 4 × 6 is true. The equal sign means "the same as".
- The unknown anywhere7 × n = 56, n ÷ 6 = 5, 3 = n ÷ 12, each turned into a known fact.
- Even or odd?Two equal groups, or a pair left over. The ones digit decides.
- The ones digit decidesFixes a mistake883 is odd, whatever its 8s say.
- Is it a multiple?Is 36 a multiple of 8? 8 × 4 = 32, 8 × 5 = 40: no.
- A number is its own multipleFixes a mistakeMultiples of 8 start at 1 × 8 = 8.
- Patterns from a ruleStart at 500, subtract 35: the 1st, 2nd, 3rd terms.
- Position is not valueFixes a mistakeIn 6, 12, 18, the 2nd term is 12. 12 is not the 12th term.
- Two-step word problemsA field trip: some drop out, the rest split into equal groups. One equation per step.
- Finish the second stepFixes a mistakeThe first step’s answer is not the question’s answer.
- One pattern, two waysTriangles made of toothpicks: 3, 6, 9. A table lists the triangles, 1, 2, 3, beside the toothpicks, 3, 6, 9. Each new triangle adds 3 toothpicks in the picture and 3 in the table.
Unit 2: Lines, shapes, area, and perimeter
Usually taught in October and November
- Points, lines, segments, raysDraw each one and name it by its ends.
- Parallel and perpendicularLines that never meet, and lines that meet at a square corner, tested with a paper corner.
- Crossing is not perpendicularFixes a mistakeTwo lines can cross without a square corner.
- Naming quadrilateralsParallelogram, rhombus, rectangle, square, trapezoid by their sides and corners.
- One shape, many namesFixes a mistakeA square is also a rectangle and a rhombus. A kite is still a quadrilateral.
- Lines of symmetryFold the letter H: both halves match, twice.
- Find every lineFixes a mistakeA square has 4 lines of symmetry, not 2.
- Area by counting squaresCover a rectangle with unit squares, no gaps, and count in square units.
- Area is rows times columns4 rows of 8 squares: 4 × 8 = 32 square units.
- Perimeter around the edgeAdd the four sides, or double each side and add.
- Same area, different perimetersEvery rectangle of 12 square feet, and each perimeter.
- Area of an L shapeSplit it into two rectangles and add the areas.
- The side you splitFixes a mistakeAfter the cut, a piece uses only part of the long side.
- The cut is not perimeterFixes a mistakeThe inside cut is not on the edge. Walk only the outside.
- Build it from cubesA picture of a solid: 2 cubes in front, 2 behind, and 1 on top at the back left. Build it one layer at a time and count the cube hidden under the top one: 5 cubes in all.
Unit 3: Place value to 10,000
Usually taught in November and December
- Numbers to 10,000 three ways2,530 in standard, expanded, and word form.
- Zeros hold empty placesFixes a mistakeNine thousand three is 9,003, not 9,300. 0 hundreds and 0 tens.
- Many names for one number5,783 as 57 hundreds and 83 ones, or 578 tens and 3 ones.
- Comparing on a number line3,790 and 3,809 on a line by 100s, then from the biggest place down.
- Read the sign as a sentenceFixes a mistake6,909 > 6,099 says "is greater than". It compares; it does not compute.
- Rounding: closer to which?428 between 400 and 500, 450 in the middle, and the closer end. Grade 4's video.
- Round at the right placeFixes a mistake923 to the nearest ten is 920, not 900.
Unit 4: Adding and subtracting
Usually taught in December and January
- Estimate first174 + 253 is near 170 + 250 = 420. Check the exact answer against it.
- Adding with regrouping12 tens become 1 hundred and 2 tens, said in place value.
- Subtracting with regrouping532 − 178: trade 1 ten for 10 ones, then 1 hundred for 10 tens.
- Subtracting across zerosFixes a mistake8,000 − 1,432: the zeros trade down one place at a time.
- Estimate or exact?Does the question need about how much, or exactly how much? Whether $20 is enough for 3 books near $6 each needs an estimate; the price to pay needs the exact total.
Unit 5: Fractions
Usually taught in February and March
- One part of equal parts1/4 as part of a shape, part of a set, and a point on a number line.
- Equal parts onlyFixes a mistakeFour parts that are not equal are not fourths.
- More parts, smaller piecesFixes a mistake1/8 is less than 1/4: the same whole cut into more parts.
- Counting unit fractions3/4 is 1/4 + 1/4 + 1/4, and 7/4 goes past one whole.
- Naming fractions4/3, 4 thirds, four-thirds, and one and one third.
- A fraction is one numberFixes a mistake8/3 is 8 thirds, one number, not "8 over 3".
- Fractions on a number lineEach jump is one unit fraction: 5/3 is five thirds from 0.
- Same bottom or same topSame denominator: count parts. Same numerator: compare piece size.
- Bigger bottom, smaller pieceFixes a mistake5/4 is less than 5/2, though 4 is more than 2.
- Same size wholesFixes a mistakeCompare fractions only on wholes the same size.
- Equal fractions, one point1/2 and 2/4 land on one point of two given lines.
- Is it really equal?Fixes a mistake2/4 and 2/6 are not equal: the pieces differ in size.
- A fraction of a set6 counters in 3 equal groups: one group is 1/3 of the set, 2 counters. Two groups are 2/3 of the set, 4 counters.
Unit 6: Data
Usually taught in April
- Scaled pictographs and bar graphsEach picture stands for 2, so the key multiplies. A bar scale starts at 0.
- Line plotsBooks each kid read: an X over each number for each kid.
- Read the keyFixes a mistake4 pictures at 5 each is 20, not 4. One grid square is not always one.
- Questions from graphsHow many more, and how many in all, from a bar graph.
- Possible outcomesA spinner with red, blue, and green can land on 3 outcomes; a coin on 2. List every outcome of a simple event.
- What the graph tells usA bar graph of books read each month: the maximum is 12 in March, the minimum is 4 in July. Kids read less in summer, so next July will likely be low too.
Unit 7: Measurement and time
Usually taught in April and May
- Measuring to the quarter inchA ruler zoomed in: halves and quarters between whole inches.
- Reading a scaleA beaker and a thermometer read like a number line standing up.
- Measurement word problemsCompare and combine grams, milliliters, or feet, with the unit in the answer.
- Telling time to the minute3:19 is 15 by fives and 4 more, then a.m. or p.m.
- Where the hour hand sitsFixes a mistakeAt 3:19 the hour hand is just past 3, not halfway to 4.
- Elapsed time on a number line9:40 to 10:25 in jumps to the next hour, then the minutes left.
- Time is not base tenFixes a mistake9:55 to 10:05 is 10 minutes, not 50. An hour has 60 minutes.
- Counting up to make changeA $3.65 snack paid with $5: count up from the cost with coins and bills, to $3.75, $4, then $5. The change is $1.35, with as few coins as possible.
- Paying now or paying laterCredit is borrowing that must be paid back. A planned buy fits the money set aside; an unplanned one can leave too little for what is needed.
- A plan to saveA $24 goal, saving $4 a week: 6 weeks. Write the plan, then sort choices into earning, spending, saving, borrowing, and giving.
- Skills earn incomeA sitter who took a first-aid class earns $8 an hour instead of $6. Skills and work, called human capital, are what a worker trades for income: more skill or more hours, more income.
- Scarce things cost moreLemons cost 25¢ each in summer, when there are plenty, and 50¢ after a frost, when few are left. When a resource is scarce, its cost goes up; when it is plentiful, its cost goes down.